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abstract: 'In this paper, a dual learning-based method in intra coding is introduced for PCS Grand Challenge. This method is mainly composed of two parts: intra prediction and reconstruction filtering. They use different network structures, the neural network-based intra prediction uses the full-connected network to predict the block while the neural network-based reconstruction filtering utilizes the convolutional networks. Different with the previous filtering works, we use a network with more powerful feature extraction capabilities in our reconstruction filtering network. And the filtering unit is the block-level so as to achieve a more accurate filtering compensation. To our best knowledge, among all the learning-based methods, this is the first attempt to combine two different networks in one application, and we achieve the state-of-the-art performance for AI configuration on the HEVC Test sequences. The experimental result shows that our method leads to significant BD-rate saving for provided 8 sequences compared to HM-16.20 baseline (average 10.24% and 3.57% bitrate reductions for all-intra and random-access coding, respectively). For HEVC test sequences, our model also achieved a 9.70% BD-rate saving compared to HM-16.20 baseline for all-intra configuration.'
author:
-
bibliography:
- 'references.bib'
title: 'Dual Learning-based Video Coding with Inception Dense Blocks '
---
Video Coding, High Efficiency Video Coding (HEVC), Reconstruction Filtering, Neural Network
Introduction
============
Recently, neural networks have shown great potential in various fields and many promising results have been achieved in video coding. Especially for intra coding [@HEVCIntra], many creative ideas have been proposed to enhance the performance of intra coding. Intra coding uses the reference pixels to predict the current block, and the residual samples calculated by original pixels minus the predicted pixels are sent to transform and quantization to obtain the compressed residual. After the inverse quantization and inverse transform process, the residual samples with distortion is obtained. The reconstructed pixels at the decoding end can thus be obtained by adding the predicted pixels and the distorted residual samples.
Many neural network-based works are carried out for the prediction [@FC1; @FC2; @FC_CONV; @HIT_CNN; @RNN] and filtering [@ARCNN; @VRCNN; @RHCNN; @MMSN], which are two key parts of intra coding. In neural network-based intra prediction, network structures mainly consist of convolutional layers and full connected layers [@FC1; @FC2; @FC_CONV; @HIT_CNN]. Li et al. [@FC1; @FC2] use full connected network to explore the capacity of prediction. PNNS is introduced by Dumas et al. [@FC_CONV] based on both fully-connected and convolutional neural networks. Cui et al. [@HIT_CNN] propose IPCNN, and this network directly applies CNNs to intra prediction, which achieves a good performance in intra coding as well. And Hu et al. [@RNN] try Progressive Spatial Recurrent Neural Network (PS-RNN) and SATD loss function, which supports variable-block-size for intra prediction.
From a perspective of reconstruction filtering, similar convolutional neural network structures can be used for tasks such as super-resolution, denoising and filtering. Dong et al. [@SRCNN] proposed SR-CNN for super-resolution, and they also design AR-CNN [@ARCNN] for compression artifacts reduction based on it. Then Dai et al. [@VRCNN] increased the width of AR-CNN and proposed VR-CNN to further improve the network performance. In more complex network design, Zhang et al. [@RHCNN] proposed an RHCNN with 3,340,000 parameters, and achieved better experimental results. And a multi-modal/multi-scale model called MMS-net [@MMSN] with 2,298,160 parameters is proposed by Kang et al., which shows a multi-scale CNN structure can effectively improve image reconstruction performance.
For filtering tasks, we believe that VR-CNN does not adequately extract the full characteristics of the data. Deeper features can help guide the network for better filtering. While RHCNN is a little complicated for practical applications, and relatively simpler design needs to be proposed. Therefore, we have designed a new reconstruction filtering network with 475,233 parameters, which based on the inception networks[@inception4]. The inception network has an excellent performance in the classification task because of its powerful feature extraction capabilities. We believe that it can also perform well in filtering tasks. Specifically, the main contributions are as follows.
- In VR-CNN, the features extracted by two convolutional blocks of the two branches. We use the improved inception dense network to replace it so that the network has stronger feature extraction ability, which leads to a better filtering performance. After using inception filtering, we can achieve 3.07%, 3.06% and 3.17% BD-rate improvement for YUV components, respectively. Moreover, the filtering network we are performing is not a frame-level but a block-level. Because block-level filtering enhance the quality of the reconstructed pixels, which directly increases the predictive performance and reduce the bit-rates of the residual in the next block.
- We combine two learning-based methods and ensure that integrating two methods together can further improve the performance. The experimental results show that this integrating method achieves extra 2.42% BD-rate saving for luminance component than using our reconstruction filtering model alone. It is thus found that the network has a superposition effect.
Proposed Methods
================
Full-connected Network-based Intra Prediction
---------------------------------------------
For the intra prediction, originally, there are 35 modes in HEVC composed of two non-directional and 33 directional modes. Based on HEVC original modes, we append one neural network mode by using the fully-connected network in [@FC1; @FC2]. The difference is that we apply the neural network mode at the prediction unit (PU) level. The mode signaling scheme is that one additional bin is consumed to represent whether the best mode of each PU is the neural network mode or not.
Convolutional Network-based Reconstruction Filtering
----------------------------------------------------
Like the structure of VRCNN in Fig.\[structure\](a), the reconstruction filtering network shown in Fig.\[structure\](c) consists of a pre-processed convolutional network shown in green rectangle, an intermediate inception dense network shown in orange rectangles and a post-processing convolutional network shown in the purple rectangle.
The first 64 feature maps convolutional layer with kernel size is 5x5 in VR-CNN pre-processed convolutional module (Fig.\[structure\](b)) is transformed into two convolutional layers of 64 feature maps with kernel size is 3x3, which helps to improve the extraction of basic features. Compared to 2 blocks in VR-CNN, our middle part uses 12 improved inception network blocks to further extract the features. The specific structure is shown in Fig. Fig.\[structure\](d). Its input is the output from the front layer, and the chunk is composed of three branches, each branch has a 32 feature maps convolutional layers with kernel size is 1x1 as the first layer. In order to extract the features from different receptive fields, the two convolution layers with kernel size are 1x3 and 3x1 are connected to the first layer in one of the branches. In addition, another branch is connected to a convolution layer with a kernel size of 3x3 serially, and then connected with two convolution layers of 1x3 and 3x1 convolution kernels in parallel. Different with the original inception, we removed the pooling layer for a more compact network structure.
[c|r|r|r|r|r|r|r|r|r|r]{} \*[Sequence]{} & &
------------------------------------------------------------------------
\
& &\*[$\Delta T_{enc}$]{}&\*[$\Delta T_{dec}$]{}& &\*[$\Delta T_{enc}$]{}&\*[$\Delta T_{dec}$]{}\
&& && & && && & \
01 & -8.31% & -14.51% & -13.22%&7219%&177661%& -3.11% & -8.56% & -7.04%&659%&17880%\
02 & -10.98% & -14.27% & -14.93%&8449%&233472%& -4.24% & -3.82% & -4.36%&708%&20839%\
03 & -10.51% & -5.46% & -14.80%&8860%&220486%& -3.65% & -6.97% & -3.47%&478%&11143%\
06 & -11.40% & -6.54% & -10.00%&8888%&226057%& -2.99% & -4.48% & -1.57%&584%&11420%\
08 & -7.91% & -12.59% & -12.35%&7480%&168358%& -2.11% & -3.37% & -3.09%&636%&10412%\
09 & -8.63% & -16.48% & -17.29%&6560%&160278%& -2.12% & -5.19% & -5.46%&623%&11474%\
10 & -11.40% & -13.24% & -13.55%&9117%&231279%& -3.24% & -1.31% & -2.20%&594%&10952%\
13 & -12.83% & -16.26% & -17.77%&7925%&185475%& -7.09% & -9.38% & -9.74%&540%&10421%\
Average & -10.24% & -12.41% & -14.24%&8015%&198322%& -3.57% & -5.38% & -4.61%&599%&12628%\
The last part is a post-processing convolution module that make the number of output feature maps return to the same number of the input. Because the inputs to our model have only one feature map, the last part is a convolutional layer with a kernel size of 3x3 and the number of feature maps is one.
The output of the post-processing convolution module and the reconstruction block will be added to obtain the filtered reconstruction block. Each convolutional layer except the last one is followed by an activation function “relu”. And all convolutional layer padding methods use “same”, which makes the input and output sizes unchanged. The input to the designed network is 32x32 reconstructed block from HM. For a CTU with the size $64\times 64$, it can be divided into four 32x32 luminance component blocks and two 32x32 chroma component blocks with pixel format “YUV420”. The output of the network is the enhanced reconstructed block. Because PSNR can reflect the performance of video coding relatively simply and effectively, we use the MSE (Mean Square Error) between enhanced reconstruction block $Y$ and the original reconstruction block $X$ as the loss function $L$ to train our network. The symbol $\Theta$ and $F$ are the parameters and outputs of our model respectively, and N is 32. $$\label{mse}
L(\Theta)=\frac{1}{N^2}\sum_{i=1}^{N}\sum_{j=1}^{N}(F(Y(i,j)|\Theta)-X(i,j))^2$$
Block-level Filtering
---------------------
Different with the previous designs, our filter network is based on block-level rather than frame-level. After each CTU encoding is complete, the reconstruction blocks can be got on both the encoding end and the decoding end. They will be sent to the network to obtain the filtered result, thereby improve the image quality of the current block on the one hand, and provide more accurate reference pixels for the next block on the other hand. So this method can reduce the bitrates and improve the picture quality at the same time. Because our filter network is block-level, de-blocking may help to improve the edge of the block as well. Therefore it can coexist with the loop filter of the HEVC.
Experimental Results
====================
Experimental Setting
--------------------
The datasets we used is DIV2K [@DIV2K], which consisted of 900 images with a resolution of 2k. Because the reconstruction quality of the chrominance components is often too high, only the luminance component is used for training. But in actual testing, chrominance components will also be tested by using our model. In order to avoid those blocks with accurate reconstruction are affected negatively by using neural network based filtering model. Different QP models are tested and finally the appropriate QPs for each target QP are selected while exporting the dataset. Specifically, we derive four datasets from different QPs to train four corresponding filtering models with different QP bands. The deep learning framework used is Keras [@keras] because of its better support for TensorFlow [@tensorflow]. We call the freeze model of TensorFlow converted from Keras model in the actual application. The GPUs used for training and the CPUs used for testing were NVIDIA GeForce RTX 2080 and Intel Xeon Gold 6134 at 3.20 GHz, respectively.
Comparison with HEVC Baseline
-----------------------------
We use the 8 PCS grand challenge short videos to test, and the test condition includes four QPs (22 27 32 37) and two configurations (AI and RA). The test results are shown in TABLE \[com\_pcs\]. In the AI configuration, for the luminance component, we get BD-rate saving of at most 12.83% and on average 10.24%. BD-rate saving of 12.41% and 14.24% for the chrominance component are obtained respectively. In the RA configuration, the YUV components obtain BD-rate saving on average of 3.57% 5.38% and 4.61%, and the luminance component obtains at most 7.09% BD-rate saving with sequence 13.
For the subjective image quality evaluation of the proposed method, Fig. \[sub\] compares the HM reconstruction image (on the left) and the enhanced image by our networks (on the right). In the face area of the blue box, we can clearly see the contouring and blocky artifacts in Fig. \[sub\](a). On the other hand, in Fig. \[sub\](b), these artifacts are well eliminated and the face is smoother and plumper. Moreover, our model provides a higher compression ratio (0.231 bpp of our models to 0.243 bpp of HM).
[c|r|r|r |r|r|r |r|r|r |r|r|r]{} \*[Sequence]{} & & & &
------------------------------------------------------------------------
\
&& &&& &&& &&& &\
01 &-2.40%&-4.24%&-1.95%&-3.65%&-10.85%&-6.51%&-6.06%&-12.51%&-9.84%&-8.63%&-16.77%&-13.57%\
02 &-2.43%&-3.70%&-2.99%&-4.53%&-6.69%&-9.71%&-8.49%&-12.65%&-14.58%&-11.81%&-19.18%&-18.68%\
03 &-1.71%&-1.93%&-2.84%&-3.07%&-6.80%&-4.18%&-5.02%&-8.56%&-8.81%&-7.19%&-13.83%&-15.96%\
06 &-3.04%&-3.23%&-2.77%&-5.70%&-6.50%&-5.02%&-9.67%&-9.33%&-6.14%&-10.72%&-12.49%&-10.46%\
08 &-2.41%&-3.29%&-2.01%&-4.26%&-4.42%&-3.49%&-7.17%&-6.76%&-6.23%&-9.31%&-11.10%&-9.94%\
09 &-1.13%&-4.05%&-2.47%&-1.88%&-10.55%&-7.07%&-3.19%&-10.58%&-9.65%&-5.33%&-14.84%&-14.36%\
10 &-2.67%&-0.50%&-4.20%&-4.95%&-2.55%&-7.42%&-8.77%&-8.73%&-8.67%&-11.05%&-12.24%&-14.16%\
13 &-2.58%&-7.43%&-1.79%&-4.61%&-15.63%&-11.19%&-8.81%&-19.34%&-16.01%&-12.53%&-23.42%&-19.16%\
Mean BD-rate &-2.30%&-3.55%&-2.63%&-4.08%&-8.00%&-6.82%&-7.15%&-11.06%&-9.99%&**-9.57%**&**-15.48%**&**-14.54%**\
Comparison with AR-CNN and VR-CNN
---------------------------------
We use the same dataset to train AR-CNN and VR-CNN and test them in the same situation. More specifically, in order not to bring in the influence of neural network-based intra prediction, the neural network prediction is turned off while testing the filtering networks. At the same time, we test them based on block-level filtering which may improve the reconstruction pixels so as to explore the potential of these filtering ways. All sequences are tested for the first frame under each QP, and the experimental results are shown in TABLE \[vrcnn\], we can find that.
- Compared with the test results of our filter model (“Our Filter.” column in the TABLE \[vrcnn\]) used alone, using our joint model (“Our All.” column in the TABLE \[vrcnn\]) achieves better performance, the BD-rate of the YUV component is reduced by 2.42%, 4.42%, and 4.55%. This means our combination of intra prediction and reconstruction filtering is successful.
- VR-CNN and our proposed method have better performance than AR-CNN because of the diversity of their convolution kernels in different subbranches. And they could get better performance in some sequences than others, such as 13. This may be related to the dataset used for training, because our dataset is a spliced sequence of static pictures (DIV2K), and those test sequences which have better performance are relatively static to some extent.
- Our filtering model has a stronger feature extraction capability than VR-CNN, and achieves up to 7.15% and on average 9.67% BD-rate saving. AR-CNN achieves 2.30%, 3.55% and 2.63% and VR-CNN achieves 4.08%, 8.00% and 6.82% respectively on the three components of YUV. Compared with VR-CNN and AR-CNN, our model has better performance in all sequences and components.
Comparison with RHCNN
---------------------
In this subsection, we test the performance of our filtering model for the HEVC test sequences and compare it with RHCNN. It can be observed from the test results TABLE \[hmsd\] that our filtering model also get results (9.70%, 11.59% and 13.35% respectively on the three components of YUV) which close to the result of 8 provided sequences. And the dataset used for training doesn’t overlap with the HEVC test sequences, thus further demonstrating the generalization ability of our model.
[l|l|r|r|r]{} \*[Class]{} & \*[Sequence]{} &
------------------------------------------------------------------------
\
& &&&\
\*[ClassA]{} & Traffic & -11.68%&-9.55%&-11.51%\
&PeopleOnStreet & -11.45%&-14.11%&-14.00%\
\*[ClassB]{} &Kimono & -5.97%&-3.42%&-3.06%\
&ParkScene & -8.65%&-7.98%&-7.43%\
&Cactus & -8.64%&-10.33%&-15.17%\
&BasketballDrive &-7.96%&-11.56%&-16.90%\
&BQTerrace &-6.61%&-9.72%&-10.92%\
\*[ClassC]{} &BasketballDrill & -11.61%&-16.27%&-19.68%\
&BQMall &-9.36%&-11.77%&-13.52%\
&PartyScene & -6.08%&-8.93%&-9.97%\
&RaceHorses & -6.91%&-11.92%&-17.54%\
\*[ClassD]{} &BasketballPass & -9.05%&-9.40%&-12.48%\
&BQSquare &-6.55%&-5.33%&-7.83%\
&BlowingBubbles & -7.42%&-11.24%&-10.46%\
&RaceHorses &-11.20%&-15.11%&-18.38%\
\*[ClassE]{} &Vidyo1 & -13.13%&-13.85%&-15.19%\
&Vidyo3 & -9.62%&-7.07%&-11.75%\
&Vidyo4 & -11.13%&-14.72%&-14.94%\
&FourPeople & -14.90%&-15.44%&-16.76%\
&Johnny & -12.95%&-19.91%&-16.10%\
&KristenAndSara & -12.74%&-15.73%&-16.78%\
& -9.70%&-11.59%&-13.35%\
[l|c|c|c]{} Sequence & & &
------------------------------------------------------------------------
\
Traffic & -6.10% & -7.72% & **-11.40%**\
PeopleOnStreet & -5.30% & -7.80% & **-11.75%**\
RaceHorses & -5.60% & -8.37% & **-12.04%**\
Vidyo1 & -7.50%& -8.59%&**-13.27%**\
Vidyo3 & -6.40% & -6.61%&**-9.68%**\
Vidyo4 & -6.20% & -7.53%&**-11.49%**\
& -6.18% & -7.77%& **-11.61%**\
The trainable parameters number of our filter model is 475,233. At the same time, The RHCNN with 3,340,000 trainable parameters is used as a comparison. We compare the test results provided in their paper [@RHCNN] with our results. It can be seen that our filter models saving 7.77% BD-rate on average and up to 11.61% BD-rate saving is obtained by our joint model. Both results are better than RHCNN’s performance while our models have fewer parameters.
[l|c|c|c|c]{} Model & VR-CNN[@VRCNN] &AR-CNN[@ARCNN] & Our filter &RHCNN[@RHCNN]\
Number &54,512 &106,448& 475,233 & 3,340,000\
Conclusion
==========
In this paper, a dual learning-based method in intra coding consists of intra prediction and reconstruction filtering was proposed for video coding.Inception dense blocks, which have strong feature extraction function is used for improving VR-CNN to make it preform better in reconstruction filtering. At the same time, we changed the frame-level filtering in previous works to block-level filtering, which can help reduce the extra bitstream caused by the error of the reference pixels. For PCS grand challenge short videos test sequences, experimental results show that our networks achieve on average 10.24% BD-rate saving for all-Intra and 3.57% BD-rate saving for random -access. Our model also obtains a 9.70% BD-rate saving compared to HM-16.20 baseline for HEVC test sequences. Finally, we compared our reconstruction filtering model with many other models, and our models achieved the best performance. In future work, we will continue to explore the more efficient and light models to achieve better coding performance.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported in part by the National Natural Science Foundation of China under Grant 61674041, in part by Alibaba Group through Alibaba Innovative Research (AIR) Program, in part by the STCSM under Grant 16XD1400300, in part by the pioneering project of academy for engineering and technology and Fudan-CIOMP joint fund.
This work was supported by the National Natural Science Foundation of China under Grant 61525401，the Program of Shanghai Academic/Technology Research Leader under Grant 16XD1400300, the Innovation Program of Shanghai Municipal Education Commission.
|
---
author:
- 'Anna Raiter, Daniel Schaerer'
- 'Robert A.E. Fosbury'
bibliography:
- 'references.bib'
date: 'Received date / Accepted date'
title: 'Predicted UV properties of very metal-poor starburst galaxies'
---
[We study the expected properties of starburst galaxies in order to provide the point of reference for interpretation of high-$z$ galaxy surveys and of very metal-poor galaxies. We concentrate mainly on the UV characteristics such as the ionizing spectra, the UV continuum, the [Ly$\alpha$]{} and [He [ii]{} $\lambda$1640]{} line and two-photon continuum emission.]{} [We use evolutionary synthesis models covering metallicities from Pop III to solar and a wide range of IMFs. We also combine the synthetic SEDs with the CLOUDY photoionization code for more accurate predictions of nebular emission, and to study possible departures from case B assumed in the synthesis models.]{} [The ionizing fluxes, UV continuum properties, and predicted [Ly$\alpha$]{} and [He [ii]{} $\lambda$1640]{} line strengths are presented for synthesis models covering a wider range of parameter space than our earlier calculations. Strong departures from case B predictions are obtained for [Ly$\alpha$]{} and 2$\gamma$ continuum at low metallicities. At low nebular densities both are shown to be enhanced proportionally to the mean energy carried by the Lyman continuum photons emitted by the ionizing source. Larger [Ly$\alpha$]{} equivalent widths are therefore predicted at low metallicity. The [He [ii]{} $\lambda$1640]{} line can be weaker than case B predicts (in terms of flux as well as the equivalent width) due to its ionization parameter dependence and to the enhanced underlying 2$\gamma$ continuum.]{} [Our results have implications for the interpretation of star-forming metal-poor and/or high redshift galaxies, for galaxies among the [Ly$\alpha$]{} emitters (LAE) and Lyman Break galaxy (LBG) populations, and for searches of Population III stars in the distant Universe.]{}
Introduction {#s_intro}
============
Over the last decade, the execution of deep multi-band imaging surveys like GOODS [@Vanzella05; @Vanzella06; @Vanzella08; @Vanzella09; @Popesso09; @Balestra10] has resulted in catalogues of significant numbers of galaxies with photometric redshifts greater than five when the Universe was only a little over a Gyr old. In some cases, these redshifts have been spectroscopically confirmed by the detection of [Ly$\alpha$]{} in emission and/or the presence of an identifiable Lyman break. The availability of an increasing sensitivity in the NIR, notably with the newly-installed WFC3 camera in the HST and with the JWST in the near future, opens the possibility of selecting high quality candidates up to and beyond a redshift of 10. This is the epoch where it can be expected that stellar populations have a very low metallicity, which may result in an excess of hot, high mass stars radiating strongly in the Lyman continuum.
To provide appropriate spectral templates for such metal-poor star-forming galaxies and to predict the observable properties of starbursts with primordial and more evolved chemical compositions @Scha02 [@Scha03] has computed new evolutionary synthesis models and has demonstrated the importance of nebular emission (lines and continua) at low metallicity. These and other studies [see e.g. @Tumlinson00; @Tumlinson01; @Bromm01] have in particular highlighted the use of strong [Ly$\alpha$]{} emission and nebular [He [ii]{}]{} emission to search for objects containing Population III (hereafter Pop III) stars. Since then, various searches for the [He [ii]{} $\lambda$1640]{} signature from Pop III have been undertaken at different redshifts, yielding so far non-detections and interesting upper limits on the Pop III star-formation rate density [see @Schaerer08; @Nagao08 and references therein]. Furthermore, among the numerous surveys for [Ly$\alpha$]{} emitters at different redshifts, some studies have found objects with apparently unusually strong [Ly$\alpha$]{} emission (high equivalent widths), which could be indicative of very metal-poor (even Pop III) stellar populations or unusual IMFs [see e.g. @Malhotra02; @Yamada05]. Other groups have invoked “unusual” IMFs, extremely metal-poor stellar populations, and/or leakage of Lyman continuum radiation to explain the apparently very blue UV slopes found for some very high redshift ($z \sim 7$) galaxies [@Bouwens10_betaz7]. However, the significance of these results is questionable, and the present data does not require such “non-standard” assumptions [@SdB10; @Fin10]. In any case, it is of interest to examine how reliable some of the major observables predicted by standard evolutionary synthesis are.
Indeed, a shortcoming of evolutionary synthesis models such as the ones mentioned above is that they calculate nebular emission in an approximate manner assuming simplified physics, such as case B recombination theory [cf. @Osterbrock06], and constant emissivities for adopted constant values of the electron temperature and density in the [H [ii]{}]{} region surrounding the starburst. In fact, as demonstrated in this paper, significant departures from case B are expected at low metallicities leading to stronger [Ly$\alpha$]{}emission, and the strength of nebular [He [ii]{}]{} emission predicted by full photoionization models can be reduced with respect to simple recombination theory. Indeed these physical effects, related to an increased importance of collisional effects at low metallicity due to lower radiative cooling and harder ionizing spectra – for [Ly$\alpha$]{} – and due to competition between H and He for ionizing photons – for the intensity of [He [ii]{}]{}/H – have been known for a while in studies of metal-poor [H [ii]{}]{} regions [cf. @Davidson85; @Stasinska99; @Luridiana03] and planetary nebulae [@Stasinska86]. @Panagia02 [@Panagia05] has recently explored photoionization models for primordial nebulae. However, the importance of the above effects for the UV emission lines has so far not been thoroughly examined, in particular in the context of metal-poor and distant starburst galaxies and using up-to-date evolutionary synthesis models. The photoionization models presented here, combined with our evolutionary synthesis models, are intended to provide a framework within which to improve our knowledge of primeval galaxies and related objects.
Another limitation of the synthesis models of @Scha02 [@Scha03] concerns the initial mass function (IMF). For simplicity, three different choices of the IMF were adopted for the bulk of the calculations in these papers. However, different IMFs have been suggested in other studies related to Pop III and early stellar generations, and considerable uncertainties remain on the true shape of the IMF in the early Universe and its dependence (or not) on physical parameters. To enable the examination of the effects of a broader choice of IMFs on the expected observable properties of starbursts, we here extend the calculations of @Scha03 to eight different IMFs. The resulting model grids, available in electronic format, should provide state-of-the-art predictions for the interpretation of high redshift galaxies, to estimate their contribution to cosmic reionization, and for other topics.
The paper is structured as follows. In Sect. \[s\_models\] we describe the input physics and the model calculations with our evolutionary synthesis code and with the photoionization code CLOUDY. The predictions from the synthesis models concerning the UV continuum, the ionizing flux, [Ly$\alpha$]{} and [He [ii]{} $\lambda$1640]{} emission are presented in Sect. \[s\_uv\_synthesis\]. In Sect. \[s\_cloudy\] we discuss the results from the photoionization models using black body spectra, explain the deviations from case B and provide simple formulae to describe these effects on [Ly$\alpha$]{}. In Sect. \[s\_sed\] we show how to connect realistic SEDs with results from photoionization models using black body ionizing spectra. Our results and several implications are discussed in Sect. \[s\_discuss\]. The main results are summarised in Sect. \[s\_conclude\].
Modeling techniques {#s_models}
===================
Synthesis models {#s_synthesis}
----------------
We have used the evolutionary synthesis code of @SV98. with the physical ingredients (stellar tracks, atmospheres, and prescriptions for nebular line and continuum emission) from @Scha03. In particular these models allow us to predict the integrated properties of stellar populations at all metallicities from zero (Population III) to “normal”, solar-like metallicity. The computations have been done for the same metallicities as in @Scha03.
[llrrrrrll]{} Model ID & colour code & & & $\alpha$ & $M_c$ & $\sigma$ & reference/comment & ID in files\
\
Salpeter & black & 1 & 100 & 2.35 & & & A in @Scha03 & S\
B & green, dashed & 1 & 500$^a$ & 2.35 & & & B in @Scha03 & B\
C & cyan, dashed & 50& 500$^a$ & 2.35 & & & C in @Scha03 & C\
Scalo & blue & 1 & 500$^a$ & 2.7$^b$ & & & @Scalo86 & Sc\
TA & red & 1 & 500$^a$ & & 10. & 1.0 & A in @Tumlinson06 & TA\
TB & magenta & 1 & 500$^a$ & & 15. & 0.3 & B in @Tumlinson06 & TB\
TE & yellow & 1 & 500$^a$ & & 60. & 1.0 & E in @Tumlinson06 & TR\
L05 & blue, dashed & 1 & 100 & & 5. & & @Larson98 & l0\
\
### Stellar initial mass function
The main extension presented here with respect to the calculations of @Scha03 concerns different assumptions regarding the stellar IMF. A wide range of IMFs has been considered, including power-law IMFs, such as the Salpeter or @Scalo86 IMFs, log-normal IMFs, and the @Larson98 IMF. The corresponding parameters are summarised in Table \[t\_imf\]. The stellar mass range is defined by the lower and upper mass cut-offs, and respectively. $\alpha$ being the slope of the power-law. The log-normal IMFs are described by the characteristic mass $M_c$ and its dispersion $\sigma$. The cases computed here correspond to the cases A, B, and E in the chemical evolution study of @Tumlinson06. The @Larson98 IMF is described by a single parameter, its characteristic mass $M_c$. We have computed one such case, assuming the same value of $M_c$ as @Ciardi2001 in their reionization calculations. Note, that at $Z \ge 0.0004 = 1/50 \zsun$ the upper mass cut-off is set to $\mup = 100$ or $120 \msun$ for all IMFs, since tracks for more massive stars are not available. The quantities discussed here are insensitive to assumptions on the IMF at low masses. Our absolute quantities may therefore simply be rescaled to other IMFs including e.g. an extension below 1 .
Current knowledge suggests that the IMF for massive stars is close to Salpeter with an upper limit of $\mup \sim$ 100–120 from solar metallicity down at least to $\sim$ 1/50 , and that a qualitative shift of the IMF towards higher characteristic masses occurs below a critical metallicity of the order of $Z_{\rm crit} \sim 10^{-5\pm 1} \zsun$ [@schne02; @schne03].
### Star formation histories
For each metallicity $Z$ and IMF we have computed evolutionary synthesis models for the two limiting cases of 1) an instantaneous burst, and 2) constant star formation (CSFR). Results for other star formation histories can be derived from the electronic files for the simple stellar population (burst) models (see Sect. \[s\_output\]). In both cases the calculations have been carried out with a small time step (0.1 Myr) to ensure the accuracy of the time integrated quantities for the CSFR case. The calculations have to be carried out up to ages of 1 Gyr. This covers the allowed ages and of galaxies at redshifts $z \ga 5.8$, of interest here, as well as sufficiently long star formation timescales to reach equilibrium in various observable properties (cf. below).
### Nebular emission
To include nebular emission (recombination lines and continuum processes) in our synthesis models we initially make the following “standard” simplifying assumptions [see @Scha02; @Scha03]: ionization bounded nebula, constant electron temperature and density ($T_e$, $n_e$), and case B. Case B in particular assumes that the recombinations to the ground-state immediately yield locally another ionization, and that photoionizations occur only from the ground-state. As we will see below, the latter may not be true in very metal-poor nebulae, leading to significant changes in the predicted spectrum of hydrogen. With these assumptions both the H and He recombination lines as well as nebular continuum emission (including free-free and free-bound emission from H, neutral He, and singly ionized He, and two-photon emission of H) are fully specified and their luminosity is proportional to the ionizing photon flux $Q$ in the appropriate energy range. To reflect to first order the changes of the conditions in the [H [ii]{}]{} regions with metallicity, the value of the line luminosity coefficient and nebular continuous emission coefficients at metallicities $Z/\zsun < 10^{-3}$ are changed as in @Scha03. More precisely we adopt $T_e=$ 30 (20) kK for lines (continua) $Z/\zsun < 10^{-3}$ and $T_e$=10 kK for higher metallicities, and a low ISM density $n_e=100$ cm$^{-3}$.
For the [Ly$\alpha$]{} luminosity we have, with the assumptions just spelled out: $$\label{eq_lyacaseb}
L_B({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}) = (1 - f_{\rm esc}) {\ifmmode Q({\rm H}) \else $Q({\rm H})$\fi}\, \times h \nu_{{\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}} \times \frac{\alpha^{\rm eff}_{2 ^2P}}{\alpha_B}$$ where the index ‘B’ stands for case B, $\alpha_B = \alpha^{\rm eff}_{2 ^2P} +\alpha^{\rm eff}_{2 ^2S}$ is the total case B recombination coefficient, and $f_{\rm esc}$ is the escape fraction of ionizing photons out of the [H [ii]{}]{} region (or galaxy). In all Figs. shown in this paper we assume $f_{\rm esc}=0$. In typical conditions $\alpha^{\rm eff}_{2 ^2P} / \alpha_B \approx$ 0.6–0.7. In other words approximately 2/3 of the Lyman continuum photons give rise to the emission of a [Ly$\alpha$]{}photon, the assumption commonly made. Similar relations also hold for other recombination lines, such as [He [ii]{} $\lambda$1640]{}, which is of special interest here. In our “standard” synthesis models we simply assume $$\begin{aligned}
\label{eq_lya}
L_B({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}) & = & Q({\rm H}) \, \times c_1, \\
\label{eq_heii}
L_B({\rm He~{\sc II} \lambda 1640}) & = & Q({\rm He^+}) \, \times c_2, \end{aligned}$$ with $c_1= 1.04 \times 10^{-11}$ erg, and $c_2= 5.67 \times 10^{-12}$ ($6.04 \times 10^{-12}$) erg for $Z \le 1/50$ ($>1/50$ ), and for $f_{\rm esc}=0$. The atomic data is from Hummer & Storey (1995) for low densities [cf. @Scha03], and [Q([H]{}) $Q({\rm H})$]{} and [Q([He\^+]{}) $Q({\rm He^+})$]{} are the ionizing photon flux (in photon s$^{-1}$) above 13.6 and 54 eV respectively.
Continuous nebular emission including free-free and bound-free emission by H, neutral He, He$^+$, and He$^{+2}$, as well as the two-photon continuum of hydrogen is included as described in @Scha02, assuming $T_e=20$ kK for $Z/\zsun \le 5.\times 10^{-4}$ and $T_e=10$ kK otherwise.
As we will show below (Sect. \[s\_cloudy\]), a proper treatment of all relevant processes leads to significant deviations from case B at very low metallicities, increasing in particular the [Ly$\alpha$]{} luminosity, $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$. In this case the $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ can be rewritten as $$L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}) = L_B({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})\times P \times \frac{\tilde{f}_{\rm coll}}{2/3},
\label{eq_caseb_corr}$$ where $P$ and $\tilde{f}_{\rm coll}$ are terms describing the mean photon energy in the Lyman continuum, and accounting for for collisional effects at high density. To compute the [Ly$\alpha$]{} equivalent width $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ we proceed as in @Scha02 [@Scha03], where we use the continuum flux at 1215.67 Å obtained from linear interpolation of the total (stellar + nebular) continuum (in $\log$) between 1190 and 1240 Å, chosen to avoid underlying stellar [Ly$\alpha$]{} absorption and other absorption lines if present. The predicted stellar [Ly$\alpha$]{} absorption is small compared to the emission, except for “post-starburst” phases [see e.g. Fig3. in @SV07].
### Model output {#s_output}
Our evolutionary synthesis code predicts a large variety of observable and related quantities derived from the detailed synthetic spectra. Here we focus on mostly on quantities describing the spectrum in the Lyman continuum, the UV (rest-frame) spectrum, as well as the [Ly$\alpha$]{}and [He [ii]{} $\lambda$1640]{} emission lines. The full set of model results, including also numerous other quantities not discussed in this paper, are available in electronic format upon request to one of the authors (DS), on [http://obswww.unige.ch/sfr]{}, or via the CDS.
Photoionization models {#s_photo}
----------------------
To predict more accurately the nebular emission from starbursts and to investigate possible departures from the simplified assumptions made in our synthesis models we use the photoionization code CLOUDY version 08 [@Ferland98]. The models we consider are ionization-bounded with a closed, spherical geometry and constant density. These assumptions imply in particular that all ionizing photons are absorbed in the [H [ii]{}]{} region, i.e.$f_{\rm esc}=0$. In certain circumstances, especially in high redshift galaxies, a fraction of the Lyman continuum photons are expected to escape [see e.g. @Gnedin08; @Wise09; @Razoumov09]. To first order the results obtained in this paper can simply be rescaled to such cases, as discussed below. The main input parameters of our models are: the spectral energy distribution (SED) of the ionizing source, the nebular density (given by $n_H$), the hydrogen number density), the ionization parameter ($U$), and the nebular metallicity (Z$_{\rm neb}$). For the SED we adopt black-body spectra described by T$_{bb}$ and SEDs from our evolutionary synthesis models. The ionization parameter (at the inner edge of the cloud) is defined as: $$U = \frac{Q(H)}{4 \pi r_{\rm in}^2 \times n(\rm H) \times c}$$ where $r_{\rm in}$ is the inner radius of the nebula which has been kept constant in our models (10$^{17}$ cm) and $c$ is the speed of light. The small inner radius used in our calculations results in a sphere-like (not shell-like) geometry of the nebulae. The ionization parameter can change throughout the nebula (decrease towards outer parts).
First, we have calculated the grid of photoionization models using the photoionization code CLOUDY covering $\log(U)$ = -4, -3, -2, -1; T$_{bb}$ = 40,000–150,000 K; $\log(n(\rm H))$ = 1, 2, 3, 4 for a primordial nebula and a number of higher metallicities. In total 192 models were computed for each metallicity. In order to keep the same ionization parameter while changing the density of the gas, we adapt the number of ionizing photons (keeping the shape of the SED). The metallicity is defined by scaling the solar abundance pattern.
Note that the electron temperature is neither constant nor fixed in our models. Its spatial distribution results from the computation of each photoionization model and it is a function of depth in the nebula. In practice it depends on all the parameters that are being investigated (T$_{bb}$, $n_H$, $U$, Z$_{neb}$). For the coolest primordial model (the coolest black body, the lowest density and the lowest ionization parameter) the temperature in the inner part of the cloud is around 12,000 K and for the hottest ones it reaches $\sim$ 38,000 K.
Selected models were subsequently computed using the SEDs from the synthesis models described above.
Predicted UV properties from synthesis models {#s_uv_synthesis}
=============================================
We now present and discuss one-by-one the main predictions of our synthesis models for different IMFs (see Table \[t\_imf\]), for metallicities from zero (Pop III) to solar, and for two different limiting star-formation histories (bursts and SFR=const). Since properties of stars below $Z \la 10^{-9}$ (i.e. $Z/\zsun \la 10^{-7.3}$) essentially converge to those of metal-free stars we assign this metallicity value to Pop III stars, as in @Scha03.
Note that all UV continuum predictions from the synthesis models described in this Section are based on the simplified assumptions spelled out above to compute nebular emission. This implies in particular that at low metallicities the contribution from the two-photon continuum process should be higher, increasing thus e.g. the predicted UV luminosity at 1500 Å, for the reasons discussed in Sect. \[s\_cloudy\].
Ionizing photon production
--------------------------
A quantity of interest, e.g. to determine the contribution of galaxies to cosmic reionization, is the the relative output of hydrogen ionizing photons to observable UV light. Here we provide $Q_H/L_{1500}$, where the Lyman continuum flux $Q_H$ is expressed in units of photon s$^{-1}$, and the UV restframe luminosity at 1500 Å is $L_{1500}$ in $L_\lambda$ ( $\AA^{-1}$) or $L_\nu$ ( Hz$^{-1}$) units [^1]. Alternatively, to derive the Lyman continuum output per unit stellar mass (or per baryon), $Q_H/L_{1500}$ discussed here can be combined with the ratio SFR$/L_{1500}$ given below, or can directly be derived from the available data files.
In Fig. \[f\_nlyc\] we show $Q_H/L_{1500}$ as a function of metallicity for constant star formation over 1 Gyr (CSFR, left panel), and very young ($\la$ 0–4 Myr) populations (right panel). As expected $Q_H/L_{1500}$ increases with decreasing metallicity, since the ionizing flux depends very strongly on the effective stellar temperature and hence increases more rapidly than the UV luminosity. The IMF dependence also behaves as expected, with the IMFs favouring the most massive stars showing also the highest the $Q_H/L_{1500}$ ratios, since $Q_H$ increases more rapidly with stellar mass than the UV luminosity. Notable is actually the increase of $Q_H/L_{1500}$ from to $\sim 10^{-4} \zsun$, where no major change of the IMF is expected (and a Salpeter IMF is favoured). For CSFR and for a fixed IMF, the increase of the relative ionizing power from solar metallicity to Pop III is typically $\sim$ 0.4–0.5 dex, i.e. a factor 2 to 3. When considering an IMF change from Salpeter to a massive IMF (i.e. all cases except Salpeter and Scalo) the increase of $Q_H/L_{1500}$ is larger, approximately 0.6 to 1 dex between solar and zero metallicity. Only for the “TB” IMF, a narrow, log-normal mass function peaked at $M =15 \msun$, we find a more extreme dependence on metallicity. This is precisely due to the fact that this IMF singles out a narrow mass range, instead of averaging the metallicity dependence of stellar properties over a larger interval in mass.
The right panel of Fig. \[f\_nlyc\] shows a narrower range of $Q_H/L_{1500}$ for zero age or very young ($\la$ 0–4 Myr) populations. This is natural, since in this case no “average” is made over populations of very different stellar ages and hence over strong variations of stellar parameters. More important is the fact that [*higher values of $Q_H/L_{1500}$ are obtained for young populations*]{}. This is mostly due to the fact that such populations emit a lower UV luminosity per unit SFR since a longer timescale is needed to reach the “equilibrium value” of the UV output (cf. below). The ZAMS values shown here correspond to the maximum of $Q_H/L_{1500}$ expected for stellar populations of different ages and SF histories.
To illustrate this dependence on the SF timescale, $Q_H/L_{1500}(t)$ is shown in Fig. \[f\_time\] for selected IMFs and metallicities. These curves show the smooth transition from the predicted “ZAMS” to the CSFR values over timescales from $\sim 10^7$ yr for massive IMFs (e.g. TE, TB) to $\sim$ 0.4–1 Gyr for the Salpeter IMF. Note also that the timescale for UV properties to reach equilibrium increases with decreasing metallicity, due to the higher effective temperatures on the ZAMS at low $Z$. In short, we caution that the relative ionizing photon to UV ratio $Q_H/L_{1500}$ may be uncertain by a factor of $\sim$ 4 depending on the SF timescale (for constant SF), or even more for bursts.
Finally, it should be noted that at low metallicity the contribution of the two-photon continuum may be larger, as shown later in Sect. \[s\_cloudy\], leading to somewhat lower values of $Q_H/L_{1500}$. For constant SF this amounts to a decrease of $Q_H/L_{1500}$ by $\sim 0.2$ dex for Pop III and the most extreme IMFs (TE, C), and smaller changes otherwise. For zero metallicity populations on the ZAMS $Q_H/L_{1500}$ should be reduced by $\sim$ 0.15–0.3 dex for all IMFs, and less at higher metallicity.
Properties of the ionizing spectra
----------------------------------
The properties of the ionizing spectra, such as their hardness, detailed shape and others have already been discussed in @Scha03 and shall not be repeated here. For example, the hardness [Q([He\^+]{})/Q([H]{}) $Q({\rm He^+})/Q({\rm H})$]{}, measured by the ratio of He$^+$ ($> 54$ eV) to hydrogen ionizing ($>13.6$ eV) photons, we predict from our new models are already bracketed by the the values predicted in @Scha03 (see their Fig. 5) for the Salpeter and the “C” IMF.
An interesting quantity describing the ionizing spectra is the average energy of the photons emitted in the Lyman continuum $\overline E(Q(\rm H))$ (see definition in Eq. \[eq\_emean\]). This quantity and its dependence on metallicity and IMF is plotted in Fig. \[f\_eq\] for constant SF (thick lines) and for the ZAMS (thin lines). The corresponding blackbody temperatures [T\_[bb]{} $T_{\rm bb}$ ]{} with the same mean ionizing photon energy are also shown for illustration. Typically $\overline E(Q(\rm H))$ is found to $\sim$ 1.5–2.5 times 13.6 eV, the ionizing potential of neutral hydrogen, and its behaviour with IMF, metallicity, and age behaves as expected. These values correspond to a range of blackbody temperatures from $\sim$ 50 to 120 kK for the hardest spectra.
On this Figure we also indicate the approximate metallicity limit [Z\_[coll]{} $Z_{\rm coll}$]{}, below which collisional effects lead to significant departures from case B, as shown below. In this metallicity range $\overline E(Q(\rm H))$ can also be used to compute more accurately the intrinsic [Ly$\alpha$]{} emission line strength (see Eq. \[eq\_lya\_final\]).
SFR calibrations from the UV continuum
--------------------------------------
Figure \[f\_sfr\] illustrates the variation of the UV luminosity for CSFR as a function of metallicity and for the different IMFs. Plotted is the conversion factor $c$, defined by SFR$=c \times
L_\nu$, where $L_\nu$ is the UV luminosity in units of Hz$^{-1}$, and SFR is the star formation rate in . As expected our model with Salpeter IMF agrees well with the widely used calibration from Kennicutt (1998) at after rescaling the latter by a factor 2.55 to account for our adopted value for the lower mass cut-off ($\mlow = 1 \msun$)[^2]. This Fig. clearly shows that in most cases the use of the Kennicutt calibration at low metallicity may overestimate the SFR, given the higher intrinsic UV output of such stellar populations. However, this may not be realistic since it relies on the assumption of CSFR over a long timescale ($\ga 10^{8.3 \ldots 9}$ yr). In younger populations the UV luminosity per unit SFR is lower, and hence $c$ and the determined SFR higher (e.g. Schaerer 2000).
In fact, the non-monotonous behaviour of SFR$/L_{1500}$ with metallicity observed in Fig. \[f\_sfr\] is due to the dependence of the stellar contribution to the total UV output at this wavelength. Indeed, the stellar UV output increases with decreasing $Z$ down to $\sim 1/50$ Z/, due to the decrease of the average temperature of stars over their lifetime. At even lower metallicities, however, their UV output (per unit SFR) [*decreases*]{} since the bulk of their flux is emitted at $\ll 1500$ Å[cf. Fig. 2 in @Scha03]. This implies, for a fixed IMF, a re-increase of SFR$/L_{1500}$ at $Z/\zsun \la 1/50$, which is only somewhat moderated by the increasing nebular contribution. Indeed, the latter contributes typically $\sim$ 10–40% of the flux at 1500 Å at SF equilibrium (see Fig. \[f\_neb\]) In other words neglecting the nebular continuum would lead to differences of $\sim$ 0.05–0.15 dex in the SFR calibrations. At low metallicity the contribution of the two-photon continuum may be larger, as shown in Sect. \[s\_cloudy\], leading to somewhat lower values of SFR$/L_{1500}$ than shown in Fig. \[f\_sfr\]. For Pop III and the most extreme IMFs (TE, C) this implies a downward revision of $\sim$ 0.2 dex.
Predicted [Ly$\alpha$]{} emission
---------------------------------
The [Ly$\alpha$]{} equivalent widths predicted by our standard models (using Eq. \[eq\_lya\]) for all IMFs and metallicities are shown with thin lines in Fig. \[f\_wlya\] for constant star-formation (left panel) and for the ZAMS (right), the latter representing the maximum of $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ for each IMF. The dependences of $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ are as expected from $Q_H/L_{1500}$ and the values shown here bracket those already presented in @Scha03 (with more limited variations of the IMF). To illustrate the departure from case B found at low metallicity from photoionization models (see Sect. \[s\_cloudy\]), we also show $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ computed from Eq. \[eq\_caseb\_corr\] in the low density regime (i.e. for $ \tilde{f}_{\rm coll}=2/3$, thick lines). Here, the equivalent widths are a factor $\sim$ 1.5–2.5 higher than our “standard” predictions (cf. Fig. \[f\_eq\]), since we assume that the continuum close to [Ly$\alpha$]{} is unchanged by this departure from case B.
A few words of caution about $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ are appropriate. First, note that for our computations of $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ we use the continuum flux at 1215.67 Å obtained from linear interpolation of the total (stellar + nebular) continuum (in $\log$) between 1190 and 1240 Å, chosen to avoid underlying stellar [Ly$\alpha$]{} absorption and other absorption lines if present (cf. above). While $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ is well defined theoretically, comparisons with observations require some caution, given the possible complexity of the continuous spectrum shortward (due to the IGM in particular) and longward of [Ly$\alpha$]{} (due to non-monotonic shape of the nebular continuum), and given different choices of broadband filters (see e.g. the simulations of Hayes & Östlin (2006)).
How much of the total radiative energy from the starburst is emitted in the [Ly$\alpha$]{} line? For constant star-formation the fraction of the [Ly$\alpha$]{}to the bolometric luminosity, $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})/L_{\rm bol}$, is shown in Fig. \[f\_lyafrac\] for all metallicities and IMFs. At solar metallicity and for a Salpeter IMF we obtain the well-known estimate of $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})/L_{\rm bol} \sim$ 3–6 % found in the first papers promoting [Ly$\alpha$]{} searches at high redshift (e.g. Partridge & Peebles 1967). The fraction of luminosity emitted in [Ly$\alpha$]{} increases with decreasing metallicity, due to the higher ionizing photon flux output per unit stellar mass. When case B departures at low metallicity are taken into account, we find that $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})/L_{\rm bol}$ can reach up to $\sim$ 20–40% depending on the IMF, i.e. up to 10 times more than expected from earlier calculations! The highest values are comparable to those from the photoionization models of @Panagia02 using very hot black body spectra. For younger populations $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})/L_{\rm bol}$ is less dependent on the IMF than for SFR=const shown here; values of $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})/L_{\rm bol}
\sim$ 0.15–0.20 (0.35–0.40) are obtained at low $Z$ with our standard (departure from case B) assumptions.
[He [ii]{}]{} line emission from very metal-poor starburst galaxies
-------------------------------------------------------------------
Our standard predictions for [He [ii]{} $\lambda$1640]{} (using Eq. \[eq\_heii\]) for constant star-formation and young bursts are shown in Fig. \[f\_wheii\]. They complement our earlier predictions in @Scha03, and show the expected behaviour. Clearly, strong nebular [He [ii]{} $\lambda$1640]{} emission from starbursts is only expected at very low metallicity and for IMFs producing enough massive stars. As we will show below, the predicted intensity of [He [ii]{} $\lambda$1640]{} (and other He$^+$ recombination lines) depends, however, also on the ionization parameter and on the ISM density to some extent. Complete photoionization models predict generally fainter [He [ii]{} $\lambda$1640]{}emission, as discussed in Sect. \[s\_cloudy\].
Importance of the nebular continuum
-----------------------------------
Figure \[f\_neb\] shows the contribution of nebular continuous emission to the total UV light at 1500 Å for all metallicities, IMFs, and for the usual limiting cases of star-formation histories. Whereas for “normal” metallicities and IMFs the contribution is relatively small ($\la$ 5% for SFR=const), the importance of the nebular continuum is larger for young bursts and/or low metallicity, as already stressed by @Scha02 and @Scha03.
Since the nebular continuum at 1500 Å is generally dominated by the two-photon continuous emission process, departures from Case B will lead to stronger nebular emission than shown here at very low metallicity (see Sect. \[s\_cloudy\]). To first order, the total two-photon emission is then enhanced by a factor $P$ for low ISM densities, increasing thus the contribution of the nebular continuum to the total (stellar + nebular) emission. At high density, the two-photon emission tends to zero. However, since the shape of the nebular continuum depends on the detailed conditions in the nebula (density, temperature, etc.) which are not constant as assumed in the synthesis models, it is not possible to predict more accurately how departures from case B affects the nebular continuum, without resort to photoionization models.
Predicted UV slope
------------------
From our synthesis models we also measured the slope of the UV continuum with and without nebular emission (cf. Schaerer & Pelló 2005). In Fig. \[f\_beta\] we show a condensed overview of various $\beta$-slopes for very young populations (ZAMS), representing the steepest slopes, i.e. the minimum for $\beta$, predicted from models. As before, the predictions are shown for all IMFs and metallicities. Thick (thin) lines show $\beta_{1550}$ ($\beta_{2000}$), defined as the slope between 1300–1800 (1800–2200) Å respectively[^3]. The upper set of lines shows the $\beta$ slopes of the total spectrum, including stellar and nebular continuum, the lower lines using the pure stellar spectrum.
Clearly, the UV slope is strongly affected by nebular emission, leading to a significant flattening of the spectrum [cf. Figs. 2 in @Scha03]. While the stellar SED becomes steeper with decreasing metallicity, the total spectrum exhibits the opposite behaviour. If we assume a varying contribution of the nebular continuum we may obtain any intermediate value of $\beta$ between the “stellar+nebular” and “pure stellar” cases. This fact and the dependence of $\beta$ on the star-formation history and age [see e.g. Fig. 1 in @Schaerer2005], shows that the UV slope cannot be used to determine metallicity. For constant star-formation (not shown here), the bulk of the models show equilibrium values of $\beta \sim$ -2.6 to -2, quite independently of metallicity and IMF.
Fig. \[f\_beta\] also shows a difference of the order of 0.2–0.3 between $\beta_{1500}$ and $\beta_{2000}$. Such differences may be relevant for comparisons of the UV slope estimated from broad-band filters. Last, but not least, the precise shape of the nebular continuum is determined by the detailed nebular structure (i.e. its detailed temperature and density structure) and can be affected by departures from case B, as the case of the two-photon continuum discussed in depth below.
Nebular predictions using photoionization models {#s_cloudy}
================================================
{height="20cm"}
For low metallicity nebulae ionized by very hot stars, the conventional case B predictions for line and continuum emission are not good approximations to the appropriate nebular astrophysics. In this Section we present the predictions of detailed photoionization modeling for such nebulae with metallicities ranging from zero (primordial = Pop III) to solar, explain the origin of departures from case B, and present a parameterisation of the results than can be readily employed for the interpretation of low metallicity nebulae. For simplicity, we first use black body spectra as ionizing sources and discuss later how to compare these with stellar SEDs. We have examined the nebular emission as a function of the stellar (black body) temperature T$_{bb}$, the hydrogen number density $n_H$, the ionization parameter (U), and the nebular metallicity (Z$_{neb}$), using the models described above (Sect. \[s\_photo\]).
Our analysis will focus mainly on the [Ly$\alpha$]{} line and the associated two-photon continuum emission, on nebular [He [ii]{}]{} emission, and on the global nebular continuum.
[Ly$\alpha$]{} line and two-photon continuum emission
-----------------------------------------------------
In Fig. \[all\_prim\] we show the deviation of the [Ly$\alpha$]{} (black symbols) and 2$\gamma$ emission (red) and other quantities as predicted from photoionization models with primordial composition and different densities with respect to their case B values. To calculate case B luminosities we adopt Eq. \[eq\_lya\] and assume $L(2\gamma)=0.5\, c_1 Q({\rm H})$ for the luminosity in the two-photon continuum. The numerical factor $0.5 \approx \alpha^{\rm eff}_{2 ^2S}/\alpha^{\rm eff}_{2 ^2P}$ is appropriate for low densities. For simplicity we always compare the results to the low density limit case B predictions.
At low density, the luminosity of both [Ly$\alpha$]{} and 2$\gamma$ emission are increased by a factor of $\sim$ 1.3 to 3 over the black body temperature range considered here. The physical reason for this strong departure from case B is due to collisional effects, which increase the population of the $n=2$ level of hydrogen from which additional ionizations can take place, leading overall to an increased ionization rate in the nebula. In equilibrium, this implies an increased recombination rate and higher [Ly$\alpha$]{} and 2$\gamma$ luminosities. Collisional excitation is significant in (very) low metallicity nebulae with hot ionizing sources because radiative cooling is reduced, leading to higher electron temperatures. Photoionization from excited states (here from $n=2$), in particular, is not taken into account under case B assumptions.
To describe quantitatively the effect of enhanced photoionization rates, including ionization from the excited $n=2$ state, it is sufficient to consider the mean energy of the ionizing photons in units of the ionization potential of hydrogen, $P$, given by $$P = \frac{\overline E}{13.6~{\rm eV}} =\left(\frac{ \int_{\rm 13.6~eV}^{\infty} F_{\nu}d\nu}{\int_{\rm 13.6~eV}^{\infty}\frac{F_{\nu} }{h\nu} d\nu}\right) / \left(13.6 ~{\rm eV}\right),
\label{eq_emean}$$ where $\overline E$ is the average ionizing photon energy in the Lyman continuum in units of eV. Indeed, as Figure \[all\_prim\] shows, the enhancement of the [Ly$\alpha$]{} and two-photon continuum emission, which both measure the effective recombination rate, scales very accurately with $P$ at low density. This scaling shows that the available energy of the Lyman continuum photons – which is in excess of the necessary minimum of 13.6 eV – is “optimally” used to maximise the number of photoionizations, leading to an effective increase of the ionization rate from $\propto
Q({\rm H})$ to $\propto P \times Q({\rm H})$, and hence to the same increase in the recombination rates, i.e. also in the [Ly$\alpha$]{} and 2$\gamma$ luminosities.
The effect of “boosted” [Ly$\alpha$]{} and 2$\gamma$ emission just discussed depends on the nebular metallicity. This is because more metals are present and more efficient cooling results in cooler gas than in the primordial metallicity case, decreasing the collisional effects for H. The range over which these effects take place is discussed below.
### Density effects
Figure \[all\_prim\] also shows that [Ly$\alpha$]{} is further enhanced at high densities at the expense of two-photon emission, the sum of the two luminosities being essentially constant if all other parameters are kept fixed. Collisional mixing of the relative populations of the $n=2$ levels between $2 ^2S$ and $2 ^2P$ of hydrogen alter the relative importance of the rates of [Ly$\alpha$]{} and two-photon transitions resulting in increased line emission relative to the 2$\gamma$ continuum.
To correct for this effect under case B assumptions, @Panagia73 [see also @sti09 [-@sti09]] writes $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}) = {\ifmmode Q({\rm H}) \else $Q({\rm H})$\fi}\, \times h \nu_{{\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}} \times f_{\rm coll}$ (cf. Eq. \[eq\_lyacaseb\]), where $$f_{\rm coll} \approx \frac{1 + a \times n_p}{1.5 + b \times n_p},
\label{eq_fcoll}$$ with $a=b=1.35\times10^{-4}$, and where $n_p$ is the proton density. In this way the factor $f_{\rm coll}$ ranges from $2/3 \approx \alpha^{\rm eff}_{2 ^2P} / \alpha_B$ for low densities to $f_{\rm coll}=1$ for high densities. Under these assumptions, the two-photon continuum luminosity is $L(2\gamma)={\ifmmode Q({\rm H}) \else $Q({\rm H})$\fi}\, \times h \nu_{{\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}} \times (1-f_{\rm coll})$.
To derive the corresponding numerical factor from our detailed photoionization models and to separate the density effect from the enhancement factor $P$ found above, we define $\tilde{f}_{\rm coll}$ through: $$L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}) = (1 - f_{\rm esc}) {\ifmmode Q({\rm H}) \else $Q({\rm H})$\fi}\, \times h \nu_{{\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi}} \times P \times \tilde{f}_{\rm coll}.
\label{eq_lya_final}$$ The corresponding values of $\tilde{f}_{\rm coll}$, obtained for our grid of models for primordial nebulae are plotted in Figure \[f\_coll\_comp\] as a function of the input hydrogen density. Also shown is, for comparison, Panagia’s expression (Eq. \[eq\_fcoll\]) assuming $n_p=n_H$. Notice that $\tilde{f}_{\rm coll}$ presented here is not derived from the [Ly$\alpha$]{} and 2$\gamma$ luminosities (which ranges from 2/3 to 1 as in Panagia’s formula) but it is the fitting formula which has been calculated by comparing the right hand side of equation \[eq\_lya\_final\] with our CLOUDY results.
For convenience, we derive the following 3-parameter fit: $$\tilde{f}_{\rm coll} \approx \frac{1+ a \times n_H}{b + c \times n_H}.
\label{f_coll_fit}$$ with $a=1.62 \times 10^{-3}$, $b=1.56$ and $c=1.78 \times 10^{-3}$. The coefficients obtained here differ from those of @Panagia73 due to the fact that we have used a suite of photoionization models covering a wide range of T$_{bb}$ and $U$ resulting in different T$_e$ and a different balance between processes in the nebula. However, the change in $\tilde{f}_{\rm coll}$ has little effect on the values of $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ for low density. We use $\tilde{f}_{\rm coll}$ defined in this way for convenience since it allows us to reproduce the CLOUDY results. As Fig. \[f\_coll\_comp\] shows, our fit formula allows us to describe density effects with an accuracy of $\pm 10 \%$ at high density and somewhat better at low $n_H$.
The black dashed lines in Figure \[all\_prim\], showing $P \times \tilde{f}_{\rm coll} / \tilde{f}_{\rm coll}(\rm log(n_H=1))$, show how well our analytical prediction for [Ly$\alpha$]{}, including also density effects, works at primordial metallicity.
### Applicable metallicity range
The effect of collisional excitations, allowing for photoionization of H from excited states, depends on the electron temperature of the nebula, and hence on its metallicity. To find the range of nebular metallicity where our analytic expressions for $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ holds, we have computed model grids at different metallicities. We can then ascertain how much our fit to the primordial case deviates from CLOUDY models as a function of metallicity.
The critical metallicity where the luminosity obtained by our fit deviates on average (for the entire grid of models) by 10 % from the correct one is $Z_{\rm coll} \approx 0.03$ . At higher metallicities, cooler nebulae are produced and hence collisional effects play a smaller role in boosting hydrogen emission. However, the effect only decreases gradually for higher metallicities and depends on the detailed condition of the gas. For instance, in our simulations for Z$_{\rm{neb}}=0.05$ , the average deviation of the results obtained with our formula (primordial case) with respect to the CLOUDY results is 13%, while for Z$_{\rm{neb}}=0.1$ it is 20%. The value of the “transition” metallicity $Z_{\rm coll} \approx 0.03$ below which [Ly$\alpha$]{} (and the 2$\gamma$ continuum) are significantly boosted (modified) by departures from case B, should therefore only be taken as an indicative value. Tailored photoionization models are necessary for more accurate predictions.
Other H lines
-------------
The luminosities of other hydrogen lines are close to their case B predictions as is shown in Figure \[all\_prim\] for H${\alpha}$ (green symbols) and H${\beta}$ (blue symbols). They are basically independent of $U$.
These lines are used to calculate the Balmer decrement which is conventionally used to measure the extinction. The Balmer decrement (H${\alpha}$/H${\beta}$) obtained from our grid of models for $\log(U)=-1$ is shown in Figure \[balmer\_dec\]. It increases towards higher [T\_[bb]{} $T_{\rm bb}$ ]{} and exceeds the values commonly used, for example 2.86 for $n_e=10^2$ and $T_e=10$ kK (Dopita & Sutherland 1996). The reason for the enhanced H${\alpha}$/H${\beta}$ ratio is the collisional excitation of hydrogen, one of the contributors to enhanced [Ly$\alpha$]{} and 2$\gamma$ emission. This effect (see Osterbrock 2006) occurs when photoelectrons carrying enough energy when colliding with H atoms, excite higher levels followed by radiative cascade. Enhanced H$\alpha$ emission increases the Balmer decrement which must be accounted for when calculating the extinction. An enhanced Balmer decrement was already found, e.g. in tailored photoionization models for the metal-poor galaxy I Zw 18 [@Davidson85; @Stasinska99] and other giant [H [ii]{}]{} regions [cf. @Luridiana03], and has also been addressed by @Luridiana09.
[He [ii]{}]{} lines {#s_heii}
-------------------
The [He [ii]{}]{} 1640 Å line is of a particular interest since it has been considered as one of the signatures of Pop III or very metal-poor stars, which are expected to have a very hard ionizing spectrum emitting copious He$^+$ ionizing photons. Figure \[all\_prim\] (orange symbols) shows [He [ii]{}]{} 1640 Å emission line luminosities relative to their case B values, calculated for the same $T_e$ and $n_e$ as the hydrogen lines. For lower ionization parameters, the line becomes weaker. This is due to an effect already discussed by @Stasinska86 in the context of planetary nebulae. Photons with sufficient energy to ionize He$^+$ are also able to ionize hydrogen. When calculating the luminosity of [He [ii]{}]{} in synthesis models, it is assumed that every photon with an energy $>$54 eV ionizes one He$^+$ ion. The absorption of some high energy photons by H atoms results in a decrease in the number of photons available for producing [He [ii]{}]{}emission. The effect becomes significant at low ionization parameters since then the He$^+$/H$^0$ fraction decreases and the probability of the high energy photon being absorbed by a H atom rather than by He$^+$ is higher. This means that the synthesis models give an upper limit for the [He [ii]{}]{} luminosity. We have not found a simple analytical prescription to account for the effect of this process on the [He [ii]{}]{} luminosity.
The behaviour of the equivalent width of [He [ii]{}]{} is discussed in Sect. \[s\_sed\].
Nebular continuum emission
--------------------------
Nebular emission arising in [H [ii]{}]{} regions can contribute significantly to the measured spectrum. The total nebular continuum is the sum of the free-free, free-bound and 2$\gamma$ continua of hydrogen and helium. Several nebular parameters affect its shape. It depends on the electron temperature, ionization parameter, nebular metallicity and particle density in the gas, as e.g. illustrated by Bottorff et al. (2006). In Figure \[cont\_hden\] we show the density dependence of the nebular continuum, which mainly affects the 2$\gamma$ emission and hence also the shape of the continuum between [Ly$\alpha$]{} and the Balmer jump. Figure \[cont\_zneb\] shows the dependence on the nebular metallicity. For higher metallicities, the increased cooling lowers the total emission in the nebular continuum, and its shape is altered as expected from atomic physics (see Bottorff et al. 2006) due to the decrease of the average electron temperature at higher metallicity. Evolutionary synthesis models such as ours, cannot properly describe the variety of shapes and strengths of the nebular continuum shown here, since they rely on simple assumptions such as constant nebular density and temperature. Again, tailored photoionization models are necessary for more accurate predictions.
Photoionization models for realistic SEDs {#s_sed}
=========================================
In this Section we show how to relate realistic SEDs obtained e.g. from evolutionary synthesis models with the results from the photoionization models discussed above, which used black body spectra. Finally, we show the updated predictions for the equivalent widths of [Ly$\alpha$]{} and [He [ii]{} $\lambda$1640]{} lines obtained from our CLOUDY models.
How to connect realistic SEDs with black body calculations
----------------------------------------------------------
{height="20cm"}
Since the number of H ionizing photons and their mean energy are the main parameters determining the luminosity of the H recombination lines and of the 2$\gamma$ continuum, $Q(H)$ and $P$ suffice to compute these quantities from arbitrary SEDs. To account also for density effects, Eqs. \[eq\_lya\_final\] and \[f\_coll\_fit\] can be used to yield the correct [Ly$\alpha$]{} luminosity. Numerical tests using SEDs described in Sect. \[s\_uv\_synthesis\] show the validity of this result, typically within 2-3 %. For black bodies the relation between $P$ and [T\_[bb]{} $T_{\rm bb}$ ]{} is given by $${\ifmmode T_{\rm bb} \else $T_{\rm bb}$ \fi}[\rm kK] = -53.42 + 69.85 \, P
\label{eq_p_bb}$$ to a good approximation. The black body temperature ${\ifmmode T_{\rm bb} \else $T_{\rm bb}$ \fi}(P)$ corresponding to the more realistic SED is thus easily determined.
For other quantities, such as the [He [ii]{}]{} line luminosity, the correspondence between black bodies and other SEDs is different. For example, since the relative line ratios of [He [ii]{}]{}/H lines depend to first order on the relative number of He$^+$/H ionizing photons, the black body with the same hardness [Q([He\^+]{}) $Q({\rm He^+})$]{}/[Q([H]{}) $Q({\rm H})$]{} is the most appropriate. For black body spectra one has, to a good approximation: $$\begin{aligned}
\lefteqn{ {\ifmmode T_{\rm bb} \else $T_{\rm bb}$ \fi}[\rm kK]=314.6 + 382.4\,x + 268.7\,x^2 +{}}
\nonumber \\
& & {} + 103.3\,x^3 + 20.12\,x^4 + 1.545\,x^5. {}
\label{eq_tx}\end{aligned}$$ where $x = \log({\ifmmode Q({\rm He^+}) \else $Q({\rm He^+})$\fi}/{\ifmmode Q({\rm H}) \else $Q({\rm H})$\fi})$. Again, we have tested a number of models and confirmed that one can get the same luminosity (within a few %) using SEDs and corresponding black body models, for otherwise identical nebular parameters. Differences of $\sim 10$% can appear in case of significantly (several eV) different values of $P$. In principle not only the stellar [Q([He\^+]{}) $Q({\rm He^+})$]{}/[Q([H]{}) $Q({\rm H})$]{} ratio determines the [He [ii]{}]{} emission, but also the conditions in the gas. Since $P$ establishes the electron temperature, significant differences in $P$ result in different luminosity coefficients (recombination rates and emissivities).
In Fig. \[cont\_comp\] we show such a comparison between a realistic SED from our synthesis models (Pop III, Salpeter IMF 1–100 M$_{\odot}$ at zero age, instantaneous burst) and the appropriate black body, chosen such as to reproduce the correct H line luminosities and nebular continuum. Here the average photon energy in the Lyman continuum is $\overline E=30.033$ eV, hence ${\ifmmode T_{\rm bb} \else $T_{\rm bb}$ \fi}=102$ kK. The input spectra, shown on the left panel, are scaled to the same total Lyman continuum flux $Q(H)$. The nebular parameters are $\log(n_H) = 1$ , $\log(U)=-1$, and primordial composition.
Several interesting points are illustrated with this figure. First, as the left panel shows, it must be remembered that realistic SEDs show a Lyman break due to the hydrogen opacity in the atmosphere of the hot stars responsible for the flux. For young ages this break corresponds typically to a flux increase by $\sim$ 0.2-0.3 dex at 912 Å, as discussed e.g. by @Scha03[^4]. In consequence, calculations relying on black body spectra underestimate the observable UV continuum, as shown in the right panel of Fig. \[cont\_comp\], leading to non-negligible differences e.g. for equivalent widths predictions of emission lines (cf. below). In Fig. \[cont\_comp\] (right panel) we also show the stellar + nebular SED predicted by our synthesis models assuming constant nebular density and temperature and case B (green line). The differences between this and the full CLOUDY model are mainly due to the boosting of the 2$\gamma$ continuum discussed earlier, and to varying $n_e$ and $T_e$.
[Ly$\alpha$]{} equivalent width predictions
-------------------------------------------
The measurement of the equivalent width of [Ly$\alpha$]{} is both observationally and interpretationally difficult, since the continuum around it is affected on one side at high redshifts by [Ly$\alpha$]{}forest absorption and on the other by an unknown combination of 2$\gamma$ continuum and starlight. Furthermore, when not measured from spectroscopy, it is common practice to measure $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ via a line to continuum ratio with the continuum estimated at another (usually longer) wavelength $\lambda_{\rm cont}$, e.g. $W_{\lambda}=L_{\rm line}/ F_{\lambda_{\rm cont}}$. Different methods to measure $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ have been reviewed by @Hayes06. To illustrate the possible impact or uncertainty related to the way the continuum is estimated we subsequently plot [Ly$\alpha$]{} equivalent widths predicted from the CLOUDY models using the continuum at both 1240 and 1500 Å as a reference.
Figure \[ew\] shows the predictions for $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ from all CLOUDY models for primordial composition, as a function of [T\_[bb]{} $T_{\rm bb}$ ]{}, ionization parameter and hydrogen density. Black symbols show the predictions using black body spectra and accounting for the nebular continuum; red symbols the same but neglecting the nebular continuum. Blue symbols show the results from CLOUDY models using SEDs from our Pop III synthesis models for zero age populations, plotted at the corresponding ${\ifmmode T_{\rm bb} \else $T_{\rm bb}$ \fi}(P)$ value (see above). Pink symbols show $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ predicted from our standard synthesis models neglecting the boost of [Ly$\alpha$]{} (case B departure).
Overall $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ increases with the [T\_[bb]{} $T_{\rm bb}$ ]{} or equivalent since the [Ly$\alpha$]{}luminosity ($\propto Q(H) \times P$) increases more rapidly than the continuum flux close to [Ly$\alpha$]{}. For very hot models and if 1500 Å is taken as a reference for the continuum, $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ tends to a maximum value (here $\sim$ 2000 Å, as shown in col. 2) since line emission and the dominating nebular continuum scale in the same manner. For the reasons discussed above (absence of the Lyman break), predictions based on black body spectra (black and red symbols) overestimate $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ compared to models including more realistic stellar SEDs (in blue). Differences between black body models including or neglecting the nebular continuum (black and red) decrease with increasing ISM density, because of the decrease of the 2$\gamma$ continuum. This also explains the decreasing difference between $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ using 1240 and 1500 Å as a reference, when $n_H$ increases. Last, but not least, $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ predicted by our standard synthesis models for the ZAMS (pink symbols) fall below the more accurate CLOUDY predictions (blue), since the latter account for case B departures. Once applying the simple correction given by Eq. \[eq\_caseb\_corr\] one obtains the result which takes into account the increase in [Ly$\alpha$]{} line flux. However, that is only the first order correction which does not include the difference coming from the continuum, mainly a similar increase in the 2$\gamma$ continuum. In cases when the continuum is measured around [Ly$\alpha$]{} (1240 Å) stellar flux is dominant so the correction described above gives a good estimate of the correct $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ value (see Fig. \[ew\_lya\_only\]).
![Equivalent widths ($W_{\lambda}$) of [Ly$\alpha$]{} at 1240 Å (first column, second row of Fig. \[ew\]). All the symbols as in Fig. \[ew\]. Additionally, the green symbols show the case B SED results (pink symbols) corrected using our formula (first order correction; only the enhancement in the line flux is taken into account, there is no correction for the continuum). The difference coming from not correcting for the nebular continuum enhancement is visible for the hottest model.[]{data-label="ew_lya_only"}](15236fg22.ps){height="10cm" width="\columnwidth"}
[He [ii]{} $\lambda$1640]{} equivalent width
--------------------------------------------
The right column in Fig. \[ew\] shows the predicted [He [ii]{} $\lambda$1640]{} equivalent widths for the same CLOUDY models except that ${\ifmmode T_{\rm bb} \else $T_{\rm bb}$ \fi}(x)$ from Eq. \[eq\_tx\] is used here . The nebular continuum predicted at 1640 Å is included in all calculations of $W(1640)$. In contrast to [Ly$\alpha$]{}, the [He [ii]{}]{} equivalent width depends on the ionization parameter, since the line luminosity changes due to the “Stasinska-Tylenda effect”. For high ISM densities $W(1640)$ increases due to the decrease of the nebular continuum. Differences between models assuming black body spectra (black symbols) and realistic SEDs (blue) are minor for $W(1640)$ as long the nebular continuum dominates at these wavelengths. This is true for sufficiently hot spectra ($\gtrsim$ 80 kK). Compared to predictions from our standard evolutionary synthesis models (pink symbols) the [He [ii]{} $\lambda$1640]{} equivalent widths predicted by CLOUDY are significantly lower for ${\ifmmode T_{\rm bb} \else $T_{\rm bb}$ \fi}\ga 100$ kK. This is due to two effects, the Stasinska-Tylenda effect reducing the line flux, and the increase of the 2$\gamma$ nebular continuum. At lower [T\_[bb]{} $T_{\rm bb}$ ]{}, case B predictions give an upper limit (corresponding to our highest ionization parameter case) for [He [ii]{} $\lambda$1640]{}. Only for a high ionization parameter and/or high density do photoionization models predict $W(1640)$ values as high as those given by @Scha03.
Summary: correction of evolutionary synthesis model results for departures from case B
--------------------------------------------------------------------------------------
In this section we give a recommended step-by-step procedure which allows the correction of the synthesis models for the departures from case B in ionization bounded regions (i.e. with no leakage of Lyman continuum photons), relevant especially for the case of very hot stars and very low metallicities.
For L(Lya):
- Take the SED (function of age, IMF, ...).
- Get (or compute) the mean energy of Lyman continuum photons, $P$, in units of 13.6 eV.
- Compute the “collisional” factor $\tilde{f}_{\rm coll}$ from our fit formula (Eq. \[f\_coll\_fit\]).
- Multiply the synthesis model result $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ by $P*\tilde{f}_{\rm coll}/(2/3)$ (Eq. \[eq\_caseb\_corr\]).
For W([Ly$\alpha$]{}): divide the equivalent width obtained by the synthesis model of interest by 2/3 and multiply it by $P*\tilde{f}_{\rm coll}$. This provides the first order correction for $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$, neglecting the increase of the nearby nebular continuum. An example following this procedure is shown by the green symbols in Fig. \[ew\_lya\_only\].
For [He [ii]{}]{} recombination lines: no simple correction is possible. Synthesis models give an upper limit for the luminosity, an upper limit for equivalent width up to log$({\ifmmode Q({\rm He^+}) \else $Q({\rm He^+})$\fi}/{\ifmmode Q({\rm H}) \else $Q({\rm H})$\fi}) \sim -1.2$ for the primordial metallicity case and overestimate it for the highest [Q([He\^+]{}) $Q({\rm He^+})$]{}/[Q([H]{}) $Q({\rm H})$]{} cases (low density) due to the lack of accounting for enhanced 2$\gamma$ emission.
Discussion and implications {#s_discuss}
===========================
Dependence on model assumptions {#s_dependence}
-------------------------------
All our CLOUDY models have been computed for ionization bounded nebulae corresponding to no escape of Lyman continuum photons, i.e. $f_{\rm esc}=0$. For density (or matter) bounded cases with $f_{\rm esc}>0$, quantities such as hydrogen recombination line luminosities and the luminosity of the nebular continuum longward of [Ly$\alpha$]{} can be rescaled to first order by scaling with $(1- f_{\rm esc})$, as already introduced in Eq. \[eq\_lya\_final\]. Such a scaling is appropriate since the corresponding emissivities are approximately constant across the nebula. However, since He$^+$ recombination lines (if present) are emitted only in the innermost parts of the nebula, [He [ii]{}]{} line luminosities will not be reduced in density bounded regions, as long the radius of the He$^+$ sphere remains smaller than that of density bounded [H [ii]{}]{} region. These behaviours are also expected in non-spherical regions, as demonstrated e.g. by @Johnson09. In consequence, both $L({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ and $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ will decrease with increasing Lyman continuum escape (although the latter not proportionally with $f_{\rm esc}$), whereas the opposite will be true for $W($[He [ii]{} $\lambda$1640]{}$)$ and for other [He [ii]{}]{} lines, where the underlying continuum is mostly of nebular origin. In other words, the decrease of $W($[He [ii]{} $\lambda$1640]{}$)$ found in this paper due to competition with H ionizing photons, may be mitigated in objects with significant leakage of Lyman continuum radiation, leading again to higher [He [ii]{}]{} equivalent widths [cf. @Scha02; @Scha03]. For other effects of “leaking”, i.e. star-forming galaxies with density bounded [H [ii]{}]{} regions, see the photoionization models of @Inoue10.
We shall now discuss some implications of our modeling for the interpretation of the hydrogen and helium emission lines and the nebular continuum observed in high redshift emission line galaxies.
Lyman continuum output
----------------------
As metallicity decreases, the Lyman continuum output of the stellar population, both relative to the observed stellar UV continuum longward of [Ly$\alpha$]{} and per unit stellar mass, increases. The amount of this increase depends obviously also on the detailed shape of the IMF. For example, for a Salpeter IMF with a uniform upper mass cut-off of 100 , the ionizing photon rate per unit UV luminosity, $Q_H/L_{1500}$, increases by a factor 3 approximately between solar an zero metallicity. For the most extreme IMF considered here this increase is up to a factor 10 (see Fig. \[f\_nlyc\]). The effect of this is to increase the line and nebular continuum emission relative to any observable stellar continuum. In extreme cases [e.g., @Raiter2010] the nebular emission may completely dominate the UV/visible/NIR spectrum. Also, if low metallicity objects and/or different IMFs are relevant for the sources of cosmic reionization, their intrinsic Lyman continuum flux, generally estimated from UV restframe observations, may need to be revised accordingly.
SFR(UV)
-------
The estimation of SFR from the level of the UV continuum [cf. @Kenn98] is based on evolutionary synthesis models assuming “standard” stellar populations and IMFs. At metallicities below solar, the output of observable UV continuum radiation (i.e. typically at 1500 Å restframe) per unit stellar mass increases somewhat, due to the increasing average temperature of the stars (cf. Fig. \[f\_sfr\]). A maximum in $L_{1500}$ is typically reached at $Z/z\sun \sim 1/100$. Below this, the average spectrum shifts more strongly into the Lyman continuum, leading to a decrease of stellar radiation in the observable UV, which is, however, compensated by increasing nebular emission. The net result, illustrated in Fig. \[f\_sfr\] for constant SFR, is that star-formation rates derived from the UV may need to be revised downward at low metallicities with respect to the “standard” calibration of @Kenn98, but typically by less than a factor 2.
[Ly$\alpha$]{}
--------------
One of the main findings of our study is the enhancement of [Ly$\alpha$]{} emission at low metallicities with respect to the commonly used case B value. This departure from case B occurs even at moderaly low metallicity, say $Z/\zsun \la 1/10$, and becomes more important for lower $Z$. In consequence, higher intrinsic [Ly$\alpha$]{} line luminosities relative to the UV continuum (to $L_{\rm bol}$, SFR, or to another measure of the rate of massive star-formation) are expected and higher [Ly$\alpha$]{} equivalent widths. Even higher values of $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ can be obtained in regions with a high ISM density, where [Ly$\alpha$]{} emission can further be “boosted” at the expense of the 2-photon nebular continuum (see Fig. \[all\_prim\]).
With respect to our earlier predictions [cf. @Scha03], $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$ is increased by a factor 2–2.5 for Pop III objects, and by more than 70% at moderately low metallicity ($Z/\zsun \la 1/20$, cf. Fig. \[f\_wlya\]), for a given IMF. This increase of [Ly$\alpha$]{} may help to understand objects with large [Ly$\alpha$]{} equivalent widths found in some surveys, without the need for recourse to unusual IMFs or to a clumpy ISM [cf. @Malhotra02; @Daws04; @Fin08].
Finally, the enhanced [Ly$\alpha$]{} emission found here should represent good news for searches for very distant/early galaxies, since the intrinsic [Ly$\alpha$]{} emission of metal-poor objects is shown to be considerably higher than previously thought, reaching up to 20–40% of $L_{\rm bol}$ in Pop III dominated objects. Of course, the intrinsic [Ly$\alpha$]{} emission (and hence also $W({\ifmmode {\rm Ly}\alpha \else Ly$\alpha$\fi})$) will be lower in objects, having a non-zero escape of flux from the Lyman continuum. Furthermore, several processes exist (e.g. dust absorption, scattering out of the line-of-sight in the ISM and/or IGM), which will reduce the [Ly$\alpha$]{} emission on the way from the source to the observer.
[He [ii]{}]{} emission
-----------------------
The reduced [He [ii]{}]{} emission found in this paper, compared to our earlier predictions assuming simple case B recombination, may indicate that ongoing searches for signatures of Pop III stars using this feature [see e.g. review by @Schaerer08] could be more difficult than anticipated so far. The corresponding upper limits of Pop III star-formation rate density derived e.g. from the survey of @Nagao08 need then to be revised upwards.
The detection of the [He [ii]{} $\lambda$1640]{} line indicates a significant number of high energy photons in the ionizing spectrum. This line has therefore been sought, but not detected so far at high-$z$ (Nagao 2008). Prescott et al. (2009) have detected this line from a $z$=1.67 spatially extended nebula (or [Ly$\alpha$]{} blob) but have not been able to determine the nature of the ionizing source. We find that the line (and therefore the equivalent width) could be fainter than expected previously, due to its dependence on the ionization parameter. This suggests that deeper observations will be needed to detect it. Additionally, another difficulty comes from the fact that for the hottest (the highest ${\ifmmode Q({\rm He^+}) \else $Q({\rm He^+})$\fi}/{\ifmmode Q({\rm H}) \else $Q({\rm H})$\fi})$ ratio) stars the strongest enhancement in 2$\gamma$ continuum is predicted which further decreases the measured $W(1640)$. However, if Pop III stars are present in objects with significant escape fractions in the Lyman continuum, the [He [ii]{} $\lambda$1640]{} equivalent width may again be stronger, due to the reduced nebular continuum (cf. Sect. \[s\_dependence\]).
Nebular continuous emission
---------------------------
The overall nebular continuum emission is dependent on several nebular parameters and can result in a variety of spectral shapes (Fig. \[cont\_zneb\] and \[cont\_hden\]). Note that our simulations have been carried out in the absence of dust which would add another parameter affecting the shape of the continuum.
Proper photoionization modeling shows the importance of the 2$\gamma$ continuum produced in a nebula ionized by very hot stars. At low ISM density and primordial/low metallicity, it is boosted in the same way as [Ly$\alpha$]{} and can completely dominate the nebular emission at 1216–1600 Å. This can affect the broadband flux measurements and reduce the equivalent widths of UV emission lines, particularly [He [ii]{}]{}. The strength of the nebular continuum and the expected variations of its spectral shape also indicate that measurements of the [Ly$\alpha$]{} equivalent width from photometry may be more uncertain than naively expected [cf. @Hayes06].
As can be seen from Fig. \[cont\_comp\], the slope of the UV continuum between $\sim$ 1300 – 2000 Å obtained from our CLOUDY model is very similar to that predicted by the evolutionary synthesis models assuming the same case B assumption, constant electron density and temperature. When the 2$\gamma$ continuum is included, one cannot obtain significantly steeper slopes than found for this Pop III simulation, since the electron temperature which affects the shape of 2$\gamma$ emissivity [cf. @Bottorff06] becomes shallower as metallicity increases.
Conclusions {#s_conclude}
===========
Building on the earlier calculations of @Scha02 [@Scha03], we have computed evolutionary synthesis models for a wide range of metallicities from Pop III (zero metallicity) to solar metallicity covering a wider range of IMFs – including power laws and log-normal IMFs with different characteristic masses – than before. Using these synthesis models, we present the expected UV properties of star-forming galaxies, including their Lyman continuum fluxes, UV luminosity, and properties of the continuum (e.g. $\beta$-slopes), as well as properties of important emission lines such as [Ly$\alpha$]{} and [He [ii]{} $\lambda$1640]{} (see Sect. \[s\_uv\_synthesis\]).
To investigate possible departures from the simple case B recombination theory assumed in many synthesis models, including ours, we have computed large grids of CLOUDY photoionization models for zero and low metallicity nebulae, using both black-body spectra and the SEDs predicted by our synthesis models (Sects. \[s\_cloudy\] and \[s\_sed\]).
Our main conclusions from the photoionization models are the following:
- Due to departures from case B (collisional excitation and ionization from excited levels), both [Ly$\alpha$]{} and 2$\gamma$ continuum emission can be significantly enhanced at low nebular densities. Their strengths are found to scale with the mean photon energy of the ionization source in the Lyman continuum.
- The equivalent width of [Ly$\alpha$]{} can be larger than expected from case B calculations due to the line flux enhancement. The measurement can also be affected by 2$\gamma$ emission if the continuum is measured at longer wavelength.
- Further enhancement of [Ly$\alpha$]{} at the expense of 2$\gamma$ emission can occur due to collisional mixing between the hydrogen $2 ^2S$–$2 ^2P$ levels at higher densities.
- [He [ii]{}]{} emission line fluxes (and consequently their equivalent widths) can be significantly decreased due to their dependence on the ionization parameter. This could make searches for the [He [ii]{} $\lambda$1640]{} line at high-$z$ more difficult.
- The enhancement of the 2$\gamma$ continuum and its dependence on nebular parameters can result in reduced equivalent widths of the UV emission lines and also change the UV restframe colours of high-$z$ galaxies.
Our results are of relevance to searches for, and interpretation of observations of, metal-poor and/or high-$z$ galaxies which may host the first stars that appeared in the Universe.
We thank Suzy Collin, Mike Fall, and Grazyna Stasińska for helpful and stimulating discussions. Some explorations of photoionization models were already undertaken earlier with Francois Ricquebourg and Michael Zamo. DS wishes to thank them here for their contributions. The work of DS is supported by the Swiss National Science Foundation. DS is thankful for support from the ESO visitor program, during which part of this work was done.
[^1]: The transformation is $\log(Q_H/L_\lambda( 1500 \AA)) =
\log(Q_H/L_\nu(1500 \AA))- 12.12$.
[^2]: More explicitely the Fig. shows $\log c$, where $\log {\rm SFR}= \log c + \log L_{1500} + \log c_M$, and $c_M=2.55$ for the IMF adopted by Kennicutt ($\mlow=0.1$ ), or $c_M=1$ for $\mlow=1$ .
[^3]: We use the standard definition $F_{\lambda} \propto \lambda^\beta$.
[^4]: The amplitudes of the Lyman break, 912$^+$/912$^-$, given in @Scha03 for very low metallicities are correct, but misleading. Indeed, with the definition adopted there, the 912$^+$ flux average (over 1080–1200 Å) includes strong absorption from the $\lambda$1084 line. This leads to values of 912$^+$/912$^-$ $< 1$ despite the fact that all models show a Lyman break in absorption.
|
---
author:
- 'Liguang Liu, Maria Vallarino, Dachun Yang'
title: |
[**Math. Z. (to appear)**]{}\
**Dyadic sets, maximal functions and applications on $ax+b$–groups**
---
[**Abstract**]{}Let $S$ be the Lie group ${\mathbb R}^n\ltimes {\mathbb R}$, where ${\mathbb R}$ acts on ${\mathbb R}^n$ by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $\rho$, which is a Lie group of exponential growth. Hebisch and Steger in \[Math. Z. 245(2003), 37–61\] proved that any integrable function on $(S,\rho)$ admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–Zygmund sets. In this paper, we show the existence of a dyadic grid in the group $S$, which has [nice]{} properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman–Stein type inequality, involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space $H^1$ and the space ${\mathop\mathrm{\,BMO\,}}$ introduced in \[Collect. Math. 60(2009), 277–295\].
[**Keywords**]{}Exponential growth group, Dyadic set, Complex interpolation, Hardy space, $\BMO$
[**Mathematics Subject Classification (2000)**]{}22E30, 42B30, 46B70
Introduction {#s: Introduction}
============
Let $S$ be the [*Lie group*]{} $\rn\ltimes \rr$ endowed with the following [*product*]{}: for all $(x, t)$, $(x', t')\in S$, $$(x, t)\cdot(x', t')\equiv (x+e^tx', t+t')\,.$$ The group $S$ is also called an [*$ax+b$–group*]{}. Clearly, $o= (0, 0)$ is the [*identity*]{} of $S$. We endow $S$ with the [*left-invariant Riemannian metric*]{} $$ds^2 \equiv e^{-2t}(dx_1^2+\cdots+dx_n^2)+dt^2\,,$$ and denote by $d$ the [*corresponding metric*]{}. This coincides with the metric on the hyperbolic space $H^{n +1}(\rr)$. For all $(x, t)$ in $S$, we have $$\label{1.1}
\cosh d\big((x, t), o\big)= \frac{ e^t+e^{-t}+e^{-t}|x|^2 }{2}.$$ The group $S$ is nonunimodular. The [*right and left Haar measures*]{} are given by $$d\rho(x, t)\equiv\,dx\,dt \quad {\rm{and}}
\quad d\lz(x, t)\equiv e^{-nt}\,dx\,dt.$$ Throughout the whole paper, we work on the [*triple $(S, d, \rho)$*]{}, namely, the group $S$ endowed with the left-invariant Riemannian metric $d$ and the right Haar measure $\rho$. For all $(x, t)\in S$ and $r>0$, we denote by $B\big((x,t), r\big)$ the [*ball centered at $(x,t)$ of radius $r$*]{}. In particular, it is well known that the right invariant measure of the ball $B(o, r)$ has the following behavior $$\begin{aligned}
\rho\big(B(o, r)\big)\sim\left\{\begin{array}{ll}
r^{n+1}\quad\quad&\text{if\quad $r<1$}\\
e^{nr}&\text{if\quad $r\ge 1$}.
\end{array}\right.\end{aligned}$$ Thus, $(S, d, \rho)$ is a space of exponential growth.
Throughout this paper, we denote by $L^p$ the [*Lebesgue space $L^p(\rho)$*]{} and by $\|\cdot\|_{L^p}$ its [*quasi-norm*]{}, for all $p\in (0, \fz]$. We also denote by $L^{1,\,\fz}$ the [*Lorentz space $L^{1,\,\fz}(\rho)$*]{} and by $\|\cdot\|_{L^{1,\, \fz}}$ its [*quasi-norm*]{}.
Harmonic analysis on exponential growth groups recently attracts a lot of attention. In particular, many efforts have been made to study the theory of singular integrals on the space $(S,d,\rho)$.
In the remarkable paper [@hs], Hebisch and Steger developed a new Calderón–Zygmund theory which holds in some spaces of exponential growth, in particular in the space $(S,d,\rho)$. The main idea of [@hs] is to replace the family of balls which is used in the classical Calderón–Zygmund theory by a suitable family of rectangles which we call Calderón–Zygmund sets (see Section \[s2\] for their [definitions]{}). We let $\crz$ denote the [*family of all Calderón–Zygmund sets*]{}.
The Hardy–Littlewood maximal function associated with $\crz$ is of weak type $(1,1)$ (see [@gs; @va1]). In [@hs], it was proven that every integrable function on $(S,d,\rho)$ admits a Calderón–Zygmund decomposition involving the family $\crz$. As a consequence, a theory for singular integrals holds in this setting. In particular, every integral operator bounded on $L^2$ whose kernel satisfies a suitable integral Hörmander’s condition is of weak type $(1,1)$. Interesting examples of singular integrals in this setting are spectral multipliers and Riesz transforms associated with a distinguished Laplacian $\Delta$ on $S$, which have been studied by numerous authors in, for example, [@as; @cghm; @gqs; @gas; @hs; @heb3; @mt; @sjo; @sv].
Vallarino [@va] introduced an atomic Hardy space $H^1$ on the group [$(S, d,\rho)$,]{} defined by atoms supported in Calderón–Zygmund sets instead of balls, and a corresponding $\BMO$ space, which enjoy some properties of the classical Hardy and $\BMO$ spaces (see [@cw2; @fest; @st93]). More precisely, it was proven that the dual of $H^1$ may be identified with $\BMO$, that singular integrals whose kernel satisfies a suitable integral Hörmander’s condition are bounded from $H^1$ to $L^1$ and from $L^{\infty}$ to $\BMO$. Moreover, for every $\theta\in (0,1)$, the real interpolation space $[H^1,L^2]_{\theta, q}$ is equal to $L^q$ if $\frac{1}{q} =1-\frac{\theta}{2}$, and $[L^2,\BMO]_{\theta, p}$ is equal to $L^p$ if $\frac{1}{p} = \frac{1-\theta}{2}$. The complex interpolation spaces between $H^1$ and $L^2$ and between $L^2$ and $\BMO$ are not identified in [@va].
In this paper, we introduce a dyadic grid of Calderón–Zygmund sets on $S$, which we denote by $\cp$ and which can be considered as the analogue of the family of classical dyadic cubes (see Theorem \[t3.1\] below). Recall that dyadic sets in the context of spaces of homogeneous type were also introduced by Christ [@c]; his construction used the doubling condition of the considered measure, so it cannot be adapted to the current setting. [In the $ax+b$–groups, the main tools we use to construct such a dyadic grid are some nice splitting properties of the Calderón–Zygmund sets and an effective method to construct a “parent" of a given Calderón–Zygmund set (see Lemma \[l3.1\] below). More precisely, given a Calderón–Zygmund set $R$, we find a bigger Calderón–Zygmund set $M(R)$ which can be split into at most $2^n$ sub-Calderón–Zygmund sets such that one of these subsets is exactly $R$ and each of these subsets has measure comparable to the measure of $R$.]{} To the best of our knowledge, this is the first time that a family of dyadic sets appears in a space of exponential growth. The dyadic grid $\cp$ turns out to be a useful tool to study the analogue of maximal singular integrals (see [@hyy]) on the space $(S,d,\rho)$, which will be investigated in a forthcoming paper [@lvy].
By means of the dyadic collection $\cp$, in Section 4 below, we prove a relative distributional inequality involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp maximal function on $S$, which implies a Fefferman–Stein type inequality involving those maximal functions; see Stein’s book [@st93 Chapter IV, Section 3.6] and Fefferman–Stein’s paper [@fest] for the analogous inequality in the Euclidean setting. The previous inequality is the main ingredient to prove that the complex interpolation space $(L^2,\BMO)_{[\theta]}$ is equal to $L^{p_{\theta}}$ if $\frac{1}{p_{\theta}}= \frac{1-\theta}{2}$ and $(H^1,L^2)_{[\theta]}$ is equal to $L^{q_{\theta}}$ if $\frac{1}{q_{\theta}}= 1-\frac{\theta}{2}$. This implies complex interpolation results for analytic families of operators (see Theorems \[t5.2\] and \[t5.3\] below). [In particular,]{} the complex interpolation result for analytic families of operators involving $H^1$ could be interesting and useful to obtain endpoint growth estimates of the solutions to the wave equation associated with the distinguished Laplacian $\Delta$ on $ax+b$–groups, as was pointed out [by Müller and Vallarino]{} [@mv Remark 6.3].
We remark that the corresponding complex interpolation results for the classical Hardy and $\BMO$ spaces were proven by Fefferman and Stein [@fest]. Recently, an $H^1$–$\BMO$ theory was developed by Ionescu [@i] for noncompact symmetric spaces of rank $1$ and, more generally, by Carbonaro, Mauceri and Meda [@cmm] for metric measure spaces which are nondoubling and satisfy suitable geometric assumptions. In those papers, the authors proved a Fefferman–Stein type inequality for the maximal functions associated with the family of balls of small radius: the main ingredient in their proofs is an isoperimetric property which is satisfied by the spaces studied in [@i; @cmm]. As a consequence, the authors in [@i; @cmm] obtained some complex interpolation results involving a Hardy space defined only by means of atoms supported in small balls and a corresponding $\BMO$ space. Notice that the space $(S,d,\rho)$ which we study here does not satisfy the isoperimetric property ([@cmm (2.2)]). Moreover, we consider atoms supported both in [“small” and “big” ]{} sets. Then we have to use different methods to obtain a suitable Fefferman–Stein inequality and complex interpolation results involving $H^1$ and $\BMO$.
Due to the existence of the dyadic collection $\cp$, it makes sense to define a dyadic $\BMO_\cp$ space and its predual dyadic Hardy space $H_\cp^1$ on $S$ (see Definitions \[dyadic-hardy\] and \[dyadic-bmo\] below). Though in Theorem \[dyadic-hardy and hardy\] below, it is proven that $H_\cp^1$ is a proper subspace of $H^1$, the complex interpolation result given by $H_\cp^1$ and $L^2$ is the same as that given by $H^1$ and $L^2$; see Remark \[dyadic-complex-interpolation\] below.
Finally, we make some conventions on notations. Set $\zz_+\equiv\{1,\,2,\,\cdots\}$ and $\nn=\zz_+\cup\{0\}$. In the following, $C$ denotes a [*positive finite constant*]{} which may vary from line to line and may depend on parameters according to the context. Constants with subscripts do not change through the whole paper. Given two quantities $f$ and $g$, by $f\ls g$, we mean that there exists a positive constant $C$ such that $f\le Cg$. If $f\ls
g\ls f$, we then write $f\sim g$. For any bounded linear operator $T$ from a Banach space $A$ to a Banach space $B$, we denote by $\|T\|_{A\to B}$ its [*operator norm*]{}.
Preliminaries {#s2}
=============
We first recall the definition of Calderón–Zygmund sets which appears in [@hs] and implicitly in [@gs]. In the sequel, we denote by $\cq$ the [*collection of all dyadic cubes in $\rn$*]{}.
\[d2.1\] A [*Calderón–Zygmund set*]{} is a set $R\equiv Q\times[t-r, t+r)$, where $Q\in\cq$ with side length $L$, $t\in\rr$, $r>0$ and $$\begin{aligned}
&&e^2\,e^tr\le L<e^8\,e^tr\quad\quad \mbox{if}\quad r<1,\\
&&e^t\,e^{2r}\le L<e^t\,e^{8r}\quad\quad\mbox{if}\quad r\ge1.\end{aligned}$$ We set $t_R\equiv t$, $r_R\equiv r$ and $x_R\equiv(c_Q, t)$, where $c_Q$ is the [*center*]{} of $Q$. For a Calderón–Zygmund set $R$, its [*dilated set*]{} is defined as $R^*\equiv \{x\in S:\, d(x, R)<r_R\}$. Denote by $\crz$ the family of all Calderón–Zygmund sets on $S$. For any $x\in S$, denote by $\crz(x)$ the family of the sets in $\crz$ which contain $x$.
\[r2.1\]For any set $R\equiv Q\times[t-r, t+r)\in\crz$, we have that $$\rho(R)=\dint_Q\dint_{t-r}^{ t+r} \,ds\,dx=2r|Q|=2rL^n,$$ where $|Q|$ and $L$ denote the Lebesgue measure and the side length of $Q$, respectively.
The following lemma presents some properties of the Calderón–Zygmund sets.
\[l2.1\] Let all the notation be as in Definition \[d2.1\]. Then there exists a constant $\kz_0\in[1, \fz)$ such that for all $R\in\crz$, the following hold:
- $B(x_R, r_R)\subset R\subset B(x_R, \kz_0r_R)$;
- $\rho(R^\ast)\le\kz_0\rho(R)$;
- every $R\in\crz$ can be decomposed into mutually disjoint sets $\{R_i\}_{i=1}^k$, with $k=2$ or $k=2^n$, $R_i\in \crz$, such that $R=\cup_{i=1}^k R_i$ and $\rho(R_i)=\rho(R)/k$ for all $i\in\{1,\cdots, k\}$.
We refer the reader to [@hs; @va] for the proof of Lemma \[2.1\]. We recall here an idea of the proof of the property (iii) from [@hs; @va]. Given a Calderón–Zygmund set $R\equiv Q\times [t-r,t+r)$, when the side length $L$ of $Q$ is sufficiently large with respect to $e^t$ and $r$, it suffices to decompose $Q$ into $2^n$ smaller dyadic Euclidean cubes $\{Q_1,\cdots,Q_{2^n}\}$ and define $R_i\equiv Q_i\times [t-r,t+r)$ for $i\in\{1,\cdots,2^n\}$. Otherwise, it suffices to split up the interval $[t-r,t-r)$ into two disjoint sub-intervals $\{I_1, I_2\}$, which have the same measure, and define $R_i\equiv Q\times I_i$ for $i\in\{1, 2\}$. This construction gives rise to either $2^n$ or $2$ smaller Calderón–Zygmund sets satisfying the property (iii) above.
The Hardy–Littlewood maximal function associated to the family $\crz$ is defined as follows.
\[d2.2\] For any locally integrable function $f$ on $S$, the [*Hardy–Littlewood maximal function $\cm f$*]{} is defined by $$\label{2.1}
\cm
f(x)\equiv\sup_{R\in\crz(x)}\dfrac1{\rho(R)}\dint_R|f|\,d\rho\qquad
{ \forall\ } x\in S\,.$$
The maximal operator $\cm$ has the following boundedness properties [@gs; @hs; @va1].
\[p2.1\] The Hardy–Littlewood maximal operator $\cm$ is bounded from $L^1$ to $L^{1,\,\fz}$, and also bounded on $L^p$ for all $p\in(1, \fz]$.
By Proposition \[p2.1\], Lemma \[l2.1\] and a stopping-time argument, Hebisch and Steger [@hs] showed that any integrable function $f$ on $S$ at any level $\alpha>0$ has a Calderón–Zygmund decomposition $f=g+\sum_ib_i$, where $|g|$ is a function almost everywhere bounded by $\kz_0 \alpha$ and functions $\{b_i\}_i$ have vanishing integral and are supported in sets of the family $\crz$. This was proven to be a very useful tool in establishing the boundedness of some multipliers and singular integrals in [@hs], and the theory of the Hardy space $H^1$ on $S$ in [@va].
Lemma [\[l2.1\](iii)]{} states that given a Calderón–Zygmund set, one can split it up into a finite number of disjoint subsets which are still in $\crz$. We shall now study how, starting from a given Calderón–Zygmund set $R$, one can obtain a bigger set containing it which is still in $\crz$ and whose measure is comparable to the measure of $R$.
\[d3.1\] For any $R\in\crz$, $M(R)\in\crz$ is called a *parent* of $R$, if
- $M(R)$ can be decomposed into $2$ or $2^n$ mutually disjointed sub-Calderón–Zygmund sets, and one of these sets is $R$;
- $3\rho(R)/2\le\rho(M(R))\le\max\{3, 2^n\}\rho(R)$.
For any $R\in\crz$, a parent of $R$ always exists, but it may not be unique. The following lemma gives three different kinds of extensions for sets $R\equiv Q\times[t{-r},\, t+r)\in\crz$ when $r\ge1$. Precisely, if $Q$ has small side length, then we find a parent of $R$ by extending $R$ “horizontally”; if $Q$ has large side length, then we find a parent of $R$ by extending $R$ either “vertically up" or “vertically down".
\[l3.1\] Suppose that $R\equiv Q\times[t{-r},\, t+r)\in\crz$, where $t\in\mathbb R$, $r\equiv r_R\ge1$ and $Q\subset\rn$ is a dyadic cube with side length $L$ satisfying $e^t\,e^{2r}\le L<e^t\,e^{8r}$. Then the following hold:
- [If]{} $e^t\,e^{2r}\le L<e^t\,e^{8r}/2$, then $R_1\equiv
Q'\times[t-r,\, t+r)$ is a parent of $R$, where $Q'\subset\rn$ is the unique dyadic cube with side length $2L$ that contains $Q$. Moreover, $\rho(R_1)=2^n\rho(R)$.
- If $e^t\,e^{8r}/2\le L<e^t\,e^{8r}$, then $R_2\equiv Q\times[t{-r},\, t+{3r})$ is a parent of $R$. Moreover, the set $R'\equiv Q\times[t+r,
t+{3r})$ belongs to the family $\crz$, $R_2=R\cup R'$ and $$\rho(R)=\rho(R')=\rho(R_2)/2.$$ The set $R_2$ is also a parent of $R'$.
- If $e^t\,e^{8r}/2\le L<e^t\,e^{8r}$, then $R_3\equiv Q\times[t{-5r},\, t+{r})$ is a parent of $R$. Moreover, the set $R''\equiv Q\times[t{-5r},
t{-r})$ belongs to the family $\crz$, $R_3=R\cup R''$ and $$\rho(R)=\rho(R'')/2=\rho(R_3)/3.$$ The set $R_3$ is also a parent of $R''$.
We first prove (i). Since $2e^te^{2r}\le 2L<e^te^{8r}$, we have that $R_1\in\crz$. Obviously $Q'\times[t{-r},
t+r)$ can be decomposed into $2^n$ sub-Calderón–Zygmund sets and one of these sets is $R$. By Remark \[r2.1\], we have $\rho(R_1)=2^n\rho(R)$. Thus, (i) holds.
To show (ii), notice that $r_{R_2}=2r$ and $t_{R_2}=t+r$. Since $r\ge 1$ and $e^t\,e^{8r}/2\le L<e^t\,e^{8r}$, we have that $e^{t+r}\,e^{4r}\le
L<e^{t+r}\, e^{16r}$, which implies that $R_2\in\crz$. If we set $$R'\equiv Q\times[t+r,\, t+3r),$$ then $t_{R'}=t+{2r}$ and $r_{R'}=r$. Since $r\ge 1$ and $e^t\,e^{8r}/2\le
L<e^t\,e^{8r}$, we obtain $e^{t+2r}\,e^{r}\le L<e^{t+2r}\,e^{8r}$, and hence $R'\in\crz$. By Remark \[r2.1\], $\rho(R)=\rho(R')=\rho(R_2)/2.$ Thus, (ii) holds.
Finally, we show (iii). Observe that $t_{R_3}=t-2r$ and $r_{R_3}=3r$. Since $r\ge 1$ and $e^t\,e^{8r}/2\le
L<e^t\,e^{8r}$, we have that $e^{t-2r}\,e^{6r}\le L<e^{t-2r}\,e^{24r}$ and hence $R_3\in\crz$. Set $R''\equiv Q\times[t{-5r},\,
t{-r})$. Notice that $t_{R''}=t{-3r}$ and $r_{R''}=2r$. Again by [$r\ge 1$]{} and $e^t\,e^{8r}/2\le L<e^t\,e^{8r}$, we obtain that $e^{t-3r}\,e^{4r}\le L<e^{t-3r}\,e^{16r}$ and hence $R''\in\crz$. It is easy to see that $R_3=R\cup R''$, $\rho(R'')=2\rho(R)$ and $\rho(R_3)=3\rho(R)$. Therefore, we obtain (iii), which completes the proof.
We conclude this section by recalling the definition of the Hardy space $H^1$ and its dual space $\BMO$ (see [@va]).
An [*$H^1$-atom*]{} is a function $a$ in $L^1$ such that
- $a$ is supported in a set $R\in\crz$;
- $\|a\|_{ L^{\infty} } \le [\rho(R)]^{-1};$
- $\int_S a \,d\rho =0$.
The [*Hardy space*]{} $H^{1}$ is the space of all functions $g$ in $ L^1$ which can be written as $g=\sum_j \lambda_j\, a_j$, where [$\{a_j\}_j$]{} are [$H^1$-atoms ]{} and [$\{\lambda _j\}_j$]{} are complex numbers such that $\sum _j |\lambda _j|<\infty$. [Denote]{} by $\|g\|_{H^{1}}$ the [*infimum of $\sum_j|\lambda_j|$ over such decompositions*]{}.
In the sequel, for any locally integrable function $f$ and any set $R\in \crz$, we denote by $f_R$ the [*average of $f$ on $R$*]{}, namely, $\frac{1}{ \rho(R)} \int_R f d\rho$.
For any locally integrable function $f$, its [*sharp maximal function*]{} is defined by $$f^{\sharp}(x)\equiv\sup_{R\in\crz(x)}\frac{1}{ \rho(R)} \int_R
|f-f_R|\,d\rho \qquad { \forall\ } x\in S\,.$$ The space $\mathcal{B}\mathcal{M}\mathcal{O}$ is the [*space of all locally integrable functions $f$ such that $f^{\sharp}\in L^
{\infty}$*]{}. The [*space $\BMO$*]{} is the quotient of $\mathcal{B}\mathcal{M}\mathcal{O}$ module constant functions. It is a Banach space endowed with the norm $\|f\|_{\BMO}
\equiv\|f^{\sharp}\|_{L^{\infty}}$.
The space $\BMO$ is identified with the dual of $H^1$; see [@va Theorem 3.4]. More precisely, for any $f$ in $\BMO$, the functional $\ell$ defined by $\ell(g)\equiv\int
fg\,d\rho$ for any finite linear combination $g$ of atoms extends to a bounded functional on $H^{1}$ whose norm is no more than $C\,\|f\|_{\BMO}.$ On the other hand, for any bounded linear functional $\ell$ on $H^{1}$, there exists a function $f^{\ell}$ in $\BMO$ such that $\|f^{\ell}\|_{\BMO}\le C\,\|\ell\|_{(H^{1})^*}$ and $\ell(g)=\int f^{\ell}g\,d\rho$ for any finite linear combination $g$ of atoms.
A dyadic grid on $(S, d, \rho)$ {#s3}
===============================
The main purpose of this section is to introduce a dyadic grid of Calderón–Zygmund sets on $(S, d,\rho)$, which can be considered as an analogue of Euclidean dyadic cubes (see [@st93 p.149] or [@g p.384]). The key tools to construct such a grid are [Lemmas \[l2.1\] and \[l3.1\].]{}
\[t3.1\] There exists a collection $\{\cp_j\}_{j\in\zz}$ of partitions of $S$ such that each $\cp_j$ consists of pairwise disjoint Calderón–Zygmund sets, and
- for any $j\in\zz$, $S=\cup_{R\in\cp_j} R$;
- if $\ell\le k$, $R\in\cp_\ell$ and $R^\pz\in\cp_k$, then either $R\subset R^\pz$ or $R\cap R^\pz=\emptyset$;
- for any $j\in\zz$ and $R\in\cp_j$, there exists a unique $R^\pz\in\cp_{j+1}$ such that $R\subset R^\pz$ and $\rho(R^\pz)\le 2^n\,\rho(R)$;
- for any $j\in\zz$, every $R\in\cp_j$ can be decomposed into mutually disjoint sets $\{R_i\}_{i=1}^k\subset\cp_{j-1}$, with $k=2$ or $k=2^n$, such that $R=\cup_{i=1}^k R_i$ and $\frac{\rho(R)}{2^n}\le\rho(R_i)\le {\frac{2\rho(R)}{3}}$ for all $i\in\{1,\cdots,k\}$;
- for any $x\in S$ and for any $j\in\zz$, let $R_j^x$ be the unique set in $\cp_j$ which contains $x$, then $\lim_{j\to -\infty}\rho(R_j^x)=0$ and $\lim_{j\to
\infty}\rho(R_j^x)=\infty$.
We write $S\equiv\boz_1\cup\boz_2$, where $\boz_1\equiv\rn\times[0,
\fz)$ and $\boz_2\equiv\rn\times(-\infty,0)$, and construct a sequence $\{\cp_j^1\}_{j\in\nn}$ of partitions of $\boz_1$ as well as a sequence $\{\cp_j^2\}_{j\in\nn}$ of partitions of $\boz_2$, respectively.
Let us first construct the desired partitions $\{\cp_j^1\}_{j\in\nn}$ of $\boz_1$ by the following four steps.
[[**Step 1.**]{} ]{} Choose a Calderón–Zygmund set $R_0\equiv
Q_0\times[t_0-r_0, t_0+{r_0})$, where $t_0=r_0\ge1$ and $Q_0=[0, \ell_0)^n\in\cq$, with $e^{t_0}\,e^{2
r_0}\le\ell_0<e^{t_0}\,e^{8 r_0}$. To find a parent of $R_0$, we consider the following two cases, separately.
[*[Case 1:]{}*]{} $e^{t_0}\,e^{8 r_0}/2\le\ell_0<e^{t_0}\,e^{8 r_0}$. In this case, by Lemma \[l3.1\](ii), $$R_1\equiv Q_0\times[t_0{-r_0}, t_0+{3r_0})$$ is a parent of $R_0$ and $\rho(R_1)=2\rho(R_0)$.
[*[Case 2:]{}*]{} $e^{t_0}\,e^{2 r_0}\le\ell_0< e^{t_0}\,e^{8 r_0}/2$. By Lemma \[l3.1\](i), $R_1\equiv Q_1\times[t_0{-r_0},
t_0+{r_0})$ is a parent of $R_0$, where $Q_1\subset\rn$ is the unique dyadic cube with side length $2\ell_0$ that contains $Q_0$, namely, $Q_1=[0, 2\ell_0)^n$.
We then proceed as above to obtain a parent of $R_1$, which is denoted by $R_2$. By repeating this process, we obtain a sequence of Calderón-Zygmund sets, $\{R_j\}_{j\in\nn}$, such that each $R_{j+1}$ is a parent of $R_j$.
Without loss of generality, for any $j\in\nn$, we may set $R_j\equiv
Q_j\times [t_j-r_j, t_j+r_j)$, where $r_{j+1}\ge r_j\ge1$, $t_j=r_j$ and $Q_j=[0, \ell_j)^n\in\cq$, with $e^{t_j}\,e^{2 r_j}\le\ell_j<e^{t_j}\,e^{8 r_j}$. Observe that $R_{j+1}$ is obtained by extending $R_j$ either “vertically up" (see [*[Case 1]{}*]{}) or “horizontally" (see [*[Case 2]{}*]{}). Notice that the definition of Calderón–Zygmund sets implies that we cannot always [extend]{} $R_j$ “horizontally” to obtain its parent $R_{j+1}$; in other words, for some $j$, to obtain $R_{j+1}$, we have to [extend]{} $R_j$ “vertically [up]{}”. Thus, $\lim_{j\to\fz}(t_j+r_j)=\fz$. This, combined with the fact that $t_j=r_j$, implies that $$\label{3.1}
\boz_1 = \bigcup_{j\in\nn} \big(\rn\times[t_j-r_j, t_j+{r_j})\big).$$
[**Step 2.** ]{} For any $j\in\nn$ and $R_j$ as constructed in [**Step 1**]{}, we set $$\label{3.2}
\cn_j\equiv\{Q\times[t_j{-r_j}, t_j+{r_j}):\, Q\in\cq,\,
\ell(Q)=\ell(Q_j)\}.$$ Then $\cn_j\subset\crz$ and we put all sets of $\cn_j$ into $\cp_j^1$. If $R_{j+1}$ is obtained by extending $R_j$ “vertically [up]{}" as in [*Case 1*]{} of [**Step 1**]{}, then we set $$\label{3.3}
\wz{\cn_j}\equiv\{Q\times[t_j+{r_j}, t_j+{3r_j}):\, Q\in\cq,\,
\ell(Q)=\ell(Q_j)\}.$$ By Lemma \[l3.1\](ii), $\wz{\cn_j}\subset\crz$. If $R_{j+1}$ is obtained by extending $R_j$ “horizontally" as in [*Case 2*]{} of [**Step 1**]{}, then we set $\wz{\cn_j}=\emptyset$. We also put all sets of $\wz{\cn_j}$ into $\cp_j^1$.
We claim that for any fixed $j\in\nn$, $$\label{Omega1unione}
\Omega_1=\bigcup_{R\in\cn_j\cup (\cup_{\ell=j}^{\infty}\wz{\cn_{\ell}})} R\,.$$ Indeed, $$\label{primaparte}
\rr^n\times [0,t_j+{3r_j})=\bigcup_{R\in \cn_j\cup \wz{\cn_j}}R.$$ Rewrite the sequence $\{\wz\cn_k:\, k>j,\, \wz{\cn}_{k}\neq\emptyset\}$ as $\{\wz{\cn}_{\ell_k}\}_{k=1}^\infty$, where $$j+1\le
\ell_1<\ell_2<\dots<\ell_k<\dots \,.$$ We have that $$\label{oss1}
t_j+{3r_j}=t_{\ell_1}+{r_ {\ell_1} } \quad {\rm{and}} \quad
t_{\ell_{k-1}}+{3r_ {\ell_{k-1}} }=t_{\ell_k}+{r_ {\ell_k} } \quad
\forall\ k\ge 1.$$ Since $$\rr^n\times [t_{\ell_k}+{r_ {\ell_k} }, t_{\ell_k}+{3r_ {\ell_k}
})= {\bigcup_ {R\in\wz{\cn} _{\ell_k} } R }$$ and $\lim_{k\to \infty} ( t_{\ell_k}+{3r_ {\ell_k}})
={\infty}$, by (\[oss1\]), we obtain that $$\label{secondaparte}
{ \rr^n\times [ t_j+{3r_j},\infty ) =\bigcup_{k\ge
1}\bigcup_{R\in\wz{\cn}_{\ell_k} } R = \bigcup_{
\ell\ge j+1} \bigcup_{R\in \wz{\cn}_{\ell} } R.}$$ The claim follows by and .
[**Step 3.** ]{} Now fix $j\in\nn$ and take $\ell\ge j+1$ such that $\wz{\cn_{\ell}}\neq\emptyset$. For any $R\in\wz{\cn_{\ell}}$, by Lemma \[l2.1\](iii), there exist mutually disjoint sets $\{R^{i}\}_{i=1}^k\subset\crz$ with $k=2$ or $k=2^n$ such that $R=\cup_{i=1}^k R_i$, and $\rho(R)/2^n\le\rho(R_{i})\le\rho(R)/2$ for all $i\in\{1,\cdots,k\}$. Denote by $\wz{\cn_{\ell}}^1$ the collection of all such small Calderón-Zygmund sets $R_i$ obtained by running $R$ over all elements in $\wz{\cn_{\ell}}$. Observe that sets in $\wz{\cn_{\ell}}^1$ are mutually disjoint. Next, we apply Lemma \[l2.1\](iii) to every $R\in\wz{\cn_{\ell}}^1$ and argue as above; we then obtain a collection of smaller Calderón–Zygmund sets, which is denoted by $\wz{\cn_{\ell}}^2$. By repeating the above procedure $i$ times, we obtain a collection of Calderón–Zygmund sets which we denote by $\wz{\cn_\ell}^{i}$. In particular, we put the collection $\wz{\cn_\ell}^{\ell-j}$ obtained after $\ell-j$ steps into $\cp_j^1$.
Thus, for any $j\in\nn$, we define $$\label{3.4}
\cp_j^1=\cn_j\bigcup\wz{\cn_j} \bigcup \lf(\bigcup_{\ell\ge
j+1}\wz{\cn_\ell}^{\ell-j}\r) .$$ By construction, the sets in $\cp_j^1$ are mutually disjoint. Moreover, since for all $j\ge 0$ and $\ell\ge j+1$, $$\bigcup_{R\in\wz{\cn}_{\ell}^{\ell-j}}R=\bigcup_{R\in\wz{\cn}_{\ell}}R\,,$$ from the formula , we deduce that $\boz_1=\cup_{R\in\cp_j^1}R$. This shows that $\cp_j^1$ satisfies the property (i).
[**Step 4.**]{} For any $0\le \ell\le k$, $R\in\cp_\ell^1$ and $R^\pz\in\cp_k^1$, by , , and the construction above, it is easy to verify that either $R\subset R^\pz$ or $R\cap R^\pz=\emptyset$, namely, the property (ii) is satisfied.
Let $R$ be in $\cp_j^1$ for some $j\in\nn$. If $R$ is in $\cn_j\cup\wz{\cn}_j$ and if $R_{j+1}$ is obtained by extending $R_j$ “horizontally", then there exists one parent of $R$ in $\cp_{j+1}$ whose measure is [$2^n\rho(R)$]{}. If $R$ is in $\cn_j\cup\wz{\cn}_j$ and if $R_{j+1}$ is obtained by extending $R_j$ “vertically [up]{}", then there exists one parent of $R$ in $\cp_{j+1}$ whose measure is [$2\rho(R)$.]{} If $R$ is in $\wz{\cn}_{\ell}^{\ell-j}$ for some $\ell\ge j+1$, then it has a parent in $\wz{\cn}_{\ell}^{\ell-j-1} \subset \cp_{j+1}$ whose measure is either $2\rho(R)$ or $2^n\rho(R)$. Thus, the property (iii) is satisfied.
So far, we have proven that there exists a sequence $\{\cp_j^1\}_{j\in\nn}$ of partitions of $\boz_1$ whose elements satisfy the properties (i)–(iii).
To obtain the desired partitions $\{\cp_j^2\}_{j\in\nn}$ on $\boz_2$, we [apply (i) and (iii) of Lemma \[l3.1\] and]{} proceed as for $\boz_1$: the details are left to the reader.
We define $\cp_j\equiv\cp_j^1\cup\cp_j^2$ for all $j\ge 0$. We now construct the partitions $\cp_j$ for $j<0$. By applying Lemma \[l2.1\](iii) to each $R\in\cp_0$, we find mutually disjoint sets $\{R_{i}\}_{i=1}^k$, with $k=2$ or $k=2^n$, such that $R_i\in\crz$, $R=\cup_{i=1}^k R_i$, and $\rho(R)/2^n\le\rho(R_{i})\le\rho(R)/2$ for all $i\in\{1,\cdots,k\}$. Then we define $\cp_{-1}$ to be the collection of all such small Calderón-Zygmund sets $R_i$ obtained by running $R$ over all elements in $\cp_0$. Clearly $\cp_{-1}$ is still a partition of $S$. Again, applying [Lemma \[l2.1\](iii)]{} to each element of $\cp_{-1}$ and using a similar splitting argument to this, we obtain a collection of smaller Calderón–Zygmund sets, which is defined to be $\cp_{-2}$. By repeating this process, we obtain a collection $\{\cp_j\}_{j<0}$, where each $\cp_j$ is a partition of $S$. By [the]{} construction of $\{\cp_j\}_{j<0}$ and by Lemma \[l2.1\](iii), it is easy to check that the sets in $\{\cp_j\}_{j<0}$ satisfy [the]{} properties [(i)–(iii).]{}
It remains to prove the properties (iv) and (v). For a set $R\in \cp_j$, with $j\le 0$, the property (iv) is easily deduced from Lemma \[l2.1\](iii). Take now a set $R$ in $\cp_j^1$ for some $j>0$. If $R$ is in $\cn_j$, then it has either $2^n$ disjoint subsets in $\cn_{j-1}$ or $2$ disjoint subsets in $\cn_{j-1}\cup \wz{\cn}_{j-1}$. If $R$ is in $\wz{\cn}_j$, then it has either $2^n$ or $2$ disjoint subsets in $\wz{\cn}_{j}^1\subset \cp_{j-1}^1$. Finally, if $R$ is in $\wz{\cn}_{\ell}^{\ell-j}$ for some $\ell\ge j+1$, then it has either $2^n$ or $2$ subsets in $\wz{\cn}_{\ell}^{\ell-j+1}\subset
\cp_{j-1}^1$. In all the previous cases, $R$ satisfies the property (iv). The case when $R$ is in $\cp_j^2$ for some $j>0$ is similar and omitted.
As far as the property (v) is concerned, given a point $x$ in $S$, for any $j\in\zz$, let $R_j^x$ be the set in $\cp_j$ which contains $x$. By the construction and the property (iv), for any $j\in\zz$, there exists a set $R_{j+1}^x\in \cp_{j+1}$ which is a parent of $R_j^x$, so that $$\rho( R_{j+1}^x)\ge \frac{3}{2}\rho(R_j^x)\ge
\lf(\frac{3}{2}\r)^j\rho(R_0^x);$$ this shows that $\lim_{j\to \infty} \rho(R_j^x)=\infty$. For any $j<0$, we have that $$\rho(R_j^x)\le {\frac23}\rho(R_{j+1}^x) \le
{\lf(\frac23\r)^j}\rho(R_0^x);$$ this shows that $\lim_{j\to -\infty}
\rho(R_j^x)=0$ and concludes the proof of the theorem.
\[r3.1\]
- It should be pointed out that a sequence $\{\cp_j\}_{j\in\zz}$ satisfying Properties (i)–(v) of Theorem \[t3.1\] is not unique.
- For any given $j\in\zz$, the measures of any two elements in $\cp_j$ may not be comparable. This is an essential difference between the collection of Euclidean dyadic cubes and of dyadic sets in spaces of homogeneous type [@c] and the dyadic sets which we introduced above.
We now choose one collection $\cp\equiv \{\cp_j \}_j$ of dyadic sets in $S$ constructed as in Theorem \[t3.1\]. In the sequel, $\cp$ always denotes this collection.
Dyadic maximal functions
========================
By using the collection $\cp$ introduced above, we define the corresponding Hardy–Littlewood dyadic maximal function and dyadic sharp maximal function as follows.
\[dyadicmaxfct\] For any locally integrable function $f$ on $(S, d, \rho)$, the [*Hardy–Littlewood dyadic maximal function*]{} $\cm_{\cp} f$ is defined by $$\label{3.6}
\cm_{\cp} f(x)\equiv\sup_{R\in\crz(x),\,
R\in\cp}\dfrac1{\rho(R)}\dint_R|f|\,d\rho\qquad {\forall\ } x\in
S\,,$$ and the [*dyadic sharp maximal function*]{} $f^{\sharp}_{\cp}$ by $$\label{fsharpd}
f^{\sharp}_{\cp}(x)\equiv\sup_{R\in\crz(x),\,
R\in\cp}\dfrac1{\rho(R)}\dint_R|f -f_R|\,d\rho\qquad {\forall\ }
x\in S\,.$$ Recall that $f_R\equiv\frac{1}{ \rho(R)} \int_R f\, d\rho$.
It is easy to see that for all locally integrable functions $f$ and almost every $x\in S$, $ f(x)\le\cm_{\cp} f(x)\le\cm f(x)$ [and $f_\cp^\#(x)\le f^\#(x)$.]{} This combined with Proposition \[p2.1\] implies the following conclusion.
\[c3.1\] The operator $\cm_{\cp}$ is bounded from $L^1$ to $L^{1,\,\fz}$, and also bounded on $L^p$ for all $p\in(1, \fz]$.
- It is obvious that $\cm_{\cp} f(x)\le \cm f(x)$ for any locally integrable function $f$ at any point $x\in S$. However, there exist functions $f$ such that $\cm f$ and $\cm_{\cp}
f$ are not pointwise equivalent. To see this, we take a set $R\equiv \,Q\times [0, {2r})$ in $\cp$ such that $Q={[0, 2^{\ell_0})}^n$ for some $\ell_0\in\zz$. Then, for all points $(y_1,\dots,y_n, s)\in S$ such that $y_j<0$ for all $j\in
\{1,\dots,n\}$ and $s<0$, we have $\cm_{\cp}(\chi_R)(y, s)=0$ and $\cm(\chi_R)(y, s)>0.$ So there does not exist a positive constant $C$ such that $\cm(\chi_R)\le C \cm_{\cp}(\chi_R)$.
- It is obvious that $f^{\sharp}_{\cp}(x)\le
f^{\sharp}(x)$ for any locally integrable function $f$ at any point $x\in S$. The same counterexample as in (i) shows that the sharp maximal function and the dyadic sharp maximal function may be not pointwise equivalent. Indeed, if we take the set $R\equiv \,Q\times [0, 2r)$ as above, then for all points $(y_1,\dots,y_n, s)\in S$ such that $y_j<0$ for all $j\in
\{1,\dots,n\}$ and $s<0$, we have $(\chi_R)^{\sharp}_{\cp} (y, s)=0$ and $(\chi_R)^{\sharp} (y, s)>0.$ So there does not exist a positive constant $C$ such that $(\chi_R)^{\sharp} \le C {(\chi_R)^{\sharp}_\cp}$.
We now state a covering lemma for the level sets of $\cm_{\cp}$, which is proven in a standard way as follows; see also [@st93 Lemma 1, p.150].
\[covering\] Let $f$ be a locally integrable function and $\alpha$ a positive constant such that $\Omega_{\alpha}\,\equiv \{x\in S:\, \cm_{\cp}f(x)>\alpha\}$ has finite measure. Then $\Omega_t$ is a disjoint union of dyadic sets, $\{R_j\}_j$, with $\alpha<\frac{1}{\rho(R_j)}\int_{R_j} |f|d\rho\le 2^n\alpha$ for all $j$.
Since the measure of $\Omega_{\alpha}$ is finite, for each $x\in\Omega_{\alpha}$ there exists a maximal dyadic set $R_x\in\cp$ which contains $x$ such that $ \alpha<\frac{1}{\rho(R_x)}\int_{R_x} |f|d\rho$. Any two of these maximal dyadic sets are disjoint. Indeed, [by Theorem \[t3.1\],]{} given two points $x,y\in \Omega_{\alpha}$, either $R_x\cap R_y=\emptyset$ or one is contained in the other; by maximality, this implies that $R_x=R_y$. We denote by $\{R_j\}_j$ this collection of dyadic maximal sets. Then it is clear that $\Omega_{\alpha}=\cup_jR_j$. Moreover, for any $j$, since $R_j$ is maximal, there exists a dyadic set $\tilde{R}_j\in\cp$ which [is a parent of $R_j$ and]{} $\frac{1}{\rho(\tilde{R}_j)}\int_{\tilde{R}_j} |f|\, d\rho\le {\alpha}$. [Thus,]{} $$\frac{1}{\rho(R_j)}\int_{R_j} |f|\,d\rho \le
2^n\frac{1}{\rho(\tilde{R}_j)}\int_{\tilde{R}_j} |f|\,d\rho\le
2^n{\alpha}.$$ This finishes the proof.
As a consequence of the previous covering lemma, following closely the proof of the inequality [@st93 (22), p.153], we obtain the following relative distributional inequality. We omit the details.
There exists a positive constant $K$ such that for any locally integrable function $f$, and for any positive $c$ and $b$ with $b<1$, $$\label{distr}
\rho\big( \{ x\in S:\,\cm_{\cp}f(x)>\alpha,\,
f^{\sharp}_{\cp}(x)\le c\alpha\}\big) \le K \frac{c}{ 1-b} \,
\rho\big( \{ x\in S:\,\cm_{\cp}f(x)>b\alpha\}\big)$$ for all $\alpha>0$. The constant $K$ only depends on $n$ and on the norm $\|\cm_{\cp}\|_{L^1\to L^{1,\infty} }$.
By the relative distributional inequality (\[distr\]) and arguing as in [[@st93 Corollary 1, p.154]]{}, we obtain the following Fefferman–Stein type inequality. We also omit the details.
\[FeffStein\] Let [$p\in(0,\infty)$.]{} There exists a positive constant $A_p$ such that for any locally integrable function $f$ such that $f^{\sharp}_{\cp}$ belongs to $L^p$ and [$\cm_{\cp}f \in L^{p_0}$]{} with $p_0\le p$, then $f$ is in $L^p$ and $$\| \cm_{\cp}f \|_{L^p}\le A_p\,\| f^{\sharp}_{\cp} \|_{L^p}.$$
Recall that for any locally integrable function $f$, $|f|\le
\cm_{\cp}f$ and $f^{\sharp}_{\cp}\le f^{\sharp}$. Thus, from Corollary \[FeffStein\], we deduce that if $p\in {(0,\infty)}$, $f^{\sharp}$ belongs to $L^p$ and $f$ belongs to some $L^{p_0}$ with $p_0\in (0, p]$, then $f$ is in $L^p$ and $$\label{FeffStein2}
\| f \|_{L^p}\le A_p\,\| f^{\sharp} \|_{L^p}\,,$$ where $A_p$ is the constant which appears in Corollary \[FeffStein\]. This generalizes the classical Fefferman–Stein inequality [[@st93 Theorem 2, p.148]]{} to the current setting.
We shall now introduce a dyadic Hardy space and a dyadic $\BMO$ space.
\[dyadic-hardy\] The [*dyadic Hardy space*]{} $H^{1}_{\cp}$ is defined to be the space of all functions $g$ in $ L^1$ which can be written as $g=\sum_j \lambda_j\, a_j$, where $\{a_j\}_j$ are $H^1$-atoms supported in dyadic sets and [$\{\lambda _j\}_j$]{} are complex numbers such that $\sum _j |\lambda
_j|<\infty$. Denote by $\|g\|_{H^{1}_{\cp}}$ the [*infimum of $\sum_j|\lambda_j|$ over all such decompositions*]{}.
\[dyadic-bmo\] The space $\mathcal{B}\mathcal{M}\mathcal{O}_{\cp}$ is the space of all locally integrable functions $f$ such that $f^{\sharp}_{\cp}\in
L^ {\infty}$. The space $\BMO_{\cp}$ is the quotient of $\mathcal{B}\mathcal{M}\mathcal{O}_{\cp}$ module constant functions. It is a Banach space endowed with the norm $\|f\|_{\BMO_{\cp}}
\equiv\|f^{\sharp}_{\cp}\|_{L^\infty}$.
It is easy to follow the proof in [@va1 Theorem 3.4] to show that the dual of $H^1_{\cp}$ is identified with $\BMO_{\cp}$. We omit the details.
Obviously, $H^1_{\cp}\subset H^1$ and $\|g\|_{H^1}\le
\|g\|_{H^1_{\cp}}$ for all $g$ in $H^1_\cp$. It is natural to ask whether the norms $\|\cdot\|_{H^1}$ and $\|\cdot\|_{H^1_{\cp}}$ are equivalent. The analog problem in the classical setting was studied by [Abu-Shammala and Torchinsky]{} [@at]. By following the ideas in [@at], we obtain the following result.
\[dyadic-hardy and hardy\] The norms $\|\cdot\|_{H^1}$ and $\|\cdot\|_{H^1_{\cp}}$ are not equivalent.
We give the details of the proof in the case when $n=1$. By the construction of $\cp$ in Theorem \[t3.1\], there exists $[0,2^{\ell_0+1})\times
[0,2{r_0})\in\cp_{k_0+1}$ for some $k_0\in\zz$, $\ell_0\in\zz$ and $r_0>0$ such that $R_0\equiv[0,2^{\ell_0})\times
[0,2{r_0})\in \cp_{k_0}$ and $E_0\equiv[2^{\ell_0}, 2\cdot
2^{\ell_0})\times [0,2{r_0})\in \cp_{k_0}$. Generally, for any $j<0$, there exist $R_j=[2^{\ell_0}-2^{\ell_j},2^{\ell_0})\times I_j\in\cp_{k_j}$ and $E_j=[2^{\ell_0},2^{\ell_0}+2^{\ell_j})\times I_j\in\cp_{k_j}$ such that $R_j\cup E_j\in \cp_{k_j+1}$, where both $\{k_j\}_{j<0}$ and $\{\ell_j\}_{j<0}$ are strictly decreasing sequences which tend to $-\infty$ as $j\to-\infty$, and each $I_j$ is an interval contained in $[0,\infty)$. Notice that for all $j\in\nn$, $\rho(R_j)=\rho(E_j)=2r_j2^{\ell_j}$ for some $r_j>0$. Set $a_j \equiv \frac 1{2\rho(R_j)} (\chi_{R_j}-\chi_{E_j})$. Obviously, each $a_j$ is an $H^1$-atom and $\|a_j\|_{H^1}\le 1$.
Take the function $\phi(x,t)\equiv\chi_{(2^{\ell_0},\infty)}(x)
\log(x-2^{\ell_0})\equiv h(x)$ for all $(x,t)\in S$. An easy calculation gives that $$\|\phi\|_{\BMO_\cp}\le \sup_{\gfz{I\subset\rr }
{ I\, \mathrm{is\, a\, dyadic\, interval}} } \frac1{|I|}\int_I
\lf|h(x)-\frac1 {|I|}\int_I h(y)\,dy\r|\,dx<\infty.$$ We then have $$\begin{aligned}
\|a_j\|_{H^1_{\cp}} & = \sup_{\psi\in \BMO_{\cp}}
\frac 1{\|\psi\|_{\BMO_{\cp}}}\lf|\int_S a_j\,\psi\, d\rho\r|\\
&\ge \frac 1{\|\phi\|_{\BMO_{\cp}}}\lf|\int_S a_j\,\phi\, d\rho\r|\\
&=\frac 1{2\|\phi\|_{\BMO_{\cp}}}\lf|2^{-\ell_j}\int_{2^{\ell_0}}^{
2^{\ell_0}+2^{\ell_j}}\,\log(x-2^{\ell_0})\,dx\r|
=\frac{|1-\log 2^{\ell_j}|}{2\|\phi\|_{\BMO_{\cp}}} \sim
|\ell_j|\,.
\end{aligned}$$ So there exists no positive constant such that $\|a_j\|_{H^1_{\cp}}\le C\|a_j\|_{H^1}$ for all $j<0$.
Notice that all the arguments of [@va Section 5] can be adapted to the dyadic spaces $H^1_{\cp}$ and $\BMO_{\cp}$ such that all results therein also hold for $H^1_{\cp}$ and $\BMO_{\cp}$. In particular, one can prove that, though $H_\cp^1$ is a proper subspace of $H^1$, the real interpolation space $[H^1_{\cp},L^2]_{\theta,q}$ is equal to $L^q$, if $\theta\in (0,1)$ and $\frac{1}{q}=1-\frac{\theta}{2}$.
Complex interpolation
=====================
We now formulate an interpolation theorem involving $H^1$ and $\BMO$. In the following, when $A$ and $B$ are Banach spaces and $\theta$ is in $(0,1)$, we denote by $(A,B)_{[\theta]}$ the complex interpolation space between $A$ and $B$ with parameter $\theta$, obtained via Calderón’s complex interpolation method (see [@Ca; @bl]).
\[int1\] Suppose that $\theta$ is in $(0,1)$. Then the following hold:
- if $\frac{1}{p_{\theta}}= \frac{1-\theta}{2}$, then $(L^2,\BMO)_{[\theta]}=L^{p_{\theta}};$
- if $\frac{1}{q_{\theta}}=1- \frac{\theta}{2}$, then $(H^1,L^2)_{[\theta]}=L^{q_{\theta}}$.
The proof of (i) is an easy adaptation of the proof of [[@fest p.156,Corollary 2]]{} and of [@cmm Theorem 7.4]. We omit the details.
The proof of (ii) follows from a duality argument (see [@cmm Theorem 7.4]). Denote by $X_{\theta}$ the interpolation space $\bigl(H^1, L^2 \bigr)_{[\theta]}$. Now by the duality theorem [@bl Corollary 4.5.2], if $\frac{1}{q_{\theta}}=1-\frac{\theta}{2}$, then the dual of $X_{\theta}$ is $\bigl(L^2,\BMO\bigr)_{[\theta]}$, which is equal to $L^{q_\theta'}$ by (i), where $\frac 1{q_\tz}+\frac 1{q_\tz'}=1$. Furthermore, $X_\theta$ is continuously included in $L^{q_\theta}$, because $H^1$ is continuously included in $L^1$ and $\bigl(L^1,L^2\bigr)_{[\theta]} =
L^{q_\theta}$. Since $L^2$ is reflexive, the interpolation space $X_\theta$ is reflexive (see [@bl Section 4.9]), so that $X_{\theta}$ is isomorphic to $X_\theta^{**} =
\bigl(L^{q_\theta'}\bigr)^*=L^{q_\theta}$. This concludes the proof.
A consequence of the previous theorem is the following.
\[t5.2\] Denote by $\Sigma$ the closed strip $\{ s\in \cc: \Re s\in [0,1] \}
$. Suppose that $\{T_s\}_{s\in\Sigma}$ is a family of uniformly bounded operators on $L^2$ such that the map $s\to \int_S
T_s(f)g\, d\rho$ is continuous on $\Sigma$ and analytic in the interior of $\Sigma$, whenever $f,g\in L^2$. Moreover, assume that there exists a positive constant $A$ such that $$\|T_{it} f\|_{L^2}\le A \,\|f\|_{L^2}\qquad {\forall\ } f\in L^2,\,
{\forall\ } t\in\rr\,,$$ and $$\|T_{1+it} f\|_{\BMO}\le A \,\|f\|_{\infty}\qquad {\forall\ } f\in
L^2\cap L^{\infty},\,{\forall\ } t\in\rr\,.$$ Then for every $\theta\in (0,1)$, the operator $T_{\theta}$ is bounded on $L^{p_{\theta}}$, with $\frac{1}{p_{\theta}}=\frac{1-\theta}{2}$ and $$\|T_{\theta} f\|_{L^{p_{\theta}}}\le A_{\theta}
\,\|f\|_{L^{p_{\theta}}}\qquad {\forall \ } f\in L^2\cap
L^{p_{\theta}}\,.$$ Here $A_{\theta}$ depends only on $A$ and $\theta$.
This follows from Theorem \[int1\](i) and [@cj Theorem 1]. Alternatively, we may follow the proof of [@st93 p.175, Theorem 4]. We leave the details to the reader.
\[t5.3\] Denote by $\Sigma$ the closed strip $\{ s\in \cc:\ \Re s\in [0,1] \}
$. Suppose that $\{T_s\}_{s\in\Sigma}$ is a family of uniformly bounded operators on $L^2$ such that the map $s\to \int_S
T_s(f)g\, d\rho$ is continuous on $\Sigma$ and analytic in the interior of $\Sigma$, whenever $f,g\in L^2$. Moreover, assume that there exists a positive constant $A$ such that $$\|T_{it} f\|_{L^1}\le A \,\|f\|_{H^1}\qquad {\forall\ } f\in
L^2\cap H^1,\,{\forall\ } t\in\rr\,,$$ and $$\|T_{1+it} f\|_{{L^2}}\le A \,\|f\|_{{L^2}}\qquad {\forall\ } f\in
L^2,\,{\forall\ } t\in\rr\,.$$ Then for every $\theta\in (0,1)$, the operator $T_{\theta}$ is bounded on $L^{q_{\theta}}$, with $\frac{1}{q_{\theta}}=1-\frac{\theta}{2}$ and $$\|T_{\theta} f\|_{L^{q_{\theta}}}\le A_{\theta}
\,\|f\|_{L^{q_{\theta}}}\qquad {\forall\ } f\in L^2\cap
L^{q_{\theta}}\,.$$ Here $A_{\theta}$ depends only on $A$ and $\theta$.
This follows from Theorem \[int1\](ii) and [@cj Theorem 1]. We omit the details.
\[dyadic-complex-interpolation\]It is easy to see that Theorems 5.1, 5.2 and 5.3 still hold if $H^1$ and $\BMO$ are replaced by $H_\cp^1$ and $\BMO_\cp$, respectively. We leave the details to the reader.
[**Acknowledgments**]{}Maria Vallarino is partially supported by PRIN 2007 “Analisi Armonica" and Dachun Yang (the corresponding author) is supported by the National Natural Science Foundation (Grant No. 10871025) of China.
Also, the authors sincerely wish to express their deeply thanks to the referee for her/his very carefully reading and also her/his so many careful, valuable and suggestive remarks which essentially improve the presentation of this article.
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abstract: 'We give a detailed exposition of the Alexandrov-Kontsevich-Schwarz-Zaboronsky superfield formalism using the language of graded manifolds. As a main illustarting example, to every Courant algebroid structure we associate canonically a three-dimensional topological sigma-model. Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action for the model.'
author:
- Dmitry Roytenberg
title: 'AKSZ-BV Formalism and Courant algebroid-induced topological field theories.'
---
Intro and Brief History.
========================
The standard procedure for quantizing classical field theories in the Lagrangian approach is by using the Feynman path integral. From the mathematical standpoint this is somewhat problematic, as it involves “integration” over the infinite-dimensional space of field configurations, on which no sensible measure has been found to exist. Nevertheless, the procedure can be made rigorous in the perturbative approach, provided the classical theory does not have too many symmetries (“too many” means, roughly speaking, an infinite-dimensional space). In the presence of these gauge symmetries, however, the procedure needs to be modified, as one has to integrate over the space of gauge-equivalence classes of field configurations. This can be accomplished by gauge-fixing (choosing a transversal slice to the gauge orbits), and in the late 60s Faddeev and Popov came up with an ingenious method of gauge-fixing the path integral by introducing extra “ghost” fields into the action functional. These ghosts, corresponding to generators of the gauge symmetries but of the opposite Grassman parity, were later incorporated into the cohomological approach of Bechi-Rouet-Stora and Tyutin (BRST), which is now considered the standard approach for quantizing gauge theories.
Unfortunately, the BRST approach fails in more complicated cases involving so-called “open” algebras of symmetries (that is, when the symmetries close under commutator bracket only modulo solutions of the classical field equations). In the early 80s Batalin and Vilkovisky [@BV1] developed a generalization of the BRST procedure which allows, in principle, to handle symmetries of arbitrary complexity. The idea is again to extend the field space by auxilliary fields (known as higher-generation ghosts, antighosts and Lagrange multipliers), as well as their “conjugate antifields”. The field-antifield space has two canonical structures. The first is an odd symplectic form, such that the fields and their respective antifields are conjugate with respect to the corresponding odd Poisson bracket $ (\cdot ,\cdot ) $ (the “antibracket”). The other is an odd second order differential operator $ \Delta $ (the so-called “BV Laplacian”) compatible with the antibracket. The original action functional is extended to a functional $ {\mathbf{S}} $ (called the master action) on this odd symplectic supermanifold, obeying the so-called quantum master equation $$({\mathbf{S}},{\mathbf{S}})-2i\hbar \Delta {\mathbf{S}}=0$$ The gauge-fixing is accomplished by choosing a Lagrangian submanifold, and the perturbative expansion is computed by evaluating the path integral over this Lagrangian submanifold. The generalized quantum BRST operator is $ Q=-i\hbar \Delta +({\mathbf{S}},\cdot ) $.
In practice the master action is computed by homological perturbation theory which involves calculating relations among the generators of the Euler-Lagrange ideal as well as the generators of the symmetries, relations among the relations, etc. This is known in homological algebra as the Koszul-Tate resolution, and can be very difficult to carry out in general. But in mid-90s Alexandrov, Kontsevich, Schwarz and Zaboronsky [@AKSZ] (referred to as AKSZ from now on) found a simple and elegant procedure for constructing solutions to the *classical master equation*[^1]: $$({\mathbf{S}},{\mathbf{S}})=0$$ Their approach uses mapping spaces of supermanifolds equipped with additional structure. Here we use a slightly refined notion of *differential graded (dg) manifold,* which is a supermanifold equipped with a compatible integer grading and a differential. The grading is needed to keep track of the ghost number symmetry important in some applications. If the source $ N $ is a dg manifold with an invariant measure, and the target $ M $ is a dg symplectic manifold, then the space of superfields $ \textrm{Maps}(N,M) $ acquires a canonical odd symplectic structure; furthermore, any self-commuting hamiltonian on $ M $ gives rise to a solution of the classical master equation. In case $ N=T[1]N_{0} $ (corresponding to the algebra of differential forms on a smooth manifold $ N_{0} $) , one gets the master action for topological field theories on $ N_{0} $ associated to various structures on the target. In particular, AKSZ show that Witten’s A and B topological sigma-models are special cases of the AKSZ construction.
Cattaneo-Felder [@CF2] and Park [@Park] further refined the AKSZ procedure by generalizing it to the case of manifolds with boundary, and produced new examples: Cattaneo and Felder studied the *Poisson sigma-model* [@SchStr] on the disk [@CF1] [@CF2], while Park considered its higher-dimensional generalizations, the *topological open p-branes*.
The Poisson sigma model is the most general 2d TFT that can be obtained within the AKSZ framework (at least if the source is $ T[1]N_{0} $), the reason being that, for two-dimensional $ N_{0} $, the symplectic form on the target must have degree 1, if ghost number symmetry is to be preserved. This immediately implies that the target is of the form $ M=T^{*}[1]M_{0} $, corresponding to the algebra of multivector fields on a manifold $ M_{0} $, with the symplectic form corresponding to the Schouten bracket. The diffrential structure on $ M $ is then necessarily given by a self-commuting bivector field on $ M_{0} $, i.e. a Poisson structure. The AKSZ procedure gives (the master action for) the Poisson sigma model. Witten’s A and B models can be obtained as special cases of this, when the Poisson tensor is invertible and there is a compatible complex structure.
Now, if we go one step further and consider three-dimensional $ N_{0} $, the AKSZ formalism requires a symplectic form of degree 2 on the target, and a self-commuting hamiltonian of degree 3. We have shown [@Roy4-GrSymp] that such structures are in canonical 1-1 correspondence with what is known as *Courant algebroids*. A Courant algebroid is given by specifying a bilinear operation on sections of a vector bundle $ E\rightarrow M_{0} $ with an inner product, satisfying certain natural properties. Thus the AKSZ procedure yields a canonical 3d TFT associated to any Courant algebroid. Its classical action is given by $$S_{0}[X,A,F]=\int _{N_{0}}F_{i}dX^{i}+\frac{1}{2}A^{a}g_{ab}dA^{b}-A^{a}P_{a}^{i}(X)F_{i}+\frac{1}{6}T_{abc}(X)A^{a}A^{b}A^{c}$$ where the fields are the membrane world volume $ X:N_{0}\rightarrow M_{0} $, an $ X^{*}E $-valued 1-form $ A $ and an $ X^{*}T^{*}M_{0} $-valued 2-form $ F $; $ g $ is the matrix of the inner product on $ E $, while $ P_{a}^{i} $ and $ T_{abc} $ are the structure functions of the Courant algebroid. This action has rather complicated symmetries, requiring the introduction of ghosts for ghosts; the master action contains terms up to degree 3 in antifields, involving up to 3rd derivatives of the structure functions of the Courant algebroid. Known special cases include the Chern-Simons theory (which is an ordinary gauge theory) and Park’s topological membrane.
The paper is organized as follows. Section 2 is a brief introduction to the general theory of differential graded manifolds. Section 3 explains the AKSZ formalism in detail. Section 4 contains a discussion of Courant algebroids and the associated closed membrane sigma model.
This paper is based on the lectures given by the author at Rencontres Mathematiques de Glanon in July 2003 to an audience of mathematicians and mathematical physicists. I would like to thank the organizers of the meeting and the people of Glanon for the excellent accomodation, friendly atmosphere and hospitality. I would also like to thank A. Cattaneo, A. Losev, J. Stasheff and T. Strobl for useful discussions and comments.
After these notes had been completed, we learned that Ikeda [@Ikeda] had obtained the Courant algebroid-valued sigma model by examining consistent BV-deformations of the abelian Chern-Simons gauge theory coupled with a zero-dimensional BF theory, rather than by immediate application of AKSZ. Also, Hofman and Park [@HP1][@HP2] considered a generalization of the topological open membrane which takes values in a quasi-Lie bialgebroid [@Roy3-QuasiLie], which is a Courant algebroid with a choice of a splitting.
Differential graded manifolds.
==============================
Here we collect the basic notions and fix the notation. The details can be found, for instance, in [@Vor3] or [@Roy4-GrSymp].
Graded manifolds.
-----------------
A *graded manifold* $ M $ over base $ M_{0} $ is a sheaf of $ {\mathbb {Z}} $-graded commutative algebras $ {\mathcal{C}}^{\cdot }(M) $ over a smooth manifold $ M_{0} $ locally isomorphic to an algebra of the form $ C^{\infty }(U)\otimes S^{\cdot }(V) $ where $ U\subset M $ is an open set, $ V $ is a graded vector space, and $ S^{\cdot }(V) $ is the free graded-commutative algebra on $ V $. Such a local isomorphism is referred to as an *affine coordinate chart* on $ M $; the sheaf $ {\mathcal{C}}^{\cdot }(M) $ is called the *sheaf of polynomial functions* on $ M $.
The generators of the algebra $ {\mathcal{C}}^{\cdot }(U)\simeq C^{\infty }(U)\otimes S^{\cdot }(V) $ are viewed as local coordinates on $ M $. Coordinate transformations are isomorphisms of algebras, hence in general non-linear. In what follows we will be mostly concerned with *nonnegatively* graded manifolds. In this case the transformation law for a coordinate of degree $ n $ is of the form $ x_{n}^{i}=A_{i'}^{i}(x_{0})x_{n}^{i'}+ $ (terms in coordinates of lower degrees, of total degree $ n $). This explains the word “affine” in the definition.
We have decomposition $ {\mathcal{C}}^{\cdot }(M)=\oplus _{k}{\mathcal{C}}^{k}(M) $ according to the degrees. Each $ {\mathcal{C}}^{k}(M) $ is a sheaf of locally free $ C^{\infty }(M_{0}) $-modules; in the nonnegative case $ C^{\infty }(M_{0})={\mathcal{C}}^{0}(M) $. We denote by $ {\mathcal{C}}_{k}(M) $ the subsheaf of algebras generated by $ \oplus _{i\leq k}{\mathcal{C}}^{i}(M) $. These form a filtration $$\cdots \subset {\mathcal{C}}_{0}(M)\subset {\mathcal{C}}_{1}(M)\subset {\mathcal{C}}_{2}(M)\subset \cdots$$ which in the nonnegative case is bounded below by $ {\mathcal{C}}_{0}(M)={\mathcal{C}}^{0}(M)=C^{\infty }(M_{0}) $. In this case there is a corresponding tower of fibrations of graded manifolds $$\label{tower}
M_{0}\leftarrow M_{1}\leftarrow M_{2}\leftarrow \cdots$$ with $ M $ being the projective limit of the $ M_{n} $’s.
If $ M=M_{n} $ for some $ n $, we say $ \textrm{deg}(M)=n $. Otherwise we say $ \textrm{deg}(M)=\infty $.
\[shift\] Given a vector bundle $ A\rightarrow M_{0} $, denote by $ A[n] $ the graded manifold obtained by assigning degree $ n $ to every fiber variable. The standard choice is $ n=1 $. Thus, $ {\mathcal{C}}^{\cdot }(A[1]) $ is the sheaf of sections of $ \wedge ^{\cdot }A^{*} $ with the standard grading.
For $ A=TM_{0} $ we have graded manifolds $ T[1]M_{0} $ corresponding to the sheaf of differential forms on $ M_{0} $, and $ T^{*}[1]M_{0} $ corresponding to the sheaf of multi-vector fields. In general, the graded manifold $ M_{1} $ in (\[tower\]) is always of the form $ A[1] $ for some $ A $, whereas the fibrations $ M_{k+1}\rightarrow M_{k} $ for $ k>0 $ are in general affine rather than vector bundles.
When working with graded manifolds it is very convenient to use the *Euler vector field* $ \epsilon $, which is defined as the derivation of $ {\mathcal{C}}^{\cdot }(M) $ such that $ \epsilon (f)=kf $ if $ f\in {\mathcal{C}}^{k}(M) $. In a local affine chart $ \epsilon =\sum _{\alpha }\textrm{deg}(x^{\alpha })x^{\alpha }\frac{\partial }{\partial x^{\alpha }} $. In particular, $ M_{0} $ is recovered as the set of fixed points of $ \epsilon $, which then acts on the normal bundle of $ M_{0} $ in $ M $; $ \textrm{deg}(M) $ defined above is simply the highest weight of this action, “the highest degree of a local coordinate”.
Graded manifolds form a category $ GrMflds $ with $ \textrm{Hom}(M,N)=\textrm{Hom}({\mathcal{C}}^{\cdot }(N),{\mathcal{C}}^{\cdot }(M)) $ in graded algebras. Any smooth manifold is a graded manifold with $ \epsilon =0 $; this gives a fully faithful embedding into $ GrMflds $. Furthermore, the assignment $ A\mapsto A[1] $ gives a fully faithful embedding of the category $ Vect $ of vector bundles into $ GrMflds $.
One also has the forgetful functor into the category $ SuperMflds $ of supermanifolds, which only remembers the grading modulo $ 2 $. The algebra $ {\mathcal{C}}^{\cdot }(M) $ is completed by allowing arbitrary smooth functions of all even variables. The Euler vector field on $ M $ is related to the parity operator on the corresponding supermanifold by $ P=(-1)^{\epsilon } $.
The roles of the $ {\mathbb {Z}}_{2} $-grading (parity) and the $ {\mathbb {Z}} $-grading are actually quite different. The former is responsible for the signs in formulas and in physics distinguishes bosons from fermions. The latter (the “ghost number” grading in physics) distinguishes physical fields from auxilliary ones (ghosts, antifields, etc.); for us the (nonnegative) integer grading has the added advantage of imposing rigid structure. But in general these two gradings are independent from one another [@Vor3]. Our requirement that they be compatible, aside from simplifying the presentation somewhat, is based on the fact that the classical field theories we consider here are bosonic. But in principle the AKSZ-BV formalism can handle more general bosonic-fermionic theories.
Vector bundles.
---------------
The notion of a vector bundle in $ GrMflds $ can be defined in two ways. Given a graded manifold $ M $, we can consider a sheaf of locally free $ {\mathcal{C}}^{\cdot }(M) $-modules. Alternatively, we can consider a graded manifold $ A $ with a surjective submersion $ A\rightarrow M $ in $ GrMflds $ and a linear structure on $ A $ given by an additional Euler vector field $ \epsilon _{\textrm{vect}} $ which assigns degree $ 1 $ to every fiber variable and degree $ 0 $ to all functions pulled back from $ M $ (consequently, $ \epsilon _{\textrm{vect}} $ necessarily commutes with the Euler vector field $ \epsilon _{A} $ defining the grading on $ A $). The $ {\mathcal{C}}^{\cdot }(M) $-module of sections of $ A $ can then be recovered by a variant of the mapping space construction described in the next section. One gets the following generalization of \[shift\]:
Given a vector bundle $ A\rightarrow M $ in $ GrMflds $, denote by $ A[n] $ the graded manifold corresponding to the Euler vector field $ \epsilon _{A}+n\epsilon _{\textrm{vect}} $. It is again a vector bundle over $ M $ (with the same $ \epsilon _{\textrm{vect}} $).
For a graded manifold $ M $, $ TM $ and $ T^{*}M $ are vector bundles in $ GrMflds $, with the Euler vector field $ \epsilon _{M} $ lifting canonically via Lie derivative (see below). Thus, we also get vector bundles $ T[n]M $ and $ T^{*}[n]M $ for various integers $ n $. For instance, in $ T^{*}[n]M $ one has $ \textrm{deg}(p_{\alpha })+\textrm{deg}(q^{\alpha })=n $. Iterating this construction gives rise to many interesting graded manifolds, such as $ T^{*}[2]T^{*}[1]M_{0} $ and so on.
Mapping spaces.
---------------
Because the structure sheaf $ {\mathcal{C}}^{\cdot }(M) $ of a graded manifold can contain nilpotents, care must be taken in defining such notions as points, maps, sections of vector bundles and so on. The most general solution is to use the categorical approach which, as far as we know, goes back to Grothendieck.
We shall assume that for any smooth manifolds $ M_{0} $, $ N_{0} $ the space of smooth maps $ \textrm{Maps}(M_{0},N_{0}) $ is endowed with a smooth structure making it an infinite-dimensional manifold.
Fix graded manifolds $ M $ and $ N $. Then the functor from $ GrMflds $ to $ Sets $ given by $ Z\mapsto \textrm{Hom}(N\times Z,M) $ is representable. In other words, there exists a graded manifold $ \textrm{Maps}(N,M) $, unique up to a unique isomorphism, such that $ \textrm{Hom}(N\times Z,M)=\textrm{Hom}(Z,\textrm{Maps}(N,M)) $. Its base $ \textrm{Maps}(N,M)_{0} $ is $ \textrm{Hom}(N,M) $, viewed as an infinite-dimensional smooth manifold containing $ \textrm{Maps}(N_{0},M_{0}) $.
(Sketch) For simplicity assume that $ M $ and $ N $ (but not $ Z $!) are nonnegatively graded, and $ \textrm{deg}(N)=1 $ (it will become clear how to handle the general case). Let $ x=\{x_{0},x_{1}\} $ be coordinates on $ N $, $ y=\{y_{0},y_{1},y_{2},\ldots \} $ (subscripts indicate the degree) on $ M $, $ z $ on $ Z $ (the latter are viewed as parameters). Then any morphism from $ N\times Z $ to $ M $ in $ GrMflds $ has a coordinate expression of the form$$\begin{array}{ccccl}
y_{0} & = & y_{0}(x,z) & = & y_{0,0}(x_{0},z)+y_{0,-1}(x_{0},z)x_{1}+\frac{1}{2}y_{0,-2}(x_{0},z)x_{1}^{2}+\cdots \\
y_{1} & = & y_{1}(x,z) & = & y_{1,1}(x_{0},z)+y_{1,0}(x_{0},z)x_{1}+\frac{1}{2}y_{1,-1}(x_{0},z)x_{1}^{2}+\cdots \\
y_{2} & = & y_{2}(x,z) & = & y_{2,2}(x_{0},z)+y_{2,1}(x_{0},z)x_{1}+\frac{1}{2}y_{2,0}(x_{0},z)x_{1}^{2}+\cdots \\
\ldots & \ldots & \ldots & \ldots & \ldots
\end{array}$$ Here we suppress the running indices, so that for instance $ \frac{1}{2}y_{0,-2}x_{1}^{2} $ actually means $ \frac{1}{2}y^{i}_{0,-2,\mu \nu }x_{1}^{\mu }x_{1}^{\nu } $. The coefficients are arbitrary expressions in $ x_{0} $ and $ z $ of total degree indicated by the second subscript. Now we let the arbitrary functions of $ x_{0} $, $ y_{p,q}(x_{0}) $ (viewed as coordinates of degree $ q $), parametrize $ \textrm{Maps}(N,M) $. As the parameter space $ Z $ is completely arbitrary, the transformation rules for the $ y_{p,q} $’s are determined by those for the $ x $’s and $ y $’s. The universal property is immediately verified: it’s just a matter of tautologically rewriting $ y_{p,q}(x_{0},z) $ as $ y_{p,q}(x_{0})(z) $. The degree $ 0 $ coordinates $ y_{p,0}(x_{0}) $ parametrize the “actual” (i.e. degree-preserving) maps from $ N $ to $ M $.
If $ N $ is just a point, we recover $ M $ as $ \textrm{Maps}(\textrm{pt},M) $. This shows the correct way to think of points in graded manifolds. In general, when dealing with points, maps, sections and so on, one must allow them to implicitly depend on arbitrary additional parameters.
In physics, the maps such as $ y_{p,0} $ appear as various kinds of physical fields, whereas the non-zero degree $ y_{p,q} $’s are referred to as “ghosts”, “antifields” and so on, and considered as non-physical, auxilliary fields. Correspondingly, the degree $ q $ in physics is called “ghost number”. Expressions such as $ y_{p}(x)=\sum y_{p,p-k}(x_{0})x_{1}^{k} $ are called *superfields*. In view of the above Proposition, there is a good reason (apart from the physical considerations) to call the non-zero degree $ y_{p,q} $’s “ghosts”: they only appear when additional parameters are introduced!
We shall not have a detailed discussion of graded (Lie) groups, but it should be clear how to define them, especially for the reader familiar with supergroups, on which there is by now extensive literature. Graded groups are just group objects in the category $ GrMflds $; the group axioms are expressed by commutative diagrams. In particular, for a graded manifold $ M $, $ \textrm{Diff}(M) $ is a graded group, constructed just as in the above Proposition. The direct product $ \textrm{Diff}(N)\times \textrm{Diff}(M) $ acts on $ \textrm{Maps}(N,M) $ in an obvious way, just like for ordinary manifolds.
The following statements are easily verified:
1. $ \textrm{Maps}(S^{1},{\mathbb {R}}[1])\simeq {\mathbb {R}}^{\infty }[1] $ (with Fourier modes as coordinates).
2. $ \textrm{Maps}({\mathbb {R}}[-1],M)\simeq T[1]M $ for any graded manifold $ M $.
Differentials.
--------------
A vector field on a graded manifold $ M $ is by definition a derivation of $ {\mathcal{C}}^{\cdot }(M) $. Vector fields form a sheaf of graded Lie algebras under the graded commutator.
A *differential graded manifold* is a graded manifold $ M $ with a self-commuting vector field $ Q $ of degree $ +1 $, i.e. $ [\epsilon ,Q]=Q $ and $ [Q,Q]=2Q^{2}=0 $.
$ \textrm{Vect}({\mathbb {R}}[-1]) $ is spanned by the Euler vector field $ \epsilon _{0}=-\theta \frac{d}{d\theta } $ and the differential $ Q_{0}=\frac{d}{d\theta } $, with commutation relations as in above definition. It acts on $ \textrm{Maps}({\mathbb {R}}[-1],M)\simeq T[1]M $ in an obvoius way, giving rise to the grading of differential forms and the de Rham differential $ d=dx^{i}\frac{\partial }{\partial x^{i}} $. In general, any differential graded manifold is, by definition, acted upon by the graded group $ \textrm{Diff}({\mathbb {R}}[-1]) $. The differential integrates to an action of the subgroup $ {\mathbb {R}}[1] $ (the “odd time flow”).
Any vector field $ v=v^{a}(x)\frac{\partial }{\partial x^{a}} $ on a graded manifold $ M $ gives rise to a vector field $ \iota _{v}=(-1)^{\textrm{deg}(v)}v^{a}(x)\frac{\partial }{\partial dx^{a}} $ on $ T[1]M $, and consequently also the Lie derivative $ L_{v}=[\iota _{v},d] $. One has $ \textrm{deg}(\iota _{v})=\textrm{deg}(v)-1 $, $ \textrm{deg}(L_{v})=\textrm{deg}(v) $ and $ L_{[v,w]}=[L_{v},L_{w}] $.
The assignment $ M_{0}\mapsto (T[1]M_{0},d) $ is a full and faithful functor from $ Mflds $ to $ dgMflds $. It is the *right* adjoint to the forgetful functor which assigns to any dg manifold $ N $ its base $ N_{0} $. In particular, the unit of adjunction gives a canonical dg map $ (M,Q)\rightarrow (T[1]M_{0},d) $ for any dg manifold $ M $ with base $ M_{0} $, called the *anchor*.
There are many interesting dg manifolds around. Those that are (as graded manifolds) of the form $ A[1] $ for some vector bundle $ A\rightarrow M_{0} $ are otherwise known as *Lie algebroids*. Those that come from Courant algebroids (see next section), however, are not of this form.
Symplectic and Poisson structures.
----------------------------------
Recall that the graded manifold structure on $ T[1]M $ is given by the Euler vector field $ \epsilon _{\textrm{tot }}=L_{\epsilon }+\epsilon _{\textrm{vect}} $ where $ \epsilon $ gives the grading on $ M $ and $ L $ denotes the Lie derivative. However, we shall speak of the degree[^2] of a differential form on $ M $ meaning only the action of the induced Euler vector field $ L_{\epsilon } $, rather than $ \epsilon _{\textrm{tot }} $ or $ \epsilon _{\textrm{vect}} $. Thus, “a $ p $-form $ \omega $ of degree $ q $” means $ \epsilon _{\textrm{vect}}\omega =p\omega $ and $ L_{\epsilon }\omega =q\omega $. In particular, a *symplectic structure of degree $ n $* is a closed non-degenerate two-form $ \omega $ on $ M $ with $ L_{\epsilon }\omega =n\omega $.
Given such a symplectic structure, one defines for any function $ f\in {\mathcal{C}}^{\cdot }(M) $ its hamiltonian vector field $ X_{f} $ (of degree $ \textrm{deg}(X_{f})=\textrm{deg}(f)-n $) via $$\iota _{X_{f}}\omega =(-1)^{n+1}df$$ and for any two functions $ f,g $ their Poisson bracket $$\{f,g\}=X_{f}\cdot g=(-1)^{\textrm{deg}(X_{f})}\iota _{X_{f}}dg=(-1)^{\textrm{deg}(f)+1}\iota _{X_{f}}\iota _{X_{g}}\omega$$ It’s easily verified that the bracket defines a graded Lie algebra structure on $ {\mathcal{C}}^{\cdot }(M)[n] $ and $ f\mapsto X_{f} $ is a homomorphism[^3].
The basic example of a graded symplectic manifold is $ T^{*}[n]M $ for some graded manifold $ M $ and an integer $ n $. The symplectic form is canonical $ \omega =(-1)^{\tilde{\omega }\tilde{\alpha }}dp_{\alpha }dq^{\alpha } $ where $ \tilde{\omega }=n\; \textrm{mod }2 $ and $ \tilde{\alpha }=\textrm{deg}(q^{\alpha })\; \textrm{mod }2 $. The signs are chosen so that we have always $ \{p_{\alpha },q^{\beta }\}=\delta _{\alpha }^{\beta } $. For $ n=1 $ and $ M=M_{0} $ an ordinary manifold, the Poisson bracket induced by $ \omega $ is just the Schouten bracket of multivector fields; it is easy to show that any nonnegatively graded symplectic manifold of degree $ 1 $ is canonically isomorphic to $ T^{*}[1]M_{0} $ for some ordinary manifold $ M_{0} $.
If $ V $ is a vector space, any nondegenerate symmetric bilinear form on $ V $ can be viewed as a symplectic structure on $ V[1] $ as follows: $ \omega =\frac{1}{2}d\xi ^{a}g_{ab}d\xi ^{b} $ where $ g_{ab}=<e_{a},e_{b}> $for some basis $ \{e_{a}\} $ of $ V $. Clearly this $ \omega $ has degree $ 2 $.
The following statements concerning a graded symplectic manifold $ (M,\omega ) $ are easily verified:
1. If $ M $ is nonnegatively graded, $ \textrm{deg}(M)\leq \textrm{deg}(\omega ) $.
2. If $ \textrm{deg}(\omega )=n\neq 0 $, then $ \omega =d\alpha $, where $ \alpha =\frac{1}{n}\iota _{\epsilon }\omega $.
3. If $ v $ is a vector field of degree $ m\neq -n $, such that $ L_{v}\omega =0 $, then $ \iota _{v}\omega =\pm d(\frac{1}{m+n}\iota _{v}\iota _{\epsilon }\omega ) $.
A corollary of the last statement is that for a graded symplectic manifold of degree $ n\neq -1 $, any differential preserving $ \omega $ is of the form $ Q=\{\Theta ,\cdot \} $ for some $ \Theta \in {\mathcal{C}}^{n+1}(M) $ obeying the Maurer-Cartan equation $ \{\Theta ,\Theta \}=0 $. A graded manifold equipped with such an $ \omega $ and $ \Theta $ will be referred to as a *differential graded symplectic manifold.* For $ n=1 $ and $ M=T^{*}[1]M_{0} $ this is the same as an ordinary Poisson structure on $ M_{0} $.
Measure and integration.
------------------------
The general integration theory on supermanifolds is quite nontrivial. Fortunately, all we need here is the notion of the integral of a function over the whole graded manifold, i.e. a volume form or a measure. By a measure on $ M $ we shall understand a functional on the space of compactly supported functions (denoted by $ f\mapsto \int _{M}\mu f $) such that locally $ \mu $ is the Berezinian measure of the form $ F(x)dx $. We call $ \mu $ *nondegenerate* if the bilinear form $ <f,g>=\int _{M}\mu fg $ is. Given a vector field $ v $ on $ M $ we say a measure $ \mu $ on $ M $ is *$ v $-invariant* if $ \int _{M}\mu (vf)=0 $ for any $ f $.
Given two graded manifolds $ M $ and $ N $ and a measure $ \mu $ on $ N $ one can define the push-forward or fiber integration of differential forms. This is a chain map $ \mu _{*}:\Omega ^{k}(N\times M)\rightarrow \Omega ^{k}(M)[\textrm{deg}\mu ] $ defined as follows: $$\mu _{*}\omega (y)(v_{1},\ldots ,v_{k})=\int _{N}\mu (x)\omega (x,y)(v_{1},\ldots ,v_{k})$$ If $ \mu $ is $ v $-invariant for some vector field $ v $ on $ N $, one has $ \mu _{*}L_{v_{1}}=0 $, where $ v_{1} $ is the lift to $ N\times M $ of $ v $.
If $ N_{0} $ is a closed oriented smooth $ n $-manifold, the graded manifold $ T[1]N_{0} $ has a canonical measure defined as follows: $$\int _{T[1]N_{0}}\mu f=\int _{N_{0}}f^{\textrm{top}}$$ where $ f^{\textrm{top}} $ denotes the top-degree component of the inhomogeneous differential form $ f $. This measure has degree $ -n $, is invariant with respect to the de Rham vector field $ d $ (Stokes’ Theorem) and also with respect to all vector fields of the form $ \iota _{v} $ (since the top degree component of $ \iota _{v}f $ is always zero); hence, it is invariant with respect to all Lie derivatives $ L_{v} $, i.e. all isotopies of $ N_{0} $ (in fact, all orientation-preserving diffeomorphisms). The induced nondegenerate pairing of differential forms is the Poincare pairing.
The AKSZ formalism.
===================
Let us fix the following data.
**The source**: a dg manifold $ (N,D) $ endowed with a nondegenerate $ D $-invariant measure $ \mu $ of degree $ -n-1 $ for a positive integer $ n $. In practice we will consider $ N=T[1]N_{0} $ for a closed oriented $ (n+1) $-manifold $ N_{0} $, with $ D=d $, the de Rham vector field, and $ \mu $ the canonical measure.
**The target**: a dg symplectic manifold $ (M,\omega ,Q) $ with $ \textrm{deg}(\omega )=n $. Then $ Q $ is of the form $ \{\Theta ,\cdot \} $ for a solution $ \Theta \in {\mathcal{C}}^{n+1}(M) $ of the Maurer-Cartan equation: $$\{\Theta ,\Theta \}=0$$ We shall assume here that both $ M $ and $ N $ are nonnegatively graded. Our space of superfields will be $ {\mathcal{P}}=\textrm{Maps}(N,M) $. This is a graded manifold, with the degree of a functional referred to as the *ghost number*. In the algebra of functions on $ {\mathcal{P}} $, generators of non-negative ghost number will be called *fields*, those of negative ghost number – the *antifields*. Among the fields we further distinguish between the *classical fields* (of ghost number zero) and *ghosts*, ghosts for ghosts, etc. (of positive ghost number). We shall put a $ QP $-structure on $ {\mathcal{P}} $, i.e. an odd symplectic form $ \Omega $ of degree $ -1 $ and a homological vector field (the BRST differential) $ {\mathbf{Q}} $ of degree $ +1 $ preserving $ \Omega $. It is not guaranteed in general that such a field is hamiltonian (since $ {\mathbf{Q}} $ and $ \Omega $ are of opposite degree), but in our case it will be since $ \Omega $ is exact. Thus we will obtain a solution $ {\mathbf{S}} $ of the classical Batalin-Vilkovisky master equation: $${(}{\mathbf{S}},{\mathbf{S}}{)}=0$$ where $ {(}\cdot ,\cdot {)} $ is the odd Poisson bracket corresponding to $ {\omega } $, otherwise known as the *BV antibracket*, while $ {\mathbf{S}} $ (of ghost number zero) is the hamiltonian of $ {\mathbf{Q}} $.
Let us first define the $ Q $-structure. As remarked above, the graded group $ \textrm{Diff}(N)\times \textrm{Diff}(M) $ acts naturally on $ {\mathcal{P}} $. Therefore, the vector fields $ D $ on $ N $ and $ Q $ on $ M $ induce a pair of commuting homological vector fields on $ {\mathcal{P}} $ which we denote by $ \hat{D} $ and $ \check{Q} $, respectively. Hence, any linear combination $ {\mathbf{Q}}=a\hat{D}+b\check{Q} $ is a $ Q $-structure. We shall fix the coefficients later to get the master action in the form we want.
Now for the $ P $-structure. There is a canonical evaluation map $ \textrm{ev}:N\times {\mathcal{P}}\rightarrow M $ given by $ (x,f)\mapsto f(x) $. This enables us to pull back differential forms on $ M $ using $ \textrm{ev} $, and then push them forward to $ {\mathcal{P}} $ using the measure $ \mu $. The resulting chain map $ \mu _{*}\textrm{ev}^{*}:\Omega ^{k}(M)\rightarrow \Omega ^{k}({\mathcal{P}}) $ has ghost number $ \textrm{deg}(\mu )=-n-1 $. So let us define $ \Omega =\mu _{*}\textrm{ev}^{*}\omega $. If the measure $ \mu $ is nondegenerate, this defines a $ P $-structure.
It remains to check that the $ P $ and $ Q $-structures are compatible and calculate the master action. We first observe that $ \mu _{*}\textrm{ev}^{*} $ applied to functions preserves the Poisson brackets. That is, we have $${(}\int _{N}\mu \phi ^{*}(\xi ),\int _{N}\mu \phi ^{*}(\eta ){)}=\int _{N}\mu \phi ^{*}(\{\xi ,\eta \})$$ for $ \xi ,\eta \in {\mathcal{C}}^{\cdot }(M) $ and a superfield $ \phi $. Moreover, if $ Q $ has hamiltonian $ \Theta $, then $ \check{Q} $ has hamiltonian $ \mu _{*}\textrm{ev}^{*}\Theta $. This gives us the interaction term of our master action: $${\mathbf{S}}_{\textrm{int}}[\phi ]=\int _{N}\mu \phi ^{*}(\Theta )$$ To see that $ \hat{D} $ also preserves $ {\omega } $, observe that, if $ \mu $ is $ D $-invariant, then $ L_{\hat{D}}\mu _{*}\textrm{ev}^{*}=0 $. This is essentially because the evaluation map is invariant under the diagonal action of $ \textrm{Diff}(N) $. Moreover, if $ \omega =d\alpha $ for some 1-form $ \alpha $ on $ M $, this implies that up to a sign, the hamiltonian of $ \hat{D} $ is $ \iota _{\hat{D}}{\alpha } $, where $ {\alpha }=\mu _{*}\textrm{ev}^{*}\alpha $. This gives the kinetic term $ {\mathbf{S}}_{\textrm{kin}} $ of the master action.
To see what this all looks like written out explicitly in coordinates, we begin with the following general remark. In any odd symplectic manifold $ {\mathcal{P}} $ with $ \Omega $ of degree $ -1 $, the ideal generated by functions of negative degree corresponds to a *Lagrangian* submanifold $ {\mathcal{L}} $, and in fact $ {\mathcal{P}} $ is canonically isomorphic to $ T^{*}[-1]{\mathcal{L}} $. In our case this $ {\mathcal{L}} $ is the space of fields (including ghosts). It contains the space of classical fields $ {\mathcal{L}}_{0} $, corresponding to the ideal generated by functions of nonzero degree (ghosts and antifields). The restriction of any $ {\mathbf{S}} $ of ghost number zero to $ {\mathcal{L}} $ depends only on the classical fields, i.e. is a pullback of a functional $ S $ on $ {\mathcal{L}}_{0} $. This way we recover the classical action. As the critical points of $ {\mathbf{S}} $ are the fixed points of $ {\mathbf{Q}} $, we see that the solutions of the classical field equations for $ S $ are the dg maps from $ (N,D) $ to $ (M,Q) $.
Let us assume from now on that $ N=T[1]N_{0} $ for some closed oriented $ (n+1) $-dimensional $ N_{0} $, with $ D=d $, the de Rham vector field, and $ \mu $ the canonical measure. In coordinates $ d=du^{\nu }\frac{\partial }{\partial u^{\nu }} $, and we denote the induced differential on superfields by $ {\mathbf{d}} $ instead of $ \hat{D} $.
As for the target, let $ \omega $ be written in Darboux coordinates as $ \omega =\frac{1}{2}dx^{a}\omega _{ab}dx^{b} $. Here $ \omega _{ab} $ are constants, and so the degrees of $ x^{a} $ and $ x^{b} $ must add up to $ n $. We shall choose $ \alpha =\frac{1}{2}x^{a}\omega _{ab}dx^{b} $ as the primitive of $ \omega $.
Now, a map $ \phi :N\rightarrow M $ is parametrized in coordinates by superfields $ \phi ^{*}(x^{a})=\phi ^{a}=\phi ^{a}(u,du) $. Then $ \Omega $ is given by$$\Omega =\int _{T[1]N_{0}}\mu (\frac{1}{2}\delta \phi ^{a}\omega _{ab}\delta \phi ^{b})=\int _{N_{0}}(\frac{1}{2}\delta \phi ^{a}\omega _{ab}\delta \phi ^{b})^{\textrm{top}}$$ To find the field-antifield splitting and compute the normal form for $ \Omega $, let us further keep track of the degree of $ x^{a} $ by writing it as a subscript: $ x_{i}^{a} $ is of degree $ 0\leq i\leq n $, then we have $$\phi _{i}^{a}=\sum _{j=0}^{n+1}\phi _{i,j}^{a}$$ where $ \phi _{i,j}^{a}=\phi _{i,j}^{a}(u)(du)^{j}=\frac{1}{j!}\phi ^{a}_{i,j,\nu _{1}\ldots \nu _{j}}(u)du^{\nu _{1}}\cdots du^{\nu _{j}} $ is the $ j $-form component of $ \phi _{i}^{a} $ whose coefficients $ \phi _{i,j}^{a}(u) $ have therefore ghost number $ i-j $. It is easy to see then that we must set, for each field $ \phi _{i,j}^{a} $ with $ i-j\geq 0 $, its conjugate antifield to be $ \phi ^{\dagger ,i,j}_{a}=(-1)^{ni}\phi _{n-i,n+1-j}^{b}\omega _{ba} $. Then we can rewrite $ \Omega $ as $$\Omega =\int _{N_{0}}(-1)^{i}\sum _{i-j\geq 0}\delta \phi ^{\dagger ,i,j}_{a}\delta \phi ^{a}_{i,j}$$ The master action is given by $${\mathbf{S}}[\phi ]=\int _{T[1]N_{0}}\mu (\frac{1}{2}\phi ^{a}\omega _{ab}{\mathbf{d}}\phi ^{b}+(-1)^{n+1}\phi ^{*}\Theta )$$ The classical action $ S $ is then recovered by setting all the antifields to zero. The sign in front of the interaction term is chosen so that the solutions of the classical field equations $ \delta S=0 $ coincided with dg maps $ \phi :(T[1]N_{0},d)\rightarrow (M,Q=\{\Theta ,\cdot \}) $. As we have remarked, $ \mu $ is invariant under all orientation-preserving diffeomorphisms of $ N_{0} $, hence $ S $ yields a topological field theory.
Courant algebroids and the topological closed membrane.
=======================================================
Specializing the above construction to various choices of the target one gets many interesting topological field theories. For instance, in case $ n=1 $ the target is necessarily of the form $ M=T^{*}[1]M_{0} $ for some manifold $ M_{0} $, and the interaction term is given by a Poisson bivector field $ \pi $ on $ M_{0} $. The BV quantization of the resulting two-dimensional TFT – the *Poisson sigma model* – was extensively studied by Cattaneo and Felder [@CF1][@CF2].
Here we would like to consider the case $ n=2 $. Symplectic nonnegatively graded manifolds $ (M,\omega ) $ with $ \textrm{deg}(\omega )=2 $ were shown in [@Roy4-GrSymp] to correspond to vector bundles $ E\rightarrow M_{0} $ with a fiberwise nondegenerate symmetric inner product $ <\cdot ,\cdot > $ (of arbitrary signature). The construction is as follows. Recall that $ \textrm{deg}(M)\leq 2 $, hence $ M $ fits into a tower of fibrations $$M=M_{2}\rightarrow M_{1}\rightarrow M_{0}$$ where $ M_{1} $ is of the form $ E[1] $ for some vector bundle $ E\rightarrow M_{0} $. Restricting the Poisson bracket to $ M_{1} $ gives the inner product. Conversely, given $ E $, $ M $ is obtained as the symplectic submanifold of $ T^{*}[2]E[1] $ corresponding to the isometric embedding $ E\hookrightarrow E\oplus E^{*} $with respect to the canonical inner product on $ E\oplus E^{*} $. If $ \{x^{i}\} $ are local coordinates on $ M_{0} $ and $ \{e_{a}\} $ is a local basis of sections of $ E $ such that $ <e_{a},e_{b}>=g_{ab}=\textrm{const}. $, we get Darboux coordinates $ \{q^{i},p_{i},\xi ^{a}\} $ on $ M $ (of degrees $ 0 $, $ 2 $ and $ 1 $, respectively), so that $$\omega =dp_{i}dq^{i}+\frac{1}{2}d\xi ^{a}g_{ab}d\xi ^{b}=d(p_{i}dq^{i}+\frac{1}{2}\xi ^{a}g_{ab}d\xi ^{b})$$ Notice that the quadratic hamiltonians $ {\mathcal{C}}^{2}(M) $ form a Lie algebra under the Poisson bracket, which is isomorphic to the Lie algebra of infinitesimal bundle automorphisms of $ E $ preserving $ <\cdot ,\cdot > $.
It was further shown in [@Roy4-GrSymp] that solutions $ \Theta \in {\mathcal{C}}^{3}(M) $ of the Maurer-Cartan equation $ \{\Theta ,\Theta \}=0 $ correspond to *Courant algebroid* structures on $ (E,<\cdot ,\cdot >) $. Such a structure is given by a bilinear operation $ \circ $ on sections of $ E $. The condition on $ \circ $ is that for every section $ e $ of $ E $, $ e\circ $ acts by infinitesimal automorphisms of $ (E,<\cdot ,\cdot >,\circ ) $. In particular, $ e\circ $ is a first-order differential operator whose symbol is a vector field on $ M_{0} $ which we denote by $ a(e) $. This gives rise to the anchor map $ a:E\rightarrow TM_{0} $. Furthermore, $ e\circ $ preserves $ <\cdot ,\cdot > $: $$a(e)<e_{1},e_{2}>=<e\circ e_{1},e_{2}>+<e_{1},e\circ e_{2}>$$ as well as the operation $ \circ $ itself: $$e\circ (e_{1}\circ e_{2})=(e\circ e_{1})\circ e_{2}+e_{1}\circ (e\circ e_{2})$$ i.e. $ \circ $ defines a Leibniz algebra on sections of $ E $. It follows also that the anchor $ a $ induces a homomorphism of Leibniz algebras. The only additional property of $ \circ $ concerns its symmetric part, namely $$<e,e_{1}\circ e_{2}+e_{2}\circ e_{1}>=a(e)<e_{1},e_{2}>$$
Now, if we introduce local coordinates as above, the corresponding $ \Theta \in {\mathcal{C}}^{3}(M) $ will be $$\Theta =\xi ^{a}P_{a}^{i}(q)p_{i}-\frac{1}{6}T_{abc}(q)\xi ^{a}\xi ^{b}\xi ^{c}$$ where $ P^{i}_{a} $ is the anchor matrix and $ T_{abc}=<e_{a}\circ e_{b},e_{c}> $. The Maurer-Cartan equation $ \{\Theta ,\Theta \}=0 $ is equivalent to the defining properties of $ \circ $. The corresponding differential $ Q $ sends a function $ f\in {\mathcal{C}}^{0}(M)=C^{\infty }(M_{0}) $ to $ a^{*}df $, and a section $ e\in \Gamma (E)={\mathcal{C}}^{1}(M) $ to $ e\circ \in {\mathcal{C}}^{2}(M) $.
Now we can write down the sigma-model. Fix a closed oriented 3-manifold $ N_{0} $, with coordinates $ \{u^{\mu }\} $. The classical fields are the degree-preserving maps $ T[1]N_{0}\rightarrow M $, consisting of a smooth map $ X:N_{0}\rightarrow M_{0} $ (the membrane world-volume), an $ X^{*}E $-valued 1-form $ A $ and an $ X^{*}T^{*}M_{0} $-valued 2-form $ F $[^4]. The superfields are written as follows: $$\begin{array}{rcl}
{\mathbf{q}}^{i} & = & X^{i}+F^{i}_{\dagger }du+\alpha ^{i}_{\dagger }(du)^{2}+\gamma ^{i}_{\dagger }(du)^{3}\\
{\xi }^{a} & = & \beta ^{a}+A^{a}du+g^{ab}A^{\dagger }_{b}(du)^{2}+g^{ab}\beta _{b}^{\dagger }(du)^{3}\\
{\mathbf{p}}_{i} & = & \gamma _{i}+\alpha _{i}du+F_{i}(du)^{2}+X_{i}^{\dagger }(du)^{3}
\end{array}$$ In particular, the $ X^{*}E $-valued scalar $ \beta $ and the $ X^{*}T^{*}M_{0} $-valued 1-form $ \alpha $ are the ghosts, while the $ X^{*}T^{*}M_{0} $-valued scalar $ \gamma $ is the ghost for ghost. The master action $${\mathbf{S}}=\int _{T[1]N_{0}}\mu ({\mathbf{p}}_{i}{\mathbf{d}}{\mathbf{q}}^{i}+\frac{1}{2}{\xi }^{a}g_{ab}{\mathbf{d}}{\xi }^{b}-\Theta ({\mathbf{q}},{\xi },{\mathbf{p}}))$$ decomposes as $ {\mathbf{S}}=S_{0}+S_{1}+S_{2}+S_{3} $ where the subscript denotes the number of antifields. Thus, $ S_{0} $ is the classical action:$$S_{0}=\int _{N_{0}}F_{i}dX^{i}+\frac{1}{2}A^{a}g_{ab}dA^{b}-A^{a}P_{a}^{i}(X)F_{i}+\frac{1}{6}T_{abc}(X)A^{a}A^{b}A^{c}$$ and the rest of the terms are as follows:$$\begin{array}{rcl}
S_{1} & = & \int _{N_{0}}(-\beta ^{a}P_{a}^{i}X_{i}^{\dagger }+(d\beta ^{c}-g^{ac}P_{a}^{i}\alpha _{i}+\beta ^{a}A^{b}T_{abr}g^{rc})A^{\dagger }_{c}+\\
& & +(-d\alpha _{j}-\beta ^{a}\partial _{j}P^{i}_{a}F_{i}-A^{a}\partial _{j}P^{i}_{a}\alpha _{i}+\frac{1}{2}\partial _{j}T_{abc}\beta ^{a}A^{b}A^{c})F^{j}_{\dagger }+\\
& & +(-d\gamma _{j}-\beta ^{a}\partial _{j}P^{i}_{a}\alpha _{i}-A^{a}\partial _{j}P^{i}_{a}\gamma _{i}+\frac{1}{2}\partial _{j}T_{abc}\beta ^{a}\beta ^{b}A^{c})\alpha ^{j}_{\dagger }+\\
& & +(\frac{1}{2}T_{abr}g^{rc}\beta ^{a}\beta ^{b}-g^{ac}P_{a}^{i}\gamma _{i})\beta ^{\dagger }_{c}+(\frac{1}{6}\partial _{j}T_{abc}\beta ^{a}\beta ^{b}\beta ^{c}-\beta ^{a}\partial _{j}P^{i}_{a}\gamma _{i})\gamma ^{j}_{\dagger })
\end{array}$$ $$\begin{array}{rcl}
S_{2} & = & \int _{N_{0}}((\frac{1}{2}\partial _{j}T_{abr}g^{rc}\beta ^{a}\beta ^{b}-g^{ac}\partial _{j}P^{i}_{a}\gamma _{i})F_{\dagger }^{j}A^{\dagger }_{c}-\frac{1}{2}(\beta ^{a}\partial _{j}\partial _{k}P^{i}_{a}\alpha _{i}+A^{a}\partial _{j}\partial _{k}P^{i}_{a}\gamma _{i}-\\
& & -\frac{1}{2}\partial _{j}\partial _{k}T_{abc}\beta ^{a}\beta ^{b}A^{c})F^{j}_{\dagger }F^{k}_{\dagger }-(\beta ^{a}\partial _{j}\partial _{k}P^{i}_{a}\gamma _{i}-\frac{1}{6}\partial _{j}\partial _{k}T_{abc}\beta ^{a}\beta ^{b}\beta ^{c})F^{j}_{\dagger }\alpha _{\dagger }^{k})
\end{array}$$ $$S_{3}=\int _{N_{0}}\frac{1}{6}(-\beta ^{a}\partial _{j}\partial _{k}\partial _{l}P^{i}_{a}\gamma _{i}+\frac{1}{6}\partial _{j}\partial _{k}\partial _{l}T_{abc}\beta ^{a}\beta ^{b}\beta ^{c})F^{j}_{\dagger }F^{k}_{\dagger }F^{l}_{\dagger }$$ It appears our sigma model has very complicated 2-algebroid gauge symmetries generated by parameters $ \alpha _{i},\beta ^{a} $ and $ \gamma _{i} $; writing down the master action without the help of AKSZ would have been extremely difficult. It would be desirable to better understand the structure of the gauge symmetries.
In conclusion, let us point out some special cases.
Let $ M_{0}=\{\textrm{pt}\} $. Then $ (E,<\cdot ,\cdot >) $ is just a vector space with an inner product. A Courant algebroid structure reduces to that of a quadratic Lie algebra with structure constants $ T_{abc}=<[e_{a},e_{b}],e_{c}> $. A quick glance reveals that in this case $ S_{0} $ is the classical Chern-Simons functional, for which the master action was written down in [@AKSZ].
Let $ M=T^{*}[2]T^{*}[1]M_{0}=T^{*}[2]T[1]M_{0} $, with coordinates $ \{q^{i},\xi ^{i},\theta _{i},p_{i}\} $ of degree 0,1,1 and 2, respectively (one thinks $ \xi ^{i}=dx^{i} $). Then $ \omega =dp_{i}dq^{i}+d\xi ^{i}d\theta _{i}=d(p_{i}dq^{i}+\xi ^{i}d\theta _{i}) $, and we consider $ \Theta =\xi ^{i}p_{i}-\frac{1}{6}c_{ijk}(q)\xi ^{i}\xi ^{j}\xi ^{k} $, where $ c=\frac{1}{6}c_{ijk}(q)\xi ^{i}\xi ^{j}\xi ^{k} $ is a 3-form on $ M_{0} $. Clearly $ \Theta $ obeys Maurer-Cartan if and only if $ dc=0 $. The corresponding Courant algebroid structure on $ E=TM\oplus T^{*}M $ (with the canonical inner product) is given by $$(X+\xi )\circ (Y+\eta )=[X,Y]+L_{X}\eta -\iota _{Y}d\xi +\iota _{X\wedge Y}c$$ The classical fields for the corresponding topological membrane action are comprised of the membrane world-volume $ X:N_{0}\rightarrow M_{0} $, an $ X^{*}TM_{0} $-valued 1-form $ A $, an $ X^{*}T^{*}M_{0} $-valued 1-form $ B $ and an $ X^{*}T^{*}M_{0} $-valued 2-form $ F $. The classical action $$S_{0}[X,A,B,F]=\int _{N_{0}}F_{i}dX^{i}+A^{i}dB_{i}-A^{i}F_{i}+\frac{1}{6}c_{ijk}(X)A^{i}A^{j}A^{k}$$ was considered by Park [@Park]. We leave it to the reader to write down the master action in this case.
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[^1]: which, for the cases they consider, also implies the quantum master equation.
[^2]: Physicists would prefer the term “ghost number”.
[^3]: Strictly speaking, one should write $ f\mapsto X_{f[-n]} $
[^4]: Strictly speaking, this is misleading as the transformation law for $ p_{i} $ is nonlinear, containing a term quadratic in the $ \xi $’s; as a graded manifold, $ M $ is isomorphic to $ E[1]\oplus T^{*}[2]M_{0} $ only after a $ g $-preserving connection has been fixed. This issue complicates a coordinate-free description of the sigma model.
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abstract: 'I study the lattice dynamics and electron-phonon coupling in non-centrosymmetric quasi-one-dimensional K$_2$Cr$_3$As$_3$ using density functional theory based first principles calculations. The phonon dispersions show stable phonons without any soft-mode behavior. They also exhibit features that point to a strong interaction of K atoms with the lattice. I find that the calculated Eliashberg spectral function shows a large enhancement around 50 cm$^{-1}$. The phonon modes that show large coupling involve in-plane motions of all three species of atoms. The $\mathbf{q}$ dependent electron-phonon coupling decreases strongly away from the $q_z = 0$ plane. The total electron-phonon coupling is large with a value of $\lambda_{\textrm{ep}} = 3.0$, which readily explains the experimentally observed large mass enhancement.'
author:
- Alaska Subedi
title: 'Strong-coupling electron-phonon superconductivity in noncentrosymmetric quasi-one-dimensional K$_2$Cr$_3$As$_3$'
---
Introduction
============
Almost all known superconductors are conventional electron-phonon superconductors, and the phenomenon of unconventional superconductivity, especially in one-dimensional (1D) materials, is relatively unexplored. Therefore, the recent report of superconductivity that is likely of an unconventional nature in non-centrosymmetric quasi-1D K$_2$Cr$_3$As$_3$ with a $T_c$ of 6.1 K by Bao *et al.* is of significant importance [@bao15].
K$_2$Cr$_3$As$_3$ is structurally similar to the $M$Mo$_3$$X$$_3$ or $M_2$Mo$_6$$X$$_6$ ($M$ = metals, $X$ = chalcogens) family of compounds first discovered in 1980 [@pote80; @honl80]. In addition to K$_2$Cr$_3$As$_3$, superconductivity has also been observed in two other isostructural compounds Rb$_2$Cr$_3$As$_3$ and Cs$_2$Cr$_3$As$_3$ with $T_c$’s of 4.8 and 2.2 K, respectively[@tang15a; @tang15b; @tang15c]. The one-dimensional element in $A_2$Cr$_3$As$_3$ ($A$ = K, Rb, Cs) compounds is a \[(Cr$_3$As$_3$)$^{2-}$\]$_\infty$ chain, which is arranged in a hexagonal lattice. This chain has the structure of a double-walled subnanotube. The inner wall is made of stacked Cr$_6$ octahedra that are face-sharing along the $c$ axis and the outer wall is made of similarly stacked face-sharing As$_6$ octahedra. The individual chains are separated by metal cations that provide the compensating charge and stabilize the structure.
The superconductivity in K$_2$Cr$_3$As$_3$ exhibits various features that are suggestive of an unconventional nature. The superconducting state arises out of a normal state that has a large specific heat coefficient of 70–75 mJ K$^{-2}$ mol$^{-1}$, which indicates strong quasiparticle mass renormalization[@bao15; @kong15]. The upper critical field exceeds the one-band BCS estimate of the Pauli limit by a factor of 3–4[@bao15; @kong15; @bala15; @wang15]. NMR experiments show that spin fluctuations grow as the temperature approaches $T_c$ [@zhi15]. Below $T_c$, the nuclear spin-lattice relaxation rate does not show Hebel-Slichter coherence peak that is characteristic of an isotropic $s$-wave superconductor, and the relaxation rate decreases with a power-law behavior suggesting the presence of zero-energy excitations in the superconducting state. The presence of zero-energy excitations is also seen in the temperature dependence of the penetration depth, which decreases linearly at temperatures well below $T_c$[@pang15]. A fit of the angular dependence of the superconducting gap to the superfluid density obtained from the penetration depth shows reasonable agreement with various $p$-, $d$- and $f$-wave models that exhibit nodes. The temperature dependence of the superfluid density obtained from transverse-field $\mu$SR measurements show equally good fit to both an isotropic $s$-wave model and a $d$-wave model with line nodes[@adro15]. In addition, zero-field $\mu$SR measurements show a presence of internal magnetic field in the superconducting state, although the magnetic field is more than 100 times smaller than that observed in Sr$_2$RuO$_4$. Raman scattering study of phonons with frequency greater than 100 cm$^{-1}$ also show very weak electron-phonon coupling that is too small to account for the observed superconductivity in this material[@zhan15]. In addition to these experimental observations, various first principles and Hubbard model based theoretical studies also find that a triplet superconducting state mediated by magnetic fluctuations to be likely present in K$_2$Cr$_3$As$_3$ [@jian14; @wu15a; @zhou15; @wu15b; @zhon15; @wu15c].
The experimental results that have so far been gathered on K$_2$Cr$_3$As$_3$ rule out an isotropic $s$-wave superconducting state in the weak-coupling limit, but they do not necessarily imply an unconventional 1D superconductivity mediated by magnetic fluctuations. Indeed, as pointed out by Balakirev *et al.*, the superconductivity in K$_2$Cr$_3$As$_3$ is inconsistent with such an unconventional superconducting state[@bala15]. They find that the anisotropy of the upper critical field $\gamma_H(T) = H_{c2}^{\perp} /
H_{c2}^{\parallel}$ is modest with $\gamma_H(T_c) \approx 0.35$ at $T_c$ and $\gamma_H(0) \approx 1.5$ at 0.7 K, which suggests a rather three-dimensional nature of superconductivity. Even though K$_2$Cr$_3$As$_3$ is structurally very one-dimensional, first principles calculations show that the electronic structure is decidedly three-dimensional with the presence of a large three-dimensional Fermi sheet in addition to two one-dimensional sheets[@jian14; @wu15a]. Furthermore, the upper critial field parallel to the chain $H_{c2}^{\parallel}$ is limited by Pauli pair breaking and is indicative of a singlet superconductivity, whereas the upper critical field perpendicular to the chain $H_{c2}^{\perp}$ shows no Pauli pair breaking effects and is limited by orbital pair breaking[@bala15]. In a quasi-1D superconductivity, $H_{c2}^{\perp}$ would have been limited by orbital and Pauli pair breaking effects yielding $H_{c2}^{\perp} < H_{c2}^{\parallel}$. More importantly, superconductivity in K$_2$Cr$_3$As$_3$ has been observed in samples with residual resistivity ratio ranging from 10 to 50[@bao15; @kong15; @bala15; @wang15], but the $T_c$ is essentially insensitive to the non-magnetic impurities. This suggest that the pairing is of $s$-wave type since scattering with non-magnetic impurities tends to average out the superconducting gap function, which would be detrimental to unconventional superconductors as their gap function has different signs in different regions of the Brillouin zone.
Motivated by these observations, I have investigated the lattice dynamics and electron-phonon coupling in K$_2$As$_3$Cr$_3$ using first principles density functional perturbation theory calculations. The phonon dispersions show the structure is stable, and there is an absence of a soft phonon behavior. Instead, they exhibit features that point to a strong coupling of K atoms with the lattice. The Eliashberg electron-phonon spectral function is strongly enhanced around 50 cm$^{-1}$. These phonons correspond to vibrations of all three constituent species and the coupling cannot be attributed to an Einstein-like rattling vibration. The phonon modes that exhibit large coupling are characterized by in-plane motion of the atoms. The $\mathbf{q}$ dependent electron-phonon coupling is highly anisotropic. In particular, the coupling strongly decreases as one moves away from the $q_z = 0$ plane. The calculated total electron-phonon coupling $\lambda_{\textrm{ep}} = 3.0$ is large and readily explains the experimentally observed large mass enhancement.
Computational approach
======================
The phonon dispersions and electron-phonon couplings presented here were obtained using density functional perturbation theory within the generalized gradient approximation (GGA) as implemented in the Quantum [espresso]{} package[@qe]. I used the ultrasoft pseudopotentials generated by dal Corso[@dalc14] and cutoffs of 50 and 500 Ry for basis-set and charge-density expansions, respectively. A $4 \times 4
\times 8$ grid was used for the Brillouin zone integration in the self-consistent calculations. The dynamical matrices were calculated on a $4 \times 4 \times 8$ grid and phonon dispersions and density of states are then obtained from Fourier interpolation. A denser $8
\times 8 \times 16$ grid and a smearing of 0.008 Ry was used in the calculation of the electron-phonon coupling. The electronic structure and structure relaxation calculations were also performed using the generalized full-potential method as implemented in the [wien]{}2k package[@wien2k] as a check.
----- ---------- ----------------- ---------------- -- -- --
Wyckoff $x$ $x$
position pseudopotential full-potential
As1 3$j$ 0.8345 0.8344
As2 3$k$ 0.1667 0.1668
Cr1 3$k$ 0.9175 0.9167
Cr2 3$j$ 0.0859 0.0864
K1 3$k$ 0.5380 0.5381
----- ---------- ----------------- ---------------- -- -- --
: \[tab:pos\] Relaxed internal parameters of K$_2$Cr$_3$As$_3$ using non-spin-polarized pseudopotential and full-potential calculations. The atomic coordinates in the $P\overline{6}m2$ space group are: As1 $(x, -x, 0)$, As2 $(x, -x,
0.5)$, Cr1 $(x, -x, 0.5)$, Cr2 $(x, -x, 0)$, K1 $(x, -x, 0.5)$, K2 $(1/3, 2/3, 0)$.
I used the experimental lattice parameters $a = 9.9832$ and $c =
4.2304$ Å but relaxed the internal atomic position parameters of the $P\overline{6}m2$ space group. The results of the geometry optimization calculations performed using both the pseudopotential and full-potential methods are summarized in Table \[tab:pos\]. These are in good agreement with each other and to the experimental values[@bao15; @tang15a]. Structural optimization leaves the As-As distances basically unchanged but reduces the Cr-Cr distances by 0.1 Å relative to the experimental values, in accord with a previous theoretical study[@wu15a].
Results and discussion
======================
![ Calculated GGA band structure of non-spin-polarized K$_2$Cr$_3$As$_3$ in the relaxed structure. []{data-label="fig:bnd"}](K2Cr3As3-stel2b.eps){width="\columnwidth"}
![ (Color online) Calculated GGA Fermi surface of non-spin-polarized K$_2$Cr$_3$As$_3$ in the relaxed structure. []{data-label="fig:fs"}](K2Cr3As3-stel2f.eps){width="0.5\columnwidth"}
The band structure and Fermi surface of K$_2$As$_3$Cr$_3$ calculated using the relaxed internal parameters are shown in Figs. \[fig:bnd\] and \[fig:fs\], respectively. These differ in details from the ones presented in Refs. and that used experimental atomic positions. The one-dimensional nature of the material is evident from the relatively small dispersion along the in-plane $\Gamma$–$K$–$M$–$\Gamma$ and $A$–$H$–$L$–$A$ paths. However, some bands do show moderate dispersion along these paths, which indicates that the bonding between \[(Cr$_3$As$_3$)$^{2-}$\]$_\infty$ chains is substantial. The bands disperse strongly in the $k_z$ direction (the $\Gamma$-$A$ path), and these reflect the short inter-atomic distances and strong bonding in the out-of-plane direction. I find substantial As $p$ character above the Fermi level and the presence of large amount Cr $d$ character in the lower part of the valence band manifold, which suggests that there is a strong covalency between Cr and As atoms. Even though the valence bands disperse relatively little in the in-plane directions, the valence band manifold extends from $-4.8$ to $0.2$ eV and has a band width of 5.0 eV. This again reflects the strong covalency between Cr and As atoms. I obtain $N(E_F) = 6.94$ eV$^{-1}$ per formula unit both spin basis for a value of the electronic density of states at the Fermi level. This corresponds to a calculated Sommerfeld coefficient of $\gamma = 16.36$ mJ/mol K$^2$. The experimental $\gamma$ obtained from specific heat capacity measurements is $\sim$70 mJ/mol K$^2$[@bao15; @kong15], and this corresponds to an ehancement by a factor of 4.3 relative to the calculated value.
There are three bands that cross the Fermi level, and these give rise to three Fermi sheets. The two upper-lying bands cross the Fermi level only along the $k_z$ direction and give rise to a pair of 1D sheets. In my calculations these are degenerate, but they are non-degenerate in the calculations that use experimental atomic positions[@jian14; @wu15a]. The lower-lying band crosses the Fermi level around the $K$ point, and this gives rise to a large three-dimensional sheet. Therefore, even though the structure of K$_2$Cr$_3$As$_3$ is quasi-1D, the material is very three-dimensional from the electronic structure point of view.
![ Calculated GGA phonon dispersions of non-spin-polarized K$_2$Cr$_3$As$_3$. []{data-label="fig:pband"}](KCA-ph0-pband.eps){width="\columnwidth"}
![ (Color online) Calculated GGA phonon density of states of non-spin-polarized K$_2$Cr$_3$As$_3$. []{data-label="fig:pdos"}](kca-pdos.eps){width="\columnwidth"}
The phonon dispersions and density of states (pDOS) of K$_2$Cr$_3$As$_3$ are shown in Figs. \[fig:pband\] and \[fig:pdos\], respectively. These again reflect the quasi-1D nature of the material as the in-plane dispersion of the optical branches is comparatively smaller than the out-of-plane dispersion. The strong bonding along the $c$ axis is also seen in the large dispersion of the out-of-plane polarized acoustic mode along the $q_z$ direction. This branch reaches more than 100 cm$^{-1}$. Although the acoustic branches have smaller dispersion along the in-plane direction, these do not exhibit soft-mode behavior. This indicates weaker bonding between the \[(Cr$_3$As$_3$)$^{2-}$\]$_\infty$ chains, but it also implies that the structure is stable and not in a very close vicinity of a structural phase transition.
The pDOS exhibits many narrow peaks that are caused by narrow band widths of the phonon branches. The partial pDOS shows that the K atoms contribute to phonon modes up to 200 cm$^{-1}$. Therefore, the K atoms do not behave like a rattler that is weakly coupled to the lattice. The interaction with K ions is also seen in the stronger dispersion around the $K$ point for branches below 200 cm$^{-1}$. The Cr and As atoms contribute to vibrations all across the phonon dispersion range. In particular, both Cr and As atoms contribute to the high-frequency phonon modes that again indicates strong covalency between these two.
The strength of the interaction between electrons and phonons is described in terms of the Eliashberg spectral function: $$\alpha^2
F(\omega)=\frac{1}{N(E_F)}\sum_{\mathbf{k},\mathbf{q},\nu,n,m}\delta(\epsilon_{\mathbf{k}}^{n})\delta(\epsilon_{\mathbf{k+q}}^{m})|g_{\mathbf{k},\mathbf{k+q}}^{\nu,n,m}|^{2}\delta(\omega-\omega
_{\nu\mathbf{q}}),
\label{eq:alpha}$$ where $N(E_F)$ is the electronic density of states at the Fermi level, $\epsilon_{\mathbf{k}}^{n}$ is the electronic energy at wavevector $\mathbf{k}$ and band index $n$, $\omega _{\nu\mathbf{q}}$ is the energy of a phonon with wavevector $\mathbf{q}$ and branch index $\nu$, and $g_{\mathbf{k},\mathbf{k+q}}^{\nu,n,m}$ is the matrix element for an electron in the state $|n\mathbf{k}\rangle$ scattering to $|m\mathbf{k+q}\rangle$ through a phonon $\omega
_{\nu\mathbf{q}}$.
The calculated Eliashberg function is shown in Fig. \[fig:eliash\]. It shows that phonon at all frequencies contribute to the coupling, although the the phonon modes around 50 cm$^{-1}$ make an especially large contribution. The pDOS also shows a peak at this frequency. Similar low-energy peaks has also been observed in $M_2$Mo$_6X_6$ compounds, and they have been attributed to the Einstein-like vibrations of $M$ atoms[@brus90; @petr10]. However, we can see from Fig. \[fig:pdos\] that the motion of all three types of atoms contribute to this low-energy peak in K$_2$Cr$_3$As$_3$. Therefore, even if these phonons have relatively small dispersion, the coupling cannot be ascribed to an Einstein-type rattling vibration of K atoms. I also investigated the polarization of the phonons that show large coupling to the electrons and find that these show strong in-plane character.
The $\mathbf{q}$ dependent total electron-phonon coupling $\lambda_{\mathbf{q}}$ ($= \int_0^{\infty} \mathrm{d}\omega
\alpha^2F(\omega,\mathbf{q})/ \omega$) is plotted for the $q_z = 0$ and $\frac{\pi}{c}$ planes of the Brillouin zone in Fig. \[fig:lambdaq\]. I find that the variance of the coupling is large within a $q_z$ plane. Furthermore, the coupling strongly decreases as one moves away from the Brillouin zone center along the $q_z$ direction. This shows that the coupling is highly anisotropic, and it should also result in an anisotropic superconducting gap function. To find out which Fermi sheets contribute more to the total electron-phonon coupling, I calculated the partial electron-phonon couplings at several $\mathbf{q}$ points by summing over the 1D and 3D Fermi sheets separately in Eq. \[eq:alpha\]. I find that the 3D sheet makes a dominant contribution to the coupling. Therefore, controlling the size of the 3D sheet by chemical doping, pressure or strain might be an effective way of enhancing or suppressing superconductivity. It will be interesting to see if magnetism or charge density wave order appears due to nesting between 1D sheets when the 3D sheet is suppressed.
![ Calculated Eliashberg spectral function of K$_2$Cr$_3$As$_3$. []{data-label="fig:eliash"}](kca-alpha2F.eps){width="\columnwidth"}
![ Calculated $\mathbf{q}$ dependent electron-phonon coupling $\lambda_{\mathbf{q}}$ shown for $q_z = 0$ and $q_z = \frac{\pi}{c}$ planes. The radius of the circle is proportional to the magnitude of $\lambda_{\mathbf{q}}$.[]{data-label="fig:lambdaq"}](elph-qz.eps){width="\columnwidth"}
I obtain a large value of $\lambda_{\textrm{ep}} = 3.0$ for the total electron-phonon coupling constant $\lambda_{\textrm{ep}}
\!=\!\sum_{\mathbf{q},\nu} \lambda_{\mathbf{q},\nu}=2
\int_{0}^{\infty} \mathrm{d}\omega \alpha^2 F(\omega) / \omega$. The logarithmically averaged phonon frequency is $\omega_{\ln} = 71$ cm$^{-1}$. These can be used to estimate the $T_c$ by using the simplified Allen-Dynes formula, even though the use of this formula is invalid in the present case of strong electron-phonon coupling that is highly anisotropic[@carb90]. The simplified Allen-Dynes formula has the form $$T_c= \frac{\omega_{\ln}}{1.2}\exp\left\{ -
\frac{1.04(1+\lambda_{\mathrm{ep}})}{\lambda_{\mathrm{ep}}-\mu^*(1+0.62\lambda_{\mathrm{ep}})}\right\}.$$ Using a value of $\mu^* = 0.12$ for the Coulomb pseudopotential parameter, I obtain $T_c = 17.8$ K, which strongly overestimates the experimental $T_c$ of 6 K. A full solution of the Migdal-Eliashberg equations, which is beyond the scope of this paper, will be required to obtain a more accurate $T_c$ based on the calculated spectral function. Furthermore, there might be additional reasons for the overestimation of $T_c$. In the phonon calculations, I have neglected the phonon anharmonicities. This can raise the phonon frequencies, which will lower the total electron-phonon coupling. Another reason for the overestimation might be because I have neglected the effects of magnetic fluctuations. First principles calculations show the presence of various magnetic interactions with in this material[@jian14; @wu15a], and this will also suppress $T_c$.
The calculated total electron-phonon coupling $\lambda_{\textrm{ep}} =
3.0$ readily explains the experimentally observed large electron mass enhancement from specific heat measurments. The electron-phonon coupling enhances the mass by a factor of $1+\lambda_{\textrm{ep}} =
4.0$, which is very close to the experimentally observed mass enhancement factor of 4.3. Therefore, the large mass enhancement in this material does not come from strong electronic correlations (in the sense of electron-electron interactions) or spin fluctuations. This is consistent with the experimentally estimated Wilson ration of $R_W \approx 1$[@kong15] that suggests a weak magnetic enhancement. The strong coupling nature of superconductivity in K$_2$Cr$_3$As$_3$ is also evident in the large jump of the dimensionless specific heat $\Delta C/\gamma T_c =
2.4$[@bao15; @kong15] at $T_c$. The strong electron-phonon coupling superconductivity proposed here can further be confirmed from tunneling experiments that should show a large value for the ratio $2\Delta(0) / T_c$, where $\Delta(0)$ is the superconducting gap.
Although the calculations presented here imply that the superconductivity in K$_2$Cr$_3$As$_3$ is caused by electron-phonon interactions, this does not necessarily mean that the superconductivity is dull or of the conventional BCS type. The crystal structure of this material lacks inversion symmetry, and parity is not a good quantum number is such a case. The pairing will likely have a dominant $s$ character mixed with some $p$ character. This can explain the presence of weak internal magnetic field below $T_c$ in $\mu$SR measurements[@adro15]. In addition, the electron-phonon coupling is highly anisotropic, which can lead to accidental gaps in the superconducting state. This would be consistent with the signatures of nodes in the gap function in the NMR[@zhi15] and penetration depth[@pang15] measurements as well as the absence of Hebel-Slichter peak in the NMR measurements.
The presence of such a large electron-phonon coupling in this material provides an opportunity to study the properties of a superconducting state in the strong coupling limit. In particular, this may help explain why the measured upper critical fields $H_{c2}^{\perp}$ and $H_{c2}^{\parallel}$ greatly exceeds the Pauli limit estimated for a weakly-coupled isotropic BCS superconductor. It would be interesting to see if the fact that the coupling strongly decreases away from the $q_z = 0$ plane can explain why $H_{c2}^{\parallel} < H_{c2}^{\perp}$ in this material. These issues require more theoretical studies based on the full solution of Migdal-Eliashberg equations to be fully clarified.
As mentioned above, superconductivity is also observed in other inorganic quasi-1D compounds including Tl$_2$Mo$_6$Se$_6$ and Li$_{0.9}$Mo$_6$O$_{17}$, and the nature of their pairing interaction has not been fully elucidated. The results of phonon dispersions and electron-phonon coupling for K$_2$Cr$_3$As$_3$ presented here provides a reference case for other quasi-1D superconductors. In particular, the present study shows that a behavior that deviates from what is expected from a weakly-coupled isotropic $s$-wave BCS superconductor does not necessarily imply that the superconductivity is mediated by magnetic interactions.
Summary and conclusions
=======================
In summary, I have investigated the lattice dynamics and electron-phonon coupling in K$_2$Cr$_3$As$_3$ using first principles calculations. The phonon dispersions do not show closeness to a lattice instability and exhibits features pointing to a strong coupling of K atoms with the lattice. The Eliashberg spectral function is strongly enhance around 50 cm$^{-1}$, but there is coupling to high frequency phonons as well. The peak in the spectral function corresponds to the vibrations of all three constituent atomic species of the material. Although the electrons do not predominantly couple to vibrations of any specific atoms, vibrations that involve in-plane motion of atoms show strong electron-phonon coupling. The electron-phonon coupling shows strong $\mathbf{q}$ dependence, especially along the out-of-plane direction. The calculated total-electron phonon coupling $\lambda_{\mathrm{ep}}$ = 3.0 is large and is consistent with the experimentally observed large mass enhancement and jump in the specific heat at $T_c$[@bao15; @kong15]. These calculations motivate tunneling experiments that can further confirm the strong coupling nature of superconductivity in this material by displaying an enhanced value of the ratio $2\Delta(0)/T_c$. The electron-phonon spectral function can also be extracted from such experiments, and they can be compared with the results presented here to validate the strong electron-phonon coupling scenario proposed here.
Acknowledgments
===============
I am grateful to David J. Singh for helpful discussions and suggestions. This work was supported by the Swiss National Supercomputing Centre (CSCS) under project ID s575 and by a grant from the European Research Council (ERC-319286 QMAC).
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|
---
abstract: |
The flawless functioning of a protein is essentially linked to its own three-dimensional structure. Therefore, the prediction of a protein structure from its amino acid sequence is a fundamental problem in many fields that draws researchers attention. This problem can be as a combinatorial optimization problem based on simplified lattice such as the hydrophobic-polar model. In this paper, we propose a new hybrid algorithm combining three different well-known heuristic algorithms: genetic algorithm, tabu search strategy and local search algorithm in order to solve the PSP problem. Regarding the assessment of suggested algorithm, an experimental study is included, where we considered the quality of the produced solution as the main quality criterion. Furthermore, we compared the suggested algorithm with state-of-the-art algorithms using a selection of well-studied benchmark instances.
<span style="font-variant:small-caps;">Keywords and phrases.</span> Protein Structure Prediction, 2D triangular Lattice, HP Model, Genetic Algorithm, Local Search, Tabu Search Strategy, Minimal Energy Conformation.
author:
- |
Nabil Boumedine$^1$, Sadek BOUROUBI$^2$\
\
$^{1,2}$ LIFORCE L,\
USTHB, F, D O R,\
P.B. 32 E, 16111, B, A, A.
bibliography:
- 'bibo.bib'
title: A new hybrid genetic algorithm for protein structure prediction on the 2D triangular lattice
---
Introduction
============
In Molecular biology, the three dimensional structure of proteins is the most crucial indicator that determines their biological activity. The prediction of the tertiary structure of proteins, also called a conformation, is a fundamental problem in the fields of biology, physics, etc. This problem is famous for protein structure prediction and denoted by PSP. A small modification in the original conformation of any given protein or an error during its folding causes many serious diseases, such as Alzheimer and cow mad. The ideal solution to treat these diseases is to predict the tertiaries structures of these proteins from their primary structure [@dill_theory_1985]. The PSP is one of the hardest problems in computational biology, molecular biology, biochemistry and physics. Furthermore, The most efficient algorithm for feasible solution determination runs in an an considerable time required; whereas the correct functioning of a protein depends essentially on its minimal energy conformation. Among the existing models in literature, the most studied in PSP problem is the H-P model (Hydrophobic Polar, H-P) [@lau_lattice_1989].In this model the free energy of a conformation is inversely proportional to the number of hydrophobic non-local bonds of H-H type (topological contacts H-H). This type of bonds occurs if two non-consecutive hydrophobic monomers occupy adjacent grid points in the lattice. Besides, each occurrence of this bond type reduces the value of global energy with one unit [@lin_efficient_2009]. The PSP is an optimization problem where the aim is to find a confirmation $c^{*}$ of a given protein sequence that minimizes the overall induced energy in all possible set of conformations $C$, i.e., a conformation $c^{*}$ such that $E(c^{*})=min\{E(c)/c\in C\}$[@shmygelska_ant_2005], where $E(c)$ represent the energy function explained later in Section 2.2. As we mentioned above, the H-H bonds reduces the induced energy. Hence, finding a minimal energy conformation (i.e., optimal conformation) amounts to find a conformation that maximizes the number of H-H contacts [@lin_efficient_2009]. As one may expect, the resolution of this problem is quite difficult due to the exponential exploration of the NP-hard problem solving when the chain size is large enough. The problem is proven to be NP-hard even when restricted to its two dimensional representation [@berger_protein_1998].Hence, it is clearly impossible to enumerate all the possible conformations when the chain of amino acids is considerably large. Several metaheuristics were proposed to solve the PSP problem. In the two-dimensional square lattice, the first genetic algorithm has been introduced by Unger and Moult [@unger_genetic_1993], and than followed by other versions, see [@dandekar_folding_1994; @konig_improving_1999; @krasnogor_protein_1999]. Similarly, an ant colony optimization (ACO) algorithm has been used by Shmygelska et al. [@shmygelska_ant_2002; @shmygelska_improved_2003; @shmygelska_ant_2005]. Moreover, the use of memetic algorithms was proposed in [@krasnogor_multimeme_2002; @pelta_multimeme_2005]. Particle Swarm Optimization algorithms has been applied in [@bautu_protein_2010]. Jiang, T. et al. proposed a hybrid approach combining tabu search and genetic algorithm [@jiang_protein_2003]. The Immune algorithms are introduced by Cutello et al. in [@cutello_immune_2005; @cutello_immune_2007]. A Clustered memetic algorithm with local heuristics have been introduced by Islam et al. in [@islam_clustered_2013].
This paper is organized as follows: in the next section, we present the two dimensional triangular lattice using the H-P model, that we are interested in, where we present a corresponding 0-1 mathematical program while focusing on: the objective function and the encoding solutions. In Section 3 we present a selection of pertinent algorithms designed to solve the PSP problem in 2D triangular lattice model. In Section 3, we present a detailed description of the suggested hybrid algorithm. In Section 4, we present the experimental study and the obtained results, where we compared our approach some existing approaches. Finally, we give, in Section 6, our main conclusions of this study.
Hydrophobic-polar model in a 2D triangular lattice Simplified
=============================================================
In H-P model, the twenty amino acids are represented by a mean of two letters H and P, in reference to their hydrophobicity chosen among the tow following options : hydrophobic or hydrophilic [@lau_lattice_1989]. For any given sequence of $n$ amino acids, the H-P model consists to transform this sequence into a chain $s = (s_1, s_2, \dots, s_n)$ such that each element of the sequence $s_{i}\in\{H,P\},\ i=\overline{1,n}$ represents the hydrophobicity of the corresponding amino acids in protein sequence: $$s_{i}=\left\{
\begin{array}{ll}
H & \hbox{if the amino acid $i$ is of hydrophobic type,} \vspace{0.1 cm}\\
P & \hbox{if the amino acid $i$ is of polar type.}
\end{array}
\right.$$ As it is shown in Figure 1, each vertex of the two dimensional triangular lattice has six neighbors. Hence, each monomer different from the first and the last element of the chain, i.e., monomers of rank , occupying any given position in the lattice can be at most in four topological contacts, in other words, it can form at most four bonds with its neighbors. Whereas maximal number of possible contacts that can occur in a monomer located either in the first or the last position is five [@agarwala_local_1997]. The symbols $1, 2, 3, 4, 5, 6$ are used to encode the following movement directions on the two dimensional triangular lattice: Right-Up, Up, Left-Up, Left-Down, Down and Right-Down, respectively.
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(9.526279441628825,8.5)
Figure \[figure2\] represents an feasible conformation in the 2D triangular lattice model for a protein sequence of 20 amino acids given in the HP model by $(HP)^{2}PH(HP)^{2}(PH)^{2}HP(PH)^{2}$. The green points represent the hydrophilic amino acids while the hydrophobic amino acids are represented in red. The energy of the conformation given in Figure \[figure2\] is $E(s)=-15$ (15 contacts of H-H type).
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Encoding Solution
-----------------
A feasible solution can be represented by the sequence of $n-1$ movements in the lattice called self-avoiding paths, (see Figure \[figure1\]) that generate this conformation. For example, the corresponding movements vector to the solution $S$, given in Figure \[figure2\] is as follow: $$mv(S)=[2,6,2,6,5,4,5,1,5,6,2,6,2,3,2,1,5,1,5].$$ The sequence $mv(S)$ allows us to deduce the position of each amino acid in the lattice.
Calculation of free energy
--------------------------
Given a feasible protein conformation of $n$ amino acids and let $s$ be its corresponding sequence in the HP model. The folding quality of this of this conformation is measured by the following energy function [@lin_efficient_2009]: $$\label{1}
E(s)=-\sum_{i=1}^{n-2}\sum_{j=i+2}^{n}\delta_{ij}\ x_{ij},$$ where $\delta_{ij}$ represents an indicator that determines whether both elements $s_{i}$ and $s_{j}$ are hydrophobic. $$\delta_{ij}=\left\{
\begin{array}{ll}
1; & \hbox{if $(s_{i}=H)\land(s_{j}=H)$}, \vspace{0.1cm}\\
0; & \hbox{otherwise,}
\end{array}
\right.$$ and $$x_{ij}=\left\{
\begin{array}{ll}
1; & \hbox{if the amino acids $i$ and $j$ form an H-H topological contact,} \vspace{0.1cm}\\
0; & \hbox{otherwise.}
\end{array}
\right.$$
Mathematical program for the PSP
--------------------------------
In the rest of this section, we present a mathematical program for the Protein Folding Problem (PSP) in its 2D triangular lattice model form. The aim here is to construct a linear mathematical program that can be implemented in modeling languages and compatible with the existing software solvers. Each node in a 2D triangular lattice blue located in a given position $(i,j)$ has six neighbors represented in a grid on a canonical basis. Considering the following possible neighboring directions of $(i,j)$ node:
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Let $n$ be the length of the protein sequence, and let $y_{ij}^{k}$ be a three dimensional variable such that: $$y_{ij}^{k} = \left\{
\begin{array}{ll}
1, & \textrm{if the position }\ (i,j)\ \textrm{contains the}\ k^{th}\ \textrm{amino acid in the protien sequence},\vspace{0.2cm}\\
0, & \textrm{else}.
\end{array}
\right.$$
### **Constraints**
First, we fix the first amino acid in the protein sequence at position $(n,n)$ as a starting point, i.e., $$y_{nn}^{1}=1.$$ Regarding the problem constraints, we can identify three different constraints that guarantees the admissibility of the resulting solution :
- A path in the grid is a solution if it occupies exactly $n$ nodes in the grid. This constraint can be written as follows: $$\sum_{k=1}^{n}\sum_{i=1}^{2n}\sum_{j=1}^{2n}y_{ij}^{k} = n.$$
- A node in the grid can contain at most one amino acid at the $k^{th}$ position, hence: $$\sum_{k=1}^{n}y_{ij}^{k} \leq 1,\ \forall i \in \{1,\dots, 2n\},\ \forall j \in \{1,\dots, 2n\}.$$
- A node in the grid can contain the amino acid in position $k+1$, if and only if at least one of its neighboring nodes contains the $k^{th}$ amino acid in the protein sequence: $$y_{ij}^{k+1} \leq y_{i-1j+1}^{k}+y_{i-1j}^{k}+y_{i-1j-1}^{k}+y_{i+1j-1}^{k}+y_{i+1j}^{k}+y_{i+1j+1}^{k},\forall i,j \in \{1,\dots, 2n\},\ \forall k \in \{1,\dots,n-1\}.\vspace{0.2cm}$$
### **The objective function**
Let $\alpha_{k}$ be a numerical interpretation of any given amino acid into a binary value (i.e., representation in $\{0,1\}$), where:
$$\alpha_{k} = \left\{
\begin{array}{ll}
1 & \mbox{if the $k^{th}$ amino acid in the protein sequence is hydrophobic, i.e., H,}\vspace{0.2cm}\\
0 & \mbox{if the $k^{th}$ amino acid in the protein sequence is hydrophilic , i.e., P.}
\end{array}
\right.$$
Thus, the objective function can be calculated as follows: $$\max(\mathcal{Z})=\frac{1}{2}\mathcal{Z^{*}}-\dsum_{k=1}^{n-1}\alpha_{k}\alpha_{k+1},$$
where
$$\hspace{-2mm}\mathcal{Z^{*}}=\max_{y}\left\{\sum_{i=1}^{2n}\sum_{j=1}^{2n}\left(\sum_{k=1}^{n}\alpha_{k}y_{ij}^{k}\right)\left(\sum_{k=1}^{n}\alpha_{k}\left(y_{i-1j+1}^{k}+y_{i-1j}^{k}+y_{i-1j-1}^{k}+y_{i+1j-1}^{k}+y_{i+1j}^{k}+y_{i+1j+1}^{k}\right)\right)\right\}.$$
This mathematical model guaranties optimal solution which is included in the grid enclosed by the points $\{(1,1),(1,2n),(2n,1),(2n,2n)\}$, with a starting point $(n,n)$. The choice of these limit points is based on the fact that in a path graph $P_n$ on the grid, the maximal distance between the fixed starting node and the rest of the nodes (i.e., basically for the end node) is at most equals to $n-1$ movements. More precisely it represents the radius of the graph, which is the number of edges in the case of a path graph $P_n$. With the view to its spatial complexity $O(n^3)$, this model clearly has a rather high cost in terms of memory. However, it provides a rather good equilibrium with the computational complexity; since all variables are binary strings/arrays.
Existing 2D H-P protein prediction algorithms
==============================================
Recently a number of metaheuristcs have been used to solve the PSP in the 2D triangular lattice model. In [@hoque_hybrid_2006], the authors suggested a new hybrid algorithm, called Hybrid Genetic Algorithm (HGA). This latter enhances the performance of a classical GA implementation by reducing the encountered through the generational process. More specifically, it is clear that when the number of generations increases, the current conformations become very compact. Hence, the application of genetic operators produces invalid conformations with considerable number of collisions. The proposed approach consists in using the generalized short pull moves strategy to produce only valid conformations to avoid blocking GA search processes. The authors have shown considerable quality improvements when compared to a simple genetic algorithm implementation SGA [@hoque_hybrid_2006]. Later on in [@bockenhauer_local_2008], the authors proposed new approach based on the tabu search algorithm using a generalized local move (i.e., pull move) as to improve the landscape exploration and the quality of the produced solutions.. Also, two approaches are proposed in [@lin_efficient_2009], including the Elite-based Reproduction Strategy-Genetic Algorithm (ERS-GA) ,and a Hybrid of Hill climbing and Genetic Algorithm called HHGA that combine the ERS-GA with a hill climbing algorithm.A new approach called IMOG has been proposed in [@yang_protein_2018], that combines ions motion optimization algorithm (IMO) with a Greedy algorithm (G), the obtained results shown that the IMOG algorithm has a good search ability and stability for the PSP problem using benchmark data sets.
Novel hybrid approach for PSP problem
=====================================
The most commonly used hybridization in literature consists of combining two metaheuristics, one based on a single solution, known as s-metaheuristics, and the other based on a population, known as p-metaheuristics. The s-metaheuristics have proved their effectiveness for intensification, while the p-metaheuristics known by their exploration capacity. Thus, this type of hybridization allows to establish a balance between the diversification and the intensification of the search process [@blum_metaheuristics_2003; @raidl_metaheuristic_2010; @talbi_metaheuristics:_2009].
In this work, we proposed a hybrid approach to solve the PSP problem, called GALSTS, in reference to the adopted combination of the following metaheuristics: genetic algorithm, local search, tabu search strategy. As all population-based approaches the first step of GALSTS is to generate an initial population $P$ of $m$ feasible solutions, in which the crossover operator is guided by a tabu list that allows to prohibiting previous movements in order to explore new regions in the search space. Each two selected parent produces a set of offsprings (i.e., $f$ solutions) by applying the crossover procedure controlled by a tabu list that contains all $k $ crossover points that are already used. This has allowed us to avoid cyclical movements.
In order to improve the quality of the children produced by the crossover process, each one of them is introduced with a probability $p_{m}$ as an initial solution of a local search algorithm. Such that the transition from a solution $s$ to one of its neighbors $s'$ is carried out by a random choice of an amino acid $i$ and replacing its direction by another one among the five other possible directions. If the quality of $s$ is better than $s'$, then it becomes the courant solution for the next iteration. This process is repeated until the satisfaction of the stops criterion. This improvement phase has allowed us to use the information provided by parents more efficiently to produce high-quality solutions. The two solutions with the highest fitness value are introduced into the new population $P_{new}$ if they not exist in this population previously, although are of lower quality than their parents to encourage exploration of the research space. The best $m$ solutions of $P\cup P_{new}$ are replaced the individuals in the population $P$ for the next generation. This approach is intended to avoid the rapid convergence towards local optima, and the wide diversification between the solutions and thus ensures the quality of the solutions during all the stages of the research space explorations, see Algorithm \[Algorithm1\].
[**Input:** ]{}A protein sequence of amino acids of size $n$.\
[**Output:** ]{}The best confirmation for the protein sequence.\
------------------------------------------------------------------------
**begin** Initialization: $P\leftarrow$ The initial population of $m$ solutions; Create a new population by the flowing steps: $k\leftarrow 0,\ P_{new}\leftarrow {\emptyset}$; Select two parents $(p_{1},p_{2})$ from $P$; $ offspring \leftarrow$ two solutions obtained by applying the algorithms (TS-LS) to the selected parents; $P_{new}\leftarrow P_{new}\cup \{offspring\}; $ $k\leftarrow k+1$; $P\leftarrow$ the $m$ best solution among $P\cup P_{new}$; **end**
\[Algorithm1\]
The Initial Population
----------------------
We start with an initial population of $m$ randomly generated individuals. An initialization algorithm was proposed that allows to generate only valid conformations for the initial population of GALSTS, we use a list T that contains all the positions used for courant solution, we put the first amino acid at one point in the lattice and save their position in T. Then, for each amino acid we choose a random direction among the six directions as shown in Figure \[figure1\]. If the selected direction generates an already occupied position (i.e., existing in T); we generate another direction different from the selected one. If all the six directions create an existing positions in T (i.e., all direction create invalid solution), we generate a new solution, see Algorithm \[Algorithm2\].
[**Input:** ]{}$n$ the number of amino acids in the protein sequence.\
[**Output:** ]{}A feasible confirmation for the protein sequence.\
------------------------------------------------------------------------
**begin** Initialization: $T\leftarrow$ Table of position for each amino acids in lattice; $T[1]\leftarrow (x_{1},y_{1}),$ i.e., put the first monomer in one vertex of the; lattice $i\leftarrow 2;$ $K\leftarrow \{1,2,3,4,6\};$ the set of all possible directions on the 2D triangular lattice $t\leftarrow true; $ $u\leftarrow$ Random direction generated from the list $K$; $K\leftarrow K\backslash\{ u \}$; $(x_{i},y_{i})\leftarrow$ Position generated by the direction $u$; $t \leftarrow false$; $T[i]\leftarrow (x_{i},y_{i})$; $ t\leftarrow false$; $i\leftarrow 2$; **end**
\[Algorithm2\]
Crossover Operator
-------------------
The crossover procedure consists to combine two or more solutions, called parents, as to create other solutions, called offsprings. There are several types of the crossover operator, here we opted for a random 1-point, which consists to swap after selecting two parents $p_{1}$ and $p_{2}$, and generating one random point $c_{1}$, $1 < c_{1} < n$, the parent subsequences limited by $c_{1}$ and $n$. As shown in Figure \[figure4\], the new two conformations (offsprings) $f_{1}$ and $f_{2}$, are obtained by combining $p_{1}$ and $p_{2}$ after generating a random cut point (here $c_{1}=5$). The energy value of offspring $f_{1}$ is $E(f_{1})=-7$, lower than the energy values of its parents, $E(p_{1})=-2$ and $E(p_{2})=-5$.
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Mutation Operator
------------------
Generally, it consists to modify some components, called genes, of an existing solution, in order to introduce more diversity into the solutions, it generally applied with low probability. In the proposed local search algorithm (see Algorithm \[Algorithm3\]), the neighbors of a given solution are defined in a similar way with the mutation operator, but the performed movement is accepted if it improves the quality of the courant solution. As shown in Figure \[figure5\], the new conformation $s_{m}$ is obtained by exciting the current solution $s$ at the mutation point 10, changing the direction from 3 to 6. The energy value of $s_{m}$ is $E(s_{m})=-7$, lower than the energy of $s$, $E(s)=-5$.
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[**Input:** ]{}A feasible solution $s$, and its associated energy value $E(s)$.\
[**Output:** ]{}The best found solution $s_{best}$.\
------------------------------------------------------------------------
**begin** Initialization: $E(s_{best})\leftarrow E(s)$; $s_{best}\leftarrow s$; $u\leftarrow$ random point of mutation; $s\leftarrow$ mutation$(s,u)$; Evaluate $s$, i.e., calculate $E(s)$; $s_{best}\leftarrow s$; $E(s_{best})\leftarrow E(s)$; **end**
\[Algorithm3\]
Selection Operator
------------------
It is a technique that favorites the best solutions to participate in the reproduction phase, in order to create new solutions with “satisfactory” quality. In this work we will use the roulette wheel selection technique (RWS). In this approach, each individual has a probability of being selected in accordance with his or her performance, so the more individuals are adapted to the problem, the more likely they are to be selected. The probability of selecting a solution $i$ among the $m$ solutions is given by the following:
$$p(i)=\dfrac{f(i)}{\dsum_{j=1}^{m}f(j)}\cdot\vspace{2mm}$$
The RWS is an iterative operator, where in each step (after the probability assignment), it withdraws a random value form the range \[0,1\] (or \[0,100\], \[0,360\] depending on the representation of the selection wheel), and then it selects the corresponding individual. Table \[table1\] presents an example of proportions assignment of four individuals according to their fitness evaluation.
Figure \[figure6\] presents the circular representation (i.e., pie chart) of the obtained selection probabilities in Table \[table1\]. As it is shown in Figure \[figure6\], the spinner indicates to select the chromosome $I_{2}$.
\[table1\]
(2.23,1.086)(8.52,4.88) (0,0)(2.23,1.086)(8.52,4.88) (5.,3.)[0.74]{} (5.,3.)[1.7640333842602893]{} (5.,3.)(5.14,3.62) (5.,3.)[0.74]{} (5.66,3.36)(6.75,3.96) (4.45,3.52)(3.55,4.367) (5.744,3.00)(6.99,3.) (4.25,3.00)(3.,3.) (6.84,4.23)[$7,5\%$]{} (2.45,4.75)[$37,5\%$]{} (2.22,3.17)[$50\%$]{} (7.067,3.16)[$0.0\%$]{} (6.096,3.46)[$I_{1}$]{} (5.056,4.36)[$I_{2}$]{} (3.70,3.59)[$I_{3}$]{} (4.85,1.90)[$I_{4}$]{}
(5.,3.)
Generating an offspring by the guided search strategy
-----------------------------------------------------
For the reproduction phase, we propose an efficient algorithm that guides the search process to produce an offsprings of “good” quality. This algorithm combines the Local Search algorithm (LS) and the Tabu Search algorithm (TS). The role of the LS algorithm is to improve the quality of the solutions obtained by the crossover process, and in order to use the parent’s information more effectively and to explore more research space; we propose to use the tabu search algorithm that memorizes the last points crossover used for some parents selected, see Algorithm \[Algorithm4\]. The working mechanisms of the suggested approach can be summarized by the following steps:
Step1
: Generate $n$ solutions for the initial population.\
Step 2
: Evaluate the fitness of each solution.\
Step 3
: Create a new population by the following steps:\
Step3.1
: Select two parents from the $n$ solutions.\
Step 3.2
: Apply Algorithm \[Algorithm1\], with a probability $p_{c}$.\
Step 3.3
: Apply Algorithm \[Algorithm2\], with a probability $p_{m}$.\
Step 3.4
: Replace the previous population, with the current population.\
Step 4
: If the stopping criterion is not checked goto **Step 2**.\
[**Input:** ]{}Two selected parents $p_{1}, p_{2}$.\
[**Output:** ]{}Two individuals produced from parents, i.e., offsprings.\
------------------------------------------------------------------------
**Begin** Initialization: $E_{1} \leftarrow E(p_{1})$; $offspring^{*}_{1}\leftarrow p_{1}$; $E_{2}\leftarrow E(p_{2})$; $offspring^{*}_{2}\leftarrow p_{2}$; $T\longleftarrow \emptyset$; // the tabu list $K leftarrow 0$; $K\leftarrow K+1 $; $u\leftarrow$ random point of crossover; $(offspring_{1}, offspring_{2})\leftarrow$ crossover $(p_{1},p_{2})$; $u_{1} \leftarrow$ random $[0,1];$ $offspring_{1}\leftarrow$ local search $(offspring_{1})$; $offspring_{2}\leftarrow$ local search $(offspring_{2})$; $ offspring^{*}_{1}\leftarrow offspring_{1}$; $E_{1}\leftarrow E(offspring_{1})$; $ offspring^{*}_{2}\leftarrow offspring_{2}$; $ E_{2}\leftarrow E(offspring_{2}) $; $ T\longleftarrow T\cup \{ u \}$; **end**
\[Algorithm4\]
Experimental Results
====================
The aim of this section is to assess the performance of the suggested approach. For the following experimental study, we used a several benchmark instances (i.e., protein sequences) presented in the H-P model of different sizes [@dandekar_folding_1994; @unger_genetic_1993]. Furthermore, we have selected the most studied in the literature instances to conduct the forthcoming experiments. Table \[table2\] is composed of 3 columns, the first one represents the number of the protein sequence, the second represents the length of the protein sequence and the third represents the protein sequence in the H-P model, where the symbol $(.)^{i}$ means $i$ fold repetitions of the respective subsequence in the brackets. For example, $(PH)^2$ is the simplified form of the sequence $PHPH$. The experimental results showed in Table \[table3\] represents a comparison between the best results obtained by the proposed approach, and the above stated algorithms used to solve the PSP in the 2D triangular lattice model, namely HGA [@hoque_hybrid_2006], TS [@bockenhauer_local_2008], ERS-GA [@su_efficient_2010], HHGA [@su_efficient_2010], IMOG [@yang_protein_2018] for each instance given in Table \[table2\]. We show that all the approaches can provide an optimal confirmation when the length of the sequence is less than 36. However, the results obtained by our approach are better than the other approaches for the sequences 4, 6 and 7 as shown in Table \[table3\] (see the respective conformation in Figure \[figure9\]). However, The best prediction obtained by the algorithms GALSTS, TS , HHGA and IMOG are better than the HGA algorithm, ERS-GA and SGA for all sequences used for this experimental study.
The experiment results, in Tables \[table3\] and \[table4\], represent a comparison between the best results obtained by GALSTS, with some existing results based on several large benchmarks found in the literature, solved by HGA and SGA [@hoque_hybrid_2006], TS [@bockenhauer_local_2008], ERS-GA [@su_efficient_2010], HHGA [@su_efficient_2010], and IMOG [@yang_protein_2018] approaches. Clearly, the number of possible conformations increases exponentially when the size of the instance increases. According to their energy values, all approaches could provide an optimal confirmation when the size of the instance is less than 36. But, as it is shown in and \[table4\], the best conformations obtained by GALSTS are better than all the cited approaches bellow, based on the instances of size up to 36. This demonstrates the ability of GALSTS to explore the search space more effectively comparing with the other approaches.
Instances larger than 64 are not covered in the literature for the 2D triangular lattice model. However, they were processed for the rectangular model [@zhao_advances_2008]. According to the obtained results in Table \[table3\], a strong improvement in energy appears clearly compared with the triangular model. Table \[table4\], graphically presented in Figure \[figure7\] and Figure \[figure8\], show a performance comparison on the stability of our approach and three other algorithms HHGA, IMOG and ERS-GA, such that the efficiency of the algorithms is measured by the best and means results in 30 independent runs for each sequence. We show that for the most instances, the proposed approach is able to find the best optimal solutions and achieves a better average solution quality than other algorithms. the average solution quality is very encouraging.
Table 5 resumes the obtained results for 30 independent runs per each of the above stated instances. The aim of this experiment is to compare the suggested algorithm GALSTS against two competing algorithms ERS-GA and SGA. The results are adduced according to the best and worst overall evaluation and their corresponding deviation from the best known value (BKV). The proposed algorithm GALSTS is shown to be more effective than the other competing algorithms; even when comparing its worst produced conformation to their best ones, with a sole exception of the first tested sequence, where it shows a slight difference. However, when increasing the size of the instance, it is clear that the suggested algorithm is incrementally taking advantage over the competing algorithms, even when comparing its worst solution to their best ones. Furthermore, the suggested algorithm is shown to be able to attain good quality conformations or even optimal, with a sole exception the sixth tested instance, where it shows a one unit deviation of the best known evaluation.
coordinates [ (20, -14.86) (24 , -15.53) (25 , -12) (36 ,-21.93) (48,-39.86) (50 ,-37.6) (60 ,-68.26) (64 ,-58.46) ]{};
coordinates [ (20, -14.73) (24 , -14.93) (25 , -11.57) (36 ,-21.27 ) (48,-37.30 ) (50 ,-34.10 ) (60 ,-61.83) (64 ,-56.53) ]{}; coordinates [ (20, -12.50) (24 , -10.20) (25 , -8.47 ) (36 ,-16.17) (48,-28.13) (50 ,-25.30) (60 ,-49.43) (64 ,-42.37) ]{};
coordinates [ (20, -15) (24 , -17) (25 , -12) (36 ,-24) (48,-43) (50 ,-40) (60 ,-70) (64 ,-67) ]{};
coordinates [ (20, -15) (24 , -17) (25 , -12) (36 ,-23 ) (48,-41 ) (50 ,-38 ) (60 ,-66) (64 ,-63) ]{}; coordinates [ (20, -15) (24 , -13) (25 , -12 ) (36 ,-20) (48,-32) (50 ,-30) (60 ,-55) (64 ,-47) ]{};
Conclusions and Perspectives
=============================
This paper presents an efficient approach called GALSTS for solving the protein structure prediction in 2D triangular model using the simplified hydrophobic-polar model. An initialization algorithm was proposed that allows to generate only valid conformations for the initial population of GALSTS. This algorithm eliminates cyclic movements during the construction of solutions. GALSTS consists in using Tabu and Local Search algorithm to explore the search space handling more efficiently. This approach allows to use the information provided by the selected parents to produce solutions with good quality. From our experimental results, GALSTS was able to find the best known solutions and it is more effective for the stability results than other existing algorithms. In terms of future scope of applications, GALSTS can be used to solve the PSP problem in the 3D cubic and 3D triangular models, it can also be used to solve other optimization problems in the combinatorial optimization framework.
Acknowledgment
==============
The authors thank N. Kantour for his generous assistance.
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---
abstract: 'This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford–Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate $L_1$ space to a weak solution to the problem. Additionally, by applying the methods and theory of operator semigroups, we are further able to show that weak solutions to the problem are unique and necessarily classical (differentiable) solutions. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence.'
address: 'Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK email: `[email protected]`, `[email protected]`'
author:
- Graham Baird
- 'Endre S[ü]{}li'
bibliography:
- 'references2.bib'
title: 'A Finite Volume Scheme for the Solution of a Mixed Discrete–Continuous Fragmentation Model'
---
Introduction
============
Fragmentation and coagulation processes occur in many physical systems, with the associated mathematical models receiving much attention in the literature. Example application areas include colloid science [@ziff80; @Costas95], population dynamics [@Degond17a; @Degond17b] and astrophysics [@johansen08; @Dullemond14]. However, analytical solutions to these models are only available for a limited number of specific cases, and we often have to resort to approximate solutions generated by an appropriate numerical scheme. A range of numerical techniques have been applied to these problems, and these broadly fall into two categories: those involving a stochastic (Monte Carlo) element, for example [@guias1997; @babovsky99; @eibeck2000] and those based around various deterministic approximation schemes [@Kumar96; @Nicmanis96; @Barrett96; @Smirnov16]. The introductory chapter of [@kumarphd10] and the references therein provide a detailed overview of a number of these approximation methods.
In the earlier work [@baird18], we presented a mixed discrete-continuous model of fragmentation in an attempt to resolve the issue of ‘shattering’ mass-loss observed in some purely continuous models [@mcgrady87]. By modelling the mass distribution amongst the smallest particles using a discrete model, whilst modelling the distribution of larger particle masses with a continuous model, the aim was to introduce a higher degree of physical fidelity thus resolving the shattering mass-loss problem, whilst also retaining the mathematical efficiency of the continuous model.
Given the similarities between this model and those existing in the literature, in addition to the added complexity of the mixed framework, we would expect in most cases to have to rely on numerical methods to obtain a solution. In this paper we present a numerical scheme for the solution of the mixed fragmentation model proposed in [@baird18]. The basis of the scheme is a finite volume discretisation of the continuous regime equation. The use of such a method would appear a reasonable choice in this case, given its conservative nature and the motivation behind the model development. Indeed, finite volume schemes have been commonly applied to the solution of coagulation and fragmentation equations, with the first such use being [@filbet04], where the case of pure coagulation was considered. For problems involving fragmentation, the article [@bourgade08] sees such a scheme employed in approximating the binary coagulation and fragmentation equation, whilst [@kumarphd10] and [@Kumar15] examine their use for the multiple fragmentation equation, with [@kumar14] extending this to include coagulation. Further works have seen these methods applied to a number of coagulation–fragmentation model variants, for example with the inclusion of spacial diffusion [@filbet08diffusion] and additional nucleation and growth processes [@QAMAR2009; @kumar2013moment]. Whilst a number of articles [@QAMAR2007; @forestier2012; @Saha19] cover the approximation of multi-dimensional coagulation or fragmentation, whereby particles may be classified by additional variables beyond their mass or volume.
Mixed Discrete-Continuous Model
-------------------------------
In the mixed model of [@baird18], a cut-off value $N\in \mathbb{N}$ is introduced; above this cut-off, particle mass is considered as a continuous variable, whilst below it, the particles are forced to take discrete integer masses. If we denote by $u_C(x,t)$ the particle mass density within the continuous mass regime ($x>N$), then the evolution of $u_C(x,t)$ is governed by the continuous multiple fragmentation equation: $$\begin{aligned}
\label{equation301}
\hspace{-8mm}\frac{\partial u_C(x,t)}{\partial t}&=-a(x)u_C(x,t)+\int_{x}^{\infty}a(y)b(x|y)u_C(y,t){\,{\mathrm{d}}}y, \hspace{1.92mm}x>N,\hspace{1.2mm}t>0,\\
u_C(x,0)&=c_{0}(x).\nonumber\end{aligned}$$ This equation is similar in form to the multiple fragmentation equation introduced in [@mcgrady87]. The function $a(x)$ provides the fragmentation rate for a particle of mass $x$, whilst $b(x|y)$ represents the distribution of particles of mass $x>N$ resulting from the break-up of a particle of mass $y>x$. The functions $a$ and $b$ are assumed to be nonnegative measurable functions, defined on $\left(N,\infty\right)$ and $\left(N,\infty\right)\times\left(N,\infty\right)$, respectively. We also require $b(x|y)=0$ for $x>y$, since no particle resulting from a fragmentation event can have a mass exceeding the original particle. The initial mass distribution within the continuous regime is given by the nonnegative function $c_{0}(x)$.
Letting $u_{Di}(t)$ denote the concentration of discrete mass $i$-mer particles ($i\leq N$) and $u_{D}(t)$ the $N$-component vector taking these values as entries, the change in the values $u_{Di}(t)$, $i=1,\dots,N$, is governed by the equation:
$$\begin{aligned}
\label{equation302}
\hspace{-80mm}\frac{{{\mathrm{d}}}u_{Di}(t)}{{{\mathrm{d}}}t} &\hspace{-.7mm}=\hspace{-.7mm}-a_iu_{Di}(t)+\hspace{-1.9mm}\sum_{j=i+1}^{N}\hspace{-1.2mm}a_jb_{i,j}u_{Dj}(t)
+\hspace{-1mm}\int_{N}^{\infty}a(y)b_i(y)u_{\small{C}}(y,t){\,{\mathrm{d}}}y,\hspace{.7mm}\hspace{1.1mm}t>0,\\
u_{D}(0)&=d_{0}.\nonumber\end{aligned}$$
In the case of $i=N$, the second term becomes an empty sum and is taken to be $0$. The values $a_i$ give the rates at which $i$-mer particles fragment, with $a_1=0$. The quantities $b_{i,j}$ give the expected number of $i$-mers produced from the fragmentation of a $j$-mer and the functions $b_i(y)$ give the expected number of $i$-mers produced from the fragmentation of a particle of mass $y>N$. The underlying physics demands that each $a_i$, $b_{i,j}$ and $b_i(y)$ be nonnegative. Finally, $d_{0}$ is the $N$-component vector of nonnegative values, specifying the initial concentrations within the discrete regime.
During each fragmentation event, mass is simply redistributed from the larger particle to the smaller resulting particles, but the total mass involved should be conserved. This gives rise to the following two conditions to supplement equations and : $$\begin{aligned}
\label{equation303}&\int_{N}^{y}xb(x|y){\,{\mathrm{d}}}x+\sum_{j=1}^{N}jb_j(y)=y\hspace{2mm} \text{for} \hspace{2mm}y>N,\\
\label{equation304}&\sum_{j=1}^{i-1}jb_{j,i}=i\hspace{2mm} \text{for} \hspace{2mm}i=2,\ldots,N.\end{aligned}$$ The condition is an expression of mass conservation upon the fragmentation of a particle from the continuous mass regime. The equation comes from the conservation of mass when a particle from the discrete mass regime breaks up.
For further details on the mixed discrete-continuous model and its properties, the reader is directed to consult [@baird18] or [@bairdphd].
Truncation and Reformulation
----------------------------
When considering the numerical solution of equations and , we encounter an issue in that the range of the continuous mass variable $x$ is an unbounded interval, which presents a computational problem. We therefore introduce a truncation parameter $R>N$, and restrict the continuous mass variable to the range $N<x<R$. Therefore, in place of equations , we consider the truncated version $$\begin{aligned}
\label{equation40001}
\hspace{-8mm}\frac{\partial u_{C}^R(x,t)}{\partial t}&=-a(x)u_{C}^R(x,t)+\int_{x}^{R}a(y)b(x|y)u_{C}^R(y,t){\,{\mathrm{d}}}y, \hspace{1.92mm}N<x<R,\hspace{1.2mm}t>0,\\
u_{C}^R(x,0)&=\chi_{(N,R)}(x)c_{0}(x), \hspace{2.5mm}N<x<R,\nonumber\end{aligned}$$\
where $\chi_{(N,R)}$ denotes the characteristic function of the interval $(N,R)$. Taking our lead from the aforementioned articles, we now rewrite equation in a conservative form, although in our case we must include an additional sink term to account for the mass leaked down to the discrete regime. Therefore we end up with the following equation for the mass quantity $xu_{C}^R(x,t)$: $$\label{equation402}
\frac{\partial \left(xu_{C}^R\right)}{\partial t}=\frac{\partial \mathcal{F}^R\left(xu_{C}^R\right)}{\partial x}-S(xu_{C}^{R}),\hspace{1.5mm}
u_{C}^R(x,0)=c_{0}(x),\hspace{1.5mm}\text{for}\hspace{1.5mm}N<\hspace{-.5mm}x\hspace{-.5mm}
< R,\hspace{.5mm}t>0,$$ where $\mathcal{F}^R$ and $S$ are a truncated flux term and sink term, respectively, given by $$\hspace{-4mm}\mathcal{F}^R(f)=\int_{x}^{R} \int_{N}^{x} \frac{y}{z} a(z) b(y|z)f(z){\,{\mathrm{d}}}y{\,{\mathrm{d}}}z,\hspace{4mm}S(f)=\frac{a(x)}{x}\sum_{i=1}^{N}ib_i(x)f(x),\hspace{2mm}\text{for}\hspace{2mm}N<x<R.$$ The equation may be recovered from by a formal application of Leibniz’s rule for differentiating under the integral. However, the equivalence of the two forms can be seen to be justified rigorously in [@bairdphd Appendix C]. Before continuing, we establish a result concerning the behaviour of the flux term $\mathcal{F}^R$ at the limits of our domain.
\[lastthing\]If the kernels $a$ and $b$ are assumed to belong to $L_{\infty,loc}$ on the domains $[N,\infty)$ and $[N,\infty)\times[N,\infty)$ respectively, which will be the case in the upcoming analysis, then for $f\in L_1(N,R)$ the flux term $\mathcal{F}^R(f)$ satisfies $$\lim_{x\rightarrow N,R}\left|\mathcal{F}^R(f)(x)\right|=0.$$
It is a straightforward matter to bound $\mathcal{F}^R(f)$ as follows: $$\label{final}
\left|\mathcal{F}^R(f)(x)\right|\leq \int_N^R \chi_{(x,R)}(z)\frac{a(z)\left|f(z)\right|}{z}\left(\int_N^x yb(y|z){\,{\mathrm{d}}}y \right){\,{\mathrm{d}}}z,$$ which holds for $x \in(N,R)$. Recalling the mass conservation condition , we deduce that $$\chi_{(x,R)}(z)\left( \int_N^x yb(y|z){\,{\mathrm{d}}}y \right)\leq \int_N^z yb(y|z){\,{\mathrm{d}}}y\leq z,$$ for all $z\in(N,R)$. Hence the integrand appearing in is bounded above by $a(z)\left|f(z)\right|$, which, thanks to $a\in L_{\infty,loc}[N,\infty)$ and $f\in L_1(N,R)$, is integrable.
Considering the limit as $x\rightarrow N$ first, if we denote by $\beta(R)$ the essential supremum of $b$ over $[N,R]\times[N,R]$, then we have $$\chi_{(x,R)}(z)\left( \int_N^x yb(y|z){\,{\mathrm{d}}}y \right)\leq x \beta(R)(x-N).$$ As such, the integrand in converges pointwise to 0 over $z\in(N,R)$ as we let $x\searrow N$. An application of the Lebesgue dominated convergence theorem then gives the required convergence of $\left|\mathcal{F}^R(f)(x)\right|$ as $x\searrow N$. Turning now to the limit as $x\nearrow R$, another application of condition provides us with $$\chi_{(x,R)}(z)\left( \int_N^x yb(y|z){\,{\mathrm{d}}}y \right)\leq\chi_{(x,R)}(z) \int_N^z yb(y|z){\,{\mathrm{d}}}y\leq \chi_{(x,R)}(z)z,$$ for $z\in(N,R)$. Therefore, the integrand from must again converge pointwise to 0 over $(N,R)$, this time as we let $x\nearrow R$. Another application of the Lebesgue dominated convergence theorem gives the convergence of $\left|\mathcal{F}^R(f)(x)\right|$ to 0, as $x\nearrow R$.
This result will be utilised later in a number of arguments, most significantly in approximating $\mathcal{F}^R$ within our numerical scheme and in establishing a weak formulation of equation .
The truncation of the continuous mass interval also has an impact on our discrete regime equation; therefore, instead of equation , we consider $$\begin{aligned}
\label{equation403}
\hspace{-80mm}\frac{{{\mathrm{d}}}u_{Di}^R(t)}{{{\mathrm{d}}}t}
&\hspace{-.7mm}=\hspace{-.7mm}-a_iu_{Di}^R(t)+\hspace{-1.9mm}\sum_{j=i+1}^{N}\hspace{-1.2mm}a_jb_{i,j}
u_{Dj}^R(t)+\hspace{-1mm}\int_{N}^{R}\hspace{-.5mm}a(y)b_i(y)u_{C}^R(y,t){\,{\mathrm{d}}}y,\\
u_{Di}^R(0)&={d_{0}}_i,\hspace{2mm}\text{for}\hspace{2mm}i=1,2,\ldots,N,\hspace{2mm}t>0.\nonumber\end{aligned}$$ In the case of $i=N$, the empty sum above is taken to be zero; this convention will be adopted in all similar cases which follow.
This truncation procedure is a standard approach when dealing with fragmentation and coagulation problems, having been applied for example in [@mcl1], where the theory and methods of operator semigroups were employed, and [@stewart89] where an alternative weak compactness style argument was adopted. The common approach of these works involves establishing the existence of solutions to a sequence of such truncated problems. A limit is then obtained as the truncation point is increased without bound, with this limit then being shown to satisfy the untruncated problem in some sense. Although in this article we restrict our attention to the numerical approximation of the truncated discrete–continuous problem, as given by equations and , it can be shown that the solutions to the truncated problems converge, in an appropriate space, to give the solutions to the untruncated and . The proof of this convergence argument follows similar lines to that set out in [@banasiak06 Section 8.3.2], with the reader being directed to [@bairdphd Chapter 6] for the specific details. Additionally, the reader may find an empirical examination of this convergence in [@bairdphd Section 7.5], where the key factors influencing the convergence, and thus the selection of an appropriate $R$ are investigated.
Preliminaries
=============
Having set out our problem in the previous section, we now present a brief outline of the key results which appear in the upcoming material and which may be considered nonstandard or which are particular to our case.
\[theorem201\] In the analysis pursued in subsequent results, we shall be working extensively in spaces of the type $L_1$. In particular we shall be working in the spaces $L_1=L_1((N,R)\times [0,T),{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t)$ and $L_1^1=L_1((N,R)\times[0,T),x {\,{\mathrm{d}}}x {\,{\mathrm{d}}}t)$, where $N$ is a positive integer and $R>N$ is a finite real value. With the associated norms, these form equivalent spaces.
First let us suppose that $f\in L_1^1$; then we have $$\|f\|_{L_1}=\int_0^T\int_N^R\left|f(x,t)\right|{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t\leq\frac{1}{N}\int_0^T\int_N^R\left|f(x,t)\right|\,x{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t=\frac{1}{N}\|f\|_{L_1^1}.$$ Therefore $f \in L_1$ also, with $\|f\|_{L_1}\leq \frac{1}{N}\|f\|_{L_1^1}$. Now let us assume that $f\in L_1$; then we have $$\|f\|_{L_1^1}=\int_0^T\int_N^R\left|f(x,t)\right|\,x {\,{\mathrm{d}}}x {\,{\mathrm{d}}}t\leq R\int_0^T\int_N^R\left|f(x,t)\right|{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t=R\|f\|_{L_1}.$$ Hence $f \in L_1^1$ with $\|f\|_{L_1^1}\leq R\|f\|_{L_1}$. Taken together, the above results show us that the spaces $L_1((N,R)\times[0,T),{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t)$ and $L_1((N,R)\times[0,t),x{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t)$ contain the same elements and have equivalent norms.
This result shall prove useful in the forthcoming analysis, allowing us to switch spaces when mathematically convenient whilst retaining convergence.
Given a sequence $\left\lbrace f_n\right\rbrace_{n=1}^\infty$ in a normed vector space $\left(X,\|\cdot\|\right)$, we assume the reader is familiar with the concept of *weak* convergence and in particular its definition in spaces of the type $L_1(\Omega,\mu)$. In our analysis when handling weakly convergent sequences we will usually find them appearing alongside other factors and we would like the product to converge weakly also. The following theorem gives us sufficient conditions for the product of two sequences to converge weakly and will be used extensively in the convergence proofs for our numerical schemes.
\[theorem203\]Let $\left(\Omega,\mathcal{A},\mu\right)$ be a measure space with $\mu$ finite. Suppose $f_h\rightharpoonup f$ in $L_1\left(\Omega,\mu\right)$, $g_h\rightarrow g$ point-wise $\mu$ a.e. in $\Omega$, and $\sup_{h}\|g_h\|_{L_\infty}<\infty$, then $f_hg_h\rightharpoonup fg$ in $L_1\left(\Omega,\mu\right)$.
The reader is referred to [@fonseca07 Proposition 2.61].
The main part of our convergence argument utilises the Dunford–Pettis theorem, which provides us with sufficient conditions to establish the weak convergence of our sequence of approximations. One such condition is that of *equiintegrability*. There are a number of equivalent characterisations of equiintegrability, which the reader may find in [@fonseca07 Theorem 2.29]. For our purposes the most important characterisation of equiintegrability is given by de la Vallée Poussin’s theorem, a refined version of which is given below.
\[theorem204\][(de la Vallée Poussin’s Theorem)]{.nodecor} Let $\mathcal{F}$ be a bounded subset of $L_1\left(\Omega,\mu\right)$, then $\mathcal{F}$ is equiintegrable if and only if there exists a nonnegative, convex function $\Phi\in C^\infty \left([0,\infty)\right)$, with $\Phi(0)=0$ and $\Phi'(0)=1$, such that $\Phi'$ is concave and $$\frac{\Phi(x)}{x}\rightarrow\infty\hspace{2mm}\text{as}\hspace{2mm}x\rightarrow\infty \hspace{3mm}\text{and}\hspace{3mm}
\sup_{f\in \mathcal{F}}\int_{\Omega}\Phi\left(\left|f\right|\right){\,{\mathrm{d}}}\mu<\infty.$$
The necessity of this condition can be derived easily from [@laurencot15 Theorem 8], which under the assumption that $\mathcal{F}$ is equiintegrable provides us with a $\Psi$ satisfying all the stated conditions with the exception that the function $\Psi$ has derivative $0$ at $0$ and is not stated to be nonnegative. Given such a $\Psi$, we set $\Phi(x)=\Psi(x)+x$. Then $\Phi$ retains the required properties of $\Psi$ but additionally $\Phi'(0)=1$. Also, by utilising the following standard inequality for $C^1$ convex functions $$\label{equation203}
\Phi(x)\geq\Phi(y)+\Phi'(y)(x-y),$$ with $x\geq 0$ and $y=0$ we can see that $\Phi(x)$ must be nonnegative on $[0,\infty)$. The sufficiency of our conditions comes straight from the standard version of the de la Vallée Poussin theorem [@fonseca07 Theorem 2.29 (iii)].
In our analysis we shall require some properties of such a function, which we set out in the following lemma.
\[lemma203\] Let $\Phi$ be as in Theorem \[theorem204\]; then for nonnegative $x$ and $y$ we have the following:
1. $x\Phi'(y)\leq \Phi(x)+\Phi(y)$,
2. $\Phi'(y)\geq 0$.
The first of these inequalities is nonstandard and the proof can be found in [@laurencot15 Proposition 13 (30)]. For the second property we return to inequality , with $x=0$ and $y\geq 0$, which gives us $$\underbrace{\Phi(0)}_{=0}\geq\underbrace{\Phi(y)}_{\geq 0}+\Phi'(y)(0-y).$$ An obvious rearrangement yields $$y\Phi'(y)\geq\Phi(y)\geq0.$$ Now in the case that $y=0$ property (ii) is given by the definition of $\Phi$. Hence we may assume that $y>0$ and divide through by it to obtain the desired result that $\Phi'(y)\geq 0$.
We now come to the Dunford–Pettis theorem, one of the most significant technical tools applied in this work. The theorem provides necessary and sufficient conditions for a subset of an $L_1$ space to be *weakly sequentially compact*. That is, any sequence in the subset must have a subsequence which is weakly convergent.
\[theorem205\][(Dunford–Pettis Theorem)]{.nodecor} Let $\left(\Omega,\mathcal{A},\mu\right)$ be a measure space and let $\mathcal{F}\subset L_1\left(\Omega,\mu\right)$. Then $\mathcal{F}$ is weakly sequentially compact if and only if the following conditions are satisfied:
1. $\mathcal{F}$ is bounded in $L_1\left(\Omega,\mu\right)$;
2. $\mathcal{F}$ is equiintegrable;
3. For every $\varepsilon >0$ there exists $A_\varepsilon\subset \Omega$ with $A_\varepsilon\in \mathcal{A}$ such that $\mu\left(A_\varepsilon\right)<\infty$ and $$\sup_{f\in \mathcal{F}} \int_{\Omega\setminus A_\varepsilon} \left|f\right|{\,{\mathrm{d}}}\mu\leq \varepsilon.$$
We note that in the case that $\mu(\Omega)<\infty$ condition iii is automatically satisfied by taking $A_\varepsilon= \Omega$ for all values of $\varepsilon$.
See [@fonseca07 Theorem 2.54].
In the later analysis of this paper we shall be relying heavily on the methods and theory of operator semigroups. In particular the concept of substochastic semigroups, the Kato–Voigt perturbation theorem and the notion of semigroup honesty. For the sake of brevity we refrain from outlining such material here, however the reader may find details of the requisite results in the preliminary sections of [@baird18] or [@bairdphd] or the text [@banasiak06].
Development of the Numerical Scheme {#section1}
===================================
We now introduce our numerical approximation scheme for the truncated system, and . First we must discretise the continuous mass variable $x$, and so we introduce the mesh $\left\lbrace x_{i-1/2}\right\rbrace_{i=0}^{I_h}$ on the interval $(N,R)$, with $$x_{-1/2}\hspace{-.2mm}=\hspace{-.2mm}N,\hspace{1.3mm}x_{I_h-1/2}\hspace{-.2mm}=\hspace{-.2mm}R,\hspace{1.3mm}x_{i}\hspace{-.2mm}=\hspace{-.2mm}
(x_{i-1/2}+x_{i+1/2})/2,
\hspace{1.3mm}h/k<\Delta x_{i}\hspace{-.2mm}=\hspace{-.2mm}x_{i+1/2}-x_{i-1/2}\hspace{-.5mm}<\hspace{-.5mm}h,$$ where $h\in(0,1)$ and $k>1$ is some constant. Additionally we denote the interval $[x_{i-1/2},x_{i+1/2})$ by $\Lambda_i$, however the (left-hand-most) interval $\Lambda_0$ is taken to be $(x_{-1/2},x_{1/2})$.
For the time variable $t$, if $T$ is the final time up to which we wish to compute an approximate solution, then we define the time step $\Delta t=T/M$ where $M$ is some large integer. The time points are then given by $t_n=n\Delta t$ for $n=0,1,\dots,M$ with corresponding time intervals $\tau_n=[t_n,t_{n+1})$ for $n=0,1,\dots,M-1$.
We restrict the choice of the mesh by assuming the existence of positive constants $k_1$ and $k_2$ so that the mesh sizes $h$ and $\Delta t$ satisfy $$\label{equation404}
k_1h\leq \Delta t \leq k_2 h.$$
The numerical scheme requires representative values for the functions $a(x)$, $b(x|y)$ and $b_i(y)$ over the appropriate intervals. This is done by taking their average value over each interval. Therefore we define $$\label{equation405}
A_i=\frac{1}{\Delta x_{i}}\int_ {\Lambda_{i}} a(x){\,{\mathrm{d}}}x\hspace{3mm}\text{for}\hspace{2mm}i=0,1,\dots,I_h-1,\nonumber$$ as our approximation of $a(x)$ over the interval $\Lambda_{i}$. We approximate $b(x|y)$ over $\Lambda_{i}\times \Lambda_{j}$ by $$\label{equation406}
B_{i,j}=\frac{1}{\Delta x_{i}\Delta x_{j}}\int_ {\Lambda_{j}} \int_ {\Lambda_{i}}\hspace{-1mm} b(x|y){\,{\mathrm{d}}}x{\,{\mathrm{d}}}y\hspace{2mm}\text{for}\hspace{2mm}
i=0,1,...,I_h-1\hspace{2mm}\text{and}\hspace{2mm}j=0,1,\ldots,I_h-1,\nonumber$$ and the functions $b_i(y)$ are approximated over $\Lambda_{j}$ by the values $$\label{equation407}
\tilde{B}_{i,j}=\frac{1}{\Delta x_{j}}\int_ {\Lambda_{j}} b_i(y){\,{\mathrm{d}}}y\hspace{2mm}\text{for}\hspace{2mm}i=1,2,\ldots,N\hspace{2mm}\text{and}\hspace{2mm}
j=0,1,\dots,I_h-1.\nonumber$$
We note by our initial assumption regarding the nonnegativity of $a$, $b$ and $b_i$, that each of the values introduced above must be nonnegative. If $\chi_I$ denotes the characteristic function of a set $I$, then we can construct piecewise constant approximations to the functions $a$, $b$ and $b_i$ as follows: $$a^h(x) \hspace{-.5mm}=\hspace{-.5mm}\sum_{i=0}^{I_h-1} \chi_{\Lambda_i}(x)A_i, \hspace{1.5mm}
b^h(x|y)\hspace{-.5mm}=\hspace{-.5mm}\sum_{j=0}^{I_h-1}\sum_{i=0}^{I_h-1}\chi_{\Lambda_i}(x)\chi_{\Lambda_j}(y)B_{i,j}, \hspace{1.5mm}
b_i^h(y)\hspace{-.5mm}=\hspace{-.5mm}\sum_{j=0}^{I_h-1} \chi_{\Lambda_j}(y)\tilde{B}_{i,j}.$$
\[convremark1\] This is a standard means of approximation and assuming the choice of kernels is suitably restricted, the approximations will converge pointwise to the desired functions almost everywhere on the appropriate domains. In our case, the kernels $a$ and $b$ will be assumed to be $L_{\infty,loc}$ on $[N,\infty)$ and $[N,\infty)\times[N,\infty)$, respectively. In addition, the restriction determines each $b_i$ as an element of $L_{\infty,loc}[N,\infty)$. Having $a$, $b$ and $b_i$ as $L_{\infty,loc}$ functions is sufficient to ensure that the approximations $a^h$, $b^h$ and $b_i^h$ converge pointwise to $a$, $b$ and $b_i$ almost everywhere in their respective domains. This is a standard result, however full details can be found in [@bairdphd Lemma 4.2.1].
We are now ready to construct the approximation scheme. Let $u_{C}^{n,i}$ denote our approximation to $u_C^R(x,t)$ over the mass interval $\Lambda_i$ for the time interval $\tau_n$. The equation is then approximated by
$$\label{equation4075}
x_i\frac{u_{C}^{n+1,i}-u_{C}^{n,i}}{\Delta t}=\frac{F_{i+1/2}^n-F_{i-1/2}^n}{\Delta x_{i}}-S_i^n,\nonumber$$
where $F_{i-1/2}^n$ is an approximation of the flux $\mathcal{F}^R(xu_C^R)$ at the point $x=x_{i-1/2}$ over the time interval $\tau_n$, and is given by $$\begin{aligned}
\label{equation408}
\left(\mathcal{F}^R(xu_C^R)\right)(x_{i-1/2})&=\int_{x_{i-1/2}}^{R} \int_{N}^{x_{i-1/2}}y a(z) b(y|z)u_C^R(z,t){\,{\mathrm{d}}}y{\,{\mathrm{d}}}z\nonumber\\
&=\sum_{j=i}^{I_h-1}\int_{\Lambda_j}\left(\sum_{k=0}^{i-1}\int_{\Lambda_k} y a(z) b(y|z)u_C^R(z,t){\,{\mathrm{d}}}y\right){\,{\mathrm{d}}}z\nonumber\\
&\approx\sum_{j=i}^{I_h-1}\sum_{k=0}^{i-1}x_{k}A_jB_{k,j}u_{C}^{n,j}\Delta x_{k}\Delta x_{j}=:F_{i-1/2}^n\nonumber,\end{aligned}$$ for $i=1,\ldots,I_h-1$, with $F_{-1/2}^n=F_{I_h-1/2}^n=0$, which can be justified by Lemma \[lastthing\]. The values $S_i^n$ approximate the sink term $S(xu_C^R)$ over $\Lambda_i$ for the time interval $\tau_n$, and are computed by $$\label{sink}
S_i^n=A_i\sum_{j=1}^{N}j\tilde{B}_{j,i}u_{C}^{n,i}\hspace{1.5mm}\text{for}\hspace{1.5mm}i=0,1,\dots,I_h-1.$$
This gives rise to the following numerical method for the computation of the approximations $u_{C}^{n,i}$: $$\label{equation409}
u_{C}^{n+1,i}=u_{C}^{n,i}+\frac{\Delta t}{x_i\Delta x_{i}}(F_{i+1/2}^n-F_{i-1/2}^n)-\frac{\Delta t}{x_i}S_i^n\hspace{2mm}\text{for}
\hspace{1.5mm}\left\lbrace \begin{array}{l} \hspace{.5mm}i=0,1,\dots,I_h-1, \\ n=0,1,\dots,M-1. \end{array}\right.$$ The sequence of approximations generated by requires us to provide an initial set of values to get started. For our starting values we simply average the initial datum over each of the mass intervals; hence $$\label{equation410}
u_{C}^{0,i}=\frac{1}{\Delta x_{i}}\int_ {\Lambda_{i}} c_0(x){\,{\mathrm{d}}}x\hspace{3mm}\text{for}\hspace{2mm}i=0,1,\dots,I_h-1.$$ Then our approximation to $u_{C}^R(x,t)$ over $(N,R)\times[0,T)$ is constructed as follows: $$\label{equation4105}
u^h_C(x,t)=\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1} \chi_{\Lambda_i}(x)\chi_{\tau_n}(t)u_{C}^{n,i}.$$
\[convremark2\] The convergence proof for our numerical scheme requires the initial approximation given by and to converge strongly in $L_1(N,R)$ to the restriction of $c_0$ to $(N,R)$. Again, that this is the case with our definition of the discretised initial datum is a standard result and details can be found in [@bairdphd Lemma 4.2.2].\
Now considering the discrete regime, let $u_{D}^{n,i}$ denote our approximation of $u_{Di}^R(t)$ over the time interval $\tau_n$. Equation is then approximated as\
$$\label{equation411}
\frac{u_{D}^{n+1,i}-u_{D}^{n,i}}{\Delta t}=-a_iu_{D}^{n,i}+\hspace{-1.9mm}\sum_{j=i+1}^{N}a_jb_{i,j}u_{D}^{n,j}+\sum_{j=0}^{I_h-1}A_j
\tilde{B}_{i,j}u_{C}^{n,j}
\Delta x_{j},\nonumber$$ giving rise to the relation $$\label{equation412}
u_{D}^{n+1,i}=(1-\Delta ta_i)u_{D}^{n,i}+\Delta t\sum_{j=i+1}^{N}\hspace{-1mm}a_jb_{i,j}u_{D}^{n,j}+\Delta t\sum_{j=0}^{I_h-1}\hspace{-1mm}A_j\tilde{B}_{i,j}u_{C}^{n,j}
\Delta x_{j}\hspace{2mm}\text{for}\hspace{1.5mm}\left\lbrace \begin{array}{l} \hspace{.5mm}i=1,\dots,N, \\\hspace{-.5mm} n=0,1,\dots,M-1. \end{array}\right.$$ The initial values for the discrete approximation are simply given by the initial condition vector $d_0$, so that $u_{D}^{0,i}={d_0}_i$ for $i=1,\ldots, N$. Then our approximations $u_{Di}^h(t)$ to $u_{Di}^R(t)$ for $t \in [0,T)$ are given by $$\label{equation4125}
u_{Di}^h(t)=\sum_{n=0}^{M-1} \chi_{\tau_n}(t)u_{D}^{n,i}\hspace{3mm}\text{for}\hspace{2mm}i=1,2,\ldots,N.$$
Properties of Numerical Solutions: Nonnegativity and Mass Conservation
======================================================================
In the article [@baird18] we proved the existence and uniqueness of a solution to our system and . This solution was shown to possess a number of properties that we would expect given the physical nature of the model, namely the solution preserved nonnegativity and conserved total mass. In the following sections we examine whether the approximate solution provided by and , also displays these properties. These properties, apart from being physically relevant, will also be utilised in the forthcoming proofs of the convergence of the approximations and to a solution to the system and , and subsequently the uniqueness and differentiability of that solution.
Nonnegativity of the Numerical Solution
---------------------------------------
\[lemma401\] For a fixed partition $(x_{i-1/2})_{i=0}^{I_h}$, suppose that $\Delta t$ is sufficiently small that the following condition is satisfied: $$0<\Delta t \leq \frac{x_i}{A_i\left(\sum_{k=0}^{i-1}x_{k}B_{k,i}\Delta x_{k}+\sum_{j=1}^{N}j\tilde{B}_{j,i}\right)},$$ for all $i \in \left\lbrace0,1,\dots,I_h-1\right\rbrace$ such that the denominator is nonzero, and $$0<\Delta t \leq \frac{1}{a_i},$$ for all $i \in \left \lbrace 2,\dots,N \right\rbrace$ such that $a_i\neq0$. Then, the approximate solutions defined by and preserve nonnegativity.
Starting with the approximation for the continuous regime, let us consider equation . By cancelling common terms we get that $$\begin{aligned}
\label{equation413}
&F_{i+1/2}^n-F_{i-1/2}^n=x_{i}\Delta x_{i}\sum_{j=i+1}^{I_h-1}A_jB_{i,j}u_{C}^{n,j}\Delta x_{j}-A_iu_{C}^{n,i}\Delta x_{i}\sum_{k=0}^{i-1}x_{k}B_{k,i}
\Delta x_{k}\end{aligned}$$ for $i=1,\dots,I_h-2$. Therefore we have
$$\begin{aligned}
&\frac{\Delta t}{x_i\Delta x_{i}}(F_{i+1/2}^n-F_{i-1/2}^n)-\frac{\Delta t}{x_i}S_i\\
&=\Delta t\sum_{j=i+1}^{I_h-1}A_jB_{i,j}u_{C}^{n,j}\Delta x_{j}-\frac{\Delta t}{x_i}A_iu_{C}^{n,i}\sum_{k=0}^{i-1}x_{k}B_{k,i}\Delta x_{k}
-\frac{\Delta t}
{x_i}A_iu_{C}^{n,i}\sum_{j=1}^{N}j\tilde{B}_{j,i} \\
&=\Delta t\sum_{j=i+1}^{I_h-1}A_jB_{i,j}u_{C}^{n,j}\Delta x_{j}-\frac{\Delta t}{x_i}A_iu_{C}^{n,i}\left(\sum_{k=0}^{i-1}x_{k}B_{k,i}\Delta x_{k}+
\sum_{j=1}^{N}j\tilde{B}_{j,i}\right).\end{aligned}$$
\
Substituting this into gives us $$\begin{aligned}
\label{equation901}
u_{C}^{n+1,i}&=\left(1-\frac{\Delta t}{x_i}A_i\left(\sum_{k=0}^{i-1}x_{k}B_{k,i}\Delta x_{k}+\sum_{j=1}^{N}j\tilde{B}_{j,i}\right)\right)u_{C}^{n,i}
+\Delta t\sum_{j=i+1}^{I_h-1}A_jB_{i,j}u_{C}^{n,j}\Delta x_{j},\end{aligned}$$ for $i=1,\dots,I_h-2$. The cases $i=0$ and $i=I_h-1$ can be handled similarly to obtain the same result, where the empty sums are taken as 0.
From this it is clear that if each of the approximations $u_{C}^{n,i}$ is nonnegative, and provided $\Delta t$ is sufficiently small such that the term within the outer brackets is nonnegative, then each of the approximations $u_{C}^{n+1,i}$, for the subsequent time step, will also be nonnegative. Hence to ensure the approximations $u_{C}^{n+1,i}$ are nonnegative we can take\
$$\label{equation414}
0<\Delta t \leq \frac{x_i}{A_i\left(\sum_{k=0}^{i-1}x_{k}B_{k,i}\Delta x_{k}+\sum_{j=1}^{N}j\tilde{B}_{j,i}\right)}\hspace{3mm}\text{for}
\hspace{2mm}i=0,1,\dots,I_h-1.$$ In the case of the above denominator being zero for some $i$, such that the bound is undefined, then $u_{C}^{n+1,i}$ can be seen from to automatically satisfy the nonnegativity requirement, for any value of $\Delta t$.
Turning to the approximation for the discrete regime, it is immediately clear from the form of that if all of the values $u_{C}^{n,i}$ and $u_{D}^{n,i}$ are nonnegative, then each $u_{D}^{n+1,i}$ will be nonnegative if for each $i=1,\dots,N$ we have that $1-\Delta ta_i$ is nonnegative. This can be ensured by taking $$\label{equation415}
0<\Delta t \leq \frac{1}{a_i} \hspace{3mm} \text{for}\hspace{2mm}i=2,\dots,N \hspace{2mm} \text{such that} \hspace{2mm} a_i\neq0.$$ Therefore if we choose a $\Delta t$ small enough that both and are satisfied, then our approximate solutions will remain nonnegative.
From now on we shall assume that conditions and are satisfied and that $c_0(x)\geq 0$ and each $d_{0,i}\geq 0$ so that our approximations remain nonnegative.\
\[remark401\]The bound is dependent on the mesh and it is perhaps not immediately apparent how this bounding value might vary as we refine the mesh. In particular, it would be advantageous to confirm that it is indeed possible to find a constant $k_1$, such that conditions and can be satisfied simultaneously, whilst $h\searrow0$. In the upcoming analysis we will place restrictions on the functions $a$ and $b$; these constraints will allow us to guarantee the existence of such a $k_1$.
The upcoming Theorem \[theorem501\] imposes the restriction $a,b\in L_{\infty}$ on the restricted domains $[N,R]$ and $[N,R]\times[N,R]$ respectively, with $\alpha(R)$ and $\beta(R)$ being the essential suprema for $a$ and $b$ on said domains. This being the case, we have $A_i\leq \alpha(R)$ and $B_{k,i}\leq\beta(R)$ for all values of $i$ and $k$ admissible in . Furthermore, from we may deduce that each $b_i(y)\leq y$, hence $\tilde{B}_{j,i}\leq R$. Finally, all mesh midpoints $x_i$ must clearly satisfy $x_i\geq N\geq1>h$. Taken together, these bounds lead, via a simple calculation, to $$\begin{aligned}
\frac{h}{\alpha(R)\left(\beta(R)R\left(R-N\right)+RN(N+1)/2\right)}
\leq \frac{x_i}{A_i\left(\sum_{k=0}^{i-1}x_{k}B_{k,i}\Delta x_{k}+\sum_{j=1}^{N}j\tilde{B}_{j,i}\right)},\end{aligned}$$ for $i=0,1,\dots,I_h-1$. Hence, we have established a possible value for $k_1$, which ensures and can be satisfied simultaneously as $h\searrow0$.
Mass Conservation by the Numerical Solutions
--------------------------------------------
In [@baird18 Lemma 6.2], the exact solution to our system of equations and was shown to conserve mass between the two regimes. We now show that this property is shared by our numerical solutions.
\[lemma402\] The approximate solutions generated by and conserve mass.
The mass associated with the approximate continuous regime solution, $u_C^h(x,t)$, is given by $$\begin{aligned}
\label{equation416}
\|u^h_C(\cdot,t)\|_{L_{1}^1(N,R)}&=\int_N^R\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1} \chi_{\Lambda_i}(x)\chi_{\tau_n}(t)u_{C}^{n,i}\,x{\,{\mathrm{d}}}x\nonumber\\
&=\sum_{n=0}^{M-1}\chi_{\tau_n}(t) \sum_{i=0}^{I_h-1} u_{C}^{n,i}\int_N^R \chi_{\Lambda_i}(x)\,x{\,{\mathrm{d}}}x\nonumber\\
&=\sum_{n=0}^{M-1}\chi_{\tau_n}(t) \sum_{i=0}^{I_h-1}x_i \Delta x_i u_{C}^{n,i},\end{aligned}$$ whilst the approximate solution $u_D^h(t)$ has associated mass given by
$$\|u_{D}^h(t)\|_{X_D}=\sum_{i=1}^N i u_{Di}^h(t)= \sum_{i=1}^N i \sum_{n=0}^{M-1} \chi_{\tau_n}(t)u_{D}^{n,i} =\sum_{n=0}^{M-1}
\chi_{\tau_n}(t) \sum_{i=1}^N i u_{D}^{n,i}.$$ Summing these two expressions gives the total mass: $$\label{equation417}
M^h(t)=\sum_{n=0}^{M-1}\chi_{\tau_n}(t)\left( \sum_{i=0}^{I_h-1}x_i \Delta x_i u_{C}^{n,i}+ \sum_{i=1}^N i u_{D}^{n,i}\right).$$ First let us examine the mass accounted for by the continuous regime. From the relation we get $$\begin{aligned}
\label{equation418}
&\sum_{i=0}^{I_h-1}x_iu_{C}^{n+1,i}\Delta x_{i}=\sum_{i=0}^{I_h-1}x_i\left(u_{C}^{n,i}+\frac{\Delta t}{x_i\Delta x_{i}}(F_{i+1/2}^n-F_{i-1/2}^n)-
\frac{\Delta t}{x_i}S_i^n\right)\Delta x_{i}\nonumber\\
&=\sum_{i=0}^{I_h-1}x_iu_{C}^{n,i}\Delta x_{i}+\Delta t\sum_{i=0}^{I_h-1}(F_{i+1/2}^n-F_{i-1/2}^n)-\Delta t\sum_{i=0}^{I_h-1}S_i^n\Delta x_{i}
\nonumber\\
&=\sum_{i=0}^{I_h-1}x_iu_{C}^{n,i}\Delta x_{i}-\Delta t\sum_{i=0}^{I_h-1}S_i^n\Delta x_{i}.\end{aligned}$$ The middle summation term is lost in going to the final line as the sum is telescoping with zero end terms. Now we consider the discrete regime mass; the generating relation gives us $$\begin{aligned}
\label{equation419}
&\sum_{i=1}^{N}iu_{D}^{n+1,i}=\sum_{i=1}^{N}i\left((1-\Delta ta_i)u_{D}^{n,i}+\Delta t\sum_{j=i+1}^{N}a_jb_{i,j}u_{D}^{n,j}
+\Delta t\sum_{j=0}^{I_h-1}A_j
\tilde{B}_{i,j}u_{C}^{n,j}\Delta x_{j}\right)\nonumber\\
&=\sum_{i=1}^{N}iu_{D}^{n,i}-\Delta t\sum_{i=1}^{N}ia_iu_{D}^{n,i}+\Delta t\sum_{i=1}^{N}i\sum_{j=i+1}^{N}a_jb_{i,j}u_{D}^{n,j}+\Delta t
\sum_{i=1}^{N}i
\sum_{j=0}^{I_h-1}A_j\tilde{B}_{i,j}u_{C}^{n,j}\Delta x_{j}\nonumber\\
&=\sum_{i=1}^{N}iu_{D}^{n,i}-\Delta t\sum_{i=2}^{N}ia_iu_{D}^{n,i}
+\Delta t\sum_{j=2}^{N}a_ju_{D}^{n,j}\hspace{-1mm}\left(\sum_{i=1}^{j-1}ib_{i,j}\right)+\Delta t\hspace{-.2mm}\sum_{j=0}^{I_h-1}\Delta x_{j}\hspace{-.2mm}\left(\hspace{-.2mm}A_ju_{C}^{n,j}\sum_{i=1}^{N}i\tilde{B}_{i,j}
\hspace{-.2mm}\right)\nonumber\\
&=\sum_{i=1}^{N}iu_{D}^{n,i}+\Delta t\sum_{j=0}^{I_h-1}\Delta x_{j}S_j^n.\end{aligned}$$ The middle two terms cancel due to the mass conservation condition . Combining equations and we obtain $$\begin{aligned}
&\sum_{i=0}^{I_h-1}x_iu_{C}^{n+1,i}\Delta x_{i}+\sum_{i=1}^{N}iu_{D}^{n+1,i}\\
&=\sum_{i=0}^{I_h-1}x_iu_{C}^{n,i}\Delta x_{i}-\Delta t\sum_{i=0}^{I_h-1}S_i^n\Delta x_{i}+\sum_{i=1}^{N}iu_{D}^{n,i}+\Delta t\sum_{j=0}^{I_h-1}
\Delta x_{j}S_j^n\\
&=\sum_{i=0}^{I_h-1}x_iu_{C}^{n,i}\Delta x_{i}+\sum_{i=1}^{N}iu_{D}^{n,i}.\end{aligned}$$ From repeated application of this equality it is easily seen that the bracketed expression appearing in is equal for all values of $n$, and hence the total mass $M^h(t)$ remains constant.
Convergence of the Scheme to a Weak Solution as
================================================
Having determined the nonnegativity and mass conservative properties of the approximate solutions provided by and , in this section we set out to prove that they converge, in some sense, to a limit as the parameter $h$, and by necessity $\Delta t$, go to zero, and show that this limit itself is an ‘exact’ solution to our truncated model.
Continuous Fragmentation Regime: Convergence
--------------------------------------------
Let us start with the continuous regime approximations $\left\lbrace u^h_C \right \rbrace$. In order to prove the (weak) convergence of this family, we employ a weak compactness argument, utilising the Dunford–Pettis theorem (Theorem \[theorem205\]), which provides necessary and sufficient conditions for weak compactness in an $L_{1}$ space. We begin by proving the equiboundedness of the set $\left\lbrace u^h_C \right \rbrace$.
\[lemma501\]The family of approximations $\left\lbrace u_C^h\right\rbrace$ is equibounded (uniformly bounded) in the space $L_1((N,R) \times [0,T),x {\,{\mathrm{d}}}x {\,{\mathrm{d}}}t)$.
Recalling equation , we have for any $t\in [0,T)$ that $$\begin{aligned}
\|u^h_C(\cdot,t)\|_{L_{1}^1(N,R)}&=\sum_{n=0}^{M-1}\chi_{\tau_n}(t) \sum_{i=0}^{I_h-1}x_i \Delta x_i u_{C}^{n,i}.\end{aligned}$$ From the analysis of Lemma \[lemma401\], each of the values $u_{C}^{n,i}$ is nonnegative, and as such the values $S_i^n$ are nonnegative. Therefore, from the last line of equation we deduce that $$\sum_{i=0}^{I_h-1}x_i \Delta x_i u_{C}^{n,i}\leq \sum_{i=0}^{I_h-1}x_i \Delta x_iu_{C}^{n-1,i}\hspace{3mm}\text{for}\hspace{2mm}n=1,\ldots,M-1.$$
Repeated application of this inequality yields $$\label{equation501}
\sum_{i=0}^{I_h-1}x_i \Delta x_i u_{C}^{n,i}\leq \sum_{i=0}^{I_h-1}x_i \Delta x_i u_{C}^{0,i}=\sum_{i=0}^{I_h-1}x_i \int_ {\Lambda_{i}} c_0(x){\,{\mathrm{d}}}x\leq\sum_{i=0}^{I_h-1}\frac{x_i}{x_{i-1/2}} \int_ {\Lambda_{i}} c_0(x)\,x{\,{\mathrm{d}}}x.$$ The quantity $\frac{x_i}{x_{i-1/2}}$ can be bounded as follows: $$\frac{x_i}{x_{i-1/2}}=\frac{x_{i-1/2}+\frac{1}{2}\Delta x_i}{x_{i-1/2}}\leq 1+ \frac{h}{2N}\leq \frac{3}{2}.$$ We note this bound as it will appear regularly in subsequent calculations. Substituting this within yields $$\sum_{i=0}^{I_h-1}x_i \Delta x_i u_{C}^{n,i}\leq\frac{3}{2}\|c_0\|_{L_{1}^1(N,R)},$$ for $n=0,\ldots,M-1$. Replacing this inequality in our calculation gives us the following, which holds for all $t\in [0,T)$: $$\begin{aligned}
\|u^h_C(\cdot,t)\|_{L_{1}^1(N,R)}&\leq\sum_{n=0}^{M-1}\chi_{\tau_n}(t)\frac{3}{2}\|c_0\|_{L_{1}^1(N,R)}= \frac{3}{2}\|c_0\|_{L_{1}^1(N,R)}.\end{aligned}$$ Integrating this inequality with respect to $t$ from $0$ to $T$ we obtain the required equiboundedness of $\left\lbrace u_C^h\right\rbrace$ in the space $L_1((N,R) \times [0,T),x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}t)$.
We now move on to prove the second of the two required conditions for the Dunford–Pettis theorem, namely equiintegrability. However, prior to this we collect all the conditions so far imposed on our model via the functions $a$, $b$ and $b_i$, and the initial distributions $c_0$ and $d_0$, and also on our mesh via the parameters $h$ and $\Delta t$.
\[conditions\]Throughout the remaining analysis, the following conditions shall be assumed to be satisfied.
1. The initial mass distributions within the continuous and discrete regimes are nonnegative, that is $c_0(x)$ for all $x>N$ and $d_{0,i}\geq0$ for $i=1,\ldots,N$.
2. The functions $a$ and $b$ are nonnegative and belong to the spaces $L_{\infty,loc}\left([N,\infty)\right)$ and\
$L_{\infty,loc}\left([N,\infty)\times[N,\infty)\right)$, respectively.
3. Each of the functions $b_i$ is assumed to be nonnegative. This nonnegativity in conjunction with condition $2$ is sufficient to guarantee that $b_i\in L_{\infty,loc}\left([N,\infty)\right)$, as per Remark \[convremark1\].
4. There exist positive constants $k_1$ and $k_2$ such that the mesh parameters $h$ and $\Delta t$ satisfy $$k_1h\leq \Delta t \leq k_2 h.$$
5. To ensure that the approximate solutions remain nonnegative, the time step $\Delta t$ is assumed to satify the following constraints: $$0<\Delta t \leq \frac{x_i}{A_i\left(\sum_{k=0}^{i-1}x_{k}B_{k,i}\Delta x_{k}+\sum_{j=1}^{N}j\tilde{B}_{j,i}\right)}\hspace{3mm}\text{for}
\hspace{2mm}i=0,1,\dots,I_h-1,$$ for all cases of the denominator being nonzero, and $$0<\Delta t \leq \frac{1}{a_i} \hspace{3mm} \text{for}\hspace{2mm}i=2,\dots,N \hspace{2mm} \text{such that} \hspace{2mm} a_i\neq0.$$
6. There exists a constant $\theta>0$ such that $$K(R) \Delta t \leq \theta<1,$$ where $K(R)=\alpha(R)\beta(R) R$, with $\alpha(R)$ and $\beta(R)$ being the essential suprema for $a$ and $b$ on the restricted domains $[N,R]$ and $[N,R]\times[N,R]$, respectively.
\[theorem501\] Under the assumptions outlined in Remark \[conditions\], the family $\left\lbrace u_C^h\right\rbrace$ is equiintegrable in $L_1((N,R) \times[0,T),x{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t)$.
Consider the constant sequence comprising solely of the initial data $c_0\in L_{1}\left((N,R),x{\,{\mathrm{d}}}x\right)$. Clearly this sequence is convergent, therefore $\left\lbrace c_0\right\rbrace$ forms a weakly sequentially compact set in $L_{1}\left((N,R),x{\,{\mathrm{d}}}x\right)$. Hence by the de la Vallee Poussin theorem (Theorem \[theorem204\]) there exists a nonnegative, convex function $\Phi \in C^\infty ([0,\infty))$, with $\Phi(0)=0$ and $\Phi'(0)=1$ such that $\Phi'$ is concave and satisfies $$\frac{\Phi(x)}{x}\rightarrow\infty\hspace{2mm}\text{as}\hspace{2mm}x\rightarrow\infty \hspace{3mm}\text{and}\hspace{3mm}\int_{N}^R\Phi(c_0)(x)\,x{\,{\mathrm{d}}}x
<\infty.$$ A standard inequality , for $C^1$ convex functions gives us $$\Phi(u_{C}^{n+1,i})-\Phi(u_{C}^{n,i})\leq\left(u_{C}^{n+1,i}-u_{C}^{n,i}\right)\Phi'(u_{C}^{n+1,i}).$$ Multiplying this by $x_i \Delta x_i$ and summing over all $i$ gives $$\sum_{i=0}^{I_h-1}x_i\Delta x_i \left(\Phi\left(u_{C}^{n+1,i}\right)-\Phi\left(u_{C}^{n,i}\right) \right)
\leq \sum_{i=0}^{I_h-1}x_i \Delta x_i\left(\left(u_{C}^{n+1,i}-u_{C}^{n,i}\right)\Phi'(u_{C}^{n+1,i})\right).$$ Utilising equation we can rewrite this as $$\begin{aligned}
\label{equation504}
&\sum_{i=0}^{I_h-1}x_i\Delta x_i \left(\Phi\left(u_{C}^{n+1,i}\right)-\Phi\left(u_{C}^{n,i}\right) \right)\nonumber\\
&\leq \sum_{i=0}^{I_h-1}x_i\Delta x_i \left(\frac{\Delta t}{x_i\Delta x_{i}}(F_{i+1/2}^n-F_{i-1/2}^n)-\frac{\Delta t}{x_i}S_i^n\right)
\Phi'\left(u_{C}^{n+1,i}\right).\end{aligned}$$
Recalling the definition of $S_i^n$ from , we see that it must be nonnegative. Additionally, Lemma \[lemma203\](ii) and Lemma \[lemma401\] give $\Phi'\left(u_{C}^{n+1,i}\right)\geq0$, hence we can drop the term involving $S_i^n$ from and the inequality will still remain valid, giving us $$\sum_{i=0}^{I_h-1}x_i\Delta x_i \left(\Phi\left(u_{C}^{n+1,i}\right)-\Phi\left(u_{C}^{n,i}\right) \right)
\leq \sum_{i=0}^{I_h-1} \Delta t(F_{i+1/2}^n-F_{i-1/2}^n)\Phi'\left(u_{C}^{n+1,i}\right).$$ With some easy modification, equation becomes the inequality $$F_{i+1/2}^n-F_{i-1/2}^n\leq x_{i}\Delta x_{i}\sum_{j=i+1}^{I_h-1}A_jB_{i,j}u_{C}^{n,j}\Delta x_{j},$$ which, if placed in the previous inequality, results in
$$\sum_{i=0}^{I_h-1}x_i \Delta x_i \left(\Phi\left(u_{C}^{n+1,i}\right)-\Phi\left(u_{C}^{n,i}\right) \right)
\leq \Delta t\sum_{i=0}^{I_h-1}\sum_{j=i+1}^{I_h-1}x_i A_jB_{i,j}u_{C}^{n,j}\Delta x_{i}\Delta x_{j}\Phi'\left(u_{C}^{n+1,i}\right).$$
Utilising Lemma \[lemma203\](i) with $x=u_{C}^{n,j}$ and $y=u_{C}^{n+1,i}$ and noting that the constants $\alpha(R)$ and $\beta(R)$ bound the average values $A_j$ and $B_{i,j}$, we get
$$\begin{aligned}
&\sum_{i=0}^{I_h-1 }x_i\Delta x_i \left( \Phi\left(u_{C}^{n+1,i}\right)-\Phi\left(u_{C}^{n,i}\right) \right)
\leq \alpha \beta \Delta t\sum_{i=0}^{I_h-1} \sum_{j=i+1}^{I_h-1}x_i\Delta x_{i} \Delta x_{j} u_{C}^{n,j}
\Phi'\left(u_{C}^{n+1,i}\right)\\
&\leq \alpha(R) \beta(R) \Delta t \left(\sum_{i=0}^{I_h-1} \left(x_i \Delta x_{i} \Phi\left(u_{C}^{n+1,i}\right)\sum_{j=i+1}^{I_h-1}\Delta x_j\right)+\sum_{i=0}^{I_h-1}\left( x_i \Delta x_{i} \sum_{j=i+1}^{I_h-1} \Delta x_j \Phi(u_{C}^{n,j})\right)\right).\end{aligned}$$
As $j$ is restricted to be greater than $i$ we have $x_j>x_i$ for admissible $j$ and $i$. This allows us to switch $x_i$ for $x_j$ in the second term and take this within the inner summation. Following this we expand the summation over $j$ to give $$\begin{aligned}
&\sum_{i=0}^{I_h-1 }x_i\Delta x_i \left( \Phi\left(u_{C}^{n+1,i}\right)-\Phi\left(u_{C}^{n,i}\right) \right)\\
&\leq \alpha(R) \beta(R) \Delta t \left(\sum_{i=0}^{I_h-1} \left(x_i \Delta x_{i} \Phi\left(u_{C}^{n+1,i}\right)\sum_{j=i+1}^{I_h-1}\Delta x_j\right)+\sum_{i=0}^{I_h-1}\left(\Delta x_{i} \sum_{j=i+1}^{I_h-1}x_j \Delta x_j \Phi(u_{C}^{n,j})\right)\right)\\
&\leq \underbrace{\alpha(R)\beta(R) R}_{=K(R)} \Delta t \left(\sum_{i=0}^{I_h-1} x_i \Delta x_{i} \Phi\left(u_{C}^{n+1,i}\right)
+ \sum_{j=0}^{I_h-1}x_j \Delta x_j \Phi(u_{C}^{n,j})\right).\end{aligned}$$
If we change the index variable from $j$ to $i$ in the second summation and re-arrange then we obtain $$\begin{aligned}
\left(1-K(R)\Delta t\right)\sum_{i=0}^{I_h-1} x_i\Delta x_{i}\Phi(u_{C}^{n+1,i})\leq \left(1+K(R)\Delta t\right)\sum_{i=0}^{I_h-1}x_i \Delta x_{i}
\Phi\left(u_{C}^{n,i}\right).\end{aligned}$$ Some further manipulations produce $$\begin{aligned}
\left(1-K(R)\Delta t\right)\sum_{i=0}^{I_h-1}x_i \Delta x_{i}\left(\Phi(u_{C}^{n+1,i})-\Phi(u_{C}^{n,i})\right)
\leq 2K(R)\Delta t\sum_{i=0}^{I_h-1}x_i \Delta x_{i}\Phi\left(u_{C}^{n,i}\right).\end{aligned}$$ By the final assumption of Remark \[conditions\], we have $1-K(R)\Delta t>0$ allowing us to divide through to get $$\begin{aligned}
\sum_{i=0}^{I_h-1}x_i \Delta x_{i}\Phi(u_{C}^{n+1,i})\leq\left(1+\frac{2K(R)\Delta t}{1-K(R)\Delta t}\right)\sum_{i=0}^{I_h-1}x_i \Delta x_{i}
\Phi\left(u_{C}^{n,i}\right).\end{aligned}$$ Repeated application of this inequality yields $$\begin{aligned}
\sum_{i=0}^{I_h-1}x_i \Delta x_{i}\Phi(u_{C}^{n+1,i})&\leq\left(1+\frac{2K(R)\Delta t}{1-K(R)\Delta t}\right)^{n+1}\hspace{1mm}\sum_{i=0}^{I_h-1} x_i
\Delta x_{i}\Phi\left(u_{C}^{0,i}\right)\\
&\leq\exp\left(\frac{2K(R)\Delta t (n+1)}{1-K(R)\Delta t}\right)\sum_{i=0}^{I_h-1}x_i \Delta x_{i}\Phi\left(u_{C}^{0,i}\right).\end{aligned}$$ For values of $t$ in the interval $\tau_n=[t_n,t_{n+1})$ this gives us $$\begin{aligned}
\int_{N}^{R}\Phi \left(u^h_C(x,t)\right)\,x{\,{\mathrm{d}}}x
&=\sum_{i=0}^{I_h-1} x_i\Delta x_{i}\Phi\left(u_{C}^{n,i}\right)\\
&\leq\exp\left(\frac{2K(R)t}{1-K(R)\Delta t} \right)\sum_{i=0}^{I_h-1}x_i \Delta x_{i}\Phi\left(u_{C}^{0,i}\right)\\
&=\exp\left(\frac{2K(R)t}{1-K(R)\Delta t} \right)\sum_{i=0}^{I_h-1} x_i \Delta x_{i}\Phi\left(\frac{1}{\Delta x_{i}}\int_ {\Lambda_{i}} c_0(x){\,{\mathrm{d}}}x
\right).\end{aligned}$$ An application of Jensen’s inequality [@lieb01 Theorem 2.2] allows us to switch the order of $\Phi$ and integration to get $$\begin{aligned}
\int_{N}^{R}\Phi \left(u^h_C(x,t)\right)\,x{\,{\mathrm{d}}}x
&\leq\exp\left(\frac{2K(R)t}{1-K(R)\Delta t} \right)\sum_{i=0}^{I_h-1} x_i \int_ {\Lambda_{i}} \Phi( c_0(x)){\,{\mathrm{d}}}x\\
&\leq\frac{3}{2}\exp\left(\frac{2K(R)t}{1-K(R)\Delta t} \right)\sum_{i=0}^{I_h-1} \int_ {\Lambda_{i}} \Phi( c_0(x))\,x{\,{\mathrm{d}}}x.\end{aligned}$$ By assumption $6$ of Remark \[conditions\], that $K(R)\Delta t \leq \theta<1$, we deduce that $$\begin{aligned}
\int_{N}^{R} \Phi \left(u^h_C(x,t)\right)\,x{\,{\mathrm{d}}}x
\leq\frac{3}{2}\exp\left(\frac{2K(R)t}{1-\theta}\right)\int_{N}^{R}\Phi( c_0(x))\,x{\,{\mathrm{d}}}x,\end{aligned}$$ which holds for all $t\in[0,T)$. Integrating the inequality with respect to $t$ from $0$ to $T$ confirms the equiintegrability of the family $\left\lbrace u_C^h\right\rbrace$ in the space $L_1((N,R) \times [0,T),x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}t)$.
By Theorem \[theorem205\] (Dunford–Pettis theorem), the sequence $\left\lbrace u_C^h\right\rbrace$ forms a weakly sequentially compact set in the space $L_{1}\left((N,R)\times[0,T),x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}t\right)$. This implies the existence of a subsequence $\left\lbrace u_C^{hj}\right\rbrace$ and a function $u_C^R\in L_{1}\left((N,R)\times[0,T),x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}t\right)$ such that $u_C^{hj}\rightharpoonup u_C^R$ in $L_{1}\left((N,R)\times[0,T),x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}t\right)$ as $j\rightarrow \infty$ and $h_{j}\rightarrow0$.\
\[remark501\] From now on this convergent subsequence will be considered implicitly, unless otherwise stated; as such we now use the notation $\left\lbrace u_C^h\right\rbrace$ to denote such a convergent subsequence, the choice of which, we note, may not be unique.
Continuous Fragmentation Regime: Weak Solution
----------------------------------------------
Having shown that our sequence of approximations converges (weakly) to a limit, we now aim to show that this limit provides a solution to our truncated equation . Precisely, we intend to show that the function $u_C^R$ satisfies the following criterion.
\[definition501\] The function $u_C^R$ is a weak solution of equation , if it satisfies $$\begin{aligned}
\label{equation505}
&\int_{0}^{T}\int_{N}^{R} xu_C^R(x,t)\frac{\partial\varphi}{\partial t}(x,t) {\,{\mathrm{d}}}x{\,{\mathrm{d}}}t+\int_{N}^{R}xc_{0}(x)\varphi(x,0){\,{\mathrm{d}}}x\nonumber\\
&=\int_{0}^{T}\int_{N}^{R}\mathcal{F}^R\left(xu_{C}^R\right)\hspace{-.6mm}(x,t)\hspace{1mm}\frac{\partial\varphi}{\partial x}(x,t) {\,{\mathrm{d}}}x{\,{\mathrm{d}}}t
+\int_{0}^{T}\int_{N}^{R}S(xu_C^R)(x,t)\varphi(x,t){\,{\mathrm{d}}}x{\,{\mathrm{d}}}t,\end{aligned}$$ for all functions $\varphi$, which are twice continuously differentiable functions on $(N,R)\times(0,T)$, such that $\varphi$ and each of its derivatives up to order $2$ may be continuously extended to $[N,R]\times[0,T)$, and such that for each fixed $x\in[N,R]$, the support of $\varphi$ with respect to $t$ is a compact subset of $[0,T)$. We denote the set of such extended functions by $C_c^2\left([N,R]\times[0,T)\right)$. Finally, we note that the weak formulation was obtained from in the usual manner, recalling the zero boundary conditions established in Lemma \[lastthing\].
\[remark502\] We now make note of a property of the function $\varphi$ and its derivatives, which we will make use of in our analysis. As $\varphi$ has compact support and is identically zero outwith this support, its derivatives, both first and second, must also be zero outwith the support. Now within this compact support, $\varphi$ and its derivatives are continuous and so must be bounded functions.
\[definition502\] In the analysis which follows we make use of the following three approximations to $x$ over the domain $(N,R)$. First we have the left endpoint approximation, defined by $$\xi^h:x \in (N,R)\rightarrow \xi^h(x)= \sum_{i=0}^{I_h-1} \chi_{\Lambda_i}(x)x_{i-1/2}.$$ Secondly we consider the midpoint approximation, defined by $$X^h:x \in (N,R)\rightarrow X^h(x)= \sum_{i=0}^{I_h-1} \chi_{\Lambda_i}(x)x_i,$$ and finally we introduce the right endpoint approximation given by $$\Xi^h:x \in (N,R)\rightarrow \Xi^h(x)= \sum_{i=0}^{I_h-1} \chi_{\Lambda_i}(x)x_{i+1/2}.$$
\[convremark3\] It is a simple exercise to show that the three approximations, introduced above, converge point-wise (uniformly) to $x$ over the domain $(N,R)$ as the mesh parameter $h$ goes to $0$. The reader may find details given in [@bairdphd Lemma 5.2.3].
We are now in a position to proceed with our proof that $u_C^R$ is a weak solution to .
\[definition503\] Let $\varphi\in C_c^2\left([N,R]\times[0,T)\right)$ , then for sufficiently small $\Delta t$, the support of $\varphi$ with respect to $t$ lies within $[0,t_{M-1}]$. We define $\varphi_{i}^{n}$ as an approximation of $\varphi$ on $\Lambda_{i}\times \tau_n$ by $$\varphi_i^n=\frac{1}{\Delta t}\int_{\tau_n}\varphi(x_{i-1/2},t){\,{\mathrm{d}}}t,$$ with $\varphi_{i}^{M-1}=\varphi_{i}^{M}=0$ for admissible $i$ and define $\varphi_{I_h}^{n}=0$ for all $n$.
Rearranging equation , multiplying by $\varphi_{i}^{n}$ and summing over $n=0,\ldots,M-1$ and $i=0,\ldots,I_h-1$, gives us $$\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}\left(x_i\Delta x_{i}\left(u_{C}^{n+1,i}-u_{C}^{n,i}\right)\varphi_{i}^{n}-\Delta t(F_{i+1/2}^n-F_{i-1/2}^n)
\varphi_{i}^{n}+\Delta t\Delta x_{i}S_i^n\varphi_{i}^{n}\right)=0.$$ Rearrangement of the summations and utilising the compact support of $\varphi$ and the zero boundary flux gives us the following equality: $$\begin{aligned}
\label{equation506}
&\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}x_i\Delta x_{i}u_{C}^{n+1,i}\left(\varphi_{i}^{n+1}-\varphi_{i}^{n}\right)
+ \sum_{i=0}^{I_h-1}x_i \Delta x_{i} u_{C}^{0,i}\varphi_{i}^{0}\nonumber\\
&-\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}\Delta t F_{i+1/2}^n(\varphi_{i+1}^{n}-\varphi_{i}^{n})
-\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}\Delta t\Delta x_{i}S_i^n\varphi_{i}^{n}
=0.\end{aligned}$$ The above equality can be seen as the discrete equivalent of the weak formulation . Our approach now involves taking the limit as $h\rightarrow 0$ of and showing that we do indeed obtain with $u_C^R$ as a weak solution. Observing the terms of we see that the integrals are with respect to the measure ${{\mathrm{d}}}x{\,{\mathrm{d}}}t$ whilst we have shown that convergence occurs in the space with weighted measure $x {\,{\mathrm{d}}}x {\,{\mathrm{d}}}t$. At this point we highlight the use of Theorem \[theorem201\] to switch spaces but retain convergence.
\[theorem502\] Under the assumptions outlined in Remark \[conditions\], the function $u_C^R$ obtained as the limit of the sequence $\left\lbrace u_C^h \right\rbrace$, is a weak solution of our equation, satisfying .
Looking initially at the first two terms of , we can express them as follows: $$\begin{aligned}
&\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}x_i\Delta x_{i}u_{C}^{n+1,i}\left(\varphi_{i}^{n+1}-\varphi_{i}^{n}\right)
+ \sum_{i=0}^{I_h-1}x_i \Delta x_{i} u_{C}^{0,i}\varphi_{i}^{0}\nonumber\\
&=\underbrace{\sum_{i=0}^{I_h-1}x_i\Delta x_{i}u_{C}^{M,i}\left(\varphi_{i}^{M}-\varphi_{i}^{M-1}\right)}_{=0}\\
&\hspace{5mm}+\sum_{n=0}^{M-2} \sum_{i=0}^{I_h-1}\int_{\tau_{n+1}} \int_{\Lambda_{i}}X^h(x)u^h_C(x,t)
\frac{\varphi( \xi^h(x),t)-\varphi( \xi^h(x),t-\Delta t)}{\Delta t}{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t\\
&\hspace{5mm}+\sum_{i=0}^{I_h-1}\int_{\Lambda_{i}}X^h(x)u^h_C(x,0)\frac{1}{\Delta t}\int_{0}^{\Delta t}\varphi( \xi^h(x),t){\,{\mathrm{d}}}t {\,{\mathrm{d}}}x\\
&=\int_{0}^{T} \int_{N}^{R} \chi_{[\Delta t,T)}(t) X^h(x)u^h_C(x,t)
\frac{\varphi( \xi^h(x),t)-\varphi( \xi^h(x),t-\Delta t)}{\Delta t}{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t\\
&\hspace{5mm}+\int_{N}^{R}X^h(x)u^h_C(x,0)\frac{1}{\Delta t}\int_{0}^{\Delta t}\varphi( \xi^h(x),t){\,{\mathrm{d}}}t{\,{\mathrm{d}}}x.\end{aligned}$$
Considering the first of the double integrals, let $(x,t)\in(N,R)\times(0,T)$, then in the case that $0<\Delta t \leq t$, Taylor expansions of the $\varphi$ terms about the point $(x,t)$ give us $$\varphi( \xi^h(x),t)=\varphi(x,t)+(\xi^h(x)-x)\frac{\partial \varphi}{\partial x}(x,t)+O(h^2),$$ $$\varphi( \xi^h(x),t-\Delta t)=\varphi(x,t)+(\xi^h(x)-x)\frac{\partial \varphi}{\partial x}(x,t)
+(t-\Delta t-t)\frac{\partial \varphi}{\partial t}(x,t)+O(h^2,h\Delta t,\Delta t^2).$$ Simple cancellations and recalling the condition relating $h$ and $\Delta t$ give us $$\label{equation507}
\chi_{[\Delta t,T)}(t)\frac{\varphi( \xi^h(x),t)-\varphi( \xi^h(x),t-\Delta t)}{\Delta t}=\left\lbrace \begin{array}{cc}&\hspace{-5mm}\dfrac{\Delta t\frac{\partial \varphi}{\partial t}(x,t)+O(\Delta t^2)}{\Delta t}, \hspace{6mm}\Delta t \leq t\\ \vspace{-3mm}\\&\hspace{10.5mm}0,\hspace{21.2mm}\Delta t > t\end{array}\right..
$$ The expression on the left-hand side of can thus be seen to converge pointwise to $\varphi_t$ on $(N,R)\times(0,T)$ as the mesh size goes to 0. Furthermore, since the derivatives of $\varphi$ are bounded as per Remark \[remark502\], we can bound the left-hand side of on $(N,R)\times(0,T)$, with a bound that is uniform w.r.t $h$, as $h\searrow 0$. As noted in Remark \[convremark3\], the functions $X^h(x)$ converge pointwise to $x$ on $(N,R)$ as $h\searrow 0$, and are clearly bounded by $R$ for all values of $x$ and $h$. Noting that $(N,R)\times[0,T) \backslash (N,R)\times(0,T)$ is of measure $0$ with respect to the measure ${\,{\mathrm{d}}}x {\,{\mathrm{d}}}t$, we see that the terms accompanying $u_C^h$ in our double integral satisfy the conditions for $\left\lbrace g_h \right\rbrace$ from Theorem \[theorem203\]. We have shown previously that $u_C^h\rightharpoonup u_C^R$ in $L_{1}\left((N,R)\times[0,T),\,x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}t\right)$ and by Theorem \[theorem201\] this is also the case in $L_{1}\left((N,R)\times[0,T),{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t\right)$ and so an application of Theorem \[theorem203\] gives us $$\begin{aligned}
\label{equation508}
\int_{0}^{T} \int_{N}^{R} \chi_{[\Delta t,T)}(t) & X^h(x)u^h_C(x,t)
\frac{\varphi( \xi^h(x),t)-\varphi( \xi^h(x),t-\Delta t)}{\Delta t} {\,{\mathrm{d}}}x {\,{\mathrm{d}}}t\rightarrow \nonumber\\
&\int_{0}^{T}\int_{N}^{R} xu_C^R(x,t)\frac{\partial\varphi}{\partial t}(x,t) {\,{\mathrm{d}}}x {\,{\mathrm{d}}}t,\end{aligned}$$ as the mesh size parameter $h$ goes to 0. Next, we consider the second term appearing above. Since $\varphi$ is $C_c^2\left([N,R]\times[0,T)\right)$, its derivatives are bounded, allowing us to deduce that $$\label{equation54322}
\frac{1}{\Delta t} \int_{0}^{\Delta t} \varphi(\xi^h(x),t){\,{\mathrm{d}}}t\rightarrow\varphi(x,0)\hspace{2mm}\text{as}\hspace{2mm}h\searrow0,$$ for all $x\in (N,R)$, as we now demonstrate. Consider the following: $$\begin{aligned}
\label{equation54321}
&\left|\frac{1}{\Delta t}\int_{0}^{\Delta t}\varphi( \xi^h(x),t){\,{\mathrm{d}}}t-\varphi(x,0)\right|=
\left|\frac{1}{\Delta t}\int_{0}^{\Delta t}\left(\varphi( \xi^h(x),t)-\varphi(x,0)\right){\,{\mathrm{d}}}t\right|\nonumber\\
&=\left|\frac{1}{\Delta t}\int_{0}^{\Delta t}\left(\varphi( \xi^h(x),t)-\varphi(x,t)+\varphi(x,t)-\varphi(x,0)\right){\,{\mathrm{d}}}t\right|\nonumber\\
&\leq\frac{1}{\Delta t}\int_{0}^{\Delta t}\left|\varphi( \xi^h(x),t)-\varphi(x,t)\right|{\,{\mathrm{d}}}t
+\frac{1}{\Delta t}\int_{0}^{\Delta t}\left|\varphi(x,t)-\varphi(x,0)\right|{\,{\mathrm{d}}}t.\end{aligned}$$ Expressing $\varphi( \xi^h(x),t)$ using a Taylor expansion about $(x,t)\in(N,R)\times(0,T)$, and recalling Remark \[remark502\] about the derivatives of $\varphi$ we get $$\varphi( \xi^h(x),t)=\varphi(x,t)+(\xi^h(x)-x)\frac{\partial\varphi}{\partial x}(x,t)+O(h^2).$$ Hence bounding the derivative $\partial\varphi/\partial x$ and noting that $\left|\xi^h(x)-x\right|\leq h$ gives us $$\left|\varphi( \xi^h(x),t)-\varphi(x,t)\right|\leq C_1h,$$ for some constant $C_1$ independent of $h$ and $\Delta t$. Similarly, expanding $\varphi(x,t)$ about $(x,0)$, where $x\in(N,R)$, produces $$\label{Tseries}
\varphi( x,t)=\varphi(x,0)+\underbrace{(t-0)}_{\leq \Delta t}\frac{\partial\varphi}{\partial t}(x,0_+)+O(h^2),$$ for $t\geq0$. The use of the notation $\tfrac{\partial\varphi}{\partial t}(x,0_+)$ signifies we are considering the right derivative of $\varphi$ with respect to $t$ at $t=0$. The expansion then leads to $$\begin{aligned}
\left|\varphi(x,t)-\varphi(x,0)\right|\leq C_2h,\end{aligned}$$ for some other constant $C_2$, independent of $h$ and $\Delta t$. Returning to we have $$\begin{aligned}
\left|\frac{1}{\Delta t}\int_{0}^{\Delta t}\varphi( \xi^h(x),t){\,{\mathrm{d}}}t-\varphi(x,0)\right|\leq (C_1+C_2)h.\end{aligned}$$ Hence does indeed hold for $x\in (N,R)$, furthermore the convergence is uniform with respect to $x$. Together with the pointwise convergence of $X^h(x)$ to $x$, and from Remark \[convremark2\], the $L_1(N,R)$ strong convergence of $u^h_C(x,0)$ to the restriction of $c_0$ to $(N,R)$, another application of Theorem \[theorem203\] yields $$\label{equation509}
\int_{N}^{R}X^h(x)u^h_C(x,0)\frac{1}{\Delta t}\int_{0}^{\Delta t}\varphi( \xi^h(x),t){\,{\mathrm{d}}}t{\,{\mathrm{d}}}x\rightarrow
\int_{N}^{R}xc_{0}(x)\varphi(x,0){\,{\mathrm{d}}}x.$$
Moving on to the third term of equation , for $t\in\tau_n$ and $x\in\Lambda_i$ we can write the numerical flux as an integral as follows: $$\begin{aligned}
F_{i+1/2}^{n}&=\sum_{j=i+1}^{I_h-1}\sum_{k=0}^{i}x_{k}A_jB_{k,j}u_{C}^{n,j}\Delta x_{k}\Delta x_{j}\\
&=\sum_{j=i+1}^{I_h-1}\sum_{k=0}^{i}\int_{\Lambda_j}\int_{\Lambda_k}X^h(w)a^h(v)b^h(w|v) u_C^h(v,t){\,{\mathrm{d}}}w{\,{\mathrm{d}}}v\\
&=\int_{x_{i+1/2}}^{R}\int_{N}^{x_{i+1/2}}X^h(w)a^h(v)b^h(w|v) u_C^h(v,t){\,{\mathrm{d}}}w{\,{\mathrm{d}}}v\\
&=\int_{N}^{R}\int_{N}^{R} \chi_{[\Xi^h(x),R]}(v)\chi_{[N,\Xi^h(x)]}(w) X^h(w)a^h(v)b^h(w|v) u_C^h(v,t){\,{\mathrm{d}}}w{\,{\mathrm{d}}}v\\
&=:\mathcal{F}^h(u_C^h)(x,t).\end{aligned}$$ Then the third term of equation is given by $$\begin{aligned}
&\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}\Delta t F_{i+1/2}^n(\varphi_{i+1}^{n}-\varphi_{i}^{n})\\
&=\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}F_{i+1/2}^n\int_{\tau_n}\varphi(x_{i+1/2},t)-\varphi(x_{i-1/2},t){\,{\mathrm{d}}}t\\
&=\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}\int_{\tau_n}\int_{\Lambda_i}F_{i+1/2}^n\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}x{\,{\mathrm{d}}}t\\
&=\int_0^T \int_{N}^R\mathcal{F}^h(u_C^h)(x,t)\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}x {\,{\mathrm{d}}}t.\end{aligned}$$ Expressed in full this gives us the following, after a switch in the order of integration: $$\begin{aligned}
\label{equation510}
&\int_0^T\int_{N}^R\int_{N}^{R}\int_{N}^{R} \chi_{[\Xi^h(x),R]}(v)\chi_{[N,\Xi^h(x)]}(w) X^h(w)a^h(v)b^h(w|v) u_C^h(v,t){\,{\mathrm{d}}}w {\,{\mathrm{d}}}v\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}x{\,{\mathrm{d}}}t\nonumber\\
&=\int_{N}^R\int_{N}^{R}\chi_{[N,\Xi^h(x)]}(w) X^h(w)\left(\int_0^T\int_{N}^{R} \chi_{[\Xi^h(x),R]}(v)a^h(v)b^h(w|v) u_C^h(v,t)\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}v{\,{\mathrm{d}}}t\right){\,{\mathrm{d}}}w{\,{\mathrm{d}}}x.\end{aligned}$$ Due to the boundedness of the partial derivative $\varphi_x$ and the $L_{\infty,loc}$ property of the functions $a$ and $b$, for almost all fixed $(x,w) \in (N,R) \times (N,R)$, the product $\chi_{[\Xi^h(x),R]}(v)a^h(v)b^h(w|v)\frac{\partial\varphi }{\partial x}(x,t)$ is a bounded (uniformly w.r.t. $h$) function of $v$ and $t$. Also, as a consequence of Remarks \[convremark1\] and \[convremark3\], it converges pointwise almost everywhere (w.r.t the measure ${\,{\mathrm{d}}}v{\,{\mathrm{d}}}t$) on $(v,t) \in (N,R) \times [0,T)$ to $\chi_{[x,R]}(v)a(v)b(w|v)\frac{\partial\varphi }{\partial x}(x,t)$ as $h\rightarrow 0$. Since $u_C^h\rightharpoonup u_C^R$ in $L_1((N,R)\times[0,T),{\,{\mathrm{d}}}x{\,{\mathrm{d}}}t)$, an application of Theorem \[theorem203\] gives us $$\begin{aligned}
&\int_0^T\int_{N}^{R} \chi_{[\Xi^h(x),R]}(v)a^h(v)b^h(w|v) u_C^h(v,t)\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}v{\,{\mathrm{d}}}t\\
&\hspace{10mm}\rightarrow \int_0^T\int_{N}^{R} \chi_{[x,R]}(v)a(v)b(w|v) u_C^R(v,t)\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}v{\,{\mathrm{d}}}t.\end{aligned}$$ Using the local boundedness of $a$ and $b$, along with the boundedness of the partial derivative $\frac{\partial\varphi }{\partial x}$, and the boundedness of the sequence $\left\lbrace u_C^h \right\rbrace$ in $L_1((N,R)\times[0,T),{\,{\mathrm{d}}}x {\,{\mathrm{d}}}t)$, the left–hand side above can be bounded by a constant. It is easily seen that $\chi_{[N,\Xi^h(x)]}(w) X^h(w)$ converges pointwise to $\chi_{[N,x]}(w)w$ as $h \rightarrow 0$ and can be bounded by $R$ over our domain of interest. Therefore applying the Lebesgue dominated convergence theorem [@lieb01 Theorem 1.8], we get that converges to $$\begin{aligned}
\label{equation511}
&\int_{N}^R\int_{N}^{R}\chi_{[N,x]}(w)w\left(\int_0^T\int_{N}^{R} \chi_{[x,R]}(v)a(v)b(w|v)
u_C^R(v,t)\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}v{\,{\mathrm{d}}}t\right){\,{\mathrm{d}}}w {\,{\mathrm{d}}}x\nonumber\\
&=\int_0^T\int_{N}^R\left(\int_{x}^{R}\int_{N}^{x}w a(v)b(w|v)u_C^R(v,t){\,{\mathrm{d}}}w{\,{\mathrm{d}}}v \right)
\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}x{\,{\mathrm{d}}}t\nonumber\\
&=\int_0^T\int_{N}^R\mathcal{F}^R\left(xu_{C}^R\right)(x,t)\frac{\partial\varphi }{\partial x}(x,t){\,{\mathrm{d}}}x{\,{\mathrm{d}}}t.\end{aligned}$$ Therefore, in the limit as $h\rightarrow 0$, the third term of coincides with the third term of .
Now the fourth term from equation is given fully by $$\begin{aligned}
&\sum_{n=0}^{M-1} \sum_{i=0}^{I_h-1}A_i\left(\sum_{j=1}^{N}j\tilde{B}_{j,i}\right)u_{C}^{n,i}\varphi_{i}^{n}\Delta x_{i} \Delta t\\
&=\sum_{n=0}^{M-1}\sum_{i=0}^{I_h-1}\int_{\tau_n}\int_{\Lambda_i}a^h(v)\left(\sum_{j=1}^{N}jb_j^h(v)\right)u_c^h(v,t)
\varphi(\xi^h(v),t){\,{\mathrm{d}}}v{\,{\mathrm{d}}}t\\
&=\int_{0}^T\int_{N}^Ra^h(v)\left( \sum_{j=1}^{N}jb_j^h(v)\right)u_c^h(v,t)\varphi(\xi^h(v),t){\,{\mathrm{d}}}v{\,{\mathrm{d}}}t.\end{aligned}$$ The pointwise convergence of $a^h$, $b_j^h$ and $\xi^h$ along with the continuity of $\varphi$ means that $$a^h(v)\left(\sum_{j=1}^{N}jb_j^h(v)\right)\varphi(\xi^h(v),t)\rightarrow a(v)\left( \sum_{j=1}^{N}jb_j(v)\right)\varphi(v,t),$$ for all $t\in[0,T)$ and almost all $v\in(R,N)$ as $h\rightarrow0$. Since $a$ and $b_i$ are in $L_{\infty,loc}([N,\infty))$ and $\varphi$ is $C^2$ on $[N,R]\times [0,T)$ with compact support (hence is a bounded function), the expressions on either side belong to $L_{\infty}((N,R)\times[0,T))$, with the left-hand side being uniformly bounded w.r.t. $h$. Hence, with $u_c^h\rightharpoonup u_C^R$ in $L_1((N,R)\times[0,T),{\,{\mathrm{d}}}v{\,{\mathrm{d}}}t)$, applying Theorem \[theorem203\], yields $$\begin{aligned}
\label{equation512}
&\int_{0}^T \int_{N}^R a^h(v) \left( \sum_{j=1}^{N}jb_j^h(v)\right) u_c^h(v,t) \varphi(\xi^h(v),t) {\,{\mathrm{d}}}v{\,{\mathrm{d}}}t\nonumber\\
&\rightarrow\int_{0}^T\int_{N}^Ra(v)\left( \sum_{j=1}^{N}jb_j(v)\right)u_c^R(v,t)\varphi(v,t){\,{\mathrm{d}}}v{\,{\mathrm{d}}}t\nonumber\\
&=\int_{0}^{T}\int_{N}^{R}S(vu_C^R)(v,t) \varphi(v,t){\,{\mathrm{d}}}v{\,{\mathrm{d}}}t.\end{aligned}$$ Taken together , , and show that $u_C^R$ satisfies for all $\varphi\in C_c^2([N,R]\times[0,T))$, and hence $u_C^R$ is a weak solution, as set out in Definition \[definition501\].
Discrete Fragmentation Regime: Convergence
------------------------------------------
Now let us consider the discrete regime approximations. This is treated by a similar approach to the one we adopted for the continuous regime equation, but as a first step we establish a bound on the values $u_{D}^{n,i}$.
\[lemma503\]There exists a constant $C$, independent of $h$ and $R$, such that for all values of $n$ and $i$ we have $$0\leq u_{D}^{n,i}\leq C.$$
The nonnegativity of $u_{D}^{n,i}$ follows from Lemma \[lemma401\]. We shall therefore concentrate on the upper bound. From Lemma \[lemma402\] we have, for all admissible $n$, that the following holds: $$\begin{aligned}
\sum_{i=0}^{I_{h-1}}x_i \Delta x_i u_{C}^{n,i} + \sum_{i=1}^N i u_{D}^{n,i}&=\sum_{i=0}^{I_{h-1}}x_i \Delta x_i u_{C}^{0,i} + \sum_{i=1}^N i u_{D}^{0,i}\\
&=\sum_{i=0}^{I_{h-1}}x_i \int_{\Lambda_i}c_0(x){\,{\mathrm{d}}}x + \sum_{i=1}^N i d_{0i}\\
&\leq\frac{3}{2}\sum_{i=0}^{I_{h-1}}\int_{\Lambda_i}c_0(x)\,x{\,{\mathrm{d}}}x + \sum_{i=1}^N i d_{0i}\\
&\leq\frac{3}{2}\int_{N}^\infty c_0(x)\,x{\,{\mathrm{d}}}x + \sum_{i=1}^N i d_{0i}=C<\infty.\end{aligned}$$ Therefore, for all $n$ and $i$ we have that $$u_{D}^{n,i}\leq C,$$ where the constant $C$ is independent of the mesh parameter $h$ and the truncation parameter $R$.
\[theorem503\]For each $i=1,\ldots,N$, the family $\left\lbrace u^h_{Di}\right \rbrace$ forms a sequentially weakly compact set in $L_{1}\left([0,T)\right)$, hence must have a weakly convergent subsequence.
The bound obtained in Lemma \[lemma503\] allows us to easily establish equiboundedness and equiintegrability in $L_{1}\left([0,T)\right)$ for each of the families $\left\lbrace u^h_{Di}\right \rbrace$ as follows:
$$\label{equation513}
\left\|u_{Di}^h(\cdot)\right\|_{L_1\left([0,T)\right)}=\sum_{n=0}^{M-1}u^{n,i}_D\Delta t \leq \sum_{n=0}^{M-1}C\Delta t=CT.$$
Now let $\Phi$ be any function of the nature described in Theorem \[theorem204\]. Since $\Phi$ is increasing, the established bound for $u^{n,i}_D$ allows us to deduce that $$\int_0^T\Phi(u_{Di}^h(t)){\,{\mathrm{d}}}t=\sum_{n=0}^{M-1}\Phi(u^{n,i}_D)\Delta t \leq \sum_{n=0}^{M-1}\Phi(C)\Delta t=\Phi(C)T.$$
Hence each of the families $\left\lbrace u^h_{Di} \right \rbrace$ is equiintegrable. By the Dunford–Pettis theorem (Theorem \[theorem205\]), each of the families form a weakly sequentially compact set in $L_{1}\left([0,T)\right)$. As such, they all contain some weakly convergent subsequence.
We note that what we seek is a collection of values $\left\lbrace h^j \right \rbrace$, such that all of the sequences $\left\lbrace u^{h^j}_{Di} \right \rbrace$, for $i=1,\ldots,N$, converge weakly, as $j\rightarrow \infty$ and $h_j\rightarrow 0$. We achieve this by means of a diagonal argument, which we now outline. Knowing that the family $\left\lbrace u^h_{D1} \right \rbrace$ has a weakly convergent subsequence, let us denote the corresponding sequence of $h$-values by $\left\lbrace h^j \right \rbrace_{j=1}^\infty$ and consider the family $\left\lbrace u^{h^j}_{D2} \right \rbrace_{j=1}^\infty$. As this set satisfies the equiboundedness and equiintegrability conditions of the Dunford–Pettis theorem, it too must have a weakly convergent subsequence. Extracting this subsequence and denoting the corresponding $h$-values by $\left\lbrace h^{j_n} \right \rbrace_{n=1}^\infty$, we then have both $\left\lbrace u^{h^{j_n}}_{D1} \right \rbrace$ and $\left\lbrace u^{h^{j_n}}_{D2} \right \rbrace$ converging (weakly) as $n\rightarrow \infty$ and $j_n\rightarrow \infty$ . We can continue this process, working through each of the families $\left\lbrace u^h_{Di} \right \rbrace$, until we have a set of common $h$-values, $\left\lbrace h_j^{\prime } \right \rbrace_{j=1}^\infty$, for which all the subsequences $\left\lbrace u^{h_j^{\prime }}_{Di} \right \rbrace_{j=1}^\infty$ are (weakly) convergent as $j\rightarrow \infty$ and $h_j^{\prime }\rightarrow 0$.
From now on these convergent subsequences are considered implicitly and we use $\left\lbrace u^h_{Di} \right \rbrace$ to denote said subsequences, unless otherwise stated. Let us denote the weak limit of $\left\lbrace u^h_{Di}\right \rbrace$ by $u_{Di}^R$ (note the upper case superscript notation for the limit).
Discrete Fragmentation Regime: Weak Solution
--------------------------------------------
Having established the convergence of our sequence of approximations we now aim to determine whether the limit produced provides a solution to the equation and if so in what sense. As such, following on from Definition \[definition501\], we introduce
\[weakdisc\] We say that the function $u_{Di}^R$ is a weak solution of equation if it satisfies $$\begin{aligned}
\label{equation515}
&\int_{0}^T u_{Di}^R(t)\frac{{{\mathrm{d}}}\phi}{{{\mathrm{d}}}t}(t){\,{\mathrm{d}}}t + d_{0_i}\phi(0) - \int_{0}^T a_iu_{Di}^R(t) \phi(t) {\,{\mathrm{d}}}t\nonumber\\
&+ \int_{0}^T \sum_{j=i+1}^{N}a_jb_{i,j}u_{Dj}^R(t) \phi(t) {\,{\mathrm{d}}}t + \int_{0}^T \int_{N}^{R} a(y)b_i(y)u_{C}^R(y,t)\phi(t){\,{\mathrm{d}}}y {\,{\mathrm{d}}}t=0\end{aligned}$$ for any $\phi \in C_c^2([0,T))$, where $C_c^2([0,T))$ is defined in an analogous fashion to $C_c^2\left([N,R]\times[0,T)\right)$ from Definition \[definition501\].
\[theorem504\]The functions $u_{Di}^R$ obtained as weak limits of the sequences $\left\lbrace u^h_{Di} \right \rbrace$ are indeed weak solutions of , satisfying equation for any $\phi \in C_c^2([0,T))$.
For such a function $\phi$, let us denote its approximation over $\tau_n$ by $\phi^n$, which is defined as $$\phi^n=\frac{1}{\Delta t}\int_{\tau_n}\phi(t){\,{\mathrm{d}}}t\hspace{2mm}\text{for}\hspace{1.5mm}n=0,\ldots,M-1,$$ and $\phi^M=0$. Multiplying by $\phi^n$ and summing over $n$ from $0$ to $M-1$, gives us the following equality: $$\begin{aligned}
&\sum_{n=0}^{M-1}\left(u_{D}^{n+1,i}-u_{D}^{n,i}\right)\phi^n= -\sum_{n=0}^{M-1}a_i u_{D}^{n,i}\phi^n \Delta t \\
&+\sum_{n=0}^{M-1} \sum_{j=i+1}^{N} a_j b_{i,j} u_{D}^{n,j} \phi^n \Delta t+ \sum_{n=0}^{M-1} \sum_{j=0}^{I_h-1} A_j \tilde{B}_{i,j} u_{C}^{n,j}\phi^n
\Delta x_{j}\Delta t.\end{aligned}$$ Since $\phi$ is compactly supported, for sufficiently small $\Delta t$ we have $\phi^{M-1}=0$; then, further manipulation of the first term yields $$\begin{aligned}
\label{equation516}
&\sum_{n=0}^{M-1} u_{D}^{n+1,i} \left( \phi^{n+1} -\phi^n \right)+ u_{D}^{0,i}\phi^{0} - \sum_{n=0}^{M-1} a_i u_{D}^{n,i} \phi^n \Delta t\nonumber \\
&+\sum_{n=0}^{M-1}\sum_{j=i+1}^{N}a_jb_{i,j}u_{D}^{n,j}\phi^n\Delta t+\sum_{n=0}^{M-1}\sum_{j=0}^{I_h-1}A_j\tilde{B}_{i,j}u_{C}^{n,j}\phi^n
\Delta x_j\Delta t=0.\end{aligned}$$ Looking more closely at the first term above we can rewrite it as $$\begin{aligned}
\sum_{n=0}^{M-1} u_{D}^{n+1,i} \left( \phi^{n+1} -\phi^n \right)
&=\underbrace{u_{D}^{M,i} \left( \phi^{M} -\phi^{M-1} \right)}_{=0}
+\sum_{n=0}^{M-2} \int_{\tau_{n+1}} u_{Di}^h(t) \frac{\phi(t)-\phi(t-\Delta t)}{\Delta t} {\,{\mathrm{d}}}t\\
&=\int_{0}^T \chi_{[\Delta t,T)}(t) u_{Di}^h(t) \frac{\phi(t)-\phi(t-\Delta t)}{\Delta t} {\,{\mathrm{d}}}t.\end{aligned}$$ Assuming that $t\in(0,T)$ and $\Delta t\leq t$, then a Taylor series expansion of $\phi(t-\Delta t)$ about $t$ gives $$\phi(t-\Delta t)=\phi(t)-\Delta t \frac{{{\mathrm{d}}}\phi}{{{\mathrm{d}}}t}(t)+O(\Delta t^2).$$ Therefore, we have $$\label{equation51515}
\chi_{[\Delta t,T)}(t)
\frac{\phi(t)-\phi(t-\Delta t)}{\Delta t}=\left\lbrace \begin{array}{cc}&\hspace{-5mm}\dfrac{\Delta t\frac{d \phi}{dl t}(x,t)+O(\Delta t^2)}{\Delta t}, \hspace{6mm}\Delta t \leq t\\ \vspace{-3mm}\\&\hspace{10.5mm}0,\hspace{21.2mm}\Delta t > t\end{array}\right..$$ As such, the left-hand side of can be seen to converge pointwise to $\phi_t$ on $(0,T)$, as $h$, and by condition , $\Delta t$ goes to $0$. By an analogous argument to that used for $\varphi$ and its derivatives, $\phi$ and its derivative $\phi_t$ must be bounded, therefore we can bound the left-hand side above, with the bound being uniform w.r.t $h$. Then, as ${u_D^h}_i\rightharpoonup {u_D^R}_i$ in $L_1\left([0,T)\right)$, applying Theorem \[theorem203\], as before, gives us $$\label{equation517}
\int_{0}^T \chi_{[\Delta t,T)}(t) u_{Di}^h(t)\frac{\phi(t)-\phi(t-\Delta t)}{\Delta t} {\,{\mathrm{d}}}t\rightarrow \int_{0}^T u_{Di}^R(t) \frac{{{\mathrm{d}}}\phi}{{{\mathrm{d}}}t}(t){\,{\mathrm{d}}}t.$$ By definition, $u_{D i}^{n}={d_0}_i$, and since $\phi$ is $C^2$ with compact support, its derivative must be bounded, from which we deduce that $$\phi^0=\frac{1}{\Delta t}\int_0^{\Delta t}\phi(t){\,{\mathrm{d}}}t\rightarrow\phi(0),$$ as $h$ goes to 0. Therefore $$\label{equation518}
u_{D}^{0 ,i}\phi^0\rightarrow{d_0}_i\phi(0)\hspace{1.5mm}\text{as}\hspace{1.5mm} h \rightarrow 0.$$ By defining $b_{i,i}$ to be $-1$, we can combine the third and fourth terms of , writing them as $$\begin{aligned}
\sum_{n=0}^{M-1} \sum_{j=i}^{N} a_j b_{i,j} u_{D}^{n,j} \phi^n \Delta t&=\sum_{n=0}^{M-1} \sum_{j=i}^{N} \int_{\tau_n}a_j b_{i,j} u_{Dj}^h(t) \phi(t)
{\,{\mathrm{d}}}t\\
&=\int_0^T\sum_{j=i}^{N} a_j b_{i,j} u_{Dj}^h(t) \phi(t) {\,{\mathrm{d}}}t,\end{aligned}$$ and since $u_{Dj}^h\rightharpoonup u_{Dj}^R$ in $L_1\left([0,T)\right)$, for each $j$, we have $$\begin{aligned}
\label{equation519}
&\int_0^T\sum_{j=i}^{N} a_j b_{i,j}u_{Dj}^h(t) \phi(t) {\,{\mathrm{d}}}t \rightarrow \int_0^T\sum_{j=i}^{N} a_j b_{i,j} u_{Dj}^R(t) \phi(t) {\,{\mathrm{d}}}t\nonumber\\
&=-\int_{0}^T a_iu_{Di}^R(t) \phi(t) {\,{\mathrm{d}}}t+ \int_{0}^T \sum_{j=i+1}^{N}a_jb_{i,j}u_{Dj}^R(t) \phi(t) {\,{\mathrm{d}}}t,\end{aligned}$$ giving us the third and fourth terms of our weak formulation . Rewriting the final term of our discrete relation, we get $$\begin{aligned}
\sum_{n=0}^{M-1} \sum_{j=0}^{I_h-1} A_j \tilde{B}_{i,j} u_{C}^{n,j}\phi^n\Delta x_j \Delta t&=\sum_{n=0}^{M-1} \sum_{j=0}^{I_h-1} \int_{\tau_n}
\int_{\Lambda_j} a^h(y)b_i^h(y)\phi(t) u_C^h(y,t){\,{\mathrm{d}}}y {\,{\mathrm{d}}}t\\
&=\int_{0}^T\int_{N}^R a^h(y)b_i^h(y)\phi(t) u_C^h(y,t){\,{\mathrm{d}}}y {\,{\mathrm{d}}}t.\end{aligned}$$ From Remark \[convremark1\] we have $a^h(y)$ and $b_i^h(y)$ converging pointwise to $a(y)$ and $b_i(y)$ respectively, and along with $\phi$ are bounded (uniformly with respect to $h$), a final application of Theorem \[theorem203\] allows us to deduce that $$\label{equation520}
\int_{0}^T\int_{N}^R a^h(y)b_i^h(y)\phi(t) u_C^h(y,t){\,{\mathrm{d}}}y {\,{\mathrm{d}}}t\rightarrow\int_{0}^T \int_{N}^{R} a(y)b_i(y)u_{C}^R(y,t)\phi(t){\,{\mathrm{d}}}y {\,{\mathrm{d}}}t,$$ as the mesh size parameter $h\rightarrow0$. Taking the results , , and , we see that by letting $h\rightarrow0$ in we obtain the weak formulation , hence $u_{Di}^R$ is indeed a weak solution of .
In this section we established the weak convergence of a subsequence of our sequence of approximate solutions as the mesh parameter was decreased to zero. The limits were shown to provide a set of weak solutions to the truncated equations and . However, there are a number of questions which remain unanswered which we seek to address in the following section.
Uniqueness and Differentiability of Solutions
==============================================
In the previous section, we formed approximate solutions to a truncated version of our system. A subsequence of these approximations was shown to converge to a weak solution to our problem, as the underlying mesh was refined. This convergence of subsequences, rather than the full sequence, raises the possibility of nonunique solutions, with each convergent subsequence possibly offering a different solution. In this section we seek to address this, showing that any limits must coincide, providing a unique solution. Further, we would like to establish whether this solution may in fact display extra regularity, as we might expect from the results in [@baird18].
Continuous Regime
-----------------
Returning to equation , we introduce the space $X_C^R=L_{1}\left((N,R),x{\,{\mathrm{d}}}x\right)$ with the aim of recasting the equation as an abstract Cauchy problem, as was carried out for in [@baird18]. Motivated by the terms appearing on the right-hand side, of we define the following linear operators $A_C^R$ and $B_C^R$ on the space $X_C^R$: $$(A_C^Rf)(x)=-a(x)f(x)\hspace{2mm} \text{and}\hspace{2mm} (B_C^Rf)(x)=\int_{x}^{R}a(y)b(x|y)f(y){\,{\mathrm{d}}}y\hspace{2mm} \text{for} \hspace{2mm}N<x<R.$$ with the respective domains $$D(A_C^R)=\left\lbrace f\in X_C^R:A_C^Rf \in X_C^R\right\rbrace \hspace{6mm}D(B_C^R)=\left\lbrace f\in X_C^R:B_C^Rf \in X_C^R\right\rbrace,$$
Assuming that the functions $a$ and $b$ retain the properties imposed in Remark \[conditions\], in particular $a\in L_{\infty,loc}\left([N,\infty)\right)$ and $b\in L_{\infty,loc}\left([N,\infty)\times [N,\infty)\right)$, then, the following property holds for the operators $A_C^R$ and $B_C^R$.
\[lemma504\]The operators $A_C^R$ and $B_C^R$ are bounded linear operators on the space $X_C^R$, with $\left\|B_C^Rf\right\|_{X_C^R}\leq\left\|A_C^Rf\right\|_{X_C^R}$ for all $f\in X_C^R$.
We will first consider the operator $A_C^R$. Let $f\in X_C^R$; then we have $$\begin{aligned}
\left\|A_C^Rf\right\|_{X_C^R}&=\int_N^R \left|a(x) f(x)\right|\,x{\,{\mathrm{d}}}x\\
&=\int_N^Ra(x)\left|f(x)\right|\,x{\,{\mathrm{d}}}x.\\
&\leq \alpha(R)\int_N^R\left|f(x)\right|\,x{\,{\mathrm{d}}}x= \alpha(R)\left\|f\right\|_{X_C^R}.\end{aligned}$$ Therefore $A_C^R$ is a bounded operator on the space $X_C^R$. The boundedness of $B_C^R$ in $X_C^R$ can be seen as follows. Let $f\in X_C^R$; then $$\begin{aligned}
\left\|B_C^Rf\right\|_{X_C^R}&=\int_{N}^{R}\left|\int_{x}^{R}a(y)b(x|y)f(y){\,{\mathrm{d}}}y\right|x{\,{\mathrm{d}}}x\nonumber\\
&\leq\int_{N}^{R}\left(\int_{x}^{R}a(y)b(x|y)\left|f(y)\right|{\,{\mathrm{d}}}y\right)x{\,{\mathrm{d}}}x\nonumber\\ &=\int_{N}^{R}a(y)\left|f(y)\right|\left(\int_{N}^{y}xb(x|y){\,{\mathrm{d}}}x\right){\,{\mathrm{d}}}y\\
&\leq \int_{N}^{R}a(y)\left|f(y)\right|y{\,{\mathrm{d}}}y=\left\|A_C^Rf\right\|_{X_C^R}\leq \alpha(R)\left\|f\right\|_{X_C^R}.\nonumber\end{aligned}$$ The change in the order of integration can be justified by the nonnegativity of the integrand along with Tonelli’s theorem. The inequality in going from the third to the fourth line comes as a result of the mass conservation condition .
Equation is then recast as the following abstract Cauchy problem in the space $X_C^R$: $$\label{equation522}
\frac{{{\mathrm{d}}}}{{\,{\mathrm{d}}}t}u_C^R(t)=\left(A_C^R+B_C^R\right)[u_C^R(t)],\hspace{3mm}t>0;\hspace{3mm}u_C^R(0)=c_{0}^R=\chi_{(N,R)}(x)c_{0}(x).$$ Here $u_C^R$ denotes an $X_C^R$-valued function rather than the scalar-valued function of two variables from the previous section. However due to the relationship between the spaces $L_1\left(I,L_1(\Omega,{\,{\mathrm{d}}}\mu)\right)$ and $L_1\left(\Omega\times I,{\,{\mathrm{d}}}\mu {\,{\mathrm{d}}}t \right)$, we may switch between the two, with each $L_1$-valued solution to providing us with a scalar-valued solution to and vice versa.
\[lemma505\] The operator $\left(A_C^R+B_C^R\right)$ generates a uniformly continuous semigroup of positive contractions on $X_C^R$.
As a bounded linear operator on the space $X_C^R$, the sum $A_C^R+B_C^R$ generates a uniformly continuous semigroup $\left(T_R(t)\right)_{t\geq0}$ on $X_C^R$, [@engel00 Chapter 1, Theorem 3.7 and Chapter 2, Corollary 1.5]. Furthermore, the Kato–Voigt perturbation theorem, [@banasiak06 Corollary 5.17] as applied in [@baird18 Theorem 3.2], is readily utilised in the case $\left(A_C^R+B_C^R,X_C^R\right)$ to give us an ‘extension’ of $\left(A_C^R+B_C^R,X_C^R\right)$ as a generator of a substochastic semigroup. Now as $A_C^R+B_C^R$ is defined and bounded on all of $X_C^R$, this extension must be $A_C^R+B_C^R$ itself and by [@pazy83 Theorem 2.6], the substochastic semigroup generated must be $\left(T_R(t)\right)_{t\geq0}$. Hence $\left(T_R(t)\right)_{t\geq0}$ is a uniformly continuous semigroup of positive contractions.
From standard results concerning strongly continuous semigroups, [@morante98 Theorem 2.40 and Theorem 2.41], the existence of the semigroup $\left(T_R(t)\right)_{t\geq0}$ on $X_C^R$, provides a unique strong solution to equation , given by $u_C^R(t)= T_R(t)c_{0}^R$. Additionally, by [@engel00 Chapter 2, Proposition 6.4] this is also a unique mild solution, satisfying an equation of the form $$\label{equation523}
u_C^R(t)=c_{0}^R+\left(A_C^R+B_C^R\right)\int_0^tu_C^R(s){\,{\mathrm{d}}}s=c_{0}^R+\int_0^t\left(A_C^R+B_C^R\right)u_C^R(s){\,{\mathrm{d}}}s, \hspace{3mm}(t\geq 0).$$ We are able to take the operator $A_C^R+B_C^R$ inside the integral as a consequence of its boundedness by applying [@arendt01 Proposition 1.1.7.]. Hence our equation corresponds with the mild solution form of [@canizo06 Definition 1.10]. The article [@canizo06] also provides a notion of an $X_C^R$-valued weak solution of equations of type , which we outline here for our specific example.
\[definition282\] The function $u_C^R:[0,T)\longrightarrow X_C^R$ is a *weak* solution of , if for all $\phi$ in $L_{\infty}\left((N,R)\right)$ (the dual space of $X_C^R$), we have that $t\rightarrow \langle u_C^R(t),\phi\rangle$ is locally integrable in $(0,T)$ and $$\label{equation282}
\int_0^T\langle u_C^R(s),\phi \rangle \frac{{{\mathrm{d}}}}{{{\mathrm{d}}}s}\psi(s){\,{\mathrm{d}}}s =-\langle f_0,\phi \rangle\psi(0) - \int_0^T \left\langle \left(A_C^R+B_C^R\right)u_C^R(s),\phi \right\rangle \psi(s){\,{\mathrm{d}}}s,$$ for all $\psi \in C^\infty\left([0,T)\right)$ with compact support, where $\langle g,\phi\rangle$ denotes the duality pairing of $g$ and $\phi$.
\[remark284\]If $D\subseteq {X_C^R}'$ is dense in the weak-$\ast$ topology, then it is sufficient to show that holds for all $\phi \in D$ to establish $u_C^R:[0,T)\rightarrow X_C^R$ as a *weak* solution of ; see [@canizo06 Definition 1.9].
The results so far have provided us with the existence of unique strong and mild solutions to . We now show that, for our case, any mild solution satisfying must necessarily be a weak solution as in Definition \[definition282\] and vice-versa, providing the existence and uniqueness of a weak solution.
\[theorem505\]The function $u_C^R:[0,\infty)\longrightarrow X_C^R$ provided by the semigroup $\left(T_R(t)\right)_{t\geq0}$ is the unique weak solution to equation , satisfying Definition \[definition282\] over any time interval $[0,T)$ where $T<\infty$.
From the analysis above we have the semigroup $\left(T_R(t)\right)_{t\geq0}$ providing a unique mild (strong) solution $u_C^R:[0,\infty)\longrightarrow X_C^R$ to equation . The result [@canizo06 Theorem 1.20] tells us that if the right-hand side of is integrable then, given either a weak or mild solution to our evolution equation , the solution can be modified on a set of measure zero to obtain a solution of the other form. Therefore in our case, if the conditions of [@canizo06 Theorem 1.20] hold, the existence of a unique mild solution will provide a unique weak solution (as given in Definition \[definition282\]) on each finite time interval $[0,T)$, with this solution being given by the semigroup $\left(T_R(t)\right)_{t\geq0}$.
Let $0<T<\infty$; and let us consider the following integral: $$\begin{aligned}
&\int_0^T\left\|(A_C^R+B_C^R)u_C^R(s)\right\|_{X_C^R}{\,{\mathrm{d}}}s\\
&=\int_0^T\hspace{-.5mm} \int_N^R \bigg|-a(x)(u_C^R(s))(x)
+\int_{x}^{R}a(y)b(x|y)(u^R_C(s))(y){\,{\mathrm{d}}}y\bigg|\,x{\,{\mathrm{d}}}x {\,{\mathrm{d}}}s\\
&\leq \alpha(R)\hspace{-.8mm}\int_0^T \hspace{-1.5mm}\int_N^R \hspace{-.5mm}\left|(u_C^R(s))(x)\right|\,x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}s
+\alpha(R)\hspace{-.7mm}\int_0^T\hspace{-1.5mm}\int_N^R\hspace{-1.5mm}\int_{x}^{R}b(x|y)\left|(u_C^R(s))(y)\right|{\,{\mathrm{d}}}y\,x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}s\\
&= \hspace{-.5mm}\alpha(R)\hspace{-1mm}\int_0^T\hspace{-1.5mm}\int_N^R \hspace{-1mm}\left|(u_C^R(s))(x)\right|\,x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}s
+\alpha(R) \hspace{-1mm}\int_0^T \hspace{-1.5mm}\int_N^R \left|(u_C^R(s))(y)\right|\int_{N}^{y}\hspace{-.7mm}b(x|y)\,x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}y{\,{\mathrm{d}}}s\\
&\leq 2\alpha(R)\int_0^T \int_N^R \left|(u_C^R(s))(x)\right|\,x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}s= 2\alpha(R)\int_0^T \left\|T_R(s)c_0^R\right\|_{X_C^R}{\,{\mathrm{d}}}s\leq 2\alpha(R)T \left\|c_0^R\right\|_{X_C^R}<\infty.\end{aligned}$$ In going to the final line, the bounding of the inner integral of the second term from the previous line comes from the mass conservation condition , whilst the subsequent norm inequality relies on the fact that the semigroup $\left(T_R(t)\right)_{t\geq0}$ consists of contractions.
By [@diestel77 Chapter 2, Theorem 2 and Theorem 4], the above bound implies that the right-hand side of equation is integrable over the interval $[0,T)$ and so by [@canizo06 Theorem 1.20], any mild or weak solutions must agree (up to sets of measure zero). Therefore equation has a unique weak solution given by $u_C^R(t)=T_R(t)c_0^R$, which is in fact a strong solution.
Let $u_C^R(x,t)$ be a scalar representation of the semigroup solution $(u_C^R(t))(x)=\left(T_R(t)c_0^R\right)(x)$. Then applying Definition \[definition282\] to our example and noting the equivalence of and as detailed fully in [@bairdphd Appendix C], we get that equation is equivalent to $$\begin{aligned}
\label{equation524}
&\int_{0}^{T}\int_{N}^{R} xu_C^R(x,t)\frac{\partial\varphi}{\partial t}(x,t){\,{\mathrm{d}}}x{\,{\mathrm{d}}}t
+\int_{N}^{R}xu_C^R(x,0)(x)\varphi(x,0){\,{\mathrm{d}}}x\nonumber\\
&=\int_{0}^{T}\int_{N}^{R}\mathcal{F}^R\left(xu_C^R(x,t)\right)\hspace{-.6mm}(x,t)\hspace{1mm}\frac{\partial\varphi}{\partial x}(x,t) {\,{\mathrm{d}}}x {\,{\mathrm{d}}}t\nonumber\\
&\hspace{45mm}+\int_{0}^{T}\int_{N}^{R}S(xu_C^R(x,t))(x,t)\varphi(x,t){\,{\mathrm{d}}}x{\,{\mathrm{d}}}t,\end{aligned}$$ for all $\varphi$ of the form $\varphi(x,t)=\phi(x)\psi(t)$ where $\phi\in C^{\infty}_c\left((N,R)\right)$ and $\psi \in C^{\infty}_c\left([0,T)\right)$, due to the weak-$\ast$ density of $C^{\infty}_c\left((N,R)\right)$ in $L_{\infty}\left((N,R)\right)$, via Remark \[remark284\].
Having determined a one-to-one correspondence between scalar-valued weak solutions satisfying and $X_C^R$-valued strong solutions of the abstract Cauchy problem , we are now in a position to establish the uniqueness and differentiability of the weak solutions of the previous section.
\[theorem506\]The weak solution obtained as the limit of the sequence of approximate solutions for the continuous regime in Theorems \[theorem501\] and \[theorem502\] is unique, continuously differentiable with respect to $t$ on any interval $[0,T)$ and satisfies equation directly, except perhaps on a set of measure zero.
It is easily seen that any scalar-valued function $u_C^R(x,t)$ satisfying Definition \[definition501\] will immediately satisfy the equation above. From any such scalar-valued function we may define a function $u_C^R:[0,T)\longrightarrow X_C^R$ via $\left(u_C^R (t) \right)(x):=u_C^R(x,t)$, for almost all $(x,t)\in (N,R)\times[0,T)$. Since the scalar function satisfies equation , the $X_C^R$-valued function must provide a weak solution, as defined in Definition \[definition282\], to the abstract Cauchy problem . By Theorem \[theorem505\], this weak solution must necessarily also be the unique (up to sets of measure zero) strong solution of . Hence for each scalar weak solution $u_C^R(x,t)$ satisfying Definition \[definition501\] there is a unique corresponding $X_C^R$-valued strong solution $u_C^R$ to the associated abstract Cauchy problem, whereby the original function $u_C^R(x,t)$ is one scalar representation of the strong solution. Since the strong solution is unique, as are its scalar representations (up to measure zero), it follows that the weak solution must be unique, up to a set of measure zero. Further, as a representation of a strongly differentiable $X_C^R$-valued function, by [@banasiak06 Theorem 2.40], $u_C^R(x,t)$ is continuously differentiable with respect to $t$, except perhaps on a set of zero measure. Further, by a similar argument as applied at the end of [@banasiak06 Theorem 8.3] or directly as in [@bairdphd Theorem 3.2.7.], $u_C^R(x,t)$ can be seen to directly satisfy equation almost everywhere.
This result greatly strengthens those of the previous section, where before we had only the existence of a (weakly) convergent subsequence and the possibility of the numerical scheme converging to multiple weak solutions. We now know that the limit solution must necessarily be unique, continuously differentiable with respect to $t$ and a solution, in the classical sense, of the truncated fragmentation equation.
Discrete Regime
---------------
Recalling the truncated discrete regime equation , we have for $i=1,\ldots,N$: $$\begin{aligned}
\label{equation550}
\hspace{-80mm}\frac{{{\mathrm{d}}}u_{Di}^R(t)}{{{\mathrm{d}}}t}
&\hspace{-.7mm}=\hspace{-.7mm}-a_iu_{Di}^R(t)+\hspace{-1.9mm}\sum_{j=i+1}^{N}\hspace{-1.2mm}a_jb_{i,j}u_{\small{D}}^R(t)_j+\hspace{-1mm}\int_{N}^{R}a(y)b_i(y)u_{\small{C}}^R(y,t){\,{\mathrm{d}}}y,\hspace{.7mm}\hspace{1.1mm}t>0,\\
u_{D}^R(0)&=d_{0}.\nonumber\end{aligned}$$ With the aim of recasting these equations as an inhomogeneous abstract Cauchy problem as in [@baird18], we introduce the space $X_D=\mathbb{R}^N$, equipped with the weighted norm: $$\left\|v\right\|_{X_D}=\sum_{j=1}^{N}j|v_j|,\hspace{2mm} \text{where} \hspace{2mm}v=(v_1,\ldots,v_N).$$ The equations then become $$\label{equation551}
\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_D^R(t)=(A_D+B_D)[u_D^R(t)]+C_R[u_C^R(t)],\hspace{3mm}t>0;\hspace{3mm}u_D^R(0)=d_{0},$$ where $A_D$ and $B_D$ are defined on $X_D$, by $$(A_Dv)_i=-a_iv_{i}\hspace{2mm}\text{and}\hspace{2mm}(B_Dv)_i=\sum_{j=i+1}^{N}a_jb_{i,j}v_{j},\hspace{2mm}\text{for}\hspace{2mm}i=1,
\ldots,N,$$ $u_C^R$ is the truncated semigroup solution of and where $C_R \hspace{-0.5mm}:\hspace{-0.5mm} D(C_R)\subseteq X_C^R \rightarrow X_D $ is given by $$(C_Rf)_i=\int_{N}^{R}a(y)b_i(y)f(y){\,{\mathrm{d}}}y, \hspace{3mm} D(C_R)=\left\lbrace f\in X_C^R:C_Rf \in X_D\right\rbrace=X_C^R,$$ for $i=1,\ldots,N$. The fact that $D(C_R)=X_C^R$ is a consequence of the $L_{\infty,loc}$ boundedness of $a$ and each $b_i$. Recall that since the space $X_D$ is finite-dimensional, the operators $A_D$ and $B_D$ are bounded and therefore by [@engel00 Chapter 1, Theorem 3.7 and Chapter 2, Corollary 1.5] their sum must generate a uniformly continuous semigroup $\left(T(t)\right)_{t\geq0}$ on $X_D$. We then consider the term $C_R[u_C^R(t)]$ as a perturbation and as an inhomogeneous abstract Cauchy problem. The following lemma establishes the differentiability of this perturbation term, a property that we will require in showing the existence of solutions of equation , along with determining their nature.
\[lemma601\] The term $C_R[u_C^R(t)]$ from is strongly differentiable (in the space $X_D$) with respect to $t$, at almost all points of $[0,T]$. Furthermore its derivative is given by $C_R[\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)]$.
Recalling the $L_{\infty,loc}$ boundedness of $a$ and the bound $b_i(y)\leq y$ for each $i\in \left\lbrace1,2,\ldots,N\right\rbrace$, which is easily derived from , we have that
$$\begin{aligned}
&\left|\frac{\left(C_R[u_C^R(t+h)]\right)_i-\left(C_R[u_C^R(t)]\right)_i}{h}-\left(C_R\left[\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)\right]\right)_i\right|\\
&=\left| \int_{N}^{R} a(y)b_i(y)\left(\frac{\left(u_C^R(t+h)\right)\hspace{-1.5mm}(y)-\left(u_C^R(t)\right)\hspace{-1.5mm}(y)}{h}
-\left[ \frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)\right]\hspace{-1mm}(y)\right) {\,{\mathrm{d}}}y\right|\\
&\leq \alpha(R)\int_{N}^{R}\left|\frac{\left(u_C^R(t+h)\right)(y)-\left(u_C^R(t)\right)(y)}{h}-\left[ \frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}
u_C^R(t)\right]\hspace{-1mm}(y)\right|\,y{\,{\mathrm{d}}}y\\
&=\alpha(R)\left\|\frac{u_C^R(t+h)-u_C^R(t)}{h}-\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)\right\|_{X_C^R},\end{aligned}$$
where $\alpha(R)$ is the essential supremum of $a$ over $[N,R]$. From the differentiability of $u_C^R$, by letting $h \rightarrow 0$ on both sides of the above calculation, we may deduce that $C_R[u_C^R(t)]$ is differentiable (in the space $X_D$) at almost all points of $[0,T]$, with derivative $C_R[\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)]$.
Having shown the differentiability of the perturbation term $C_R[u_C^R(t)]$, we now look at its derivative more closely, showing that it is integrable, belonging to the space $L_1((0,T),X_D)$ and in doing so establish the existence of a unique strong solution to equation .
\[theorem602\]The derivative of $C_R[u_C^R(t)]$ belongs to the space $L_1((0,T),X_D)$. As such equation has a unique strong solution, which is given by $$\label{solutionform}
u_{D}^R(t)=T(t)d_{0}+\int_0^t T(t-s)C_R[u_C^R(s)]{\,{\mathrm{d}}}s.$$
If we take the $X_D$-norm of the derivative $\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}C_R[u_C^R(t)]$ established in the previous lemma and integrate from $0$ to $T$, then we obtain $$\begin{aligned}
\label{equation604}
&\int_0^T \left\|\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}C_R\left[u_C^R(t)\right]\right\|_{X_D}\,\hspace{-2mm}{\,{\mathrm{d}}}t
=\int_0^T \left\|C_R\left[\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)\right]\right\|_{X_D}\hspace{-1mm}{\,{\mathrm{d}}}t\nonumber\\
&=\int_0^T \sum_{i=1}^Ni\left|\int_N^R a(y)b_i(y)\left[\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)\right](y){\,{\mathrm{d}}}y\right|
{\,{\mathrm{d}}}t\nonumber\\
&\leq\alpha(R)N^2\int_0^T\left\lbrace \int_N^R \left|\left[\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)\right](y)\right|\,y{\,{\mathrm{d}}}y
\right\rbrace{\,{\mathrm{d}}}t\nonumber\\
&=\alpha(R)N^2 \int_0^T \left\| \frac{{{\mathrm{d}}}} {{{\mathrm{d}}}t}u_C^R(t) \right\|_{X_C^R}\hspace{-2mm}{\,{\mathrm{d}}}t.\end{aligned}$$ Recalling the abstract Cauchy problem , since the operators $A_C^R$ and $B_C^R$ are bounded and since $u_C^R$ is given by a contraction semigroup, we have $$\begin{aligned}
\left\|\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}u_C^R(t)\right\|_{X_C^R}\leq\left\|A_C^R+B_C^R\right\|\left\|u_C^R(t)\right\|_{X_C^R}\leq M \left\|c_0\right\|_{X_C^R},\end{aligned}$$ where $M$ is a constant such that $\left\|A_C^R+B_C^R\right\|\leq M$. Inserting this into gives us $$\begin{aligned}
\int_0^T\left\|\frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t}C_R[u_C^R(t)]\right\|_{X_D}\,\hspace{-4mm}{\,{\mathrm{d}}}t&\leq \alpha(R)N^2\int_0^T M \left\|c_0\right\|_{X_C^R}{\,{\mathrm{d}}}s\leq \alpha(R)N^2TM \left\|c_0\right\|_{X_C^R}<\infty.\end{aligned}$$ Therefore the derivative of $C_R[u_C^R(t)]$ belongs to the space $L_1((0,T),X_D)$; hence, by [@pazy83 Chapter 4, Corollary 2.2 and Corollary 2.10], the equation has a unique strong solution $u_{D}^R:[0,T)\rightarrow X_D$ given by .
Having established the existence of a unique strong solution to equation given by , this solution must also provide us with a unique mild solution to our equation. Now we consider the possibility of weak solutions, as defined in Definition \[definition282\]. We aim to show that any weak solution of equation must also be a mild solution (permitting changes on sets of measure zero), and hence the weak solution must be unique and differentiable.
\[theorem603\]Given an integrable weak solution $u_{D}^R:[0,T)\rightarrow X_D$ of equation as defined in Definition \[definition282\], then it must also be a strong solution. Therefore any integrable weak solution must be unique up to sets of measure zero and differentiable in $X_D$.
Considering the right-hand side of equation , taking the norm in $X_D$ and integrating from $0$ to $T$ gives us $$\begin{aligned}
&\int_0^T \left\|(A_D+B_D)[u_D^R(t)]+C_R[u_C^R(t)]\right\|_{X_D} {\,{\mathrm{d}}}t\\
&\leq \int_0^T \left\|(A_D+B_D)[u_D^R(t)]\right\|_{X_D} {\,{\mathrm{d}}}t+\int_0^T \left\|C_R[u_C^R(t)]\right\|_{X_D} {\,{\mathrm{d}}}t\\
&= \int_0^T \left\|(A_D+B_D)[u_D^R(t)]\right\|_{X_D} {\,{\mathrm{d}}}t+\int_0^T \sum_{i=0}^Ni\left|\int_N^R a(y)b_i(y)(u_C^R(t))(y){\,{\mathrm{d}}}y
\right|{\,{\mathrm{d}}}t\\
&\leq \int_0^T\left\|A_D+B_D\right\|\left\|u_D^R(t)\right\|_{X_D}{\,{\mathrm{d}}}t
+\alpha(R)N^2\int_0^T \int_N^R \left|(u_C^R(t))(y)\right|\,y{\,{\mathrm{d}}}y
{\,{\mathrm{d}}}t\\
&\leq \left\|A_D+B_D\right\| \int_0^T\left\|u_D^R(t)\right\|_{X_D}{\,{\mathrm{d}}}t+\alpha(R)N^2\int_0^T \left\|u_C^R(t)\right\|_{X_C^R} {\,{\mathrm{d}}}t.\end{aligned}$$ The assumption that $u_D^R(t)$ is integrable and [@diestel77 Chapter 2, Theorem 2 and Theorem 4] allow us to deduce that the first of these integrals must be finite, whilst recalling that $u_C^R(t)$ was given by a contraction semigroup immediately enables us to bound the second integral above. Hence, [@diestel77 Chapter 2, Theorem 2 and Theorem 4], the right-hand side of equation is integrable and therefore [@canizo06 Theorem 1.20] tells us that the weak solution $u_D^R(t)$ (allowing for changes on sets on measure zero) must also be a mild solution. Since, by Theorem \[theorem602\], equation has a unique mild solution which is in fact a strong solution, the weak solution $u_D^R(t)$ we started with must agree with the strong solution (up to sets of measure zero) and therefore is unique and differentiable.
Having established that any integrable weak solution of equation , in the sense of Definition \[definition282\], is also a strong solution, we now set out to prove that the solutions of equation obtained previously as the limit of our numerical scheme, provide us with such a weak solution and in the process establish their uniqueness and differentiability.
\[lemma602\]The weak solutions $u_{Di}^R(t)$ to the equations , obtained from our numerical scheme, when taken as the components of $u_{D}^R:[0,T)\rightarrow X_D$, produce a $u_{D}^R$ which is integrable.
The weak solutions $u_{Di}^R$ to the equations , constructed in the previous section, were obtained as the weak limits in $L_1(0,T)$ of the sequences $\left\lbrace u^h_{Di} \right \rbrace$ as we let $h\searrow 0$. By the weak lower semicontinuity of the norm [@lieb01 Theorem 2.11], and using the bound , we obtain $$\|u_{Di}^R\|_{L_1(0,T)}\leq\liminf_{h\rightarrow 0}\left\|u^h_{Di} \right\|_{L_1(0,T)}\leq CT,$$ where $C$ denotes the constant from . If we take the components of the function $u_{D}^R:[0,T)\rightarrow X_D$ to be given by $u_{Di}^R$ for $i=1,\ldots,N$ then we get $$\begin{aligned}
\int_0^T \|u_{D}^R(t)\|_{X_D}{\,{\mathrm{d}}}t=\int_0^T \sum_{i=1}^Ni\left|u_{Di}^R(t)\right|{\,{\mathrm{d}}}t
\leq N\sum_{i=1}^N \underbrace{\int_0^T \left|u_{Di}^R(t)\right|{\,{\mathrm{d}}}t}_{=\|u_{Di}^R\|_{L_1(0,T)}}
\leq N^2CT<\infty.\end{aligned}$$ Therefore the function $u_{D}^R:[0,T)\rightarrow X_D$ formed by taking $u_{Di}^R$ as its $i^{th}$ component is integrable.
\[theorem604\]The weak solutions $u_{Di}^R:[0,T)\rightarrow \mathbb{R}$ for $i=1,\ldots,N$ of the equations obtained in the previous section agree with the components of the strong solution established in Theorem \[theorem602\] and hence are unique (up to sets of measure zero) and differentiable.
Let us consider our abstract equation with the aim of rewriting it in a weak formulation as in . First let us note that the dual space of $X_D$ is $\mathbb{R}^N$ and the duality pairing $\langle \cdot,\cdot\rangle$ appearing in is given by the standard inner product on $\mathbb{R}^N$. Let $u_{D}^R:[0,T)\rightarrow X_D$ be a weak solution of equation with the components $u_{Di}^R:[0,T)\rightarrow \mathbb{R}$ for $i=1,\ldots,N$. Then, in this case, the function $F:(0,T)\rightarrow X_D$ appearing in is given componentwise by $$F_i(t)=-a_iu_{Di}^R(t)+\hspace{-1.9mm}\sum_{j=i+1}^{N}\hspace{-1.2mm}a_jb_{i,j}u_{Dj}^R(t)
+\hspace{-1mm}\int_{N}^{R}a(y)b_i(y)(u_C^R(t))(y){\,{\mathrm{d}}}y,$$ for $i=1,\ldots,N$. Therefore the weak formulation of equation becomes: $$\begin{aligned}
&\sum_{i=1}^N \phi_i \int_{0}^T \hspace{-1.5mm}u_{Di}^R(t) \frac{{{\mathrm{d}}}}{{{\mathrm{d}}}t} \psi(t) {\,{\mathrm{d}}}t
=-\sum_{i=1}^N \phi_i{d_{0}}_i\psi(0)
+\sum_{i=1}^N \phi_i\int_{0}^T\hspace{-1.5mm} a_iu_{Di}^R(t) \psi(t) {\,{\mathrm{d}}}t\\
&-\sum_{i=1}^N \phi_i\int_{0}^T\hspace{-1.5mm} \sum_{j=i+1}^{N}\hspace{-1.5mm}a_jb_{i,j}u_{Dj}^R(t) \psi(t) {\,{\mathrm{d}}}t
-\sum_{i=1}^N \phi_i\int_{0}^T \hspace{-1mm}\int_{N}^{R}a(y)b_i(y)u_{C}^R(y,t)\psi(t){\,{\mathrm{d}}}y {\,{\mathrm{d}}}t,\end{aligned}$$ for any $\phi=\left(\phi_1,\ldots,\phi_N\right)\in \mathbb{R}^N$ and $\psi \in C_c^\infty\left([0,T)\right)$, where $u_{C}^R(\cdot,t)$ is the (unique) real-valued representation of the $X_C^R$-valued $u_{C}^R:[0,T)\rightarrow X_C^R$.
Comparing this with equation of the previous section, it is easily seen that the $u_{Di}^R$ obtained there provide us with a solution to the above equation and so taking these $u_{Di}^R:[0,T)\rightarrow \mathbb{R},$ $i=1,\ldots,N$, as the components of an $X_D$-valued function $u_{D}^R:[0,T)\rightarrow X_D$ provides us with an integrable weak solution to equation . As any such $u_{D}^R:[0,T)\rightarrow X_D$ must be unique and differentiable in $X_D$, the components $u_{Di}^R:[0,T)\rightarrow \mathbb{R}$ must be unique (up to sets of measure zero) and differentiable in the traditional sense.
In this article we have shown the convergence of our approximate solutions to a weak solution of the truncated problem given by equations and , and the equivalence of this weak solution to the unique strong/classical solution of the truncated problem. However, it is possible using standard arguments along the lines of [@banasiak06 Section 8.3.2] to show these truncated solutions converge, in the sense of the appropriate space, to the unique strong/classical solution of the untruncated problem as given by and , whose existence was established in [@baird18]. For further details of this convergence in the specific case of the mixed discrete-continuous model, the reader is directed to consult [@bairdphd Chapter 6]. Furthermore, for an experimental study of this convergence, the factors influencing it and therefore the selection of a suitable value for the truncation parameter $R$, the reader is directed to [@bairdphd Section 7.5].
Numerical Experiments {#sectionsexperiments}
=====================
To assess the efficacy of our numerical scheme, we tested it on the power law model as set out in [@baird18 Section 7], where the continuous equation was defined by $$a(x)=x^\alpha,\hspace{4mm}\alpha\in\mathbb{R},\hspace{5mm}\text{and}\hspace{5mm}b(x|y)=(\nu+2)\frac{x^\nu}{y^{\nu+1}},\hspace{4mm}-2<\nu\leq0.$$ The discrete equation was specified by the following values for $a_i$ and $b_{i,j}$ $$a_i=\left\{ \begin{aligned}
\hspace{2mm} 0\hspace{4mm} &\textup{for} \hspace{2mm} i=1, \\
\hspace{2mm} i^\alpha\hspace{4mm} &\textup{for} \hspace{2mm} i=2,\ldots,N,
\end{aligned} \right.\hspace{5mm}
b_{i,j}=\frac{2}{j-1}, \hspace{5mm}i=1,\ldots,N-1,\hspace{1.2mm}j=i,\ldots,N,$$ and the continuous to discrete distribution functions $b_i(y)$ were given by $$\begin{aligned}
b_i(y)=\frac{i^{\nu+2}-(i-1)^{\nu+2}}{iy^{\nu+1}}, \hspace{5mm}y>N,\hspace{1.2mm}i=1,\ldots,N.\end{aligned}$$
\
\
\
\
It can be easily verified that conditions and are satisfied by these choices. The parameters $N$ and $R$ were set at $5$ and $15$ respectively, and an initial state assumed, with $$c_0(x)=\left\{ \begin{aligned}
\hspace{2mm} 1\hspace{4mm} &\textup{for} \hspace{2mm} 5<x<15, \\
\hspace{2mm} 0\hspace{4mm} &\textup{for} \hspace{2mm} x\geq 15,
\end{aligned} \right.$$ and $d_0$ being the $N-$vector consisting entirely of $1'$s. The parameters $\alpha$ and $\nu$ were varied, taking all possible combinations of $\alpha\in\left\lbrace0.5,0.1,-0.5,-1,-2\right\rbrace$ and $\nu\in\left\lbrace0,-0.5,-1,-1.5\right\rbrace$, with the final time $T$ selected in each case to allow the system to reach a near equilibrium state.
The approximate solutions generated by the numerical scheme were compared to the exact solutions derived in [@baird18 Section 7.1], with the discrepancy being measured by taking the relative error with respect to the norm on $L_1\left([0,T),X_D\right)\times L_1\left([0,T),X_C^R\right)$. That is, supposing $u^h=(\underline{u}_D^h(t),u_C^h(x,t))$ is our approximation of an exact solution $u=(\underline{u}_D(t),u_C^R(x,t))$, then we measure the error via $$\textrm{Error}(u^h|u)=\frac{\|u^h-u\|}{\|u\|},$$ where the norm $\|\cdot\|$, is given by $$\begin{aligned}
\|u\|&=\int_0^T\sum_{i=1}^N i \left|u_{Di}(t)\right|{\,{\mathrm{d}}}t+\int_0^T\int_N^R\left|u_C^R(x,t)\right|\,x{\,{\mathrm{d}}}x{\,{\mathrm{d}}}t.\end{aligned}$$ For each model configuration, we computed approximate solutions over a sequence of uniform meshes, refining at each step by halving the mesh parameter $h$. The charts in Figure \[figure705\] plot the observed relative error against the mesh parameter $h$, for all possible parameter configurations. From even the briefest examination of the charts it is clear that as the mesh is refined, the relative error of the approximations is reduced. Whilst if we were to examine the gradients of the lines appearing in Figure \[figure705\], then they appear generally to be getting closer to 1, as the mesh is refined. With the gradients between the most refined mesh pairings having a mean value of 1.0301, across all configurations. This would suggest that our numerical scheme has order $\gamma\approx1$, with the error in the approximations being $\mathcal{O}(h)$. The full numerical details of the errors and the associated convergence rates underlying Figure \[figure705\] may be found in [@bairdphd Appendix A].
Conclusions
===========
In this article we introduced a numerical scheme for the approximate solution of a truncated version of a mixed discrete-continuous fragmentation model. The scheme was based upon a finite volume discretisation of the modelling equation for the continuous component.
The resulting numerical approximations were first shown to be nonnegative and to conserve mass, provided the underlying mesh satisfied certain constraints. Following which we established the weak convergence of a subsequence of our approximations, as the mesh size parameter $h$ was decreased to zero. The resulting limits were then shown to provide a weak solution to the truncated model.
By relating the scalar-valued weak formulation of our truncated model to an equivalent weak formulation within a Banach space setting, we were able to establish a one-to-one relationship between any scalar and Banach-space-valued weak solutions. Under suitable constraints, these Banach-space-valued weak solutions were shown to provide the unique strong solution to the associated abstract Cauchy problem, in the process establishing the uniqueness of the original scalar-valued weak solutions. Additionally, as a further consequence of this linkage, the scalar weak solutions were shown to be differentiable classical solutions.
Finally, by conducting a range of experiments with a test model, under varying model parameter choices and mesh refinements, we experimentally established that the error in our numerical solutions was $\mathcal{O}(h)$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the UK Engineering and Physical Sciences Research Council \[EP/J500495/1 03\].
|
---
abstract: 'An exactly solvable large $N$ model is constructed which reduces at low energies to the Standard Model plus the dimension six Higgs-gauge operators $g_1^2 H^\dagger H B_{\mu \nu} B^{\mu \nu}$, $g_2^2 H^\dagger H W^a_{\mu \nu} W^{a\mu \nu}$, $g_1 g_2 H^\dagger \tau^a H W^a_{\mu \nu} B^{\mu \nu}$, and $\epsilon^{abc} W^a_{\mu}{}^\nu W^b_{\nu}{}^\rho W^c_{\rho}{}^\mu$. All other dimension six operators are suppressed by powers of $1/N$. The Higgs-gauge operators lead to deviations from the Standard Model $h \to \gamma\gamma$ and $h \to \gamma Z$ rates. A simple variant of the model can be used to also generate the Higgs-gluon operator $g_3^2 H^\dagger H G^A_{\mu \nu} G^{A\mu \nu}$ which contributes to the Higgs production rate via gluon fusion.'
author:
- 'Aneesh V. Manohar'
bibliography:
- 'paperrefs.bib'
title: 'An Exactly Solvable Model for Dimension Six Higgs Operators and $h \to \gamma \gamma$'
---
A scalar boson with a mass of $\sim 125$GeV has recently been discovered at the LHC, and it is important to study its properties in a model-independent way. The Standard Model provides a good description of the LHC data so far, with no evidence for any new particles with masses below $\sim 1$TeV. A widely used starting point for analysis is to assume that the theory at $125$GeV is the Standard Model including a fundamental scalar doublet, and all new physics effects are characterized by higher dimension operators involving Standard Model fields.
A recent paper [@Grojean:2013kd] considered the impact of dimension-six operators on the Higgs decay rate. The theory considered was the Standard Model plus the dimension six Hamiltonian $$\begin{aligned}
\hsix = -\mathcal{L}^{(6)} &= c_{G} \mathcal{O}_{G}+c_{B} \mathcal{O}_{B} + c_{W} \mathcal{O}_{W} + c_{W\!B} \mathcal{O}_{W\!B}\label{h6}\end{aligned}$$ generated by new physics at some scale $\Lambda$, where $$\begin{aligned}
\mathcal{O}_{G} &= \frac{g_3^2}{2 \Lambda^2} H^\dagger H G_{\mu \nu}^A G^{A \mu \nu},
&\mathcal{O}_{B} &= \frac{g_1^2}{2 \Lambda^2} H^\dagger H B_{\mu \nu} B^{\mu \nu}, \nn
\mathcal{O}_{W} &= \frac{g_2^2}{2 \Lambda^2} H^\dagger H W^a_{\mu \nu} W^{a\mu \nu},
& \mathcal{O}_{W\!B} &= \frac{g_1 g_2}{2 \Lambda^2} H^\dagger \tau^a H W^a_{\mu \nu} B^{\mu \nu} .
\label{ops}\end{aligned}$$ using the notation of Refs. [@Manohar:2006gz; @Manohar:2006ga]. These operators give amplitudes which can interfere constructively or destructively with the Standard Model amplitudes for $gg \to h$, $h \to \gamma \gamma$, $h \to Z\gamma$, etc.. The phenomenology of the operators in Eq. (\[ops\]), including constraints from recent LHC measurements of the Higgs decay rates, and from precision electroweak constraints, was studied in Ref. [@Grojean:2013kd].
This paper constructs an exactly soluble model which generates the dimension six Higgs operators in Eq. (\[ops\]), with arbitrary coefficients consistent with the effective theory power counting. It also provides an explicit realization of the Lagrangian given in the appendix of Ref. [@Jenkins:2013fya].
The set of all dimension-six operators in the Standard Model was classified in Ref. [@Grzadkowski:2010es]. There are 59 independent ones after redundant operators are eliminated by the equations of motion. The operators not involving fermions are the ones listed in Eq. (\[ops\]), their $CP$-odd partners $\widetilde O_{G},\widetilde O_{B},\widetilde O_{W},\widetilde O_{W\!B}$, four pure gauge operators of which two are $CP$ even and two are $CP$ odd, $$\begin{aligned}
O_{G^3}&= f^{ABC} G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu,
&\widetilde O_{G^3}&= f^{ABC} \widetilde G^A_{\mu}{}^\nu G^B_{\nu}{}^\rho G^C_{\rho}{}^\mu, \nn
O_{W^3}&= \epsilon^{abc} W^a_{\mu}{}^\nu W^b_{\nu}{}^\rho W^c_{\rho}{}^\mu,
&\widetilde O_{W^3}&=\epsilon^{abc} \widetilde W^a_{\mu}{}^\nu W^b_{\nu}{}^\rho W^c_{\rho}{}^\mu,\end{aligned}$$ and three more operators involving the Higgs field, $$\begin{aligned}
O_H &=\left(H^\dagger H\right)^3 ,\nn
O_{H \Box {}} &=\left(H^\dagger H\right) \partial^2 \left(H^\dagger H \right), \nn
O_{HD} & =\left(H^\dagger D_\mu H\right)^\dagger \left( H^\dagger D_\mu H\right) \,.
\label{1.7}\end{aligned}$$ The exactly soluble model given here is a large-$N$ version of that constructed in Ref. [@Manohar:2006ga]. It produces the $CP$ conserving operators in Eq. (\[ops\]) with arbitrary order one coefficients, and the operator $O_{W^3}$, and does not generate any other dimension six operators at leading order in $1/N$.
Consider the Standard Model plus an additional scalar field $S^\alpha$ which is a weak $SU(2)$ doublet with hypercharge $Y_S$, and transforms as the $N$ dimensional representation of an internal $SU(N)$ global symmetry. $S_\alpha$ is a two component column vector, and $\alpha=1,\ldots, N$. The theory is given by $$\begin{aligned}
\mathcal{L} &= \mathcal{L}_{\rm SM} + D_\mu S^{\dagger \alpha} D^\mu S_{\alpha} - V,\end{aligned}$$ the usual Standard Model Lagrangian $L_{\rm SM}$, the $S_\alpha$ kinetic energy term, and the potential $$\begin{aligned}
V &= m_S^2 S^{\dagger \alpha} S_{\alpha} +
\frac{\lambda_1}{N} H^{\dagger} H \, S^{\dagger \alpha} S_{\alpha}
+ \frac{\lambda_2}{N} H^{\dagger}\tau^a H \, S^{\dagger \alpha} \tau^a S_{\alpha}\nn
& + \frac{\lambda_3}{N} S^{\dagger \alpha} S_{ \alpha}\, S^{\dagger \beta} S_{\beta} + \frac{\lambda_4}{N} S^{\dagger \alpha}\tau^a S_{ \alpha}\, S^{\dagger \beta}\tau^a S_{\beta}
\label{pot}\end{aligned}$$ where the Standard Model Higgs potential $ \lambda\left(H^{\dagger} H- {v^2}/{2}\right)^2$ is part of $\mathcal{L}_{\rm SM}$. The Lagrangian is the most general renormalizable one consistent with the symmetries. Yukawa couplings of $S$ to the Standard Model fermions are forbidden by $SU(N)$ invariance. We assume that $m_S^2 >0$, so that $SU(N)$ is not spontaneously broken and the scalar mass $m_S$ is larger than the electroweak scale $v \sim 246$ GeV, so that the new interactions can be treated as higher dimension operators at the electroweak scale.
The large-$N$ limit of the theory is taken in the standard way [@Coleman:1980nk]. One treats the theory in a perturbative expansion in the electroweak couplings $g \sim g_{1,2}$, i.e. one first expands in powers of $g$ and then takes the $N \to \infty$ limit. This is the usual method of computing weak decays in QCD using the $1/N$ expansion [@Manohar:1998xv].[^1]
The method of Refs. [@Coleman:1980nk; @Coleman:1974jh; @Abbott:1975bn] is used to solve the theory. Add to $V$ the dimension two auxiliary fields $\Phi$ and $\Psi^a$ which are real $SU(2)$ singlet and triplet fields with $Y=0$, $$\begin{aligned}
V &\to V -\frac{\lambda_3 N}{4} \left( \frac{2}{N} S^{\dagger \alpha}S_{\alpha}+\frac{m_S^2}{\lambda_3} +\frac{\lambda_1}{\lambda_3N} H^\dagger H-\frac{\Phi}{\lambda_3} \right)^2\nn
& -\frac{\lambda_4 N}{4} \left( \frac{2}{N} S^{\dagger \alpha}\tau^a S_{\alpha}+\frac{\lambda_2}{\lambda_4N} H^\dagger \tau^a H
-\frac{\Psi^a}{\lambda_4} \right)^2
\label{npot}\end{aligned}$$ The auxiliarly field equations of motion are $$\begin{aligned}
\Phi &=m_S^2+\frac{\lambda_1}{ N} H^\dagger H ++ \frac{2 \lambda_3}{N} S^{\dagger \alpha}S_{\alpha}\nn
\Psi^a &= \frac{\lambda_2}{ N} H^\dagger \tau^a H + \frac{2 \lambda_4 }{N} S^{\dagger \alpha}\tau^a S_{\alpha} \end{aligned}$$ which can be used to eliminate them and give back the original Lagrangian Eq. (\[pot\]).
In weak coupling, the scalar mass $m_S^2$ is $\vev{\Phi}$. We will therefore use $\vev{\Phi}$ as the scale of new physics $\Lambda$ in Eq. (\[h6\]) and to normalize the operators in Eq. (\[ops\]). Using $(H^\dagger \tau^a H)^2 = (H^\dagger H)^2$, the new potential Eq. (\[npot\]) is $$\begin{aligned}
V &=\frac{N}{2\lambda_3} m_S^2 \Phi-\frac{N}{4 \lambda_3} \Phi^2 - \frac{N}{4 \lambda_4} \Psi^a \Psi^a + \frac{\lambda_1}{2\lambda_3} H^\dagger H \Phi \nn
& + \frac{\lambda_2}{2 \lambda_4} H^\dagger \tau^a H \Psi^a +\Phi S^{\dagger \alpha} S_\alpha + \Psi^a S^{\dagger \alpha} \tau^a S_\alpha -\frac{N m_S^4}{4\lambda_3} \nn
&-\frac{\lambda_1}{2\lambda_3}m_S^2 H^\dagger H - \left( H^\dagger H\right)^2 \left(\frac{\lambda_1^2}{4 \lambda_3 N} +\frac{\lambda_2^2}{4 \lambda_4 N}\right)
\label{new}\end{aligned}$$ The last term, which is subleading in $1/N$, as well as the cosmological constant term, can be dropped.
The field $S_\alpha$ is now integrated out. This can be done exactly in the large-$N$ limit [@Coleman:1980nk] to give an effective action which is an expansion in powers of $H$, $\Phi$ and $\Psi^a$. The Higgs field $H$ does not couple directly to $S$ in Eq. (\[new\]), so the $S$ functional integral generates terms which only depend on $\Phi$ and $\Psi^a$. The effective action has a derivative expansion in inverse powers of $m_S$, which will turn into a derivative expansion in inverse powers of $\vev{\Phi}$. The infrared divergences are controlled by $\vev{\Phi}$, since the theory is in the phase where the $SU(N)$ symmetry is unbroken and the $S$-sector is massive.
At zero derivatives, one gets the Coleman-Weinberg effective potential [@Coleman:1973jx] in the $\overline{\hbox{MS}}$ scheme $$\begin{aligned}
V_{\rm CW} &= \frac{1}{64\pi^2}\text{Tr}\, \left(M^2\right)^2 \left[-\frac32+\log \frac{M^2}{\mu^2}\right]\end{aligned}$$ where $$\begin{aligned}
\left[M^2\right]_{ab} &= \frac{\partial^2 V}{\partial \phi_a \partial \phi_b}\end{aligned}$$ and $\phi_a=\text{Re}\,S_{\alpha,i}, \text{Im}\,S_{\alpha,i}$ are the scalar fields. For the interaction in Eq. (\[new\]), $M^2$ has eigenvalues $\Phi \pm \bm{\Psi}$, $\bm{\Psi}^2=\Psi^a \Psi^a$, each twice, so that $$\begin{aligned}
V &= \frac{N}{2\lambda_3} \left(m_S^2+\frac{\lambda_1}{N}H^\dagger H\right) \Phi-\frac{N}{4 \lambda_3} \Phi^2 - \frac{N}{4 \lambda_4} \Psi^a \Psi^a \nn
&+ \frac{\lambda_2}{2 \lambda_4} H^\dagger \tau^a H \Psi^a \nn
&+ \sum_{\pm} \frac{N}{32 \pi^2} \left(\Phi \pm \bm{\Psi}\right)^2\left[-\frac{3}{2}+\log \frac{ \Phi \pm \bm{\Psi} }{\mu^2}\right]
\label{new.d}\end{aligned}$$ on integrating out the $S_\alpha$ field. This potential is exact in the large $N$ limit. $V$ is quadratic in $\bm{\Psi}$. From the renormalization group (RG) equation for $V$, one finds that $$\begin{aligned}
&\Phi, &&\Psi^a, &&\frac{1}{\lambda_3}\left(m_S^2+\frac{\lambda_1}{N}H^\dagger H\right), &&\frac{\lambda_2}{ \lambda_4} H^\dagger \tau^a H
\label{RGinv}\end{aligned}$$ are $\mu$ independent[^2] and the exact $\beta$-functions are $$\begin{aligned}
\mu \frac{\rd \lambda_3}{\rd \mu} &= \frac{\lambda_3^2}{2\pi^2}, &\mu \frac{\rd \lambda_4}{\rd \mu} &= \frac{\lambda_4^2}{2\pi^2} \,.\end{aligned}$$ Introduce the parameters $\Lambda_{3,4}$ in place of $\lambda_{3,4}(\mu)$, $$\begin{aligned}
\frac{1}{\lambda_3(\mu)} &= \frac{1}{4\pi^2} \log \frac{\Lambda_3^2}{\mu^2},
&\frac{1}{\lambda_4(\mu)} &= \frac{1}{4\pi^2} \log \frac{\Lambda_4^2}{\mu^2},
\label{landau}\end{aligned}$$ which are RG invariant. They are the scales at which $\lambda_{3,4}$ have a Landau pole, and at which the scalar theory breaks down. For consistency, we need $\Lambda_{3,4} > \vev{\Phi}$. New physics has to enter below $\Lambda_{3,4}$ for the theory to be valid to arbitarily high energies. For example $S_\alpha$ could be scalar fields generated by strong dynamics, or new interactions could enter which make the scalar couplings asymptotically free. We also define [@Abbott:1975bn] $$\begin{aligned}
\ks^2 &= \frac{m_S^2}{\lambda_3}
\label{kappa}\end{aligned}$$ which is RG invariant, from Eq. (\[RGinv\]).
The effective action can be computed exactly in the large $N$ limit [@Iliopoulos:1974ur; @Chan:1985ny; @Cheyette:1985ue; @Zuk:1986xz; @Chan:1986jq] in a derivative expansion. The terms which generate operators with dimension $\le 6$ in the Standard Model with coefficients that are non-vanishing in the $N \to \infty$ limit are $$\begin{aligned}
\mathcal{L}_{S} &= \frac{N}{96 \pi^2 \left(\Phi \right)}\left[ \partial_\mu \Phi \partial^\mu \Phi + D_\mu \Psi^a D^\mu \Psi^a \right] \nn
& +\frac{N}{384\pi^2} \left( \log \frac{\Phi}{\mu^2}\right)\biggl[ W_{\mu \nu}^a W^{a \mu \nu}
+4 Y_S^2 B_{\mu \nu}B^{\mu \nu} \biggr] \nn
&+\frac{N}{192\pi^2 \left(\Phi\right)} Y_S \Psi^a W_{\mu \nu}^a B_{\mu \nu}
+ \mathcal{O}(g^3)
\label{l1}\end{aligned}$$ where the gauge fields have been normalized so that the covariant derivative is $D_\mu = \partial_\mu + i W_\mu^a T_a + i B_\mu Y$. $\Phi$ is a gauge singlet, and has an ordinary derivative. $\Psi^a$ is in the $I=1$ representation of $SU(2)_W$ with $Y=0$, and has a covariant derivative $$\begin{aligned}
D_\mu \Psi^a &= \partial_\mu \Psi^a + i \left(T^c\right)_{ab} W^c_\mu \Psi^b, &\left(T^c\right)_{ab} &= -i \epsilon_{cab}\end{aligned}$$
It is instructive to analyze Eq. (\[l1\]) at weak coupling. Let $$\begin{aligned}
\Phi &= m_S^2 + \frac{4 \sqrt{3}\, \pi m_S}{ \sqrt{N}} \sigma, & \Psi^a &= \frac{4 \sqrt{3}\, \pi m_S}{ \sqrt{N}} \Sigma^a\end{aligned}$$ The $\sigma$ and $\Sigma^a$ are dimension one fields with a canonically normalized kinetic energy term, and $$\begin{aligned}
\mathcal{L}_{S} &= \frac12 \partial_\mu \sigma \partial^\mu \sigma + \frac12 D_\mu \Sigma^a D^\mu \Sigma^a
\nn
& +\frac{N}{384\pi^2} \left( \log \frac{m_S^2}{\mu^2}\right) \biggl[ W_{\mu \nu}^a W^{a \mu \nu}
+4 Y_S^2 B_{\mu \nu}B^{\mu \nu} \biggr] \nn
& +\frac{\sqrt{N}}{32 \sqrt{3}\, \pi m_S}\sigma \biggl[ W_{\mu \nu}^a W^{a \mu \nu}
+4 Y_S^2 B_{\mu \nu}B^{\mu \nu} \biggr] \nn
&+\frac{\sqrt{N} Y_S }{16 \sqrt{3}\, \pi m_S} \Sigma^a W_{\mu \nu}^a B_{\mu \nu}
+ \mathcal{O}(g^3)\,.
\label{l2}\end{aligned}$$ The effective action can be computed at weak coupling from the graphs in Fig. \[fig\], which add up to give the gauge invariant structure in Eq. (\[l2\]).
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The one-loop effective action generates kinetic energy terms for $\sigma,\Sigma^a$, given in the first line. The second line gives the threshold correction between the gauge couplings $g_h$ in the theory above $m_S$ and $g_l$ in the theory below $m_S$, $$\begin{aligned}
-\frac{1}{4 g_{l,2}^2(\mu)} &= -\frac{1}{4 g_{h,2}^2(\mu)} + \frac{N}{384\pi^2} \log \frac{m_S^2}{\mu^2}, \nn
-\frac{1}{4 g_{l,1}^2(\mu)} &= -\frac{1}{4 g_{h,1}^2(\mu)} + \frac{N}{384\pi^2} 4Y_S^2\log \frac{m_S^2}{\mu^2},
\label{threshold}\end{aligned}$$ for the $SU(2)$ and $U(1)$ coupling constants. The discontinuity in coupling matches the $S_\alpha$ contribution to the $\beta$-functions, which exists above $m_S$, but not below. The remaining terms are $\sigma W_{\mu \nu}^a W^{a \mu \nu}$, $\sigma B_{\mu \nu}B^{\mu \nu}$ and $\Sigma^a W_{\mu \nu}^a B_{\mu \nu}$ interactions.
The scalar potential Eq. (\[new.d\]) becomes $$\begin{aligned}
V &= -{12 \pi^2 m_S^2}\left( \frac{\sigma^2}{\lambda_3}+\frac{\Sigma^a \Sigma^a}{\lambda_4} \right) \nn
&+\frac{2 \sqrt {3}\, \pi}{\sqrt N} \left( \frac{\lambda_1}{\lambda_3} \sigma H^\dagger H + \frac{\lambda_2}{\lambda_4} \Sigma^a H^\dagger \tau^a H\right)\end{aligned}$$ which has mass terms for $\sigma$ and $\Sigma^a$ (with the wrong sign, but it does not matter, they are auxiliary fields), and couplings of $\sigma$ and $\Sigma^a$ to the Higgs doublet. Integrating out the auxilary fields generates the operators Eq. (\[ops\]) via the graph in Fig. \[fig:ops\]
![\[fig:ops\] Graph generating the $h \to \gamma \gamma$ amplitude.](fd30){width="4cm"}
We can now integrate out $\Phi,\Psi^a$ exactly, by doing the functional integral using the method of steepest descent [@Iliopoulos:1974ur]. The minimum $\vev{\Phi}$ is at $$\begin{aligned}
\Phi \log \frac{\Phi}{\mu^2}-\Phi + \frac{4\pi^2}{\lambda_3} \left(m_S^2+\frac{\lambda_1}{N}H^\dagger H\right)
- \frac{4\pi^2}{\lambda_3} \Phi &=0.\end{aligned}$$ Dropping the $1/N$ term, and using Eq. (\[landau\],\[kappa\]), gives[^3] $$\begin{aligned}
\frac{\vev{\Phi}}{e \Lambda_3^2} \log \frac{\vev{\Phi}}{e \Lambda_3^2}+ \frac{4\pi^2\ks^2}{e \Lambda_3^2}&=0
\label{min}\end{aligned}$$ in terms of RG invariant parameters. Instead of choosing $\ks$ as the Lagrangian parameter, and solving Eq. (\[min\]) for $\vev{\Phi}$, we can use $\vev{\Phi}$ as the free parameter and then determine $\ks$ from Eq. (\[min\]). $\ks \to 0$ as $\vev{\Phi} \to 0$. As $\vev{\Phi}$ increases, so does $\ks$, and $2 \pi \ks \to \Lambda_3$ as $\vev{\Phi} \to \Lambda_3^2$. Eq. (\[min\]) implies that $\ks$ decreases again for $\vev{\Phi} > \Lambda_3$[@Coleman:1974jh; @Abbott:1975bn], but this is above the Landau pole, and the theory is not valid in this regime.
Evaluating the functional integral around $\vev{\Phi}$ gives $$\begin{aligned}
\mathcal{L}_{S} &= \frac{N}{384\pi^2} \biggl[ \left( \log \frac{\vev{\Phi}}{\mu^2}\right) \left(W_{\mu \nu}^a W^{a \mu \nu}
+4 Y_S^2 B_{\mu \nu}B^{\mu \nu} \right) \biggr]\nn
& +\frac{\lambda_1}{2\lambda_3} \left(\lambda_3 \ks^2-\vev{\Phi}\right)H^\dagger H \nn
&+ \frac{\lambda_1}{96 \vev{\Phi} \lambda_3 \log \frac{\Lambda_3^2}{\vev{\Phi}}} H^\dagger H \left(W_{\mu \nu}^a W^{a \mu \nu}
+4 Y_S^2 B_{\mu \nu}B^{\mu \nu} \right) \nn
& +\frac{\lambda_2 Y_S}{48 \vev{\Phi} \lambda_4 \log \frac{\Lambda_4^2} {\vev{\Phi}}} H^\dagger \tau^a H W_{\mu \nu}^a B_{\mu \nu}\nn
&+\frac{N g_2^3}{2880 \pi^2 \vev{\Phi}}\epsilon^{abc} W^a_{\mu}{}^\nu W^b_{\nu}{}^\rho W^c_{\rho}{}^\mu\,.
\label{ans1}\end{aligned}$$ There are also terms at higher order in the derivative expansion which have not been computed here. The first term in Eq. (\[ans1\]) gives the strong-coupling version of the threshold correction Eq. (\[threshold\]). The second term is a shift in the Higgs mass proportional to the $S_\alpha$ mass, and can be absorbed into the $v^2$ term in the Higgs potential in $\mathcal{L}_{\rm SM}$.
Using $\Lambda=\vev{\Phi} \sim m_S^2 > v$ as the scale in Eq. (\[ops\]), we see that we have generated the Standard Model Lagrangian plus the three $CP$-even dimension six operators in Eq. (\[h6\]) with coefficients $$\begin{aligned}
c_W &= \frac{(\lambda_1/\lambda_3)}{48 \log \frac{\Lambda_3^2}{\vev{\Phi}}}\,, \nn
c_B &= \frac{(\lambda_1/\lambda_3) Y_S^2 }{12 \log \frac{\Lambda_3^2}{\vev{\Phi}}}\,,\nn
c_{W\!B} &= \frac{(\lambda_2/\lambda_4)Y_S}{24 \log \frac{\Lambda_4^2}{\vev{\Phi}}}\,,
\label{result}\end{aligned}$$ and the $O_{W^3}$ operator with coefficient $$\begin{aligned}
c_{W^3} &=\frac{N g_2^3}{2880 \pi^2 \vev{\Phi}}\,.\end{aligned}$$ All other dimension six operators are subleading in $1/N$. The ratios $(\lambda_1/\lambda_3)$ and $(\lambda_2/\lambda_4)$ are RG invariant under $S_\alpha$ dynamics with the Standard Model fields treated as background fields, from Eq. (\[RGinv\]).
The linear combinations of coefficients relevant for $h \to \gamma \gamma$ and $h \to \gamma Z$ decays are $$\begin{aligned}
c_{\gamma \gamma} &=c_{W}+c_{B} -c_{W\!B}\,, \nn
c_{\gamma Z} &= c_{W} \cot \theta_W -c_{B} \tan \theta_W -c_{W\!B} \cot 2 \theta_W\,.\end{aligned}$$ The operator $c_{W\!B}$ is constrained by the $S$-parameter [@Grinstein:1991cd; @Hagiwara:1993ck; @Alam:1997nk; @Han:2004az], $$\begin{aligned}
S &= - {8 \pi^2 v^2} \frac{c_{W\!B}}{\Lambda^2}\,.\end{aligned}$$ From Eq. (\[result\]), we see that we can get order unity values of $c_W$, $c_B$ and $c_{W\!B}$. The phenomenology of the Higgs-gauge operators was discussed in detail in Refs. [@Manohar:2006gz; @Manohar:2006ga; @Grojean:2013kd].
There is one relation that follows from Eq. (\[result\]), $$\begin{aligned}
c_B &= 4 Y_S^2 c_W\,,\end{aligned}$$ if we restrict to the model considered here with a single scalar multiplet with hypercharge $Y_S$. One can construct trivial generalizations of the large $N$ model with multiple heavy scalar fields, which can have different hypercharges, and can also be colored. In this case, one can also generate the gluon term $c_G$, as in the octet scalar model of Ref. [@Manohar:2006ga], and the $c_B-c_W$ relation no longer holds.
The large $N$ calculation drops terms of order $1/N$, as well as higher order radiative corrections of order $g_2^2 N/(16\pi^2)$. For finite $N$, the neglected terms are small if $1 \ll N \ll 400$. It would be interesting to explore the full parameter space of scalar couplings and masses where the potential is stable and $m_S$ is below the Landau pole, to determine the allowed region for $c_W$, $c_B$ and $c_{W\!B}$.
The $S_\alpha$ interactions break custodial $SU(2)$ symmetry, since $S_\alpha$ is in a complex representation of $SU(N)$, and the real and imaginary parts of $S_\alpha$ cannot be combined to form an $O(4)$ vector, as is possible for the Higgs field. This does not affect the standard relations such as $M_W=M_Z \cos \theta_W$ that follow from custodial $SU(2)$ symmetry in the Higgs sector. Custodial $SU(2)$ symmetry violation due to $S_\alpha$ interactions only arise from higher dimension operators. One can also study variants of the theory with $SO(N)$ symmetry, or double the $S_\alpha$ fields to have a $O(4) \times SU(N)$ symmetry. In these variants, custodial $SU(2)$ can be incorporated in the $S$ potential.
The model has threshold corrections to the gauge couplings, Eq. (\[threshold\]), which affects gauge unification. This was studied in Ref. [@Manohar:2006gz]. In general, all theories that introduce new dynamics will modify the standard unification scenario, which has perturbative unification with a desert up to the GUT scale.
Finally, the model needs a fine tuning of order $1\%$ to keep $m_H$ small compared to the scale $\Lambda \sim m_S$, since there is contribution to $m_H^2 \propto m_S^2$ in Eq. (\[ans1\]). While not desirable, this is not worse than fine-tunings required in many models proposed to solve the hierarchy problem. The $H$ mass term and dimension six operators have different dependence on the RG invariant parameters, so theories with additional $S$ multiplets can cancel the $m_H$ contribution without cancelling the Higgs-gauge operators, if the parameters satisfy $$\begin{aligned}
\sum_i \lambda_{1,i} {\ks}_{,i}^2-\frac{\lambda_{1,i}}{\lambda_{3,i}} \vev{\Phi_i} &=0\,.
\label{31}\end{aligned}$$ This cancellation condition is not adjusted order-by-order in perturbation theory, since Eqs. (\[ans1\],\[31\]) are exact at leading order in $1/N$. The Higgs mass is then light because it is $1/N$ suppressed.
I would like to thank E. Jenkins and M. Trott for helpful discussions.
[^1]: The expansion has terms of order $(g^2 N)^r$, so we take the limit $g^2 \to 0$ first, followed by $N \to \infty$. Equivalently, $N \gg 1$, and $g^2 N \ll 1$.
[^2]: RG invariance refers to the dynamics of $S_\alpha$. The Standard Model fields are treated as external background fields.
[^3]: $e=2.71828\ldots$.
|
---
abstract: 'We address the problem of distributing approximation errors in large-scale quantum programs. It has been known for some time that when compiling quantum algorithms for a fault-tolerant architecture, some operations must be approximated as they cannot be implemented with arbitrary accuracy by the underlying gate set. This leads to approximation errors which often can be grouped along subroutines that the given quantum algorithm is composed of. Typically, choices can be made as to how to distribute approximation errors so that the overall error is kept beneath a user- or application-defined threshold. These choices impact the resource footprint of the fault-tolerant implementation. We develop an automatic approximation error management module to tackle the resulting optimization problems. The module is based on annealing and can be integrated into any quantum software framework. Using the benchmark of simulating an Ising model with transverse field, we provide numerical results to quantify the benefits and trade-offs involved in our approach.'
author:
- Thomas Häner
- Martin Roetteler
- 'Krysta M. Svore'
title: Managing Approximation Errors in Quantum Programs
---
Introduction
============
Quantum computing promises to solve certain problems much faster than classical devices. These problems include factoring large numbers, which is possible using Shor’s algorithm [@shor94], unstructured search problems [@grover1996fast], and simulation of quantum mechanical systems. The latter area is where the idea of quantum computing originally emerged [@Feynman1982]. While only few quantum algorithms are known that offer super-polynomial speedups over their classical counterparts, many polynomial speedups have been found over the recent years. However, most of these algorithmic advances take place at a high level of abstraction and often do not take into account constants that may not be explicit in asymptotic analyses. Therefore, while desirable, actual practical resource estimates of complex, large-scale quantum algorithms remain scarce, although they could be used in various ways: to determine the first practical applications of quantum computers, to identify and remedy bottlenecks in existing quantum algorithms, to determine cross-over points (i.e., the smallest problem size for which it is beneficial to use quantum computers), and for hardware/software co-design in general.
In order to estimate the required resources of a given quantum algorithm, the high-level representation of the algorithm must be translated to a universal low-level set of operations that can be realized in practice. One of the standard gate sets that is often considered is the so-called Clifford+$T$ gate set, which can be generated from a few single-qubit operations and the CNOT (controlled-NOT) gate. In particular, arbitrary single-qubit rotations must be translated to this discrete gate set employing a *rotation synthesis* algorithm, where the resulting gate sequences get longer as the desired accuracy is increased. Therefore, besides the problem of compiling abstract high-level functions to the native gate set, the resulting approximation errors need to be managed in a way that ensures that the resulting code performs the desired overall operation within a certain (user-specified) tolerance. We address this problem by introducing a method capable of handling these errors automatically.
**The need for approximation.**\[sec:origin\] While it is not possible to perform error correction over a continuous set of quantum operations (gates), this can be achieved over a discrete gate set such as the aforementioned Clifford+$T$ gate set. As a consequence, certain operations must be approximated using gates from this discrete set. An example is the operation which achieves a rotation around the z-axis, $$\text{Rz}_\theta=\left(\begin{matrix}
e^{-i\theta/2} & 0\\0 & e^{i\theta/2}
\end{matrix}\right).$$ To implement such a gate over Clifford+$T$, synthesis algorithms such as the ones in Refs. [@ross2014optimal; @KMM1] can be used. Given the angle $\theta$ of this gate, such a rotation synthesis algorithm will produce a sequence of $\mathcal O(\log{\varepsilon_R^{-1}})$ Clifford+$T$ gates which approximate Rz$_\theta$ up to a given tolerance $\varepsilon_R$. We measure approximation error $\varepsilon_R$ with respect to distance in operator norm. As we focus on unitary channels this is equivalent to diamond distance, i.e., approximation errors can be composed safely.
In most error correction protocols, the $T$-gate is the most expensive operation to realize, as it cannot be executed natively but requires a distillation protocol to distill many noisy magic states into one good state, which can then be used to apply the gate. As a consequence, it is crucial to reduce the number of these $T$-gates as much as possible in order to allow executing a certain quantum computation.
**Compilation of quantum programs.** The job of a quantum program compiler is to translate a high-level description of a given quantum program to hardware-specific machine-level instructions. As in classical computing, such compilation frameworks can be implemented in a hardware-agnostic fashion by introducing backend-independent intermediate representations of the quantum code [@haener2018software].
During the compilation process, it is crucial to optimize as much as possible in order to reduce the overall depth of the resulting circuit to keep the overhead of the required quantum error correction schemes manageable. Optimizations include quantum versions of constant-folding (such as merging consecutive rotation gates, or even additions by constants) and recognition of compute/action/uncompute sections to reduce the number of controlled gates [@haener2018software]. To allow such optimizations, it is important to introduce multiple layers of abstractions instead of compiling directly down to low-level machine instructions [@haener2018software; @Steiger2018projectqopensource], which would make it impossible to recognize, e.g., two consecutive additions by constants. Even canceling a gate followed by its inverse becomes computationally hard, or even impossible once continuous gates have been approximated.
To translate an intermediate representation to the next lower level of abstraction, a set of decomposition rules is used, some of which introduce additional errors which can be made arbitrarily small at the cost of an increasing circuit size or depth, which in turn implies a larger overhead when applying quantum error correction. Therefore, it is important to choose these error tolerances such that the computation succeeds with high probability given the available resources (number and quality of qubits). See Fig. \[fig:compilation\] for an abstract depiction of the compilation process of a quantum phase estimation on a given unitary $U$. At each level of abstraction, the compiler introduces additional accuracy parameters (in the figure denoted by $\varepsilon_i$) which must be chosen such that
1) the overall error lies within the specifications of the algorithm and
2) the implementation cost is as low as possible while 1) is satisfied.
As mentioned above, it is important to measure approximation errors in a way that is composable to avoid potential issues in underreporting actual approximation errors [@MPGC13; @GSVB13] which could be devastating when composing complex quantum algorithms. This leads to diamond distance as the preferred way to measure closeness to the target operation as it composes. As unraveling a complex quantum algorithm eventually leads to primitive gates that are unitary—such as the mentioned $\text{Rz}_\theta$ rotations which are implemented on subsystem of a constant number of qubits—bounding the approximation error in operator norm implies error in diamond norm, i.e., estimates of approximation error can be composed.
Error-propagation in quantum circuits {#sec:error}
=====================================
The time-evolution of a closed quantum system can be described by a unitary operator. As a consequence, each time-step of our quantum computer can be described by a unitary matrix of dimension $2^n\times 2^n$ (excluding measurement), where $n$ denotes the number of quantum bits (qubits). When decomposing such a quantum operation $U$ into a sequence of lower-level operations $U_M\cdots U_1$, the resulting total error can be estimated from the individual errors $\varepsilon$ of the lower-level gates as follows:
\[lm:unitaryerror\]Given a unitary decomposition of $\,U$ such that $U=U_M\cdot U_{M-1}\cdots U_1$ and unitaries $V_i$ which approximate the unitary operators $U_i$ such that $\|V_i-U_i\|<\varepsilon_i\;\forall i$, the total error can be bounded as follows: $$\|U-V_M\cdots V_1\|\leq\sum_{i=1}^M\varepsilon_i.$$
This lemma can be shown straightforwardly from the ‘hybrid argument’ [@BV97] based on triangle inequality and submultiplicativity of $\|\cdot\|$ with $\|U\|\leq 1$.
Note that using only this Lemma in the compilation process to automatically optimize the individual $\varepsilon_i$ would make the resulting optimization problem infeasibly large. What is even worse is that the number of parameters to optimize would vary throughout the optimization process since the number of lower-level gates changes when implementing a higher-level operation at a different accuracy, which in turn changes the number of distinct $\varepsilon_i$. To address these two issues, we introduce Theorem \[thm:decomp\] which generalizes Lemma \[lm:unitaryerror\]. First, however, we require a few definitions.
Let $V_{M(\varepsilon)}\cdots V_1$ be an approximate decomposition of the target unitary $U$ such that $\|V_{M(\varepsilon)}\cdots V_1\|\leq\varepsilon$. A set of subroutine sets $\mathcal S(U, \varepsilon)=\{S_1,...,S_K\}$ is a *partitioning of subroutines of $\,U$* if $\forall i\exists !k : V_i\in S_k$ and we denote by $S(V)$ the function which returns the subroutine set $S$ such that $V\in S$.
Such a partitioning will be used to assign to each $V_i$ the accuracy $\varepsilon_{S(V_i)}=\varepsilon_{S_k}$ with which all $V_i\in S_k$ are implemented. In order to decompose the cost of $\,U$, however, we also need the notion of a *cost-respecting partitioning of subroutines of $\,U$* and the costs of its subsets:
Let $\mathcal S(U, \varepsilon)=\{S_1,...,S_K\}$ be a set of subroutine sets. $\mathcal S(U, \varepsilon)$ is a *cost-respecting partitioning of subroutines of $\,U$* w.r.t. a given cost measure $C(U, \varepsilon)$ if $\forall \varepsilon,i,j,k: ( V_i\in S_k\land V_j\in S_k\Rightarrow C(V_i,\varepsilon)=C(V_j,\varepsilon))$. The cost of a subroutine set $S$ is then well-defined and given by $C(S, \varepsilon):=C(V, \varepsilon)$ for any $V\in S$.
With these definitions in place, we are now ready to generalize Lemma \[lm:unitaryerror\].
\[thm:decomp\] Let $\mathcal S(U, \varepsilon)=\{S_1,...,S_K\}$ be a cost-respecting partitioning of subroutines for a given decomposition of $\,U$ w.r.t. the cost measure $C(U, \varepsilon)$ denoting the number of elementary gates required to implement $\,U$. Then the cost of $\,U$ can be expressed in terms of the costs of all subroutine sets $S\in\mathcal S(U, \varepsilon_U)$ as follows $$\begin{aligned}
C(U, \varepsilon)&=\sum_{S\in \mathcal S(U, \varepsilon_U)} C(S,\varepsilon_{S})f_S(\varepsilon_U)\\
&\text{with } \sum_{S\in \mathcal S(U, \varepsilon_U)} \varepsilon_S f_S(\varepsilon_U)\leq\varepsilon-\varepsilon_U,\end{aligned}$$ where $f_S(\varepsilon_U)$ gives the number of subroutines in the decomposition of $\,U$ that are in $S$, given that the decomposition of $\,U$ would introduce error $\varepsilon_U$ if all subroutines were to be implemented exactly and $\varepsilon_S$ denotes the error in implementing subroutines that are in $S$.
It is easy to see that the cost $C(U,\varepsilon)$ can be decomposed into a sum of the costs of all subroutines $V_i$. Furthermore, since $\varepsilon_{V}=\varepsilon_{S}$ $\forall V\in S$, $$\begin{aligned}
C(U,\varepsilon)&=\sum_i C(V_i, \varepsilon_{V_i})\\&=\sum_i C(V_i, \varepsilon_{S(V_i)})\\&=\sum_{S\in\mathcal S}|\{i:V_i\in S\}|C(S, \varepsilon_{S})\end{aligned}$$ and $f_S(\varepsilon_U):=|\{i:V_i\in S\}|$ $\forall S\in\mathcal S(U,\varepsilon_U)$.
To prove that the overall error remains bounded by $\varepsilon$, let $\tilde U$ denote the unitary which is obtained by applying the decomposition rule for $U$ with accuracy $\varepsilon_U$, i.e., $\|U-\tilde U\|\leq\varepsilon_U$ (where all subroutines are implemented exactly). Furthermore, let $V$ denote the unitary which will ultimately be executed by the quantum computer, i.e., the unitary which is obtained after all decomposition rules and approximations have been applied. By the triangle inequality and Lemma \[lm:unitaryerror\], $$\begin{aligned}
\|U-V\|&\leq\|U-\tilde U\|+\|\tilde U-V\|\\&\leq \varepsilon_U + \sum_{S\in\mathcal S(U,\varepsilon_U)}\varepsilon_S f_S(\varepsilon_U)\\&\leq \varepsilon\end{aligned}$$
In Fig. \[fig:compilation\], for example, the left-most $^cU$ box gets $\varepsilon_1$ as its error budget. Depending on the implementation details of $^cU$, some of this budget may already be used to decompose $^cU$ into its subroutines, even assuming that all subroutines of $^cU$ are implemented exactly. The remaining error budget is then distributed among its subroutines, which is exactly the statement of the above theorem.
{width=".9\linewidth"}
The decomposition of the cost can be performed at different levels of granularity. This translates into, e.g., having a larger set $\mathcal S(U,\varepsilon)$ and more functions $f_S(\varepsilon_U)$ that are equal to 1. The two extreme cases are
1. $f_S(\varepsilon)=1\;\forall S\in\mathcal S(U,\varepsilon)$, $|\mathcal S(U,\varepsilon)|=\#$gates needed to implement $U$:
A different $\varepsilon_U$ for each gate
2. $f_S(\varepsilon)=\#$gates needed to implement $U\;\forall S\in\mathcal S(U,\varepsilon)$, $|\mathcal S(U,\varepsilon)|=1$:
The same $\varepsilon_\varnothing$ for all gates
Therefore, this solves the first issue of Lemma \[lm:unitaryerror\]: In a practical implementation, the size of the set $\mathcal S(U,\varepsilon)$ can be adaptively chosen such that the resulting optimization problem which is of the form $$\begin{aligned}
&{ }(\varepsilon_{S_1}^\star,\cdots,\varepsilon_{S_N}^\star)\in\arg\min C_\text{Program}(\varepsilon_{S_1}^{},\cdots,\varepsilon_{S_N}^{})\\
&\text{such that }\; { }\varepsilon_\text{\tiny Program}(\varepsilon_{S_1}^{\star},\cdots,\varepsilon_{S_N}^{\star})\leq \varepsilon\end{aligned}$$ for a user- or application-defined over-all tolerance $\varepsilon$, can be solved using a reasonable amount of resources. Moreover, the costs of optimization can be reduced by initializing the initial trial parameters $\varepsilon_{S_i}^{}$ to the corresponding solution accuracies of a lower-dimensional optimization problem where $\mathcal S(U,\varepsilon)$ had fewer distinct subroutines. This approach is similar to multi-grid schemes which are used to solve partial differential equations.
The second issue with a direct application of Lemma \[lm:unitaryerror\] is the varying number of optimization parameters, which is also resolved by Theorem \[thm:decomp\]. Of course one can simply make $\mathcal S(U,\varepsilon)$ tremendously large such that most of the corresponding $f_S(\varepsilon)$ are zero. This, however, is a rather inefficient solution which would also be possible when using Lemma \[lm:unitaryerror\] directly. A better approach is to inspect $\mathcal S(U,\varepsilon)$ for different $\varepsilon$ and to then choose $A$ auxiliary subroutine sets $S^a_1,...,S^a_A$ such that each additional subroutine $V_k^a$ which appears when changing $\varepsilon$ (but is not a member of any $S$ of the original $\mathcal S(U,\varepsilon)$) falls into exactly one of these sets. The original set $\mathcal S(U,\varepsilon)$ can then be extended by these auxiliary sets before running the optimization procedure. Again, the level of granularity of these auxiliary sets and thus the number of such sets $A$ can be tuned according to the resources that are available to solve the resulting optimization problem.
Example application {#sec:tfim}
===================
As an example application, we consider the simulation of a quantum mechanical system called the *transverse-field Ising model* (TFIM), which is governed by the Hamiltonian $$\hat H=-\sum_{\langle i,j\rangle} J_{ij}\sigma_z^{i}\sigma_z^{j}-\sum_i\Gamma_i\sigma_x^i,$$ where $J_{ij}$ are coupling constants and $\Gamma_i$ denotes the strength of the transverse field at location $i$. $\sigma_x^i$ and $\sigma_z^i$ are the Pauli matrices, i.e., $$\sigma_x=\left(\begin{matrix}0 & 1\\1 & 0\end{matrix}\right)\;\;\text{ and }\;\;
\sigma_z=\left(\begin{matrix}1 & 0\\0 & -1\end{matrix}\right)$$ acting on the $i$-th spin. The sum over $\langle i,j\rangle$ loops over all pairs of sites $(i,j)$ which are connected. In our example, this corresponds to nearest-neighbor sites on a one-dimensional spin chain (with periodic boundary conditions) of length $N$. Given an approximation $\ket{\tilde{\psi_0}}$ to the ground state $\ket{\psi_0}$ of $\hat H$, we would like to determine the ground state energy $E_0$ such that $$\hat H\ket{\psi_0}=E_0\ket{\psi_0}.$$ It is well-known that quantum phase estimation (QPE) can be used to achieve this task which leads to a general circuit structure as in Fig. \[fig:qpe\].
**Individual compilation stages.** We now analyze the QPE algorithm for TFIM ground state estimation and the resulting optimization problem for approximation errors. First note that if the overlap between $\ket{\psi_0}$ and $\ket{\tilde\psi_0}$ is large, a successful application of QPE followed by a measurement of the energy register will collapse the state vector onto $\ket{\psi_0}$ and output $E_0$ with high probability (namely $p=|\braket{\tilde\psi_0|\psi_0}|^2$).
There are various ways to implement QPE [@nielsen2002quantum], but the simplest to analyze is the coherent QPE followed by a measurement of all control qubits, see Fig. \[fig:qpe\] for an illustration of the circuit. This procedure requires ${16\pi}/{\varepsilon_\text{QPE}}$ applications of (the controlled version of) the time-evolution operator $U_\delta=\exp(-i\delta\hat H)$ for a success probability of $1/2$, where $\varepsilon_\text{QPE}$ denotes the desired accuracy (bit-resolution of the resulting eigenvalues) [@Reiher7555]. Using a Trotter decomposition of $U_\delta$, i.e., for large $M$ $$\begin{aligned}
U_\delta&\approx \left( U^J_\frac \delta M U^\Gamma_\frac\delta M\right)^M\\&=\left( e^{-i\frac\delta M\sum_{i}J_{i,i+1}\sigma_z^i\sigma_z^{i+1}} e^{-i\frac\delta M\sum_i \Gamma_i\sigma_x^i}\right)^M\\
&=\left( \prod_ie^{-i\frac\delta MJ_{i,i+1}\sigma_z^i\sigma_z^{i+1}} \prod_ie^{-i\frac\delta M\Gamma_i\sigma_x^i}\right)^M,\end{aligned}$$ allows to implement the global propagator $U_\delta$ using a sequence of local operations. These consist of z- and x-rotations in addition to nearest-neighbor CNOT gates to compute the parity (before the z-rotation and again after the z-rotation to uncompute the parity). The rotation angles are $\theta_z=2\frac{\delta}{M}J_{i,i+1}$ and $\theta_x=-2\frac{\delta}{M}\Gamma_i$ for z- and x-rotations, respectively. The extra factor of two arises from the the definitions of the Rz and Rx gates, see Sec. \[sec:origin\].
In order to apply error correction to run the resulting circuit on actual hardware, these rotations can be decomposed into a sequence of Clifford+$T$ gates using rotation synthesis. Such a discrete approximation up to an accuracy of $\varepsilon_R$ features $\mathcal O(\log\varepsilon_R^{-1})$ T-gates if the algorithms in [@ross2014optimal; @KMM1] are used, where even the constants hidden in the $\mathcal O$ notation were explicitly determined.
**Casting the example into our framework.** The first compilation step is to resolve the QPE library call. In this case, it is known that the cost of QPE applied to a general propagator $U$ is $$C(\text{QPE}_U, \varepsilon) = \frac{16\pi}{\varepsilon_\text{QPE}} C(^cU,\varepsilon_U),$$ where $^cU$ denotes the controlled version of the unitary $U$, i.e., $$^cU := \Ket 0\Bra 0\otimes\mathbbm 1 + \Ket 1\Bra 1\otimes U.$$ Furthermore, the chosen tolerances must satisfy $$\frac{16\pi}{\varepsilon_\text{QPE}}\varepsilon_U \leq \varepsilon - \varepsilon_\text{QPE}.$$ The next step is to approximate the propagator using a Trotter decomposition. Depending on the order of the Trotter formula being used, this yields $$\begin{aligned}
C(^cU,\varepsilon_U) &= M(\varepsilon_\text{Trotter}) (C(^cU_1,\varepsilon_{U_1}) + C(^cU_2,\varepsilon_{U_2}))\\&\text{ with }\; M(\varepsilon_\text{Trotter})(\varepsilon_{U_1}+\varepsilon_{U_2})\leq\varepsilon_U-\varepsilon_\text{Trotter}.\end{aligned}$$ In the experiments section, we will choose $M(\varepsilon_{\text{Trotter}})\propto\frac 1{\sqrt{\varepsilon_{\text{Trotter}}}}$ as an example. Finally, approximating the (controlled) rotations in $^cU_1$ and $^cU_2$ by employing rotation synthesis, $$\begin{aligned}
C(^cU_i,\varepsilon_{U_i}) = 2N\cdot 4\log\varepsilon_R^{-1}\\\text{ with }\; 2N\varepsilon_R\leq\varepsilon_{U_i}\;\text{ for }i\in\{1,2\}.\end{aligned}$$ Collecting all of these terms and using that $C(^cU_1,\cdot) = C(^cU_2,\cdot)$ yields $$\begin{aligned}
C(\text{QPE}_U,\varepsilon) = \frac{16\pi}{\varepsilon_\text{QPE}}M(\varepsilon_\text{Trotter})\cdot 2\cdot 2N\cdot 4\log\varepsilon_R^{-1},\\
\text{ with }\varepsilon_\text{QPE} + \frac{16\pi}{\varepsilon_\text{QPE}}(2M(\varepsilon_\text{Trotter})\cdot 2N\varepsilon_R + \varepsilon_\text{Trotter})\leq\varepsilon.\end{aligned}$$ Note that this example is a typical application for a quantum computer which at the same time can serve as a proxy for other, more complex simulation algorithms.
While the individual compilation stages may be different for other applications, the basic principle of iterative decomposition and approximation during compilation is ubiquitous. In particular, a similar compilation procedure would be employed when performing quantum chemistry simulations, be it using a Trotter-based approach [@Wecker14a] or an approach that is based on a truncated Taylor series [@berry2015simulating].
(*$\beta$*) = 0
(*$\overline\varepsilon$*) = (*$\left(0.1, 0.1, \cdots, 0.1\right)$*)
cost = get_cost((*$\overline\varepsilon$*))
error = get_total_error((*$\overline\varepsilon$*))
for step in range(num_steps):
i = floor(rnd() * len(eps))
old_(*$\overline\varepsilon$*) = (*$\overline\varepsilon$*)
if rnd() < 0.5:
(*$\overline\varepsilon_i$*) *= 1 + (1 - rnd()) * (*$\delta$*)
else:
(*$\overline\varepsilon_i$*) /= 1 + (1 - rnd()) * (*$\delta$*)
if error <= goal_error:
# reduce cost
(*$\Delta E$*) = get_cost((*$\overline\varepsilon$*)) - cost
else:
# reduce error
(*$\Delta E$*) = get_total_error((*$\overline\varepsilon$*)) - error
(*$p_\text{accept}$*) = min(1, (*$e^{-\beta \Delta E}$*))
if rnd() > p(*$_\text{accept}$*):
(*$\overline\varepsilon$*) = old_(*$\overline\varepsilon$*)
(*$\beta$*) += (*$\Delta\beta$*)
Implementation and numerical results {#sec:results}
====================================
In this section, we present implementation details and numerical results of our error management module. While the optimization procedure becomes harder for fine-grained cost and error analysis, the benefits in terms of the cost of the resulting circuit are substantial.
**Optimization methodology.** We use a two-mode annealing procedure for optimization, in which two objective functions are reduced as follows: The first mode is active whenever the current overall error is larger than the target accuracy $\varepsilon$. In this case, it performs annealing until the target accuracy has been reached. At this point, the second mode becomes active. It performs annealing-based optimization to reduce the circuit cost function. After each such step, it switches back to the error-reduction subroutine if the overall error increased above $\varepsilon$.
Both annealing-based optimization modes follow the same scheme, which consists of increasing/decreasing a randomly chosen $\varepsilon_i$ by multiplying/dividing it by a random factor $f\in(1,1+\delta]$, where $\delta$ can be tuned to achieve an acceptance rate of roughly $50\%$. Then, the new objective function value is determined, followed by either a rejection of the proposed change in $\varepsilon_i$ or an acceptance with probability $$p_{\text{accept}}=\min(1, e^{-\beta\Delta E}),$$ where $\beta=T^{-1}$ and $T$ denotes the annealing temperature. This means, in particular, that moves which do not increase the energy, i.e., $\Delta E \leq 0$ are always accepted. The pseudo-code of this algorithm can be found in Listing \[lst:annealingcode\].
**Results.** Using the example of a transverse-field Ising model which was discussed in Sec. \[sec:tfim\], we determine the benefits of our error management module by running two experiments. The first experiment aims at assessing the difference between a feasible solution, i.e., values $\varepsilon_i$ which produce an overall error that is less than the user-defined tolerance, and an optimized feasible solution. In the first case, we only run the first mode until a feasible solution is obtained and in the latter, we employ both modes as outlined above. Fig. \[fig:erroronly\] depicts the costs of the resulting circuit as a function of the desired overall accuracy $\varepsilon$.
The second experiment aims to show the benefit of an increased number of $\varepsilon_i$ parameters in the same example. The difference between the circuit costs when using just two such parameters (i.e., setting $\varepsilon_R=\varepsilon_{\text{Trotter}}$) versus using all three is depicted in Fig. \[fig:numparams\].
Finally, we measure the robustness of the optimization procedure by introducing redundant parameters, i.e., additional rotation gate synthesis tolerances $\varepsilon_{R_i}$, where the optimal choice would be $\varepsilon_R=\varepsilon_{R_i}=\varepsilon_{R_j}$ for all $i,j$. However, because the resulting optimization problem features more parameters, it is harder to solve and the final circuit cost is expected to be higher. In addition, the time it takes to find an initial feasible solution will grow. See Figs. \[fig:rob1\] and \[fig:rob2\] for the results which indicate that this approach is scalable to hundreds of variables if the goal is to find a feasible solution. However, as the number of parameters grows, it becomes increasingly harder to simultaneously optimize for the cost of the circuit. This could be observed, e.g., with 100 additional (redundant) parameters, where further optimization of the feasible solution reduced the cost from $1.65908\cdot 10^{12}$ to $1.10752\cdot 10^{12}$, which is far from the almost $2x$ improvement which was observed for smaller systems in Fig. \[fig:erroronly\]. Also, the scaling of the runtime in Fig. \[fig:rob2\] can be explained since new updates are proposed by selecting $i\in [0,...,N-1]$ uniformly at random (followed by either increasing or decreasing $\varepsilon_i$). Due to this random walk over $i\in[0,...,N-1]$, the overall runtime is also expected to behave like the expected runtime of a random walk and, therefore, to be in $\mathcal O(N^2)$.
Summary and future work {#sec:outlook}
=======================
We have presented a methodology for managing approximation errors in compiling quantum algorithms. Given that the way in which the overall target error is distributed among subroutines greatly influences the resource requirements, it is crucial to optimize this process, in particular for large-scale quantum algorithms that are composed of many subroutines. We leverage an annealing-based procedure to find an initial feasible solution which then is optimized further.
Our scheme for error management only addresses errors that occur in approximations during the compilation process into a fault-tolerant gate set. Future work might include hardware errors, e.g., systematic over- or under-rotations of gates performed by the target device. Furthermore, additional numerical studies for various quantum algorithms can be performed in order to arrive at heuristics for choosing the number of optimization parameters. Moreover, building on our error management methodology, one can automate the entire process of resource estimation for certain subclasses of quantum algorithms. This would yield a useful tool for assessing the practicality of known quantum algorithms, similar to the analysis carried out manually in [@Reiher7555].
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|
---
abstract: 'We study the prospects of using the electric quadrupole transitions from the ground states of Cu, Ag and Au to the metastable state $^2{\rm D}_{5/2}$ as clock transitions in optical lattice clocks. We calculate lifetimes, transition rates, systematic shifts, and demonstrate that the fractional uncertainty of the clocks can be similar to what is achieved in the best current optical clocks. The use of these proposed clocks for the search of new physics, such as time variation of the fine structure constant, search for low-mass scalar dark matter, violation of Local Position Invariance and violation of Lorenz Invariance is discussed.'
author:
- 'V. A. Dzuba$^1$, Saleh O. Allehabi$^1$, V. V. Flambaum$^1$, Jiguang Li$^2$ and S. Schiller$^3$'
title: 'Time keeping and searching for new physics using metastable states of Cu, Ag and Au'
---
Introduction
============
Using optical clock transitions for searching for new physics beyond the standard model is a promising area of research. A hypothetical manifestation of new physics at low energy is expected to be very small. Therefore, the highest possible accuracy of the measurements is needed. Fractional uncertainty of the best optical clocks currently is around $1\times$ $10^{-18}$ [@Ludlow; @Chou; @Beloy1; @Beloy2; @Ushijima; @Nicholson; @Katori], the highest accuracy so far achieved in the history of measurements. However, apart from a few exceptions (Hg$^+$, Yb$^+$) the best optical clocks are not sensitive to new physics such as time variation of the fine structure constant, violation of Local Position Invarance (LPI) and violation of Local Lorentz Invariance (LLI), etc. [@qdot; @EEP; @Lia; @Yb+LLI]. LPI, LLI and the Weak Equivalence Principle form the Einstein Equivalence Principle, the foundation of General Relativity. Several ideas were proposed to combine high accuracy of optical clocks with high sensitivity to new physics. These include the use of the highly charged ions (HCI) [@H-likeHCI; @HCIa; @HCIb; @HCIc], nuclear clocks [@th-clock], and metastable atomic states with large value of the total angular momentum $J$ ($J > 1$) [@qdot; @Hg+clock0; @Hg+clock; @Yb-DFS; @Yb+clock]. These states are connected to the ground state via transitions which correspond to single-electron transitions with large change of the single-electron total angular momentum $j$. The large $\Delta j$ is what makes the transition to be sensitive to the variation of the fine structure constant (see e.g. [@DFW99]). For example, in the present work we consider transitions between the $nd^{10}(n+1)s \ ^2$S$_{1/2}$ ground state and the $nd^{9}(n+1)s^2 \ ^2$D$_{5/2}$ excited metastable state. This is roughly the $s_{1/2}$ to $d_{5/2}$ transition with $\Delta j$=2.
The energy diagrams for the five lowest states of Cu, Ag and Au studied in this work are presented in Fig. \[f:EL\]. The metastable state of interest ($^2$D$_{5/2}$) is the first excited state for Cu and Au. In Ag the $^2$P$^{\rm o}_{1/2}$ state lies below the $^2$D$_{5/2}$ clock state. However, this has no significance since the states are very weakly connected (by the E3, M2 or hyperfine-induced E1 transitions). The clock transition in Ag was studied experimentally in Ref. [@AgZ2].
Sensitivity of the above metastable states to variation of the fine structure constant was studied before [@qdot]. In this work we further study the states in terms of their suitability for high accuracy measurements and sensitivity to other manifestations of new physics, such as LPI violation and LLI violation.
Calculations
============
We are mostly interested in five lowest states of Cu, Ag and Au shown on Fig. \[f:EL\]. Two out of the five states have excitations from a $d$-shell. This means that the $d$-shell is open and $d$-electrons should be treated as valence ones. The total number of valence electrons, eleven, is too large for most of standard calculational approaches. We use a version of the configuration interaction (CI) method specifically developed for such systems (the CIPT method [@CIPT]). In this method off-diagonal matrix elements of the CI Hamiltonian between highly excited states are neglected. This allows to reduce the CI matrix to an effective matrix of a small size in which contribution from high states is included perturbatively (see Ref. [@CIPT] for details).
We perform the calculations in the $V^{N-1}$ approximation, with one electron removed from initial relativistic Hartree-Fock (HF) calculations to obtain the potential for calculating single-electron basis states. Given the ground electronic configurations of the atoms here discussed, at first sight the best choice is the removal of the valence $s$ electron. It turns out, however, that much better accuracy is achieved if a $d$ electron is removed instead. Thus, we perform the HF calculations for the \[Ca\]$3d^94s$ configuration of Cu, the \[Sr\]$4d^95s$ configuration of Ag, and the \[Yb\]$5d^96s$ configuration of Au. The B-spline technique [@B-spline] is used to construct single-electron basis states above the core. Many-electron states for the CIPT calculations are constructed by exciting one or two electrons from a reference configuration and then using the resulting configurations to build all corresponding many-electron states of definite value of the total angular momentum $J$ and its projection $J_z$. States corresponding to about a hundred lowest non-relativistic configurations go into the effective CI matrix, while higher states are treated perturbatively. Note that our calculations are completely relativistic. We only use non-relativistic configurations to simplify the procedure of generating many-electron basis states. In the list of non-relativistic configurations each of them is subsequently replaced by a corresponding set of relativistic configurations. E.g., the $5d^96s6p$ configuration is replaced by four relativistic ones, the $5d_{3/2}^45d_{5/2}^56s6p_{1/2}$, $5d_{3/2}^35d_{5/2}^66s6p_{1/2}$, $5d_{3/2}^45d_{5/2}^56s6p_{3/2}$, and $5d_{3/2}^35d_{5/2}^66s6p_{3/2}$ configurations.
To calculate transition amplitudes we use the well-known random phase approximation (RPA, see, e.g. [@CPM]). The RPA equations for a single-electron state have the form $$\label{e:RPA}
(H^{\rm HF} - \epsilon_c)\delta \psi_c = -(\hat F + \delta V_{F}^{N-1})\psi_c.$$ Here $H^{\rm HF}$ is the relativistic Hartree-Fock Hamiltonian, index $c$ numerates single-electron states, $\hat F$ is the operator of an external field, $\delta \psi_c$ is a correction to the state $c$ due to an external field, $\delta V_{F}^{N-1}$ is the correction to the self-consistent Hartree-Fock potential due to the external field. The same $V^{N-1}$ potential is used in RPA and HF calculations. The RPA equations (\[e:RPA\]) are solved self-consistently for all states $c$ in the core. Transition amplitudes are found as matrix elements between many-electron states found in the CIPT calculations for the effective operator of an external field $$\label{e:Amp}
A_{ab} = \langle b| \hat F + \delta V^{\rm core}_{F} | a \rangle.$$ Note that valence states are included in $V^{N-1}_F$ in (\[e:RPA\]) but non included in the effective operator for valence states $\delta V^{\rm core}_{F}$ in (\[e:Amp\]).
The rates of spontaneous emission are given in atomic units by $$\label{e:Td}
\Gamma_{\rm E1,M1} = \frac{4}{3}(\alpha\omega)^3 \frac{A^2_{\rm E1,M1}}{2J+1},$$ for electric dipole (E1) and magnetic dipole (M1) transitions, and by $$\label{e:Tq}
\Gamma_{\rm E2,M2} = \frac{1}{15}(\alpha\omega)^5 \frac{A^2_{\rm E2,M2}}{2J+1},$$ for electric quadrupole (E2) and magnetic quadrupole (M1) transitions. In these formulas, $\alpha$ is the fine structure constant, $\omega$ is the energy difference between the lower and upper states, $A$ is the amplitude of the transition (\[e:Amp\]), $J$ is the total angular momentum of the upper state. The magnetic amplitudes $A_{\rm M1}$ and $A_{\rm M2}$ are proportional to the Bohr magneton $\mu_B =|e|\hbar/2mc$. Its numerical value in Gaussian-based atomic units is $\mu_B=\alpha/2 \approx 3.65 \times 10^{-3}$. The lifetimes of excited states are calculated by $\tau_a = 2.4189 \times 10^{-17}/\sum_b \Gamma_{ab}$, where $\tau_a$ is the lifetime of atomic state $a$ in seconds, the summation goes over all possible transitions to lower states $b$, the transition probabilities $\Gamma_{ab}$ are given by (\[e:Td\]) or (\[e:Tq\]).
Energy levels and lifetimes of the five lowest states of Cu, Ag and Au are presented in Table \[t:ECI\]. Lifetimes were calculated using transition amplitudes and probabilities from Table \[t:TR\]. [Given the complexity of the considered systems the agreement between theory and experiment for the energies can be regarded as very good. However,]{} the theoretical values for the lifetimes and transition rates are less accurate. This is partly due to the fact that the current version of the computer code includes perturbation correction to the energy but does not include a corresponding correction to the wave function [@CIPT]. States with no excitation from the upper $d$-shell (e.g., the $5f^{10}6p_{1/2,3/2}$ states of Au) can be treated more accurately within different approaches, for example with the use of the correlation potential method [@CPM]. The main advantage of the current approach is that it can be used for any state of considered atoms, including states with excitations from the upper $d$-shell where most of other methods would not work.
[lllccc]{} && &\
& & &
& &\
\
1 & $3d^{10}4s$&$^2$[S]{}$_{1/2}$&0&0&$\infty$\
2 &$3d^{9}4s^{2}$&$^2$[D]{}$_{5/2}$&11203&10521&44.9 s\
3&$3d^{9}4s^{2}$& $^2$[D]{}$_{3/2}$ &13245&12270&7.3 s\
4 & $3d^{10}4p$& $^2$[P]{}$^{\rm o}$$_{1/2}$ &30535&29489&7.1 ns\
5 & $3d^{10}4p$& $^2$[P]{}$^{\rm o}$$_{3/2}$ &30784&31115&6.9 ns\
\
1 & $4d^{10}5s$& $^2$[S]{}$_{1/2}$&0&0&$\infty$\
2 &$4d^{10}5p$ & $^2$[P]{}$^{\rm o}$$_{1/2}$ &29552&29495&6.6 ns\
3&$4d^{10}5p$&$^2$[P]{}$^{\rm o}$$_{3/2}$ &30473&30451&6.1 ns\
4 & $4d^{9}5s^{2}$ & $^2$[D]{}$_{5/2}$ &30242&32480&192 ms\
5 & $4d^{9}5s^{2}$& $^2$[D]{}$_{3/2}$ &34714&36430&174 $\mu$s\
\
1 & $5d^{10}6s$& $^2$[S]{}$_{1/2}$&0&0&$\infty$\
2 &$5d^{9}6s^{2}$ & $^2$[D]{}$_{5/2}$ &9161&10671&43.7 s\
3&$5d^{9}6s^{2}$ & $^2$[D]{}$_{3/2}$ &21435&22096&33 ms\
4 & $5d^{10}6p$& $^2$[P]{}$^{\rm o}$$_{1/2}$ &37359&38853&4.1ns\
5 &$5d^{10}6p$& $^2$[P]{}$^{\rm o}$$_{3/2}$&41175&43028&3.3ns\
[llrdc]{}
& & &
&\
\
2-1 & E2&11203&2.603&2.23$\cdot$10$^{-2}$\
3-1 &M1&13245&0.0002\_B&6.27$\cdot$10$^{-7}$\
4-1&E1& 30535 &2.217&1.42$\cdot$10$^{+8}$\
5-1 & E1&30784&3.140&1.45$\cdot$10$^{+8}$\
3-2 & M1&2043&1.549\_B&0.138\
5-2 &E1&19581&0.546&1.13$\cdot$10$^{+6}$\
4-3&E1& 17290&0.404&8.48$\cdot$10$^{+5}$\
5-3 & E1&17538 &0.174&8.23$\cdot$10$^{+4}$\
5-4 & M1&248&1.154\_B&1.38$\cdot$10$^{-4}$\
\
2-1 & E1&29552&2.410&1.52$\cdot$10$^{+8}$\
3-1 &E1&30473&3.382&1.64$\cdot$10$^{+8}$\
4-1&E2& 30242 &3.325&5.22\
5-1 & M1&34714&0.0002\_B&1.13$\cdot$10$^{-5}$\
3-2 & M1&921&1.154\_B&7.01$\cdot$10$^{-3}$\
5-2 &E1&5162&0.274&1.04$\cdot$10$^{+4}$\
4-3&E1&230&0.368&0.839\
5-4 & M1&4472&1.549\_B&1.45\
5-3 & E1&4242&0.119&5.45$\cdot$10$^{+2}$\
\
2-1 & E2&9161&4.359&2.29$\cdot$10$^{-2}$\
3-1 &M1&21435&0.0008\_B&4.25$\cdot$10$^{-5}$\
4-1&E1& 37359 &2.153&2.45$\cdot$10$^{+8}$\
5-1 & E1&41175&2.923&3.02$\cdot$10$^{+8}$\
3-2 & M1&12274&1.549\_B&29.9\
5-2 &E1&32013&0.983&1.61$\cdot$10$^{+6}$\
4-3&E1& 15924&0.504&1.04$\cdot$10$^{+6}$\
5-3 & E1&19739 &0.243&2.30$\cdot$10$^{+5}$\
5-4 & M1&3816&1.141\_B &0.488\
Analysis
========
Clock transitions
-----------------
Cu has two long-lived metastable states ($N=2,3$ in Tab.\[t:ECI\]), Ag has one ($N=4$), and Au has one ($N=2$). Only the states of Cu and Au have lifetimes substantially larger than 1s, comparable to those of the currently used Sr and Yb lattice clocks. The 0.2s lifetime of the Ag upper clock state may turn out to be a competitive disadvantage for achieving top performance. In the following we consider the clock transitions between these four states and their respective ground states. [Note that the two clocks states of Cu are very similar and therefore in most of cases we present the data for only the $^2$[D]{}$_{5/2}$ state.]{}
Hyperfine structure
-------------------
The atoms considered here all exhibit hyperfine structure in the ground state, in the clock state and in the excited state addressed in laser cooling. The nuclear spins are given in Tab.\[t:isotopes\]. The hfs splitting is given by [@Landau] $$\begin{aligned}
\label{e:hfs}
E_{\rm HFS}(F)&=&\frac{A}{2}F(F+1) + \frac{B}{2}\left[ F^2(F+1)^2 + \right.\\
&+&\left. F(F+1)[1-2J(J+1)-2I(I+1)]\right]. \nonumber \end{aligned}$$ The total angular momentum is $\mathbf{F}=\mathbf{J}+\mathbf{I}$, where $I$ is nuclear spin. $A$ and $B$ are magnetic dipole and electric quadrupole hfs constants, respectively. They are reported in Tab.\[t:ABhfs\].
-------------------------- ----------------------------- ----- ----- ---------
$^{63,65}$Cu, $^{197}$Au $^2$S$_{1/2} - ^2$D$_{5/2}$ 3/2 1,2 1,2,3,4
$^{63,65}$Cu $^2$S$_{1/2} - ^2$D$_{3/2}$ 3/2 1,2 0,1,2,3
$^{107,109}$Ag $^2$S$_{1/2} - ^2$D$_{5/2}$ 1/2 0,1 2,3
-------------------------- ----------------------------- ----- ----- ---------
: \[t:isotopes\] Stable isotopes with non-zero nuclear spin ($I$) and possible values of total angular momentum $F$ ($\mathbf{F}=\mathbf{I}+\mathbf{J}$) for ground and clock states of Cu, Ag, Au.
------------ --------------- ------- --- --------------- -------- ------- ---------------------
$^{63}$Cu $^2$S$_{1/2}$ 5863 0 $^2$D$_{5/2}$ 749.1 186.0 [@Cu-hfs; @Cu-hfs1]
$^{63}$Cu $^2$S$_{1/2}$ 5863 0 $^2$D$_{3/2}$ 1851.0 137.4 [@Cu-hfs; @Cu-hfs1]
$^{107}$Ag $^2$S$_{1/2}$ -1713 0 $^2$D$_{5/2}$ -126 0 [@AgZ2]
$^{197}$Au $^2$S$_{1/2}$ 3050 0 $^2$D$_{5/2}$ 80.2 -1049 [@Au-hfs]
------------ --------------- ------- --- --------------- -------- ------- ---------------------
: \[t:ABhfs\] Magnetic dipole ($A$) and electric quadrupole ($B$) hfs constants (MHz) used in the calculation of the second-order Zeeman shift.
For example, the hyperfine structure of Au has been studied experimentally with high precision in the 1960s, and has also been calculated [@Itano; @Bieron]. The hyperfine splitting between $F=1,2$ in the ground state amounts to 6.10GHz[@Dahmen]. The splittings in $^2$D$_{5/2}$ are [@Au-hfs]
$F=1\leftrightarrow F=2$: 1.00GHz,
$F=2\leftrightarrow F=3$: 0.71GHz, and
$F=3\leftrightarrow F=4$: 0.52GHz.
The splittings in $^2$D$_{3/2}$ are [@Blachman]
$F=0\leftrightarrow F=1$: 1.11GHz,
$F=1\leftrightarrow F=2$: 1.31GHz, and
$F=2\leftrightarrow F=3$: 0.31GHz.
Laser cooling of Cu, Ag and Au
------------------------------
### Silver
Silver has been laser-cooled [@Aglasercooling]. Here, the cooling scheme is straightforward: the cooling transition is between ground and second excited state, $^2$S$_{1/2}\rightarrow^2$P$_{3/2}$, so that there are no leakage channels to other electronic states. No repumper laser fields are needed.
### Gold
A scheme for laser cooling of Au is presented in Fig. \[f:Au-cool\]. The main cooling transition is the electric dipole transition between the ground state and the excited odd-parity $^2$P$^{\rm o}_{1/2}$ state. Compared to using $^2$P$^{\rm o}_{3/2}$ as upper level, the advantage is that only one repumper is needed and that the longer cooling wavelength is experimentally advantageous. There is leakage from the $^2$P$^{\rm o}_{1/2}$ to the $^2$D$_{3/2}$ state by another electric dipole transition ($4\rightarrow3$). Therefore, without repumping only $\sim$ 250 cycles are possible. With repumping (628nm) the cooling may go for as long as needed. Another leakage channel is too weak to affect the scheme.
### Copper
A cooling scheme similar to silver can be considered for copper: $^2$S$_{1/2}(F_g=2)\rightarrow ^2$P$_{3/2}(F_e=3)$.
### Additional remarks
Optical lattice clocks require the cooling of atoms to the $\mu$K level for efficient loading of the optical lattice with the atoms. Therefore, after cooling on the strong E1 transition to a temperature on the order of 1mK, a second cooling process utilizing a weak transition should follow (“narrow-linewidth cooling”). One option is to cool on the $^2$S$_{1/2}\rightarrow$ $^2$D$_{3/2}$ transition ($1-3$ for Cu, $1-5$ for Ag, $1-3$ for Au). These are M1 transitions and are very weak. However, the strengths could be increased and the lifetime of the $^2$D$_{3/2}$ states shortened by E1 coupling them to the respective $^2$P states using appropriate waves.
The hyperfine structure in both lower and upper laser cooling levels will typically require additional repumper fields to optimize cooling efficiency. We shall not discuss such experimental details here.
Finally, we note that copper and silver atoms have been cooled using buffer-gas cooling [@CuAg-cool].
Polarizabilities, black-body radiation shifts and magic frequencies
-------------------------------------------------------------------
Knowledge of the atomic polarizabilities for both states of the clock transition is important for estimation of the frequency shift caused by black-body radiation and for finding the so called [*magic*]{} frequency of the lattice laser field, i.e. the frequency at which the dynamic polarizabilities of both states are equal, causing no frequency shift.
The static scalar polarizability $\alpha_v(0)$ of an atom in state $v$ is given by $$\label{e:pol0}
\alpha_v(0) = \frac{2}{3(2J_v+1)}\sum_n\frac{|\langle v||D||n\rangle |^2}{E_n-E_v},$$ where $D$ is the electric dipole operator with the RPA correction (see the previous section), and the summation goes over the complete set of excited many-electron states.
Static scalar polarizabilities of the ground states of Cu, Ag and Au are known from a number of calculations and measurements [@pol0]. Table \[t:pol\] presents the recommended values taken from Ref. [@pol0]. In contrast, to the best of our knowledge, there is no similar data for the upper clock states of Cu, Ag and Au. Therefore, we performed the calculations using two different approaches.
In the first approach we stay within the CIPT method and calculate twenty odd-parity states for each value of the total angular momentum $J$ which satisfies the electric dipole selection rules for the transitions from the ground and clock states ($J=1/2, 3/2, 5/2, 7/2$). Then we use the formula (\[e:pol0\]) to perform the calculations for both states. These calculations show three important things: (a) there is good agreement with other data for the ground state, (b) there is good saturation of the summation in (\[e:pol0\]), (c) the summation for the clock states is strongly dominated by the transitions to the states of the $5d^96s6p$ configuration (we use the Au atom as an example).
The last fact implies that a different approach can be used, previously suggested for atoms with open $f$-shells [@AKozlov]. In this second approach we use the fact that the sum (\[e:pol0\]) is dominated by the $6s$ - $6p$ transitions while the open $5d^9$ subshell remains unchanged. Therefore, the open $d$-shell is attributed to the core and treated as a closed shell with an occupational number of 0.9. The atom is treated as a system with two external electrons above the closed-shell core and an appropriate CI+MBPT [@CI+MBPT] method is used (see Ref. [@AKozlov] for more details). The advantage of this approach is the efficient completeness of the basis with two electron excitations. The shortcoming is the omission of the transition amplitudes involving excitations from the $d$-shell. In contrast, the CIPT approach includes all amplitudes; however the summation in (\[e:pol0\]) is truncated much earlier.
In the end, both approaches give similar results. The results for the clock states are presented in Table \[t:pol\] together with estimated uncertainties. For these estimations we used a comparison of the two approaches for the clock states as well as comparison of the CIPT calculation with other data for the ground states.
The results of the calculations indicate that the values of the polarizabilities of the clock states of Cu, Ag and Au are similar to those of the ground state. This is a non-standard situation. More often, the polarizabilities of excited states are larger. Indeed, the higher is the state on the energy scale, the smaller is the energy denominator in (\[e:pol0\]). Present results can be explained by the fact that summation in (\[e:pol0\]) is dominated by the states of the $5d^{10}np$ configurations for the ground state (we use Au again as an example) and by the states of the $5d^96s6p$ configuration for the clock state. The later states are higher on the energy scale.
The BBR shift is given by (see, e.g. [@BBR]) $$\label{e:BBR}
\delta \nu_{\rm BBR} = -\frac{2}{15}(\alpha \pi)^3 T^4 \left(\alpha_c(0) - \alpha_g(0)\right),$$ where $\alpha$ is the fine structure constant, $T$ is temperature, $\alpha_c(0)$ and $\alpha_g(0)$ are static scalar polarizabilities of the clock and ground states. For simplicity, we do not include the dynamic correction to the BBR shift. For the more complete formula see, e.g. [@BBR].
The BBR shift is proportional to the difference between the polarizabilities of the two states. The similarity of the polarizabilities implies a substantial cancellation of the black-body radiation (BBR) frequency shift, a very favourable effect. Since the differences are smaller or close to the uncertainty of both numbers we can only give upper limits for the BBR shifts. The results are presented in Table \[t:pol\]. These limits are lower than the shifts in the standard ytterbium and strontium lattice clocks. More accurate estimations might be possible if the polarizabilities are measured or calculated to higher accuracy.
Magic frequencies can be found in the vicinity of every resonance for one of the polarizabilities, i.e. when the frequency of the lattice laser field is approximately equal to the excitation energy (energy denominator in (\[e:pol0\])). The first magic frequency is near the first resonance for the ground state polarizability, i.e. $\hbar \omega_m \simeq 30535\,{\rm cm}^{-1}$ for Cu, $\hbar \omega_m \simeq 29552\,{\rm cm}^{-1}$ for Ag and $\hbar \omega_m \simeq 37359\,{\rm cm}^{-1}$ for Au. Note that since the clock states have large values of the total angular momentum ($J=5/2$), the magic frequencies would also depend on the quadrupole contribution to the polarizabilities. The current level of computational accuracy does not allow to find accurate values of the magic frequencies. Having more experimental data may help. In the vicinity of a resonance or a few resonances a semi-empirical formula can be used $$\label{e:pw}
\alpha_a(\omega) \approx \alpha^{\prime}_a(0) + \frac{2}{3(2J_v+1)}\sum_b \frac{A_{ab}^2}{\omega - \Delta E_{ab}},$$ where $\alpha^{\prime}_a(0)$ is chosen in such a way that $\alpha_a(\omega=0)$ is equal to known (e.g. experimental) static polarizability of state $a$. Summation in (\[e:pw\]) goes over close resonances. If the static polarizability is known to sufficient accuracy and amplitudes $A_{ab}$ of E1-transitions are extracted from experimental data or from accurate atomic calculations, then (\[e:pw\]) can be used to find magic frequencies.
---- ------- ------ ------ ------ ------- ------- ---------- -----------------------
Cu 47(1) 54.5 46.8 42.9 45(8) 2(8) $ <0.09$ $<2.6 \cdot 10^{-16}$
Ag 55(8) 51.8 45.9 49.5 47(2) -8(8) $ <0.14$ $<1.5 \cdot 10^{-17}$
Au 36(3) 35.7 38.9 33.2 36(3) 0(4) $ <0.03$ $<5.6 \cdot 10^{-17}$
---- ------- ------ ------ ------ ------- ------- ---------- -----------------------
Stark, quadrupole and Zeeman shifts
-----------------------------------
Interaction of atomic electrons with external electric field and its gradient lead to Stark and electric quadrupole shifts of transition frequencies. These shifts are tiny in optical lattice clocks [^1]. We consider the shifts in more details in the Appendix.
The linear Zeeman shift is given by the expression $$\label{e:z1}
\Delta E_{F,F_z} = g_F \mu_B B F_z,$$ where $g_F$ is the $g$-factor of a particular hfs state. It is related to the electron $g_J$-factor by $$\label{e:f}
g_F=g_J \langle F,F_z=F,I,J|\hat J_z|F,F_z=F,I,J \rangle/F.$$ Electron $g$-factors have approximate values $g_{1/2} \approx 2$, $g_{3/2} \approx 0.8$, $g_{5/2} \approx 1.2$. More accurate values for Cu, Ag and Au can be found in NIST tables [@NIST]. For a clock state with $J=5/2$ and $F=2$ we have $g_2= (11/12)g_{5/2}=1.1$. For a clock state with $J=3/2$ and $F=2$ we have $g_2= (1/2)g_{3/2}=0.4$.
The linear Zeeman shift can be avoided if only transitions between states with $F_z = 0$ are considered, as sug- gested in the past for clock operation. Alternatively, one can average over the transition frequencies with positive and negative $F_z$ in order to cancel the linear shift. However, the large individual shifts will make it difficult to achieve an accurate cancellation.
A second-order Zeeman shift is unavoidable. Therefore, it is important to know its value. If we consider transitions between definite hfs components, then the shift is strongly dominated by transitions within the same hfs multiplet. The total shift is the difference between the second-order shifts in the clock and in the ground state. Both shifts are given by $$\begin{aligned}
\label{e:z2}
&&\delta E_{F,F_z} = \\
&&\sum_{F'=F \pm 1,F'_z}\frac{|\langle F'F'_zIJ|\hat J_z|FF_zIJ \rangle g_J \mu_B B |^2}{\Delta E_{\rm{HFS}}(F,F')}. \nonumber\end{aligned}$$ Here $\Delta E_{\rm{HFS}}(F,F')=E(F\,I\,J)-E(F'\,I\,J)$ is the hfs interval. It has a different sign depending on whether this is an up or down transition. It follows from (\[e:hfs\]) that $$\begin{aligned}
&& \Delta E_{\rm HFS}(F,F+1) = -A(F+1) - \nonumber \\
&&B \left( 2(F+1)^2 +1-2J(J+1)-2I(I+1)\right), \nonumber \end{aligned}$$ and $$\begin{aligned}
&&\Delta E_{\rm HFS}(F,F-1) = \nonumber \\
&&AF + B \left( 2F^2 +1-2J(J+1)-2I(I+1)\right).\nonumber
\end{aligned}$$ Using experimental values for $A$ and $B$ (see Table \[t:ABhfs\]) we calculate the second-order Zeeman shift for Cu, Ag and Au. The results are presented in Tables \[t:AgZ2\] and \[t:CuAuZ2\]. The shift for Ag was studied before [@AgZ2]. Our result differs from theirs; this may be due to a simple calculational error. Table \[t:AgZ2\] presents separate contributions from the shifts in the ground and excited states. One can see that the disagreement may come from the sing error in a particular contribution. Different signs are caused by energy denominators. For example, when we move from first to the second line of the table, the sign of the energy denominator for the ground state contribution changes and so does the contribution itself. Since other contribution remains the same, the total shift must change.
Table \[t:CuAuZ2\] shows the second-order Zeeman shift for $^{63}$Cu and $^{197}$Au. As in the case of $^{107}$Ag the shift is small. Note that Cu has one clock transition with both tiny quadratic shift coefficient and no linear shift. By measuring two or more $F_{gz}=0\rightarrow F_{ez}=0$ Zeeman components and taking appropriate combinations of the corresponding transition frequencies the second-order shift may be substantially reduced.
[The quadratic shift vanishes in the considered approximation for transitions between states with maximum value of $F$ and its projection $F_z$ (see bottom lines of Table \[t:CuAuZ2\]). This is because there are no terms in (\[e:z2\]) which would satisfy the selection rules. Note also that the (non-zero) numbers in Table \[t:CuAuZ2\] should be considered as rough estimations only. This is because of uncertainties of the experimental data for the electric quadrupole hfs constant $B$, in particular for Cu [@Cu-hfs1]. The numbers can change several times depending on which set of data is used.]{}
[cc dddd]{} & & & & &\
2 & 0 & 0.186 & 0.114 & 0.072 & 0.07\
2 & 1 & 0.186 & -0.114 & 0.301 & 0.07\
3 & 1 & -0.186 & -0.114 & -0.072 & -0.3\
[cccc ddd]{} & & & &\
&&&&& &\
&&&&& &\
1 & 0 & 0 & 0 & & -0.759 &\
1 & 0 & 1 & 0 & 0.087 & 0.743 & 0.023\
1 & 0 & 2 & 0 & 0.193 & 0.058 & 0.027\
1 & 0 & 3 & 0 & -0.247 & 0.025 & 0.050\
2 & 0 & 0 & 0 & & -0.792 &\
2 & 0 & 1 & 0 & 0.053 & 0.710 & -0.041\
2 & 0 & 2 & 0 & 0.160 & 0.024 & -0.037\
2 & 0 & 3 & 0 & -0.281 & -0.009 & -0.014\
2 & 0 & 4 & 0 & 0.001 & & -0.037\
2 & $\pm2$ & 3 & $\pm2$ & -0.044 & 0.004 & 0.002\
2 & $\pm2$ & 3 & $\pm3$ & -0.017 & 0.0 & 0.004\
2 & $\pm2$ & 4 & $\pm4$& 0.0 & & 0.0\
Search for new physics
======================
An exceptionally high accuracy of atomic clocks is a great advantage for using them in a search for new physics. The search is conducted by monitoring relative values of different atomic frequencies over a significant time interval. Hypothetical time-variation of the frequency ratio allows multiple interpretations. E.g., the interaction between low-mass scalar dark matter and ordinary matter may lead to oscillation of the fine structure constant and transient variation effect [@DM1; @DM2; @DM3]. In this section we consider time variation of the fine structure constant $\alpha$ ($\alpha = e^2/\hbar c$), Local Position Invariance (LPI) violation and Local Lorentz Invariance (LLI) violation.
Time variation of the fine structure constant
---------------------------------------------
It is convenient to present to parametrise the $\alpha$-dependence of atomic frequencies by the formula $\omega = \omega_0 + q[(\frac{\alpha}{\alpha_0})^2-1]$ [@CJP], where $\alpha_0$ and $\omega_0$ are present-day values of the fine structure constant and the frequency of the transition, $q$ is the sensitivity coefficient which comes from the calculations. To monitor possible frequency change one atomic frequency is measured against another. Then $$\label{e:w1w2}
\frac{\partial}{\partial t}\ln \frac{\omega_1}{\omega_2} = \frac{\dot \omega_1}{\omega_1} - \frac{\dot \omega_2}{\omega_2} =
\left(\frac{2q_1}{\omega_1} - \frac{2q_2}{\omega_2}\right) \frac{\dot \alpha}{\alpha}.$$ The value $K=2q/\omega$ is called an enhancement factor. It shows that if $\alpha$ changes in time then $\omega$ changes $K$ times faster. Calculated values of $q$ and $K$ for different optical clock transitions are presented in Table \[t:q\]. Note that we include not all known clock transitions but only those which are sensitive for $\alpha$-variation searches. There are six transitions where $|K|>1$. The largest values of $|K|$ correspond to the smallest values of transition frequency $\omega$. It would be wrong to say that all these transitions are good for searching for $\alpha$-variation. This is because the accuracy of the measurements is equally important (see also discussion in Ref. [@Yb-DFS]). The true figure of merit is the ratio of the relative frequency shift due to variation of $\alpha$ and the fractional uncertainty of the measurements, $(q/\omega)/(\delta \omega/\omega) = q/\delta \omega$. This ratio does not depend on $\omega$. Therefore, looking for a large value of $K$ caused by the small value of $\omega$ brings no benefit. The value of the relativistic energy shift $q$ is more important. Comparing the values of $q$ for different clock transitions (see Table \[t:q\]) we see that the E2 clock transition in Au is essentially as good as the recently proposed new transitions in neutral ytterbium and only 30% smaller than the octupole transition in the ytterbium ion (YbII).
[l cccdddd dd]{} & & & & & &\
& && & & && & &\
Cu & $3d^{10}4s \ ^2$S$_{1/2}$ &-& $3d^{9}4s^2 \ ^2$D$_{5/2}$ & 11202.565 & -4000 & -0.71 &-48 & 0.98 &\
Cu & $3d^{10}4s \ ^2$S$_{1/2}$ &-& $3d^{9}4s^2 \ ^2$D$_{3/2}$ & 13245.443 & -1900 & -0.29 &-37 & 0.99 &\
Ag & $4d^{10}5s \ ^2$S$_{1/2}$ &-& $4d^{9}5s^2 \ ^2$D$_{5/2}$ & 30242.061 & -11300 & -0.75 & -41 & 0.93 &\
Au & $5d^{10}6s \ ^2$S$_{1/2}$ &-& $5d^{9}6s^2 \ ^2$D$_{5/2}$ & 9161.177 & -38550 & -8.4 & -45 & 0.67 &\
Hg II& $5d^{10}6s \ ^2$S$_{1/2}$ &-& $5d^{9}6s^2 \ ^2$D$_{5/2}$ & 35514.624 & -52200 & -2.94 && 0.68 & 0.2\
Yb& $4f^{14}6s^2 \ ^1$S$_{0}$ &-& $4f^{14}6s6p \ ^3$P$^{\rm o}_{0}$ & 17288.439 & 2714 & 0.31 && 1.12 & 1.20\
Yb& $4f^{14}6s^2 \ ^1$S$_{0}$ &-& $4f^{13}5d6s^2 J=2$ & 23188.518 & -44290 & -3.82&& 0.65 & 1.40\
Yb& $4f^{14}6s6p \ ^1$P$^{\rm o}_{0}$ &-& $4f^{13}5d6s^2 J=2$ & 5900.079 & -43530 & -15 && &\
Yb II& $4f^{14}6s \ ^2$S$_{1/2}$ &-& $4f^{13}6s^2 \ ^2$F$^{\rm o}_{7/2} $ & 21418.75 & -56737 & -5.3 &-135& 0.58 & -1.9\
Yb II& $4f^{14}6s \ ^2$S$_{1/2}$ &-& $4f^{14}5d \ ^2$D$_{3/2} $ & 22960.80 & 10118 & 0.88 &10&1.42& 1.48\
Yb II& $4f^{13}6s^2 \ ^2$F$^{\rm o}_{7/2} $ &-& $4f^{14}5d \ ^2$D$_{3/2} $ & 1542.06 & -66855 & -87 &&&\
LPI violation
-------------
In the standard model extension, the term in the Hamiltonian responsible for the LPI violation can be presented in the form (see, e.g., Ref. [@EEP]) $$\label{e:EEP}
\hat H_{\rm EEP} = c_{00}\frac{2}{3}\frac{U}{c^2}\hat K,$$ where $c_{00}$ is the unknown parameter characterizing the magnitude of the LPI violation, $U$ is the gravitational potential, $c$ is the speed of light, $\hat K= c\gamma_0\gamma^j p_j/2$ is the relativistic operator of kinetic energy in which $\gamma_0$ and $\gamma^j$ are Dirac matrices, and $\mathbf{p}=-i\hbar \mathbf{\nabla}$ is electron momentum operator.
The presence of the term (\[e:EEP\]) in the Hamiltonian would manifest itself via a dependence of the atomic frequencies on the time in the year, caused by the changing Sun-Earth distance leading to change of the Sun’s gravitational potential $U$. As in the case of the $\alpha$-variation, at least two clock transitions are needed to measure one clock frequency against the other. The interpretation of the measurements is based on the formula [@EEP] $$\label{e:REEP}
\frac{\Delta \omega_1}{\omega_1} - \frac{\Delta \omega_2}{\omega_2} = - (R_1 - R_2)\frac{2}{3}c_{00}\frac{\Delta U}{c^2},$$ where $\Delta \omega$ and $\Delta U$ are the change of atomic frequencies and gravitational potential between the measurements, respectively. $R$ in (\[e:REEP\]) is the relativistic factor which describes the deviation of the kinetic energy $E_K$ from the value given by the non-relativistic virial theorem (which states that $E_K=-E$, where $E$ is the total energy), $$\label{e:R}
R_{ab} = - \frac{E_{K,a} - E_{K,b}}{E_a-E_b}.$$ The values of the factor $R$ are calculated in computer codes by varying the value of the kinetic energy operator in the Dirac equation (see Ref. [@EEP] for details).
The results are very sensitive to the many-body effects which means that the effects should be treated very accurately or avoided. Otherwise the results are unstable. A good criterion for the reliability of the results is the achievement of the non-relativistic limit $R=1$. This can be done by setting to zero the value of the fine structure constant $\alpha$ in the computer codes. It turns out that for complicated systems like those considered in present work the best results are obtained by simple estimations based on single-electron consideration. Namely, all clock transitions in Cu, Ag and Au can be considered as a $ns \rightarrow (n-1)d_{5/2}$ single-electron transitions ($n=4,5,6$ for Cu, Ag, Au respectively). Therefore, we just use single-electron energies of theses states in (\[e:R\]). We use the same approach for Hg$^+$ and Yb$^+$.
The results are presented in Table \[t:q\] together with results obtained earlier for other systems. Note, that the results for the transitions involving excitation from the $5d$ shell in Hg$^+$ and the $4f$ shell in Yb$^+$ are different from what was published before. The old calculations were based on a version of the CI method [@CI1; @CI2] which contained a fitting parameter responsible for correct energy interval between states of different configurations. It was assumed that this parameter does not change under variation of the kinetic energy operator. We believe that present results are more reliable because they are free from any assumptions and because they reproduce the non-relativistic limit $R=1$. Note that the values of $R$ for transitions in Yb and Yb$^+$, which do not involve excitation from the $4f$ shell are in good agreement with previous calculations ($R=1.12$ and $R=1.42$, see Table \[t:q\]). This means that present single-electron estimations work well and that accurate many-body calculations are possible for simple systems.
To study the LPI violation one needs to compare two clocks with different values of the relativistic factors $R$ (see formula (\[e:REEP\])) over at least half of a year. Table \[t:q\] shows that there is a wide range of choices for such clock pairs. In particular, the Au clock is practically as good as the Hg$^+$ clock which was used before for this purpose in combination with the Al$^+$ clock [@Hg+clock0; @EEP]. The Cu and Ag clocks, which have $R$ values close to unity, can be used in combination with clocks with large relativistic effects, e.g. Yb or Yb$^+$ clocks.
LLI violation
-------------
The LLI violation term is a tensor operator $$\label{e:LLI}
\hat H_{\rm LLI} = -\frac{1}{6}C_0^{(2)} T_0^{(2)},$$ where $C_0^{(2)}$ is unknown constant and the relativistic form of the $T_0^{(2)}$ operator is given by $T_0^{(2)} = c\gamma_0(\gamma^jp_j-3\gamma^3p_3)$. To study the effect of the LLI violating term (\[e:LLI\]) one needs long-lived atomic states with large value of the total electron angular momentum $J$, $J>1/2$. All clock states of Cu, Ag and Au satisfy this requirement. This term should cause a dependence of the atomic frequencies on the apparatus orientation in space (e.g., due to Earth rotation). Interpretation of the measurements requires knowing the values of the reduced matrix elements of the operator $T_0^{(2)}$ for the clock states. We calculate these matrix elements using the CIPT method to obtain wave functions and the RPA method to obtain effective operator for valence electrons.
The results are presented in Table \[t:q\]. The results of earlier calculations for Yb II [@Yb+LLI] are also presented for comparison. In contrast to the search of the $\alpha$-variation and LPI violation, one clock state is sufficient for the search of the LLI violation. The comparison of frequencies is done for states with different projections of the total angular momentum $J$ [@Lia; @Yb+LLI]. The large value of the matrix element is important but it is not the most important parameter, e.g. the lifetime of the metastable state is even more important (see, e.g. [@Yb+LLI] for more discussion). The calculations for Cu and Au show that these systems are also suitable for the search of the LLI violation. Ag appears less well suited because its upper clock state has a significantly shorter lifetime.
Conclusion
==========
We studied electric quadrupole transitions between ground and excited metastable states of Cu, Ag and Au and demonstrated that the transitions have all features of optical clock transitions. Important systematic effects such as black-body radiation shift, Stark and Zeeman shifts, etc. are similar to or smaller than in current top-performing optical clocks. On the other hand, some of the transitions are more sensitive to new physics beyond the standard model than the currently used neutral-atom optical clocks, or of complementary value. The studied effects included time variation of the fine structure constant, Local Lorentz Invariance violation and Local Position Invariance violation.
This work was supported by the National Natural Science Foundation of China (Grant No. 11874090) and the Australian Research Council. V.A.D. would like to express special thanks to the Institute of Applied Physics and Computational Mathematics in Beijing for its hospitality and support. This research includes computations using the computational cluster Katana supported by Research Technology Services at UNSW Sydney.
Stark and electric quadrupole shifts.
=====================================
Stark shift of the frequency of the transition between atomic states $a$ and $b$ due to interaction with residual static electric field $\varepsilon$ is $$\label{eq:St}
\delta \omega_{ab} = -\Delta \alpha_{ab}(0)\left(\frac{\varepsilon}{2}\right)^2,$$ where $\Delta \alpha_{ab}(0)$ is the difference between static scalar polarizabilities of states $a$ and $b$. The shift is quadratic in electric field and usually small. It is further suppressed for considered clock transitions due to small difference in the polarizabilities (see Table \[t:pol\]).
The energy shift due to a gradient of a residual static electric field $\varepsilon$ is described by a corresponding term in the Hamiltonian $$\label{eq:HQ}
\hat H_Q = -\frac{1}{2}\hat Q\frac{\partial \varepsilon_z}{\partial z},$$ where $\hat Q$ is the atomic quadrupole moment operator ($\hat Q = |e|r^2 Y_{2m}$, the same as for E2 transitions). The energy shift of a state with total angular momentum $J$ is proportional of the atomic quadrupole moment of this state. It is defined as twice the expectation value of the $\hat Q$ operator in the stretched state $$\label{eq:Q}
Q_J = 2\langle J,J_z=J|\hat Q| J,J_z=J \rangle.$$
Calculations using the CIPT method for wave functions and the RPA method for the operator give the values $Q_J=0.431$ a.u. for the $^2$D$_{5/2}$ clock state of Cu, $Q_J=0.296$ a.u. for the $^2$D$_{3/2}$ clock state of Cu, $Q_J=0.966$ a.u. for the clock state of Ag, and $Q_J=1.47$ a.u. for the clock state of Au. Quadrupole moments of the ground states of these atoms are zero due to small value of the total electron angular momentum ($J$=1/2).
Consider transitions between hyperfine structure (hfs) components of the ground and clock states with definite values of the total angular momentum $F$. The quadrupole shift is proportional to $3F_z^2-F(F+1)$, where $F_z$ is the projection of $\mathbf{F}$. For $F=3$ and $F_z=\pm 2$ this factor is zero and the quadrupole shift vanishes. Note that clock states with $F=3$ exist for all stable isotopes of all three considered atoms (see Table \[t:isotopes\]). [Using these states would lead to a linear Zeeman shift. It cancels out by averaging over the transition frequencies to the states with $F_z=-2$ and $F_z=2$.]{}
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[^1]: On the inner surface of a metallic vacuum chamber there can be spatial variations of the electrostatic potential of order 0.1 V. The typical size of a vacuum chamber may be 10 cm. Thus, the electric field gradient is smaller than 0.1 V/(10 cm$)^2$. Assuming a typical quadrupole moment $Q \sim 1$ a.u. leads to a negligible quadrupole shift $\sim 10^{-5}$ Hz. The corresponding Stark shift is $\sim 10^{-7}$ Hz.
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abstract: 'The homogeneity degree of a topological space $X$ is the number of orbits of the action of the homeomorphism group of $X$ on $X$. We initiate a study of dendroids of small homogeneity degree, beginning with fans. We classify all smooth fans of homogeneity degree $3$, and discuss non-smooth fans and prove some results on degree $4$.'
address:
- 'Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, D.F. 04510, Mexico'
- 'Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, Box 5002, North Bay, Ontario, Canada, P1B 8L7'
- 'Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, Box 5002, North Bay, Ontario, Canada, P1B 8L7'
author:
- Gerardo Acosta
- 'Logan C. Hoehn'
- Yaziel Pacheco Juárez
bibliography:
- 'FansHomogeneity.bib'
title: Homogeneity degree of fans
---
[^1]
Introduction
============
For a topological space $X$ we denote by ${ \mathcal{H}(X)}$ the group of homeomorphisms of $X$ onto itself. Given $x \in X$, we denote by ${\mathrm{Orb}_X}(x)$ the orbit of $x$ under the action of ${ \mathcal{H}(X)}$ on $X$; that is, ${\mathrm{Orb}_X}(x) = \{h(x): h \in { \mathcal{H}(X)}\}$. The *homogeneity degree* of $X$ is the number of orbits for the action of ${ \mathcal{H}(X)}$ on $X$. Alternatively, we say $X$ is *$\frac{1}{n}$-homogeneous* to mean that $X$ has homogeneity degree $n$.
A space is *homogeneous* if its homogeneity degree is $1$. Homogeneity is a classical and well-studied notion in continuum theory; however, some important classes of spaces (e.g. dendroids) include no homogeneous spaces, and thus do not interact with this theory. In this paper, we aim to demonstrate that an appropriate notion of homogeneity for dendroids, especially for fans, is $\frac{1}{3}$-homogeneous.
An emerging theme in the study of continua (or more generally compact metric spaces) is that spaces of low homogeneity degree tend to be rare and remarkable, and conversely many well-known and interesting compact metric spaces have a low degree of homogeneity. This can be seen in recent work, e.g. in the classification of all homogeneous plane continua (more generally compact spaces in the plane) completed in [@HoehnOversteegen], and the classification of all $\frac{1}{3}$-homogeneous dendrites [@13-dendrites].
In this paper we initiate a study of dendroids of low homogeneity degree, beginning with fans. We classify all $\frac{1}{3}$-homogeneous smooth fans, and prove that there are no smooth fans with homogeneity degree $4$. We also consider some properties of non-smooth fans of homogeneity degrees $3$ or $4$. It is not yet known whether these exist – see Problem \[nonsmooth exist\].
Definitions and notation
------------------------
For a topological space $X$ and $A \subset X$ the symbol ${\mathrm{Cl}_X}(A)$ denotes the closure of $A$ in $X$. In this paper, all spaces considered will be metric spaces, and the metric will always be denoted by $d$. When we refer to the distance or convergence of closed subsets of $X$, it will be understood that we are considering the Hausdorff metric.
A *continuum* is a compact connected metric space. An *arc* is a space homeomorphic to the interval $[0,1] = I$. We call any space homeomorphic to the standard middle-third Cantor set a *Cantor set*. A *dendroid* is an arcwise connected and hereditarily unicoherent continuum. Given a dendroid $X$ and points $p,q \in X$, we denote by $pq$ the unique arc in $X$ having $p$ and $q$ as its endpoints, and put $(pq) = pq - \{p,q\}, [pq) = pq - \{q\}$ and $(pq] = pq - \{p\}$.
For a dendroid $X$ we say that $p$ has *order* $r$ (in the classical sense), in symbols ${\mathrm{Ord}_X}(p) = r$, if $p$ is a common endpoint of exactly $r$ arcs in $X$ which are disjoint from one another beyond $p$. A point $p \in X$ is an *endpoint* of $X$ if ${\mathrm{Ord}_X}(p) = 1$, a *ramification point* of $X$ if ${\mathrm{Ord}_X}(p) > 2$, and an *ordinary point* if ${\mathrm{Ord}_X}(p) = 2$. We denote by $E(X)$, $R(X)$, and $O(X)$ the set of endpoints, the set of ramification points, and the set of ordinary points of $X$ respectively. It is clear that any homeomorphism of $X$ onto itself preserves these three sets.
A *fan* is a dendroid $X$ with exactly one ramification point called the *top* of $X$. Note that in a fan $X$ the sets $E(X)$, $O(X)$, and $R(X)$ are all nonempty, hence any fan has homogeneity degree greater than or equal to $3$. If $X$ is a fan with top $t$, then $X = \bigcup_{e \in E(X)} te$ and $te \cap td = \{t\}$ for any two distinct $e,d \in E(X)$. It is clear that $O(X)$ is dense in $X$. From [@Lelek Theorem 7.5, p.311], we have that $E(X)$ is a $G_\delta$ subset of $X$.
A dendroid $X$ is *smooth* if there is a point $p \in X$ such that for each sequence $\{a_n\}_n$ of points of $X$ which converges to a point $a \in X$, the sequence of arcs $\{pa_n\}_n$ converges to the arc $pa$. In a smooth fan, the top of the fan can always be used for the point $p$ in the previous definition.
Effros’ Theorem
---------------
We recall some fundamental notions from descriptive set theory. A *Polish space* is a separable, completely metrizable space. A subset of a Polish space $X$ is itself Polish if and only if it is a $G_\delta$ subset of $X$. If $X$ is a compact metric space, then ${ \mathcal{H}(X)}$ is a *Polish group*, i.e. a separable, completely metrizable topological group.
It is a consequence of a theorem of Effros [@Effros Theorem 2.1, p.39] that if $X$ is a Polish space and $G$ is a Polish group acting transitively on $X$, then for any $x \in X$ and any neighborhood $U$ of the identity in $G$, the set $Ux = \{g \cdot x: g \in U\}$ is an open neighborhood of $x$ in $X$. This result has seen extensive use in the study of homogeneous continua since it was first applied in the area by Ungar in [@Ungar]. We will demonstrate in this paper that it is also useful in the study of spaces with homogeneity degree greater than $1$. We will use the following consequence ($\dagger$) of the theorem.
An *$\varepsilon$-homeomorphism* is an element $h \in { \mathcal{H}(X)}$ such that $d \big( x,h(x) \big) < \varepsilon$ for all $x \in X$.
- Let $X$ be a compact metric space and let $x \in X$. Suppose ${\mathrm{Orb}_X}(x)$ is a $G_\delta$ subset of $X$. Then for any $\varepsilon > 0$ there exists $\delta > 0$ such that for any $v \in X$ with $d(x,v) < \delta$ there exists an $\varepsilon$-homeomorphism $h: X \to X$ such that $h(x) = v$.
Preliminary results
===================
In [@Lelek Theorem 2.1, p.302], it is proved that if $X$ is a dendroid, $E(X)$ does not contain any non-degenerate continua.
When we say that a subset $A$ of $X$ is an *orbit of $X$*, we mean that $A$ is an orbit under the action of ${ \mathcal{H}(X)}$ on $X$.
\[lemanlc cantor\] Let $X$ be a smooth fan which is not locally connected. If $E(X)$ is an orbit of $X$, then $O(X) \cap {\mathrm{Cl}_X}\big( E(X) \big) \neq \emptyset$ or ${\mathrm{Cl}_X}\big( E(X) \big)$ is a Cantor set.
Assume that $O(X) \cap {\mathrm{Cl}_X}\big( E(X) \big) = \emptyset$.
Denote by $N(X)$ the set of points where $X$ is not locally connected. By [@Nadler92 Theorem 5.12, p.76], $N(X)$ contains a *continuum of convergence*, i.e. a non-degenerate continuum $K \subset N(X)$ for which there is a sequence $\{A_n\}_n$ of pairwise disjoint subcontinua of $X$ converging to $K$ such that, for each $n \in {\mathbb{N}}$, we have $K \cap A_n = \emptyset$. Since $E(X)$ does not contain any non-degenerate continuum, neither does $E(X) \cup \{t\}$, and so $K \cap O(X) \neq \emptyset$.
Choose $x \in K \cap O(X)$. We can assume $t \notin A_n$ for all $n$. Then for each $n$ there is $e_n \in E(X)$ such that $A_n \subset te_n$. We can assume (by taking a subsequence if needed) that the sequence $\{e_n\}_n$ converges to a point $e \in X$. Because $X$ is smooth, $te_n \to te$. Note that $x \in (te)$ and so $e \neq t$. Also, $e \notin O(X)$ since $O(X) \cap {\mathrm{Cl}_X}\big( E(X) \big) = \emptyset$. Therefore, $e \in E(X)$. Since $E(X)$ is an orbit, this means that each point in $E(X)$ is a limit of a sequence of endpoints.
Note that by our initial assumption ${\mathrm{Cl}_X}\big( E(X) \big) \subset E(X) \cup \{t\}$. Since $E(X)$ does not contain any non-degenerate continuum, neither does $E(X) \cup \{t\}$. Hence, ${\mathrm{Cl}_X}\big( E(X) \big)$ is totally disconnected and perfect, and consequently it is a Cantor set.
\[abaCantor\] Let $C$ denote the standard middle-thirds Cantor set and $F_C$ the *Cantor fan* $(C \times [0,1]) / (C \times \{1\})$. It is known that a fan is smooth if and only if it is embeddable in $F_C$ (see [@Onfans Theorem 9, p.27], [@Imacantorfan Proposition 4, p.165] and [@Eberhart Corollary 4, p.90]).
\[F\_CW\] We now construct the *shrinking Cantor fan* $F_{C_\omega}$. For each $i \in {\mathbb{N}}$ put $$F_i = \bigcup \left\{ tp_c: c \in C \cap \left[ \frac{(3)^{i-1}-1}{(3)^{i-1}}, \frac{(3)^i - 2}{(3)^i} \right] \textrm{ and } p_c = \left( c, \frac{1}{2^{i-1}} \right) \right\} .$$ Then $\{F_i\}_i$ is a sequence of fans homeomorphic to the Cantor fan tending to $\{t\}$ such that the intersection of any two of them is $\{t\}$. Define $$F_{C_\omega} = \bigcup_{i \in {\mathbb{N}}} F_i .$$ Note that $F_{C_\omega}$ is a smooth fan, and ${\mathrm{Cl}_{F_{C{\omega}}}\!\left(E(F_{C_\omega})\right)} = E(F_{C_\omega}) \cup \{t\}$.
The following theorem gives us a characterization of $F_C$ and $F_{C_\omega}$. We use this characterization to classify $\frac{1}{3}$-homogeneous smooth fans in Theorem \[aba13\] and to prove that there is no $\frac{1}{4}$-homogeneous smooth fan in Theorem \[nosmooth1/4\].
\[Cantor-ShrinkCantor\] Let $X$ be a smooth fan not locally connected such that $E(X)$ is an orbit of $X$. Then:
1. $X$ is homeomorphic to $F_C$ if and only if $E(X)$ is closed in $X$; and
2. $X$ is homeomorphic to $F_{C_\omega}$ if and only if ${\mathrm{Cl}_X}\big( E(X) \big) = E(X) \cup \{t\}$.
In both 1) and 2) the left to right implication follows at once. Now, for 1) assume that $E(X)$ is closed in $X$. By Lemma \[lemanlc cantor\], $E(X)$ is a Cantor set. We can suppose that $X \subset F_C$ and $E(X) \subset E(F_C) = C \times \{0\}$ (see [@FansKelley Theorem 1 (2), p.74]). Now it is easy to see that $X$ is homeomorphic to $F_C$.
For 2) suppose that ${\mathrm{Cl}_X}\big( E(X) \big) = E(X) \cup \{t\}$. As a consequence of Lemma \[lemanlc cantor\], ${\mathrm{Cl}_X}\big( E(X) \big)$ is a Cantor set. So there is a sequence $\{E_i\}_i$ of disjoint, open and closed sets of ${\mathrm{Cl}_X}\big( E(X) \big)$ such that $$E(X) = {\mathrm{Cl}_X}\big( E(X) \big) - \{t\} = \bigcup_{i \in {\mathbb{N}}} E_i \hspace{0.3cm} \textrm{and} \hspace{0.3cm} \lim_{i \to \infty} E_i = \{t\} .$$
For each $i \in {\mathbb{N}}$, define $X_i = \bigcup_ {e \in E_i} te$. Observe that for each $i \in {\mathbb{N}}$, the set $E(X_i) = E_i$ is a Cantor Set. Hence $X_i$ is homeomorphic to the Cantor fan.
Given that $\{E_i\}_i$ converges to $\{t\}$ and $X$ is smooth, the sequence of fans $\{X_i\}_i$ converges to $\{t\}$. Note that $X = \bigcup X_i$. It is now straightforward to see that $X$ is homeomorphic to $F_{C_\omega}$.
$\frac{1}{3}$-homogeneous smooth fans
=====================================
Given $n \in {\mathbb{N}}- \{1,2\}$, the *simple $n$-od* is the unique (up to homeomorphism) fan with exactly $n$ endpoints. Simple $n$-ods are sometimes called *finite fans*. Clearly any simple $n$-od is $\frac{1}{3}$-homogeneous.
\[fomegad\] We denote by $F_\omega$ the unique (up to homeomorphism) locally connected fan whose top has infinite order. That is, $F_\omega$ is homeomorphic to $\bigcup_{i=1}^\infty ob_i$, constructed in ${\mathbb{R}}^2$, where $o = (0,0)$, $b_i = \left( \frac{1}{i},\frac{1}{i^2} \right)$, and $ob_i$ is the straight line segment joining $o$ and $b_i$ for each $i \in {\mathbb{N}}$. Clearly $F_\omega$ is $\frac{1}{3}$-homogeneous as well.
Because the Cantor set is homogeneous, it is straightforward to see that the Cantor fan $F_C$ and the shrinking Cantor fan $F_{C_\omega}$ are both $\frac{1}{3}$-homogeneous.
In 1961 A. Lelek constructed a fan as an example of a dendroid whose set of endpoints is dense and 1-dimensional ([@Lelek 9, p.314]). Such a fan is called the *Lelek fan*, and it is characterized as the only smooth fan whose set of endpoints is dense (see [@Lelekunique Corollary, p.33] and [@BulaOversteegen Theorem, p.529]).
In [@Aarts-Oversteegen93] Jan M. Aarts and Lex G. Oversteegen constructed the *hairy arc* $H$. This is a smooth dendroid constructed as the intersection of a sequence of subsets of $[0,1]^2$ containing the base $B = [0,1] \times \{0\}$ in such a way that the closure of each component of $H - B$ is an arc, called a *hair*, these hairs are pairwise disjoint, the set of endpoints $E(H)$ is dense in $H$, and $E(H) \cup B$ is connected ([@Aarts-Oversteegen93 Corollary 2.5, p.905]). The following result is well known.
\[H/B Lelek\] The Lelek fan is homeomorphic to $H/B$.
Let $X = H/B$ and $q$ be the quotient function from $H$ to $X$. Then $q$ identifies the base $B$ into a point $x_0 \in X$, and $q|_{H-B}$ is a homeomorphism from $H - B$ into $X - \{x_0\}$. Hence $q$ is monotone and by [@SmoothDendroids Corollary 10, p.309], $X$ is a smooth dendroid. Since $R(H) \subset B$, we have $q(R(H)) = \{x_0\}$ and so $X$ is a smooth fan with top $x_0$.
We have $E(X)$ is dense in $X$, because $q|_{H-B}$ is a homeomorphism and $E(H)$ is dense in $H-B$. Thus $X$ is homeomorphic to the Lelek fan.
In [@Aarts-Oversteegen95 Corollary 1.5, p.285] it is proved that there are exactly five homeomorphism types of points in $H$, namely
1. endpoints of the base $B$,
2. endpoints of hairs,
3. hairless base points,
4. base points with hair attached,
5. interior points of hairs.
It follows easily from this and Lemma \[H/B Lelek\] that the Lelek fan is $\frac{1}{3}$-homogeneous.
We now prove that the fans discussed above comprise the complete list of $\frac{1}{3}$-homogeneous smooth fans.
\[aba13\] A smooth fan is $\frac{1}{3}$-homogeneous if and only if it is homeomorphic to one of the following fans.
1. A simple $n$-od, for some $n \in {\mathbb{N}}-\{1,2\}$,
2. the dendrite $F_\omega$,
3. the Cantor fan,
4. the Lelek fan,
5. the shrinking Cantor fan $F_{C_\omega}$.
It has been observed above that each of the fans in this list is $\frac{1}{3}$-homogeneous.
Suppose that $X$ is a $\frac{1}{3}$-homogeneous fan. It is not difficult to see that the only locally connected $\frac{1}{3}$-homogeneous fans are $F_\omega$ and simple $n$-ods.
Assume, then, that $X$ is not locally connected. If $E(X)$ is closed, from Theorem \[Cantor-ShrinkCantor\] part 1), we have that $X$ is homeomorphic to the Cantor fan. If $E(X)$ is dense, $X$ is homeomorphic to the Lelek fan by the characterization mentioned above. Finally, suppose that $E(X)$ is neither closed nor dense in $X$. We want to prove that ${\mathrm{Cl}_X}\big( E(X) \big) = E(X) \cup \{t\}$. Suppose that ${\mathrm{Cl}_X}\big( E(X) \big) \cap O(X) \neq \emptyset$. This implies $O(X) \subset {\mathrm{Cl}_X}\big( E(X) \big)$, because $E(X)$ and $O(X)$ are orbits of $X$. It follows that $E(X)$ is dense, which contradicts our assumption. Consequently, ${\mathrm{Cl}_X}\big( E(X) \big) \subset E(X) \cup \{t\}$ and since $E(X)$ is not closed, the equality holds. By Theorem \[Cantor-ShrinkCantor\] part 2), $X$ is homeomorphic to $F_{C_\omega}$.
$\frac{1}{4}$-homogeneous smooth fans
=====================================
In this section we prove the following result.
\[nosmooth1/4\] There is no $\frac{1}{4}$-homogeneous smooth fan.
We first establish two auxiliary results, which apply to both smooth and non-smooth fans.
\[aba1/4\] Let $X$ be a $\frac{1}{4}$-homogeneous fan with top $t$. Then $O(X) = O_1 \cup O_2$, where $O_1$ and $O_2$ are two orbits of $X$ and for each $e \in E(X)$, both $t$ and $e$ belong to ${\mathrm{Cl}_X}\big( O_1 \cap (te) \big)$ and to ${\mathrm{Cl}_X}\big( O_2 \cap (te) \big)$.
It is easy to see that $E(X)$ must be one of the orbits of $X$ and $O(X)$ is the union of two orbits of $X$, $O(X) = O_1 \cup O_2$. Let $e \in E(X)$. Since $E(X)$ is an orbit of $X$, the two orbits $O_1$ and $O_2$ of ordinary points must both intersect $(te)$.
Suppose for a contradiction that ${\mathrm{Cl}_X}\big( O_1 \cap (te) )$ does not contain $e$. Then there is a point $x \in (te)$ such that $O_1 \cap (xe) = \emptyset$, but for any $x' \in (tx)$ we have $O_1 \cap (x'x] \neq \emptyset$. Let $y \in (xe)$, which implies $y \in O_2$.
If $x \in O_2$, then we can choose a homeomorphism $h: X \to X$ such that $h(y) = x$. But then $h(x) \in (tx)$, and $h \big( (xe) \big) = \big( h(x)e \big) \subset O_2$, contradicting the choice of $x$. Thus $x \in O_1$. By a similar argument, we can see that $(tx] \not \subset O_1$. But then if $z \in O_2 \cap (tx)$, we can choose a homeomorphism $h: X \to X$ such that $h(z) = y$, and then we have $h(x) \in O_1 \cap (ye) \subset O_1 \cap (xe) = \emptyset$, again a contradiction. Therefore $e \in {\mathrm{Cl}_X}\big( O_1 \cap (te) )$.
The proofs that $t \in {\mathrm{Cl}_X}\big( O_1 \cap (te) \big)$ and $t,e \in {\mathrm{Cl}_X}\big( O_2 \cap (te) \big)$ are similar.
\[T,O-T\] If $X$ is a $\frac{1}{4}$-homogeneous fan, then either $O(X) \cap {\mathrm{Cl}_X}\big( E(X) \big) = \emptyset$ or $O(X) \subset {\mathrm{Cl}_X}\big( E(X) \big)$.
Suppose for a contradiction that $X$ is a $\frac{1}{4}$-homogeneous fan such that $O_1 = O(X) \cap {\mathrm{Cl}_X}\big( E(X) \big)$ and $O_2 = O(X) - {\mathrm{Cl}_X}\big( E(X) \big)$ are both nonempty. This means that these sets $O_1$ and $O_2$ must be orbits of $X$.
Take a point $x \in O_1$ and a sequence $\{e_n\}_n$ of points in $E(X)$ converging to $x$. Denote by $e$ the endpoint such that $x \in te$. From Lemma \[aba1/4\], we can choose $y \in (tx) \cap O_2$. Note that $tx \subset \lim_{n \to \infty} te_n$. Hence, for each $n$ we can choose $y_n \in te_n$ such that $y_n \to y$. Since $O_2 = X - \big( \{t\} \cup {\mathrm{Cl}_X}\big( E(X) \big) \big)$ is open in $X$, we can assume that $y_n \in O_2$ for all $n$.
Put $\varepsilon = \frac{1}{2} d(x,e)$. Since $O_2$ is an open orbit, by Effros’ Theorem ($\dagger$) there exists $\delta > 0$ such that, if $v \in O_2$ with $d(x,v) < \delta$, then there is an $\varepsilon$-homeomorphism of $X$ onto itself sending $y$ to $v$. Choose $n \in {\mathbb{N}}$ such that $d(y,y_n) < \delta$ and $d(x,e_n) < \varepsilon$. Then there is an $\varepsilon$-homeomorphism $h: X \to X$ such that $h(y) = y_n$. Observe that $h(e) = e_n$. But $d(e,e_n) \geq d(e,x) - d(x,e_n) > 2\varepsilon - \varepsilon = \varepsilon$, which is a contradiction since $h$ is a $\varepsilon$-homeomorphism.
Suppose for a contradiction that $X$ is a $\frac{1}{4}$-homogeneous smooth fan with top $t$. It is easy to see that $X$ cannot be locally connected, as the only locally connected fans are simple $n$-ods and $F_\omega$. Since $E(X)$ is an orbit of $X$ and $F_C$ and $F_{C_\omega}$ are $\frac{1}{3}$-homogeneous, by Theorem \[Cantor-ShrinkCantor\], it follows that ${\mathrm{Cl}_X}\big( E(X) \big)$ is not contained in $E(X) \cup \{t\}$. Note that $E(X)$ is not dense, because in that case $X$ would be the Lelek fan which is also $\frac{1}{3}$-homogeneous. It follows that $O(X) \cap {\mathrm{Cl}_X}\big( E(X) \big) \neq \emptyset$ and $O(X) \not \subset {\mathrm{Cl}_X}\big( E(X) \big)$, which contradicts Lemma \[T,O-T\].
Non-smooth fans
===============
For the remainder of this section, let $X$ be a fan with top $t$.
Given $a \in X$, we say that $X$ is *smooth at $t$ with respect to* $a$ if for every sequence $\{a_n\}_n$ converging to $a$, we have $ta_n \to ta$. Define the set $$S(X) = \{a \in X: X \textrm{ is smooth at } t \textrm{ with respect to } a \} .$$ From the definition, it is clear that $X$ is smooth if and only if $S(X) = X$, and $S(X)$ is invariant under homeomorphisms. In [@SmoothandKelley Corollary 10, p.124] it is proved that $S(X)$ is a dense $G_\delta$-subset of $X$.
\[Etsmooth\] If for each $x \in X - \{t\}$ we have $(tx) \cap S(X) \neq \emptyset$, then $E(X) \subset S(X)$.
Suppose contrary to the claim that there is a point $e \in E(X) - S(X)$. Then there is a sequence $\{e_n\}_n$ of points converging to $e$ such that the arcs $te_n$ converge to a continuum $Y \neq te$. Observe that $Y \supsetneq te$. Take $a \in Y - te$ and $b \in (te)$. From our hypothesis we can assume that $a$ and $b$ belong to $S(X)$. Since $a,b \in Y$, for each $n$ there are $a_n,b_n \in te_n$ such that $a_n \to a$ and $b_n \to b$. Note that $ta_n \to ta$ and $tb_n \to tb$. Since $b \notin ta$, we can suppose that $b_n \notin ta_n$ for all $n$. This means that $ta_n \subset tb_n$ for each $n$. Hence $ta \subset tb$. This contradicts the choice of $a$ and $b$.
In particular, Lemma \[Etsmooth\] implies that if $O(X) \subset S(X)$ then $E(X) \subset S(X)$ as well. It is easy to prove in this case that $t \in S(X)$ too. Hence, we obtain the following result:
\[Xsmooth\] If $X$ is a fan with $O(X) \subset S(X)$, then $X$ is smooth.
A space $X$ is *colocally connected* at a point $p$ in $X$, provided that each neighborhood of $p$ contains a neighborhood $V$ of $p$ such that $X - V$ is connected.
\[13-homo nonsmooth\] If $X$ is a $\frac{1}{3}$-homogeneous non-smooth fan, then:
1. $X$ is colocally connected at each point of $E(X)$;
2. $E(X)$ is dense in $X$; and
3. $S(X) = E(X) \cup \{t\}$.
Let $X$ be a $\frac{1}{3}$-homogeneous non-smooth fan. The three orbits of $X$ are $\{t\}$, $O(X)$, and $E(X)$. According to [@DendroidsK-Minc Theorem 3.5, p.235 and Theorem 4.1, p.237], the set of endpoints at which $X$ is colocally connected is non-empty. Because $E(X)$ is an orbit, we obtain 1).
From Corollary \[Xsmooth\] and the fact that $O(X)$ is an orbit, we have that $O(X) \cap S(X) = \emptyset$. Thus $S(X) \subset E(X) \cup \{t\}$. Since $S(X)$ is dense, it intersects $E(X)$. Hence, $$E(X) \subset S(X) \subset E(X) \cup \{t\} .$$ It follows that $E(X)$ is also dense.
Suppose for a contradiction that $t \notin S(X)$. Then there is a sequence $\{t_n\}_n$ converging to $t$ such that the arcs $\{tt_n\}_n$ do not converge to $\{t\}$. In this case we may assume, by taking a subsequence, that there exists a point $b \in O(X)$ and a sequence $\{b_n\}_n$ such that $b_n \in (tt_n)$ and $b_n \to b$. Since $O(X)$ is an orbit, it follows that the same is true for every ordinary point; that is, for every $c \in O(X)$, there exist sequences $\{s_n\}_n$ and $\{c_n\}_n$ such that $c_n \in (ts_n)$, $s_n \to t$, and $c_n \to c$. Since $O(X)$ is dense in $X$, we have the same property for endpoints $c \in E(X)$. It is straightforward to see that this contradicts the fact that $X$ is colocally connected at each of its endpoints. Therefore $t \in S(X)$. This completes the proof of 3).
\[14-homo nonsmooth\] If $X$ is a $\frac{1}{4}$-homogeneous non-smooth fan, then:
1. $X$ is colocally connected at each point of $E(X)$;
2. $E(X)$ is dense in $X$; and
3. $S(X) \supset E(X) \cup \{t\}$.
Let $X$ be a $\frac{1}{4}$-homogeneous non-smooth fan. Since $E(X)$ is an orbit of $X$, we obtain 1) in exactly the same way as in the proof of Theorem \[13-homo nonsmooth\].
To prove $t \in S(X)$, we can use a similar argument as in the proof Theorem \[13-homo nonsmooth\]. The only difference is that in the present situation $O(X)$ is not an orbit of $X$, so instead we must use the result of Lemma \[aba1/4\], that for each $e \in E(X)$ the two orbits of ordinary points in $(xe)$ both accumulate on $e$.
It remains to prove that $E(X)$ is dense and $E(X) \subset S(X)$.
If $S(X) \cap O(X) = \emptyset$, then as in the proof of Theorem \[13-homo nonsmooth\], since $S(X)$ is dense it follows that $E(X) \subset S(X)$, and hence $E(X)$ is dense as well.
Assume for the remainder of the proof that $S(X) \cap O(X) \neq \emptyset$. In this case the four orbits of $X$ are: $$\{t\}, \; E(X), \; O_1 = O(X) \cap S(X), \textrm{ and } O_2 = O(X) - S(X) .$$
From Lemmas \[aba1/4\] and \[Etsmooth\], we conclude that $E(X) \subset S(X)$. To prove $E(X)$ is dense, according to Lemma \[T,O-T\] it suffices to prove that ${\mathrm{Cl}_X}\big( E(X) \big) \cap O(X) \neq \emptyset$, for in this case $O(X) \subset {\mathrm{Cl}_X}\big( E(X) \big)$ and hence ${\mathrm{Cl}_X}\big( E(X) \big) = X$.
Suppose for a contradiction that ${\mathrm{Cl}_X}\big( E(X) \big) \cap O(X)$ is empty. Since $S(X)$ is dense, it follows that $O(X) \subset {\mathrm{Cl}_X}(O_1)$. Hence, $O_1$ is dense in $X$.
We make the following claim:
- For any $x \in O_2$, there exist $a,b \in O_1$ such that $x \in (ab) \subset O_2$.
To see this, suppose $x \in O_2$ and let $e \in E(X)$ be such that $x \in (te)$. Since $x \notin S(X)$, there is a sequence of arcs $\{tx_n\}_n$ such that $x_n \to x$ but $tx_n \to Y \neq tx$, where $Y$ is a subcontinuum of $X$. Since $t \in S(X)$, we must have $Y = tx'$ for some $x' \in (te)$ with $x \in (tx')$. Clearly this implies that $[xx') \subset O_2$. Since $O_2$ is an orbit, it follows that every point of $O_2$ is part of an interval in $O_2$. The claim ($\star$) then follows from Lemma \[aba1/4\].
Now take any $x \in O_2$. Because $O_1$ is dense, there is a sequence $\{a_n\}_n$ of points in $O_1$ such that $a_n \to x$. For each $n$ let $e_n \in E(X)$ such that $a_n \in (te_n)$. We can assume that $\{e_n\}_n$ is convergent to a point $e \in {\mathrm{Cl}_X}\big( E(X) \big) \subset E(X) \cup \{t\}$. It follows easily from the fact that $E(X) \cup \{t\} \subset S(X)$ that $e \in E(X)$, and $x \in te = \lim_{n \to \infty} te_n$.
Recall that by [@Lelek Theorem 7.5, p.311], $E(X)$ is a $G_\delta$ orbit, so we can apply Effros’ theorem ($\dagger$) to it. Put $\varepsilon = \frac{1}{2} d \big( x, O_1 \cap te) \big)$. Observe that $\varepsilon > 0$ by ($\star$). Let $\delta$ be given by Effros’ theorem ($\dagger$) and choose $n \in {\mathbb{N}}$ such that $d(x,a_n) < \varepsilon$ and $d(e,e_n) < \delta$. Then there is an $\varepsilon$-homeomorphism $h: X \to X$ such that $h(e_n) = e$. Note that $h(te_n) = te$ and $d(x,h(a_n)) \geq 2\varepsilon$, because $h(a_n) \in O_1 \cap te$. So $$d \big( a_n,h(a_n) \big) \geq d \big( x,h(a_n) \big) - d(x,a_n) \geq 2\varepsilon - \varepsilon = \varepsilon .$$ This is a contradiction since $d \big( p,h(p) \big) < \varepsilon$ for all $p \in X$. Hence, $O(X) \cap {\mathrm{Cl}_X}\big( E(X) \big) \neq \emptyset$.
Now, from Lemma \[T,O-T\] we have that $O(X) \subset {\mathrm{Cl}_X}\big( E(X) \big)$. Therefore $E(X)$ is dense.
Questions
=========
Theorems \[13-homo nonsmooth\] and \[14-homo nonsmooth\] provide some information about how a $\frac{1}{3}$-homogeneous or $\frac{1}{4}$-homogeneous non-smooth fan would have to look, but as yet we are not aware of any examples of such objects. To complete the classification of all $\frac{1}{3}$-homogeneous or $\frac{1}{4}$-homogeneous fans, one must answer the following question:
\[nonsmooth exist\] Does there exist a $\frac{1}{3}$-homogeneous non-smooth fan? a $\frac{1}{4}$-homogeneous non-smooth fan?
We have confined our attention here to fans, as they comprise a simple class of dendroids, and hence make a good starting point for the study of homogeneity degree of dendroids. In the classification of all $\frac{1}{3}$-homogeneous dendrites given in [@13-dendrites], many interesting dendrites appear, including the locally connected fans, the universal ($n$-branching) dendrites, and the ($n$-branching) Gehman dendrites. It would be interesting to extend this to a classification of all $\frac{1}{3}$-homogeneous dendroids, perhaps beginning with smooth dendroids.
\[13-homo dendroids\] What are all the $\frac{1}{3}$-homogeneous (smooth) dendroids?
[^1]: The second named author was supported by NSERC grant RGPIN 435518. The third named author was partially supported by CONACYT grant 316891.
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abstract: 'Thanks to the rapid development of information technology, the size of wireless network is becoming larger and larger, which makes spectrum resources more precious than ever before. To improve the efficiency of spectrum utilization, game theory has been applied to study efficient spectrum sharing for a long time. However, the scale of wireless network in existing studies is relatively small. In this paper, we introduce a novel game called aggregative game and model spectrum sharing in a large-scale, heterogeneous, and dynamic network using such game concept. Meanwhile, the massive usage of spectrum leads to easier divulgence of privacy of spectrum users, which calls for privacy and truthfulness guarantees. In a large decentralized scenario, each user has no priori about other users’ channel access decisions, which forms an incomplete information game. A “weak mediator”, e.g., the base station or licensed spectrum regulator, is introduced and turns this game into a complete one, which is essential to reach a Nash equilibrium (NE). By utilizing past experience on the channel access, we propose an online learning algorithm to improve the utility of each user. We show that the learning algorithm achieves NE over time and provides no regret guarantee for each user. Specifically, our mechanism admits an approximate *ex-post*. The mechanism is also joint differentially private and is incentive-compatible. Efficiency of the approximate NE is evaluated, and innovative scaling law results are disclosed. We also provide simulation results to verify our analysis.'
author:
- 'Pan Zhou, *Member, IEEE*, Wenqi Wei, *Student Member, IEEE*, Kaigui Bian, *Member, IEEE*, Dapeng Oliver Wu, *Fellow, IEEE*, Yuchong Hu, *Member, IEEE*, Qian Wang, *Member, IEEE* [^1]'
title: 'Private and Truthful Aggregative Game for Large-Scale Spectrum Sharing'
---
Spectrum sharing, aggregative game, differential privacy, online learning, truthfulness, heterogeneous.
Introduction
============
Motivation
----------
Dynamic spectrum sharing has been considered as a promising technique that allows unlicensed secondary users (SUs) to opportunistically access idle channels owned by legacy spectrum holders. In general, the increasing spectrum demand and frequent spectrum usage often lead to large and uncertain network dynamics. Competition among users makes spectrum utilization inefficient due to potential interferences and massive data packet collisions.
To improve individual efficiency, each user may speculatively conduct a mixed strategy of channel access over multiple channels. However, such a selfish behavior may result in more severe channel contentions and degrade the performance of the entire network. Given that the size of the network keeps growing nowadays, it gets more and more challenging to resolve this issue.
Classic spectrum sharing problem is usually studied for small scale networks. In this aspect, numerous efforts have been made to design efficient spectrum utilization mechanisms based on game theory [@MY][@CH]. However, large-scale spectrum sharing game with the heterogeneous individual impact is less understood, where different users have different interference impacts on the aggregative contention probability. The different actions of users, i.e., the mixed channel access probabilities, may contribute to such heterogeneous impacts. Moreover, it is implausible to study spectrum sharing in a large-scale wireless network with a priori, or with complete information. Users’ decisions of channels are so decentralized that any user is incapable of collecting all of them, which leads to an incomplete information setting. The need to improve the efficiency of such heterogeneous large-scale spectrum sharing motivates us to explore a novel equilibrium solution.
Aggregative Game with Weak Mediator
-----------------------------------
To study the impact on the contention from heterogeneous individuals in a large-scale network, it requires a completely different view and some new equilibrium concepts. Therefore, we model spectrum sharing in wireless network as an aggregative game. Informally, aggregative game means that the payoff of each user is a function of the user’s own action and the aggregator of all users’s actions [@JENSEN]. With such a game model, we are able to identify users’ heterogeneous actions and learn their effects on the contention. Since there are multiple channels, we arrive at a multi-dimensional aggregative game. To the best of our knowledge, this is the first work to study large-scale spectrum sharing under the aggregative game.
Similar to [@NZD], we model a network in which a couple of users share multiple channels. In the network, users try to occupy channels with maximum effort to meet their spectrum demands. Such a competition makes spectrum sharing in large-scale wireless network a typical non-cooperative game [@NH1]. To form a complete information game and hence achieve a NE where every user can get his desired utility, we introduce a “weak mediator" as it did in [@RR]. The weaker mediator can be a base station (BS), a spectrum service anchor point or a licensed spectrum regulator. Although users still lack the knowledge of others’ choices of channels, the mediator can collect these channel access information and process them. As shown in Fig. \[fig:digraph1\], the BS works as the weaker mediator. We also assume that the mediator only has the power to collect users’ reported mixed multi-channel access strategies, to calculate a mixed strategy profile and to provide non-binding channel access suggestions for each user. Such limited power enables us to call the mediator “weak”.
Owing to the weak mediator, users can either opt-in using the BS as their advisor or opt-out neglecting its existence. For those users who are willing to opt-in, BS requires them to report their mixed channel access strategies. As for those who opt-out, e.g., devices which do not send their channel access “requests" to BS in Fig. \[fig:digraph1\], they access the channel as they have intended. When BS gathers reported mixed strategies, it then calculates a mixed channel access strategy profile and suggests it to all users according to their submitted mixed strategies. Noted that even opting-out users receive the suggestions, but their suggestions are fixed because the action of opting-out is considered as a constant input. At last, all users access channels referring to or disregard the suggestion. Due to the weak mediator, even those who opt-in are able to neglect the suggestion. As a practical illustration, the opt-in users could be high power wireless multimedia users who are willing to report with little energy cost while the opt-out users could be low power energy-constrained sensor nodes.
![Mediated large-scale multi-channel wireless network[]{data-label="fig:digraph1"}](pic1.eps)
Truthfulness and Privacy
------------------------
The mediated large-scale spectrum sharing forms a complete information game, but it raises two issues. On the one hand, the introduction of weak mediator calls for the *truthfulness* guarantee. When BS cannot force any user to opt-in or strictly follow its suggestions, users have the tendency to opt-out or opt-in telling lies. Two reasons account for such cheating. One reason is the incentive to get a higher payoff. The other reason is the fear of revealing their private information.
On the other hand, the *privacy* issue in spectrum sharing arises when selfish users are competing [@HUANGQ]. When users are frequently accessing channels, they have a high risk of exposing their *types*. A user’s type, which can be inferred from his action, is his private information. Such type can be the location of spectrum user, the purpose of spectrum usage, e.g. voice services, or watching a video, etc. The action is determined by his transmission power, contention probability and his mixed strategy over multiple channels. We will explain these concepts in details later. Since a user’s action is associated with his mixed strategy over multiple channels, it can be easily learned using historical statistics. Moreover, in a busy wireless network, a high demand of spectrum require that users cannot hold channels for long. A user may only keep one channel for a certain period and need to compete for next usage. This makes inference about one’s *type* even easier.
Previously, cryptography was the main tool for preserving privacy in wireless networks [@Naor99]. But it often incurs high computation complexity. The recently proposed differential privacy [@DR] offers us an illuminating perspective. Differential privacy, intuitively, means that a single variation in the input data set can only have a limited impact on the output. The fact that differential privacy has superior performance in the large-scale data sets (i.e., user requests data sets) enlights an application in large-scale spectrum sharing. Besides, differential privacy is often implemented in a truthful manner [@ZLWS]. Thus, it is practical to achieve both truthfulness and privacy preservation via a differentially private mechanism. In this paper, we apply joint differential privacy [@KPRU], which is adapted from standard differential privacy, to guarantee the privacy of spectrum users.
Necessity of Online Learning
----------------------------
The efficiency of NE based on one or a few sets of users’ mixed channel access strategies varies. It sometimes suffers from users’ speculations. In a large-scale spectrum sharing game, users need to repeatedly decide which channel to use. Therefore, we can utilize past accumulated information of channel choices of users to improve the efficiency of NE. We propose an online learning algorithm based on [@ARORA] to compute a mixed channel access strategy suggestion. The algorithm learns by maintaining weights on former mixed strategy experience. Through delicate design, we achieve an *ex-post* NE over time. The algorithm also gives no-regret guarantee to each user.
In fact, if users not only opt-in using the BS as the mediator but also truthfully report their types and subsequently follow the mediated instruction, the suggested strategy profile forms an approximate *ex-post* NE of the mediated game. The resulting game achieves an approximate NE of the original complete information game. Although we can only achieve an approximate NE, we solve the equilibrium in strong sense if the spectrum sharing is in large scale. The approximation loss becomes less with the growing number of spectrum users, which is promising in future large-scale networks.
Main Contributions and Organizations
------------------------------------
In this paper, we design an efficient, truthful and privacy-preserving mechanism for large-scale spectrum sharing using aggregative game. The heterogeneous individual impact is also considered in the aggregative game model. The main contributions of this work are described below.
- General game formulation: We formulate large-scale spectrum sharing in wireless network as an *aggregative* game, with the heterogeneous individual impact on users’ contention probability.
- Truthfulness and privacy guarantee: The proposed mechanism preserves ($\varepsilon$,$\delta$)-joint differential privacy for each user and such privacy is implemented in a truthful manner. Algorithm $1$ prevents cheating learning in the BS and adversary users cannot distinguish a user’s action by the NE he achieves. Besides, algorithm $3$ provides a privacy-preserving mixed strategy solution, which makes malicious statistics learning of actions through difficult.
- Online learning for achieving NE: We propose an online learning algorithm. The output of the algorithm, which is the mixed strategy suggestion profile, leads to an *ex-post* NE over time.
- Large-scale performance: The *ex-post* NE achieved by online learning has better performance with the growing number of users, which demonstrates promising scaling law results in the large-scale spectrum sharing.
The rest of this paper is organized as follows. We discuss related work in Section II. Section III gives aggregative game model of multi-channel wireless network. Section IV presents several techniques of independent interest. Section V details the privacy-preserving and truthful design of our mediated spectrum sharing game. Section VI gives online learning algorithm to compute NE. Numerical results are available in Section VII and the paper is concluded in Section VIII.
Related Work
============
Numerous efforts have been made to design an efficient spectrum utilization mechanism based on game theory [@HUANG; @NH2; @Huang06; @NH; @MY; @CH]. [@HUANG] showed that a pure strategy equilibrium exists both for spatial spectrum access games on directed acyclic graphs and games satisfying the congestion property on directed trees and directed forests. [@NH] studied effective wireless network selection via evolutionary game approach while the author of [@MY] induced congestion game[@ROSE] into spectrum sharing. The classic congestion game assumes that a user’s payoff depends on the number of users who share the same channels. The impact of individual users, which is related to the number of users competing for the same channel, is heterogeneous with regard to different users. [@CH] studied spatial congestion game and considered the heterogeneous individual impact. However, large-scale spectrum game with the heterogeneous individual impact taken into account is less understood. We here implement aggregative game [@JENSEN] to study heterogeneous spectrum sharing game. To the best of our knowledge, this is the first to study large-scale heterogeneous spectrum sharing game under aggregative game.
Truthfulness has long been studied in terms of incentive compatibility in mechanism design [@AGT], [@AMD]. We use both terms without difference. In spectrum sharing game, [@CLW] studied truthfulness in peer-to-peer network under spatial evolutionary game theory while [@FCZ] studied the truthful double auction mechanism for heterogeneous spectrums. The author of [@EPT] studied truthfulness in spectrum sharing problem where multiple systems coexist in an unlicensed band and interfere with each other.
As for privacy, McSherry and Talwar [@SMKT] first incorporated the techniques of differential privacy into mechanism design. Recently, Differential privacy has already been used in privacy-preserving wireless communications. Zhu and Shin [@ZS] first incorporated the techniques of exponential scheme [@SMKT] of differential privacy into mechanism design and studied cognitive radio spectrum auction. They implemented differential privacy in an auction in a truthful manner. In our large-scale wireless network, we use a number of tools developed from differential privacy [@DR][@DMNA] to design a privacy-preserving spectrum sharing mechanism. We also adapt the online learning from [@ZLWS] to ensure that the mixed strategy output of every user is joint differential private [@KPRU].
Network Model
=============
This section presents an aggregative game model for the mediated spectrum sharing in wireless networks. Consider a wireless network where $n$ aggressive users are competing for $k$ channels. The user set is $\mathcal{N}=\{1, ...,i,..., n\}$ and the channel set is $\mathcal{K}=\{1, ...,d,..., k\}$. We also give the action set $\mathcal{M}=\{1, ...,j,..., m\}$, where a user’s action is an application related index that specifies its concrete transmission power $\phi _{ij}^{d}$ and its contention probability $p_{i}$. Since transmission power varies when terminal users are performing different tasks on their devices, we associate the concept with the term action. e.g, transmission power needed for watching videos on the phone is different from that of making a phone call. Meanwhile, contention probability can reflect what the user is doing as well. A formal constitution of action is provided later. According to Shannon capacity, each user $i$ with action $j$ over channel $d$ has a data rate $C_{ij}^d$: $$\vspace{-.1cm}
C_{ij}^d = B_{i}^{d} \log _2 (1 + \frac{{\phi _{ij}^{d} \emph{g}^d_{i} }}{{ \omega _{i}^d }}),$$ where $B_{i}^{d}$ is the spectrum bandwidth that user $i$ can get from channel $d$. $\phi _{ij}^{d}$ is the fixed transmission power of user $i$’s action $j$ on channel $d$. $\omega^d _{i}$ denotes the background noise power while $\emph{g}^d_{i}$ is the channel gain of user $i$.
W.l.o.g., we define following individual throughput as the utility function of user $i$ on channel $d$ under action $j$: $$\vspace{-.1cm}
u_{ij}^d(p_i, \vec p)= C_{ij}^dp_i \prod\nolimits_{l \ne i} {(1 - p_{l} )},$$ where $\vec p=\{ p_1, ..., p_i,..., p_n\}$ is the channel contention probability vector for all users. Taking fairness in communication into consideration, we define and write individual utility function $U_{ij}^d(p_i,\vec p)$ in a proportional-fair way: $$U_{ij}^d(p_i,\vec p) = \log u_{ij}^d(p_i, \vec p) = \log C_{ij}^dp_{i} \prod\nolimits_{l \ne i} {(1 - p_{l} )}=\log C_{ij}^d + \log p_{i} + \sum\nolimits_{l \ne i}\log{(1 - p_{l} )}.
\label{aggre1}$$
From equation (\[aggre1\]), it is observed that user $i$’s utility function on channel $d$ is related to $\log p_{i} + \sum\nolimits_{l \ne i}\log{(1 - p_{l} )}$, which is an aggregative function of every competing user’s contention probability on the channel. Noted that the contention probability of users who are not competing for the channel is “0". Then we can define following aggregative function: $$\vspace{-.2cm}
Q_{i}^d(\vec p) = \sum\nolimits_{l = 1}^n {q_{i}^d(p_l)},$$ where $q_{i}^d(p_l)$ is the individual function for contention probability from user $i$’s perspective. We here normalize these $q_{i}^d(p_l)$ to be within $[0,1]$, and write it as: $$q_{i}^d(p_l) = \left\{ \begin{array}{l}
\frac{{\log p_{i}}}{{\log p_{i} + \sum\nolimits_{l \ne i} {\log (1 - p_l )} }},\quad l=i,\\
\frac{{\log (1 - p_{l} )}}{{\log p_{i} + \sum\nolimits_{l \ne i} {\log (1 - p_l )} }},\quad l \ne i, \quad \forall l\in \mathcal{N}.\\
\end{array} \right.$$
It is explicit that each user has a heterogeneous impact on the aggregative contention probability function. Thus, the utility function can be written as $U_{ij}^d(p_i,Q_{i}^d(\vec p)).$
![Mediated process within one period[]{data-label="fig:digraph"}](pic2.eps)
Due to the limitation of power, the contention that a single user can cause to other users should be limited for scalable network planning in large-scale spectrum sharing. Accordingly, we introduce $\gamma$-aggregative game [@Babi]. This means that the greatest change a user can unilaterally cause to the aggregative contention probability is bounded by $\gamma$, which is determined as a scaling law index in later sections. We can then revise the aggregative contention probability function by $$\vspace{-.2cm}
Q_{i}^d(\vec p) = \gamma \sum\nolimits_{l = 1}^n {q_{i}^d(p_l)}.$$ With the normalization and the assumption of $\gamma$-aggregative game, the range of each channel’s aggregative contention probability function $Q_{i}^d(\vec p) $ is bounded by $Q_{i}^d(\vec p) \in [0,n\gamma]$.
According to formal game theoretical notation, we define $Q_{i}^d(p_i',\vec p_{-i})$, where $p_i'$ means that user $i$ deviates from his original action given the unchanged contention probabilities of other users, i.e., we have $\vec p_{-i}= \vec p-p_i$. Then, the following property is immediate: $$|U_{ij}^d(p_i, Q_{i}^d(\vec p)) - U_{ij}^d(p_i', Q_{i}^d(p_i',\vec p_{-i}))| = |Q_{i}^d(\vec p) - Q_{i}^d(p_i', \vec p_{-i}) )|.\vspace{-.1cm}$$ With the assumption of $\gamma$ aggregative game, utility functions are $1$-Lipschitz with regard to the aggregative contention probability function, $$\! |U_{ij}^d(p_i,\! Q_{i}^d(\vec p))\! - \! U_{ij}^d(p_i,\! Q_{i}^d(p_i',\vec p_{-i}))| \!\le\! ||Q_{i}^d(\vec p) \! - \!Q_{i}^d(p_i',\vec p_{-i}) )||_\infty.\vspace{-.1cm}$$
[|C[1.3cm]{}|L[3.2cm]{}|C[1.9cm]{}|L[4.7cm]{}|]{} $\mathcal{N}$ & user set & $\vec p$ & contention probability set\
$\mathcal{M}$ & action set & $\phi _{ij}^{d}$ & fixed transmission power\
$\mathcal{K}$ & channel set & $\beta$ & confidence parameter\
$\emph{g}^d_{i}$ & channel gain & $\omega^d _{i}$ & background noise power\
$B_{i}^{d}$ & spectrum bandwidth & $\alpha$ & discrete gap of aggregative functions\
$C_{ij}^d$ & data rate & $\zeta$ & Lipschitz game approximation\
$\gamma$ & aggregative bound& $T$ & cost threshold in Alg.1\
$\vec P_d$ & mixed strategy set & $\varepsilon_0$, $\varepsilon$, $\delta$ & privacy parameters\
[|C[2.3cm]{}|L[9.65cm]{}|]{}
$\{P_{ij}^1,...,P_{ij}^k\}$ & mixed strategy set of user $i$ with action $j$\
$\pi$ & specific action defined by $\{\phi _{ij}^{d}$, $p_i$, $P_{ij}^{d}\}$\
$q_{i}^d(p_l)$ & the individual function for contention probability from user $i$’s perspective\
$Q_{ij}^d( \vec p,\vec P_d)$ & aggregative contention probability function of user $i$ whose action $j$ on channel $d$\
$ U_{ij}^d(p_i,Q_{i}^d(\vec p, \vec P_d)) $ & utility function for user $i$ with action $j$ on channel $d$\
$U_{ij} (p_i ,Q_i^d (\vec p,\vec P))$ & expected utility for user $i$\
$\xi \!\!- \!\!\vec {BA}_{i}(Q_{i}^d(\vec p, \vec P_d)) $ & $\xi$-aggregative best response\
$\widehat Q_{ij}^d( \vec p,\vec P_d)$ & optimal set-valued aggregative function\
$\xi$ & approximation of aggregative best response\
$a(\widehat Q_{ij}^d( \vec p,\vec P_d))$ & difference between user $i$’s aggregative functions and set-valued function\
$E1,E2$ & approximation brought by Alg. 1 and Alg. 3\
$\vec {P_i^t}$ & user $i$’s mixed channel access strategy $(P_{ij}^1,...,P_{ij}^k) $ at period $t$\
${\vec P}^t$ & mixed strategy profile $({\vec P}_1^t ,...,{\vec P}_n^t )$ of all users at period $t$\
When consider mixed strategy over multiple channels, we note that a user’s action is also determined by his mixed strategy. Specifically, a user who performs action $j$ conducts mixed strategy over all channels $\{P_{ij}^1,...,P_{ij}^k\}$ each with a probability $P_{ij}^d, \forall i \in \mathcal{N}, \forall j \in \mathcal{M}, \forall d \in \mathcal{K}$. We then give user $i$’s mixed strategy aggregative contention probability function: $$\label{MixProb}
Q_{ij}^d( \vec p,\vec P_d) = \gamma \sum\nolimits_{l = 1}^n {\sum\nolimits_{j = 1}^m {q_{i}^d(p_l)P_{lj}^d } }, \quad \forall d \in \mathcal{K}.
\vspace{-.1cm}$$ where $\vec P_d$={{$P_{11}^d,...,P_{n1}^d$},..., {$P_{1j}^d,...,P_{nj}^d$}..., {$P_{1m}^d,...,P_{nm}^d$}} is the mixed strategy set of all users on channel $d \in \mathcal{K} $. Noted that a user has $k$ such aggregative contention probability functions in total. Under mixed strategy, a user’s action $\pi$ is determined as $\pi$ = $\{\phi _{ij}^{d}, p_i, P_{ij}^{d}\}, \forall i \in \mathcal{N}, \forall j \in \mathcal{M}, \forall d \in \mathcal{K}$. From the perspective of applications, we can refer each user as a “type” of specific application, which can be inferred by his action profile. Accordingly, we have following utility function for user $i$ under action $j$ and channel $d$: $$\vspace{-.1cm}
U_{ij}^d(p_i,Q_{i}^d(\vec p, \vec P_d)) =\sum\nolimits_{d \in \mathcal{K}} {\log C_{ij}^d + Q_{ij}^d( \vec p,\vec P_d)}.$$ This equation presents the utility a user $i$ can get by using channel $d$ under action $j$. In the assumption of multiple channels and mixed strategy, we need to consider and define the expected utility on all channels for user $i$: $$\vspace{-.1cm}
U_{ij} (p_i ,Q_i^d (\vec p,\vec P)) = \mathop \mathbb{E}\nolimits_{\vec P} U_{ij}^d (p_i ,Q_i^d (\vec p,\vec P_d )).$$
Moreover, since the high demand of spectrum limits the channel usage, we divide time into small periods. Consider a periodic mediated spectrum sharing game, three steps occur in a period as shown in Fig. 2. At the beginning of the period, users access channels with or without the suggested mixed strategies. Then, users send their mixed strategies on subsequent period’s channel demands to BS. Meanwhile, BS computes a mixed strategy profile according to submitted requests. Before the end of the period, BS suggests the mixed strategy profile to each user. A user may only keep one channel for one period and have to compete for next channel usage.
In our model, there are two kinds of common target adversaries who try to learn actions of spectrum users. One adversary is the cheater who picks up information in BS. Since spectrum users submit their mixed strategies to BS and BS chooses the ideal NE for each user, it is not very hard for those cheaters to infer a user’s action by the NE he achieves. The other adversary is malicious competing users who try to learn other users’ actions through statistics. The fact that users conduct mixed strategy, together with our assumption that different actions contribute to different mixed strategies, makes statistics an effective way to learn users’ actions. Since a user’s private type can be inferred from his action, we preserve privacy of spectrum users by ensuring that a user’s action is less likely to be studied. We list important notation in this paper in **Table 1**.
Aggregative Game and Privacy Model
==================================
In this section, we review some preliminary concepts in algorithmic game theory and differential privacy. We also briefly explain how we adopt these concepts in our model.
$\gamma$-aggregative Game
-------------------------
Since a user’s utility function on one channel is related to the aggregative function of every competing user’s contention probability on the channel, we bring in the concept of aggregative game[@JENSEN]. A multi-dimension $\gamma$-aggregative game is a class of games represented by $\Gamma = (\mathcal{N},\mathcal{M}, \mathcal{K},\{ S(\mathcal{N})_i^d\},\{ U_i^d\},\gamma)$. Specifically, $\mathcal{N}$ is the set of players, $\mathcal{M}$ is the set of actions and $\mathcal{K}$ is the dimension set of the game. Player $i$ is in the player set $\mathcal{N}$ and the aggregative function is aggregated by all users in $\mathcal{N}$. $S(\mathcal{N})_i^d$ is a player $i$’s aggregative function, or aggregator, on dimension $d$ and $\{ S(\mathcal{N})_i^d\}$ is the aggregative function set. $U_i^d$ is player $i$’s utility function on the dimension $d$ while $\{ U_i^d\}$ is the set of utility functions. $\gamma$ is the impact bound, limiting the greatest difference a player can make to the aggregative function by changing his action.
In the wireless network, the dimension in the multi-dimension $\gamma$-aggregative game is instantiated as the cardinality of the available channel set. User $i$’s aggregator on channel $d$ is the aggregative contention probability function $Q_{i}^d(\vec p)$ for the channel from his perspective. We use “user" and “player", “aggregator" and “aggregative contention probability functions" interchangeably.
Let $Q_{i}^d(p_i',\vec p_{-i})$ be an aggregative contention probability when any single user deviates from his original action. Then, we have the following definition.
*[@Babi] An aggregative game whose every aggregator satisfies $||Q_{i}^d(\vec p) -Q_{i}^d(p_i',\vec p_{-i})||_\infty \le \gamma$ is a $\gamma$-aggregative game.*
The fact that individual impact on aggregative contention probability function is bounded by $\gamma$ makes it a typical Lipschitz game. Here $\gamma$ is called Lipschitz constant of a game [@AS]. Noted that this game-theoretical definition of Lipschitz is different from 1-Lipschitz mentioned above, which is a mathematical definition. We summarize the property of $\gamma$ in the Lipschitz game as follows.
(Lipschitz Game [@AS]) A game with $n$ players, $m$ actions and a Lipschitz constant $\gamma$ admits $\gamma \sqrt {{\rm{8}}n\log (2mn)}$-approximate Nash equilibria.
Theorem 1 guarantees that our aggregative model in large-scale wireless network always converges to Nash equilibria.
Aggregative Best Response
-------------------------
Aggregative best response, which is a very important concept in aggregative game [@Babi], is also introduced to describe Nash equilibria in the wireless network. Let $\mathcal{P}$ be the contention probability set and $\vec p$ be the contention probability profile of all users. $p_i'$ denotes that user $i$ deviates from his original action and thus changes his contention probability. A user $i$ is playing an $\eta$-best response to $\vec p$ if his utility function satisfies $U_{ij}^d(p_i,Q_{i}^d(\vec p)) \ge \mathop {\max }\nolimits_{p_i ^\prime } U_{ij}^d(p_i',Q_{i}^d(p_i',\vec p_{-i}))-\eta$, for all $p'_{i} \in \mathcal{P}$. In the random access network, if all users are randomly playing an $\eta$-best response, the situation is defined as an $\eta$-approximate NE. Similarly, we define the aggregative best response $\vec {BA}_{i}(Q_{i}^d(\vec p))$ for user $i$ with aggregative contention probability $Q_{i}^d(\vec p)$ on each channel $d$ as $$\label{BA3}
U_{ij}^d(p_i,Q_{i}^d(\vec p)) \ge \mathop {\max }\nolimits_{p_i ^\prime } U_{ij}^d(p_i',Q_{i}^d(\vec p))-\eta.$$ The aggregative best response indicates that the difference between user $i$’s worst case $\vec {BA}_{i}(Q_{i}^d(\vec p))$ and his exact aggregative best response is at most $\eta$.
**Remark 1:** Noted that a user is playing best response when his action is against other users’ actions $\vec p_{-i}$, while aggregative best response is acted against aggregative contention probability function $Q_{i}^d(\vec p)$. In large-scale spectrum sharing, the effect of individual’s action on the aggregative contention probability function, which is bounded by $\gamma$, is relatively small comparing to the aggregator. So in function (\[BA3\]), we ignore the effect of the user’s action on the aggregative function in the aggregative best response. The user plays aggregative best response as if the aggregative function were promised to be $Q_{i}^d(\vec p)$.
In the aggregative spectrum sharing game, we use aggregative best response instead of best response to describe a NE. In fact, an aggregative best response can translate into a best response with regard to our assumption of $\gamma$-aggregative game and the property of 1-Lipschitz, vice versa.
Let $\vec p$ be the contention probability profile such that user $i$’s contention probability $p_{i}$ is an $\eta$-best response. Then, $p_{i}$ is an $(\eta + \gamma)$-aggregative best response to $U_{ij}^d(p_i,Q_{i}^d(\vec p))$.
Let $Q_{i}^d(p_i',\vec p_{-i})$ be the aggregative function if any user deviates from his original action. Since $p_{i}$ is an $\eta$-best response, we know that $U_{ij}^d(p_i,Q_{i}^d(\vec p)) \ge U_{ij}^d(p_i',Q_{i}^d(p_i',\vec p_{-i}))-\eta$ . By the bounded influence of user $i$, we know that $||Q_{i}^d(\vec p) - Q_{i}^d(p_i',\vec p_{-i})||_\infty\le\gamma$. Also, combining Lipschitz property of $u_{ij}^d$ and our assumption of $\gamma$ aggregative game, we have that $|U_{ij}^d(p_i',Q_{i}^d(\vec p)) - U_{ij}^d(p_i',Q_{i}^d(p_i',\vec p_{-i})|\le\gamma$. It follows that $ U_{ij}^d(p_i',Q_{i}^d(p_i',\vec p_{-i}) \ge U_{ij}^d(p_i',Q_{i}^d(\vec p))- \gamma$, and therefore $U_{ij}^d(p_i,Q_{i}^d(\vec p)) \ge U_{ij}^d(p_i',Q_{i}^d(\vec p))- \gamma-\eta$.
If one user is playing an aggregative best response in the spectrum sharing network, we are able to know that another user is also playing an aggregative best response as long as the aggregative contention probability functions of two users are close. We state such relation formally in the following lemma.
Suppose user $i$’s contention probability $p_{i}$ is an $\eta$-aggregative best response to the aggregator function $Q_{i}^d(\vec p)$ for user $i$. Let $Q_{i}^d(p_i',\vec p_{-i})$ be the aggregative contention probability such that $||Q_{i}^d(\vec p) - Q_{i}^d(p_i',\vec p_{-i})||_\infty\le\alpha$. Then, $p_{i}$ is an $(\eta + 2\alpha)$-aggregative best response to $Q_{i}^d(p_i',\vec p_{-i})$.
Let $Q_{i}^d(p_i',\vec p_{-i})$ be the aggregative contention probability if any user deviates from his original action. Since $p_{i}$ is an $\eta$-aggregative best response to $Q_{i}^d(\vec p)$, we have $U_{ij}^d(p_i,Q_{i}^d(\vec p)) \ge U_{ij}^d(p_i',Q_{i}^d(\vec p))- \eta$. By 1-Lipschitz property and the assumption of $\gamma$-aggregative game of $U_{ij}^d(p_i,Q_{i}^d(\vec p))$, $U_{ij}^d(p_i,Q_{i}^d(p_i',\vec p_{-i})) \ge U_{ij}^d(p_i,Q_{i}^d(\vec p))- \alpha$ and also $u_{i}(a ,q) \ge u_{i}(a,q')- \alpha$. Combining these inequalities, we have $U_{ij}^d(p_i,Q_{i}^d(p_i',\vec p_{-i})) \ge U_{ij}^d(p_i',Q_{i}^d(p_i',\vec p_{-i}))- \eta-2\alpha$.
The relationship among different aggregative best responses shown in Lemma 1, together with the relationship between the aggregative best response and the best response shown in Lemma 2, is utilized to realize an approximate NE in the spectrum sharing. We provide the following lemma to show the connection.
Let $\vec p$ be the contention probability profile such that every user is playing $\eta$-aggregative best response to $Q_{i}^d(\vec p)$. Then, we know that each user is playing ($\gamma + \eta$)-best response, and hence $\vec p$ forms a ($\gamma + \eta$)-NE.
Let $Q_{i}^d(p_i',\vec p_{-i})$ be the aggregative contention probability if any user deviates from his original action. Since $p_{i}$ is an $\eta$-aggregative best response, we know $U_{ij}^d(p_i,Q_{i}^d(\vec p)) \ge U_{ij}^d(p_i',Q_{i}^d(\vec p))-\eta$. We know that $||Q_{i}^d(\vec p)- Q_{i}^d(p_i',\vec p_{-i})||_\infty\le\gamma$ by bounded influence of $i$. Then by Lipschitz property and the assumption of $\gamma$-aggregative game, $U_{ij}^d(p_i',Q_{i}^d(\vec p))\ge U_{ij}^d(p_i',Q_{i}^d(p_i',\vec p_{-i}))- \gamma$. It follows that $U_{ij}^d(p_i,Q_{i}^d(\vec p)) \ge U_{ij}^d(p_i',Q_{i}^d(p_i',\vec p_{-i}))- \gamma-\eta$.
These relations among best response, aggregative best response and NE bridge the gap between an aggregative best response and a NE. Therefore, we can achieve a NE when all users are randomly playing aggregative best response in the spectrum sharing network.
Mediated Game
-------------
The introduction of a mediator transforms the incomplete information game into a complete one. In the mediated game, a user can opt-in using the mediator or neglect it. If a user opt-in, he will send his mixed channel access to the mediator. Let $P$ denote mixed strategies of opting-in users. For those who opt-out, we use notation “$\bot$" to indicate their mixed strategies. We also define $\mathcal{A}^n$ as the output of the mediator. Intuitively, a mediator is a mechanism $ (P \cup {\rm{\{ }} \bot {\rm{\} )}}^n \to \mathcal{A}^n$ that takes users’ mixed strategies as input, computes a solution according to the corresponding contention probabilities and outputs a suggested mixed strategy profile for every user.
Since we assume a weak mediator in the spectrum sharing game, users have several options: they can opt-out leaving out of the BS and report $\bot$. Alternatively, they can opt-in using the BS as the mediator. Besides, opting-in users are free to decide whether to follow the suggestion or not when they receive the suggested mixed strategies.
Regardless of users’ options, the mediator always gives out suggestions to all users. Therefore, there are two kinds of outputs $\mathcal{A}_i'$ for opting-in users and $\mathcal{A}_i''$ for opting-out users. $\mathcal{A}_i=\mathcal{A}_i' \cup \mathcal{A}_i''$, where $$\mathcal{A}_i'=\{\mathcal{A}_i|P_i\to \mathcal{A}_i\}, \mathcal{A}_i''=\{\mathcal{A}_i|\bot\to \mathcal{A}_i,\mathcal{A}_i\quad is\quad constant\}.
\vspace{-.2cm}$$ $\mathcal{A}_i'$ is the user-specific suggestion for opting-in users and $\mathcal{A}_i''$ is the fixed suggestion for opting-out users.
Differential Privacy
--------------------
We then present some tools from differential privacy, which is an ideal tool to study *large-scale* spectrum sharing in wireless network. Intuitively, differential privacy means that a single change in the input data set only has a limited impact on the output. This impact diminishes with the size of users growing large.
*(Differential Privacy [@DMNA]) A randomized algorithm $\mathcal{M}$ satisfies ($\epsilon,\delta$)-differential privacy if for any two input sets $A$ and $B$ with only a single input difference, and for any set of outcomes $\mathcal{R}$, we have: $$\Pr(\mathcal{M}(A) \in \mathcal{R}) \le e^{\epsilon}
\Pr(\mathcal{M}(B) \in \mathcal{R}) + \delta.
\vspace{-.3cm}$$*
An adapted version of differential privacy, which is called “joint differential privacy", is applied to suit the case of wireless network. The joint differential privacy denotes that a single change in the input data other than user $i$’s can only have a limited impact on the output to user $i$, i.e.,
*(Joint Differential Privacy [@KPRU]) Let $\mathcal{R}^{n}$ be the output profile for $n$ users. Two input sets $A$ and $B$ with size $n$ are $i$ - neighbors if they differ only in the $i$-th component. An algorithm $\mathcal{M}$ is ($\epsilon,\delta$)-joint differentially private if for any user $i$, for any pair of $i$-neighbors input $A$ and $B$, and for any subset of outputs $S\subseteq \mathcal{R}^{n-1}$, $$\Pr(\mathcal{M}(A)_{-i}) \in \mathcal{S}) \le e^{\epsilon}
\Pr(\mathcal{M}(B)_{-i}) \in \mathcal{S}) + \delta.
\vspace{-.4cm}$$*
One important property of differential privacy is the adaptive composition theorem. In the proposed online learning algorithm, BS needs as many as $T$ periods to learn users’ strategies. This makes the situation $T$-fold adaptive.
(Adaptive Composition [@DRV]) Let $\mathcal{A}$ be the input set and mechanism be $\mathcal{M}: \mathcal{A}^{n} \to \mathcal{R}^{T}$ in $T$ periods. If $\mathcal{M}$ is an adaptive composition of as many as $T$ individual ($\epsilon,\delta$)-differentially private mechanisms, then $\mathcal{M}$ satisfies ($\epsilon',T\delta+\delta'$)-differential privacy for $$\vspace{-.4cm}
\epsilon' = \epsilon\sqrt{2Tln(1/\delta')} + T\epsilon(e^{\epsilon} - 1).$$
An immediate and useful lemma provides us with a ($\epsilon,\delta$)-differentially private mechanism.
For any $\epsilon \le 1$ and $\delta \ge 0$ , if $M$ is a $T$-fold adaptive composition of ($\epsilon/\sqrt{8Tln(1/\delta)}$, 0)-differentially private mechanisms, then $M$ satisfies ($\epsilon,\delta$)-differential privacy.
Truthfulness
------------
Intuitively, a mechanism is incentive compatible, or called “truthful", when players can get the highest utility only by acting truthfully. In other words, truth telling is one’s dominant strategy.
*(Incentive Compatible [@AGT]) A mechanism is called incentive compatible (IC) if every user can achieve the best outcome just by telling the true information.*
We slightly modify this definition by assuming that a mechanism is incentive compatible if the user only has the incentive to opt-in rather than opt-out.
$\gamma$-approximate *Ex-post* NE
---------------------------------
Finally, we provide the definition of $\gamma$-approximate *ex-post* NE, which is the equilibrium we concern in this paper. Intuitively in game theory, NE is a solution concept in which no player can get a better payoff by unilaterally changing his strategy or action.
Compared to the standard NE, our $\gamma$-approximate *ex-post* NE has two major differences. The first one is that an *ex-post* NE is achieved without the knowledge of priori distribution of strategies. The second one is that the $\gamma$-approximate NE reaches an equilibrium solution that deviates from the exact NE by most $\gamma$.
*($\gamma$-approximate *ex-post* NE [@MS]) Assuming a collection of strategies $\{ \sigma _i :\mathcal{P} \to \mathcal{A}_i \} _{i = 1}^n$. A $\gamma$-approximate *ex-post* NE is formed if for any mixed strategy in $
p\in \mathcal{P}^n$, and for any player $i$ and action $a_i \in \{\mathcal{A}_i\}$: $$\vspace{-.4cm}
u_i ^\prime (\sigma _i (p_i ),\sigma _{ - i} (p_{ - i} )) \ge u_i ^\prime (a_i ,\sigma _{ - i} (p_{ - i} )) - \eta.$$*
Due to the fact that users are not able to know the prior distribution of strategies, the *ex-post* NE is a very strong solution concept for incomplete information games. In the following two sections, we detail private and truthful equilibrium computation carried out in the mediator.
Mediated Private and Truthful Game
==================================
Searching for an Ideal Equilibrium
----------------------------------
In a mediated spectrum sharing game, if all non-cooperative users conduct pure strategy to compete for channels, pure strategy Nash equilibria exist. In terms of the best response, a network performs $\eta$-approximate NE when all users are randomly playing $\eta$-best responses in contention probability profile $\vec p$ under pure strategy. However, real world selfish users conduct mixed channel access strategies to get better utility. With mixed strategy, whether a mixed strategy NE exists in the network is unknown. Consequently, designing new computation schemes to search for an ideal NE is essential.
Since we focus on the aggregative mediated spectrum sharing, users send their mixed channel access strategies ${\vec P} = ({\vec P}_1 ,...,\vec {P_i},...,{\vec P}_n )$, where $\vec {P_i}= (P_{ij}^1,...,P_{ij}^k) $ to the BS. With these submitted channel access strategies, we need to find the aggregative contention probability for every user so that they can randomly achieve $\xi$-aggregative best response: $\xi \!\!- \!\!\vec {BA}_{i}(Q_{i}^d(\vec p, \vec P_d))$ with mixed strategy. As mentioned earlier, we reach aggregative best response through best response.
We first define the ideal aggregative contention probability $\widehat Q_{ij}^d( \vec p,\vec P_d)$ as the optimal set-valued aggregative function for each user when he accesses the channel $d$. To find the set-valued aggregative contention probability function for each user, we round the whole range of aggregative contention probability $[ 0, n\gamma] ^k$ for all $k$ channels. In the process, we search through the discretized grid of the whole possible space $\{ 0,\alpha,..., n\gamma\} ^k$ of aggregative function for all users as well as all channels. Since there are several equilibria for user $i$ on channel $d$ in the assumption of pure strategy, all the corresponding set-valued aggregative functions can be found. However, with our goal to provide an efficient mechanism, we need to select the set-valued aggregative function $\widehat Q_{ij}^d( \vec p,\vec P_d)$ that has the minimum difference $a(\widehat Q_{ij}^d( \vec p,\vec P_d))$ with users’ aggregative probability function on that channel. See the objective function in (\[eq:Apart1\]) and constraints (\[equcross1\]) and (\[equcross2\]). This difference is the cost brought by the mixed strategy. Noted that the mixed strategies submitted to the BS for who opt-out are considered $\frac{1}{k}$ in (\[equcross1\]) and (\[equcross2\]). With the assumption of 1-Lipschitz condition, the minimum difference between two aggregative functions leads to the minimum difference between the corresponding utilities.
Noted that our model is a general model of spectrum sharing network, which contains models of spectrum reuse [@CH] and conflicting graph [@ZS]. On the one hand, a user will allocate probability “0" to those channels in $\mathcal{K}$ that are too far away for transmission. Obviously, faraway channels can only be utilized by its surrounding users. On the other hand, if two interfering (or conflicting) channels are in the transmission range of a user, we can only assign a probability to one channel and set the other one to be zero, and set the rest of non-zero mixed probabilities on other accessible non-conflicting channels. However, we do not plan to go into details for the issues of spatial reuse and channel conflict in this paper. In short, if a user cannot play aggregative best response on channel $d$, his mixed strategy on it will be zero.
Summarizing above discussions, we can define following formulation for each user $i$: $$\vspace{-.2cm}
\min \quad a(\widehat Q_{ij}^d( \vec p,\vec P_d)),\IEEEyesnumber \label{eq:Apart1}$$ $$\vspace{-.2cm}
\label{equcross1}
\gamma \sum\limits_{l = 1}^n {\sum\limits_{j = 1}^m {q_{i}^d(p_l)P_{lj}^d } } \le \widehat Q_{ij}^d( \vec p,\vec P_d) + a(\widehat Q_{ij}^d( \vec p,\vec P_d)),\quad\forall d \in \mathcal{K},$$ $$\vspace{-.2cm}
\label{equcross2}
\gamma \sum\limits_{l = 1}^n {\sum\limits_{j = 1}^m {q_{i}^d(p_l)P_{lj}^d } } \ge \widehat Q_{ij}^d( \vec p,\vec P_d) - a(\widehat Q_{ij}^d( \vec p,\vec P_d)),\quad\forall d \in \mathcal{K},$$ $$\vspace{-.2cm}
\label{equaindi}
0 \le P_{ij}^d \le 1,\quad \forall i\in\mathcal{N}, \quad j \in \xi\!-\!\vec {BA} _i(\widehat Q_{ij}^d( \vec p,\vec P_d)),$$ $$\vspace{-.2cm}
\label{BA1}
P_{ij}^d = 0,\quad\forall i\in\mathcal{N},\quad j \notin \xi\!-\! \vec {BA} _i(\widehat Q_{ij}^d( \vec p,\vec P_d)),$$ $$\vspace{-.2cm}
\label{BA2}
\sum\limits_{d = 1}^k {P_{ij}^d } = 1,\quad\forall i\in\mathcal{N},j\in \mathcal{M}.$$ where $\xi=\gamma+2\alpha+\zeta$. $\xi$ is the result of lemma 1, 2 and 3. Noted that $\zeta \ge
\gamma \sqrt {{\rm{8}}n\log (2mn)}$, which is introduced on account that the Lipschitz game admits $\gamma \sqrt {{\rm{8}}n\log (2mn)}$-approximate Nash equilibria. $\gamma$ comes from the assumption of $\gamma$-aggregative game. According to lemma 1, 2, we add up these two parameters. Since we discretize the aggregative contention probability space by $\alpha$, $2\alpha$ is added to compensate the gap among discretized aggregative functions due to lemma 3.
An Efficient Private-Preserving Equilibrium Selection Algorithm
---------------------------------------------------------------
Now, we show how to devise an efficient privacy-preserving equilibrium selection algorithm. With our goal to find the minimum difference $a$, we need to compare every possible value of contention probability function to the aggregative contention probability functions constituted by users’ submitted mixed strategies. Since there are as many candidates as $
\left( {\frac{{n\gamma }}{\alpha }} \right)^k$ to check, together with our goal to prevent cheating learning in the BS, the design of efficient privacy-preserving algorithm can be very challenging.
However, since we only need to output one contention probability function on one channel for each user, we introduce the *sparse vector mechanism* to deal with such large aggregative contention probability candidates data flows. The mechanism is originated from differential privacy [@DR]. To fit it in the aggregative spectrum sharing game, we devise a similar mechanism and name it the Sparse Cost for Aggregative Contention Probability (**SparCost**), shown in Algorithm 1.
To find the lowest cost, our algorithm first takes in a sequence of costs $a(\widehat Q_{ij}^d( \vec p,\vec P_d))$ for aggregative contention probability functions and a cost threshold $T$ which is chosen according to past channel usage. The cost threshold is used as a benchmark in our algorithm. Then, the **SparCost** adds noise subject to Laplace distribution to the cost threshold $T$ and the cost $a(\widehat Q_{ij}^d( \vec p,\vec P_d))$ respectively. Noted that adding noises to two inputs is essential to protect user’s mixed strategy as well as their types. The privacy concern brings about an error bound $e_1$, which we will explain in details in following Theorem 3. Next, we compare the noisy cost ${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over a_n(Q_i^d)} }}$ with the noisy cost threshold ${{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over T} }}$. By comparing the input costs and the threshold, we can find a cost that is fairly small and thus obtain the corresponding aggregative contention probability as well as the ideal equilibrium. Finally, the **SparCost** outputs the first noisy cost that is below the noisy threshold.
The corresponding aggregative function is the set-valued aggregative contention probability we are looking for. Before the ideal cost is found, the **SparCost** reports $\bot$ for all costs that are above the threshold. Noted that the algorithm runs at most $k$ times for each user and so runs $nk$ times in total. In the algorithm, $v_n$ means the $n$-th noise add to the $n$-th cost when every user needs to check $N=\frac{n\gamma}{\alpha}$ costs for every channel.
**Let** $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T} = T + Lap(\frac{{2\gamma }}{\varepsilon })$.
**let** $\sigma = \frac{{4\gamma }}{\varepsilon }$.
Let $e_1$ be the error bound for the output aggregative function. For any sequence of $N$ costs $a_1(\widehat Q_{ij}^d( \vec p,\vec P_d)$), ..., $a_N(\widehat Q_{ij}^d( \vec p,\vec P_d))$ for user $i$ on channel $d$ such that$|n: a_n(\widehat Q_{ij}^d( \vec p,\vec P_d)) \le T +e_1| \le c$, **SparCost** satisfies $\varepsilon$-differential privacy and, with probability at least $1-\frac{\beta}{2}$ releases cost answers such that $|\theta_n - a_n(\widehat Q_{ij}^d( \vec p,\vec P_d))| \le e_1$, and for all $\theta_n$=$ \bot $, $a_n(\widehat Q_{ij}^d( \vec p,\vec P_d)) \ge T - e_1,$ where $$e_1 = \frac{{8\gamma (\log N + \log (4/\beta )}}{\varepsilon }.$$ In the **SparCost**, $N$ is $
\left( {\frac{{n\gamma }}{\alpha }} \right)$ for one user on one channel. Since a user has $k$ channels to choose from , each channel can bring an $e_1$. Also, for different users, their aggregative contention functions have different $e_1$. For simplicity, we take the biggest of them and give the overall error bound brought by **SparCost**: $$E_1= \max _{i\ \to e_1 } \frac{{8\gamma (k\log \frac{{n\gamma }}{\alpha } + \log (4/\beta ))}}{\varepsilon }.$$ Combining the approximation brought by discretion, the total approximation brought by **SparCost** is $\alpha + E_1$.
The proof is given in Appendix A.
The algorithm **SparCost** guarantees that the output cost and the corresponding aggregative contention probability are differentially private and have low sensitivity to users’ submitted strategies. Hence, any malicious user or opting-out user can only have little effect on the search for ideal aggregative contention probability function. Besides, **SparCost** protects the privacy of individual mixed strategy, which constitutes the aggregative contention probability function. It prevents cheating learning in BS. Therefore, the algorithm provides us with privacy guarantee.
As illustrated in Fig. 3, there are different Nash equilibria with different amount of users. The left part of Fig. 3 shows that aggregative contention probabilities of freely competing users can hardly get close to any of set-valued aggregative functions. Meanwhile, the right side demonstrates the performance of **SparCost**, showing that resulting aggregative contention probabilities are much more closer to those set-valued aggregative functions for constraint (\[equcross1\]). Even though we have obtained the expected aggregative contention probability for each user, we aim to give out a strategy profile solution so that all users can get their optimal utilities. In the next section, we propose the learning algorithm that fulfills our goal to provide a private and truthful NE solution for all users.
Private and Truthful NE Computation
===================================
In this section, we discuss how to compute the strategy profile solution via an online learning algorithm. We also show that the solution achieves a NE and has better performance when the spectrum sharing is in large scale. The solution guarantees privacy by satisfying joint differential privacy and ensures truthfulness.
![**SparCost**’s performance on searching for objective aggregative function from original aggregative function[]{data-label="fig:digraph"}](pic5.eps)
Rounded Exponential Mechanism for Preserving Privacy
----------------------------------------------------
Although **SparCost** guarantees privacy of the users’ submitted mixed strategies, it is not enough to ensure that the output strategy solution has low sensitivity to the input submitted mixed strategy. We introduce differential privacy via *Exponential Mechanism* to induce joint differential privacy to the output strategy solution.
(Exponential Mechanism [@DR]) Let $f(x,r)$ be a function of the input $x$ and the result of the mechanism $r$. The exponential mechanism $\emph{EXP}(x, f(x, r), \varepsilon)$ selects and outputs an element $r$ with probability proportional to $\exp (\frac{{\varepsilon f(x,r)}}{{2\Delta f}})$.
The exponential mechanism has following property:
[@DR] *EXP*(x, f, $\varepsilon$) satisfies $\varepsilon$-differential privacy and, with probability at least 1-$\beta$, outputs an outcome r such that $$\vspace{-.2cm}
f(x,r) \ge \mathop {\max }\nolimits_{r'} f(x,r') - \frac{{2\Delta (f)(\log |R|/\beta )}}{\varepsilon },$$ where $|R|$ denotes the output space range, which is $\frac{n\gamma}{\alpha}$ in our case.
To satisfy the cross constraint (\[equcross1\]) well partitioned among all $n$ users , we regard the cross constraint as the score function $f$ in the *EXP* for each user $i$ under channel $d$: $$\vspace{-.1cm}
f(\vec P_d, Q_{ij}^d( \vec p,\vec P_d)) = \gamma \sum\limits_{l = 1}^n {\sum\limits_{j = 1}^m {q_{i}^d(p_l)P_{lj}^d } } - \lambda_d, \forall d \in \mathcal{K}, \forall i \in \mathcal{N}.$$ The parameter $\lambda_d$, which equals to $\widehat Q_{ij}^d( \vec p,\vec P_d) + \alpha + E_1$, is the upper difference bound between user $i$’s aggregative contention probability on channel $d$ and the set-valued aggregative function. Noted that $\widehat Q_{ij}^d( \vec p,\vec P_d)$ is obtained from **SparCost** and $a$ is replaced by $\alpha + E_1$.
Every user $i$ has as many as $k$ heterogeneous impacts on the aggregative contention probability and so possesses $k$ set-valued aggregative contention probabilities. We implement above exponential mechanism in $k$ rounds for each user and devise the following Algorithm 2 of Rounded Exponential Mechanism (REM) for preserving privacy. We name it **REXP**.
The Algorithm 2 proceeds in rounds and we only provide one round here. For each user, BS feeds mixed strategies ${\vec P}$ to *EXP*$({\vec P} ,f,\varepsilon)$ and selects an aggregative contention probability for each channel respectively. Then, the *EXP* outputs these $k$ selected aggregative contention probability functions $\{Q_{ij}^d( \vec p,\vec P_d), \forall d\in \mathcal{K} \}=\{Q_{ij}^1( \vec p,\vec P_1),...,\{Q_{ij}^d( \vec p,\vec P_d)\},...,\{Q_{ij}^k( \vec p,\vec P_k)\}\}$ for the user. For user $i$, his individual contention probabilities $q_{i}^d(p_i)$=$\frac{{\log p_{i}}}{{\log p_{i} + \sum\nolimits_{l \ne i} {\log (1 - p_l )} }}$ from selected functions are the key to update his own part of mixed strategy suggestion. Such update is carried out by following online learning algorithm.
**Initialize** ${\vec P} = ({\vec P}_1 ,...,{\vec P_n} )$.
**Let** $\varepsilon=\varepsilon_0\sqrt{8Tln(1/\delta)}$.
The Rounded Exponential Mechanism for preserving privacy described in Algorithm 2 satisfies ($\varepsilon$, $\delta$)-differential privacy.
We first prove that *EXP*(${\vec P}$, $f(\vec P_d, Q_{ij}^d( \vec p,\vec P_d))$, $\varepsilon_0$) is $\varepsilon_0$ differentially private. For each user, there are at most $k$ aggregative contention probabilities. For user $i$, *EXP* outputs some aggregative contention probabilities $Q_{ij}^d( \vec p,\vec P_d)$ on two neighboring inputs ${\vec P}$ and ${\vec P}'$. For users that form ${\vec P}$, if one of them changes his mixed strategy, the resulting mixed strategy distribution forms ${\vec P}'$. We use $Q_{ij}^d$ for simplicity of expression. $$\frac{Pr[\emph{EXP}({\vec P}, f(\vec P_d, Q_{ij}^d), \varepsilon_0)=Q_{ij}^d]}{Pr[\emph{EXP}({\vec P}', f(\vec P_d', Q_{ij}^d), \varepsilon_0)=Q_{ij}^d]}
= \frac{\big(\frac{exp(\frac{{\varepsilon_0} f(\vec P_d, Q_{ij}^d)}{2{\Delta}f})}{\sum_{d \in \mathcal{K}} exp(\frac{{\varepsilon_0} f(\vec P_d, Q_{ij}^d)}{2{\Delta}f})}\big)}{\big(\frac{exp(\frac{{\varepsilon_0} f(\vec P_d', Q_{ij}^d)}{2{\Delta}f})}{\sum_{d \in \mathcal{K}} exp(\frac{{\varepsilon_0} f(\vec P_d', Q_{ij}^d)}{2{\Delta}f})}\big)}
= \big( \frac{exp(\frac{{\varepsilon_0} f(\vec P_d, Q_{ij}^d)}{2{\Delta}f})}{exp(\frac{{\varepsilon_0} f(\vec P_d', Q_{ij}^d)}{2{\Delta}f})}) \cdot \big( \frac{\sum_{d \in \mathcal{K}} exp(\frac{{\varepsilon_0} f(\vec P_d', Q_{ij}^d)}{2{\Delta}f})}{\sum_{d \in \mathcal{K}} exp(\frac{{\varepsilon_0} f(\vec P_d, Q_{ij}^d)}{2{\Delta}f})}\big)$$ $$\hspace{3.5cm}= exp( \frac{\varepsilon_0 (f(\vec P_d, Q_{ij}^d)- f(\vec P_d', Q_{ij}^d))}{2{\Delta}u} )
\cdot \big( \frac{\sum_{d \in \mathcal{K}} exp(\frac{{\varepsilon_0} f(\vec P_d', Q_{ij}^d)}{2{\Delta}f})}{\sum_{d \in \mathcal{K}} exp(\frac{{\varepsilon_0} f(\vec P_d, Q_{ij}^d)}{2{\Delta}f})}\big)$$ $$\hspace{1.2cm}\leq exp(\frac{\varepsilon_0}{2}) \cdot exp(\frac{\varepsilon_0}{2}) \cdot \big( \frac{\sum_{d \in \mathcal{K}} exp(\frac{{\varepsilon_0} f(\vec P_d', Q_{ij}^d)}{2{\Delta}f})}{\sum_{d \in \mathcal{K}} exp(\frac{{\varepsilon_0} f(\vec P_d, Q_{ij}^d)}{2{\Delta}f})}\big)$$ $$\hspace{-4.3cm}= exp(\varepsilon_0).$$
Similarly, $ \frac{Pr[\emph{EXP}({\vec P}', f(\vec P_d, Q_{ij}^d), \varepsilon_0)=Q_{ij}^d]}{Pr[\emph{EXP}({\vec P}, f(\vec P_d', Q_{ij}^d), \varepsilon_0)=Q_{ij}^d]} \geq exp(-\varepsilon_0)$ by symmetry. Therefore, *EXP*(${\vec P}$, $f(\vec P_d, Q_{ij}^d( \vec p,\vec P_d))$, $\varepsilon_0$) satisfies $\varepsilon_0$ differential privacy. Since the following over time learning goes through $T$ periods, **REXP** restarts every $T$ periods. It satisfies ($\varepsilon$, $\delta$)-differential privacy when we set $\varepsilon=\varepsilon_0\sqrt{8Tln(1/\delta)}$ according to Theorem 2 of the adaptive composition and the corresponding lemma 4.
We have got the aggregative contention probability $Q_{ij}^d( \vec p,\vec P_d)$ on each channel for each user and proved that the aggregative contention probability is ($\varepsilon$, $\delta$)-differentially private. Then, we show how to compute the private channel access strategy solution via an online learning algorithm.
![Online learning procedure for each user[]{data-label="fig:digraph"}](pic4.eps)
Online Learning: Performance of our result
------------------------------------------
Due to the speculation of selfish users, the efficiency of NE based on one or a few sets of users’ mixed strategies varies. The performance of one time slot NE may suffer. In large-scale spectrum sharing, users need to repeatedly make channel choice decisions. Therefore, we utilize these past accumulated information of mixed channel access strategies to improve the efficiency of the approximate NE.
The key idea of online learning is to adapt a user’s mixed strategy decision based on its accumulated experiences. We achieve our goal to suggest a mixed channel access strategy profile for all users by maintaining weights on former mixed strategy experience. We here assume that our online learning algorithm refresh itself every $T$ periods. Using a doubling trick, we provide a specific period number $T$ to show the optimal result. Similar to [@ARORA], we name our algorithm Multiplicative Weights for strategy online learning, or **MW**.
To implement **MW**, we need to define some notations at first. For each period $t\in\{1,...,T\}$, let ${\vec P}^t = ({\vec P}_1^t ,...,{\vec P}_n^t )$ be the overall mixed strategy profile of all users at period $t$, while $\vec {P_i^t}= (P_{ij}^1,...,P_{ij}^k) $ is the individual mixed channel access strategy over all channels at period $t$. Also, we use $\overline q_{i}^d(p_i)=\{ q_{i}^1(p_i),..., q_{i}^k(p_i)\}$ to represent user $i$’s normalized channel connection probability on all $k$ channels.
At the first initialization period, our algorithm initializes mixed strategies of all users ${\vec P}^1$ by $\frac{1}{k}$ and feed them into **REXP**. Then for each user $i$, **REXP** processes these mixed strategies and output $k$ aggregative contention probabilities $Q_{ij}^d( \vec p,\vec P_d))$ for $k$ channels. Next, the user’s weight on the mixed strategy for each channel is updated by multiplying them with his normalized channel connection probabilities $\overline q_{i}^d(p_i)$ from the output of **REXP**. And then mixed strategy is updated. Next, our algorithm projects user $i$’s updated mixed strategy into the $P_{ij}^d$, which is from his submitted individual constraint set (\[equaindi\]). The submitted individual set is associated with the user’s *type* and represented by his mixed strategy. Noted that for those who opt-out, their submitted mixed strategies are considered $\frac{1}{k}$ for each channel in their individual sets. After the update, the first period is then finished.
In the following period, the updated mixed strategy from last period is taken as the input to **REXP**, and other procedures remain the same. Since there are $T$ successive periods, the weight for each user $i$, as well as the corresponding mixed strategy, is updated by $T$ times. Finally, the algorithm averages the mixed strategies for user $i$ at each period and suggests it to the user. Intuitively, with higher weight on higher connection probability, this learning method works out better in the long run. The whole process is illustrated in Fig. 4. Noted that algorithm **REXP** needs the set-valued aggregative contention probability from **SparCost**, which is the results of users’ submitted mixed strategies.
**Initialize** ${P^1}$: $P_{ij}^d = \frac{1}{k}$ for all $i\in\mathcal{N}$ and all $d\in\mathcal{K}$.
Initialize number of periods $T$ and update parameter $\eta$.
**Let** $T = \frac{{16{n^2}{\gamma ^2}\log k}}{{{E_2 ^2}}},$ $\eta = \frac{E_2}{4n\gamma}.$
The output of **MW** forms the suggestion strategy distribution for all users. With regard to such distribution, we present user $i$’s aggregative contention probability function on each channel $d$: $$\vspace{-.1cm}
Q_{ij}^d( \vec p,\overline P_d)) =\gamma \sum\nolimits_{l = 1}^n {q_{i}^d(p_l)P_{lj}^{d}} .$$ In our result, one’s mixed strategy is no longer specifically associated with action $j$. A user conducts the suggested mixed strategy only to satisfy aggregative best response $\xi\!-\!\vec {BA} _i(\widehat Q_{ij}^d( \vec p,\vec P_d))$ (constraints (\[BA1\]) and (\[BA2\])). Here $\overline P_d$ denotes each user’s suggested mixed strategies on channel $d$. Therefore, the corresponding throughput, or utility, is
$$\vspace{-.2cm}
U_{ij}^d(p_i,Q_{i}^d(\vec p, \overline P_d)) = \log C_{ij}^d + Q_{ij}^d( \vec p,\overline P_d)= \log B_{i}^{d} \log _2 (1 + \frac{{\phi _{ij}^d \emph{g}^d_{i} }}{{\omega _{i}^d }}) + \gamma \sum\nolimits_{l = 1}^n {q_{i}^d(p_l)P_{lj}^{d}}.$$
For each user $i$, his expected utility is then: $$\vspace{-.2cm}
U_{ij} (p_i ,Q_i^d (\vec p,\overline P)) = \mathop \mathbb{E}\nolimits_{\overline P} U_{ij}^d (p_i ,Q_i^d (\vec p,\overline P_d )).$$
(billboard lemma[@HHRRW]). Suppose $
M:X^n \to R$ is ($\varepsilon$,$\delta$)-differentially private. Consider any set of functions $
F_i = X_i \times R \to R'
$, where $X_i$ is user $i$’s input data. The composition {$F_i (\Pi _i X,M(X))$} is ($\varepsilon$,$\delta$)-joint differentially private, where $\Pi _i$ is the projection to user $i$’s data.
The billboard lemma enables us to achieve joint differential privacy. We then have following theorem:
**MW**( $\{\widehat Q_{ij}^d( \vec p,\vec P_d)\}$, $E_2$, $\beta$, $\varepsilon_0$, $\delta$) satisfies ($\varepsilon$,$\delta$)-joint differential privacy.
In particular, the algorithm gets aggregative contention probability on certain channel $d$ for each user at each period via **REXP**, which satisfies ($\varepsilon$,$\delta$)-differentially privacy. All these aggregative contention probabilities form the billboard where each user can know his own impact on others. Each user only utilizes his connection probability on each channel from his aggregative contention probability to update his mixed strategy on the corresponding channel. And then, his updated mixed strategy is projected to his constraint set. With regard to the *billboard lemma*, it is safe to conclude that **MW** satisfies ($\varepsilon$,$\delta$)-joint differential privacy.
Since the output strategy solution is ($\varepsilon$,$\delta$)-joint differentially private, a single change in the input data other than user $i$’s can only have a limited impact on the output to user $i$, which prevents deliberate coalition from malicious users. Competing users who try to learn actions of other users can hardly make any inference through statistics.
We then show that our computation solution is effective. In particular, our result achieves $\eta$-approximate *ex-post* NE in the multi-channel wireless network, where $\eta$ is the approximation brought by the computation. The large-scale spectrum sharing and the fear of leaking types prevent users from getting the knowledge of others’ strategies, giving rise to the absence of priori distribution. This makes the situation *ex-post*. Recall the verdict of Lipschitz game, a Lipschitz game always converges to approximate Nash equilibria. Consequently, we conclude that our computation satisfies $\eta$-approximate *ex-post* NE.
Let $\zeta \ge \gamma \sqrt {{\rm{8}}n\log (2mn)}$, $
\varepsilon$, $\delta$, $\beta$ $\in$ (0,1). Our mechanism satisfies (2$\varepsilon$, $\delta$)-joint differential privacy. With probability at least $1-\beta$, it computes an $\eta$-approximate NE, where $\eta=\zeta+\alpha+E_1+E_2$. Here, the $E_2$ is the approximation brought by **MW** and $$E_2 = \tilde {\rm O}{\left( {\frac{{n{\gamma ^2}}}{\varepsilon }\log \left( {\frac{{2kn}}{\beta }} \right)\sqrt {\log \left( k \right)\ln \left( {1/\delta } \right)} } \right)^{1/2}}.$$
The proof of $\eta$-approximate *ex-post* NE is given in Appendix B.
We now explore the efficiency of our approximate NE. Intuitively, our approximate parameters $\zeta, E_1,$ and $E_2$ diminish with the increase of number of users with our motivation to study large-scale spectrum sharing. It is apt to set $\alpha$ and $\gamma$ by $\frac{1}{n}$ due to the fact that individual contention probability $q_{i}^d(p_l)$ is not larger than 1. Then, $$\hspace{-4.5cm}\zeta {\rm{ = }}\sqrt {\frac{{{\rm{8}}\log (2mn)}}{n}},$$ $$\vspace{-.2cm}
\hspace{-2.8cm}E_1 = \max _{i\; \to e_1 } \frac{{8(k\log n + \log (4/\beta ))}}{{n\varepsilon }},$$ $$\vspace{-.1cm}
\hspace{-0.7cm}E_2 = \widetilde{\rm{O}}\left( {\frac{1}{{n\varepsilon }}\log \left( {\frac{{2kn}}{\beta }} \right)\sqrt {\log \left( k \right)\ln \left( {1/\delta } \right)} } \right)^{1/2}.$$
**Remark 2:** In future network, the number of spectrum users tends to be very large. However, our approximations $\zeta, E_1$, and $E_2$ have good performance with large user number $n$. For approximation $\zeta$ brought by Lipschitz game and $E_2$ from **MW**, their scaling law on user number is in $\widetilde{\rm{O}}(\sqrt {\frac{{\log n}}{n}} )$, showing that they become relatively smaller with $n$ growing larger. Similar for the error bound of **SparCost**, its scaling law on user number is in $\widetilde{\rm{O}}(\frac{{\log n}}{n})$. All these results lead to superior performance in large-scale spectrum sharing. Then, we focus on how number of channels can affect individual utility. The approximation $\zeta, \alpha, E_1,$ and $E_2$ define total error caused by the computation. However, since a user can access only one channel at a time, we measure average approximation of channels. $$\hspace{-2.3cm}E_1 = \max _{i\; \to e_1 } \frac{{8(k\log n + \log (4/\beta ))}}{{kn\varepsilon }},$$ $$E_2 = \widetilde{\rm{O}}\left( {\frac{1}{{kn\varepsilon }}\log \left( {\frac{{2kn}}{\beta }} \right)\sqrt {\log \left( k \right)\ln \left( {1/\delta } \right)} } \right)^{1/2}.$$ It is explicit that the scaling law in terms of channel number $\widetilde{\rm{O}}(\sqrt {\frac{{\log k}{\sqrt{\log k}}}{k}} )$ is smaller with $k$ growing large.
**Remark 3:** It should be noted that even if there are not many users who are willing to opt-in, we still reach an approximate NE. Our result of approximate NE does not depend on the number of users who opt-in. Only the efficiency of the approximation can be affected by them. When **SparCost** chooses a set-valued aggregative function whose corresponding equilibrium is not so close to the expected equilibrium, the approximation suffers.
If we carefully choose the confidence parameter $\beta$ and privacy parameters ($\varepsilon,\delta$), the result can be even better. Hence, we have proved that our approximate NE result performs better in larger-scale spectrum sharing, which is promising for future wireless network.
Finally, we state that our privacy-preserving mechanism is incentive compatible, which means that users are encouraged to opt-in and thus they can get a better payoff.
The output of **MW** is incentive compatible.
The proof is given in Appendix C.
Further Discussion on User Dynamics
-----------------------------------
We further discuss the impact of user dynamics on the performance. Users could be mobile or experiences spectrum mobility [@Akyildiz1] in spectrum sharing environments. We here consider the mobility of users in an uncertain environment. Assume a user has a limited transmission range, his location determines how many channels he can access. We then define spatial $\gamma$-aggregative game $\Gamma = (\mathcal{N},\mathcal{M},\mathcal{K_D},\mathcal{D},\{ S(\mathcal{N'})_i^d\},\{ U_i(\mathcal{D})\},\gamma,)$. $\mathcal{D}$ is the location set of each user and $\mathcal{K_D}$ is the channels set according to the location set. $\mathcal{N}'$ denotes the conflicting user set. In our result, users’ expected utilities are related to the amount of channels. Therefore, a user’s utility can be associated with his location.
As noticed, users could be mobile to seek for better channel access opportunities. Because the location could affect the utility, users tend to move to places with higher payoff, which are places with more channels or better channel qualities. However, individual throughput declines with the increase of user number. Although we have shown that more users lead to better performance in the large-scale spectrum sharing scenario, the result is in terms of NE and is based on overall utility. Consequently, these users tend to find a balance between the number of channels and the number of conflicting users.
Intuitively from our analysis, the private and truthful scaling results in **Theorem 3, 5, 6, 7, 8** still hold and only have some constant loss. Simulation results support our conclusion, i.e., a user can have better utility when there are more channels in the transmission range and individual usage degrades with the increase number of competing users. It is also shown that users tend to change their location all the time searching for the balance point. Note that users’ changing their location can be usually modeled by a typical Markov chain [@CH]. Its stationary distribution and properties need further investigation and are our future work.
Simulation Results
==================
We now evaluate the proposed mechanism and its dynamics by simulations. All results are averaged over 1000 runs. We first show our mechanism’s performance with different amounts of users and channels, respectively. As for the number of users, we vary it from 500 to 2000 with a step of 100. These conflicting users are randomly deployed in a hexagonal area with side length 500m (Fig. 5). The hexagon represents the base station’s signal coverage. Without loss of generality, only one base station exists in the middle of the hexagon and has multiple channels. We set the number of actions $m$ by 50 and the number of channels by 15 on the base station. We also assume the confidence constant $\beta$ to 0.25, the privacy constants $\varepsilon$ to 0.1 and $\delta$ to 0.25. As it did in previous section, $\gamma$ and $\alpha$ are set by $\frac{1}{n}$. We evaluate the efficiency by showing the trends of $\eta=\zeta+\alpha+E_1+E_2$.
As shown in Fig. 6, the approximation decreases in a speed of $ \frac{\log n}{n}$ with the growing number of users. Since $\eta$ is the worst case approximation in our mechanism, we show that longer learning periods of $T$ can lead to better performance using the worst case $\eta$ as a benchmark. When we set the learning period $T$ by 10, 20, 30 respectively, Fig. 6 shows that both the starting points and the ends of approximation gradually become smaller with longer periods. Note that the optimal approximation cannot be 0 because of the privacy concern.
We also compare 1000-user hexagon areas whose number of channels varies from 5 to 20, and measure the average effect of channels $\frac{\eta}{k}$. As shown in Fig. 7, the average approximation decreases in a scaling law of $\widetilde{\rm{O}}(\sqrt {\frac{{\log k}{\sqrt{\log k}}}{k}} )$, which indicates that a channel can be better utilized when there are more channels. Fig. 7 also shows the average approximation becomes smaller when learning period of $T$ is larger.
![Performance on the number of users[]{data-label="fig:digraph"}](pic6.eps)
Then, we show the benefit of mediator. When there are 1000 users in one hexagon, we set the ratio of opting-in users to opting-out users by 2:8, 5:5, 8:2, respectively. Consider a Rayleigh fading channel environment, the channel gain $\emph{g}^d_{i}$ is an exponentially distributed random variable. In following simulation, we set user $i$’s spectrum bandwidth $B_{i}^{d}$ by 20MHz and his fixed transmission power $\phi^{d} _{ij}$ by 100mW. The background noise power $\omega _{i}^d$ is -100dBm. Users’ contention probabilities are randomly selected from the set {0.1,0.2,...0.9} while their mixed strategies are randomly selected from {0.01,0.02,...,0.99}. As shown in Fig. 8, the average utility $\mathop \mathbb{E}\nolimits_{\vec P} U_{ij}^d (p_i ,Q_i^d (\vec p,\vec P_d ))$ for opting-out users is less than that $\mathop \mathbb{E}\nolimits_{\overline P} U_{ij}^d (p_i ,Q_i^d (\vec p,\overline P_d ))$ of opting-in users. With more opting-in users, the gap between the two utilities becomes larger, showing that our mechanism is incentive compatible. Noted that the introduction of differential privacy slightly lowers individual utility to prevent users’ *types* from being exposed .
![Performance on the number of channels[]{data-label="fig:digraph"}](pic7.eps)
Finally, we show the dynamics of spectrum users. The left of Fig. 5 is a random distribution of users when there are many base stations. Each base station has its hexagon and users in different hexagons can use the spectrum simultaneously. Such phenomenon is called spatial reuse and is studied in [@CH] via a congestion game. The number inside the hexagon denotes the number of channels in the area. We have shown that the average approximation of channels diminishes with the increase of channel number. So users can get a better payoff if the hexagon he is in possesses more available channels. With this in mind, users tend to move to places with more channels. However, individual usage deteriorates with the growth of conflicting users. In Fig. 5, the mobility of spectrum users with learning period $T=20$ is shown. After $T$=100 periods, the left graph turns into the right graph. We can see from Fig. 5 that some of the users choose to stay in his original hexagon while others change their locations. Similarly, Fig. 9 shows the dynamics over time, indicating the impact of the number of channels on the dynamics. The stationary distribution of whether a user moves or not needs further investigation.
![Performance of the mediator[]{data-label="fig:digraph"}](pic8.eps)
Conclusion and Future Works
===========================
In this paper, we modeled spectrum sharing of large-scale wireless network as a multi-dimensional aggregative game by taking users’ heterogeneous impact into account. To the best of our knowledge, this is the first study in large-scale spectrum sharing via aggregative game while concerning privacy and truthfulness. We designed a mediated privacy-preserving and truthful game which admits an $\eta$-approximate *ex-post* NE. We also showed that the equilibrium has better performance when the spectrum sharing is in larger scale. We demonstrated that the approximation of NE decreases in a speed of $ \frac{\log n}{n}$ with the growing number of users, and decreases in a scaling law of $\widetilde{\rm{O}}(\sqrt {\frac{{\log k}{\sqrt{\log k}}}{k}} )$ on the channel. We proved that our game satisfies joint differential privacy and is incentive compatible. Numerical result supports our conclusion. In the future, we plan to investigate the dynamic channel selection game and find the stationary distribution of users’ dynamics.
Appendix A Proof of the accuracy bound for SparCost {#appendix-a-proof-of-the-accuracy-bound-for-sparcost .unnumbered}
===================================================
We first prove that $e_1 = \frac{{8\gamma (\log N + \log (4/\beta )}}{\varepsilon }$, then we prove **SparCost** satisfies $\varepsilon$-differential privacy.
![User dynamics with user fluctuation under different number of channels[]{data-label="fig:digraph"}](pic9.eps)
For error bound $e_1$, $e_1 = \frac{{8\gamma (\log N + \log (4/\beta )}}{\varepsilon }$.
Let $[N]$ be the set of sequence number $\{1,...,N\}$. Observe that we can prove the theorem by showing that except with probability at most $\frac{\beta}{2}$: $$\mathop {\max}\nolimits_{n\in[N]} |v_n| + |T - \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}| \leq e_1.$$ If this is the case, then for any output $\theta_n$, $$a_n(\widehat Q_{ij}^d( \vec p,\vec P_d))+v_n \le \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T} \le T+|T-\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}|,$$ or in other words: $$a_n(\widehat Q_{ij}^d( \vec p,\vec P_d)) \le T+|T-\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}|+|v_n| \le T+e_1.$$ Similarly, for any output $\bot$, $$a_n(\widehat Q_{ij}^d( \vec p,\vec P_d)) > \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T} \ge T-|T-\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}|-|v_n| \ge T-e_1.$$ For any $n< N$ : $a_n(\widehat Q_{ij}^d( \vec p,\vec P_d)) > T-e_1 > T+|v_n|+|T-\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}|$, and also : $a_n(\widehat Q_{ij}^d( \vec p,\vec P_d))+v_n \ge \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}$, meaning the output is $\bot$. Therefore the algorithm does not halt before all $N$ candidates are checked.
Recall that if $X$ satisfies Laplace distribution: $X \sim Lap(b)$, then: $Pr[|X| \ge t \cdot b] = exp(-t)$. Therefore, $$Pr[|T-\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}| \ge \frac{e_1}{2}] = exp(- \frac{\epsilon e_1}{4\gamma}).$$ We set this quantity by at most $\beta/4$, we then find $e_1 \ge \frac{4 \log{(4/\beta)}}{\epsilon}$. Similarly, by a union bound, we have: $$Pr[ \mathop{\max}\nolimits_{i \in |N|}|v_n| \ge \frac{e_1}{2} ] \le N \cdot exp(- \frac{\epsilon e_1}{8\gamma}).$$ We set this quantity by at most $\beta/4$ as well, we then find $ e_1 \ge \frac{8\gamma(\log{(4/\beta)}+\log{N})}{\epsilon}. $ These two claims combine to prove $e_1 = \frac{{8\gamma (\log N + \log (4/\beta )}}{\varepsilon }.$
Now, we turn to prove that **SparCost** satisfies $\varepsilon$-differential privacy.
At first, we need to redefine some of the notations. Let $a_n(Q, \widehat Q)=a_n(\widehat Q_{ij}^d( \vec p,\vec P_d))$. $Q$ is the aggregator from submitted mixed strategy while $\widehat Q$ is still set-valued aggregative function. Let $Q'$ be the aggregator when one of the users submits another mixed strategy. Let $\theta$ and $\theta'$ be the output of **SparCost** when the input is $Q$ and $Q'$ respectively. Noted that $\theta$ here includes both desired cost and $\bot$.
In following analysis, we fixed values of Laplace noise $v_1$,...,$v_{N-1}$. For $v_N$ and noisy threshold $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}$, we take probabilities over the randomness of them. Define $g(Q)$ as the minimum noisy value of $a_1(Q, \widehat Q)$,..., $a_{N-1}(Q, \widehat Q)$ and so: $$\vspace{-0.1cm}g(Q)=\min _{n < N} (a_n(Q, \widehat Q)-v_n).\vspace{-.1cm}$$ We also use $\mathop{\Pr}[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}=t]$ as the probability distribution function of $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}, \mathop {\Pr }[v_N=v]$ for the noise and $\mathop{\Pr}[a_n(Q, \widehat Q)=a]$ for the output. Let $I[x]$ be the indicator of event $x$. We then have: $$\hspace{-0.cm} \mathop {\Pr }\limits_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T},v_N } [\theta = a] = \mathop {\Pr }\limits_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T},v_N }[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}\le g(Q) \quad and \quad a_N(Q, \widehat Q)-v_N \le \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T} ]
=\mathop {\Pr }\limits_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T},v_N }[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}\in (a_N(Q, \widehat Q)-v_N,g(Q))]$$ $$\hspace{-0.25cm}= \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {\mathop {\Pr }[v_N=v]} } {\mathop {\Pr }[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}=t]}
*I[t\in (a_N(Q, \widehat Q)-v,g(Q))]dvdt$$ $$\hspace{-10.cm}= \omega.
\vspace{-.3cm}$$
We now change some of the variables: $$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over v}=v-g(Q)+g(Q')-a_N(Q', \widehat Q)+a_N(Q, \widehat Q), \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over t}=t-g(Q)+g(Q').
\vspace{-.2cm}$$ Then we have: $$\omega= \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {\mathop {\Pr }[v_N=\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over v}]} } {\mathop {\Pr }[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}=\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over t}]}*I[t-g(Q)+g(Q')\!\!\in\!\! (a_N(Q',\! \widehat Q)-v-g(Q)+g(Q')\!,\!g(Q))]dvdt$$ $$\hspace{-4.0cm}=\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {\mathop {\Pr }[v_N=\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over v}]} } \mathop {\Pr }[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}=\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over t}]*I[t \in (a_N(Q', \widehat Q)-v,g(Q'))]dvdt$$ $$\hspace{-1.7cm}\le \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty }\exp(\frac{\varepsilon}{2}) {\mathop {\Pr }[v_N=\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over v}]} }\exp(\frac{\varepsilon}{2}) \mathop {\Pr }[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}=\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over t}]*I[t \in (a_N(Q', \widehat Q)-v,g(Q'))]dvdt$$ $$\hspace{-7.3cm}=\exp(\varepsilon)\mathop {\Pr }\limits_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T},v_N }[\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T} <g(Q') and a_N(Q', \widehat Q)-v_N \le \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T}]$$ $$\hspace{-11.5cm}=\exp(\varepsilon)\mathop {\Pr }\limits_{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over T},v_N }[\theta'=a].$$ Thus completes our proof that **SparCost** satisfies $\varepsilon$-differential privacy.
Appendix B Proof of $\eta$-approximate NE {#appendix-b-proof-of-eta-approximate-ne .unnumbered}
=========================================
We now prove that our game is $\eta$-approximate, where $\eta=\zeta+\alpha+E_1+E_2$.
In the **MW**, the channel connection probabilities on the billboard is proceeded by both **SparCost** and **REXP**. According to composition theorem [@DR], these probabilities possess (2$\varepsilon$, $\delta$)-differential privacy. With regard to Billboard lemma, our result satisfies (2$\varepsilon$, $\delta$)-joint differential privacy.
According to Theorem 1 of Lipschitz Game, the assumption that our game has $n$ players, $m$ actions, and satisfies Lipschitz constant of $\gamma$ ensures that our game converges to $\gamma \sqrt {{\rm{8}}n\log (2mn)}$ -approximate NE. This NE does not concern any accuracy issue from privacy and algorithm.
Now we consider the inaccuracy induced from privacy and algorithm. From Theorem 3, we have $$E_1= \max _{i\ \to e_1 } \frac{{8\gamma (k\log \frac{{n\gamma }}{\alpha } + \log (4/\beta ))}}{\varepsilon }.$$ We then explore the accuracy of **MW** which is $$E_2 = \tilde {\rm O}{\left( {\frac{{n{\gamma ^2}}}{\varepsilon }\log \left( {\frac{{2kn}}{\beta }} \right)\sqrt {\log \left( m \right)\ln \left( {1/\delta } \right)} } \right)^{1/2}}.$$
The **MW** algorithm gives a no-regret guarantee for each user $i$: $$\frac{1}{T}\sum\nolimits_t {\left\langle {P_i^t,q_{i}(p_i)} \right\rangle}\le \mathop {\min }\nolimits_{{P_i} } \frac{1}{T}\sum\nolimits_t {\left\langle {P_i^t,q_{i}(p_i)} \right\rangle } + \eta + \frac{{\log (k)}}{{{T_\eta }}}= \mathop {\min }\nolimits_{{P_i}} \frac{1}{T}\sum\nolimits_t {\left\langle {P_i^t,q_{i}(p_i)} \right\rangle } + \frac{E_2 }{{2n\gamma }},$$ where ${\left\langle {P_i^t,q_{i}(p_i)} \right\rangle }$ is the input and output of **REXP**. We study the effect of $q_{i}^d(p_i)$ for all channels and omit $d$. $\lambda^t$ in following proof is from the score function in **REXP**. Let $q(p)$ be the set for channel contention probability of all users on all channels, i.e., $q(p)=\{\overline{q}_{1}^d(p_1),...,\overline{q}_{n}^d(p_n)\}$. Thus, the joint play of all $n$ agents satisfies $$\vspace{-.1cm}
\frac{1}{T}\sum\nolimits_t {\left( {\gamma \left\langle {P^t,q(p)} \right\rangle - {\lambda^t}} \right)}\le \mathop {\min }\nolimits_{P} \frac{1}{T} \sum\nolimits_t {\left( {\gamma \left\langle {P^t,q(p)} \right\rangle - {\lambda^t}} \right)} + E_2 /2.$$ It is easy to see that $$\vspace{-.2cm}
\mathop {\min }\nolimits_{P} \frac{1}{T}\sum\nolimits_t {\left( {\gamma \left\langle \vec {P^t,q(p)} \right\rangle - {\lambda^t}} \right)} \le 0.$$ So from above we know, $$\label{proof1}
\vspace{-.1cm}
\frac{1}{T}\sum\nolimits_t {\left( {\gamma \left\langle {P^t,q(p)} \right\rangle -{\lambda^t}} \right)} \le E_2 /2.$$
By Theorem 4, with probability at least $1-\frac{\beta}{2}$, exponential mechanism gives $$\hspace{-.cm}\sum\nolimits_t {\left( {\gamma \left\langle {P^t,q(p)} \right\rangle - {\lambda^t}} \right)} \ge \mathop {\max }\nolimits_{\left( {q,\lambda} \right)} \sum\nolimits_t {\left( {\left( {\gamma \left\langle {P^t,q(p)} \right\rangle - {\lambda_d^t}} \right) - \frac{{2\gamma \log \left( {\frac{{2kT}}{\beta }} \right)}}{{{\varepsilon _0}}}} \right)}.
\vspace{-.2cm}$$ And so, $$\label{proof2}
\vspace{-.1cm}
\frac{1}{T}\sum\nolimits_t {\left( {\gamma \left\langle {P^t,q(p)} \right\rangle - {\lambda^t}} \right)}\ge \mathop {\max }\nolimits_{\left( {q,\lambda} \right)} \frac{1}{T}\left[ {\sum\nolimits_t {\left( {\gamma \left\langle {P^t,q(p)} \right\rangle - {\lambda^t}} \right)} } \right] - \frac{{2\gamma \log \left( {\frac{{2kT}}{\beta }} \right)}}{{{\varepsilon _0}}}.$$
Combining above (\[proof1\]) and (\[proof2\]) with the definition of $\overline {P}$, we get $$\mathop {\max }\nolimits_{(q,\lambda)} \left( {\gamma \left\langle {\overline {P},q(p)} \right\rangle - \lambda} \right) = \mathop {\max }\nolimits_{\left( {q,\lambda} \right)}
\frac{1}{T}\sum\nolimits_t {\left( {\gamma \left\langle {P^t,q(p)} \right\rangle \!- \! {\lambda^t}}\right)}\le \frac{{2\gamma \log \left( {\frac{{2kT}}{\beta }} \right)}}{{{\varepsilon _0}}} + E_2 /2 \le E_2 ,$$ as long as $\frac{2\gamma \log \left(\frac{2kT}{\beta} \right)}{\varepsilon _0} \leq E_2 /2$. Plugging in parameter $\varepsilon _0$ from differential privacy, we then have $$\vspace{-.1cm}
{E_2 ^2} \ge \frac{{32\sqrt 2 n{\gamma ^2}\log \left( {\frac{{2kT}}{\beta }} \right)\sqrt {\log k\ln \left( {\frac{1}{\delta }} \right)} }}{\varepsilon }.$$
Plugging in $T$, we get our desired bound $$\vspace{-.4cm}
E_2 = \tilde {\rm O}{\left( {\frac{{n{\gamma ^2}}}{\varepsilon }\log \left( {\frac{{2kn}}{\beta }} \right)\sqrt {\log \left( m \right)\ln \left( {1/\delta } \right)} } \right)^{1/2}}.$$
Combining above, we have completed the proof. Now we have $\eta=\zeta+\alpha+E_1+E_2$.
Appendix C Proof of Incentive Compatible {#appendix-c-proof-of-incentive-compatible .unnumbered}
========================================
The large-scale spectrum sharing game we are discussing here contains two undesirable deviations. One is opting out of the mediated game. The other one is lying or not faithfully following the suggestion. However, it is incentive compatible. In fact, our mechanism encourages users to opt-in and to behave themselves, and the corresponding proof is as follows.
We first prove why users who tend to lie or not to truthfully follow the solution should play faithfully. Let $\overline P$ be the mixed strategy set for all users under NE. $P_i$ and $\overline P_{-i}$ are corresponding mixed strategies for user $i$ and users other than $i$, respectively. Let $x_i$ be user $i$’s type. We concern the type in the utility in the proof and assume that type $x_i$ needs the corresponding $P_i$ to get the desired utility. Since our game converges to $\eta$-approximate NE, we have, $$\label{APP1}
\vspace{-.2cm}
\mathop \mathbb{E}\nolimits_{x_i ,x_{ - i} }U_{ij} (p_i ,Q_i^d (\vec p,\overline P)) \ge\mathop \mathbb{E}\nolimits_{x_i ,x_{ - i} } U_{ij} (p_i' ,Q_i^d ((p_i',\vec p_{-i}),(P_i',\overline P_{-i})) - \eta,$$ where $p'_i$ denotes the action with the best utility and $ p_i$ is an action of aggregative best response. $P_i'$ and $P_i$ are corresponding mixed strategies. In other words, action $p'_i$ leads to exact NE while action $ p_i$ leads to a NE in terms of aggregative best response. The latter is no worse than the former by at most $\eta$. (\[APP1\]) provides the reason why users are willing to opt-in. Only in this way can they get a better payoff via reaching a NE.
Then, we invoke the privacy condition to explain why users are willing to opt-in telling the true mixed strategies and follow the recommendation. Our result satisfies (2$\varepsilon$, $\delta$)-joint differential privacy, so $$\mathop \mathbb{E}\nolimits_{x_i ,x_{ - i} }U_{ij} (p_i ,Q_i^d (\vec p,\overline P))$$ $$\hspace{-2cm}\ge\mathop \mathbb{E}\nolimits_{x_i ,x_{ - i} } U_{ij} (p_i' ,Q_i^d ((p_i',\vec p_{-i}),(P_i',\overline P_{-i})) - \eta$$ $$\ge \exp ( - 2\varepsilon )\mathop \mathbb{E}\nolimits_{x_i ',x_{ - i} } U_{ij} (p_i' ,Q_i^d ((p_i',\vec p_{-i}),(P_i',\overline P_{-i})) - \eta - \delta$$ $$\hspace{-0.5cm}\ge \mathop \mathbb{E}\nolimits_{x_i ',x_{ - i} } U_{ij} (p_i' ,Q_i^d ((p_i',\vec p_{-i}),(P_i',\overline P_{-i})) - \eta - \delta - 4L\varepsilon,\vspace{-0.2cm}$$ where $L$ is the upper bound of the utility function. Noted that $x_i '$ means user $i$’s type is not mapped into aggregative best-response to $x_{-i} $. This property comes from joint differential privacy rather than the standard. The last inequality is the result of inequality $e^\varepsilon \le 1 + 2\varepsilon.$ Then, consider the analysis of $\beta$-probability event that the mechanism fails to output an approximate NE, we have: $$\mathop \mathbb{E}\nolimits_{x_i ,x_{ - i} }U_{ij} (p_i ,Q_i^d (\vec p,\overline P)) \ge \mathop \mathbb{E}\nolimits_{x_i ',x_{ - i} } U_{ij} (p_i' ,Q_i^d ((p_i',\vec p)_{-i}),(P_i',\overline P_{-i})) - \eta - \delta - 4L\varepsilon - L\beta.$$ This completes the proof.
[1]{}
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[^1]: Pan Zhou and Wenqi Wei, these authors contributed equally to this work and are considered co-first authors, are with School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, China (email: [email protected], [email protected]).
Kaigui Bian is with the Institute of Network Computing and Information Systems, School of EECS, Peking University, Beijing 100871, China (email: [email protected]).
Dapeng Oliver Wu is with Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA (email: [email protected]).
Yuchong Hu, School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, China (email: [email protected]).
Qian Wang is with the Key Lab of Aerospace Information Security and Trusted Computing, School of Computer, Wuhan University, Wuhan 430072, China (email: [email protected]).
This work was supported by the National Science Foundation of China under Grant 61401169, Grant CNS-1116970, and Grant NSFC 61529101.
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abstract: 'It is shown that the ‘arrow of time’ operator, $\hat{M}_F$, recently suggested by Strauss et al., in arXiv:0802.2448v1 \[quant-ph\], is simply related to the sign of the canonical ‘time’ observable, $T$ (apparently first introduced by Holevo). In particular, the monotonic decrease of $\langle \hat{M}_F\rangle$ corresponds to the fact that $\langle \,{\rm sgn~}T\rangle$ increases monotonically with time. This relationship also provides a physical interpretation of the property $\hat{M}_F\leq \hat{1}$. Some further properties and generalisations are pointed out, including to almost-periodic systems.'
author:
- 'Michael J. W. Hall'
title: 'Comment on “An Arrow of Time Operator for Standard Quantum Mechanics” (a sign of the time!)'
---
Introduction
============
Strauss et al. have recently given an interesting example of a ‘Lyapunov operator’ applicable to a wide class of quantum systems [@strauss], i.e., an operator which has a monotonically decreasing expectation value for all initial states. In particular, for any system with a Hamiltonian operator $\hat{H}$ with eigenstates of the form $$\label{hform}
\hat{H}|E,j\rangle = E |E,j\rangle,~~~~~E\in [0,\infty),~~~j=1,2,\dots,d$$ for some fixed $d$ that is independent of $E$, Strauss et al. define the operator $$\label{mf}
\hat{M}_F := \frac{i}{2\pi}\sum_j \int_0^\infty dE \int_0^\infty dE'\, \frac{|E,j\rangle\,\langle E',j|}{ E-E' +i0^+} ,$$ and demonstrate that for any state $|\psi_t\rangle$ the expectation value $\langle \hat{M}_F\rangle_{\psi_t}$ is monotonic decreasing with time [@strauss]. Strauss et al. further give some numerical examples, and determine the eigenstates of $\hat{M}_F$.
The physical origin of $\hat{M}_F$ above is not particularly obvious, and earlier work of Strauss only provides a rather mathematical motivation, related to a Hardy space representation of the Schrödinger equation [@s1; @s2]. Here it will be shown that the operator $\hat{M}_F$ is in fact closely related to the canonical time observable, $T$, apparently first introduced by Holevo [@holevo]. In particular, one has the general relation $$\label{sign}
\langle \hat{M}_F \rangle \equiv \frac{1}{2}\left( 1- \langle \,{\rm sgn}~ T \rangle \right).$$ Hence, [*the Lyapunov operator is closely related to the sign of the canonical time observable*]{}, providing a simple physical interpretation for the former.
It is important to note that the canonical time observable $T$ is a probability operator measure (POM) [@holevo], and hence is described by a set of positive operators which sum to the identity operator [@holevo; @nielsen], i.e., $$T \equiv \{ \hat{T}_t \}, ~~~~~\hat{T}_t\geq0,~~~~~~\int_{-\infty}^\infty dt\,\hat{T}_t = \hat{1} ,$$ with the probability density for a measurement of $T$ to give result $t$ for state $|\psi\rangle$ given by $$\label{pt}
p_T(t|\psi) = \langle \psi|\hat{T}_t|\psi\rangle .$$ Such POM observables are well known to be essential for describing all possible measurements that may be made on a given quantum system, and may always be represented in terms of measurement of a Hermitian operator on an ‘apparatus’ system which has interacted with the system [@holevo; @nielsen]. The main advantage of the POM formalism is that one does not have to describe such apparatus systems explicitly, when considering the possible measurements on a given system, which is particularly useful when determining optimal measurements for extracting information in various scenarios.
In the next section the origin and basic properties of the canonical time observable $T$ are reviewed. In particular, $T$ is the optimal observable for covariantly estimating time translations, i.e., it is the optimal ‘clock’ observable for the system. The relation (\[sign\]) between $\hat{M}_F$ and $T$ is demonstrated in section III, and the corresponding physical interpretation of $\hat{M}_F$ is discussed. Some further properties and possible generalisations are discussed in Sec. IV.
Canonical time observable
=========================
For a quantum system with energy eigenstates as per equation (\[hform\]), define the corresponding ‘time’ kets $$|t,j\rangle := (2\pi)^{-1/2} \int_0^\infty dE\, e^{-iEt} |E,j\rangle~~~~~~~j=1,2,\dots,d.$$ Note that natural units with $\hbar=1$ have been adopted, in keeping with [@strauss]. The corresponding canonical time observable $T$ is then defined as the POM observable $\{\hat{T}_t\}$, with $$\label{tt}
\hat{T_t} := \sum_j |t,j\rangle\,\langle t,j| .$$ It is easily checked that $$\int_{-\infty}^\infty dt\,\hat{T}_t = \sum_j \int_0^{\infty} dE\,|E,j\rangle\,\langle E,j| = \hat{1} ,$$ as required. This ‘canonical’ time observable appears to have first been considered in some detail by Holevo, primarily for the case of a free particle [@holevo].
It is worth noting some properties of $T$ here, to indicate why it is a natural time observable to consider at all. First, observe that if $E$ and $t$ were replaced by momentum and position coordinates $p$ and $x$ in the above definition of $|t,j\rangle$, with the range of integration extended over the whole real line, then one would obtain the usual Fourier relation between conjugate position and momentum kets. Hence, by analogy, $T$ can be said to be conjugate to the energy observable $\hat{H}$. Indeed, the (truncated) Fourier relation between $|E,d\rangle$ and $|t,d\rangle$ immediately implies the entropic uncertainty relation $$\label{ent} L_H L_T \geq \pi e\hbar ,$$ precisely as for the case of position and momentum observables [@bbm], where the ‘ensemble length’ $L_A$ is the natural geometric measure of the spread of observable $A$, given by the exponential of the entropy of $A$ [@ensemble]. Hence, the energy and time uncertainties cannot simultaneously be arbitrarily small.
Second, it follows immediately from Eqs. (\[pt\]) and (\[tt\]) that $$\label{cov}
p_T(t'|\psi_t) = p_T(t'-t|\psi_0) ,$$ i.e., the probability distribution simply translates under time evolution of the system. This time-covariance property is of course expected of any good ‘clock’ observable [@holevo]. Note in particular it implies that $$\label{clock}
\langle T\rangle_{t} := \int_{-\infty}^{\infty}dt'\,t'\,p_T(t'|\psi_t) = \langle T\rangle_{0} + t .$$
Third, Holevo has shown that the canonical time observable provides the best estimate of an unknown time shift of the system, for a particular figure of merit [@holevo]. It may further be shown that $T$ is optimal in the sense that measurement of any [*other*]{} time-covariant observable is equivalent to first subjecting the system to some ‘noise’ process, and then making a measurement of $T$ - see, for example, the analogous property in Ref. [@hallqcm] for optical phase.
Finally, it should be remarked that the ‘time’ kets $|t,j\rangle$ are [*not*]{} mutually orthogonal, due to the semiboundedness of the energy spectrum. Hence, $T$ cannot correspond to some Hermitian operator on the Hilbert space of the system. Indeed, the truncated Fourier transform defining $|t,j\rangle$ implies, via the Paley-Wiener theorem, that $p_T(t|\psi)$ cannot vanish on any non-zero finite interval [@paley], and so the canonical time distribution is always ‘fuzzy’, with support over the entire real axis. Of course, for each real function $f(t)$, the corresponding average value of $f(T)$ follows from (\[pt\]) as $$\label{ft}
\langle f(T)\rangle_\psi = \int_{-\infty}^{\infty}dt\,f(t)\,p_T(t|\psi) = \int_{-\infty}^{\infty}dt\,f(t)\, \langle \psi|\hat{T}_t|\psi\rangle,$$ and hence one can define a corresponding Hermitian operator $$\label{ftop} \widehat{f(T)} := \int_{-\infty}^\infty dt\, f(t)\hat{T}_t$$ satisfying $$\langle f(T)\rangle_\psi = \langle\psi |\widehat{f(T)}|\psi\rangle$$ for all states $\psi$. In general, however, these operators are not simply related algebraically - for example, one does not have $\widehat{(T^2)}=(\widehat{T})^2$. Hence, it is the POM $T$ which is of fundamental significance, rather than any particular Hermitian operator $\widehat{f(T)}$. This has a bearing on the interpretation of the Lyapunov operator in Eq. (\[mf\]), which corresponds to a particular choice of $f(t)$.
‘Arrow of time’ vs ‘canonical’ time
===================================
Consider now the observable corresponding to the [*sign*]{} of the canonical time observable, where ${\rm sgn~}t$ is defined to be $-1$, $0$ and $+1$ for $t<0$, $t=0$ and $t>0$ respectively. By definition this observable can only have measured values in $\{-1, 0, 1\}$, and hence one must have $$\label{bound} -1 \leq \langle\,{\rm sgn~}T \rangle_\psi \leq 1 .$$ Moreover, recalling from Eq. (\[cov\]) that the probability distribution of $T$ moves to the ‘right’ as $t$ increases, one expects that the sign of $T$ must increase monotonically on average, just as $T$ itself does as per Eq. (\[clock\]). Indeed, noting that, trivially, ${\rm sgn~}(t'+t)\geq {\rm sgn~}t'$ for $t\geq 0$, one has from Eq. (\[cov\]) that $$\label{mono}
\langle\,{\rm sgn~}T \rangle_{\psi_t} = \int dt' \,{\rm sgn~}t'\, p_T(t'-t)|\psi_0) = \int dt' \,{\rm sgn~}(t'+t)\, p_T(t'|\psi_0) \geq \langle \,{\rm sgn~}T \rangle_{\psi_0} .$$ In fact, strict inequality holds for $t>0$, since $p_T(t|\psi)$ cannot vanish on any non-zero finite interval as noted in the previous section.
It is seen from Eqs. (\[clock\]) and (\[mono\]) that both $T$ and ${\rm sgn~}T$ have monotonic increasing expectation values. It then follows immediately that, for example, $-T$ and $-{\rm sgn~}T$ both have monotonic [*decreasing*]{} expectation values for all initial states. Hence, in particular, the corresponding Hermitian operators $-\widehat{T}$ and $-\widehat{{\rm sgn~}T}$ defined via Eq. (\[ftop\]) are ‘Lyapunov’ operators in the sense of Strauss et al. [@strauss]. Clearly there are many more such operators, but what is of interest here is the connection between $T$ and the particular Lyapunov operator $\hat{M}_F$ in Eq. (\[mf\]).
In particular, from Eqs. (\[tt\]) and (\[ftop\]), and using a standard table of Fourier transforms, one has the explicit expression $$\begin{aligned}
\widehat{{\rm sgn~}T } & = & (2\pi)^{-1}\sum_j \int dE\int dE' |E,j\rangle\langle E',j| \int dt\,({\rm sgn~}t)\,e^{-i(E-E')t}\\
& = & (2\pi)^{-1} \sum_j \int dE\int dE' |E,j\rangle\langle E',j|~~ {\rm P.V.}\left[ -2i(E-E')^{-1}\right] \\
& = & 1 - 2 \hat{M}_F ,\end{aligned}$$ where P.V. denotes the principal value, and the last line follows via the definition of $\hat{M}_F$ in Eq. (\[mf\]). One hence obtains the desired simple relation in Eq. (\[sign\]) between $\hat{M}_F$ and ${\rm sgn~}T$.
Note that the property $0\leq \hat{M}_F\leq 1$, and the monotonic decrease of the expectation value of $\hat{M}_F$ with time, both proved by Strauss et al. [@strauss], follow immediately from Eqs. (\[sign\]), (\[bound\]) and (\[mono\]). They are seen to correspond to (i) the property $|\,{\rm sgn~}t|\leq 1$ and (ii) the monotonicity of ${\rm sgn~}t$. Further, the [*strict*]{} monotonic decrease of $\langle \hat{M}_F \rangle$ follows from the property that the canonical time distribution has support over the whole real axis (up to a set of measure zero). However, the eigenfunctions of $\hat{M}_F$, determined by Strauss et al. [@strauss], are seen not to have any particular fundamental significance - they are merely eigenfunctions of an operator corresonding to a particular function of the canonical time observable, where different eigenfunctions would be obtained by choosing a different function of $T$.
Discussion
==========
It has been shown that the particular Lyapunov operator $\hat{M}_F$, investigated by Strauss et al. [@strauss], has a simple relationship to the sign of the canonical time observable $T$, as per Eq. (\[sign\]), and that its main properties can be easily obtained from general properties of $T$. Further, many other Lyapunov operators can be constructed from $T$ - in particular, the operator $\widehat{g(T)}=\int dt\,g(t)\,\hat{T}_t$, for any monotonic decreasing function $g(t)$.
It is worth noting that in practice it is simplest to determine $\langle\hat{M}_F\rangle$, for a given state $|\psi\rangle$, by first calculating $$p_T(t|\psi_0) = \sum_j \left| \langle t,j|\psi_0\rangle \right|^2 ,$$ and then using Eqs. (\[sign\]) and (\[cov\]) to calculate $$\label{simp}
\langle \hat{M}_F\rangle_{\psi_t} = \int_{-\infty}^0 dt'\,p_T(t'-t|\psi_0)= \int_{-\infty}^{-t} dt'\,p_T(t'|\psi_0) .$$ Note it follows that $\langle M_F\rangle$ is the cumulative probability distribution for the canonical time observable, thus providing a physical interpretation for Eq. (7) in Ref. [@strauss].
For example, for an initially stationary free particle of mass $m$ in one dimension, with Gaussian momentum representation $$\psi_0(p) = e^{-p^2/(4\sigma^2)}$$ up to a normalisation factor, one finds the Gaussian integral $$\langle t,\pm|\psi_0\rangle = \int_0^\infty dp\,p^{1/2} e^{ip^2t/(2m)} e^{-p^2/(4\sigma^2)}$$ up to a normalisation factor (where $p=\pm (2mE)^{1/2}$), thus yielding a distribution of the form $$p_T(t|\psi_0) = N \left[ t^2 + (m/2\sigma^2)^2\right]^{-3/2}$$ for the canonical time observable, for some normalisation constant $N$. One therefore obtains from Eq. (\[simp\]), restoring general units, the explicit result $$\langle \hat{M}_F\rangle_{\psi_t}= 1/2 - (1/2)\,t\, [t^2 + (\hbar m/2\sigma^2)^2]^{-1/2} .$$
Note that the case of a free particle does not quite satisfy the assumed spectral condition in Eq. (\[hform\]), as the degeneracy breaks down at $E=0$. While not affecting the above results, this property leads, for example, to a divergence in the variance of the canonical time observable when the wavefunction has a non-zero component corresponding to $E=0$ [@holevo], which can be seen to occur for the free-particle example above. Hence, no Heisenberg type uncertainty relation can be written down in this case. In contrast, the ensemble lengths $L_H$ and $L_T$ in the time-energy uncertainty relation (\[ent\]) are perfectly well-defined for the above example [@ensemble], and can be explicitly calculated using standard tables of integrals (allowing evaluation of how close their product is to the lower bound $\pi e\hbar$).
It is of interest to consider how the canonical time observable may be generalised to quantum systems with energy spectra different to that of Eq. (\[hform\]). This is quite simple for an evenly spaced discrete energy spectrum, such as a harmonic oscillator, where essentially the Fourier transform defining the ‘time’ kets $|t,j\rangle$ is replaced by a discrete Fourier transform, yielding a periodic canonical time observable [@holevo; @helstrom; @hallqcm]. This replacement also applies to any energy spectra which is a subset of an evenly spaced discrete set. More generally, for an arbitrary continuous or uniform discrete energy spectrum, with a possibly continuous degeneracy which may depend on $E$, one may formally extend the energy spectrum to the form $\{|E,j\rangle\}$ with $j$ ranging over some sufficiently large measurable set $J$, with $|E,j\rangle=0$ for some values of $j$ if the degeneracy is not uniform, and define the corresponding canonical time observable $T\equiv \{\hat{T}_t\}$ as per Eq. (\[tt\]), where the $|t,j\rangle$ are defined as before (with integration replaced by summation for discrete energy spectra).
The cases of mixed continous and discrete energy spectra, and of discrete non-uniform energy spectra, appear to be more difficult - such systems simply may not make good ‘clocks’ (certainly this would be the case for chaotic classical systems). Note, however, that if the state of the system only has support on the continuous portion of the spectrum, or on a uniformly spaced subset of the discrete portion, then a time observable may be defined as above on the corresponding restricted Hilbert space.
Finally, it is of interest to note, assuming uniform degeneracies for convenience, that for a general discrete energy spectrum $\{ |E_k\rangle\} $ one may be able to at least speak of ‘clocks’ relative to a certain resolution. In particular, one can certainly always define a POM $T(\tau):=\{ \hat{T}_t(\tau);\hat{P}(\tau)\}$, with $t$ taking values in $[0,\tau)$, by $$\hat{T}_t(\tau):= \sum_j \hat{N}_\tau^{-1/2} |t,j\rangle\langle t,j| \hat{N}_\tau^{-1/2},$$ where $|t,j\rangle$ is defined as in Sec. II (with integration replaced by summation), $N_\tau$ is the positive operator $$\hat{N}_\tau := \sum_j \int_{0}^\tau dt\,|t,j\rangle\langle t,j| ,$$ $\hat{N}_\tau^{-1/2}$ is defined to be $0$ when acting outside the support of $N_\tau$, and $\hat{P}_\tau$ is the projection operator onto the zero eigenspace of $\hat{N}_\tau$. The limit $\tau\rightarrow\infty$ is well defined when the energy spectrum is uniform spaced, yielding a periodic time observable. More generally, the evolution will be almost-periodic, and for a given resolution $\epsilon$ there will be a period $\tau_\epsilon$ for which the system state will evolve arbitrarily close to its (arbitrary) initial state, to within a distance defined by $\epsilon$. Hence, it appears that one may define a time observable for such systems relative to a given resolution parameter $\epsilon$. This would be of interest for further investigation.
[99]{}
Y. Strauss, S. Silman, S. Machnes and L.P. Horwitz, Eprint arXiv:0802.2448v1 (2008). Y. Strauss, Eprint arXiv:0706.0268 \[math-ph\] (2007). Y. Strauss, Eprint arXiv:0710.3604 \[math-ph\] (2007). A.S. Holevo, [*Probabilistic and Statistical Aspects of Quantum Theory*]{} (North-Holland, Amsterdam, 1982). M.A. Nielsen and I.L. Chuang, [*Quantum Computation and Quantum Information*]{} (Cambridge University Press, Cambridge, U.K., 2000). I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. [**44**]{} 129 (1975). M.J.W. Hall, Phys. Rev. A [**59**]{} 2602 (1999); Eprint arXiv:physics/9903045. M.J.W. Hall, J. Mod. Opt. [**40**]{} 809 (1993). A. Papoulis, [*Signal Analysis*]{} (McGraw-Hill, New York, 1977). C.W. Helstrom, Int. J. Theoret. Phys. [**11**]{} 357 (1974); J.H. Shapiro and S.R. Shephard, Phys. Rev. A [**43**]{} 3795 (1991).
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---
abstract: 'The reuse at the component level is generally more effective than the one at the object-oriented class level. This is due to the granularity level where components expose their functionalities at an abstract level compared to the fine-grained object-oriented classes. Moreover, components clearly define their dependencies through their provided and required interfaces in an explicit way that facilitates the understanding of how to reuse these components. Therefore, several component identification approaches have been proposed to identify components based on the analysis object-oriented software applications. Nevertheless, most of the existing component identification approaches did not consider co-usage dependencies between API classes to identify classes/methods that can be reused to implement a specific scenario. In this paper, we propose an approach to identify reusable software components in object-oriented APIs, based on the interactions between client applications and the targeted API. As we are dealing with actual clients using the API, dynamic analysis allows to better capture the instances of API usage. Approaches using static analysis are usually limited by the difficulty of handling dynamic features such as polymorphism and class loading. We evaluate our approach by applying it to three Java APIs with eight client applications from the DaCapo benchmark. DaCapo provides a set of pre-defined usage scenarios. The results show that our component identification approach has a very high precision.'
author:
- Anas Shatnawi
- Hudhaifa Shatnawi
- Mohamed Aymen Saied
- Zakarea Al Shara
- Houari Sahraoui
- Abdelhak Seriai
bibliography:
- 'main.bib'
title: 'Identifying Software Components from Object-Oriented APIs Based on Dynamic Analysis'
---
Introduction {#Introduction}
============
Decades of research have shown that developing software applications based on Application Programming Interfaces (APIs) improves software reuse by offering pre-implemented and tested functionalities [@shatnawi2016reverse] [@zibran2011useful] [@saied2015could] [@moritz2013export]. For Object-Oriented (OO) APIs, the basic unit is a class, which encapsulates its functionalities and specifies the public interface for using them [@moritz2013export]. Reusing and understanding large OO APIs, e.g., JDK and .NET framework APIs, are complex tasks due to large numbers of included classes and methods [@shatnawi2016reverse] [@moritz2013export] [@saied2016cooperative] [@robillard2011field]. Considering JDK 1.8.0 as an example, we have approximately 4240 classes used to provide its functionalities to software applications [@liguori2014java]. Thus, it is a challenge to understand this large number of classes to identify needed functionalities [@uddin2012temporal].
On the other hand, API classes/methods are reused based on reuse scenarios represented by the combinations of API classes or methods that offer the required functionalities to software applications [@shatnawi2016reverse] [@moritz2013export]. However, different applications rely on different scenarios, depending on their needs of API functionalities [@uddin2012temporal]. This leads to the possibility to have a large number of scenarios, corresponding to different combinations of API classes/methods [@uddin2012temporal]. To help support understanding these reuse scenarios, several research approaches have been proposed to abstract high level views of these scenarios in terms of frequent co-usage patterns between API classes/methods. The frequent co-usage patterns are identified based on the analysis of how software applications (called client applications) have (re)used API classes/methods [@shatnawi2016reverse] [@saied2015mining] [@uddin2012temporal] [@montandon2013documenting] [@saied2015observational] [@Monperrus:2010:ECOOP] [@salman2017identification]. These abstracted co-usage patterns of API classes/methods are used to support software engineers to perform several engineering tasks; API reengineering [@shatnawi2016reverse], API documenting [@montandon2013documenting], API understanding [@saied2015observational], API policy enforcing [@riganelli2017policy] [@riganelli2017verifying], etc.
Identifying components in OO APIs is another alternative to improve APIs’ documentation by providing an abstract high-level view of the provided funcltionalities. In addition to the documentation motivation, the identified components can also help migrating OO APIs into component-based ones if desired. Such a reegineering process does not only support feeding component-based repositories, but it also allows one to get the benefits of the component-based software engineering by developing large-scale software applications using selected components and integrating them together based on flexible architectures [@shatnawi2013mining]. It has been admitted that software components are more reusable and understandable software modules than object-oriented classes. This admission is based on the granularity level where components are coarse-grained modules compared to the fine-grained object-oriented classes [@adjoyan2014service]. Furthermore, components clearly define their dependencies through their provided and required interfaces in an explicit way that facilitates the understanding of how to reuse these components. These are the motivations of several component identification approaches that have been proposed to identify components based on the analysis of source code of OO software [@seriai2014enactment] [@shatnawi2013mining] [@allier2011object].
Existing component identification approaches are designed to identify components from OO source code of software applications. They only rely on OO dependencies (e.g., method invocations, sharing types, etc.) to identify dependencies between classes. However, in the context of software OO APIs, co-usage dependencies between API classes should also be considered to identify classes/methods that can be reused to implement a specific scenario. This requires the analysis of source code of APIs and their client applications. Therefore, approaches designed for OO applications cannot be applied directly to OO APIs. To the best of our knowledge, the only approach proposed to identify components from OO APIs is one described in [@shatnawi2016reverse] and [@shatnawi2014mining]. This approach is based on the static analysis of the source code of OO APIs and their client applications. As we are dealing with actual clients using the OO API, dynamic analysis allows to better capture the instances of API usage. Approaches using static analysis are usually limited by the difficulty of handling dynamic features such as polymorphism and class loading.
In this paper, we propose an approach to identify reusable components in OO APIs. Our approach is based on the dynamic analysis of interactions between client applications and the targeted OO API. Thus, we generate the execution traces for the different usage scenarios implemented in client applications. These execution traces realize dependencies between API classes through their method invocations. We assume that components are identified in terms of API classes that are frequently reused together by client applications. Therefore, we consider that groups of methods frequently appearing together in execution traces form provided interfaces of components, and the owner API classes of these methods constitute the structure of that candidate component. To evaluate our approach, we applied it to three Java APIs with their clients from the DaCapo benchmark. The results show that the precision of our component identification approach is 98%.
The rest of this paper is organized as follows. Section \[sec2:ApproachFramework\] presents the framework behind our approach. Section \[sec3:IdentifyingExecutionTraces\] describes the process of identifying execution traces related to usage scenarios of client applications of APIs. In Section \[sec4:qualityFunction\], we identify graph representations of APIs methods based on the relationships appeared in execution scenarios using a proposed quality function. Section \[sec5:identifingGroups\] shows how classes composing components are identified based on a graph-based clustering algorithm. Evaluation results are presented in Section \[sec5:Experimentation\]. Related works are discussed in Section \[sec6:RelatedWork\]. Conclusion and future work are presented in Section \[sec8:ConclusionFutureWork\].
The Proposed Approach Framework {#sec2:ApproachFramework}
===============================
In this section, we provide an overview, summarize the principles and propose the process of the proposed approach.
Approach Overview
-----------------
Our approach identifies reusable components based on the dynamic analysis of interactions between client applications and the targeted API. We define a software component as a collection of classes that participate to implement one or more functionalities for client applications of an API. The provided interfaces, of this component, are API methods that have been invoked together frequently by client applications. Conversely, the required interfaces are API methods that have been invoked by the component classes and belong to other API components’ classes.
Classes composing a component are identified based on their invoked methods (provided interfaces). To determine which methods are invoked jointly, we analyze execution traces of client applications using the API. As for dynamic analysis, we need representative usage scenarios, we rely on use cases of client applications to identify such representative scenarios, and execute them to produce the traces. The execution traces highlight the dependencies between API methods. We consider methods that appear frequently together in execution traces as provided interfaces of a component. The owner classes of these methods define the structure of the component implementation.
We want to package each collection of API methods frequently used together to form a provided interface of a reusable component, without changing the internal structure of the API (i.e., we do not aim to identify architectural view of the API). Therefore, we allow a class to be a part of more than one component as different subsets of its methods can participate with various groups of methods related to other classes to implement different functionalities.
To evaluate the quality of a candidate collection of API methods to form a component provided interface, we define quality function that analyzes dependencies between API methods based on the identified execution traces. We cluster API methods using this quality function.
Approach Principles
-------------------
The principles of our approach are summarized as follows.
- A component is defined as a group of classes identified through its potential provided interfaces.
- A provided interface of a component is a set of methods used frequently together in execution traces.
- A required interface of a component is a set of methods used by this component and belonging to other components’ classes.
- A class can belong to several components since different subsets of its methods can be included in different component interfaces.
- Execution traces are used to identify the dependencies between API methods. Thus, they guide the component identification process.
![The process of component identification from object-oriented APIs[]{data-label="fig:mining_process"}](mining_process_4.png){width="47.00000%"}
Approach Process
----------------
To implement our approach, we propose a process, presented in Figure \[fig:mining\_process\], based on the following three steps:
1. **Identifying execution traces**: execution traces are identified based on usage scenarios related to client applications of APIs. Each executing trace is realized in terms of a call tree that represents the dynamic relationships between API methods corresponding to a usage scenario.
2. **Building graph representations of APIs:** to facilitate the problem definition, we rely on graph representations to highlight dependencies between methods of an API. These dependencies are identified based on the strength of relationships between methods in the call trees.
3. **Identifying classes composing components:** we apply a graph-based clustering algorithm to partition the graph into sub-graphs. Each sub-graph represents a group of methods forming a provided interface of a component, while their classes represent the component implementation.
Identifying Execution Traces {#sec3:IdentifyingExecutionTraces}
============================
In this section, we discuss how to identify execution traces related to APIs based on usage scenarios of client applications.
An execution trace is a set of methods that represent the execution of a usage scenario of a client application of the targeted API. These methods can be represented in terms of a call tree defined as a *directed tree T = $\langle V, E\rangle$*, where *V* is a set of vertices referring to a set of API methods and *E* is a set of edges. An edge $\langle V1, V2\rangle$ refers to that a method *V1* invokes a method *V2*. The root of the tree represents the starting point of the execution trace.
Executing Usage Scenarios to Identify Call Trees
------------------------------------------------
We execute each usage scenario to identify the set of methods corresponding to its implementation. A method representing the entry of a usage scenario is considered as the call tree root. Then, we follow method calls to identify the other nodes. When a method invokes a method, a new node is added to the tree with an edge from the first node to the class node, and so on.
Methods related to the call tree’s nodes can be classified into three categories: methods implemented in application’s classes, method of API classes used by application’s classes to access API functionalities and method of API classes that have got called by the second category’s methods (from API method used by method of application’s classes). Figure \[fig:api\_tree\] shows an example of a call tree that is composed of 4 application methods, which are *A, C, D* and *E*, and 6 API methods, which are *B, F, G, H, L* and *K*. The *B* and *F* methods have been invoked from an application class, while *G* and *K* methods have been called from an API method.
![Example of call tree corresponding to usage scenario[]{data-label="fig:api_tree"}](app_api_tree.png){width="47.00000%"}
Removing Application’s Methods from Call Trees
----------------------------------------------
Since our goal is to analyze interdependencies between API methods, we prune the identified call trees by removing methods related to application classes. For each call tree, we perform a *Breadth First Search* algorithm starting from the root node. If a child is an API method, then we leave it. Otherwise, we remove the node. The children of the removed node are attached to the parent of the removed node. This process is continued until verifying all tree levels. For example, Figure \[fig:norm\_api\_tree\] shows the pruned tree of the call tree presented in Figure \[fig:api\_tree\]. As it is noticed, the tree size is significantly reduced which decreases the complexity of analyzing call trees. For instance, it is reduced from 21 to 13 nodes.
![Pruned call tree of one in Figure \[fig:api\_tree\][]{data-label="fig:norm_api_tree"}](norm_api_tree.png){width="47.00000%"}
Building Graph Representations of APIs {#sec4:qualityFunction}
======================================
We represent an API in terms of an *undirected weighted graph G = $\langle V, E\rangle$*, where *V* is the set of API methods and *E* is the set of edges between the vertices *V*. An edge links the vertices *(u,v)* if they appeared together in call trees. The weight associated to each edge is based on the strength of relationship of its vertices in call trees. We distinguish three attributes that should be analyzed in call trees to measure the strength of these relationships. These are call frequency, call distance and call weight.
Call Frequency
--------------
This attribute refers to the relationship between numbers of co-occurrence of a group of methods in call trees and their cohesive. Methods frequently appeared in call trees are likely participating to provide related functionalities. For instance, in the call trees presented in Figure \[fig:call\_trees\_example\], the call frequency of *A* and *B* is 2 times, and it is 1 for *A* and *L*. Thus, *A* and *B* are likely cohesive than *A* and *L*.
The call trees are identified from different applications developed by different development teams. Thus, we propose to measure the call frequency based on two cases related to methods frequently used together in call trees of the same applications by the same team or different applications of different teams. We consider that methods used by different development teams will provide a global view of their reuse frequency (i.e., global frequency) compared to methods used by the same development team (i.e., local frequency).
Local frequency measures how many methods are used together in the same applications (the same developers). It is measured based on the average number of appearances of a group of methods in one application. We calculate the ratio between the number of call trees containing the methods to the total number of call trees in each application (c.f. Equation \[eq:Lfreq\]).
Global frequency measures how many methods are used together in different applications. Global frequency is calculated based on the average number of applications that contain the methods to the total number of applications (c.f. Equation \[eq:Gfreq\]). The call frequency of a group of methods is the mean value of local and global frequencies of all pairs of methods.
Equation \[eq:callFreq\] measures the call frequency for a set of methods *E*, where *Co-occur(c,v)* returns 1 if the methods *c* and *v* exist in the call tree *t*, otherwise it returns 0. *T(a)* refers to the set of call trees of the application *a* in the set of applications *apps*.
\[eq:callFreq\] CallFreq(E)=
\[eq:Lfreq\] LFreq(c,v)=
\[eq:Gfreq\] GFreq(c,v)=
![An example of three call trees[]{data-label="fig:call_trees_example"}](example_call_trees.png){width="50.00000%"}
Call Distance
-------------
This attribute is related to the *distance* between methods in call trees, which refers to the strength of their interactions. As much as methods are closer in call trees they have high probabilities to provide same functionalities. For example, in the first call tree in Figure \[fig:call\_trees\_example\], the distance between *A* and *B* is 1, while it is 3 between *A* and *E*. Thus, *A* and *B* have higher probability to be cohesive than *A* and *E*.
We define Equation \[eq:CallDist\] to measure the call distance of a group of methods *E*. For each pair of methods, it calculates the mean distance of this pair in all call trees using Equation \[eq:Distance\]. Then, the call distance is the average value of the mean values of all pairs.
\[eq:CallDist\] CallDist(E)=
\[eq:Distance\] Distance(c,v)=
\[eq:dis\] dis(c,v,t)= 1-()
Where *avgDistance(c,v,t)* finds the average distance between the methods *c* and *v* in a call tree *t*. It returns *2\*getTreeDepth(t)* if the methods *c* and *v* do not appear in a call tree.
Call Weight
-----------
This attribute is related to the weight of the co-occurrence of a group of methods in call trees. Methods that are co-located in the same call trees have different weight of co-occurrences. This affects to the degree of their interaction, and consequently their cohesive. The weight depends on the number of calls in the call trees. For example, in Figure \[fig:call\_trees\_example\], *D* and *E* appear together in call tree (1) and call tree (3), but with different weights. This means that their appearance has different impact to their cohesive. In fact, it is clear that their appearance in call tree (3) has a higher impact than in call tree (1).
We measure the call weight based on the number of edges in call trees. For a group of methods, the call weight is the average call weight of all pairs in this group. Since the call weight of a pair of methods in such a call tree is the fraction between the number of direct call between these methods and the total number of edges in the call tree. For our example (D and E), it is 20% in call tree (1) and 100% in call tree (3), thus it is 60% in average. Equation \[eq:callWght\] measures the call weight of a collection of methods *E*:
\[eq:callWght\] CallWeight(E)=
\[eq:wight\] Weight(c,v)=
Where *Wei(c,v,t)* is the wight between the methods *c* and *v* in the call tree *t*. It is calculated using Equation \[eq:wi\].
\[eq:wi\] Wei(c,v,t)=
For evaluating the quality of a group of methods *E*, we define Equation \[eq:qualityFunction\] as a linear combination of the three functions evaluating each attribute.
\[eq:qualityFunction\] Q(E)=(\_1CallFreq(E) + \_2CallDist(E) + \_3CallWght(E))
Where $\lambda_1$, $\lambda_2$ and $\lambda_3$ are weight values, situated in \[0-1\]. These are used by the API expert to weight each attribute as needed.
Identifying Components Through their Provided Interfaces {#sec5:identifingGroups}
========================================================
As we mentioned previously, methods composing provided interfaces of components are grouped based on a graph-based clustering algorithm. We select a clustering algorithm that allows overlapping between clusters. The idea behind that is to allow a method to appear in different clusters (component provided interfaces) as when it is used frequently with different groups of method.
Identifying Overlapping Clustering of Methods
---------------------------------------------
The identification of methods composing component interfaces is based on the identification of a set of subgraphs, where they have accepted quality values. We define the component provided interface as a subgraph $G^*$ = $\langle V^*$, $E^* \rangle$ where the average weight of edges in $E^*$ maximizes the quality function, the set of methods corresponding to the vertices $V^*$ represents the component provided interfaces. The identification of an optimal set of subgraphs needs to identify all possible candidate subgraphs that can be extracted from the graph. Then, the selection of subgraphs is made by choosing ones that maximize the quality function. Nevertheless, this is considered as *NP-Complete* problem since the time complexity is exponential. Thus, we propose to use a heuristic clustering algorithm to find a near optimal set of subgraphs. We use the *OClustR* [@perez2013oclustr] clustering algorithm. We select this algorithm as it is a graph-based clustering algorithm that is able to identify overlapping clusters. In addition, there is no need to identify the number of the needed clusters as we do not have any idea about the number of components that should be identified.
Based on *OClustR*, the problem is to identify a set of subgraphs *W = $\{G_{1}^*, G_{2}^*, ..., G_{k}^*\}$*, where the graph *G* is covered by *W* (i.e. *$ \cup_{i=1}^k V_{i}^*$ = V*, such that *$G_{i}^*$* covers a vertex *v* if *v* belongs to *$V_{i}^*$*). Finding the minimum number of subgraphs, i.e. the set *W*, is known as a vertex cover problem. A vertex cover is a smallest subset of vertices *V\^* $\subseteq$ *V*, such that each vertex *v* $\in$ *V* is either a part of *V\^* or directly connected, i.e. an adjacent vertex, to another vertex *u* $\in$ *V\^*. Nevertheless, a vertex cover is classified as *NP-Complete* problem \[Ref\]. Thus, *OClustR* proposed an approximation algorithm that aims at mining a near optimal set of vertices *V\^*, such that graphs corresponding to *V\^* cover *G* and maximizing the quality of the corresponding component as well. This algorithm identifies the subgraphs based on two steps. The first step aims at mining the initial subgraphs (clusters). The second step goals at refining the initial subgraphs by minimizing the number of clusters, since they may include useless subgraphs, and the overlapping between them.
### Mining initial clusters
This step aims at identifying the set of initial subgraphs *W*, such that each subgraph *$G_{i}^*$* is a *weighted star subgraph, (ws-graph)* for short, in G. A *ws-graph* *$G_{i}^* = \langle V^*,E^*\rangle$* have a vertex *c* $\in$ *$V^*$*, *c* is called the *center* of *$G_{i}^*$*, where there is an edge connecting *c* with the other vertices in *$V^*$*, the other vertices are called *satellites*. Here, the problem is to identify a set of *center* vertices *$Z = \{c_{1}, c_{2}, ..., c_{k}\}$*, where each vertex *$c_{i} \in Z$* is the center of the subgraph *$G_{i}^* \in W$*. In order to identify the set *Z*, all vertices should be investigated. The investigation is based on an iterative process. At each iteration, one vertex *v* is added to *Z*. This is continued until Z reaches the condition of covering *G*. The selection of a vertex *v* to be added to *Z* is based on an evaluation criterion. This criterion depends on two factors. The first one aims at maximizing the cover of *G* at each iteration and controlling the overlapping between the *ws-graphs*. This factor is called *Relative Density*, *(RD)* for short. The second factor goals at keeping the quality of the component corresponding to the *ws-graph* related to a vertex *v*, since a vertex that provides a high coverage of G could produce a low quality component corresponding to our quality fitness function. This factor is called *Relative Compactness*, *(RC)* for short.
*RD* is related to the number of *satellite* vertices that could be covered by *ws-graph* corresponding to a *center* vertex *v*. Since vertices having higher degree, i.e. vertices having more *satellites*, produce *ws-graphs* that contribute more to cover *G*, the selection of a vertex should take the vertex degree into account. In some cases, this way is not sufficient since some of the *satellite* vertices maybe already covered by another *center* vertex *r* that is added to *Z* in a previous iteration. Thus, we need to take the number of *satellite* vertices that will be covered by selecting a vertex *v* in the current iteration, i.e. we exclude *satellite* vertices already covered by another *center* vertices. We measure *RD* of a vertex *$v \in V$* based on Equation \[algo.RD\]. The higher value of *RD* the better vertex to be selected, where its value is situated in \[0-1\].
\[algo.RD\] RD(v)=
*RC* is related to the quality of a *ws-graph* corresponding to *v* compared to its *satellite* vertices. This means that we investigate if a vertex *v* is the best vertex that maximizing the quality among its *satellite* vertices. Therefore, we measure *RC* based on the ratio between the number of *satellite* vertices that produces *ws-graphs* having lower quality values than the *ws-graph* corresponding to *v* and the total number of *satellite* vertices. Where the quality of a *ws-graph* is the average weight values between all pairs of vertices included by the *ws-graph*. We calculate *RC* based on Equation \[alg.RC\], where *SatelliteCompactness* is the number of *satellites s* such that the quality of *ws-graph* of *s* is grater than the quality of the *ws-graphs* of *v*. The higher value of *RC* the better vertex to be selected, where its value is situated in \[0-1\].
\[alg.RC\] RC(v)=
Based on these factors, we firstly sort the vertices in a decreasing order based on their relative quality; the average of their *RD* and *RC*, denoted by RQ. Then, these vertices are iteratively added to the set *Z* with respect to one of these conditions: (1) a vertex is not covered and (2) it is covered but at lease there is one of its *satellites* that is not covered. This continues until covering graph *G*.
\[algo:InitialClusters\] *L* $\longleftarrow$ *G(V)* Sort *L* in a decreasing order based on *RQ*
### Refining the initial clusters
This step aims at enhancing the initial clusters, identified in the previous step. The goal is at reducing the number of resulted clusters as well as the overlapping between them. The process of identifying the set *Z* is a greedy one since it selects the vertex having the highest quality at each iteration. Thus, this may lead to the situation of adding a useless vertex *u* to *Z*. The identification of a useless subgraph *$G_{u}^*$* is based on how much its *satellites* are shared with other *$G^*$s*. The worst case is that *u* $\in$ *Z*, such that *G* is still covered by the *$G^*$s* corresponding to *$Z - \{u\}$*, i.e. *u* and its *satellites* are covered by other *$G^*s$*. The average case is that *u* $\in$ *Z*, such that *$G_{u}^*$* shares most of its *satellites* with the other *$G^*s$* corresponding to *$Z - \{u\}$*. In these case, the *$G_{u}^*$* is considered as a useless subgraph and *u* should be deleted from *Z*.
According to that, *$G_{u}^*$* is considered as a useless if it meets two conditions. The first one is that *u* is a *satellite* vertex of at lease another *$G_{v}^*$*. The second condition is that *$V_{u}^*$* shares more than half of its *satellite* vertices with another *$G^*s$*. Once a useless *$G_{u}^*$* is identified, *u* is removed from *Z*, and the non-shared *satellites* are distributed to another *$G^*$*. We select the *$G^*$* that covers *u* as it covers the center of *$G_{u}^*$*. In case that there is more than one *$G^*$* covering *u*, we select *$G^*$* having the grater number of . By doing this, we allow producing clusters having many vertices.
Algorithmically, we refine the initial clusters *$G^*$* as follows. It firstly sorts the vertices of *Z* in a descending order based on their degrees and sign all of them as unvisited. Then, starting from the vertex having the highest degree, each vertex is analyzed. The analysis consists of removing from *Z* any vertex *$u \in Z$* such that *u* is a *satellite* of *$G_{v}^*$* (*$u \in V_{v}^*$*) and *$G_{u}^*$* is considered as useless *$G^*$*. Then, all *satellite* verices of *$G_{u}^*$* is added to *$G_{v}^*$* since *$G_{v}^*$* is the one having the grater number of among *u* adjacent vertices. Once all *satellite* verices in *$G_{v}^*$* are analyzed, *v* is signed as visited and *$G_{v}^*$* is considered as a final cluster.
\[algo:RefiningClusters\] *L* $\longleftarrow$ *G(V)* Sort *Z* in a decreasing order based on their degree
\[tb:dataSet\]
Evaluation Results {#sec5:Experimentation}
==================
Description of APIs and their Client Applications
-------------------------------------------------
We evaluate our approach using real usage scenarios from DaCapo Benchmarks [@blackburn2006dacapo]. We select DaCapo because it is composed of a set of open-source real Java applications coupled with collections of *pre-defined usage scenarios*.
To show the applicability of our approach in several context, we select from DaCapo three widely used Java APIs of three different size (i.e., 23, 99 and 302 classes) and from three different domains. The first one is the Apache Commons Logging (ACL) API[^1] that contains 23 classes, and offers a log interfaces and a middleware/tooling developer with a simple logging abstraction. The second one is the ASM API [^2] consists of 99 classes. It provides functionalities related to Java bytecode manipulation and analysis. The third API is the XML API[^3] that is composed of 302 classes. It offers functionalities for processing XML files.
DaCapo only provides a limited number of usage scenarios of software applications developed based on these three APIs. We are able to identify only eight client applications that use methods of classes of the APIs. These applications are as follows.
- **Tomcat** implements J2EE technologies like Servlet and JavaServer Pages.
- **Fop** is an output-independent print formatter that parses and formats XSL-FO files.
- **Xalan** is an XSLT processor for transforming XML documents.
- **Batik** is a Scalable Vector Graphics (SVG) toolkit that renders a number of SVG files. Xalan, Fop and Batik use classes from the XML API.
- **Pmd** is a source code analyzer for Java code. It uses two APIs; XML and ASM ones.
- **Jython** is a python interpreter written in Java to execute and interpret Python programs. It is a client for the ASM API.
- **Lusearch** and **Luindex** are respectively a text search tool and a text indexing for a corpus of data comprising the works of Shakespeare and the King James bible. Both applications are clients of the Licene API.
Table \[tb:dataSet\] shows the description of the collected dataset. For each API, we provide the number of included classes, the number of applications considered as clients of this API and the names corresponding to the client applications.
Evaluation Process
------------------
Our evaluation process is based on four steps. First, we have discussed the results of identifying call trees related to execution traces of usage scenarios of client applications of APIs. Then, we have shown the results of identifying groups of methods that are considered as provided interfaces of components. Next,we have presented how we evaluate the resulting components based on functionalities provided by methods representing their provided interfaces. Next, we have provided a discussion about dynamic and static analysis component identification approaches. Finally, we have discussed threats to validity related to the results of our approach.
\[tb:componentIdentification\]
\[tb:exampleComponent\]
Results of Identifying Call Trees Based on Executing Usage Scenarios
--------------------------------------------------------------------
We rely on BTrace[^4] dynamic tracing tool to collect execution traces related to usage scenarios. BTrace supports the execution of a Java program based on its bytecode and dynamically collect the methods related to this execution. We configure BTrace to run each client application only once since we are not interested in performance analysis. Table \[tb:allcalltrees\] shows the results related to the identified call trees of execution traces of usage scenarios of client applications of APIs. For each call tree, we present the tree’s size in terms of the number of included nodes, the number of distinct API methods included in this call tree, the tree’s height, the minimum, maximum and average method’ repetition in this call tree.
The results show that the call trees have small heights compared to the large number of nodes included in the trees. The average height of call trees related to ACL, XML and ASM APIs is respectively 12.5 ((12+13)/2), 7.5 ((7+4+14+5)/4) and 18.5 ((10+27)/2), while the average trees size is respectively 472676 and 1946555 nodes. This means that execution traces go deeply by 12.5, 7.5 and 18.5 methods in average respectively for ACL, XML and ASM APIs.
Indeed, the root of a call tree represents an execution scenario, and its children nodes represents the set of methods directly invoked in the source code of client applications to access API’s functionalities. Each sub-tree that corresponds to a root child explains internal dependencies related to method invocation between API methods to provide the invoked functionality by the corresponding root child. For example, the height of a call tree of 18 nodes refers to the maximum number of API methods that are invoked corresponding to an API method invocation directly invoked by client applications.
Another observation is that the call trees are widely distributed in a horizontal way, i.e., each node in a call tree has a large number of children relatively to its height. Therefore, API methods have dense interactions with each others such that each API method invokes a bunch of other API methods. In addition to that, we find that some API methods have been invoked only once compared to some other ones which have been invoked 198972.5 times in average (average max node repetition). Many groups of methods are invoked in similar frequently at the same time. These are interested to be in a same provided interface, even if they belong to different classes.
Results of Component Identification
-----------------------------------
The results of our clustering algorithm are shown in Table \[tb:componentIdentification\]. For each API, we present the number of identified components, the average size of provided interfaces in terms of methods and the average component size in terms of included classes.
Respectively for ACL, ASM and XML APIs, the results show that the average number of methods that need to be used together to access an API functionality is 9, 24 and 22 methods. These methods offer functionalities of 2.14, 2.13 and 3.25 owner classes in average.
For each API, Table \[tb:exampleComponent\] shows an example of a group of object-oriented API classes representing the implementation of identified component provided interfaces. To enable their reuse for software developers, these groups need to be transformed to confirm an exiting component model. It worths to mention some works that can be used by the user of our approach to transform these object-oriented implementation of identified components to be confirmed to component models. Alshara et al. [@alshara2016materializing] [@alshara2015migrating] provided initial approaches for transforming object-oriented component implementation to be confirmed to several component models (e.g., OSGi [@alliance2003osgi] and Fractal [@bruneton2006fractal]).
Results of Evaluating Identified Components
-------------------------------------------
The identified components are evaluated based on the functionalities offered by their provided interfaces. For each component, we evaluate how much of the methods composing its provided interface are related to provide the same API functionalities. The quality of each provided interface is calculated based on the ratio between the number of related methods to the total number of methods composing this interface (c.f. Equation \[eq:evaluation\]). We rely on the API documentations to identify each method’s functionality. Then, among a group of methods composing the provided interface, we select ones that have related functionalities to at least one method in this group.
For a pair of methods, we consider them as functional-related if one of these two cases is applied. The first one refers to identify direct indication(s) in the API documentations stated the correlation. The second one is based on human experts where their experiences allow them to decide if the two methods is functional-related. The number of selected methods represents the numerator in Equation \[eq:evaluation\]. The authors perform the evaluation themselves. To avoid bias, each component was evaluated by at least three authors. The average represents the final evaluation result of this component.
\[eq:evaluation\] Evaluation(Component)=
The precision of our approach is the average of the resulting values by applying Equation \[eq:evaluation\] to components of an API. Figure \[fig:precision\] shows the precision. The results show that our approach is able to correctly identify components where their provided interfaces are composed of functional-related API methods with 98% precision ((100%+93%+100%)/3). During our evaluation, we note that a component can be composed of one class and a subset of its methods forms the provided interface of this component. The same class forms another component with different subset of methods as provided interface. We consider these as correctly identified components that offer (sub)functionalities related to their provided interfaces. Each component can be (re)used as stand-alone or composed with other ones to generate higher level components (composite components).
Moreover, we observe that the same set of classes can form different components using different subsets of their methods as provided interfaces, i.e., component having the same classes but different provided interfaces. By checking the functionality of each method in these components, we find that they are related to same functionalities, but in different contexts. As it is noticed, some classes are parts of several components since different subset of their methods are part of these components. We look to the position of these classes as correct when the corresponding methods are harmonic with other methods in a given component. In XML API, we identify a component that its provided interface consists of 14 methods belonging to 11 different classes. After checking the functionality offered by these methods, we find that methods of 7 classes are related to the same functionality, while the other 4 classes are not related to each other since they are helper classes.
![The precision of component identification[]{data-label="fig:precision"}](precision.png)
Dynamic vs Static Analysis
--------------------------
The dynamic analysis approach is very expensive (run-time startup, collect usage scenarios ...), if we compare it to the static analysis which can only rely on source code files. However, dynamic analysis is more accurate as it contributes to provide real dependencies that cannot be recovered only by static analysis. Such dependencies are those related to late-dynamic binding, method overriding polymorphism and implicit dependencies in configuration files. These dependencies can exist extensively or rarely depending on the application context and framework.
On the other hand, the dynamic analysis approach relies on usage scenarios to cover API functionalities. However, some API functionalities may not be covered by the usage scenarios implemented in the selected client applications (thus, they will not be covered by the identified components). We recommend to select client applications that cover the maximum number of possible usage scenarios, which can be a strong constraint in some cases.
We compare the two approaches by applying our approach and the static approach presented in [@shatnawi2016reverse] and [@shatnawi2014mining] into the collected APIs and client applications. We find that both approaches construct different abstractions of the same functionalities. Components identified by dynamic analysis can be composed to form components resulting from static analysis. In terms of functionality-related, the two approaches provide similar results. We interpret this similarity by the fact that the studied APIs do not rely on the dependencies that are not detected based on the static analysis approach (e.g., implicit dependencies). In this context, we would like to recommend to rely on the dynamic analysis approach for APIs that do rely on late-dynamic binding, methods overriding polymorphism and implicit dependencies in configuration files to codify dependencies.
Threats to Validity {#ch6:threats}
-------------------
We identify two types of threats to validity concerning the proposed approach; internal validity and external validity.
### Threats to Internal Validity
Two aspects concern the internal threats to validity:
1. We need to rely on human expert opinions to evaluate the results. However, it is not easy to access the experts of the APIs. The authors considered themselves as experts since each author has at least more than 8 years of experience in software development.
2. Our evaluation does not consider the recall as we are not able to identify how many API methods are actually related to a functionality and not extracted by the approach. However, the recall depends on the coverage of the usage scenarios and does not rely on our approach. Any API method used in at least one usage scenario will be a part of at least one component.
### Threats to External Validity
We identified two aspects related to the external threats to validity:
1. We evaluated our approach based on APIs and client applications written using *Java*. The proposed approach can be generalized not only for other object-oriented languages (e.g., C++), but also for procedural ones (e.g, C, Pascal). The idea is that our approach does not care about the implementing programing language since its input is a set of call trees. The nodes in these call trees can refer either to methods or procedures/routines. The main difference is that identified components will be groups of procedures.
2. The selection of API client applications may impact identified components as different applications use API methods following different scenarios. This may impact the reusability of identified components for new independent applications. However, we assume API functionalities are (re)used following similar patterns by developers. This assumption was successfully utilized and proven in several research papers [@shatnawi2016reverse] [@uddin2012temporal] [@montandon2013documenting] [@Monperrus:2010:ECOOP]. We recommend selecting as much as possible of API client applications to minimize the influence of domain-specific API usages.
Related Work {#sec6:RelatedWork}
============
To the best of our knowledge, there is only one approach proposed to identify components from object-oriented APIs [@shatnawi2016reverse] [@shatnawi2014mining]. It identifies components as groups of API classes that are frequently used together by the client applications of an API and structurally dependent. To identify co-usage relationships between classes, Frequent-Pattern Growth algorithm was used where transactions are collections of API classes used by a client application. However, the approach presented in [@shatnawi2016reverse] [@shatnawi2014mining] does not trace the API method calls from client applications to deepen in the API as the proposed approach does to identify relationships between API classes. Instead, it relies on a static analysis technique to analyze structural dependencies in the source code of APIs. Recently developed software systems contains implicit dependencies resulted form the use of late dynamic-binding, Java-reflection and container-services offered by the frameworks. These implicit dependencies are difficult to be detected using static analysis techniques [@shatnawi2017analyzing].
All of these existing component identification approaches that are designed to identify components from OO software applications are not be directly applied to OO APIs. The reason is that they only relied on OO dependencies (e.g., method invocations, sharing types, etc.) to identify dependencies between classes, while co-usage dependencies between API classes should also be considered to identify classes/methods that can be reused to implement a specific scenario.
In the following three paragraphs, we discuss the related work on component identification in standalone applications using dynamic and static analysis, as well as the identification of usage patterns.
**Dynamic Analysis Component Identification from Applications.** In [@allier2011object], the authors presented an approach to identify component-based architectures from object-oriented applications. The identification is based on execution traces derived from use case scenarios. Classes frequently appearing in traces represent a candidate component. Grouping these classes is done by a clustering algorithm and a heuristic search, simulated annealing. Both techniques rely on a quality function. This quality function only considers the call frequency metric, which is insufficient without including metrics like our call weight and call distance. In [@cornelissen2007dynamic], the authors presented an approach to extract an architecture view of object-oriented software using the analysis of execution traces. This research contribution lacks details on how such a process can be automated, and only the visualization aspect is discussed. In [@dugerdil2013dynamic], the authors relied on the execution traces to recover two artifacts: 1) information that is used to update use cases’ documentations, and 2) component-based architecture by mapping the traced classes into clusters. Execution traces are identified from the business tasks of end-users, i.e., the records of what classes have been executed during a user usage.
**Static Analysis Component Identification from Applications.** In [@Boussaidi2009], the authors used the Knowledge Discovery Meta-model to represent the software elements, at different levels of abstraction, and their structural dependencies. They used a vertical clustering algorithm to identify software components as groups of classes, and a horizontal clustering algorithm to recover the layered architecture view of the identified components. In [@seriai2014deriving], the authors proposed an approach to identify component interfaces of object-oriented components identified using reverse engineering approaches. Classes of a component are analyzed to structure the required and provided interfaces. They extract a set of methods invoked by other components. These methods are then grouped by means of formal concept analysis. Some other approaches relied on the analysis of several software applications at the same time to identify components cross these applications [@shatnawi2017recovering] [@shatnawi2015recovering] [@shatnawi2013mining]. For example, in [@shatnawi2013mining], components are identified as groups of classes frequently presented in different applications and structural dependent.
**Frequent Usage Patterns of API entities.** Several approaches have been proposed to identify abstract reuse scenarios in terms of frequent usage patterns of API entities. These frequent usage patterns are not themselves direct reusable entities, but they help improving the reusability and understandability of APIs. We classified these existing approaches following their goals, usage-order consideration, the granularity of the API entities under the study, and the algorithm used to identify usage patterns. Generally the goal of usage-pattern identification approaches can be: (i) to provide examples to support recommendation systems [@Montandon:2013:WCRE] [@Uddin:2012:TAA], (ii) to support the documentation of APIs at different levels of abstraction [@Montandon:2013:WCRE] [@Wang:2013:MSH] [@saied2016cooperative], (iii) to predict bugs resulted from incorrect usage scenarios [@Monperrus:2010:ECOOP], etc. The patterns were identified with respect to the order in which the API elements are used in some approaches [@Montandon:2013:WCRE] [@Wang:2013:MSH], while other approaches do not take into account such an order [@Monperrus:2010:ECOOP] [@Bruch:2006:FIS] [@saied2016cooperative]. The granularity of the API elements that compose the identified patterns is at the method-level [@Montandon:2013:WCRE] [@Wang:2013:MSH] [@saied2016cooperative] or the class-level [@shatnawi2016reverse][@Bruch:2006:FIS]. In [@salman2017identification], multi-level patterns are identified. As for the algorithms to identify the patterns, the approaches used association rules mining [@Bruch:2006:FIS], frequent-pattern growth [@shatnawi2016reverse], clustering algorithms [@Wang:2013:MSH] [@saied2016cooperative] or a heuristic defined by the authors [@Montandon:2013:WCRE] [@Monperrus:2010:ECOOP]. Some approaches relied on a combination of many algorithms like Principle Component Analysis and clustering algorithm [@Uddin:2012:TAA].
Conclusion and Future Work {#sec8:ConclusionFutureWork}
==========================
We propose an approach that aims to identify reusable software components based on the dynamic analysis of interactions between client applications and the targeted API. The approach relies on co-usage and objected-oriented dependencies to define relationships between classes of APIs. To do so, we execute usage scenarios of client applications of APIs to collect execution traces. Our approach packages groups of methods frequently appearing together in the collected traces as component provided interfaces, and their owner classes define the structure of components’ implementation. These groups of methods are identified based on a graph-based clustering algorithm from the information in execution traces.
To evaluate our approach, we experimented with the DaCoPo benchmarks, focusing on three of its APIs and eight client applications. We considered pre-defined usage scenarios that are already provided by DaCoPo. The evaluation results show that the precision of our approach is up to 98%.
We will consider three future directions to investigate
1. We plan to transform the object-oriented implementation of the identified component provided interfaces into service-oriented interfaces to have truly reusable interfaces.
2. We want to extend the approach evaluation by considering more APIs and client applications to generalize the approach results.
3. We will also provide a visualization framework, that developers can use to identify reusable components of an API. They will have to load a set of client application using the API of interest, run the client applications while the framework is collecting the execution traces, and produces reusable components.
[^1]: Available at: https://commons.apache.org/proper/commons-logging/guide.html
[^2]: Available at: http://www-etud.iro.umontreal.ca/ saiedmoh/asm-3.3.1/doc/javadoc/user/index.html
[^3]: Available at: https://xerces.apache.org/xerces2-j/javadocs/api/index.html
[^4]: Available at: https://kenai.com/projects/btrace
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author:
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*Mansoureh Hosseinpour*$^{\ 1}$[^1] and *Hassan Hassanabadi*$^{\ 1}$[^2]\
*$^{\ 1}$ Faculty of Physics, Shahrood University of Technology, Shahrood, Iran*\
*P.O. Box 3619995161-316*
title:
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Introduction
============
The Duffin-Kemmer-Petiau (DKP) equation has been used to describe relativistic spin-0 and spin-1 bosons [@1; @2; @3; @4]. The DKP equation has five and ten dimensional representation respectively for spin-0 and spin-1 bosons[@5]. This equation is compared to the Dirac equation for fermions [@6]. The DKP equation has been widely investigated in many areas of physics. The DKP equation has been investigated in the momentum space with the presence of minimal length [@7; @8] and for spins 0 and 1 in a noncommutative space [@9; @10; @11; @12]. Also, the DKP oscillator has been studied in the presence of topological defects [@13]. Recently, there has been growing interest in the so-called DKP oscillator [@14; @15; @16; @17; @18; @19; @20; @21; @22; @23] in particular in the background of a magnetic cosmic string [@13]. The cosmic strings and other topological defects can form at a cosmological phase transition[@24]. The conical nature of the space-time around the string causes a number of interesting physical effects. Until now, some problems have been investigated in the gravitational fields of topological defects including the one-electron atom problem.[@25; @26; @27]. Spinning cosmic strings similar usual cosmic string, characterized by an angular parameter $ \alpha$ that depends on their linear mass density $ \mu$. The DKP oscillator is described by performing the non-minimal coupling with a linear potential. The name distinguishes it from the system called a DKP oscillator with Lorentz tensor couplings of Refs. [@7; @8; @9; @10; @11; @12; @14; @15; @16]. The DKP oscillator for spin-0 bosons has been investigated by Guo et al in Ref. [@10] in noncommutative phase space. The DKP oscillator with spin-0 hase been studied by Yang et al. [@11]. Exact solution of DKP oscillator in the momentum space with the presence of minimal length has been analysed in[@8]. De Melo et al. construct the Galilean DKP equation for the harmonic oscillator in a non-commutative phase space. [@28]. Falek and Merad investigated the DKP oscillator of spins 0 and 1 bosons in non- commutative space [@9]. Recently, there has been an increasing interest on the DKP oscillator [@13; @14; @15; @16; @28; @29; @30; @31]. The non-relativistic limit of particle dynamics in curved space-time is considered in Refs. [@32; @33; @34; @35; @36]. Also, the dynamics of relativistic bosons and fermions in curved space-time is considered in Refs[@17; @20; @31].\
The influence of topological defect in the dynamics of bosons via DKP formalism has not been established for spinning cosmic strings. In this way, we consider the quantum dynamics of scalar bosons via DKP formalism embedded in the background of a spining cosmic string. We solve DKP equation in presence of the spinning cosmic string space time whose metric has off diagonal terms which involves time and space. The influence of this topological defect in the energy spectrum and DKP spinor presented graphically.\
The structure of this paper is as follows: Sec. 2 describes the covariant form of DKP equation in a spining cosmic string background. In Sec. 3, we introduce the DKP oscillator by performing the non-minimal coupling in this space time, and we obtain the radial equations that are solved. We plotted the DKP spinor, density of probability and the energy spectrum for different conditions involving the deficit angle, the oscillator frequency. In the Sect. 4 we present our conclusions.
Covariant form of the DKP equation in the spinning cosmic string background
============================================================================
We choose the cosmic string spacetime background, where the line element is given by $$\begin{aligned}
\label{1}
ds^2=-dT^2+dX^2+dY^2+dZ^2\end{aligned}$$ The spacetime generated by a spinning cosmic string without internal structure, which is termed ideal spinning cosmic string can be obtain by coordinate transformation as $$\begin{aligned}
\label{2}
\begin{gathered}
T = t + a{\alpha ^{ - 1}}\varphi \hfill \\
X = rCos(\varphi ) \hfill \\
Y = rSin(\varphi ) \hfill \\
\varphi = \alpha \varphi ' \hfill \\
\end{gathered} \end{aligned}$$
With this transformation, the line element () becomes[@37; @38; @39; @40; @41; @42; @43]
$$\begin{aligned}
\label{3}
\begin{split}
d{s^2} = - {(dt + ad\varphi )^2} + d{r^2} + {\alpha ^2}{r^2}d{\varphi ^2} + d{z^2}\\
= - d{t^2} {\color{black}{+ d{r^2}}} - 2adtd\varphi + ({\alpha ^2}{r^2} - {a^2})d{\varphi ^2} + d{z^2}{\text{ }}
\end{split}
\end{aligned}$$
with $- \infty < z < \infty$ , $\rho \ge 0$ and $0 \le \varphi \le 2\pi$. From this point on, we will take $c=1$. The angular parameter $ \alpha $runs in the interval $ \left( {0,1} \right] $ is related to the linear mass density $\mu$ of the string as $\alpha = 1 - 4\mu$ and corresponds to a deficit angle $ \gamma=2\pi(1-\alpha) $. We take $a = 4Gj$ where $G $ is the universal gravitation constant and $j$ is the angular momentum of the spinning string; thus $a$ is a length that represents the rotation of the cosmic string. Note that in this case, the source of the gravitational field relative to a spinning cosmic string possesses angular momentum and the metric has an off diagonal term involving time and space.
The DKP equation in the cosmic string spacetime reads[@13; @17; @31] $$\begin{aligned}
\label{4}
({\rm{i}}{\beta ^\mu }\left( x \right){\nabla _\mu } - M)\Psi \left( x \right) = 0.\end{aligned}$$ The covariant derivative in is $$\begin{aligned}
\label{5}
{\nabla _\mu } = {\partial _\mu } + {\Gamma _\mu }\left( x \right)\end{aligned}$$ Where ${\Gamma _\mu }$ are the spinorial affine connections given by $$\begin{aligned}
\label{6}
{\Gamma _\mu } = \frac{1}{2}{\omega _{\mu ab}}\left[ {{\beta ^a},{\beta ^b}} \right].\end{aligned}$$ The matrices ${\beta ^a}$ are the standard Kemmer matrices in Minkowski spacetime. $$\begin{aligned}
\label{7}
{\beta ^\mu } = e_a^\mu {\beta ^a}\end{aligned}$$ The Kemmer matrices are an analogous to Dirac matrices in Dirac equation. There has been an increasing interest Dirac equation for spin half particles [@44; @45; @46; @47]. The matrices ${\beta ^a}$ satisfies the DKP algebra, $$\begin{aligned}
{\beta ^\mu }{\beta ^\nu }{\beta ^\lambda } + {\beta ^\lambda }{\beta ^\nu }{\beta ^\mu } = {g^{\mu \nu }}{\beta ^\mu } + {g^{\lambda \nu }}{\beta ^\mu }.\nonumber\end{aligned}$$ The conserved four-current is given by $$\begin{aligned}
{J^\mu } = \frac{1}{2}\bar \Psi {\beta ^\mu }\Psi, \nonumber\end{aligned}$$ and the conservation law for ${J^\mu }$ takes the form $$\begin{aligned}
\label{8}
{\nabla _\mu }{J^\mu } + \frac{i}{2}\bar \Psi (U - {\eta ^ 0 }{U^\dag }{\eta ^ 0 })\Psi = \frac{1}{2}\bar \Psi ({\nabla _\mu }{\beta ^\mu })\Psi \end{aligned}$$ The adjoint spinor ${\bar \Psi }$ is defined as $\bar \Psi = {\Psi ^\dag }{\eta ^ 0 }$ with ${\eta ^ 0 } = 2{\beta ^ 0 }{\beta ^ 0 } - 1$, in such a way that ${({\eta ^ 0 }{\beta ^\mu })^\dag } = {\eta ^ 0 }{\beta ^\mu }$. The factor $\frac{1}{2}$ which multiplies $\bar \Psi {\beta ^\mu }\Psi $, is of no importance for the conservation law, ensures the charge density is compatible with the one used in the Klein-Gordon theory and its non-relativistic limit. Thus, if $U$ is Hermitian with respect to $\eta ^ 0$ and the curved- space beta matrices are covariantly constant, then the four-current will be conserved if [@30] $$\begin{aligned}
{\nabla _\mu }{\beta ^\mu } = 0.\nonumber\end{aligned}$$ The algebra expressed by these matrices generates a set of 126 independent matrices whose irreducible representations comprise a trivial representation, a five-dimensional representation describing the spin-zero particles and a ten-dimensional representation associated to spin-one particles. We choose the $5 \times 5$ beta-matrices as follows [@31] $$\begin{aligned}
\label{9}
\begin{array}{l}
{\beta ^0} = \left( {\begin{array}{*{20}{c}}
\theta &{{0_{2 \times 3}}}\\
{{0_{3 \times 2}}}&{{0_{3 \times 3}}}
\end{array}} \right),\vec \beta = \left( {\begin{array}{*{20}{c}}
{{0_{2 \times 2}}}&{\vec \tau }\\
{ - {{\vec \tau }^T}}&{{0_{3 \times 3}}}
\end{array}} \right),\\
\theta = \left( {\begin{array}{*{20}{c}}
0&1\\
1&0
\end{array}} \right),{\tau ^1} = \left( {\begin{array}{*{20}{c}}
{ - 1}&0&0\\
0&0&0
\end{array}} \right),{\tau ^2} = \left( {\begin{array}{*{20}{c}}
0&{ - 1}&0\\
0&0&0
\end{array}} \right),{\tau ^3} = \left( {\begin{array}{*{20}{c}}
0&0&{ - 1}\\
0&0&0
\end{array}} \right).
\end{array}\end{aligned}$$ In eq. , $e_a^\mu $ denote the tetrad basis, that we can choose as $$\begin{aligned}
\label{10}
e_a^\mu = \left( {\begin{array}{*{20}{c}}
1&{\frac{{a\operatorname{Sin} (\varphi )}}{{r\alpha }}}&{\frac{{ - a\operatorname{Cos} (\varphi )}}{{r\alpha }}}&0 \\
0&{\operatorname{Cos} (\varphi )}&{\operatorname{Sin} (\varphi )}&0 \\
0&{\frac{{ - \operatorname{Sin} (\varphi )}}{{r\alpha }}}&{\frac{{\operatorname{Cos} (\varphi )}}{{r\alpha }}}&0 \\
0&0&0&1
\end{array}} \right).\end{aligned}$$ For the specific tetrad basis given by Eq. , we find from Eq. that the curved-space beta-matrices read
$$\begin{aligned}
\label{11}
\begin{gathered}
{\beta ^{(0)}} = e_a^t{\beta ^a} = {\beta ^0} - \frac{a}{{r\alpha }}{\beta ^\varphi } \hfill \\
{\beta ^{(1)}} = e_a^r{\beta ^a} = {\beta ^r} \hfill \\
{\beta ^r} = \cos \varphi {\beta ^1} + \sin \varphi {\beta ^2} \hfill \\
{\beta ^{(2)}} = e_a^2{\beta ^a} = \frac{{{\beta ^\varphi }}}{{r\alpha }} \hfill \\
{\beta ^\varphi } = - \sin \varphi {\beta ^1} + \cos \varphi {\beta ^2} \hfill \\
{\beta ^{(3)}} = e_a^z{\beta ^a} = {\beta ^3} = {\beta ^z} \hfill \\
\end{gathered}
\end{aligned}$$
and the spin connections are given by $$\begin{aligned}
\label{12}
{\Gamma _\varphi} = (1 - \alpha)[\beta^1,\beta^2]
\end{aligned}$$ We consider only the radial component in the non-minimal substitution. Since the interaction is time-independent, one can write $\Psi \left( {r,t} \right) \propto {e^{im\varphi }}{e^{ik_zz}}{e^{-iEt}} \Phi (r)$, where $E$ is the energy of the scalar boson, $m$ is the magnetic quantum number and $k_z$ is the wave number. The five-component DKP spinor can be written as ${\Phi ^T} = ({\Phi _1},{\Phi _2},{\Phi _3},{\Phi _4},{\Phi _5})$, and the DKP equation leads to the five $$\begin{aligned}
\label{13}
\begin{gathered}
(r\alpha \left( { - {\rm M}{\Phi _1}(r) + {\rm E}{\Phi _2}(r) + kz{\Phi _5}(r)} \right) + \cos \varphi \left( {\left( {a{\rm E} + m} \right){\Phi _4}(r) - {\text{i}}\left( {\left( { - 1 + \alpha } \right){\Phi _3}(r) + r\alpha {{\Phi '}_3}(r)} \right)} \right) - \hfill \\
\sin \varphi \left( {\left( {a{\rm E} + m} \right){\Phi _3}(r) + {\text{i}}\left( {\left( { - 1 + \alpha } \right){\Phi _4}(r) + r\alpha {{\Phi '}_4}(r)} \right)} \right)) = 0 \hfill \\
\left( {{\rm E}{\Phi _1}(r) - {\rm M}{\Phi _2}(r)} \right) = 0 \hfill \\
\left( {\left( {a{\rm E} + m} \right)\sin \varphi {\Phi _1}(r) + r\alpha \left( { - {\rm M}{\Phi _3}(r) + {\text{i}}\cos \phi {{\Phi '}_1}(r)} \right)} \right) = 0 \hfill \\
\left( { - \left( {a{\rm E} + m} \right)\cos \varphi {\Phi _1}(r) + r\alpha \left( { - {\rm M}{\Phi _4}(r) + {\text{i}}\sin \varphi {{\Phi '}_1}(r)} \right)} \right) = 0 \hfill \\
\left( {kz + {\rm M}\psi {\Phi _5}(r)} \right) = 0 \hfill \\
\end{gathered} \end{aligned}$$ Then we obtain the following equation of motion for the first component ${\Phi _1}$ of the DKP spinor : $$\begin{aligned}
\label{14}
\left(-\frac{(a \text{E}+m )^2}{\alpha ^2 r^2}+\text{E}^2-\text{kz}^2-\text{M}^2\right){{{\Phi }_1}(r)}+\frac{(\alpha -1) {{{\Phi '}_1}(r)}}{\alpha r}+{{\Phi ''}_1}(r)=0\end{aligned}$$ $$\begin{aligned}
\label{15}
\Phi _1= {r^{ \frac{1}{2\alpha}}}{R_{n,\ell }}(r).\end{aligned}$$ Then Eq. changes to $$\begin{aligned}
\label{16}
{{R''}_{n,m}}(r) + \frac{{{{R'}_{n,m}}(r)}}{r} + \left( {{E^2} - {k_z}^2 - {M^2} - \frac{{1 + 4{{\left( {aE + m} \right)}^2}}}{{4{r^2}{\alpha ^2}}}} \right)R(r) = 0\end{aligned}$$
By the change of variable $ r=x\eta $, we can write Eq. in the form
$$\begin{aligned}
\label{17}
{{R''}_{n,m}}(x) + \frac{{{{R'}_{n,m}}(x)}}{x} + \left( {1 - \frac{{{\lambda ^2}}}{{{x^2}}}} \right)R(x) = 0\end{aligned}$$
where $ \lambda = {(\frac{{1 + 4{{\left( {aE + m} \right)}^2}}}{{4{\alpha ^2}}})^{\frac{1}{2}}} $ and $ \eta = {\left( {{E^2} - {k_z}^2 - {M^2}} \right)^{\frac{-1}{2}}} $. The physical solution of Eq. are $Bessel J $ function. Therefore the general solution to eq. (17) is given by\
$$\begin{aligned}
\label{18}
R(r) = {A_{\lambda \eta }}{J_\lambda }(\frac{r}{\eta }) + {B_{\lambda \eta }}{Y_\lambda }(\frac{r}{\eta })\end{aligned}$$\
where ${Y_\lambda }(\frac{r}{\eta })$ is the Bessel function of the second kind. Sometimes this family of functions is also called Neumann functions or Weber functions. ${J_\lambda }(\frac{r}{\eta })$ is the Bessel function of the first kind, given by\
$$\begin{aligned}
{J_\nu }(x) = \sum\limits_{k = 0}^\infty {\frac{{{{( - 1)}^k}}}{{k!\Gamma (k + \nu + 1)}}} {\left( {\frac{x}{2}} \right)^{\nu + 2k}}\end{aligned}$$\
and ${Y_\lambda }(\frac{r}{\eta })$ is the Bessel function of the second kind, given by\
$$\begin{aligned}
{Y_\nu }(x) = \frac{{{J_\nu }(x)\operatorname{Cos} (\nu \pi ) - {J_{ - \nu }}(x)}}{{\operatorname{Sin} (\nu \pi )}}\end{aligned}$$\
By considering the boundary condition for Eq. such that ${B_{\lambda \eta }}= 0$ , we find\
$$\begin{aligned}
R(r) = {A_{\lambda \eta }}{J_\lambda }(\frac{r}{\eta }) \end{aligned}$$\
The DKP oscillator in spinning cosmic string background
=========================================================
The DKP oscillator is introduced via the non-minimal substitution [@17; @30; @31] $$\begin{aligned}
\frac{1}{i}{\vec \nabla _\alpha } \to \frac{1}{i}{\vec \nabla _\alpha } - iM\omega {\eta _ 0 }\vec r\end{aligned}$$ where $\omega $ is the oscillator frequency, $M$ is the mass of the boson already found in Eq. , and ${\vec \nabla }$ is defined in Eq. . We consider only the radial component in the non-minimal substitution. The DKP equation leads to the five equations: $$\label{23}
\begin{gathered}
(r\alpha \left( { - M{\Phi _1}(r) + E{\Phi _2}(r) + kz{\Phi _5}(r)} \right) + (\cos \varphi \left( {aE + m} \right){\Phi _4}(r)) + \hfill \\
({\text{i}}\cos \varphi \left( {{\Phi _3}(r) + \alpha \left( { - 1 + {r^2}{\text{M}}\omega } \right){\Phi _3}(r) - r\alpha {{\Phi '}_3}(r)} \right)) - \hfill \\
\sin \varphi \left( {\left( {aE + m} \right){\Phi _3}(r) - {\text{i}}\left( {1 - \alpha + {r^2}\alpha {\text{M}}\omega } \right){\Phi _4}(r) + {\text{i}}r\alpha {{\Phi '}_4}(r)} \right)) = 0 \hfill \\
\left( {\left( {aE + m} \right)\sin \varphi {\Phi _1}(r) - Mr\alpha {\Phi _3}(r) + {\text{i}}r\alpha \cos \varphi \left( {r{\rm M}\omega {\Phi _1}(r) + {{\Phi '}_1}(r)} \right)} \right) = 0 \hfill \\
\left( { - \left( {aE + m} \right)\cos \varphi {\Phi _1}(r) - Mr\alpha {\Phi _4}(r) + {\text{i}}r\alpha \sin \varphi \left( {r{\rm M}\omega {\Phi _1}(r) + {{\Phi '}_1}(r)} \right)} \right) = 0 \hfill \\
\left( {E{\Phi _1}(r) - M{\Phi _2}(r)} \right) = 0 \hfill \\
\left( {kz{\Phi _1}(r) + {\rm M}{\Phi _4}(r)} \right) = 0 \hfill \\
\end{gathered}$$ By solving the above system of equations in favour of $\Phi_1$we get\
$$\begin{aligned}
\begin{gathered}
{\Phi _2}(r) = \frac{E}{M}{\Phi _1}(r) \hfill \\
{\Phi _5}(r) = - \frac{{{k_z}}}{M}{\Phi _1}(r) \hfill \\
{\Phi _4}(r) = \frac{{ - aECos\varphi {\Phi _1}(r) - m\operatorname{Sin} \varphi {\Phi _1}(r) + {\text{i}}\left( {{r^2}\alpha {\rm M}\omega \operatorname{Sin} \varphi {\Phi _1}(r) + r\alpha \operatorname{Sin} \varphi {{\Phi '}_1}(r)} \right)}}{{Mr\alpha }} \hfill \\
{\Phi _3}(r) = \frac{{aE\operatorname{Sin} \varphi {\Phi _1}(r) + m\operatorname{Sin} \varphi {\Phi _1}(r) + {\text{i}}\left( {{r^2}\alpha {\rm M}\omega Cos\varphi {\Phi _1}(r) + r\alpha Cos\varphi {{\Phi '}_1}(r)} \right)}}{{Mr\alpha }} \hfill \\
\end{gathered} \end{aligned}$$
Combining these results we obtain an equation(20) of motion for the first component of the DKP spinor: $$\begin{aligned}
\label{25}
&{{\Phi ''}_1}(r) + \frac{{\left( { - 1 + \alpha } \right){{\Phi '}_1}(r)}}{{r\alpha }} \nonumber+\\
& \left( {{E^2} - k{z^2} - {M^2} + 2M\omega - \frac{{M\omega }}{\alpha } - \frac{{{{\left( {aE + m} \right)}^2}}}{{{r^2}{\alpha ^2}}} - {r^2}{{\rm M}^2}{\omega ^2}} \right){\Phi _1}(r) = 0
\end{aligned}$$ Let us take $\Phi _1$as $$\begin{aligned}
\label{26}
\Phi _1= {r^{ \frac{1}{2\alpha}}}{R_{n,\ell }}(r).\end{aligned}$$ Then Eq. changes to $$\begin{aligned}
\label{}
&{R''_{n,m}}(r) + \frac{{{{R'}_{n,m}}(r)}}{r} \nonumber+\\
&\left( {{E^2} - {k_Z}^2 - {M^2} + 2M\omega - \frac{{M\omega }}{\alpha } - \frac{{1 + 4{{\left( {aE + m} \right)}^2}}}{{4{r^2}{\alpha ^2}}} - {r^2}{{\rm M}^2}{\omega ^2}} \right){R_{n,m}}(r) = 0\end{aligned}$$ In order to solve the above equation, we employ the change of variable: $s = r^2$, thus we rewrite the radial equation in the form $$\label{}
{R_{n,m}}^{\prime \prime }(s) + \frac{1}{{s}}{R_{n,m}}^\prime (s) + \frac{1}{{{s^2}}}\left( { - \,\,{\xi _1}{s^2} + {\xi _2}s - {\xi _3}} \right){R_{n,m}}(r) = 0$$ If we compare with this second-order differential equation with the Nikiforov-Uvarov (NU) form, given in Eq. of Appendix A, we see that $$\begin{aligned}
\label{}
\begin{gathered}
{\xi _1} = \frac{{{M^2}{\omega ^2}}}{4} \hfill \\
{\xi _2} = \frac{1}{4}({E^2} - k{z^2} - {M^2} + 2M\omega - \frac{{M\omega }}{\alpha }), \hfill \\
{\xi _3} = \frac{{1 - {\alpha ^2} + 4{{\left( {aE + m} \right)}^2}}}{{16{\alpha ^2}}} \hfill \\
\end{gathered} \end{aligned}$$ which gives the energy levels of the relativistic DKP equation from $$\begin{aligned}
\label{26}
(2n + 1)\sqrt {{\xi _1}} - \xi _2+ 2\sqrt {{\xi_3\xi_1}} = 0,\end{aligned}$$ where $$\begin{aligned}
\label{27}
\begin{array}{l}
{\alpha_1} = 1,{\alpha_2} = {\alpha_3} ={\alpha_4} ={\alpha_5} = 0 , {\alpha_6} = {\xi_1},\\
{\alpha_7} = - {\xi_2},{\alpha_8} = {\xi_3},{\alpha_9} = {\xi_1},{\alpha_{10}} =1+ 2\sqrt { {\xi_3}} ,{\alpha_{11}} = 2\sqrt {{ {\xi_1}}} \\
{\alpha_{12}} = \sqrt {{ {\xi_3}}} ,{\alpha_{13}} = - \sqrt {{ {\xi_1}}}
\end{array}.\end{aligned}$$ As the final step, it should be mentioned that the corresponding wave function is $$\begin{aligned}
\label{28}
{R_{n,m}}(r) = {N r^{2{\alpha_{12}}}}{e^{{\alpha_{13}}{r^2}}}L_n^{{\alpha_{10}} - 1}({\alpha_{11}}{r^2}).\end{aligned}$$ where $N$ is the normalization constant. In figure. 1, $ \Phi_1(r) $, is plotted vs. $ r $ for different quantum number with the parameters listed under it .The density of probability $ {\left| {{\Phi _1}} \right|^2}$ is shown in figure 2. The negative and positive solution of energy vs. $ \alpha $ is shown in figure 3 and 4 for $n=1,5$ and 10. As in Fig. 3 and 4, we observe that the absolute value of energy decreases with $ \alpha $. Also in figure. 5 energy is plotted vs. $ \omega $ for quantum numbers. We see that absolute value of energy increases with $ \omega $. The negative and positive solution of energy vs. $ n $ is shown in figure 6 for different parameter $ \alpha $. We obtained the energy levels of the DKP oscillator in that background and observed that the energy increases with the level number. In figures. 7, energy is plotted vs. $a$ for different quantum numbers. We see energy increases with parameter $a$. Also We observed that the energy levels of the DKP oscillator in that background increases with the level number.
Conclusion
==========
The overall objective of this paper is the study of the relativistic quantum dynamics of a DKP oscillator field for spin-0 particle in the spinning cosmic string space-time. The line element in this background obtained by coordinate transformation of cartesian coordinate. The metric has off diagonal terms which involves time and space. We considered the covariant form of DKP equation in the spinning cosmic string background and obtained the solutions of DKP equation for spin-0 bosonse. Second we introduced DKP oscillator via the non-minimal substitution and considered DKP oscillator in that background. From the corresponding DKP equation, we obtained a system of five equations. By combining the results of this system we obtained a second order differential equation for first component of DKP spinor that the solutions are Laguerre polynomials. We see the results are dependent on the linear mass density of the cosmic string. In the limit case of $a=0$ and $\alpha$ = 1, i.e., in the absence of a topological defect recover the general solution for flat space- time. We plotted $ \Phi_1(r) $, for $n=1,2$. We examined the behaviour of the density of probability $\left|\Phi_1\right|^2$ . We observed that $\left|\Phi_1\right|^2$ for any parameter by increasing $ r $ have a very small peak at beginning and then have a taller peak and then by increasing $r$ it tend to zero. We obtained the behaviour of energy spectrum as a function of $ \alpha $. We see that the absolute value of energy decreases as $ \alpha $ increasing.\
APPENDIX {#appendix .unnumbered}
========
Nikiforov-Uvarov (NU) method {#nikiforov-uvarov-nu-method .unnumbered}
=============================
The Nikiforov-Uvarov method is helpful in order to finde igenvalues and eigenfunctions of the Schrödinger equation, as well as other second-order differential equations of physical interest. More details can be found in Refs[@48; @49]. According to this method, the eigenfunctions and eigenvalues of a second- order differential equation with potential $$\label{a.1}
{\Phi ^{\prime \prime }}(s) + \frac{{{\alpha _1} - {\alpha _2}s}}{{s(1 - {\alpha _3}s)}}{\Phi ^\prime }(s) + \frac{1}{{{{(s(1 - {\alpha _3}s))}^2}}}\left( { - \,\,{\xi _1}{s^2} + {\xi _2}s - {\xi _3}} \right)\Phi (s) = 0$$ According to the NU method, the eigenfunctions and eigenenergies, respectively, are
$$\begin{aligned}
\label{26}
\Phi (s) = {s^{{\alpha _{12}}}}{(1 - {\alpha _3}s)^{ - {\alpha _{12}} - ({{{\alpha _{13}}} \mathord{\left/
{\vphantom {{{\alpha _{13}}} {{\alpha _3}}}} \right.
\kern-\nulldelimiterspace} {{\alpha _3}}})}}P_n^{({\alpha _{10}} - 1,({{{\alpha _{11}}} \mathord{\left/
{\vphantom {{{\alpha _{11}}} {{\alpha _3}}}} \right.
\kern-\nulldelimiterspace} {{\alpha _3}}}) - {\alpha _{10}} - 1)}(1 - 2{\alpha _3}s)
\end{aligned}$$
and $$\begin{aligned}
{\alpha _2}n - (2n + 1){\alpha _5} + (2n + 1)(\sqrt {{\alpha _9}} + {\alpha _3}\sqrt {{\alpha _8}} ) + n(n - 1){\alpha _3} + {\alpha _7} + 2{\alpha _3}{\alpha _8} + 2\sqrt {{\alpha _8}{\alpha _9}} = 0\end{aligned}$$\
where $$\begin{aligned}
\begin{gathered}
{\alpha _4} = \frac{1}{2}(1 - {\alpha _1}),\,\,\,{\alpha _5} = \frac{1}{2}({\alpha _2} - 2{\alpha _3}),\,\,\,\,{\alpha _6} = \alpha _5^2 + {\xi _1},\,\,\,\,\,\,{\alpha _7} = 2{\alpha _4}{\alpha _5} - {\xi _2}, \hfill \\
{\alpha _8} = \alpha _4^2 + {\xi _3},\,\,\,\,\,{\alpha _9} = {\alpha _3}{\alpha _7} + \alpha _3^2{\alpha _8} + {\alpha _6},\,\,\,{\alpha _{10}} = {\alpha _1} + 2{\alpha _4} + 2\sqrt {{\alpha _8}} , \hfill \\
{\alpha _{11}} = {\alpha _2} - 2{\alpha _5} + 2(\sqrt {{\alpha _9}} + {\alpha _3}\sqrt {{\alpha _8}} ),\,\,\,{\alpha _{12}} = {\alpha _4} + \sqrt {{\alpha _8}} , \hfill \\
{\alpha _{13}} = {\alpha _5} - (\sqrt {{\alpha _9}} + {\alpha _3}\sqrt {{\alpha _8}} ) \hfill \\
\end{gathered} \end{aligned}$$
In the rather more special case of $\alpha =0$, $$\begin{aligned}
\begin{gathered}
\mathop {\lim }\limits_{{\alpha _3} \to 0} P_n^{({\alpha _{10}} - 1,({\alpha _{11}}/{\alpha _3}) - {\alpha _{10}} - 1)}(1 - 2{\alpha _3}s) = L_n^{{\alpha _{10}} - 1}({\alpha _{11}}s) \hfill \\
\mathop {\lim }\limits_{{\alpha _3} \to 0} {(1 - {\alpha _3}s)^{ - {\alpha _{12}} - ({\alpha _{13}}/{\alpha _3})}} = {e^{{\alpha _{13}}s}} \hfill \\
\end{gathered}
\end{aligned}$$ and, from equation (11), we find for the wave function $$\begin{aligned}
\Phi(s) = {s^{{\alpha _{12}}}}{e^{{\alpha _{13}}s}}L_n^{{\alpha _{10}} - 1}({\alpha _{11}}s)\end{aligned}$$ where $L_n^{\alpha_{10}-1 }$ denotes the generalized Laguerre polynomial.\
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[99]{}
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[^1]: E-mail:[email protected](corresponding author)
[^2]: E-mail: [email protected]
|
---
abstract: 'The anisotropic frustrated three dimensional (3D) XY model with disorder in the coupling constants is simulated as a model of a point disordered superconductor in an applied magnetic field. From a finite size scaling analysis of the helicity modulus it is concluded that the data is consistent with a finite temperature transition with isotropic scaling and the correlation length exponent is found to be $\nu=1.50\pm0.12$, consistent with 3D gauge glass universality.'
author:
- Peter Olsson
title: Vortex glass transition in a frustrated 3D XY model with disorder
---
The hypothesis of a vortex glass in disordered high temperature superconductors[@Fisher:89; @Fisher_Fisher_Huse] has spurred much research and many discussions during more than one decade and continues to be a very controversial issue[@Reichhardt_Otterlo_Zimanyi]. The essence of this suggestion is that random point disorder in superconductors may conspire with the vortex line interaction to pin the vortices and that this takes place through a sharp transition into a phase with vanishing linear resistance.
Computer simulations have for quite some time played an important role in the examination of critical phenomena and have also recently been used in the study of some vortex glass models. One important such model is the three dimensional (3D) gauge glass, which is an isotropic 3D XY model with randomness included through a random vector potential added to the phase difference of the superconducting order parameter. The evidence has for quite some time pointed at a finite temperature transition in this model[@Huse_Seung; @Reger_TYF], but strong evidence for a real phase transition has been obtained only recently[@Olson_Young] through the use of the exchange Monte Carlo (MC) technique[@Hukushima_Nemoto]. The value of the correlation length exponent was then found to be $\nu = 1.39\pm0.20$. It is however generally recognized that the 3D gauge glass models is too much of a simplification to allow for any safe conclusions regarding the behavior of disordered superconductors in applied magnetic fields[@Huse_Seung]. Most seriously, the gauge glass is an isotropic model with no net field which means that the possibility of anisotropic scaling is excluded at the outset.
Two studies with the necessary ingredients of disorder and applied field have so far been reported in the literature. The first is an examination of a frustrated 3D XY model with randomness in the couplings[@Kawamura:00], and the data was there interpreted as evidence for a phase transition with $\nu\approx 2.2$. The crossing of the data for different sizes expected from finite size scaling, was however not entirely convincing, possibly because of the open boundary conditions employed in the simulation. The second study gives results for a 3D random pinning model with strong disorder[@Vestergren_Lidmar_Wallin]. The value of the correlation length exponent was there found to be $\nu\approx 0.7$, undistinguishable from $\nu\approx 0.67$ in the pure zero field 3D XY model. This is at odds with the common expectation that a vortex glass transition should be in a different universality class than the pure model.
In this paper we present results from large scale simulations on a frustrated 3D XY model with disorder in the coupling constants. The quantity in focus is the helicity modulus and we find that the data is consistent with a finite temperature glass transition with isotropic scaling and obtain the correlation length exponent $\nu=1.50\pm0.12$. The agreement with $\nu=1.39\pm0.20$ for the 3D gauge glass model[@Olson_Young] suggests a common universality class.
The model we simulate is given by the Hamiltonian[@Olsson_Teitel:xy3fp] $${\cal H} = -\sum_{{\rm bonds}\,i\mu}J_{i\mu}\cos
(\theta_i-\theta_{i+\hat\mu}-A_{i\mu} + \delta_\mu),
\label{Hamiltonian}$$ where $\theta_i$ is the phase of the superconducting wave function at site $i$ of a periodic $L_x\times L_y\times L_z$ lattice and the sum is over all bonds in directions $\mu = x$, $y$, $z$. An applied magnetic field in the $z$ direction corresponding to $1/5$ flux quantum per plaquette is obtained through the quenched vector potential with the choice $A_{ix} = y_i\; 2\pi/5$, and $A_{iy} =
A_{iz} = 0$. The randomness is included through disorder in the coupling constants, $$\begin{array}{ll}
J_{i\mu}=J_\perp(1+p\epsilon_{i\mu}),\qquad & \mu=x,y, \\
J_{i\mu}={J_\parallel}, & \mu=z,
\end{array}$$ where $\epsilon_{i\mu}$ are independent variables from a Gaussian distribution with $\langle\epsilon_{i\mu}\rangle = 0$ and $\langle\epsilon_{i\mu}\rangle^2 = 1$. The disorder strength $p=0.4$ together with the anisotropy ${J_\parallel}= J_\perp/40$ were choosen since they were found sufficient to prohibit the formation of Abrikosov lattices. The simulations are performed with fluctuating twist boundary conditions[@Olsson:self-cons.long]; the $\delta_\mu$ in [Eq. (\[Hamiltonian\])]{} are the twist variables and the total twist in the respective directions are $\Delta_\mu = L_\mu\delta_\mu$. The simulations are performed with $L = L_x = L_y$ and a fixed aspect ratio, $L/L_z = 5/3$. The temperature is given in units of $J_\perp$.
The quantity in focus in our analysis is the helicity modulus which is defined from the change in free energy density, $f$, due to an applied twist, $\delta_\mu$: $\Upsilon_\mu = \partial^2 f/\partial
\delta_\mu^2$ [@Fisher_Barber_Jasnow]. To use the helicity modulus as a signal of the stiffness of the system the derivative should be evaluated at the twist that minimizes the free energy. In ordered systems this minimum is always at zero twist and the helicity modulus may then be evaluated by means of a correlation function determined with periodic boundary conditions, $\Delta_\mu=0$. For disordered systems, however, the minimizing twist will in general be different from zero and one then needs to make simulations with the twist variables $\Delta_\mu$ as additional dynamical variables and collect histograms $P_\mu(\Delta_\mu)$. The helicity modulii are then determined from the free energies $F_\mu = -T\ln P_\mu$, as discussed below. To analyze the critical behavior we use the standard scaling relation for the helicity modulus in 3D, $$L\Upsilon \sim g\left(t L^{1/\nu}\right),
\label{scaling}$$ where $t=(T-T_c)/T_c$ and $g$ is a scaling function[@Li_Teitel:89]. This expression presumes isotropic scaling; the more general scaling relations are gives in Ref. [@Vestergren_Lidmar_Wallin]. A naive analysis of $\Upsilon(L)$ rather than the correct scaling quantity $L\Upsilon$, led to an erroneous conclusion regarding the existence of a vortex glass phase in Ref. [@Olsson_Teitel:xy3fp].
$L$ $L_z$ $N_d$ $N_T$ ${T_\mathrm{min}}$ $X_\mathrm{acc}$ (%) ${\tau_\mathrm{eq}}$ ${\tau_\mathrm{max}}$ sweeps$/10^6$
------ ------- ------- ------- -------------------- ---------------------- ---------------------- ----------------------- ---------------
$10$ $6$ 600 12 0.09 30 1 15 0.3 + 3.9
$15$ $9$ 600 24 0.09 30 4 12 1.0 + 3.1
$20$ $12$ 600 36 0.09 32 11 21 2.9 + 5.5
$25$ $15$ 200 36 0.115 27 17 31 4.5 + 8.1
: Parameters describing the simulations. For systems of size $L\times L\times L_z$ we simulated $N_d$ disorder configurations with $N_T$ temperatures in the range ${T_\mathrm{min}}\leq T < {T_\mathrm{max}}$, cf. [Eq. (\[Tm\])]{}. The acceptance ratio for the exchange step is given by $X_\mathrm{acc}$. Of the bins corresponding to $2^{18}=262144$ sweeps ${\tau_\mathrm{eq}}$ are first discarded and the remaining ${\tau_\mathrm{max}}$ are used for calculating averages. The same information is also given in terms of the number of sweeps for equilibration and for collecting data.[]{data-label="Param"}
Our simulations are performed with exchange MC which is a method for simultaneously simulating multiple copies of a particular configuration of disorder with each copy at a different temperature. According to certain rules these copies may now and then interchange temperature[@Hukushima_Nemoto] and therefore effectively perform random walks in temperature space. These random changes in temperature greatly help the different copies avoid getting trapped in restricted parts of the phase space and therefore makes it possible to sample the whole phase space and obtain the true thermodynamic averages. In our simulations the temperatures were chosen according to the equation $$T_m = {T_\mathrm{min}}\left(\frac{{T_\mathrm{max}}}{{T_\mathrm{min}}}\right)^{m/N_T},\quad
m=0,\ldots,N_T-1,
\label{Tm}$$ with ${T_\mathrm{max}}= 0.24$ and ${T_\mathrm{min}}$ as given in Table \[Param\]. Before doing the actual exchange MC the initial spin configurations for the $N_T$ temperatures were obtained by slowly cooling the system with standard MC simulations. The number of temperatures, the acceptance ratio for the exchange step, and the number of disorders simulated for the different system sizes are shown in Table \[Param\]. The exchange steps are attempted once every 16 sweep.
The exchange MC method ensures that all the different copies remain at thermal equilibrium as soon as equilibrium has been reached. The approach to equilibrium may however be very slow since information and configurations have to propagate all the way from high to low temperatures. We have carefully examined the approach to equilibrium and especially for our largest sizes the times for equilibration are indeed very long. For the next largest size, $L=20$, equilibration is only reached after about $2.9\times10^6$ sweeps, and to make thermalization at all possible for our largest size, $L=25$, we chose not to go to quite that low temperatures for the largest size, cf. Table \[Param\]. This was decided since the time required for thermalization may increase very rapidly with decreasing temperature. Still, about $4.5\times 10^6$ sweeps were necessary to reach equilibrium for $L=25$.
![$\overline{P_x}(\Delta_x)$ for a certain disorder realisation. Note that the peaks of the histograms become higher and sharper as the temperature is lowered. This kind of data is used for the determination of $\Upsilon_x$ through [Eq. (\[Upsilon\])]{}. The inset gives the corresponding quantity for the $z$ direction.[]{data-label="PDelta"}](PDelta2.ps){width="8cm"}
For each system size, disorder configuration and temperature the main output from the simulations are histograms $P_\mu(\Delta_\mu; \tau)$, where $\tau$ enumerates bins corresponding to $2^{18} = 262144$ sweeps over the lattice. The further analysis is then based on the average $$\overline{P_\mu}(\Delta_\mu) = \frac{1}{{\tau_\mathrm{max}}}
\sum_{\tau=1}^{{\tau_\mathrm{max}}} P_\mu(\Delta_\mu; \tau),
\label{overlineP}$$ and the helicity modulus is determined from the curvature at the minimum of the associated free energy by fitting a second order polynomial to the free energy in a narrow interval around the minimum, $\Delta_\mu^0$, which is also determined in the fit, $$\overline{F_\mu}(\Delta_\mu) \sim \Upsilon_\mu \frac{V}{2L_\mu^2}
(\Delta_\mu - \Delta_\mu^0)^2.
\label{Upsilon}$$ However, it turns out that the values of $\Upsilon_\mu$ obtained in this way are biased towards too large values. The origin of this bias as well as the method employed to eliminate it from the data is discussed below after the discussion of the results.
We now focus on disorder averaged quantities for which the bias mentioned above has already been eliminated. With ${[\ldots]_\mathrm{av}}$ denoting the average over disorder configurations we define
\[DisAver\] $$\begin{aligned}
{\Upsilon_\perp}& = & \frac{1}{2}{[\Upsilon_x + \Upsilon_y]_\mathrm{av}}, \\
{\Upsilon_\parallel}& = & {[\Upsilon_z]_\mathrm{av}}.\end{aligned}$$
Our results for $L{\Upsilon_\perp}$ are shown in [Fig. \[Ups\]]{}. To a very good accuracy the data for the different sizes cross at a single temperature. To further verify the scaling according to [Eq. (\[scaling\])]{} we fit our data for $L\Upsilon_\perp$ near $T_c$ to a fourth order polynomial expansion of $g(tL^{1/\nu})$ and obtain the values $\nu =
1.50\pm0.12$ and $T_c = 0.123\pm 0.003$. The collapse which is shown in [Fig. \[Ucoll\]]{} is excellent and holds in a surprisingly large temperature interval. The error estimates are obtained with a resampling technique and correspond to one standard deviation.
![$L{\Upsilon_\perp}$ versus temperature for four different system sizes. The curves cross at a single point, which is an indication of critical behavior. The inset shows $L{\Upsilon_\parallel}/{J_\parallel}$ which shows a similar crossing at almost the same temperature.[]{data-label="Ups"}](Ups2.ps){width="8cm"}
![The scaling collapse of $L{\Upsilon_\perp}$ gives $\nu = 1.50\pm0.12$ and $T_c = 0.123\pm 0.003$. The inset is an attempt to collapse $L{\Upsilon_\parallel}/{J_\parallel}$ with the same values for $\nu$ and $T_c$. The nice collapse gives additional support for a vortex glass transition with isotropic scaling.[]{data-label="Ucoll"}](Coll2.ps){width="8cm"}
A vortex glass transition should also be seen in the parallel component of the helicity modulus and since the anisotropy exponent enters the scaling relation in different ways for ${\Upsilon_\parallel}$ and ${\Upsilon_\perp}$[@Vestergren_Lidmar_Wallin], a scaling analysis of $L{\Upsilon_\parallel}$ constitutes an additional test that the scaling actually is isotropic. The crossing of $L{\Upsilon_\parallel}$ at $T_c$ for different sizes and the scaling collapse are shown in the insets of Figs. \[Ups\] and \[Ucoll\], respectively. From the not so smooth curves it is clear that the precision in ${\Upsilon_\parallel}$ is much worse than for ${\Upsilon_\perp}$. This may be traced back to the less good quality of the histograms $P_z(\Delta_z)$, as shown in the inset of [Fig. \[PDelta\]]{}. Nevertheless, the data for $L{\Upsilon_\parallel}$ (see inset in [Fig. \[Ucoll\]]{}) collapses nicely when using the same values of $\nu$ and $T_c$ as in the main figure. This therefore constitutes additional evidence for a vortex glass transition in the model. We now turn to the bias mentioned above and also discuss similarities and differences with the more common method to determine the root mean square current ${I_\mathrm{rms}}$, in analyses of models with disorder.
As mentioned above the determination of $\Upsilon$ from a twist histogram suffers from a bias towards too large values. To investigate the reason for this bias we performed additional simulations with fluctuating twist boundary conditions of the pure zero-field 3D XY model and collected histgrams $P(\Delta,\tau)$ where $\tau$ enumerates the bins. This data was then used to calculate averages over ${\tau_\mathrm{aver}}$ consecutive bins $\overline{P}(\Delta;{\tau_\mathrm{aver}})$, which in turn were used to determine the helicity modulus. By repeating this procedure for several values of ${\tau_\mathrm{aver}}$ the dependence of $\Upsilon$ on ${\tau_\mathrm{aver}}$ was determined. From this kind of analysis it was found that the bias decays as $1/{\tau_\mathrm{aver}}$ to a very good precision. Another finding is that the bias may be made to vanish altogether by making use of $\Delta^0 = 0$ in [Eq. (\[Upsilon\])]{} instead of using $\Delta^0$ as a free parameter. ($\Delta^0 = 0$ is the known value of the minimizing twist in the pure system.) The latter observation suggests that the bias is related to the usual complication in determining the width (variance) of a distribution when the true average is not known[@Variance]. Since $\Upsilon$ is inversely related to the variance of the distribution $P(\Delta)$ around the maximum it follows that the estimates based on runs of length ${\tau_\mathrm{aver}}$ would be expected to decay towards the true value as[@Variance] $$\Upsilon({\tau_\mathrm{aver}}) = \frac{1}{1-b/{\tau_\mathrm{aver}}} \Upsilon(\infty),
\label{Ut}$$ where $b$ is a free parameter related to the decorrelation time in the simulations. For small values of $b/{\tau_\mathrm{aver}}$ the expected behavior is therefore entirely in accordance with the $1/{\tau_\mathrm{aver}}$ decay discussed above.
Returning to our data for the vortex glass model, the procedure used to determine the data points in [Fig. \[Ups\]]{} consists of three steps: (i) Determine $\Upsilon_\mu({\tau_\mathrm{aver}})$ for each disorder configuration and several values of ${\tau_\mathrm{aver}}$ by fitting histogram $\overline{P}_\mu(\Delta_\mu;{\tau_\mathrm{aver}})$ based on ${\tau_\mathrm{aver}}$ consecutive bins, $P_\mu(\Delta_\mu,\tau)$ to [Eq. (\[Upsilon\])]{}. (ii) Calculate the disorder averaged quantities ${\Upsilon_\perp}({\tau_\mathrm{aver}})$ and ${\Upsilon_\parallel}({\tau_\mathrm{aver}})$, cf. [Eqs. (\[DisAver\])]{}. (iii) Fit this data to [Eq. (\[Ut\])]{} to obtain the unbiased estimates ${\Upsilon_\perp}\equiv {\Upsilon_\perp}(\infty)$ and ${\Upsilon_\parallel}\equiv
{\Upsilon_\parallel}(\infty)$. The last step is illustrated in [Fig. \[Uperp-tau\]]{} for $T=0.125$ close to $T_c$. The error bars on the last point for each size are the errors associated with the disorder average.
![The figure shows the elimination of the bias in ${\Upsilon_\perp}$ by fitting ${\Upsilon_\perp}({\tau_\mathrm{aver}})$ to [Eq. (\[Ut\])]{} with $b$ and ${\Upsilon_\perp}(\infty)$ as free parameters. The time is in units of the bin size which is $2^{18} = 262144$ sweeps.[]{data-label="Uperp-tau"}](Uperp-tau.ps){width="8.5cm"}
We now shortly discuss the relation between the helicity modulus calculated in the present paper and the more common method to determine the root mean square current ${I_\mathrm{rms}}$[@Reger_TYF]. Since the current and the helicity modulus are first and second derivatives, respectively, of the same function $F(\Delta)$, both quantities effectively probe the roughness of the function $F(\Delta)$. The low temperature phase is characterized by large energy barriers growing with system size whereas the free energy above the transition temperature becomes a flat function of $\Delta$ in the limit of large $L$. However, $\Upsilon$ turns out to be more efficient in measuring the roughness of $F(\Delta)$. The reason is that $\Upsilon$ measures a property at an extremum (the minimum of the free energy) whereas the current at $\Delta=0$ for a given shape of the function may be large or small depending on the location of the structure in $F(\Delta)$. ${I_\mathrm{rms}}$ is therefore only *on the average* a good measure of the properties of $F(\Delta)$ and this has to be compensated for by using a larger number of disorder configurations. This is the reason for the good precision in our data in spite of the rather small number of disorder configurations.
Note also that there is a bias in the determination of the root mean square current ${I_\mathrm{rms}}$ that is very similar to the bias in $\Upsilon$ discussed above. For ${I_\mathrm{rms}}= \sqrt{{[I^2]_\mathrm{av}}}$ the origin of this bias is that the statistical error $\delta I$ gives a term $(\delta I)^2
\geq 0$. Since $\delta I$ would be expected to vanish with simulation time as $\sim 1/\sqrt{{\tau_\mathrm{aver}}}$ the bias in ${I_\mathrm{rms}}$ would vanish as $1/{\tau_\mathrm{aver}}$, which is essentially the same as the behavior of $\Upsilon({\tau_\mathrm{aver}})$. There is however a difference in that the bias in ${I_\mathrm{rms}}$ is easily eliminated by performing two independent simulations, $\alpha$ and $\beta$, for each disorder and measuring the quantity ${[I_\alpha I_\beta]_\mathrm{av}}$[@Olson_Young]. Since the bias in $\Upsilon$ is of a very different origin it cannot be eliminated with such methods.
To summarize, we have performed a finite size scaling analysis of the helicity modulus in a frustrated 3D XY model with disorder in the coupling constants. The data is consistent with isotropic scaling and the correlation length exponent is found to be $\nu=1.50\pm 0.12$. The good agreement with $\nu=1.39\pm0.20$ of the 3D gauge glass suggests that the two models actually do belong to same universality class.
The author would like to acknowledge helpful discussions with S. Teitel and M. Wallin. This work has been supported by the Swedish Research Council, contract No. E 5106-1643/1999, and by the resources of the Swedish High Performance Computing Center North (HPC2N).
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---
abstract: 'We investigate strong nonlinear damping effects which occur during high amplitude oscillations of neutron stars, and the gravitational waves they produce. For this, we use a general relativistic nonlinear hydrodynamics code in conjunction with a fixed spacetime (Cowling approximation) and a polytropic equation of state (EOS). Gravitational waves are estimated using the quadrupole formula. Our main interest are $l=m=2$ $f$ modes subject to the CFS (Chandrasekhar, Friedman, Schutz) instability, but we also investigate axisymmetric and quasiradial modes. We study various models to determine the influence of rotation rate and EOS. We find that axisymmetric oscillations at high amplitudes are predominantly damped by shock formation, while the nonaxisymmetric $f$ modes are mainly damped by wave breaking and, for rapidly rotating models, coupling to nonaxisymmetric inertial modes. From the observed nonlinear damping, we derive upper limits for the saturation amplitude of CFS-unstable $f$ modes. Finally, we estimate that the corresponding gravitational waves for an oscillation amplitude at the upper limit should be detectable with the advanced LIGO and VIRGO interferometers at distances above $10\usk\mega{\ensuremath{\mathrm{pc}}}$. This strongly depends on the stellar model, in particular on the mode frequency.'
author:
- Wolfgang Kastaun
- Beatrix Willburger
- 'Kostas D. Kokkotas'
title: 'On the saturation amplitude of the f-mode instability'
---
Introduction {#sec:introduction}
============
As a consequence of Einstein’s theory of relativity, it is predicted that the violent nonradial pulsations excited immediately after the birth of a rotating proto-neutron star result in the emission of significant amounts of gravitational radiation. In addition, it has been discovered by Chandrasekhar [@1970PhRvL..24..611C] and Friedman & Schutz [@1978ApJ...221..937F; @1978ApJ...222..281F] that certain pulsation modes of rotating relativistic stars may grow exponentially due to gravitational radiation backreaction, especially if the proto-neutron star is rapidly rotating; this is the so-called CFS instability.
The exact amount of energy emitted by the stable oscillations of neutron stars in the form of gravitational waves is actually unknown and depends uniquely on the initial conditions. The stable stellar pulsations will emit gravitational waves which are detectable only if the neutron star is in our galaxy or in the most optimal case in the nearby ones [@2009arXiv0912.0384A]. Thus, the cases in which these oscillations become unstable are of special interest for gravitational wave astronomy, since the corresponding gravitational waves will be detectable even for sources in the nearby galactic clusters. It has been suggested that the detection of gravitational waves from pulsating neutron stars will allow the study of their interior, see [@1998MNRAS.299.1059A; @2001MNRAS.320..307K; @2010arXiv1005.5228G]. It is expected that the identification of specific pulsation frequencies in the observational data will reveal the properties of matter at densities that currently cannot be probed by any other experiment.
The study of the dynamics of fast rotating neutron stars in a general relativistic framework was practically impossible until recently. The main reason why linear theory failed is the form and the size of the relevant perturbation equations. Thus, it is not surprising that the first results for the oscillations of fast rotating stars were derived by using evolutions of the nonlinear equations, see [@SAF2004; @DSF2006; @Kastaun06; @Kastaun08]. Still most of these studies were purely axisymmetric and thus there was a significant influence of rotation on the oscillation spectra only for very high rotation rates. The CFS instability on the other hand is active only for nonaxisymmetric perturbations. In the last two years there was a significant progress in the study of nonaxisymmetric perturbations, for the first time in a general relativistic framework, using both perturbation theory [@2008PhRvD..78f4063G; @2009PhRvD..80f4026G; @2010PhRvD..81h4019K; @2010arXiv1005.5228G] and nonlinear evolutions of coupled hydrodynamic and Einstein equations [@2010PhRvD..81h4055Z].
For the case of $r$ modes, the CFS instability is active for any rotation rate [@1998ApJ...502..708A; @1998ApJ...502..714F; @1998PhRvL..80.4843L; @1998PhRvD..58h4020O; @1999ApJ...510..846A]. In practice the unstable $r$ modes will be excited even for slowly rotating stars if they are hot enough ($T > 10^7\usk\kelvin$), so that the instability growth time is shorter than the shear viscosity damping time. Initially it was considered as a prime source for gravitational waves, but later more detailed studies on the effect of the magnetic field on the instability [@2000ApJ...531L.139R], or the presence of hyperons in the core [@2002PhRvD..65f3006L], seriously questioned the potential of the instability. Detailed studies suggested that the $r$ mode is limited to very small amplitudes due to energy transfer to a large number of other inertial modes, in the form of a cascade, leading to an equilibrium distribution of mode amplitudes [@2002PhRvD..65b4001S]. The small saturation values for the amplitude are supported by recent nonlinear estimations [@2004PhRvD..69h4001S; @2005PhRvD..71d4007S] based on the drift, induced by the $r$ modes, causing differential rotation. On the other hand, hydrodynamical simulations of limited resolution showed that an $r$ mode of large initial amplitude does not decay appreciably over several dynamical timescales [@2001PhRvL..86.1148S], but on a longer timescale a catastrophic decay occurs [@2002PhRvD..66d1303G]; this indicates a transfer of energy to other modes due to nonlinear mode couplings, and suggests that a hydrodynamical instability may be operating. A specific resonant 3-mode coupling was identified in [@2006MNRAS.370.1295L] as the cause of the instability and a perturbative analysis of the decay rate suggests a maximum dimensionless saturation amplitude $\alpha_\text{max} < 10^{-2}-10^{-3}$. A new computation using second-order perturbation theory finds that the catastrophic decay seen in the hydrodynamical simulations [@2002PhRvD..66d1303G; @2006MNRAS.370.1295L] can indeed be explained by a parametric instability operating in 3-mode couplings between the $r$ mode and two other inertial modes [@2004PhRvD..70l1501B; @2004PhRvD..70l4017B; @2005PhRvD..71f4029B; @2007PhRvD..76f4019B; @2009PhRvD..79j4003B].
The CFS instability of the $f$ modes (nonradial modes with no radial nodes, restored by pressure) can potentially be the strongest source for gravitational waves from isolated neutron stars. Although it was the first to be studied, already 30-years ago, it was forgotten because at that time it was nearly impossible to study rapidly rotating neutron stars in a general relativistic framework. Studies using Newtonian theory came to the disappointing conclusion that (in Newtonian theory) the $l=m=2$ $f$ mode is stable for rotation rates below the Kepler limit. The $l=m=4$ $f$ mode does become unstable, but the growth time is so long that viscosity severely limits the amplitudes, rendering the process unimportant for gravitational wave astronomy. Only in the late ’90s, Stergioulas & Friedman [@1998ApJ...492..301S] showed that for certain EOS the $l=m=2$ $f$ mode can become unstable in full GR (general relativity). Still, this study was left aside since no proper linear or nonlinear codes were available to deal with the perturbations of rapidly rotating neutron stars in GR.
If such an instability sets in, the star will emit copious amounts of gravitational radiation. First studies suggest that the signal may be detectable from distances as far as $15\usk\mega{\ensuremath{\mathrm{pc}}}$ for the current sensitivity of Virgo & LIGO, and from distances greater than $100\usk\mega{\ensuremath{\mathrm{pc}}}$ for the sensitivities of Advanced Virgo & LIGOs [@1995ApJ...442..259L; @2004PhRvD..70h4022S; @2004ApJ...617..490O]. In the latter case the event rate of supernovas resulting in the creation of a proto-neutron star can be quite high, i.e. more than thousand per year. However, it should be noted that it is not clear what will be the initial rotation period of the collapsing core; it seems to depend strongly on the profile of the angular momentum distribution, the initial mass and angular momentum of the collapsing star, and also the strength of the magnetic field, which can slow down the newly born compact object quite efficiently. The hope is that still a number of supernova events, maybe 10%, will produce rapidly rotating cores which can be subject to the CFS instability of the $f$ modes. In any case, the next generation of gravitational wave detectors, such as Einstein Telescope (ET) [@2010CQGra..27h4007P] will be ideal for the detection of this type of instability [@2009arXiv0912.0384A].
The two most recent hydrodynamical simulations [@2004PhRvD..70h4022S; @2004ApJ...617..490O] (in the Newtonian limit and using an artificially increased post-Newtonian radiation-reaction potential) essentially confirm this picture. In [@2004PhRvD..70h4022S] a differentially rotating, $N=1$ polytropic model with a large $\beta=T/|W| \sim 0.2 \dots 0.26$ was chosen as the initial equilibrium state ($T$ is the rotational kinetic energy and $|W|$ the gravitational binding energy). The main difference of this simulation compared to the ellipsoidal approximation [@1995ApJ...442..259L] lies in the choice of the EOS. For $N=1$ Newtonian polytropes it was argued that the secular evolution cannot lead to a stationary Dedekind-like state. Instead, the $f$-mode instability will continue to be active until all nonaxisymmetries are radiated away and an axisymmetric shape is reached. In another recent simulation [@2004ApJ...617..490O], the initial state was chosen to be a uniformly rotating, $N=0.5$ polytropic model with $T/|W|\sim 0.18$. Again, the main conclusions reached in [@1995ApJ...442..259L] were confirmed, however the assumption of uniform initial rotation limits the available angular momentum that can be radiated away, leading to a detectable signal only out to about $40\usk\mega{\ensuremath{\mathrm{pc}}}$. The star appears to be driven towards a Dedekind-like state, but after about 10 dynamical periods, the shape is disrupted by growing short-wavelength motions, which are explained by a shearing type instability such as the elliptic flow instability [@1993ApJ...408..603L].
The recent progress [@2008PhRvD..78f4063G; @2009PhRvD..80f4026G; @2010PhRvD..81h4019K; @2010arXiv1005.5228G; @2010PhRvD..81h4055Z], demonstrates that the problem can be handled properly, which should allow to answer the following questions: First, has the unstable mode any chance to be excited? Second, what is the instability window of the mode? Third, what are the exact rotation rates at the onset of the instability for the various EOS? And finally, what is the maximum amplitude that the mode may reach before saturated by nonlinear phenomena, e.g. mode coupling, mass loss, or shock formation? In this work we try to answer the last question. In particular, we investigate nonlinear effects active at oscillation amplitudes sufficiently large to allow a detection of sources at distances greater than $1\usk\mega{\ensuremath{\mathrm{pc}}}$. Since our methods are not restricted to a certain mode, we map out nonlinear effects occurring for axisymmetric modes as well. The latter also serves as a numerically less expensive test case to calibrate our code.
The question about the saturation amplitude of the $f$-mode instability is a major one that has to be answered since it may completely eliminate the importance of this instability. In this article we present a systematic study of nonlinear damping effects for different amplitudes, rotation rates, and equations of state. For this, we use a nonlinear general relativistic evolution code named [<span style="font-variant:small-caps;">pizza </span>]{}[@Kastaun06; @Kastaun08]. Up to now it has been successfully tested on small amplitude stable oscillations of rapidly rotating neutron stars and self-gravitating tori. After a number of improvements the code is well suited to follow even large amplitude oscillations of rotating stars, such as those excited during the saturation phase of the $f$-mode instability. A similar study has been performed for the $r$-mode instability in [@2002PhRvD..66d1303G], where an artificially excited $r$ mode was left to evolve. The amplitude of the $r$ mode seemed initially to decrease slowly but then it decayed catastrophically. The decay time was found to be amplitude dependent and the effect attributed to three mode coupling.
The structure of the paper is as follows. First we describe the numerical methods in [Sec. \[sec\_meth\]]{}. In [Sec. \[sec\_models\]]{}, we introduce the stellar models and physical problems we studied. The numerical results are presented in [Sec. \[sec\_results\]]{}. The main results on the damping and detectability of the CFS-unstable $f$ mode can be found in [Sec. \[sec\_damp\_m2\]]{} and \[sec\_detect\], and a summary is given in [Sec. \[sec\_summary\]]{}.
Throughout the paper we use geometrical units $c=G=1$. Greek indices run from $0\dots 3$, Latin indices from $1\dots 3$. We denote the lowest order quasi radial mode by [$F$]{}, the axisymmetric fundamental pressure modes ($f$ modes) by $^l\!f_0$, and nonaxisymmetric ones by $^l\!f_m$, where $m>0$ refers to the counter-rotating modes.
Numerical method {#sec_meth}
================
Time evolution
--------------
For our simulations, we use the [<span style="font-variant:small-caps;">pizza </span>]{}code described in [@Kastaun06]. It evolves the general relativistic nonlinear hydrodynamic evolution equations for an ideal fluid without magnetic fields, while keeping the spacetime metric fixed (Cowling approximation).
The hydrodynamic equations in covariant form are $$\begin{aligned}
\nabla_\mu T^{\mu\nu} &= 0 , \label{eq_div_tmunu}\\
\nabla_\mu \left({\rho}u^\mu \right) &=0 .\end{aligned}$$ The stress energy tensor $T^{\mu\nu}$ of an ideal fluid is given by $$\begin{aligned}
T^{\mu\nu} = {\rho}h u^\mu u^\nu + P g^{\mu\nu} ,\end{aligned}$$ where ${\rho}$, $h$, and $u^\mu$ are the rest mass density, relativistic specific enthalpy, and 4-velocity of the fluid; $g^{\mu\nu}$ is the metric tensor of signature $(-,+,+,+)$.
The covariant equations can be written as a first order system of hyperbolic evolution equations in conservation form with source terms: $$\begin{aligned}
{\partial}_0 q &= -{\partial}_i f^i(q,x^i) + s(q,x^i) , \label{eq_qconslaw} \\
q &\equiv \left({D},{E},{S}_j \right) . \end{aligned}$$ For our study, it is important to use a conservative formulation, because it ensures correct shock wave propagation speeds when evolved by means of a finite volume scheme.
The evolved hydrodynamic variables are $$\begin{aligned}
{D}&\equiv {\sqrt{\gamma}}{W}{\rho},\\
{E}&\equiv {\sqrt{\gamma}}\left( {W}^2 {\rho}h - P -{W}{\rho}\right) ,\\
{S}_i &\equiv {\sqrt{\gamma}}{W}^2 {\rho}h v_i , \label{def_Cmom}\end{aligned}$$ where $\gamma$ is the determinant of the 3-metric, $v^i$ the 3-velocity, and ${W}$ the corresponding Lorentz factor. In the Newtonian limit using Cartesian coordinates, ${D}$, ${E}$, and ${S}^i$ reduce to mass density, energy density, and momentum density.
The flux terms $f^i=(f^i_{{D}},f^i_{{E}},f^i_{{S}_j})$ are given by $$\begin{aligned}
f^i_{{D}} &= w^i {D}, \\
f^i_{{E}} &= w^i {E}+ {\alpha}{\sqrt{\gamma}}v^i P , \\
f^i_{{S}_j} &= w^i {S}_j + {\alpha}{\sqrt{\gamma}}P \delta^i_j ,\end{aligned}$$ where $w^i=u^i/u^0$ is the coordinate velocity of the fluid and ${\alpha}$ the lapse function. The evolution equations need to be completed by an equation of state (EOS) to compute the pressure. We choose a polytropic EOS defined by $$\begin{aligned}
P({\rho}) = {\rho_p}\left(\frac{{\rho}}{{\rho_p}}\right)^\Gamma \equiv K {\rho}^\Gamma ,
\label{eq_eos_poly}\end{aligned}$$ where ${\rho_p}$ is a constant density scale, and the constant $\Gamma$ is the polytropic exponent.
When assuming a one-parametric EOS, the evolution equations (\[eq\_qconslaw\]) are over-determined. Therefore, we do not evolve the energy density ${E}$, but recompute it from the remaining variables and the EOS. The physical implications will be discussed in the next subsection.
The above formulation is used by many modern relativistic codes. For a review, we refer to [@FontLRR]. Like most codes, the [<span style="font-variant:small-caps;">pizza </span>]{}code is based on a HRSC (high resolution shock capturing) scheme, which was however optimized for quasistationary simulations. The difference to standard HRSC schemes is that source and flux terms are treated more consistently, making use of a special formulation of the source terms in [Eq. (\[eq\_qconslaw\])]{}. As a consequence, the code is able to preserve a stationary star to high accuracy. For details, see [@Kastaun06].
One of the weak points of current relativistic hydrodynamic schemes is the treatment of the stellar surface, where the numerical schemes designed for the interior would break down without further measures. The most common remedy is to enforce an artificial atmosphere of low density. The original [<span style="font-variant:small-caps;">pizza </span>]{}code described in [@Kastaun06] used a different workaround, which works perfectly for low amplitude oscillations. Unfortunately, we found that it yields unsatisfactory results for high amplitude oscillations in conjunction with a stiff EOS.
For the results in this article, we used yet another method of treating the surface. In short, it is based on a smooth transition of the numerical flux, between the formally correct expression at and above a certain density, and a flux which causes no acceleration at zero density. The local density scale for the beginning of the transition is computed from the local gravity, using an expression which corresponds to the average density gradient of the stationary model near the stellar surface times the grid spacing. No artificial atmosphere is used. To prevent negative densities, the flux is limited. The details of the scheme will be described in a forthcoming publication. We just note that the evolution below the aforementioned density scale is still plain wrong, like it is the case with other common schemes. Due to the low densities involved, the huge local errors at the surface have only limited impact on the global evolution. Nevertheless, the surface is an important source of numerical errors for this study, in particular with regard to the numerical damping.
The code has been tested on shock tube problems, linear oscillations of neutron stars, see [@thesis; @Kastaun06; @Kastaun08], and recently with self-gravitating relativistic tori. On the timescales of the simulations presented here, the code is able to evolve the stationary background model without significant changes in density or rotation profile. In contrast to codes using artificial atmospheres, our code exactly conserves the total mass.
We use Cartesian grids for three-dimensional simulations, cylindrical coordinates for two-dimensional ones, and spherical coordinates for one-dimensional problems. The rigidly rotating models are evolved in the corotating frame. To save computation time, we further assume equatorial symmetry. Finally, we apply vacuum outer boundary conditions, such that any material hitting the outer boundary is lost, and monitor the total mass to detect this case.
Treatment of shock waves {#sec_polyshock}
------------------------
For some of our results, shock formation plays an important role. The use of a polytropic EOS then becomes one of the main limitations, as will be explained in the following.
For any ideal fluid system with a two-parametric EOS, the physically correct solution evolves adiabatically as long as there are no shock waves present. For initial data which is isentropic, the evolution in the absence of shocks is thus the same as for an one-parametric EOS corresponding to a curve of constant specific entropy of the two-parametric EOS.
The polytropic EOS used in our simulations are the curves of constant specific entropy of the ideal gas EOS given by $$\begin{aligned}
P({\rho},{\epsilon}) &= \left(\Gamma-1\right) {\rho}{\epsilon},\end{aligned}$$ where ${\epsilon}$ is the specific internal energy and $\Gamma$ matches the polytropic exponent. Our simulations are therefore equivalent to the case of an ideal gas EOS with isentropic initial data, until shock waves form. Hence we can accurately predict the onset of shock formation.
After the shocks form, using the polytropic EOS becomes unphysical. Formally, the reduced system of evolution equations for the polytropic case admits solutions with discontinuities. However those differ from realistic shock waves in two aspects: there is no entropy production, i.e. shock heating, and the local conservation of energy is violated.
Nevertheless, we also present results based on the evolution after shock formation. For this, we make the assumption that the main difference between shock solutions for hot and cold EOS is that the kinetic energy converted into heat for the correct solution is simply lost when using the cold EOS, while the evolution of the density profile remains similar. Although this is clearly a strong simplification, the results may serve as a first estimate.
The reason why we do not simply use the ideal gas EOS is purely technical: the version of our code which is capable of evolving two-parametric EOSs is using a method of treating the surface which is not well suited for high amplitude oscillations. Fortunately, the only results depending on evolutions beyond shock formation are the damping times of axisymmetric oscillations, but not the critical amplitudes for the onset of shock formation; for the nonaxisymmetric oscillations, no significant shocks occur, and hence the use of the polytropic EOS is perfectly valid there.
Estimating the gravitational wave amplitude {#sec_gwmultipole}
-------------------------------------------
To extract gravitational waves, we use the multipole formalism for Newtonian sources, as described in [@Thorne80]. In this formalism, the radiation field at distance $r$ far away from the source is expanded in terms of spin tensor harmonics $$\begin{gathered}
h^{TT}_{jk} \left(t+r\right) = \frac{1}{r} \sum_{l=2}^\infty \sum_{m=-l}^l \left[
{A^{E2}}_{lm}\left(t\right) T^{E2,lm}_{jk} \right. \\
+ \left. {A^{B2}}_{lm} \left(t\right) T^{B2,lm}_{jk} \right] , \label{eq_gwstrain}\end{gathered}$$ and the total gravitational luminosity is given by $$\begin{aligned}
L &= \frac{1}{32\pi} \sum^\infty_{l=2} \sum_{m=-l}^l
\left( |\partial_t {A^{E2}}_{lm} |^2 + |\partial_t {A^{B2}}_{lm} |^2 \right) .\end{aligned}$$ In the Newtonian limit, the coefficients of the radiation field multipole expansion can be expressed in terms of the mass- and current-multipoles of the source $$\begin{aligned}
{A^{E2}}_{lm} &= \frac{4\sqrt{2\pi}}{(2l+1)!!}\sqrt{\frac{(l+1)(l+2)}{(l-1)l}(2l+1)} \: \partial_t^l q_{lm} ,\\
{A^{B2}}_{lm} &= \frac{32\pi}{(2l+1)!!} \sqrt{\frac{l+2}{2(l-1)}} \: \partial_t^l J_{lm} ,\end{aligned}$$ with the multipole moments given by $$\begin{aligned}
q_{lm} &= \sqrt{\frac{4\pi}{2l+1}}\int \rho r^l Y^{lm*} \,d^3x , \label{eq_mass_mult}\\
J_{lm} &= \int \rho r^l \vec{v} \cdot \vec{Y}^{lm*}_B \,d^3x . \label{eq_curr_mult}\end{aligned}$$ The above integrals are defined in the inertial frame, and $\vec{Y}^{lm}_B$ denotes the vector spherical harmonics of magnetic type.
The error induced by using the above formulas for sources as relativistic as neutron stars is difficult to estimate analytically. In [@Shib03], results using the quadrupole formula are compared with direct wave extraction methods for the case of a pulsating neutron star model, finding an error around 10-20% in the strain amplitude.
However, this error does not directly carry over to our results. The reason is that besides neglecting relativistic corrections, another error arises from the ambiguity of choosing a coordinate system for strongly curved spacetimes. This depends not only on the model, but also on the gauge choices made when computing the initial data. To get a rough estimate for the deviation from Euclidean geometry, we compute $$\begin{aligned}
\eta_1 &= \frac{R_{pe}}{R_e}, &
\eta_2 &= \frac{R_{pp}}{R_p} , \\
\eta_3 &= \frac{R_\text{circ}}{R_e}, &
\eta_4 &= \frac{ \int_0^{R_e} {\sqrt{\gamma}}x^2 \,dx }{ \int_0^{R_e} x^2 \,dx } ,\end{aligned}$$ where $R_e$ and $R_p$ are the equatorial and polar coordinate radius, while $R_{pe}$, $R_{pp}$, and $R_\text{circ}$ are the proper equatorial radius, proper polar radius, and equatorial circumferential radius.
From this average quantities we estimate the error of the multipole moments by assuming that the radius $r$ in [Eq. (\[eq\_mass\_mult\])]{} and [Eq. (\[eq\_curr\_mult\])]{} is wrong by a constant factor $\eta_r$, in the sense that the correct value is in the range $[r/\eta_r,r \eta_r]$, and that the volume element $d^3x$ is wrong by a factor of $\eta_4$. We further set $\eta_r$ to the maximum over $\eta_{1\dots 3}$ and the reciprocal values.
In consequence, the strain amplitude (which is mainly due to the $l=2$ mass multipoles) would be wrong by a factor $\eta_t=\eta_r^2 \eta_4$. While for most of our models, $\eta_t$ is in the range 1.2–2, it is as big as 13 in one case. This shows how dangerous it is to generalize error bounds for the quadrupole formula computed for a “typical” neutron star model.
Our estimate does not take into account the possibility that large cancellations occur in the integrals for the multipole moments. In this study, this affects the quasiradial [$F$]{}-mode oscillations, in particular for slowly rotating stars. For those, we do not compute the error; to obtain robust results, fully relativistic studies are needed.
To implement the above formalism, we compute the multipole moments in the corotating frame used in our simulations. For the current multipoles, we nevertheless use the inertial frame velocity in the integrals. It is straightforward to show that the multipole moments in the inertial frame are given by $$\begin{aligned}
q_{lm} &= e^{-im\Omega t} q_{lm}' , \\
J_{lm} &= e^{-im\Omega t} J_{lm}' ,\end{aligned}$$ where $q_{lm}'$ and $J_{lm}'$ are the multipole moments computed in the corotating frame defined by $\phi'=\phi - \Omega t$, with $\Omega$ being the angular velocity of the star.
The multipole moments are computed during the evolution with a sampling rate of $75\usk\kilo\hertz$. When evaluating time derivatives of 2nd order or higher, special care has to be taken to avoid amplification of high frequency numerical noise. For this, we first apply numerical smoothing by convolution of the time series with a Blackman window function. Derivatives up to 3rd order are then computed using cubic splines. For derivatives of 4th and 5th order, we first compute the 3rd derivative and than apply the whole scheme again.
We computed the frequency response function of the resulting scheme, which effectively cuts off contributions to the gravitational wave (GW) signal above $10\usk\kilo\hertz$. In the frequency range of the actual signal, the loss of strain amplitude due to the smoothing stays below 15%.
Although only the $l=2$ mass multipoles are important for our problems, we generally compute all mass multipole moments up to $l=4$, and the current multipoles with $m \ge 2, l\le4$,
The second important error for the strain, beside the use of the multipole formula, is due to the Cowling approximation, which is known to cause a significant error in the oscillation frequencies. Given an estimate for the inertial frame frequency $\hat{f}_i$ of a harmonic oscillation in full GR, one can approximate the strain amplitude $\hat{A}^{E2}_{20}$ by $$\begin{aligned}
\hat{A}^{E2}_{20} &\approx {A^{E2}}_{20} \left(\frac{\hat{f}_i}{f_i}\right)^2 ,\end{aligned}$$ where $f_i$ and ${A^{E2}}_{20}$ are frequency and strain amplitude in the Cowling approximation.
Measuring dissipation
---------------------
In order to measure the damping of oscillations, we monitor the evolution of an average corotating velocity defined by $$\begin{aligned}
\bar{v} &= \sqrt{\frac{1}{M} \int {D}v_c^i v_c^j g_{ij} \, d^3x} ,\\
M &= \int {D}\, d^3x ,\\
v_c^i &= \frac{u^i}{{\alpha}u^0} = v^i - \frac{\beta^i}{{\alpha}} ,\end{aligned}$$ where ${\alpha}$ and $\beta^i$ are lapse function and shift vector. Since we are working in the corotating frame, $v_c^i$ is a corotating velocity.
This measure is zero if and only if the fluid velocity is everywhere the same as for the unperturbed stationary star. For the nonrotating case in the Newtonian limit, $\bar{v}$ is also directly related to the total kinetic energy. As long as there is only one dominant oscillation mode, the decay of $\bar{v}$ is a measure for the decay of the mode amplitude. On the other hand, it is not sensitive to energy transfer from one oscillation mode to another.
From the evolution, we extract three quantities.
- The initial amplitude $A_I=\bar{v}(0)$.
- The final amplitude. Since $\bar{v}$ is oscillating, we use the maximum amplitude during the last oscillation cycle, $$\begin{aligned}
\label{eq_def_afinal}
A_F &= \max\{\bar{v}(t) | T_e - T_0 \le t \le T_e \} ,\end{aligned}$$ where $T_e$ is the time over which we evolved the system and $T_0$ is the period of the oscillation mode used to perturb the system.
- The timescale $\tau$ of the initial decay of $\bar{v}$, which we define as $$\begin{aligned}
\label{eq_def_tau}
e^{-\frac{\Delta T}{\tau}} &= \frac{\bar{v}_a(\Delta T)}{\bar{v}_a(0)} ,
&
\bar{v}_a(t) &= \sqrt{\int_t^{t+T_0} \bar{v}^2 \,dt} ,\end{aligned}$$ where $T_0$ is the oscillation period of the mode used for perturbation, and $\Delta T = 4 T_0$.
To detect the presence of shocks, we make use of the fact that there exists a conserved energy when using the Cowling approximation, given by $$\begin{aligned}
E_c &= \int \rho_e {\sqrt{\gamma}}\,d^3x , \label{eq_def_econs}\\
\rho_e &= -\alpha T^0_0 - {W}{\rho}.\end{aligned}$$ More precisely, $E_c$ is conserved for smooth and weak solutions of [Eq. (\[eq\_div\_tmunu\])]{}. As mentioned, shock solutions of the reduced system of equations used with a one-parametric EOS violate the local energy conservation, and thus the conservation of $E_c$. Thus, any change of $E_c$ points to the existence of shock waves.
Computing eigenfunctions and eigenfrequencies
---------------------------------------------
To extract eigenfunctions, we use the mode recycling method described in [@DSF2006]. In short, we perturb the star using a generic perturbation to excite different modes and extract their frequencies using Fourier analysis. Then we extract a first guess of the eigenfunction by evaluating at each point of the numerical grid the Fourier integral of specific energy and velocity perturbations at the frequency of the desired mode.
The estimate of the eigenfunction obtained in this way is in general still contaminated with other modes, due to the finite evolution time. Therefore, additional simulations are performed, using the eigenfunction obtained in the previous step as initial perturbation, until only the desired oscillation mode is present in the evolution.
We further improved this scheme by making use of the fact that for any axisymmetric star, the eigenfunctions of ${\epsilon}, v^d$, $v^z$, and $v^\theta$ can be written in the form $$\begin{aligned}
\delta {\epsilon}(d,z,\phi) = e^{i\beta} \delta\hat{{\epsilon}}(d,z) e^{im\phi} .\end{aligned}$$ The two-dimensional eigenfunction $\delta\hat{{\epsilon}}(d,z)$ is real-valued and has a well defined $z$-parity. The constant $\beta$ is an irrelevant complex phase.
The Fourier analysis of the numerical evolution yields an estimate for the complex-valued $\delta{\epsilon}(d,z,\phi)$. To obtain $\delta\hat{{\epsilon}}$, we first compute the integral $$\begin{aligned}
\label{eq_ef_phi_avg}
\delta\hat{{\epsilon}}_c &=
\int_0^{2\pi} \delta {\epsilon}(d,z,\phi) e^{-im\phi}\, d\phi .\end{aligned}$$ Since we use Cartesian coordinates for nonaxisymmetric problems, this step involves interpolation. For this, a multidimensional cubic spline interpolation method is applied. Note we also increase the resolution by evaluating the above integral at finer intervals than the original grid spacing.
Errors due to the presence of unwanted modes with different $\phi$-dependency are strongly suppressed during the computation of $\delta\hat{{\epsilon}}_c$. This step is absolutely necessary to separate co- and counter-rotating modes for slowly rotating stars, since their frequencies approach each other.
It seems plausible to take the real or imaginary part of $\delta\hat{{\epsilon}}_c$ as estimate for the eigenfunction. However, this is a bad idea since the phase is arbitrary, which can lead to strong suppression of the actual eigenfunction with respect to numerical errors. It is preferable to remove the complex phase factor, using the expression $$\begin{aligned}
\label{eq_phase_avg}
e^{2 i \beta} &=
\frac{\int \left({\rho}\delta\hat{{\epsilon}}_c\right)^2 \,d^2x}{\int \left|{\rho}\delta\hat{{\epsilon}}_c\right|^2 \,d^2x} ,&
\delta \hat{{\epsilon}} &= e^{-i\beta} \delta\hat{{\epsilon}}_c .\end{aligned}$$ It is easy to see that this equation holds for the correct eigenfunction, which has constant complex phase. In case of numerical errors, the above expression yields an averaged phase which is insensitive to the unavoidable phase errors near the nodes of the eigenfunction, as well as the errors at the stellar surface.
Since analytically the complex phase should be constant, we can convert it’s variation into a measure for the quality of the numerical eigenfunction $$\begin{aligned}
q &= \frac{\int \left| C^2 - |C|^2 \right| \,d^2x}{2\int |C|^2 \,d^2x}, &
C &= {\rho}\delta\hat{{\epsilon}} . \label{eq_ef_qual}\end{aligned}$$ It is easy to see that $q=0$ for an exact eigenfunction and $q=1$ in the worst case. Due to the density weight, $q$ is a measure for the error of bulk motion, while errors at the surface are ignored. The complex phase of the velocity components is removed using the phase computed from $\delta\hat{{\epsilon}}_c$, taking into account the phase shift by $\pi/2$ of $\delta v^d, \delta v^z$ with respect to $\delta {\epsilon}, \delta v^\phi$. The quality measure is computed for each of the velocity components separately.
Axisymmetric eigenfunctions are extracted from two-dimensional simulations, in which case we do not need to evaluate [Eq. (\[eq\_ef\_phi\_avg\])]{}. Eigenfunctions of radial modes of spherical stars are computed using one-dimensional simulations at high resolutions. The eigenfunctions become one-dimensional and the integrals (\[eq\_phase\_avg\]), (\[eq\_ef\_qual\]) are replaced by integrals over the radial coordinate.
The eigenfunctions used in this study are only extracted once with a good resolution, and then used in all simulations, regardless of grid resolution or dimensionality. For this, linear interpolation is used to map the numerical eigenfunction to the desired resolution and coordinate system. This way, we can do convergence tests and comparisons between 2D and 3D simulations without worrying about differences in the eigenfunctions itself.
The frequencies are extracted using the time evolution of density and velocity at some sample point in the corotating frame, and of the multipole moments in the inertial frame defined in [Sec. \[sec\_gwmultipole\]]{}. For this, we fit an exponentially damped sinusoidal, which is usually more accurate than using the Fourier transform. The comparison between inertial and corotating frame is useful to identify $m$ for unknown modes, and serves as a consistency check.
The frequencies in the inertial and corotating frame are related by $$\begin{aligned}
f_i = \left| f_c - m F_R \right| ,\end{aligned}$$ where $f_c$ is the frequency in the corotating frame, $f_i$ in the inertial frame, and $F_R$ is the rotation rate of the star. All frequencies are defined as positive and measured with respect to coordinate time. For our setups, $F_R$ and $f_i$ are also identical to the rotation rate and oscillation frequency observed at infinity, see [Sec. \[sec\_initial\_data\]]{}. We define $m$ such that $m>0$ if the wave-patterns of the mode appear counterrotating in the corotating frame, and $m<0$ for modes which are corotating in the corotating frame. Note a mode counter-rotating in the corotating frame appears corotating in the inertial frame if $f_c < m F_R$.
Setting up initial data {#sec_initial_data}
-----------------------
Our stellar models are rigidly rotating (and nonrotating) stationary configurations of an ideal fluid in general relativity, with a polytropic EOS defined by [Eq. (\[eq\_eos\_poly\])]{}. The models are characterized uniquely by central density ${\rho}_c$, rotation rate $F_R$, polytropic exponent $\Gamma$ and polytropic density scale ${\rho_p}$.
To compute the spacetime describing a rigidly rotating relativistic star, we use the code described in [@Ansorg03; @Ansorg08]. This code is able to solve the full set of stationary Einstein and hydrostatic equations with high precision even for models rotating near the Kepler (mass shedding) limit.
For all simulations of a given model, we use one and the same spacetime which is computed once with a resolution of at least 100 points per stellar radius. The data is mapped onto the computational grid used in our simulations using linear interpolation. The shift vector is initialized such that we obtain coordinates corotating with the (rigidly) rotating star. For three-dimensional simulations, we apply the standard transformation from cylindrical to Cartesian coordinates.
To set up spherical stars, we use our own code to solve the ordinary differential equations derived by [@OV39] (TOV-equations). For two- or three-dimensional simulations, a standard transformation from spherical coordinates to cylindrical or Cartesian coordinates is applied.
We note that the line element found by the two methods is not exactly the same for a given nonrotating star due to different gauge choices. The line elements for the rotating model can be found in [@Ansorg03; @Ansorg08]. For the simulation itself the choice of coordinates doesn’t matter, since it is gauge invariant (up to numerical errors). It will however have an impact on the error of the GW extraction, as discussed in [Sec. \[sec\_gwmultipole\]]{}.
The time coordinate, which is only fixed up to a global factor, is normalized by the initial data codes such that ${\alpha}=1$ at infinity. Since the spacetime is stationary, any frequency measured with respect to coordinate time at a fixed point in the inertial coordinate frame is identical to the frequency observed at infinity.
To excite oscillations, we perturb specific energy ${\epsilon}$ and 3-velocity $v^i$. Since we are using the Cowling approximation during evolution, we neither perturb the metric nor reinforce the constraint equations. In case the density becomes negative, which can happen near the surface, it is simply reset to zero. When perturbing with axisymmetric eigenfunctions, we have the freedom to choose the phase of the oscillation such that the initial density perturbation is zero, and perturb only the velocity. For nonaxisymmetric modes, both specific energy and velocity are perturbed.
Problem setup {#sec_models}
=============
In order to study nonlinear effects, we perturb stationary neutron star models with various eigenfunctions, which are scaled to amplitudes ranging from the linear regime to the strongly nonlinear one, and let the system evolve long enough to observe strong damping effects. In detail, we study the [$F$]{}, [${}^2\!f_0$]{}, [${}^2\!f_{2}$]{}, and [${}^2\!f_{-2}$]{} modes.
We use models with three different polytropic EOSs, which are summarized in [Tab. \[tab\_eos\]]{}. The motivation behind our choice is to cover a wide range of stiffness, in order to demonstrate it’s influence. EOS A was introduced in [@2004PhRvD..70h4026S] as a rough polytropic approximation to the more realistic Pandharipande EOS as tabulated in [@Arnett77]. It is the stiffest of our EOSs. The models with EOS B and C (not to be confused with B and C in [@Arnett77]) are generic toy models; EOS C is very soft, while EOS B is of medium stiffness. EOS B is often used as a reference point in numerical studies.
For each EOS, we investigate a nonrotating model as well as rigidly rotating models with various rotation rates, some close to the Kepler limit. Our models have gravitational (ADM) masses 1.4–1.9$\usk M_\odot$, which is in the range of observed neutron star masses. We checked that the nonrotating models are on the stable branch of the mass-radius diagram, and expect the same for the rotating ones. The central sound speed is $0.75\usk c$ for model MA100, $0.45\usk c$ for MB100, and $0.21 \usk c$ for MC100. All our models are summarized in [Tab. \[tab\_models\]]{}.
Model MA65 is of particular astrophysical interest. As shown in [Sec. \[sec\_freq\_ef\]]{}, the counter-rotating [${}^2\!f_{2}$]{} mode is most probably subject to the CFS instability in full GR. We stress that the CFS instability is not active in the Cowling approximation. Even in full GR, it’s growth timescale is on the order of seconds, and therefore irrelevant on the timescales of our simulations. However, it could provide a mechanism of exciting the high amplitudes investigated in this study, provided the instability is not suppressed already at much lower amplitudes due to viscosity or mode coupling effects.
The maximum amplitude of the initial perturbation is chosen such that during the evolution, the fluid stays inside a given region, which is usually twice as big as the bounding box of the star. Only for 3D simulations of model MA65, we had to restrict ourselves to an expansion factor of 1.6, because otherwise the corners of the corotating coordinate system would move with superluminal speed.
To excite high amplitude oscillations, we linearly scale the eigenfunctions of specific energy and 3-velocity, and add them to the background model. For axisymmetric simulations, the sign of the perturbation is chosen such that the first maximum of the $x$-velocity along the $x$-axis is positive. For nonaxisymmetric perturbations, the sign is irrelevant since changing the sign corresponds to a rotation. In the rest of this paper we refer to such a high amplitude perturbation based on the eigenfunction e.g. of the [$F$]{} mode simply as an [$F$]{}-mode perturbation.
Note that this choice is not unique. For example, one could scale the momentum density instead of the velocity, or change the sign of the perturbation. In the nonlinear regime, this will lead to small differences in the results. Also, our setups probably differ slightly from what one would obtain by letting a mode grow to high amplitudes by means of some physical instability or artificial backreaction force.
Name $n$ $\Gamma$ $\rho_p / \gram\usk\centi\meter\rpcubed$ $K$
------ -------- ---------- ------------------------------------------ -------
C 2 1.5 7.000[$\cdot 10^{16}$]{} 2.970
B 1 2 6.176[$\cdot 10^{15}$]{} 100.0
A 0.6849 2.46 4.070[$\cdot 10^{15}$]{} 11.65
: \[tab\_eos\] The polytropic EOS used in our models, specified by polytropic constant $\Gamma=1+1/n$ and polytropic density scale $\rho_p$. The more commonly used polytropic constant $K = {\rho}_p^{1-\Gamma}$ is given in geometric units $G=c=M_\odot=1$.
Name EOS $M / M_\odot$ $F_R / \hertz$ $R_c / \kilo\meter$ $R_p/R_e$ $\rho_c / \gram\usk\centi\meter\rpcubed$
------- ----- --------------- ---------------- --------------------- ----------- ------------------------------------------
MA100 A 1.615 0 9.529 1 2.065[$\cdot 10^{15}$]{}
MA65 A 1.910 1687 11.61 0.65 2.065[$\cdot 10^{15}$]{}
MB100 B 1.400 0 14.16 1 7.905[$\cdot 10^{14}$]{}
MB85 B 1.503 590.9 15.38 0.85 7.905[$\cdot 10^{14}$]{}
MB70 B 1.627 792.1 17.27 0.70 7.905[$\cdot 10^{14}$]{}
MC100 C 1.400 0 47.60 1 7.618[$\cdot 10^{13}$]{}
MC95 C 1.400 56.97 51.17 0.95 6.660[$\cdot 10^{13}$]{}
MC85 C 1.400 83.47 59.41 0.85 5.222[$\cdot 10^{13}$]{}
MC65 C 1.400 93.11 80.80 0.65 4.007[$\cdot 10^{13}$]{}
: \[tab\_models\] Details of the stellar models. $M$ is the gravitational mass, $F_R$ the rotation rate as observed from infinity, $R_c$ the equatorial circumferential radius, $R_p$ and $R_e$ the polar and equatorial coordinate radius, and $\rho_c$ the central rest mass density.
Numerical results {#sec_results}
=================
Mode frequencies and eigenfunctions {#sec_freq_ef}
-----------------------------------
As a prerequisite for our studies, we extracted the frequencies and eigenfunctions of various low-order axisymmetric and nonaxisymmetric pressure modes. All frequencies can be found in [Tab. \[tab\_freq\_modes\]]{}. Based on convergence tests, including those in [@Kastaun06], we are confident that the numerical error of the frequency is generally below 2%.
We also compared our results to those obtained by [@2008PhRvD..78f4063G], where a linearized evolution code and Cowling approximation was used. The differences are below 1.2% for models MA100, MA65, and below 0.6% for models MB100, MB85, MB70, which is an excellent agreement. For the nonrotating models MC100 and MA100, the frequencies we extracted for the [${}^2\!f_0$]{}and [${}^2\!f_{2}$]{} modes (which are exactly the same analytically) agree better than 0.1%.
For the extraction of nonaxisymmetric modes, the mode-recycling method is computationally very expensive, since it requires repeated 3D simulations over many oscillation periods. Therefore, we did not repeat the recycling step as often as for axisymmetric modes, and used a resolution of only 50 points per stellar radius, in contrast to 100–200 points for 2D simulations.
As a consequence, when using the nonaxisymmetric numeric eigenfunctions as a perturbation, other modes are sometimes excited as well, at amplitudes up to a few % of the desired mode. We believe that regarding the dynamics of the evolution, this error can be neglected in comparison to other numerical errors. When interpreting the GW signal on the other hand, it has to be taken into account.
In agreement with [@2008PhRvD..78f4063G], we find that the counter-rotating [${}^2\!f_{2}$]{} mode of model MA65 becomes corotating in the inertial frame. Although we are working in the Cowling approximation, this should be the case in full GR as well. As shown in [@2010PhRvD..81h4055Z] for similar models, dropping the Cowling approximation seems to lower the neutral point, i.e. the critical rotation rate where the counter-rotating $f$ mode becomes corotating in the inertial frame. Therefore, the counter-rotating [${}^2\!f_{2}$]{} mode of model MA65 is most probably CFS-unstable.
The eigenfunction of the [${}^2\!f_{2}$]{} mode is shown in [Figure \[fig\_ef\_jega65\_fl2m2\]]{}. Although model MA65 is rotating quite rapidly, the eigenfunction is only moderately deformed in comparison to the nonrotating model MA100 shown in [Figure \[fig\_ef\_ega1\_fl2m2\]]{}. The main difference is that, compared to the inner region, the oscillation amplitude near the equator is increased. This is to be expected since the material is bound less strongly with increasing centrifugal force.
The eigenfunctions we found for stars with moderate rotation rates are well described by the slow-rotation limit, where the eigenfunctions are given by a spherical harmonic times some radial function. For the purpose of our study, it is more interesting to look at the rapidly rotating models.
One important observation with regard to GW emission is that for rapidly rotating models, the quasiradial [$F$]{} mode possesses a considerable quadrupole moment due to the oblateness of those stars. As an example, the quasiradial [$F$]{} mode of model MA65 is shown in [Figure \[fig\_ef\_jega65\_fl0m0\]]{}. We have to stress that the figure shows the eigenfunction in the coordinate system set up by the initial data code. It would be easy to construct an asymptotically Euclidean coordinate system in which even a spherical star looks oblate. However, given that MA65 is rotating near the Kepler limit, we are confident that the deformation of the star and the eigenfunction is not mainly a coordinate artifact.
Another effect becomes important very close to the Kepler limit. As shown in [Figure \[fig\_ef\_jsft7\_fl2m0\]]{}, the [${}^2\!f_0$]{}-mode eigenfunction of model MC65 is not only strongly deformed in comparison to the slow-rotation limit, it also develops very pronounced peaks near the equator. Looking at the bulk properties of mode, we find that the dynamics of the oscillation is still determined by the inner regions of the star, but a given amplitude in the interior induces much higher amplitudes near the equator. This is not surprising since the material there is only marginally bound for this model. As a consequence, oscillation modes of MC65 can store only little energy before nonlinear effects set in, as will be shown in [Sec. \[sec\_damp\_axi\]]{}. In fact, we did not even succeed to obtain a clean eigenfunction of the [$F$]{} mode of this model, due to strong nonlinear couplings.
Since the GW luminosity depends strongly on the frequency, we have to estimate the influence of the Cowling approximation. Frequencies in Cowling approximation and full GR, computed by means of fully relativistic simulations, are given in [@2010PhRvD..81h4055Z], for model MB100 and models similar to MA100 and MA65, amongst others. After adding an additional safety margin to account for the different models, we assume that we over-estimate the frequencies of the [$F$]{} modes by a factor of less than 2.5 and the [${}^2\!f_0$]{} modes by a factor less than 1.5. Further, we assume that the frequency of the [${}^2\!f_{2}$]{} mode of in the *corotating* frame is over-estimated by a factor $< 1.3$. For model MA65, this implies that the frequency in the inertial frame is *under-estimated* by a factor up to 2.3. For model MB70, we find that the [${}^2\!f_{2}$]{} mode could become CFS-unstable in full GR, with an inertial frame frequency in the range 0–90.
Model Mode $f_c / \hertz$ $f_i / \hertz$ q
------- ---------------------------------------------------------------- ---------------- ---------------- ----------------------
MC100 [$F$]{} 472 472 2[$\cdot 10^{-4}$]{}
MC100 [${}^2\!f_0$]{} 418 418 5[$\cdot 10^{-4}$]{}
MC100 ${\ensuremath{{}^2\!f_{2}}}\equiv {\ensuremath{{}^2\!f_{-2}}}$ 418 418 4[$\cdot 10^{-4}$]{}
MC95 [$F$]{} 437 437 8[$\cdot 10^{-5}$]{}
MC95 [${}^2\!f_0$]{} 392 392 2[$\cdot 10^{-3}$]{}
MC85 [$F$]{} 383 383 5[$\cdot 10^{-5}$]{}
MC85 [${}^2\!f_0$]{} 337 337 2[$\cdot 10^{-4}$]{}
MC85 [${}^2\!f_{2}$]{} 371 205 3[$\cdot 10^{-3}$]{}
MC65 [${}^2\!f_0$]{} 239 239 2[$\cdot 10^{-4}$]{}
MB100 [$F$]{} 2686 2686 3[$\cdot 10^{-5}$]{}
MB100 [${}^2\!f_0$]{} 1883 1883 5[$\cdot 10^{-4}$]{}
MB85 [${}^2\!f_0$]{} 1893 1893 4[$\cdot 10^{-4}$]{}
MB70 [${}^2\!f_0$]{} 1785 1785 3[$\cdot 10^{-4}$]{}
MB70 [${}^2\!f_{2}$]{} 1948 364 4[$\cdot 10^{-3}$]{}
MA100 [$F$]{} 4606 4606 4[$\cdot 10^{-5}$]{}
MA100 [${}^2\!f_0$]{} 3038 3038 5[$\cdot 10^{-4}$]{}
MA100 ${\ensuremath{{}^2\!f_{2}}}\equiv {\ensuremath{{}^2\!f_{-2}}}$ 3037 3037 2[$\cdot 10^{-3}$]{}
MA65 [$F$]{} 3997 3997 3[$\cdot 10^{-4}$]{}
MA65 [${}^2\!f_0$]{} 2662 2662 2[$\cdot 10^{-3}$]{}
MA65 [${}^2\!f_{2}$]{} 2856 518 6[$\cdot 10^{-3}$]{}
MA65 [${}^2\!f_{-2}$]{} 1179 4553 2[$\cdot 10^{-2}$]{}
: \[tab\_freq\_modes\] Frequencies $f_i$ of the different oscillation modes as observed from infinity in the inertial frame, and corresponding frequencies $f_c$ in the corotating frame. $q$ is the quality factor of the numerical eigenfunction for $\delta \hat{\epsilon}$ defined in [Eq. (\[eq\_ef\_qual\])]{}.
![\[fig\_ef\_jega65\_fl2m2\] Eigenfunction of the counter-rotating [${}^2\!f_{2}$]{} mode of model MA65. The left half shows the two-dimensional eigenfunction $\delta\hat{{\epsilon}}(d,z)$ of specific energy as a contour plot. The eigenfunction is zero on the rotation axis. The arrows in the right half correspond to the eigenfunction of the velocity in the meridional plane. ](figure1){width="\columnwidth"}
![\[fig\_ef\_ega1\_fl2m2\] Like [Figure \[fig\_ef\_jega65\_fl2m2\]]{}, but showing the [${}^2\!f_{2}$]{} mode of nonrotating model MA100. The specific energy eigenfunction is zero only on the rotation axis. ](figure2){width="\columnwidth"}
![\[fig\_ef\_jega65\_fl0m0\] Eigenfunction of the quasiradial [$F$]{} mode of model MA65, plotted like in [Figure \[fig\_ef\_jega65\_fl2m2\]]{}. The additional thick solid line marks the node. ](figure3){width="\columnwidth"}
![\[fig\_ef\_jsft7\_fl2m0\] Eigenfunction of the axisymmetric [${}^2\!f_0$]{} mode of model MC65. TOP: Eigenfunction $\delta\hat{{\epsilon}}(d,z)$ as gray-scale plot. The thick solid line marks the node position. The arrows show the velocity eigenfunction. BOTTOM: Same eigenfunctions scaled by $\rho d$, to visualize the bulk properties of the mode. The node near the equator is very likely a numerical artifact. ](figure4a "fig:"){width="\columnwidth"}\
![\[fig\_ef\_jsft7\_fl2m0\] Eigenfunction of the axisymmetric [${}^2\!f_0$]{} mode of model MC65. TOP: Eigenfunction $\delta\hat{{\epsilon}}(d,z)$ as gray-scale plot. The thick solid line marks the node position. The arrows show the velocity eigenfunction. BOTTOM: Same eigenfunctions scaled by $\rho d$, to visualize the bulk properties of the mode. The node near the equator is very likely a numerical artifact. ](figure4b "fig:"){width="\columnwidth"}
Damping of axisymmetric modes {#sec_damp_axi}
-----------------------------
For all axisymmetric oscillations of all models, we find a common behavior at high amplitudes: if the initial amplitude exceeds a certain threshold, shock waves form in the outer layers of the star, which dissipate energy during a few oscillation cycles until the oscillation amplitude falls below the threshold again. After this phase, we usually observe only the numerical damping. In some cases, we also see mode coupling effects, which are however small compared to the strongest damping due to shocks.
The final amplitude can be smaller than the threshold, in particular for strong initial excitation. The final amplitude as a function of the initial one thus has a maximum for some modes. [Figure \[fig\_shock\_vm2\]]{} shows a typical evolution of the mean velocity in presence of shocks.
To detect shocks, we monitor the conserved energy $E_c$ defined by [Eq. (\[eq\_def\_econs\])]{}. [Figure \[fig\_shock\_econs\]]{} shows an example of energy dissipation in the presence of shocks. To investigate the details of shock formation, we produced movies from several of our simulations, showing the evolution of the density and velocity in the meridional plane. The shock formation was clearly visible for simulations with significant energy loss. For one-dimensional simulations, we also verified that shock formation and energy dissipation occur simultaneously. The shocks we observed formed in the outer layers of the star, but not directly at the surface.
![\[fig\_shock\_vm2\] Evolution of mean velocity $\bar{v}$ for an [$F$]{}-mode perturbation of model MC95 at high amplitudes. ](figure5){width="\columnwidth"}
![\[fig\_shock\_econs\] Violation of energy conservation due to shocks, for an [$F$]{}-mode perturbation of model MC95. Plotted is the loss of conserved energy $E_c$ in units of the total stellar mass versus the time in units of the oscillation period $T_0$. Only the phase of strong decay is shown. ](figure6){width="\columnwidth"}
To get an overview on the magnitude of nonlinear damping effects, we plot the final amplitude $A_F$ defined in [Eq. (\[eq\_def\_afinal\])]{} versus the initial amplitude $A_I$. As an example, [Figure \[fig\_vm\_jsft3\_fl2m0\]]{} shows the results for the [${}^2\!f_0$]{} mode of model MC85.
In general, we choose different evolution times for different sequences, such that the strong damping phase at high amplitudes (compare [Figure \[fig\_shock\_vm2\]]{}) is included, but not much longer. We choose short evolution times because we are interested mainly in strong damping effects. In any case, we can only accurately measure damping effects acting on timescales significantly shorter than the timescale of numerical damping.
For each sequence, we extract the amplitude $A_D$ at which nonlinear damping effects become stronger than the numerical damping, given in [Tab. \[tab\_onset\_nl\]]{}. This amplitude is independent from the exact value of the evolution time. As discussed in [Sec. \[sec\_polyshock\]]{}, the onset of shock formation is predicted accurately despite the use of a cold (one-parametric) EOS. Therefore, the onset of nonlinear damping is not affected by this approximation neither.
The values we found for the [$F$]{}and [${}^2\!f_0$]{} modes of the different models cover a wide range $A_D=0.003\dots 0.13 \usk c$. We will discuss the dependence on the model parameters in sections \[sec\_damp\_eos\] and \[sec\_damp\_rot\]. Naturally, the values of $A_D$ provide upper limits for the applicability of linear perturbation theory.
The final amplitude contains only information on how much the system is damped, but not how fast. To quantify the damping speed, we use the timescale $\tau$ of the initial decay defined in [Eq. (\[eq\_def\_tau\])]{}. [Figure \[fig\_tau\_jega65\_fl2m0\]]{} shows $\tau$ versus the initial amplitude $A_I$ for the case of the axisymmetric [${}^2\!f_0$]{} mode of model MA65. We stress that the values of $\tau$ *are* affected by the use of a cold EOS, as discussed in [Sec. \[sec\_polyshock\]]{}, and should be regarded as an educated guess.
Not surprising, the damping becomes faster with increasing amplitude. By definition, $\tau$ is a measure for time averaged decay. More detailed investigation reveals that for the highest amplitudes, the larger part of the energy is dissipated on timescales smaller than the oscillation period during the formation of a strong, but short-lived shock.
The decay timescale can be used to estimate the required growth timescale of a hypothetical instability saturating at a given amplitude. We are not aware of any axisymmetric instability acting on timescales shorter than the numerical damping. For any process with longer growth times, the value $A_D$ provides an upper limit for the saturation amplitudes.
To estimate the numerical errors, we computed several sequences using 3 different resolutions. Note it is important to choose the sign of the perturbation consistently, since we observed differences in the nonlinear regime comparable to the numerical error at low resolution.
By far the greatest errors are found for the stiff EOS A. [Figure \[fig\_tau\_jega65\_fl2m0\]]{} shows the convergence of the nonlinear damping timescale of the [${}^2\!f_0$]{} mode of model MA65 as a representative example. To explain the behavior at low amplitudes, we note that when approaching the linear regime, the timescale of physical damping goes to infinity. In that case, we only see the numerical damping, which becomes weaker with increasing resolution. The stronger the damping, the more accurate is the value we find for $\tau$.
For the soft EOS C, the numerical damping is extremely low. For a resolution of 100, we find timescales $\tau_N>800\usk\milli\second$ in the linear regime. Also the nonlinear results match extremely well, as shown in [Figure \[fig\_vm\_jsft3\_fl2m0\]]{}. The accuracy of models with EOS B is in between the other two cases. Since the numerical damping is mainly given by surface effects, as shown in [@Kastaun06], and the steepness of the density gradient at the surface increases with stiffness, this behavior is not surprising.
To get a rough error estimate for simulations which we only ran at one resolution, we assume that the numerical damping observed in the linear regime for single oscillations is also operational at high amplitudes, acting on the same timescale $\tau_D$. We then scale our time series by $\exp(t/\tau_D)$ to obtain corrected values for final amplitude $A_F$ and damping time $\tau$.
To estimate the accuracy of three-dimensional simulations we also performed 3D simulation of axisymmetric modes at high amplitudes. We generally found a good agreement with the axisymmetric results for the same setup; see [Figure \[fig\_vm\_jsft3\_fl2m0\]]{} and [Figure \[fig\_tau\_jega65\_fl2m0\]]{}.
![\[fig\_vm\_jsft3\_fl2m0\] Nonlinear damping of the [${}^2\!f_0$]{} mode of model MC85. Plotted is the final amplitude $A_F$ after an evolution time of $20\usk\milli\second$ over the initial amplitude $A_I$. We show results of axisymmetric simulations with resolutions of $N=50$, 100, and 200 points per stellar radius, as well as a three-dimensional simulation with resolution $N=50$. ](figure7){width="\columnwidth"}
![\[fig\_tau\_jega65\_fl2m0\] Timescale of initial decay $\tau$ due to shock damping versus the initial amplitude $A_I$, for an [${}^2\!f_0$]{}-mode perturbation of model MA65. Shown are results from axisymmetric simulations with three different resolutions as well as one 3D-simulation of the same problem. The resolution $N$ is given in points per stellar equatorial radius. ](figure8){width="\columnwidth"}
Model Mode $A_D / (10^{-3}\usk c)$
------- ----------------- -------------------------
MC100 [$F$]{} $11 \pm 1 $
[${}^2\!f_0$]{} $4 \pm 1 $
MC95 [$F$]{} $9 \pm 2 $
[${}^2\!f_0$]{} $5 \pm 2 $
MC85 [$F$]{} $7 \pm 2 $
[${}^2\!f_0$]{} $4 \pm 1 $
MC65 [${}^2\!f_0$]{} $<0.1 $
MB100 [$F$]{} $53 \pm 7 $
[${}^2\!f_0$]{} $20 \pm 5$
MA100 [$F$]{} $100 \pm 25$
[${}^2\!f_0$]{} $45 \pm 15$
MA65 [$F$]{} $40 \pm 15$
[${}^2\!f_0$]{} $8 \pm 3 $
: \[tab\_onset\_nl\] Onset of nonlinear damping. $A_D$ is the initial amplitude $A_I$ at which nonlinear damping is strong enough to be distinguishable from numerical damping, read off from plots of initial versus final amplitude.
Influence of the EOS on damping {#sec_damp_eos}
-------------------------------
To study the influence of the EOS, we use the nonrotating models MC100, MB100, MA100, which have similar masses, but EOSs of different stiffness.
The amount of nonlinear damping for the different models is shown in [Figure \[fig\_vm\_eos\]]{}. The amplitudes $A_D$ at which nonlinear damping effects become visible is given in [Tab. \[tab\_onset\_nl\]]{}.
As one can see, the nonlinearity sets in one order of magnitude earlier for the EOS C, which is the softest one, than for EOS A, the stiffest one. Also the maximum final amplitude is the largest for the stiff EOS. Note that MA100 is 13% heavier than MC100, which might account for part of the differences. Nevertheless, the masses of models MC100 and MB100 are exactly the same, and still the damping differs strongly. We conclude that with increasing stiffness of the EOS a neutron star (of a given mass) can store more energy in axisymmetric oscillations before strong damping sets in.
We believe that the onset of shock formation is determined mostly by the stiffness of the EOS in the density range of the outer layers of the star, since this is the region where the shocks form. However, this requires further investigation.
In [Figure \[fig\_tau\_eos\]]{}, we plot the damping timescale for the different nonrotating models. At a given damping timescale, the possible oscillation amplitudes differ by roughly one order of magnitude. Further, the damping timescale at the highest possible amplitudes (before material starts leaving the computational domain) decreases with increasing stiffness, i.e. the damping is faster.
![\[fig\_vm\_eos\] Onset of nonlinearity for the [$F$]{}and [${}^2\!f_0$]{} modes of the nonrotating models MB100, MC100, and MA100. Plotted is the final amplitude $A_F$ versus the initial one $A_I$. The solid gray lines are values corrected for the numerical damping. The [$F$]{} mode results are extracted from one-dimensional simulations with a resolution of 200 points per stellar radius, the [${}^2\!f_0$]{} modes are computed with two-dimensional simulations at resolution of $100^2$. ](figure9){width="\columnwidth"}
![\[fig\_tau\_eos\] Initial damping timescale $\tau$ versus initial amplitude $A_I$, for the same simulations shown in [Figure \[fig\_vm\_eos\]]{}. The solid gray lines are values corrected for the numerical damping. We only show amplitudes for which the difference is less than 50%. For slower damping, the error of $\tau$ blows up quickly. ](figure10){width="\columnwidth"}
Influence of rotation on damping {#sec_damp_rot}
--------------------------------
To investigate the influence of rotation on axisymmetric modes, we first compare the models MC100, MC95, and MC85, which have same mass and EOS, but different rotation rates up to 90% of the Kepler limit. The amount of damping for the [$F$]{}and [${}^2\!f_0$]{} modes is shown in [Figure \[fig\_vm\_rot\]]{}, and the damping timescales in [Figure \[fig\_tau\_rot\]]{}. For the stiff EOS A, we compared nonrotating model MA100 with the rapidly rotating model MA65. Note however that MA65 is also 18% heavier. The results are shown in [Figure \[fig\_vm\_rot\_eosa\]]{}.
The differences between models MC100 and MC95 are very small. For MC85 on the other hand, the nonlinear damping is significantly faster and stronger in comparison. The onset of nonlinearity occurs at slightly lower amplitudes for the [$F$]{} mode, while for the [${}^2\!f_0$]{} modes we found no significant difference. Since the fluid is bound less strongly with increasing centrifugal force, an increase of nonlinearity at given amplitude is to be expected.
An extreme case is given by model MC65, which is rotating almost at breakup velocity. For this case nonlinear effects start much earlier than for MC85. For the [${}^2\!f_0$]{} mode, nonlinear effects are definitely present for $A_I > 10^{-4}$. At an amplitude as low as $A_I=1.5{\ensuremath{\cdot 10^{-3}}}$ a fraction of $2{\ensuremath{\cdot 10^{-5}}}$ of the total mass leaves the computational domain. We obviously observe what is called mass shedding induced damping, i.e. even small oscillations liberate mass near the equator. The same mechanism has already been observed in [@SAF2004; @DSF2006] for models with EOS B. Since stars rotating so close to the Kepler limit are unlikely to occur in nature, we did not investigate this case further.
![\[fig\_vm\_rot\] Influence of rotation on nonlinear damping of axisymmetric modes. Shown are final versus initial amplitude for a perturbation with the [$F$]{} mode (TOP) and the [${}^2\!f_0$]{} mode (BOTTOM) of models MC100, MC95, MC85. The evolution time was $20\usk\milli\second$ for the [${}^2\!f_0$]{} mode and $15\usk\milli\second$ for the [$F$]{} mode. ](figure11a "fig:"){width="\columnwidth"} ![\[fig\_vm\_rot\] Influence of rotation on nonlinear damping of axisymmetric modes. Shown are final versus initial amplitude for a perturbation with the [$F$]{} mode (TOP) and the [${}^2\!f_0$]{} mode (BOTTOM) of models MC100, MC95, MC85. The evolution time was $20\usk\milli\second$ for the [${}^2\!f_0$]{} mode and $15\usk\milli\second$ for the [$F$]{} mode. ](figure11b "fig:"){width="\columnwidth"}
![\[fig\_tau\_rot\] Influence of rotation on damping timescale $\tau$ of [$F$]{} modes (TOP) and [${}^2\!f_0$]{} modes (BOTTOM). The values are already corrected for the numerical damping. Only points are shown for which the correction is less than 50%. ](figure12a "fig:"){width="\columnwidth"} ![\[fig\_tau\_rot\] Influence of rotation on damping timescale $\tau$ of [$F$]{} modes (TOP) and [${}^2\!f_0$]{} modes (BOTTOM). The values are already corrected for the numerical damping. Only points are shown for which the correction is less than 50%. ](figure12b "fig:"){width="\columnwidth"}
![\[fig\_vm\_rot\_eosa\] Nonlinear damping of [$F$]{} mode (TOP) and [${}^2\!f_0$]{} mode (BOTTOM) for nonrotating model MA100 and rapidly rotating model MA65. Evolution time was $4\usk\milli\second$. ](figure13a "fig:"){width="\columnwidth"} ![\[fig\_vm\_rot\_eosa\] Nonlinear damping of [$F$]{} mode (TOP) and [${}^2\!f_0$]{} mode (BOTTOM) for nonrotating model MA100 and rapidly rotating model MA65. Evolution time was $4\usk\milli\second$. ](figure13b "fig:"){width="\columnwidth"}
Damping of nonaxisymmetric modes {#sec_damp_m2}
--------------------------------
To study the evolution of nonaxisymmetric modes, we performed three-dimensional simulations of the counter-rotating [${}^2\!f_{2}$]{} mode of the rotating models MA65, MA100, MB70, and MC85. For model MA65, we also studied the corotating [${}^2\!f_{-2}$]{} mode.
In contrast to the axisymmetric case, we did not observe any steep decrease of the conserved energy $E_c$ that would point to formation of strong shocks. We did however observe a continuous decrease of the mean velocity and the $l=m=2$ multipole moment. The dissipation of energy was considerably slower than the one observed in case of shock formation in axisymmetric oscillations, even for high perturbation amplitudes.
Figures \[fig\_tau\_jsft3\_fl2m2\]–\[fig\_sat\_rot\_m2\] show the damping timescale. For models MA65 and MB70, it proved difficult to disentangle physical and numerical damping, which are of comparable strength at the resolutions we could afford.
To estimate the error of the damping, we assume that the timescale of the numerical damping in the nonlinear regime is equal to the one observed in the linear regime. For the axisymmetric case described in [Sec. \[sec\_damp\_axi\]]{}, this estimate agreed well with the convergence test results. For axisymmetric setups, we also proved that the code is as accurate in 3D as in 2D. As a further check, we compared resolutions of 50 and 75 points per stellar radius. The results are shown in [Figure \[fig\_tau\_jsft3\_fl2m2\]]{} to \[fig\_tau\_jega65\_fl2m2\].
To identify the damping mechanism, we produced animations showing the density and velocity evolution in the equatorial plane as well as the meridional planes. For high amplitudes, we found that the wave-patterns traveling around the star become more and more nonsinusoidal towards the surface, where we observe effects similar to wave breaking in ocean waves. A snapshot is shown in [Figure \[fig\_wave\_jbu6\]]{} and \[fig\_wave\_jega65\]. We do not find any formation of shocks in the interior of the star, only at the surface in regions with densities around $10^{-4}\dots 10^{-3}$ times the central one. Note this also justifies the approximation of adiabatic evolution implied by the use of a polytropic EOS. For low amplitudes, the wave-patterns looked exactly like the eigenfunctions we used to excite the oscillation. One important dissipative mechanism at high amplitudes thus seems to be wave breaking at the surface. As will be shown in the next section, the [${}^2\!f_{2}$]{} modes can also participate in mode coupling.
Since wave breaking is very difficult to resolve numerically, our results are probably not as accurate as the axisymmetric ones. More important, the surface of realistic neutron stars does not consist of a boundary between a cold fluid and vacuum, like in our models; there is either a solid crust or a hot envelope, and also a magnetic field. To compute the damping of the nonaxisymmetric modes correctly, it will be necessary to add a more detailed description of the surface to the numerical models.
Taking into account the strong deformation of the star, one also has to question the accuracy of the Cowling approximation. On the other hand, the gravitational field is mainly determined by the denser regions of the star, for which we observe much smaller deformations. We therefore assume that the error on the bulk motion of the star is comparable to the linear regime. This should be verified by means of a fully relativistic study.
![\[fig\_tau\_jsft3\_fl2m2\] Timescale $\tau$ of initial damping for the [${}^2\!f_{2}$]{} mode of model MC85. Shown are numerical results obtained at resolutions 50 and 75 points per stellar equatorial radius. The error bars are our estimate for the influence of numerical damping. ](figure14){width="\columnwidth"}
![\[fig\_tau\_jbu6\_fl2m2\] Like [Figure \[fig\_tau\_jsft3\_fl2m2\]]{}, but for the [${}^2\!f_{2}$]{} mode of model MB70. Error bars leaving the plot are of infinite size, i.e. the numerical results are compatible with no physical damping. ](figure15){width="\columnwidth"}
![\[fig\_tau\_jega65\_fl2m2\] Like [Figure \[fig\_tau\_jsft3\_fl2m2\]]{}, but for the [${}^2\!f_{2}$]{} mode of model MA65. ](figure16){width="\columnwidth"}
![\[fig\_sat\_rot\_m2\] Nonlinear damping of nonaxisymmetric modes for nonrotating model MA100 and rapidly rotating model MA65. ](figure17){width="\columnwidth"}
![\[fig\_wave\_jbu6\] Snapshot of the evolution of density and velocity in the equatorial plane, for an [${}^2\!f_{2}$]{}-mode perturbation of model MB70 with amplitude $A_I=0.04$. Shown are the lines of constant density ${\rho}$, spaced at regular intervals of $\sqrt{{\rho}}$. The outermost line corresponds to a density of $10^{-4}$ times the central density. The arrows correspond to the velocity in the corotating frame. The wave-patterns are rotating clockwise, leaving behind low density regions which fall back. Note those regions are not well resolved numerically. ](figure18){width="\columnwidth"}
![\[fig\_wave\_jega65\] TOP: Like [Figure \[fig\_wave\_jbu6\]]{}, but for the [${}^2\!f_{2}$]{} mode of model MA65. BOTTOM: Same data, but only densities above $10^{-2}$ times the central one are shown, and the arrows correspond to the momentum density instead of velocity. ](figure19a "fig:"){width="\columnwidth"} ![\[fig\_wave\_jega65\] TOP: Like [Figure \[fig\_wave\_jbu6\]]{}, but for the [${}^2\!f_{2}$]{} mode of model MA65. BOTTOM: Same data, but only densities above $10^{-2}$ times the central one are shown, and the arrows correspond to the momentum density instead of velocity. ](figure19b "fig:"){width="\columnwidth"}
Mode coupling {#sec_mode_coupling}
-------------
For the [${}^2\!f_{2}$]{} modes of models MA65 and MB70, we found that beside wave breaking, mode coupling has a significant influence on the damping.
In the spectrogram shown in [Figure \[fig\_spg\_jega65\]]{}, one can easily spot mode coupling between the [${}^2\!f_{2}$]{} mode and some unidentified mode, called mode X in the following. For low amplitudes, only the [${}^2\!f_{2}$]{} mode is excited. With increasing amplitude, mode X is growing faster. At the highest amplitude, it becomes dominant in the $v^\phi$-spectrum on a timescale as low as $4\usk\milli\second$.
The Fourier spectrum for amplitude $A_I=0.04\usk c$ is shown in [Figure \[fig\_spec\_jega65\]]{}. Mode X is located at $1458 \pm 50\usk\hertz$, the [$F$]{} mode at $3997\pm 50\usk\hertz$, the [${}^2\!f_0$]{} mode at $2590\pm 50\usk\hertz$, and the [${}^2\!f_{2}$]{} mode at $2912\pm 50\usk\hertz$.
From this, we find two interesting resonances for the unknown mode. First, mode X is located at half the frequency of the [${}^2\!f_{2}$]{} mode, measured in the corotating frame. Second, the frequency also equals (up to the resolution of the Fourier spectrum) the difference between the [$F$]{}and the [${}^2\!f_0$]{} mode. The latter could be interpreted as a hint that the unidentified peak does not belong to a proper oscillation mode, but is a nonlinear harmonic instead, i.e. a typical feature of Fourier transforms in case of nonharmonic waveforms and nonlinear superpositions. However, the [$F$]{} mode is not present in the spectrum of $v^\phi$, and the [${}^2\!f_0$]{} mode is neither present in the spectrum of $v^\phi$ nor of $v^r$, while the unknown peak is present in the spectra of $v^\phi$ and $v^r$. This supports the interpretation that the peak belongs to an actual oscillation mode, because the amplitude of a nonlinear harmonic in the Fourier spectrum of some quantity scales with the product of the amplitudes (in the same spectrum) of the modes producing the harmonic. However, it is still possible that the unidentified peak in the spectrum of the multipole $q_{20}$ is a nonlinear harmonic of the [$F$]{}and [${}^2\!f_0$]{} modes, and thus unrelated to mode X.
Since the frequency of the unidentified mode is in the inertial mode range, and well below the frequencies of the lowest order axisymmetric pressure modes, it is most probably an inertial mode. In order to identify the modes appearing for models MB70 and MA65, we first attempted mode recycling with axisymmetric simulations. However, the method did not converge. This does not necessarily mean that the mode is nonaxisymmetric. As described in [@Kastaun08], extracting inertial modes using mode recycling is more difficult than for pressure modes. Nevertheless, several low-order inertial modes of model MB70 were identified in [@Kastaun08]; none of them matches the frequency of the unknown mode. Since extracting inertial modes typically requires more recycling steps than pressure modes (due to the dense spectrum), and because the frequencies are lower, extracting nonaxisymmetric inertial modes is computationally expensive. Therefore, we only performed the first step, using an [${}^2\!f_{2}$]{}-mode perturbation of model MA65 at amplitude $A_I=0.04$, for which we know that the unknown mode is strongly exited. The resulting three-dimensional numerical eigenfunction had predominantly a $|m|=2$ $\phi$-dependency. We tried extracting a two-dimensional eigenfunction assuming either $m=+2$ or $m=-2$, but obtained a big phase error ($q\approx 0.5$), which typically points to a superposition of different modes of almost the same frequency. In our case, it seems that the unidentified peak actually consists of two high order inertial modes with $m=\pm 2$, although a clear identification will require additional mode recycling steps.
![\[fig\_spg\_jega65\] Mode coupling at different amplitudes of the [${}^2\!f_{2}$]{} mode of model MA65 to an unknown mode at half the frequency. Shown is time evolution of the Fourier spectrum of the $\phi$-velocity in the corotating frame, at $r=\frac{1}{2}R$ on the space diagonal. The spectra were computed using a running window of width $4\usk\milli\second$. The perturbation amplitudes $A_I$ are 0.01 (upper left), 0.02 (upper right), 0.04 (lower left), and 0.08 (lower right). The frequency resolution is $250\usk\hertz$. ](figure20){width="\columnwidth"}
![\[fig\_spec\_jega65\] Fourier spectra of $v^\phi$, $v^r$, and $q_{20}$, in arbitrary units, for a [${}^2\!f_{2}$]{}-mode oscillation of model MA65, at amplitude $A_I=0.04$. All frequencies are given in the corotating frame. ](figure21){width="\columnwidth"}
We note that the mode coupling is significantly stronger for resolution 75 than for resolution 50, which means the process is not well resolved numerically. In the aforementioned low-quality numerical eigenfunctions, we found structures smaller than one fifth of the stellar radius. Therefore, the excited inertial modes probably suffer from strong numerical damping, which suppresses the coupling at resolution $N=50$. Further, the time evolution in presence of mode coupling effects typically depends strongly on the initial amplitudes of both modes, even if one amplitude is small. In our case, the unknown mode is probably seeded only by numerical errors; in real stars, the situation might be different. It is hence possible that the [${}^2\!f_{2}$]{} mode decays even faster in reality than in our simulations.
Beside the inertial modes, the [${}^2\!f_{2}$]{} mode seems to couple to the [${}^2\!f_{-2}$]{} mode as well, albeit less dramatic. This coupling will be discussed in [Sec. \[sec\_gw\_lum\]]{}, since it interferes with the estimation of the GW strain. It is still unclear what the exact nature of the observed couplings is, and whether the mode coupling picture is adequate at such high amplitudes at all.
To measure the decay in presence of mode coupling, $\bar{v}$ is not useful because it is quite insensitive to transfer of energy between modes. However, we can use the dominant multipole moment of the mode, which for the [${}^2\!f_{2}$]{} modes is the $l=m=2$ mass multipole. In the Fourier spectra of $q_{22}$, the dominant contribution is due to the [${}^2\!f_{2}$]{} mode. By fitting a function $q^0 \exp(-t/\tau_m + i \omega t)$ to the initial evolution of the multipole moment $q_{22}$, we obtain another damping timescale $\tau_m$.
While $\tau$ is a good measure for the overall energy dissipation, $\tau_m$ is a measure for the decay of the [${}^2\!f_{2}$]{} mode. [Figure \[fig\_tau\_multi\]]{} shows a comparison between the two. As one can see, mode coupling is even more important for the decay of the [${}^2\!f_{2}$]{} mode than dissipative effects, e.g. wave breaking.
At least this is the case for models MA65 and MB70; for the [${}^2\!f_{2}$]{} mode of model MA100 and the [${}^2\!f_{-2}$]{} mode of model MA65, we do not observe mode coupling, while for model MC85 we observe weak coupling to a mode at one quarter of the (corotating) frequency of the [${}^2\!f_{2}$]{} mode. What is the difference? In model MB70 and MA65, the coupled mode is an inertial mode which oscillates at $\frac{1}{2} f_c$, with $f_c$ being the frequency of the [${}^2\!f_{2}$]{} mode in the corotating frame. Since the inertial mode range is bounded by $2 F_R$ (see [@Kastaun08]), where $F_R$ is the rotation rate of the star, this is only possible if $$\begin{aligned}
4 F_R > f_c .\end{aligned}$$ Model MC85 (and MA100 anyway) rotates too slowly to satisfy this condition. The 1:4 resonance to an inertial mode on the other hand is possible for model MC85.
Note the above condition automatically holds for CFS-unstable [${}^2\!f_{2}$]{} modes, for which $f_c<2F_R$. It seems plausible that the coupling is also present at lower amplitudes, but evolves at longer timescales. To compute this timescale and thus a more stringent limit on the saturation amplitude of CFS-unstable [${}^2\!f_{2}$]{} modes, perturbative studies are required.
The upper limits for the onset of nonlinearity and the corresponding damping timescales for the nonaxisymmetric modes are summarized in [Tab. \[tab\_onset\_nl\_m2\]]{}. The mode coupling effects for models MA65 and MB70 are taken into account. Since the CFS instability of the [${}^2\!f_{2}$]{} mode of model MA65 has a growth time in the order of seconds, our result yields an upper limit for the saturation amplitude.
![\[fig\_tau\_multi\] Decay timescales $\tau$ and $\tau_m$ of $\bar{v}$ and $q_{22}$ versus initial amplitude $A_I$, for models MB70 and MA65 perturbed with the [${}^2\!f_{2}$]{} mode. The results are obtained at resolution 75. The error bars estimate the influence of the numerical damping. ](figure22){width="\columnwidth"}
Model Mode $A_D / (10^{-3}\usk c)$ $\tau / \milli\second$
------- ---------------------------------------------------------------- ------------------------- ------------------------
MC85 [${}^2\!f_{2}$]{} 10 90
MB70 [${}^2\!f_{2}$]{} 35 9
MA100 ${\ensuremath{{}^2\!f_{2}}}\equiv {\ensuremath{{}^2\!f_{-2}}}$ 130 20
MA65 [${}^2\!f_{2}$]{} 18 20
MA65 [${}^2\!f_{-2}$]{} 30 40
: \[tab\_onset\_nl\_m2\] Upper limits on the onset of nonlinearity, for nonaxisymmetric oscillation modes. $A_D$ is the initial amplitude $A_I$ at which nonlinear damping (including mode coupling effects) can be safely distinguished from the numerical damping. $\tau$ is an upper limit for the decay timescale of the mode at amplitude $A_D$.
Gravitational luminosity {#sec_gw_lum}
------------------------
To estimate the GW production in our simulations, we use the multipole formalism described in [Sec. \[sec\_gwmultipole\]]{}. For each simulation, we plot the time evolution of the magnitude of the strain coefficients ${A^{E2}}_{lm}$ and ${A^{B2}}_{lm}$, as shown in [Figure \[fig\_ts\_gwstr\]]{}.
In general, the only significant contribution is due to the $l=2$ mass multipoles $q_{22}$ or $q_{20}$. Unless noted otherwise, one of the strain amplitudes ${A^{E2}}_{20}$ or ${A^{E2}}_{22}$ is strongly dominant, and will be denoted by $A$ in the following. For the axisymmetric case, terms with $m \neq 0$ vanish.
For oscillations in the linear regime, the strain is proportional to the amplitude. We introduce a normalized mode strain $\mathcal{A}$ and luminosity $\mathcal{L}$ by $$\begin{aligned}
A &= \frac{\bar{v}}{10^{-3} c} \mathcal{A}, &
L &= \left(\frac{\bar{v}}{10^{-3} c}\right)^2 \mathcal{L} .\end{aligned}$$ From the simulations with the lowest amplitudes, which are in the linear regime, we extracted $\mathcal{A}$ and $\mathcal{L}$ for each mode. The results are given in [Tab. \[tab\_mode\_strain\]]{}.
The strain produced by the [${}^2\!f_{2}$]{} mode of model MA65 is the lowest one of all modes. This is mainly due to the low frequency in the inertial frame. The corotating [${}^2\!f_{-2}$]{} mode produces a strain larger by a factor of 143 at the same amplitude; the $\omega^2$ frequency dependency accounts for a factor of 77.
Since model MA65 is already close to the Kepler limit, the strain cannot be increased significantly by uniformly spinning up the star. It might however be possible to construct differentially rotating models for which the CFS-unstable [${}^2\!f_{2}$]{} modes have higher frequencies.
The strain of the [${}^2\!f_{2}$]{} mode of model MC85 is surprisingly high given that it’s frequency is even lower than for model MA65. On the other hand, the radius of model MC85 is $5.1$ times the one of MA65, which enhances the $l=2$ multipole moments by a factor 25. We thus consider the high strain amplitude of model MC85 as a consequence of the unrealistically large radius.
Another noteworthy result is that the strain amplitudes of [$F$]{} modes of rapidly rotating stars are comparable to those of the [${}^2\!f_0$]{} modes. As discussed in [Sec. \[sec\_gwmultipole\]]{}, cancellations in the integrands of the multipoles prevented us from computing a meaningful error estimate for the strain produced by the [$F$]{} modes. We nevertheless believe that at least for model MA65, the high strain amplitude of the [$F$]{} mode is not just an artifact due to the use of the quadrupole formula, but a genuine effect of the rapid rotation rate. However, this can only be verified by means of a fully relativistic study, using direct wave extraction methods. The question is relevant for GW astronomy because strong [$F$]{}-mode oscillations are likely to occur in a number of astrophysical scenarios.
![\[fig\_ts\_gwstr\] Evolution of GW strain amplitudes for model MA65 perturbed with the corotating [${}^2\!f_{-2}$]{} mode at amplitude $A_I=0.01\usk c$. Shown are only the 5 biggest contributions. ](figure23){width="\columnwidth"}
Model Mode $\mathcal{A} / 10\usk\mega{\ensuremath{\mathrm{pc}}}$ $\mathcal{L} / \watt$ q
------- -------------------- ------------------------------------------------------- ------------------------- -----
MA65 [$F$]{} 2.96[$\cdot 10^{-25}$]{} 1.06[$\cdot 10^{40}$]{} ?
MA65 [${}^2\!f_0$]{} 1.24[$\cdot 10^{-24}$]{} 8.28[$\cdot 10^{40}$]{} 13
MA65 [${}^2\!f_{2}$]{} 4.09[$\cdot 10^{-26}$]{} 1.37[$\cdot 10^{37}$]{} 13
MA65 [${}^2\!f_{-2}$]{} 5.84[$\cdot 10^{-24}$]{} 2.13[$\cdot 10^{43}$]{} 13
MA100 [${}^2\!f_0$]{} 7.18[$\cdot 10^{-24}$]{} 3.59[$\cdot 10^{42}$]{} 2
MA100 [${}^2\!f_{2}$]{} 4.96[$\cdot 10^{-24}$]{} 6.85[$\cdot 10^{42}$]{} 2
MB100 [${}^2\!f_0$]{} 7.79[$\cdot 10^{-24}$]{} 1.63[$\cdot 10^{42}$]{} 1.5
MB70 [${}^2\!f_{2}$]{} 1.10[$\cdot 10^{-25}$]{} 4.92[$\cdot 10^{37}$]{} 4
MC100 [${}^2\!f_0$]{} 5.28[$\cdot 10^{-24}$]{} 3.67[$\cdot 10^{40}$]{} 1.2
MC100 [${}^2\!f_{2}$]{} 3.74[$\cdot 10^{-24}$]{} 7.39[$\cdot 10^{40}$]{} 1.2
MC95 [$F$]{} 6.11[$\cdot 10^{-25}$]{} 5.38[$\cdot 10^{38}$]{} ?
MC95 [${}^2\!f_0$]{} 4.08[$\cdot 10^{-24}$]{} 1.93[$\cdot 10^{40}$]{} 1.5
MC85 [$F$]{} 1.79[$\cdot 10^{-24}$]{} 3.54[$\cdot 10^{39}$]{} ?
MC85 [${}^2\!f_0$]{} 3.41[$\cdot 10^{-24}$]{} 9.95[$\cdot 10^{39}$]{} 1.4
MC85 [${}^2\!f_{2}$]{} 8.65[$\cdot 10^{-25}$]{} 9.44[$\cdot 10^{38}$]{} 1.4
: \[tab\_mode\_strain\] Gravitational luminosity and strain produced by oscillation modes in the linear regime, normalized to an amplitude $\bar{v}=10^{-3}$. The values are computed from a fit to the multipole moments. The error bounds due to the use of the multipole formalism (see [Sec. \[sec\_gwmultipole\]]{}) are $\mathcal{A}q^{-1} \dots \mathcal{A}q$ for the strain, and $\mathcal{L} q^{-2} \dots \mathcal{L} q ^2$ for the luminosity. For the [$F$]{} modes, the error is unknown; the values are given only for completeness.
An overview over the maximum GW strain observed in the nonlinear simulations of axisymmetric modes can be found in [Figure \[fig\_gwstr\_axi\]]{}. As one can see, the maximum strain roughly scales with the amplitude; by simply extrapolating from the linear regime, one obtains an estimate which is correct up to a factor of 2 even for the highest amplitudes.
![\[fig\_gwstr\_axi\] Maximum gravitational wave amplitude $A=|{A^{E2}}_{20}|$ versus initial amplitude $A_I$, for axisymmetric perturbations. The contribution of other multipole moments to the strain are negligible in all cases. The dotted lines are an extrapolation from the lowest amplitude, assuming linearity. ](figure24){width="\columnwidth"}
For each simulation, we computed the Fourier spectra of the strain to determine the frequency of the dominant contribution. Usually, the only significant contribution is due to the mode used for perturbation. For the [${}^2\!f_{2}$]{} mode of MA65 on the other hand, we find that the GW strain is mainly due to the corotating [${}^2\!f_{-2}$]{} mode, as well as the axisymmetric [$F$]{}and [${}^2\!f_0$]{} modes. Those modes have a lower amplitude than the [${}^2\!f_{2}$]{} mode. But since the [${}^2\!f_{2}$]{} mode of this model is a relatively inefficient GW emitter, due to the low oscillation frequency, the other modes dominate the GW strain.
To understand the strain of the [${}^2\!f_{2}$]{} mode, we thus need to understand why the other modes are present. [Figure \[fig\_spg3\_jega65\_fl2m2\]]{} shows a spectrogram of the corresponding strain amplitudes. By further comparing the absolute amplitudes of the multipole moments $q_{22}$ and $q_{20}$, we find the following picture:
The [${}^2\!f_{-2}$]{} mode is excited due to a small contamination of the numerical eigenfunction; it’s initial amplitude is around 4% of the [${}^2\!f_{2}$]{} mode in the linear regime. For high amplitudes however, it grows up to 20 % during the evolution. The reason is unclear, it might be another mode coupling effect.
The [$F$]{}- and [${}^2\!f_0$]{} modes are present from the beginning as well, but in the linear regime their amplitude becomes negligible compared to the amplitude of the [${}^2\!f_{2}$]{} mode. For a high initial amplitude $A_I=0.04\usk c$, the amplitude of $q_{20}$ is still less than 20% of $q_{22}$. We explain the presence of the [$F$]{}and [${}^2\!f_0$]{} modes by the ad hoc way in which we scale the eigenfunctions to high amplitudes; it would be surprising if only one mode was excited. The relative amplitudes could be totally different for the case of a [${}^2\!f_{2}$]{} mode slowly grown to high amplitudes by the CFS mechanism.
To quantify the strain produced by the [${}^2\!f_{2}$]{} mode alone, we fit a function $q_0 \exp(-t/\tau_m + i \omega t)$ to the $q_{22}$ multipole and then compute the strain from the multipole moment given by the fit. This way, we effectively filter out contributions of frequencies other than the one of the [${}^2\!f_{2}$]{} mode. Note all the strains in the linear regime, given in [Tab. \[tab\_mode\_strain\]]{}, have been computed in the same way (for the axisymmetric modes, we use a damped sinusoidal since $q_{20}$ is real-valued).
The results for different amplitudes are shown in [Figure \[fig\_strain\_jega65\_fl2m2\]]{}. Only 3–24% of the total strain is due to the main [${}^2\!f_{2}$]{} mode. We stress that the total strain found for our setups is probably higher than the strain of a [${}^2\!f_{2}$]{}-oscillation grown to high amplitude by the CFS-mechanism. To accurately compute the latter, one needs to model carefully the growth of secondary coupled modes, even if they are dynamically unimportant.
![\[fig\_spg3\_jega65\_fl2m2\] Spectrogram of the gravitational wave amplitude, for the [${}^2\!f_{2}$]{}-mode oscillation of model MA65, at perturbation amplitudes $A_I=0.01$ (TOP) and $A_I=0.04$ (BOTTOM). Shown is the Fourier spectrum of ${A^{E2}}_{20}$ (LEFT) and $\Re({A^{E2}}_{22})$ (RIGHT) inside a running time window of width $4\usk\milli\second$, versus the time at the center of the window. ](figure25){width="\columnwidth"}
![\[fig\_strain\_jega65\_fl2m2\] Initial GW amplitude $A$ versus initial oscillation amplitude $A_I$, for model MA65 perturbed with the counter-rotating [${}^2\!f_{2}$]{} mode. Shown is the total strain as well as the strain due to the [${}^2\!f_{2}$]{} mode alone (see main text). The dotted lines are extrapolations from the linear regime. ](figure26){width="\columnwidth"}
For the [${}^2\!f_{2}$]{} mode of model MB70, we qualitatively find the same picture as for model MA65. For the [${}^2\!f_{2}$]{} mode of models MC85 and MA100, as well as the [${}^2\!f_{-2}$]{} mode of MA65, contribution of other modes to the GW strain are small. The strain at high amplitudes is shown in [Figure \[fig\_gwstr\_m2\]]{}. Again, the extrapolation from the linear regime yields a rough estimate, correct up to a factor of 3.
![\[fig\_gwstr\_m2\] Maximum gravitational wave amplitude $A=|{A^{E2}}_{22}|$ versus initial amplitude $A_I$, for $m=2$ perturbations. For the [${}^2\!f_{2}$]{} modes of models MB70 and MA65, only the contribution of the [${}^2\!f_{2}$]{} mode itself is plotted (see main text). The dotted lines are an extrapolation from the lowest amplitude, assuming linearity. ](figure27){width="\columnwidth"}
Gravitational wave detectability {#sec_detect}
--------------------------------
In the following, we estimate the prospects of detecting GW in the case of either a one-time strong excitation or a continuous excitation at lower amplitudes.
First we investigate the continuous oscillation of one mode at saturation amplitude of some instability. Our primary interest here is the CFS instability of the [${}^2\!f_{2}$]{} mode of model MA65. This instability has a growth time of seconds, which is longer than the timescale of the numerical damping. Therefore, we can only give upper limits on the saturation amplitude, namely the amplitude at which we definitely see nonlinear damping. The actual saturation amplitude may be much smaller.
[Figure \[fig\_detect\_onset\]]{} shows the gravitational strain at the observed onset of nonlinear damping for several oscillation modes. We plot the $h_{\theta\theta}$ component given by [Eq. (\[eq\_gwstrain\])]{}, assuming an optimal viewing angle. The general expression for the spin tensor harmonics can be found in [@Mathews62] (note the standard notation of magnetic and electric type radiation is reversed there).
The detectability of sinusoidal GW signals depends not only on the strain amplitude, but also on the integration time. Effectively, the detectable amplitude scales with $\sqrt{N}$, where $N$ is the number of available wave cycles. For a continuous signal, this only depends on the detector and the computational resources for data analysis. If we assume at least a few hundred cycles, the [${}^2\!f_{2}$]{} mode at the onset of strong damping is detectable with advanced LIGO at a distance of 10[$\mathrm{pc}$]{}. Under the assumption that the saturation amplitude of the CFS instability of the [${}^2\!f_{2}$]{} mode is only limited by the strong damping effects investigated in this work, it would be possible to detect sources as far away as the Virgo galaxy cluster. To compute the actual saturation amplitude however, perturbative studies at lower amplitudes are needed.
The numbers above are valid for model MA65, which is an extreme case. Models rotating slightly slower will produce considerably lower strain. The reason is that the strain scales with the square of the mode frequency in the inertial frame, which for the [${}^2\!f_{2}$]{} modes crosses zero when the rotation rate falls below the neutral point. Keeping mass, EOS, and mode amplitude $\bar{v}$ fixed, we can make an expansion of the strain amplitude $A$ in terms of rotation rate $F_R$ near the neutral point: $$\begin{aligned}
A &= A_0 \left(\frac{F_R-F_R^N}{F_R^0-F_R^N}\right)^2 ,\end{aligned}$$ where $A_0$, $F_R^0$ are the strain amplitude and rotation rate of MA65, and $F_R^N$ is the rotation rate at the neutral point. Further, it is safe to assume that the amplitude marking the onset of nonlinearity behaves smooth near the neutral point, since the rotation rate influences the dissipation effects mainly via the centrifugal force, and the inertial modes participating in mode coupling depend on the Coriolis force; both forces depend on the absolute rotation rate.
For frequencies in the range 0.1–2 times the [${}^2\!f_{2}$]{}-mode frequency of model MA65, the sensitivity curve of advanced LIGO is quite flat. In this range, the *upper limit* we find for the distance at which we can observe the [${}^2\!f_{2}$]{} mode roughly scales with $(F_R-F_R^N)^{2}$.
Since model MA65 is close to the Kepler limit, the strain cannot be increased significantly by increasing the rotation rate; on the contrary, we expect that nonlinear effects set in much earlier when approaching the Kepler limit further, as observed for model MC65. Further, we doubt that one can construct models where the frequency of the CFS-unstable [${}^2\!f_{2}$]{} modes is more than three times higher without resorting to extremely unrealistic EOSs. On the other hand, this statement only applies to uniformly rotating stars; it might be fruitful to investigate differentially rotating models as well.
Another restriction for detection of continuous signals is currently given by the limited computing resources for data analysis, which prevents to search the whole parameter space for signals. In particular, searches [@LIGO09; @LIGO09b] for continuous signals in early data from the 5th science run of LIGO assumed a maximum time derivative of the signal frequency. Those searches where targeted to signals from spinning nonaxisymmetric neutron stars, based on estimates on the maximum sustainable ellipticity. The luminosity produced by such sources is orders of magnitude below the upper limit we computed for the CFS-unstable [${}^2\!f_{2}$]{} mode of model MA65. Accordingly, our models spin down much faster. Assuming that the mode frequency $f_c$ in the corotating frame stays constant, the decrease of the signal frequency, i.e. the mode frequency in the inertial frame, is $\dot{f}_i = m \dot{F}_R$, provided that $f_c<m F_R$. The spin-down rate can be computed from the expressions for the angular momentum lost by gravitational waves given in [@Thorne80] and the moment of inertia computed by the initial data code. For the [${}^2\!f_{2}$]{} mode of model MA65 at the upper limit, we compute a signal frequency decrease of $\dot{f} = -5 \cdot 10^{-3} \usk\hertz\per\second$, which is orders of magnitude outside the ranges searched in [@LIGO09; @LIGO09b]. For models closer to the neutral point, luminosity and spin-down rate would be smaller. As a rough estimate, we assume that only the frequencies change, while maximum amplitude and oscillation pattern stay constant. The spin-down rate then scales with $f_i^5$. It is still above the maximum for the whole frequency range $50\dots 1100\usk\hertz$ investigated in [@LIGO09; @LIGO09b]. Those searches are clearly only sensitive to neutron star oscillations with amplitudes well below the onset of strong damping effects investigated here.
![\[fig\_detect\_onset\] Gravitational wave strain $h_{\theta\theta}$ and mode frequency, for [${}^2\!f_{2}$]{}and [${}^2\!f_{-2}$]{} modes at amplitudes where we are confident to observe nonlinear damping. The strain is given for a source at distance $10\usk\mega{\ensuremath{\mathrm{pc}}}$, assuming optimal viewing angle. The lines are estimates for the error due to the Cowling approximation, assuming that only the frequency changes. The error bars correspond to the error due to the use of the quadrupole formula. Other errors are negligible. For comparison, we plot the sensitivity curves of advanced LIGO and the proposed Einstein Telescope (ET). ](figure28){width="\columnwidth"}
For modes other than the [${}^2\!f_{2}$]{} mode of model MA65, we are not aware of any instability. Nevertheless, we provide limits for a hypothetical instability. [Figure \[fig\_detect\_onset\]]{} and \[fig\_detect\_onset\_axi\] shows upper limits for the strain amplitudes assuming that the instability acts on timescales slower than the numerical damping. [Figure \[fig\_detect\_20pf\]]{} gives the strain at saturation amplitude for the optimistic assumption that the hypothetical instability has growth time $\tau_I=20\usk T_F$, where $T_F$ is the period of the [$F$]{} mode of the corresponding stellar model.
![\[fig\_detect\_onset\_axi\] Like [Figure \[fig\_detect\_onset\]]{}, but for axisymmetric modes. The errorbar of the [$F$]{} mode is only correct under the assumption that cancellation effects do not increase the error in comparison to nonradial modes; see [Sec. \[sec\_gwmultipole\]]{}. ](figure29){width="\columnwidth"}
![\[fig\_detect\_20pf\] Like [Figure \[fig\_detect\_onset\]]{}, but for amplitudes such that the damping timescale satisfies $\tau=20\usk T_F$, where $T_F$ is the oscillation period of the [$F$]{} mode of the corresponding model. ](figure30){width="\columnwidth"}
Besides slowly growing instabilities, oscillations can be excited by sudden violent events, e.g. glitches, magnetar giant flares, phase-change induced collapse ([@Ernazar09]), and mergers forming a metastable rapidly rotating star. It is beyond the scope of this article to estimate the amplitude of oscillations excited by such events. We investigate the detectability for a given amplitude, assuming no further excitation. In that case, the amplitude starts decaying due to strong nonlinear damping, until it reaches a threshold where only slow damping occurs.
In order to detect such an event, one could either search in short segments of the detector data for the initial strong signal, or search for the tail by investigating long time intervals. Since any long tail has an amplitude below the onset of strong damping, we already know the maximum strain amplitudes, which are given in [Figure \[fig\_detect\_onset\]]{} and \[fig\_detect\_onset\_axi\]. The big unknown is the number of available cycles, which depends on the residual damping timescale of the tail, and thus cannot be computed by nonlinear evolution.
To estimate the detectability of the initial peak, we neglect the tail, assume a pure exponential decay, and limit the integration time to $T_a=1\usk\second$. We define an effective strain amplitude by $$\begin{aligned}
\label{eq_def_eff_str}
A_\text{eff} &= A \sqrt{\frac{1-e^{-2T_a/\tau}}{1-e^{-2/(f_i\tau)}}} ,\end{aligned}$$ where $f_i$ is the mode frequency in the inertial frame, and $\tau$ is the initial damping timescale, corrected for the numerical damping. The enhancement factor in [Eq. (\[eq\_def\_eff\_str\])]{} was taken from [@Ernazar09]. For low amplitudes where we cannot disentangle numerical and physical damping, we set $\tau=1000\usk \second$.
The results are plotted in [Figure \[fig\_eff\_strain\]]{}. As one can see, the increase in amplitude after the onset of nonlinearity competes with the rapid decrease in signal duration, and in most cases the latter wins. The effect becomes more pronounced when increasing the integration time. When considering the tail again, we find that the detectability does not increase significantly for perturbation amplitudes above the onset of nonlinearity.
![\[fig\_eff\_strain\] Effective GW strain amplitude $A_\text{eff}$ defined by [Eq. (\[eq\_def\_eff\_str\])]{} versus perturbation amplitude $A_I$ for the nonaxisymmetric oscillations. ](figure31){width="\columnwidth"}
Summary and discussion {#sec_summary}
======================
In our study, we investigated strong nonlinear damping effects in high amplitude oscillations of various neutron star models, and the limits imposed on the gravitational wave signal. For this, we used a relativistic evolution code, but kept the spacetime fixed.
Our neutron star models are rigidly rotating ideal fluid configurations with polytropic EOSs of different stiffness, and typical neutron star masses in the range 1.4–1.9 solar masses. The oscillation modes we considered are the axisymmetric [$F$]{}and [${}^2\!f_0$]{} modes, the corotating [${}^2\!f_{-2}$]{} mode, and the counter-rotating [${}^2\!f_{2}$]{} mode.
Our main interest is the [${}^2\!f_{2}$]{} mode of one model (MA65), which can be excited by the CFS instability in full GR, and which is therefore a candidate for detectable gravitational waves from rapidly rotating neutron stars.
We found different damping mechanisms for axisymmetric and nonaxisymmetric modes. Axisymmetric oscillations of high initial amplitude are rapidly damped by formation of shocks in the outer layers of the star, until the amplitude falls below a certain threshold. Below the threshold we observed no further energy dissipation on the timescales of our evolutions.
Although we are using a one-parametric EOS, our results accurately determine the onset of shock formation, and provide a first estimate for the magnitude of energy dissipation by shocks. We find that the damping timescale decreases with increasing amplitude, down to a few oscillation cycles. The damping timescale and the threshold vary by one order of magnitude for the different EOSs we investigated. The stiffer the EOS, the higher amplitudes are possible before shocks form, and also the damping observed at the highest amplitudes is faster for stiffer EOSs.
With increasing rotation rate, nonlinear effects start at lower amplitudes. However, the influence of rotation on the damping of axisymmetric modes is relatively weak until close to the Kepler limit, where the damping threshold decreases to zero. At the Kepler limit, damping by mass shedding occurs. We explain this behavior by the fact that the material at the equator is bound less strongly for faster rotation.
For the nonaxisymmetric oscillations, damping is mainly caused by wave breaking and nonlinear mode coupling effects, while shock formation only occurs in low density regions at the stellar surface. As for axisymmetric modes, nonlinear damping sets in at lower amplitudes when increasing the rotation rate or decreasing the stiffness of the EOS.
Since wave breaking is a surface effect, we feel that an accurate description of the damping at high amplitudes requires an improved numerical treatment of the surface and probably a better physical model as well, considering e.g. a hot envelope for a proto-neutron star, a solid crust for a cold star (although in that case, one would also have to add superfluidity), and the influence of the magnetic field.
Significant mode coupling was only observed for rapidly rotating models, where the main mechanism seems to be a 1:2 resonance with high order $m=\pm 2$ inertial modes. For CFS-unstable models, such a resonance is always possible. Further, the coupling is most probably active already at lower amplitudes, but on timescales too long for nonlinear study. Since the growth time of CFS-unstable modes is longer as well, it will be necessary to investigate the coupling at lower amplitudes perturbatively to determine the saturation amplitudes. Nevertheless, we were able to provide upper limits.
Next, we computed the gravitational wave strain and luminosity of several modes in the linear regime. The CFS-unstable [${}^2\!f_{2}$]{} mode was found to be a weak emitter; at the same amplitude, the strain is thirty times lower than for the axisymmetric [${}^2\!f_0$]{} mode. This is mainly due to the low frequency of the [${}^2\!f_{2}$]{} mode in the inertial frame. To construct models for which this mode is more luminous, one needs to further separate the Kepler limit from the neutral point (where the [${}^2\!f_{2}$]{} mode has zero inertial frame frequency). This can either be done by increasing the stiffness of the EOS, or probably by introducing differential rotation. Since our CFS-unstable model already has a relatively stiff EOS ($\Gamma=2.46$), we consider the latter more realistic.
We also found that the quasiradial [$F$]{} modes of rapidly rotating models emit gravitational waves almost as efficient as the [${}^2\!f_0$]{} modes. For the CFS-unstable model MA65, the [$F$]{} mode emits stronger gravitational waves than the [${}^2\!f_{2}$]{} mode at the same amplitude. The reason is that the eigenfunction of the [$F$]{} mode has a considerable quadrupole moment due to the oblateness of the star, as well as the high oscillation frequency.
Further we investigated the gravitational waves produced by oscillations in the nonlinear regime. The maximum strain we observed usually agrees with the values extrapolated from the linear regime up to a factor of 3 (2 for the axisymmetric modes). For the CFS-unstable [${}^2\!f_{2}$]{} mode however, we found that the signal is dominated by secondary modes. We consider those as a feature of our particular setup; their amplitudes most likely depends on the exact way the main mode was excited. To compute the strain for the case of a [${}^2\!f_{2}$]{} mode grown to high amplitude by the CFS instability, it will be necessary to model carefully the growth of secondary modes, especially at higher frequencies. Secondary modes dominate the strain only for the CFS-unstable [${}^2\!f_{2}$]{} mode, because it is the weakest emitter. For models where it has a higher frequency, secondary modes are probably less important.
Finally, we combined the results on strong damping and luminosity to establish upper limits on the GW detectability of various modes. There are two astrophysically relevant cases: a continuous emission by CFS-unstable [${}^2\!f_{2}$]{} modes at saturation amplitude, and stable modes excited by some violent event.
From the upper limit for the saturation amplitude of the CFS instability, we obtain an upper limit on corresponding gravitational strain. The detectability of a continuous signal depends on the integration time. If we consider searches using time windows containing at least a few hundred cycles, the [${}^2\!f_{2}$]{} mode (of model MA65) oscillating at the upper limit will be detectable at distances around $10\usk\mega{\ensuremath{\mathrm{pc}}}$ with advanced LIGO. This distance scales approximately quadratically with the mode frequency in the inertial frame. Current searches for continuous signals in the 5th LIGO science run are not sensitive to the above source since they exclude the high spindown rates caused by gravitational waves of such strength. We stress again that the actual saturation amplitude may be considerably lower than the upper limits we provided. We also note that we did not take into account the secondary excited modes; a careful study of the couplings during the growth phase might reveal an enhanced detectability.
For stable modes excited by a sudden violent event, we find that the detectability does not significantly increase for amplitudes above the onset of strong damping, since the oscillation is quickly damped below the threshold and stays there. The detectability thus depends on the effective number of wave cycles in the tail, and thus on the residual slow damping at amplitudes near the threshold. If this information becomes available for a given mode, e.g. using perturbative techniques, the detectability can be computed from the luminosities we provided. However, we find that the [${}^2\!f_0$]{} modes oscillating near the strong damping threshold are detectable with advanced LIGO at least up to distances of $10 \usk\mega{\ensuremath{\mathrm{pc}}}$. The strong damping effects thus impose only weak limits on the detectability of those; for most processes, the important factor will probably be the strength of the excitation.
It is worth mentioning that although some of the questions we rise can only be computed with perturbation theory, only a nonlinear study like ours can provide the necessary limits for the amplitudes at which perturbation theory can be safely applied. Given that the physically most interesting amplitudes are at the transition from linear to the nonlinear regime, we feel that it is necessary to combine linear and nonlinear methods to obtain accurate results on the detectability of oscillation modes.
This work was supported by the Deutsche Forschungsgemeinschaft DFG (German Research Foundation) via SFB/TR7, B.W. received funding from Cusanuswerk. Computations have been carried out on the <span style="font-variant:small-caps;">hpc</span> cluster of the University of Tübingen and the <span style="font-variant:small-caps;">blade</span> cluster at SISSA.
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|
---
abstract: |
The Self-Consistent Field theory of brushes of neutral water-soluble polymers described by two-state models is formulated in terms of the effective Flory interaction parameter $\chi _{eff}(T,\phi)$ that depends on both temperature, $T$ and the monomer volume fraction, $\phi$. The concentration profiles, distribution of free ends and compression force profiles are obtained in the presence and in the absence of a vertical phase separation. A vertical phase separation within the layer leads to a distinctive compression force profile and a minimum in the plot of the moments of the concentration profile vs. the grafting density. The analysis is applied explicitly to the Karalstrom model. The relevance to brushes of Poly(N-isopropylacrylamide)(PNIPAM) is discussed.
[**Accepted for publication in the Journal of Chemical Physics**]{}
author:
- 'Vladimir A. Baulin, Ekaterina B. Zhulina,$^\dag$ Avi Halperin'
title: |
Self-Consistent Field Theory\
of Brushes of Neutral Water-Soluble Polymers
---
Introduction\[introduction\]
============================
A number of “two-state” models were proposed to rationalize the phase behavior of Poly(ethylene oxide) (PEO) and its solution thermodynamics.[Karlstrom,Matsuyama,n-Cluster1,n-Cluster2,Bekiranov,DormidPhase]{} Within these models the monomers are in dynamic equilibrium involving two interconverting states (Fig. \[fig1\]). The Flory-Huggins lattice and the mixing entropy of the chains are retained. Additional contributions are due to the mixing entropy of the different monomeric states and their interactions. While the two-state models were proposed for aqueous solutions of PEO they are also candidates for the description of other neutral water-soluble polymers such as Poly(N-vinylpyrrolidone) (PVP) and Poly(N-isopropylacrylamide) (PNIPAM).[@water2] The *equilibrium* free energy obtained from these models can be expressed as[@BH] $%
F/kT=N^{-1}\phi \ln \phi +(1-\phi )\ln (1-\phi )+\chi _{eff}\phi (1-\phi )$ where $N$ is the polymerization degree. In distinction to the familiar Flory free energy, the effective Flory interaction parameter $\chi _{eff}$ is a function of both the monomer volume fraction, $\phi $, and the temperature, $%
T$. In this formulation the specific features of a particular model are grouped into $\chi _{eff}(T,\phi )$. In the following we consider the Self-Consistent Field (SCF) theory of brushes of “two-state polymers” in terms of $\chi _{eff}(T,\phi )$. A significant part of our discussion is devoted to brushes of polymers capable of undergoing a second type of phase separation.[@footXX; @n-Cluster1; @n-Cluster2; @Solc1; @Solc2] Within a brush, this type of phase separation can lead to a vertical phase separation associated with a discontinuous concentration profile.[@Wagner; @BH2] Our analysis focuses on the signatures of such phase separation. These include the non-monotonous variation of the brush thickness with the grafting density and the appearance of distinct regimes in the compression force profiles.
![Schematic picture of a brush of a “two-state polymer”. Open and filled circles depict monomers in the different, interconverting, monomeric states.[]{data-label="fig1"}](Fig1.eps){width="6cm"}
This approach is of interest because of a number of reasons: (i) $\chi
_{eff}(T,\phi )$ determines a number of important characteristics of the brush among them the concentration profile, the distribution of free-ends and the force profile associated with the compression of the brush. Thus, a description of the brush behavior in terms of $\chi _{eff}(T,\phi )$ accounts for the leading brush properties and facilitates the comparison of the predictions of the different models. The specific features of the individual models and their parameters come into play when the distribution of the monomer states is of interest. However, as we shall discuss, even in this case it is convenient to first specify the brush characteristics in terms of $\chi _{eff}(T,\phi )$. (ii) The formulation of the theory in terms of $\chi _{eff}(T,\phi )$ underlines the relationship to the measurable $$\overline{\chi }(T,\phi )=\chi _{eff}-(1-\phi )\partial \chi _{eff}/\partial
\phi$$as obtained from the study of the colligative properties of the polymer solutions.[@Flory; @wolfBook; @Flory1; @rev] $\overline{\chi
}(T,\phi )$ is helpful in determining the parameters of the models. In the context of brushes, the behavior of $\overline{\chi
}(T,\phi )$ provides, as we shall discuss, a useful diagnostic for systems expected to exhibit a vertical phase separation within the brush. (iii) While our discussion focuses on the two-state models cited above, the analysis can be extended to other models[@sanchez; @Freed] that yield a $\phi $ dependent $\chi
_{eff}$. (iv) The SCF theory of brushes characterized by $\chi
_{eff}(T,\phi )$ suggests useful tests for the occurrence of vertical phase separation. This is of interest, as we shall discuss in the final section, because of experimental indications that such behavior occurs in brushes of PNIPAM. (v) The concentration profiles obtained from the SCF theory are essentially identical to those derived[@BH2] from the Pincus approximation where the distribution of free-ends is assumed rather than derived.[@Pincus; @Pincus1] In marked contrast, the compression force profiles are sensitive to the distribution of free-ends and the two methods yield different results.
The two-state models differ in their identification of the interconverting states. Within the $n$-cluster model[@n-Cluster1; @n-Cluster2] one is a bare monomer while the second is a monomer incorporated into a stable cluster of $n$ monomers. In the remaining models one of the monomeric states is hydrophilic and the other is hydrophobic. The hydrophilic state is preferred at low $T$ while the hydrophobic state is favored at high $T$. In the Karlstrom model[@Karlstrom] the two states differ in their dipole moment and their interconversion involves an internal rotation. The models of Matsuyama and Tanaka,[@Matsuyama] Bekiranov *et al*[@Bekiranov] and of Dormindotova[@DormidPhase] assume that the hydrophilic monomeric state forms a H-bond to a water molecule while the hydrophobic state does not. The brush structure within the Karlstrom model was studied using numerical SCF theory of the Scheutjens-Fleer type[@Linse; @Bjorling] and allowed to rationalize the aggregation behavior of copolymers incorporating PEO blocks.[@Linse1] The brush structure within the $n$-cluster model was studied using SCF theory[@Wagner] and by simulations.[@Mattice] These reveal the possibility of a vertical phase separation within the brush giving rise to a discontinuity in the concentration profile. In turn, this was invoked in order to rationalize observations about the collapse of PNIPAM brushes.[Napper]{} The force profiles due to the compression of brushes described by the $n$-cluster model were also analyzed.[@AHcluster; @Sevick] The studies of brushes of ”two-state polymers” focused on a particular model and were formulated in terms of the corresponding free energy. This obscured common features between the different models and hampered the comparison between them. For example, while a vertical phase separation is possible within all two-state models, this scenario was mainly studied for the $n$-cluster model thus creating a misleading impression about the physical origins of this phenomenon.
Our discussion concerns a brush of flexible two-state chains, terminally grafted to a planar surface. We assume that the chains are monodisperse and that each chain incorporates $%
N$ monomers. The surface area per chain, $\sigma $, is constant and the surface is assumed to be non-adsorbing for the two monomeric states. The free energy per lattice site is $$f_{\infty }(\phi ,T)/kT=(1-\phi )\ln (1-\phi )+\chi _{eff}(\phi ,T)\phi
(1-\phi ) \label{Ffint}$$This form corresponds to the $N\rightarrow \infty $ limit. It is appropriate for brushes of any $N$ because the grafted chains lose their mobility and thus have no translational entropy. The application of our analysis to a particular case is illustrated for the Karlstrom model. However, most of our analysis is model independent in that $\chi _{eff}(T,\phi )$ is not specified explicitly. The only assumption made is that $\overline{\chi }%
(T,\phi )$ can be expanded in powers of $\phi $ $$\overline{\chi }(T,\phi )=\sum_{i=0}\overline{\chi }_{i}(T)\phi ^{i}$$where the $\overline{\chi }_{i}(T)$ are specific to a given model. For simplicity we further limit the discussion to systems where the first three terms provide an accurate description of $\overline{\chi }(T,\phi )$. As we shall discuss, the presence of a third order term is the minimal condition for the possibility of a vertical phase separation within the brush. The power series expansion of $\overline{\chi }(T,\phi )$ is clearly related to the virial expansion of the osmotic pressure, $\pi $. The two differ in that the second incorporates terms originating in the translational entropy. In discussions of the SCF theory $f_{\infty }(\phi )/kT$ was often approximated by the second and third terms in the virial expansion of the Flory-Huggins free energy. From this point of view it is important to note two points: (i) in the power series of $\overline{\chi }(T,\phi )$ all the coefficients are $%
T$ dependent and can change sign. (ii) Use of $\overline{\chi }(T,\phi )$ series expansion including a $\phi ^{3}$ term corresponds to a virial expansion incorporating a $\phi ^{4}$ term.
The next five sections, \[SCFtheory\]–\[distributionKarlsrom\], are devoted to the model independent aspects of the SCF theory based on $%
f_{\infty }(\phi ,T)/kT$ with $\phi $ dependent $\chi _{eff}$. In section \[SCFtheory\] we formulate the analytical SCF model for $\chi _{eff}(\phi
,T)$ as a generalization of the familiar SCF theory of brushes.[Semenov,Milner,MilnerBrush,Skvortsov,Zhulina]{} The concentration profiles and their moments are discussed in section \[concentrationProf\]. The technical details corresponding to section \[concentrationProf\] are described in Appendix \[appendixCalc\]. Section \[distribution\] discusses the distribution of free ends while the technical details are given in Appendix \[appendixg\]. The force profiles associated with the compression of the brush are analyzed in section \[compression\]. In every case we distinguish between brushes characterized by a continuos concentration profile and those exhibiting a vertical phase separation. Finally, in section \[distributionKarlsrom\] we illustrate the implementation of our analysis to the Karlstrom model. In particular, we obtain the corresponding $\chi _{eff}(T,\phi )$, $\overline{\chi }(T,\phi )$ and the distribution of monomeric states in the brush.
The SCF theory for $\protect\chi _{eff}(T,\protect\phi )$[SCFtheory]{}
======================================================================
Consider a brush of neutral and flexible polymers comprising $N$ monomers of size $a$. Each chain is grafted by one end onto an impermeable, non-adsorbing, planar surface. The area per chain is denoted by $\sigma $ and $H$ is the maximal height of the brush. Following refs. , the free energy per chain, $F_{chain}$, consists of two terms: an interaction free energy, $F_{int}$, and an elastic free energy, $F_{el}$, $F_{chain}=F_{int}+$ $F_{el}$. The interaction free energy per chain is
$$\frac{F_{int}}{kT}=\frac{\sigma }{a^{3}}\int_{0}^{H}f_{\infty }(\phi )dz
\label{Fint}$$
where the interaction free energy density $f_{\infty }(\phi )$ is given by (\[Ffint\]). In a strong stretching limit, when the chains are extended significantly with respect to Gaussian dimensions, the elastic free energy is$^{\mathbf{27}}$
$$\frac{F_{el}}{kT}=\frac{3}{2a^{2}}\int_{0}^{H}g(z^{\prime })dz^{\prime
}\int_{0}^{z^{\prime }}E(z,z^{\prime })dz. \label{Fel}$$
Here $E(z,z^{\prime })=dz/dn$ characterizes the local chain stretching at height $z$ when the free end is at height $z^{\prime }$. $g(z^{\prime })$ specifies the height distribution of the free ends and obeys the normalization condition $\int_{0}^{H}g(z^{\prime })dz^{\prime }=1$.
The concentration of monomers, $\phi (z)$, at height $z$ is specified by
$$\phi (z)=\frac{a^{3}}{\sigma }\int_{z}^{H}\frac{g(z^{\prime })dz^{\prime }}{%
E(z,z^{\prime })} \label{ConFi}$$
Since each chain consists of $N$ monomers we have
$$N=\frac{\sigma }{a^{3}}\int_{0}^{H}\phi (z)dz \label{ConfN1}$$
At the same time
$$N=\int_{0}^{z^{\prime }}\frac{dz}{E(z,z^{\prime })} \label{ConN}$$
which can be regarded as a normalization condition for the function $%
E(z,z^{\prime })$.
The equilibrium $\phi (z)$ in the brush is determined by the variation of the functional $F_{chain}$ with respect to $E(z,z^{\prime })$ and $%
g(z^{\prime })$ subject to the constraints (\[ConfN1\]) and (\[ConN\]) yielding
$$E(z,z^{\prime })=\frac{\pi }{2N}\sqrt{z^{\prime 2}-z^{2}} \label{E}$$
and
$$\mu (\phi )=\lambda -Bz^{2} \label{profile}$$
where $B=3\pi ^{2}/8N^{2}a^{2}$, $\lambda $ is the Lagrange multiplier associated with constraint (\[ConfN1\]) and $\mu (\phi )=\partial
f_{\infty }(\phi )/\partial \phi $ is the exchange chemical potential. Up to this point the SCF theory is identical to the familiar versions, as obtained for $\chi =\chi (T)$.
For dilute brushes, $\sigma >>1,$ immersed in a good solvent and when $\chi $ is independent of$\phi $, $\chi _{eff}(\phi ,T)=\chi
(T)\ll 1/2$, the chemical potential is linear in $\phi $, $\mu
(\phi )\sim \phi $ . In this case, when binary interactions are dominant, eq. (\[profile\]) leads to a *parabolic* concentration profile.[@MilnerBrush; @Zhulina] At higher grafting densities, $\sigma \geq 1,$ higher order terms become significant. These were typically handled by incorporation of the third virial term.[@MilnerBrush; @Zhulina] However, as discussed in the introduction, deviations from these scenarios are expected when $\chi
_{eff}(\phi )$ varies with $\phi $ and $\mu (\phi )$, as obtained from ([Ffint]{}), assumes the form
$$\begin{aligned}
\mu (\phi )&=&-\ln \left( 1-\phi \right) -1+\chi _{eff}(\phi
)-2\chi _{eff}(\phi )\phi + \nonumber \\
&&\phi (1-\phi )\frac{\partial \chi _{eff}(\phi )}{\partial \phi }\end{aligned}$$
Since colligative measurements yield $\overline{\chi }(\phi )=\chi
_{eff}(\phi )-(1-\phi )\partial \chi _{eff}(\phi )/\partial \phi $[foot2]{} rather than $\chi _{eff}(\phi )$ it useful to express $\mu (\phi )$ as
$$\mu (\phi )=-\ln (1-\phi )-1+\chi _{eff}(0)-\int_{0}^{\phi }\overline{\chi }%
(\phi )d\phi -\phi \overline{\chi }(\phi ) \label{muint}$$
Finally, the Lagrange multiplier $\lambda $ is determined by the concentration at the outer edge of the brush, $\phi _{H}\equiv \phi (z=H)$
$$\begin{aligned}
\lambda &=&BH^{2}+\mu (z=H) \nonumber \\
&=&BH^{2}-\ln (1-\phi _{H})-1+\chi _{eff}(0)-\nonumber \\
&&\int_{0}^{\phi _{H}}\overline{%
\chi }(\phi )d\phi -\phi _{H}\overline{\chi }(\phi _{H}),\end{aligned}$$
In turn, $\phi _{H}$ of a free brush is set by the osmotic pressure at $H$ that is, $\pi _{osm}(\phi _{H})=\left. \phi ^{2}\partial \left[ f_{\infty
}(\phi )/\phi \right] /\partial \phi \right\vert _{\phi =\phi _{H}}=0$, leading to
$$-\ln (1-\phi _{H})-\phi _{H}-\overline{\chi }(\phi _{H})\phi _{H}^{2}=0
\label{Pi}$$
In a good solvent $\phi _{H}=0$.
The Concentration Profiles and Their Moments[concentrationProf]{}
=================================================================
In order to obtain $\phi (z)$ it helpful to express (\[profile\]) as
$$\Delta \mu (\phi )=B(H^{2}-z^{2}) \label{mustar}$$
where
$$\Delta \mu (\phi )=-\ln \frac{1-\phi }{1-\phi _{H}}-\int_{\phi _{H}}^{\phi }%
\overline{\chi }(\phi )d\phi +\phi _{H}\overline{\chi }(\phi _{H})-\phi
\overline{\chi }(\phi ) \label{mu1}$$
Equation (\[mustar\]) does not specify $\phi (z)$ directly. Rather, it yields $z(\phi )=\sqrt{H^{2}-\Delta \mu (\phi )/B}$. The brush height is determined in terms of the monomer volume fraction at the surface, $\phi
_{0}=\phi (z=0)$, leading to $H=\sqrt{\Delta \mu (\phi _{0})/B}$ and
$$z(\phi )=\sqrt{\frac{\Delta \mu (\phi _{0})-\Delta \mu (\phi )}{B}}
\label{zphi}$$
$\phi (z)$ is determined by equation (\[zphi\]) together with the normalization condition (\[ConfN1\]), which relates $\phi _{0}$ to the grafting density, $1/\sigma $.
![The state diagram of a brush with $\overline{%
\chi }(\phi )=1/2+\chi _{2}\phi ^{2}$ and $N=200$ in the $\chi _{2}$, $%
1/\sigma $ plane. Vertical phase separation occurs in the hatched region. At $\bullet$ $\phi _{0}=\phi_{-}(T)$. The dashed line, at higher $1/\sigma $, corresponds to brushes exhibiting an inflection point at altitudes that increase with $1/\sigma $. The boundary for lower $1/\sigma $ corresponds to $\phi _{-}$ $\phi
_{+}$ coexistence at the grafting surface.[]{data-label="fig2"}](Fig2.eps){width="6cm"}
We now distinguish between two cases. In one the concentration profile is continuous while in the second a discontinuity occurs due to vertical phase separation. In the first case (\[ConfN1\]) may be expressed as $$N=\frac{\sigma }{a^{3}}\int_{\phi _{0}}^{\phi _{H}}\phi \frac{\partial z}{%
\partial \phi }d\phi \label{sig}$$ The concentration profile for a given $\sigma $ is fully specified by ([zphi]{}) and (\[sig\]). A vertical phase separation in the brush results in a discontinuity at height $H_{t}$. At this altitude two phases coexist: a dense inner phase with a monomer volume fraction $\phi _{+}(H_{t})$ and a dilute outer phase with $\phi _{-}(H_{t})$. In this case the normalization condition (\[ConfN1\]) assumes the form $$N=\frac{\sigma }{a^{3}}\int_{\phi _{0}}^{\phi _{+}(H_{t})}\phi \frac{%
\partial z}{\partial \phi }d\phi +\frac{\sigma }{a^{3}}\int_{\phi
_{-}(H_{t})}^{\phi _{H}}\phi \frac{\partial z}{\partial \phi }d\phi
\label{sigdis}$$ where $\phi _{+}(H_{t})$ and $\phi _{-}(H_{t})$ are determined by $\mu (\phi
_{+})=\mu (\phi _{-})$ and $\pi (\phi _{+})=\pi (\phi _{-})$. $\phi (z)$ is now determined by equation (\[zphi\]) together with the normalization condition (\[sigdis\]),
It is of interest to consider the phase behavior when $\overline{\chi }(\phi
)$ is described by $\overline{\chi }(\phi )=\chi _{0}+\chi _{1}\phi +\chi
_{2}\phi ^{2}$. In the case of $\overline{\chi }(\phi )=\chi _{0}$ or $%
\overline{\chi }(\phi )=\chi _{0}+\chi _{1}\phi $ the critical point, as specified by $\partial ^{2}f_{\infty }(\phi )/\partial \phi ^{2}=\partial
^{3}f_{\infty }(\phi )/\partial \phi ^{3}=0$
![$\phi (z)$ for different areas per chain $\sigma $ when $\overline{\chi }(\phi )=1/2+\chi _{2}\phi ^{2}$. (a) $\chi
_{2}=1$ (b) $\chi _{2}=1.05$. In every case $N=200$.[@FootFig][]{data-label="fig3"}](Fig3.eps){width="6cm"}
$$\begin{aligned}
\frac{1}{1-\phi }-2\overline{\chi }(\phi )-\phi \frac{\partial \overline{%
\chi }(\phi )}{\partial \phi } &=&0 \label{CritP} \\
\frac{1}{\left( 1-\phi \right) ^{2}}-3\frac{\partial \overline{\chi }(\phi )%
}{\partial \phi }-\phi \frac{\partial ^{2}\overline{\chi }(\phi )}{\partial
\phi ^{2}} &=&0 \label{CritP2}\end{aligned}$$
occurs at $\phi _{c}=0$. This corresponds to the familiar case of a polymer rich phase in coexistence with a neat solvent. In this situation there is no vertical phase separation within the brush and the concentration profile is continuous. A second type of phase separation,[@n-Cluster1; @n-Cluster2; @Solc1; @Solc2] associated with a discontinuous $\phi (z)$, is possible when higher order terms are involved. For $\overline{\chi }(\phi )=1/2+\chi
_{2}\phi ^{2}$ a critical point occurs at $\phi _{c}=1/2$ and $\chi _{2c}=1$. In the vicinity of the critical point, for $\chi _{2}\gtrsim 1$ and $\phi
\gtrsim $ $1/2$ the coexistence curve is well approximated by the spinodal line $\partial ^{2}f_{\infty }(\phi )/\partial \phi ^{2}=0$
$$\frac{1}{1-\phi }-1-4\chi _{2}\phi ^{2}=0 \label{spinodal}$$
leading to $\phi _{\pm }=\frac{1}{2}\pm \frac{1}{2}\sqrt{1-1/\chi _{2}}$. The state diagram of a brush in the $\chi _{2},1/\sigma $ plane when eq. (\[spinodal\]) applies is shown in Fig. \[fig2\]. Concentration profiles obtained from $\overline{\chi }(\phi )$ of this form are depicted in Fig. \[fig3\].
![Plots of $\phi (z)$ (a) and $g(z)$ (b) above and below the critical point for $\overline{\chi }(\phi )=1/2+\chi _{2}\phi ^{2}$, $\sigma =17$ and $N=200 $. In all cases, the outer phase is swollen.[FootFig]{}[]{data-label="fig4"}](Fig4.eps){width="6cm"}
Experimentally, the brush thickness $H$ is inaccessible. Certain experimental technique yield $\phi (z)$ allowing one to obtain moments of $%
\phi (z)$
$$\langle z\rangle =\frac{\int_{0}^{H}z\phi (z)dz}{\int_{0}^{H}\phi (z)dz}=%
\frac{\sigma }{Na^{3}}\int_{0}^{H}z\phi (z)dz \label{firstmom}$$
$$\langle z^{2}\rangle =\frac{\int_{0}^{H}z^{2}\phi (z)dz}{\int_{0}^{H}\phi
(z)dz}=\frac{\sigma }{Na^{3}}\int_{0}^{H}z^{2}\phi (z)dz. \label{secondmom}$$
Other techniques, such as ellipsometry, measure $\langle z\rangle
$.[@charmet] As we shall discuss, the $\sigma $ dependence of the moments provides useful information on the brush structure. The details of the calculation of these moments are described in Appendix \[appendixCalc\].
When $\phi (z)$ is continuous, both moments increase smoothly with the grafting density. In marked contrast, vertical phase separation within the brush gives rise to a non-monotonic behavior. In particular, both $\langle
z\rangle $ and $\sqrt{\langle z^{2}\rangle }$ exhibit a minimum at intermediate $\sigma $. A vertical phase separation gives rise to a plateau in the $H$ vs. $\sigma $ plot (Fig. \[fig8\]) while in the plots of $%
\langle z\rangle $ and $\sqrt{\langle z^{2}\rangle }$ vs. $\sigma $ it is associated with a minimum (Fig. \[fig6\] and Fig. \[fig7\]). The physical origin of this behavior is the partitioning of the monomers between the inner dense phase and the outer dilute one. The minima are traceable to the higher weight give to the inner phase. Since the inner phase is denser, the onset of vertical phase separation is associated with a decrease $%
\langle z\rangle $ and $\sqrt{\langle z^{2}\rangle }$. These features provide a useful diagnostic for the occurrence of a vertical phase separation in the brush. The SCF analysis in this section confirms earlier results[@BH2] obtained by utilizing the Pincus approximation.[@Pincus; @Pincus1] As we shall discuss this is the case for properties that are insensitive to the precise form of $g(z)$. In marked contrast, the compression force profile (section V) does depend on $g(z)$ and the SCF result differ from the one obtained from the Pincus approximation.
![Plots of $\phi (z)$ (a) and $g(z)$ (b) for the case of two coexisting dense phases with $\overline{\chi }(\phi
)=0.51+\chi _{2}\phi ^{2}$, $\sigma =18$ and $N=200$.[@FootFig][]{data-label="fig5"}](Fig5.eps){width="6cm"}
The Distribution of Free Ends\[distribution\]
=============================================
The SCF formalism allows to obtain the height distribution of free ends, $%
g(z)$. Current experimental techniques do not allow to probe $g(z)$ directly. However, $g(z)$ is of interest because it plays a role in the calculation of the compression force profile. When $\chi =const$ the brush structure is dominated by the contributions of the second and third virial terms of $F_{int}$. Three scenarios emerge. In a good solvent the ends are distributed throughout the brush and $g(z)$ is a smooth function vanishing at $z=0$ and $z=H$. When the brush is collapsed in a poor solvent the ends reside preferentially at the outer edge of the brush and $g(z)$ diverges at $%
H$. In a $\theta $ solvent $g(z)$ increases smoothly with $z$ but does not diverge.[@Skvortsov; @Zhulina] As we shall see, a new scenario emerges when a vertical phase separation occurs. In particular, $g(z)$ will then diverge at the phase boundary indicating localization of the ends at the boundary. We will obtain $g(z)$ from the integral equation (\[ConFi\]). The details of the calculation are described in Appendix \[appendixg\].
![$\langle z\rangle $ as a function of $\sigma $ for different $\chi _{2}$ values and $N=300$.[@FootFig][]{data-label="fig6"}](Fig6.eps){width="8cm"}
$g(z)$ of a brush with a continuous profile (Fig. \[fig4\]a) is specified by
$$\begin{aligned}
g(z)&=&z\frac{\sigma }{Na^{3}}\left( \frac{\phi
_{H}}{\sqrt{H^{2}-z^{2}}}+\right. \nonumber \\
&&\left. \sqrt{B}\int_{\phi _{H}}^{\phi }\frac{d\phi ^{\prime
}}{\sqrt{\Delta \mu (\phi )-\Delta \mu (\phi ^{\prime })}}\right)
\label{gCon}\end{aligned}$$
where $\phi $ and $z$ are related by (\[mustar\]). When a vertical separation occurs within the brush, equation (\[ConFi\]) yields now two expressions, for $\phi (z)$. At the outer edge, $H_{t}< z< H$
$$\phi (z)=\frac{a^{3}}{\sigma }\int_{z}^{H}\frac{g(z^{\prime })dz^{\prime }}{%
E(z,z^{\prime })} \label{phim}$$
while at the inner dense phase, $0< z< H_{t}$
$$\phi (z)=\frac{a^{3}}{\sigma }\int_{z}^{H_{t}}\frac{g(z^{\prime })dz^{\prime
}}{E(z,z^{\prime })}+\frac{a^{3}}{\sigma }\int_{H_{t}}^{H}\frac{g(z^{\prime
})dz^{\prime }}{E(z,z^{\prime })} \label{phip}$$
In the outer region only free ends with $H_{t}< z$ contribute while for the inner phase all free ends are involved.
The expression for $g(z)$ in the two regions are given below while the details of the derivation are presented in appendix \[appendixg\]. At the outer phase, $H_{t}< z< H$
$$\begin{aligned}
g(z)&=&z\frac{\sigma }{Na^{3}}\left( \frac{\phi
_{H}}{\sqrt{H^{2}-z^{2}}}+\right. \nonumber \\
&& \left. \sqrt{B}\int_{\phi _{H}}^{\phi }\frac{d\phi ^{\prime
}}{\sqrt{\Delta \mu (\phi )-\Delta \mu (\phi ^{\prime })}}\right)
\label{gm}\end{aligned}$$
while in the inner phase, $0< z< H_{t}$ $$\begin{aligned}
&&g(z)=\frac{z\sigma }{Na^{3}}\left[ \frac{\phi _{+}(H_{t})-\phi _{-}(H_{t})%
}{\sqrt{H_{t}^{2}-z^{2}}}+\frac{\phi _{H}}{\sqrt{H^{2}-z^{2}}}+\right. \nonumber \\
&&\sqrt{B}\int_{\phi _{+}(H_{t})}^{\phi }\frac{d\phi ^{\prime
}}{\sqrt{\Delta \mu
(\phi )-\Delta \mu (\phi ^{\prime })}}+\frac{\sqrt{B}}{\pi }\sqrt{H_{t}^{2}-z^{2}}\times \nonumber \\
&& \int_{\phi _{H}}^{\phi _{\_}(H_{t})}\frac{\frac{d\Delta \mu
(\phi
^{\prime })}{d\phi ^{\prime }}d\phi ^{\prime }}{\left( H^{2}-z^{2}-\frac{%
\Delta \mu (\phi ^{\prime })}{B}\right)
\sqrt{H^{2}-H_{t}^{2}-\frac{\Delta \mu (\phi ^{\prime })}{B}}}
\times \nonumber \\
&&\left. \int_{\phi _{H}}^{\phi ^{\prime }}\frac{d\phi ^{\prime
\prime }}{\sqrt{\Delta \mu (\phi ^{\prime })-\Delta \mu (\phi
^{\prime \prime })}}\right] . \label{gp(z)}\end{aligned}$$ The first integral allows for the contribution of the inner phase and the second for the contribution of the outer phase. $g(z)$ (\[gp(z)\]) at the interval $0<z<H_{t}$ diverges at the phase boundary $z=H_{t}$. $g(z)$ ([gm]{}) at the interval $H_{t}<z<H$ diverges at $H$ when the outer phase is collapsed and $\phi _{H}>0$. In this case the two coexisting phases are dense (Fig. \[fig5\]). When $\phi _{H}=0$ the outer phase is swollen and $%
g(z)$ does not diverge at $H$ (Fig. \[fig4\]b). A rough approximation yielding closed form expressions for $g(z)$ for discontinuous brushes is described in Appendix \[appendixModel\].
![$\sqrt{\langle z^{2}\rangle}$ as a function of $%
\sigma $ for different $\chi _{2}$ values and $N=300$.[@FootFig][]{data-label="fig7"}](Fig7.eps){width="8cm"}
The Compression Force Profile of a “Two-State” Brush[compression]{}
===================================================================
The surface force apparatus allows to measure the restoring force arising upon compression of a brush. For brushes of polymers characterized by a constant $\chi $ the force increases smoothly with the compression and the force profile is essentially featureless. When the brush consists of polymers characterized by $\overline{\chi }(\phi )$ the compression can induce a vertical phase separation even if the concentration profile of the brush is initially continuous. The existence of vertical phase separation, be it compression induced or not, gives rise to distinctive regimes in the force profile. In particular, the slope of the force vs. distance curve in different compression regimes can be markedly different. In such experiments $H$ is determined by the compressing surface rather than by $\sigma $. Accordingly, $\phi _{H}$ is set by the normalization condition (\[ConfN1\]) and not by $\pi (\phi _{H})=0$. The compression increases $F_{chain}$ and the restoring force per area $\sigma $ is $$f(H)=-\frac{\partial F_{chain}}{\partial H} \label{Force}$$In the following we obtain this force law for the case of compression by impenetrable, non-adsorbing surface.
![$H$ as a function of $\sigma $ for different $%
\chi _{2}$ values and $N=300$.[@FootFig][]{data-label="fig8"}](Fig8.eps){width="7cm"}
For a brush with a continuous $\phi (z)$
$$\frac{F_{chain}}{kT}=\int_{\phi _{0}}^{\phi _{H}}\left[ \frac{\sigma }{a^{3}}%
f_{\infty }(\phi )+\frac{3}{2a^{2}}\frac{\pi ^{2}}{8N}z^{2}(\phi )g(\phi )%
\right] \frac{\partial z}{\partial \phi }d\phi$$
obtained by invoking $\int_{0}^{z^{\prime }}E(z^{\prime },z)dz=z^{\prime
2}\pi ^{2}/(8N)$. Here $g(z)$ is given by (\[gCon\]), while $z(\phi )$ and $\partial z/\partial \phi $ are specified by (\[zphi\]). $f(H)$ is calculated numerically subject to the constraint (\[sig\]). When the concentration at the wall, $\phi _{0}$, exceeds $\phi _{-}$, the brush undergoes a vertical phase separation and $\phi (z)$ is no longer continuous. In this case $F_{chain}$ assumes the form
![The compression force profile for a brush with $%
\overline{\chi }(\phi )=1/2+1.05\phi ^{2}$, $\sigma =120$ and $N=300$. The uncompressed brush is in a single phase state ($\phi _{0}<\phi _{-})$.[FootFig]{}[]{data-label="fig9"}](Fig9.eps){width="7cm"}
$$\begin{aligned}
\frac{F_{chain}}{kT} &=&\frac{\sigma }{a^{3}}\int_{\phi
_{0}}^{\phi _{H}}f_{\infty }(\phi )\frac{\partial z}{\partial \phi
}d\phi +\nonumber \\
&&\frac{3}{2a^{2}}\frac{\pi ^{2}}{8N}\left[ \int_{\phi _{0}}^{\phi
_{+}(H_{t})}z^{2}(\phi )g(\phi )\frac{\partial z}{\partial \phi }d\phi +\right. \nonumber \\
&& \left. \int_{\phi _{-}(H_{t})}^{\phi _{H}}z^{2}(\phi )g(\phi
)\frac{\partial z}{\partial \phi }d\phi \right]\end{aligned}$$
where $g(\phi )$ is specified by eqs. (\[gp(z)\]) for the inner phase and by (\[gm\]) for the outer one. The conservation of monomers is enforced by the constraint (\[sigdis\]).
![$\phi (z)$ plots corresponding to the three regimes in Fig. \[fig9\]. (a) single phase swollen phase, $H=12.9$, (b) a coexistence of a dense and a dilute phase, $H=6.1$, (c) single dense phase, $H=3.7$. In every case $\sigma =120$ and $N=300$.[@FootFig][]{data-label="fig10"}](Fig10.eps){width="3.2cm"}
When the conditions permit a vertical phase separation within the brush, it can take place in two ways. It can occur when the grafting density exceeds a certain critical value thus causing $\phi _{0}>\phi _{-}$. Alternatively, it can also take place as a result of compression when the grafting density does not lead to phase separation in the unperturbed brush. The development of $\phi (z)$ and $f(H)$ for this second case is depicted in figures [fig9]{} and \[fig10\] respectively. Initially, the brush retains the single phase structure and the associated force law. When the compression enforces $%
\phi _{0}>\phi_{-}$, a vertical phase separation occurs and is signalled by a weaker slope of the $f(H)$ vs. $H$ curve. Stronger compression causes complete conversion to a dense phase thus causing an abrupt increase in $%
f(H) $.
The Karlstrom Model and the distribution of monomeric states [distributionKarlsrom]{}
======================================================================================
Thus far, our discussion concerned brushes characterized by an arbitrary $%
\chi _{eff}(\phi )$. We now illustrate these considerations for the case of the Karlstrom model.[@Karlstrom] We focus on this model because of its simplicity and its semiquantiative agreement with the phase diagram of aqueous solutions of PEO at atmospheric pressure[@Karlstrom] and the measured $\overline{\chi }(T,\phi )$.[@BH] Within this model the monomers exist in two states: a polar, hydrophilic state (A) and an apolar, hydrophobic state (B). The two intercovert via internal rotations. In the $%
N\rightarrow \infty $ limit the corresponding Flory-type free energy is $$\begin{aligned}
\frac{f_{\infty }(\phi )}{kT} &=&(1-\phi )\ln (1-\phi )+ \nonumber \\
&& \phi \lbrack p\ln p+(1-p)\ln
(1-p)]+ \nonumber \\
&&\phi p\Delta \epsilon +\phi (1-\phi )[p\chi _{AS}+(1-p)\chi
_{BS}]+\nonumber \\
&&\phi ^{2}\chi _{AB}p(1-p) \label{K}\end{aligned}$$ where $p$ denotes the fraction of monomers in the A state. The first term allows for the mixing entropy of the solvent while the second reflects the mixing entropy of the A and B states along the chain. $\Delta \epsilon $ is the energy difference between two states and $\phi p\Delta \epsilon $ allows for the effect of interconversion between them. The fourth term is the generalization of the $\chi \phi (1-\phi )$ in the Flory free energy to allow for the interactions of the two states with the solvent S. The interactions between $A$ and $B$ states gives rise to the last term.
The equilibrium value of $p$ for a given $\phi $ is specified by $\partial
f_{\infty }/\partial p=0$ leading to $$\begin{aligned}
\frac{p}{1-p}&=&\exp \left[ -\Delta \epsilon -(1-\phi )(\chi
_{AS}-\chi _{BS})-\right. \nonumber \\
&& \left. \phi \chi _{AB}(1-2p)\right] \label{eqP}\end{aligned}$$ The parameters used by Karlstrom[@Karlstrom] to fit phase diagram of PEO in water are $\chi _{As}=80.0/T$, $\chi _{Bs}=684.5/T$, $\chi _{AB}=155.6/T$, $\Delta \varepsilon =-625.2/T+\ln 8$. In the remainder of this section we consider brushes at $T=60^{o}$ C. The equilibrium $\chi _{eff}(\phi )$ is obtained by equating (\[K\]) and (\[eqP\])
$$\begin{aligned}
\chi _{eff}(\phi ) &=&p\chi _{AS}+(1-p)\chi _{BS}+ \nonumber \\
&&\frac{\phi }{1-\phi }\left[ \chi _{AB}p(1-p)+
p\Delta \epsilon \right] + \nonumber \\
&& \frac{p\ln p+(1-p)\ln (1-p)}{1-\phi } \label{chieff}\end{aligned}$$
where $p(\phi )$ is specified by (\[eqP\]). $\ \overline{\chi }=-\frac{%
\partial }{\partial \phi }\frac{f_{\infty }(\phi )}{\phi }$ together with (\[eqP\]) yield the equilibrium value $$\overline{\chi }(\phi )=p\chi _{AS}+(1-p)\chi _{BS}-\chi _{AB}p(1-p)
\label{chiK}$$ Both $\chi _{eff}(\phi )$ and $\overline{\chi }(\phi )$ can be expanded in powers of $\phi $. The coefficients in the expansion depend on the parameters, $\Delta \epsilon $, $\chi _{AS}$, $\chi _{BS}$, $\chi _{AB}$. High order terms are of negligible importance and $\overline{\chi }(\phi
)\approx 0.48+0.31\phi +0.07\phi ^{2}$ provides a good approximation for $%
\overline{\chi }(\phi )$.
![A plot of the fraction of hydrophilic states, $%
p $, vs. $z$, in the Karlstrom model[@Karlstrom] for different $\sigma $. In every case $\chi _{As}=80.0/T$, $\chi _{Bs}=684.5/T$, $\chi
_{AB}=155.6/T$, $\Delta \varepsilon =-625.2/T+\ln 8$, $N=300$ $T=60^{o}$ C. [@FootFig][]{data-label="fig11"}](Fig11.eps){width="8cm"}
Equation (\[eqP\]) allows to relate the volume fraction $\phi $ to $p$, the fraction of hydrophilic A states, as
$$\phi (p)=\frac{\ln \frac{p}{1-p}+\Delta \epsilon +\chi _{AS}-\chi _{BS}}{%
\chi _{AS}-\chi _{BS}-\chi _{AB}(1-2p)}$$
Accordingly, the exchange chemical potential can be specified in terms of $p$, *i.e.* $\Delta \mu (\phi (p))$. In turn, eq. (\[mustar\]) enables us to obtain $z=z(p)$. To this end we invoke two boundary conditions: (i) In this range of parameters the brush is swollen and $\phi $ vanishes at the outer edge, $\phi _{H}=\phi (p=p_{H})=0$ where $p_{H}$ is the value of $p$ at the height $H$. (ii) At the grafting surface we have $\Delta \mu
(z=0,p=p_{0})=BH^{2}$, where $p_{0}$ is the value of $p$ at $z=0$. In addition we utilize the conservation of monomers as given by (\[sig\]). The corresponding plots of $p=p(z)$ as well as the concentration profiles of the two states are depicted, for different $\sigma $, in Fig. \[fig11\] and in Fig. \[fig12\]. Since all brushes considered are swollen, with $%
\phi _{H}=0$, the $p$ values at the outer edge of the brush, $z=H$ are identical, $p=p_{H}$. Increasing grafting density leads to higher concentration at the grafting surface. This favors the hydrophobic B state and lower $p$ at the surface. For the chosen parameters, the minimal value of $p$, corresponding to a PEO melt ($\phi =1$), is $p_{\ast }=0.45$.
![The overall $\phi (z)$ and the corresponding concentration profiles of the monomeric states $\phi
_{A}(z)=p(z)\phi (z)$ and $\phi _{B}(z)=[1-p(z)]\phi (z)$ within the Karlstrom model for the conditions specified in Fig. \[fig11\].[@FootFig][]{data-label="fig12"}](Fig12.eps){width="8cm"}
Discussion\[discussion\]
========================
In this article we presented a common framework for the analysis of the structure of brushes of neutral water-soluble polymers (NWSP) that own their solubility to the formation of H-bonds with water molecules. Our analysis concerned a family of two-state models developed for PEO but applicable, in principle, to other NWSP. The particular aspects of the models were grouped into $\chi _{eff}(T,\phi )$ thus allowing for a unified discussion of the brush structure within these models. Significant part of the discussion concerned brushes exhibiting vertical phase separation that can occur for polymers capable of a second type of phase separation. In particular, we examined the distinctive behavior of plots of $\langle z\rangle $ and $\sqrt{%
\langle z^{2}\rangle }$ vs. $\sigma $ and the compression force profiles associated with such brushes. $\phi (z)$ and its moments are insensitive to the precise form of $g(z)$ and the SCF analysis recovers the results obtained earlier[@BH2] by the use of the Pincus approximation.[@Pincus; @Pincus1] In marked contrast, the compression force law does depend on $g(z)$ and a full SCF analysis is necessary in order to obtain the correct results. These features are useful criteria for the occurrence of vertical phase separation. Such criteria are of interest because of indirect experimental indications that brushes of PNIPAM exhibit this effect. Among these indications the following are especially noteworthy. First, is an early study by Zhu and Napper[@Napper] of the collapse of PNIPAM brushes grafted to latex particles immersed in water. This revealed a collapse involving two stages. An early collapse, took place below $30^{o}$ C, at better than $\theta $-conditions, and did not result in flocculation of the neutral particles. Upon raising the temperature to worse than $\theta $-conditions the collapse induced flocculation. This indicates that the colloidal stabilization imparted by the PNIPAM brushes survives the early collapse. It lead to the interpretation of the effect in terms of a vertical phase separation within the brush associated with a second type of phase separation as predicted by the $n$-cluster model. More recently, Balamurugan *et al*[Balamurugan]{} used Surface Plasmon Resonance (SPR) and contact angle measurements with water to characterize PNIPAM brushes grafted from a self-assembled monolayer on gold. The brush properties were studied at between $10^{o}$ C and $40^{o}$ C. The SPR measurements indicated gradual variation of the brush properties. In marked contrast, the contact angle measurements revealed an abrupt change at $\sim 32^{o}$ C. Both experiments are consistent with a brush undergoing vertical phase separation such that the outer phase is hydrophilic. Within this picture, the abrupt changes reported in the wetting and aggregation behavior correspond than to the onset of a single, dense and hydrophobic phase. This picture is also supported by recent study of the phase behavior of PNIPAM by Afroze *et al*.[@A] Early study of the phase behavior of PNIPAM in water, by Heskins and Guillet,[@HG] identified a LCST at $\phi _{c}\simeq 0.16$ and $T_{c}\simeq 31.0^{o}$ C. In marked contrast, the work of Afroze *et al*[@A] identified PNIPAM as a polymer undergoing a second type of phase separation. In particular: (i) While the LCST of PNIPAM depends on $N$ the LCST occurs around $T_{c}\simeq 27-28^{o}$ C and $\phi
_{c}\simeq 0.43$ (ii) In the limit of $\phi \rightarrow 0$, the phase separation occurs, depending on $N$, between $30^{o}$ C and $34^{o}$ C. Thus, the phase diagram of Afroze *et al* suggests that a vertical phase separation is indeed expected in brushes of PNIPAM.[@BH2] Systematic Neutron Reflectometry (NR) studies of PNIPAM brushes will eventually provide clearer picture of the situation. Early studies were hampered by high polydispersity as well as difficulties in determining $N$ and $\sigma $.[@Yim] With this in mind, the NR results revealed that the structure of PNIPAM brushes in acetone was very different from their structure in water, both at $20^{o}$ C and at $55^{o}$ C. In acetone the concentration profile was smoothly decaying while in water it consisted of a narrow, inner, dense region and an outer, extended and dilute region. More recent work utilized NR,[@Yim1] to study samples with lower polydispersity and higher grafting density between $20^{o}$ C and $40^{o}$ C. Importantly, the results indicate that most of the conformational change occurred between $28^{o}$ C and $34^{o}$ C though the corresponding concentration profiles were not reported. The results suggest a repartitioning of the monomers between a dilute outer tail and an inner dense region.[@Yim2] Finally, recent experimental results of Hu *at al*[@Hurec] are suggestive of the predictions obtained above concerning the variation of $\langle z\rangle$ upon decreasing $\sigma$ (Fig. \[fig6\]). Hu *et al* studied the thickness of a PNIPAM brush grafted to spherical microgels of copolymers of PNIPAM and acrylic acid 2-hydroxyethyl ester (HEA). The microgels shrink as $T$ increases from $24^o$ C to $36^o$ C thus inducing a decrease in $\sigma$. Remarkably, the thickness of the brush initially decreases in the range $27^o$ C to $32^o$ C but subsequently increases upon further heating in the range $27^o$–$32^o$ C. Unfortunately, in this experiment it is impossible to separate the effects due to change in $T$ from those to the change in $\sigma$ because the $T$ is used to tune $\sigma$.
In confronting experimental results with theoretical predictions concerning the vertical phase separation it is important to note two points. First, polydispersity in $N$ and in $\sigma $ can smooth out the discontinuity in $%
\phi (z)$. These factors give rise to domains with different $\phi _{0}$. In turn, the altitude of the discontinuity in these domains will also differ. NR measurements will reflect the weighted average of these domains. Polydispersity in $N$ can lead to further strengthening of this effect if $%
\chi _{eff}$ depends on $N$. Second, the vertical phase separation involves a first order phase transition. Accordingly, nucleation dynamics may play a role and it is necessary to allow for the possibility of long relaxation times.
Calculation of the average thickness\[appendixCalc\]
====================================================
For a $\phi (z)$ (\[firstmom\]) is evaluated using integration by parts utilizing $d\Delta \mu =-2Bzdz$:
$$\begin{aligned}
\langle z\rangle &=&\frac{\sigma
}{Na^{3}}\frac{1}{2B}\int_{0}^{BH^{2}}\phi (z)d\Delta \mu
\nonumber \\
&=&\frac{\sigma }{Na^{3}}\left( H^{2}\frac{\phi _{0}}{2}-\frac{1%
}{2B}\int_{\phi _{H}}^{\phi _{0}}\Delta \mu (\phi )d\phi \right)
\label{mom1}\end{aligned}$$
For a discontinuous $\phi (z)$ (\[firstmom\]) this procedure leads to
$$\begin{aligned}
\langle z\rangle &=&\frac{\sigma }{Na^{3}}\left[ \int_{0}^{H_{t}}z\phi
(z)dz+\int_{H_{t}}^{H}z\phi (z)dz\right] \nonumber \\
&=&\frac{\sigma }{Na^{3}}\frac{1}{2B}\left(
\int_{B(H^{2}-H_{t}^{2})}^{BH^{2}}\phi (z)d\Delta \mu
+ \right. \nonumber \\
&&\left. \int_{0}^{B(H^{2}-H_{t}^{2})}\phi (z)d\Delta \mu \right) \nonumber \\
&=&\frac{\sigma }{Na^{3}}[H^{2}\frac{\phi _{0}}{2}-\left(
H^{2}-H_{t}^{2}\right) \frac{\phi _{+}(H_{t})-\phi _{-}(H_{t})}{2} \nonumber
\\
&-&\frac{1}{2B}\int_{\phi _{+}(H_{t})}^{\phi _{0}}\Delta \mu (\phi )d\phi -%
\frac{1}{2B}\int_{\phi _{H}}^{\phi _{-}(H_{t})}\Delta \mu (\phi
)d\phi ] \label{mom11}\end{aligned}$$
The evaluation of (\[secondmom\]) in the case of a continuous $\phi (z)$ involves introducing the variable $\Delta \mu =B(H^{2}-z^{2})$ and invoking $%
z=\sqrt{H^{2}-\Delta \mu /B}$. Integration by parts leads to
$$\begin{aligned}
\langle z^{2}\rangle &=&\frac{\sigma }{Na^{3}}\frac{1}{2B}
\int_{0}^{BH^{2}}z\phi (z)d\Delta \mu \nonumber \\
&=&-\frac{\sigma }{Na^{3}}\frac{1}{2B}\int_{\Delta \mu =0}^{\Delta
\mu =BH^{2}}\Delta \mu d(z\phi (z)) \nonumber \\
&=&-\frac{\sigma }{Na^{3}}\frac{1}{2B}\int_{\phi _{H}}^{\phi
_{0}}\left( \phi \frac{\partial z}{\partial \phi }+z\right) \Delta
\mu d\phi \label{mom2}\end{aligned}$$
Using
$$\frac{\partial z}{\partial \phi }=-\frac{1}{B\sqrt{H^{2}-\Delta \mu /B}}%
\frac{\partial \Delta \mu }{\partial \phi }$$
we obtain the final expression
$$\begin{aligned}
\langle z^{2}\rangle &=&\frac{\sigma
}{Na^{3}}\frac{1}{2B^{2}}\int_{0}^{\phi
_{0}}\frac{\phi \Delta \mu \frac{\partial \Delta \mu }{\partial \phi }}{%
\sqrt{H^{2}-\Delta \mu /B}}d\phi -\nonumber \\
&&\frac{1}{2B}\int_{0}^{\phi _{0}}\Delta \mu \sqrt{H^{2}-\Delta
\mu /B}d\phi \label{mom22}\end{aligned}$$
where $\Delta \mu $ is specified by (\[mu1\]).
Following a similar procedure for a discontinuous $\phi (z)$ (\[secondmom\]) yields
$$\begin{aligned}
\langle z^{2}\rangle &=&\frac{\sigma }{Na^{3}}\left[ \int_{0}^{H_{t}}z^{2}%
\phi (z)dz+
\int_{H_{t}}^{H}z^{2}\phi (z)dz\right] \nonumber \\
&=&\frac{\sigma }{Na^{3}}\frac{1}{2B}\left(
\int_{B(H^{2}-H_{t}^{2})}^{BH^{2}}z\phi (z)d\Delta \mu
+ \right. \nonumber \\
&& \left. \int_{0}^{B(H^{2}-H_{t}^{2})}z\phi (z)d\Delta \mu \right) \nonumber \\
&=&\frac{\sigma }{Na^{3}}\left[ \frac{1}{2B^{2}}\int_{\phi
_{+}(H_{t})}^{\phi _{0}}\frac{\phi \Delta \mu \frac{\partial \Delta \mu }{%
\partial \phi }}{\sqrt{H^{2}-\Delta \mu /B}}d\phi - \right.\nonumber \\
&&\frac{1}{2B}\int_{\phi _{+}(H_{t})}^{\phi _{0}}\Delta \mu
\sqrt{H^{2}-\Delta \mu /B}d\phi
\nonumber \\
&&+\frac{1}{2B^{2}}\int_{0}^{\phi _{-}(H_{t})}\frac{\phi \Delta \mu \frac{%
\partial \Delta \mu }{\partial \phi }}{\sqrt{H^{2}-\Delta \mu /B}}d\phi
- \nonumber \\
&&\frac{1}{2B}\int_{0}^{\phi _{-}(H_{t})}\Delta \mu \sqrt{H^{2}-\Delta \mu /B}%
d\phi \nonumber \\
&&\left. -H_{t}\left( H^{2}-H_{t}^{2}\right) \frac{\phi _{+}(H_{t})-\phi
_{-}(H_{t})}{2}\right] \label{mom222}\end{aligned}$$
Calculation of the distribution of ends function\[appendixg\]
=============================================================
Upon introducing the variables $\rho =H^{2}-z^{2}$, $t=H^{2}-z^{\prime 2}$ and $g(z^{\prime })dz^{\prime }=-f(t)dt$ eq. (\[ConFi\]) assumes the form of an Abel integral equation
$$v(\rho )=\int_{0}^{\rho }\frac{f(t)dt}{\sqrt{\rho -t}}, \label{Abel}$$
where $v(\rho )=\frac{\pi \sigma }{2Na^{3}}\phi (z)$ whose solution ([Abel]{}) is
$$f(\rho )=\frac{1}{\pi }\left( \frac{v(0)}{\sqrt{\rho }}+\int_{0}^{\rho }%
\frac{1}{\sqrt{\rho -t}}\frac{dv(t)}{dt}dt\right) . \label{solAbel}$$
It is convenient to rewrite the integral in (\[solAbel\]) as
$$\begin{aligned}
&&\int_{0}^{\rho }\frac{1}{\sqrt{\rho -t}}\frac{dv(t)}{dt}dt=\int_{v(H)}^{v(z)}%
\frac{dv}{\sqrt{\rho -t(v)}}\nonumber \\
&=&\frac{\pi \sigma
}{2Na^{3}}\sqrt{B}\int_{\phi _{H}}^{\phi }\frac{d\phi ^{\prime
}}{\sqrt{\Delta \mu (\phi )-\Delta \mu (\phi ^{\prime })}}\end{aligned}$$
where $\Delta \mu (\phi )$ is given by (\[mustar\]). Substituting this integral into (\[solAbel\]) while noting that $v(0)=\frac{\pi \sigma }{%
2Na^{3}}\phi _{H}$ and $g(z)=2zf(\rho )$, leads to $g(z)$ in the form ([gCon]{}). This equation yields a simple form $z(\phi )$ rather than for $\phi
(z)$. Thus, it is naturally to consider $g(z)$ as a parametric function
$$\begin{aligned}
g(\phi ) &=&\frac{\sigma }{Na^{3}}\sqrt{H^{2}-\Delta \mu (\phi
)/B}\left( \frac{\phi _{H}}{\sqrt{\Delta \mu (\phi
)/B}}+\right.\nonumber
\\&& \left.\sqrt{B}\int_{\phi _{H}}^{\phi }\frac{d\phi ^{\prime
}}{\sqrt{\Delta \mu (\phi )-\Delta \mu (\phi ^{\prime
})}}\right) \label{g(z)} \\
z(\phi ) &=&\sqrt{\frac{\Delta \mu (\phi _{0})-\Delta \mu (\phi )}{B}}\end{aligned}$$
These two equations are augmented by (\[sig\]) relating $\sigma $ to $\phi
_{0}$.
In the case of discontinuous $\phi (z)$ it is necessary to obtain $g(z)$ in the outer and inner phases separately. For the outer phase the introduction of the variables $\rho =H^{2}-z^{2}$, $t=H^{2}-z^{\prime 2}$ and $%
g_{-}(z^{\prime })dz^{\prime }=-f_{-}(t)dt$ transforms (\[phim\]) into an Abel integral equation (\[Abel\]) whose solution is (\[gm\]).
To obtain $g_+(z)$ at the inner phase we substitute (\[gm\]) into ([phip]{}) and transform the first term of (\[phip\]) into an Abel integral (\[Abel\]) by introducing the variables $\rho ^{\prime }=H_{t}^{2}-z^{2}$, $%
t^{\prime }=H_{t}^{2}-z^{\prime 2}$ and $g_{+}(z^{\prime })dz^{\prime
}=-f_{+}(t)dt$. This leads to
$$\frac{\pi \sigma }{2Na^{3}}\phi _{+}(z)-\int_{0}^{H^{2}-H_{t}^{2}}\frac{%
f_{-}(t)dt}{\sqrt{\rho -t}}=\int_{0}^{\rho ^{\prime }}\frac{f_{+}(t^{\prime
})dt^{\prime }}{\sqrt{\rho ^{\prime }-t^{\prime }}},$$
where $f_{-}(t^{\prime })=g_{-}(z^{\prime })/2z^{\prime }$, and $\rho =\rho
^{\prime }+H^{2}-H_{t}^{2}$. Thus, $v(\rho ^{\prime })$ in the solution of the Abel equation (\[Abel\]) is
$$v(\rho ^{\prime })=\frac{\pi \sigma }{2Na^{3}}\phi
_{+}(z)-\int_{0}^{H^{2}-H_{t}^{2}}\frac{f_{-}(t)dt}{\sqrt{\rho -t}}
\label{vrho1}$$
and its value at $\rho ^{\prime }=0$ is
$$\begin{aligned}
v(0)&=&\frac{\pi \sigma }{2Na^{3}}\phi _{+}(H_{t})-\int_{0}^{H^{2}-H_{t}^{2}}%
\frac{f_{-}(t)dt}{\sqrt{H^{2}-H_{t}^{2}-t}}\nonumber\\
&=&\frac{\pi
\sigma }{2Na^{3}}\left[ \phi _{+}(H_{t})-\phi _{-}(H_{t})\right] ,
\label{v0}\end{aligned}$$
where eq. (\[Abel\]) with $\rho =H^{2}-H_{t}^{2}$ was used in order to calculate the second term in (\[v0\]). In addition
$$\begin{aligned}
\frac{dv(\rho ^{\prime })}{d\rho ^{\prime }}&=&\frac{\pi \sigma }{2Na^{3}}%
\frac{d\phi _{+}(z)}{d\rho ^{\prime }}+ \nonumber \\
&&\frac{1}{2}\int_{0}^{H^{2}-H_{t}^{2}} \frac{f_{-}(t)dt}{\left(
H^{2}-H_{t}^{2}+t^{\prime }-t\right) ^{3/2}} \label{dvrho1}\end{aligned}$$
All together, upon substituting (\[vrho1\]), (\[v0\]) and (\[dvrho1\]) into (\[solAbel\]) we obtain
$$\begin{aligned}
&&f_{+}(\rho ^{\prime }) =\frac{\sigma }{2Na^{3}}\left[ \frac{\phi
_{+}(H_{t})-\phi
_{-}(H_{t})}{\sqrt{H_{t}^{2}-z^{2}}}+\right.\nonumber
\\
&&\left. \int_{0}^{\rho ^{\prime }}\frac{1}{\sqrt{\rho ^{\prime
}-t^{\prime }}}\frac{d\phi
_{+}(z^{\prime })}{dt^{\prime }}dt^{\prime }\right] \nonumber \\
&&+\frac{1}{2\pi }\int_{0}^{\rho ^{\prime }}\frac{dt^{\prime }}{\sqrt{\rho
^{\prime }-t^{\prime }}}\int_{0}^{H^{2}-H_{t}^{2}}\frac{f_{-}(t)dt}{\left(
H^{2}-H_{t}^{2}+t^{\prime }-t\right) ^{3/2}}\end{aligned}$$
In the first integral we express $\rho ^{\prime }$ and $t^{\prime }$ in terms of (\[mustar\]) and we change the order of integration in the second integral obtaining
$$\begin{aligned}
&&f_{+}(\rho ^{\prime }) =\frac{\sigma }{2Na^{3}}\left[ \frac{\phi
_{+}(H_{t})-\phi _{-}(H_{t})}{\sqrt{H_{t}^{2}-z^{2}}}+\right.
\nonumber \\
&&\left.\sqrt{B}\int_{\phi _{+}(H_{t})}^{\phi _{+}}\frac{d\phi
_{+}^{\prime }}{\sqrt{\Delta \mu (\phi
_{+})-\Delta \mu (\phi _{+}^{\prime })}}\right] \nonumber \\
&&+\frac{1}{2\pi }\int_{0}^{H^{2}-H_{t}^{2}}f_{-}(t)dt \times
\nonumber \\
&&\int_{0}^{\rho ^{\prime }}\frac{dt^{\prime }}{\sqrt{\rho
^{\prime }-t^{\prime }}\left( H^{2}-H_{t}^{2}+t^{\prime }-t\right)
^{3/2}} \label{fp(rho1)}\end{aligned}$$
The inner integral in (\[fp(rho1)\]) is
$$\begin{aligned}
&&\int_{0}^{\rho ^{\prime }}\frac{dt^{\prime }}{\sqrt{\rho
^{\prime
}-t^{\prime }}\left( H^{2}-H_{t}^{2}+t^{\prime }-t\right) ^{3/2}}\nonumber \\
&=&\frac{2%
\sqrt{\rho ^{\prime }}}{\left( \rho ^{\prime }+H^{2}-H_{t}^{2}-t\right)
\sqrt{H^{2}-H_{t}^{2}-t}}\end{aligned}$$
leading to
$$\begin{aligned}
&&f_{+}(\rho ^{\prime }) =\frac{\sigma }{2Na^{3}}\left[ \frac{\phi
_{+}(H_{t})-\phi
_{-}(H_{t})}{\sqrt{H_{t}^{2}-z^{2}}}+\right.\nonumber
\\ &&\left.\sqrt{B}\int_{\phi _{+}(H_{t})}^{\phi _{+}}\frac{d\phi
_{+}^{\prime }}{\sqrt{\Delta \mu (\phi
_{+})-\Delta \mu (\phi _{+}^{\prime })}}\right] \nonumber \\
&&+\frac{\sqrt{\rho ^{\prime }}}{\pi }\int_{0}^{H^{2}-H_{t}^{2}}\frac{%
f_{-}(t)dt}{\left( \rho ^{\prime }+H^{2}-H_{t}^{2}-t\right) \sqrt{%
H^{2}-H_{t}^{2}-t}}\end{aligned}$$
Substitution of $f_{-}(t)=g_{-}(z)/2z$ from (\[gm\]) yields the final expression for $g_{+}(z)$ in the form
$$\begin{aligned}
&&g_{+}(z)=\frac{z\sigma }{Na^{3}}\left[ \frac{\phi
_{+}(H_{t})-\phi _{-}(H_{t})}{\sqrt{H_{t}^{2}-z^{2}}}+\frac{\phi
_{H}}{\sqrt{H^{2}-z^{2}}}+\right. \nonumber \\
&&\left. \sqrt{B}\int_{\phi _{+}(H_{t})}^{\phi _{+}}\frac{d\phi
_{+}^{\prime }}{\sqrt{ \Delta \mu (\phi _{+})-\Delta \mu (\phi
_{+}^{\prime })}}\right. \nonumber
\\
&& +\frac{\sqrt{B}}{\pi }\sqrt{H_{t}^{2}-z^{2}} \times
\nonumber \\
&&\int_{0}^{H^{2}-H_{t}^{2}}\frac{dt}{\left( H^{2}-z^{2}-t\right) \sqrt{%
H^{2}-H_{t}^{2}-t}} \times \nonumber \\
&&\left.\int_{\phi _{H}}^{\phi _{\_}^{\prime }(t)}\frac{d\phi
_{-}^{\prime \prime }}{\sqrt{\Delta \mu (\phi _{\_}^{\prime
})-\Delta \mu (\phi _{\_}^{\prime \prime })}}\right] \label{gp}\end{aligned}$$
It is convenient to express $g_{+}(z)$ in terms of the concentration of the dense phase, $\phi _{+}$ utilizing $z(\phi
_{+})\-=\sqrt{H^{2}-\Delta \mu (\phi _{+})/B}$. Introducing the variables $t\-=\Delta \mu (\phi _{\_}^{\prime })/B$ and $dt\-=1/B(d\Delta \mu (\phi _{\_}^{\prime })/d\phi _{\_}^{\prime
})d\phi _{\_}^{\prime }$ in the last integral leads to (\[gp(z)\]).
Model concentration profiles\[appendixModel\]
=============================================
A rough approximation allows to obtain analytical expression for $g(z)$ of a brush in the presence of a vertical phase separation. In the dense phase the variation of $\phi $ is slow and we can approximate it as constant, $\phi
_{+}=\phi _{0}$. In the outer phase dilute phase $\phi $ is rather low and we can neglect nonconstant terms in $\overline{\chi }(\phi )$. Thus
$$\phi (z)=\left\{
\begin{array}{c}
\phi _{0},0<z<H_{t} \\
\phi (z),H_{t}<z<H%
\end{array}
\right.$$
where $\phi (z)$ is determined by (\[mustar\]) with $\chi =const$, while the value of $\phi _{0}$ is set by(\[ConfN1\]). As before, $g(z)$ in the outer phase is determined by (\[gm\]) while in the inner phase it is determined by (\[gp\]).
First, consider the case of $\chi =1/2$ leading to $\Delta \mu (\phi )=-\ln
(1-\phi )-\phi \approx \phi ^{2}/2$ and eq. (\[gm\]) yields
$$g(z)=z\frac{\sigma \pi }{2Na^{3}}\sqrt{2B} \label{gmm}$$
Substitution of (\[gmm\]) into (\[gp\]) gives the expression for $g(z)$ in the inner phase
$$\begin{aligned}
g(z)&=&\frac{z\sigma }{Na^{3}}\left[ \frac{\phi _{+}(H_{t})-\phi _{-}(H_{t})}{%
\sqrt{H_{t}^{2}-z^{2}}}+\right.\nonumber \\
&&\left.\sqrt{2B}\arctan \sqrt{\frac{H^{2}-H_{t}^{2}}{%
H_{t}^{2}-z^{2}}}\right] \label{gpm}\end{aligned}$$
where $\phi _{-}(H_{t})=\sqrt{2B(H^{2}-H_{t}^{2})}$ and $\phi
_{+}(H_{t})=\phi _{0}$. This function is discontinuous and diverges at the phase boundary $z=H_{t}$. The value of $\phi _{0}$ can be found from ([ConfN1]{})
$$\phi _{0}=\frac{Na^{3}}{\sigma H_{t}}-\sqrt{\frac{B}{2}}\left[ \frac{H^{2}}{
H_{t}}\left( \frac{\pi }{2}-\arcsin \frac{H_{t}}{H}\right) -\sqrt{
H^{2}-H_{t}^{2}}\right]$$
When $\chi =0$ eq. (\[gm\]) specifies $g(z)$ at the outer phase, $%
H_{t}<z<H $
$$g(z)=z\frac{\sigma }{Na^{3}}B\sqrt{H^{2}-z^{2}}.$$
and eq. (\[gp\]) for $g(z)$ in the inner phase, $0<z<H_{t}$, yields
![Comparison between the exact $\phi (z)$ and $%
g(z)$ and their approximate values as calculated from for the case of $%
\overline{\chi}(\phi )=1/2+1.05\phi ^{2}$, $\sigma =120$ and $N=300$.[FootFig]{}[]{data-label="fig13"}](Fig13.eps){width="7cm"}
$$\begin{aligned}
g(z)&=&\frac{z\sigma }{Na^{3}}\left[ \frac{\phi _{+}(H_{t})-\phi _{-}(H_{t})}{%
\sqrt{H_{t}^{2}-z^{2}}}+\right.\nonumber \\
&&\left. 2B\left( 2\sqrt{H^{2}-H_{t}^{2}}+\sqrt{%
H_{t}^{2}-z^{2}}-\sqrt{H^{2}-z^{2}}\right) \right]\end{aligned}$$
Again, $g(z)$ of the inner phase diverges at the phase boundary. The performance of this approximation is illustrated in Fig \[fig13\]. It captures the main features of $\phi (z)$ and the behavior of $g(z)$ in the inner region. However $g(z)$ at the outer region increases rather than decrease.
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All calculation utlize equation (\[sig\]) or ([sigdis]{}) and begin with choosing $\phi_{0}$. This is then used to determine $H$ and then $\sigma $. For numerical convenience the calculation of $\sigma
$ is carried out by using discretized version of equation (\[sig\]) or (\[sigdis\]). The integral is approximated by a sum of $100$ equidistant terms. It is thus difficult to control $\sigma $ precisely. The $\sigma $ stated values are the integr values within $.2$ of the real value.
|
---
abstract: 'Given a star product with separation of variables on a pseudo-Kähler manifold, we obtain a new formal (1,1)-form from its classifying form and call it the phase form of the star product. The cohomology class of a star product with separation of variables equals the class of its phase form. We show that the phase forms can be arbitrary and they bijectively parametrize the star products with separation of variables. We also describe the action of a change of the formal parameter on a star product with separation of variables, its formal Berezin transform, classifying form, phase form, and canonical trace density.'
address: 'Department of Mathematics, Abilene Christian University, ACU Box 28012, Abilene, TX 79699-8012'
author:
- Alexander Karabegov
date: 'March 16, 2015'
title: On the phase form of a deformation quantization with separation of variables
---
Introduction
============
Given a manifold $M,$ we denote by $C^\infty(M)[\nu^{-1},\nu]]$ the space of formal Laurent series with a finite principal part, $$f = \sum_{r = k}^\infty \nu^r f_r,$$ where $f_r \in C^\infty(M)$ and $k \in {\ensuremath{\mathbb{Z}}}$. If $k > 0$, we will say that the formal function $f$ vanishes at $\nu=0$.
Deformation quantization on a Poisson manifold $(M, \{\cdot,\cdot\})$ is an associative product (named a star product) $\ast$ on $C^\infty(M)[\nu^{-1},\nu]]$ given by the following $\nu$-adically convergent series, $$f \ast g = fg + \sum_{r = 1}^\infty \nu^r C_r(f,g),$$ where $C_r$ are bidifferential operators and $C_1(f,g) - C_1(g,f) = i\{f,g\}$. We assume that the unit constant is the unity for the star product, $f \ast 1 = 1 \ast f = f$. A star product can be restricted (localized) to an open subset $U \subset M$. We denote by $L_f$ and $R_f$ the operators of left and right star multiplication by a function $f$, respectively, so that $f \ast g = L_f g = R_g f$. The operators $L_f$ and $R_g$ commute for any functions $f, g$.
Nondegenerate Poisson bivectors bijectively correspond to symplectic forms. We will call a star product nondegenerate if the corresponding Poisson structure is nondegenerate.
Two star products $\ast$ and $\ast'$ on a Poisson manifold $M$ are called equivalent if there exists a formal differential operator $B = 1 + \nu B_1 + \ldots$ on $M$ such that $$f \ast' g = B^{-1}(Bf \ast Bg).$$ Deformation quantization was introduced in [@BFFLS]. It was proved by a number of authors ([@D],[@F2],[@NT2]) that the equivalence classes of star products on a symplectic manifold $(M, \omega_{-1})$ are bijectively parametrized by the formal cohomology classes in $$\frac{1}{\nu}\left[\omega_{-1}\right] + H^2(M)[[\nu]].$$ Fedosov gave a simple geometric construction of star products in each equivalence class on an arbitrary symplectic manifold in [@F1]. Kontsevich proved in [@K] that star products exist on arbitrary Poisson manifolds and gave a classification of star products up to equivalence in terms of formal deformations of the Poisson structure.
If $\ast$ is a star product on a $2m$-dimensional symplectic manifold $(M, \omega_{-1})$, there exists a canonically normalized formal trace density $$\label{E:mu}
\mu = \frac{1}{m!}\left(\frac{1}{\nu}\, \omega_{-1}\right)^m e^{\varkappa},$$ where $\varkappa$ is a globally defined formal function on $M$ which vanishes at $\nu =0$ (see [@F1], [@LMP2]). On a compact $M$, the index theorem for deformation quantization ([@F2],[@NT]) gives a topological formula for the total volume of $\mu$ in terms of the cohomology class of the star product.
If $M$ is a complex manifold with a Poisson bracket $\{\cdot,\cdot\}$ of type $(1,1)$ with respect to the complex structure, a star product $\ast$ on $(M, \{\cdot,\cdot\})$ has the property of separation of variables (of the anti-Wick type) if $a \ast f = af$ and $f \ast b = bf$ for any locally defined holomorphic function $a$ and antiholomorphic function $b$, i.e., the operators $L_a = a$ and $R_b = b$ are pointwise multiplication operators. Equivalently, every bidifferential operator $C_r$ of the star product $\ast$ differentiates its first argument in antiholomorphic directions and the second argument in holomorphic ones.
On a coordinate chart on $M$, the Poisson bracket is given by a Poisson tensor $g^{\bar lk}$ of type (1,1), $$\{f,g\} = i g^{\bar lk}\left(\frac{{\partial}f}{{\partial}z^k}\frac{{\partial}g}{{\partial}\bar z^l} - \frac{{\partial}g}{{\partial}z^k}\frac{{\partial}f}{{\partial}\bar z^l}\right).$$ For any star product with separation of variables on $(M,\{\cdot,\cdot\})$ the operator $C_1$ is given locally by the formula $$C_1(f,g) = g^{\bar lk} \frac{{\partial}f}{{\partial}\bar z^l} \frac{{\partial}g}{{\partial}z^k}.$$ Given a star product with separation of variables $\ast$ on $M$, there exists a formal differential operator $I = 1 + \nu I_1 + \nu^2 I_2 + \ldots$ globally defined on $M$ such that for a local holomorphic function $a$ and an antiholomorphic function $b$, $$I(ab) = b \ast a.$$ In particular, $Ia = a$ and $Ib = b$ and therefore $I_ra =0$ and $I_rb =0$ for $r \geq 1$. The operator $I_1$ is the Laplace-Beltrami operator $\Delta$ given by the local formula $$\Delta = g^{\bar lk}\frac{{\partial}^2}{{\partial}z^k {\partial}\bar z^l}.$$
If $\lambda$ is a closed global (1,1)-form on $M$, its local potential $\varphi$ on an open subset $U \subset M$ is defined up to a summand $a + b$ and therefore the function $I_r \varphi$ does not depend on the choice of the potential. Such functions on a contractible covering glue to a global function on $M$ which depends only on the form $\lambda$ and the operator $I_r$. This observation will be used in the proof of Theorem \[T:existuniq\].
A star product with separation of variables $\ast$ on a complex manifold $M$ is completely determined by its formal Berezin transform $I$. It was proved in [@CMP3] that the inverse operator $I^{-1}$ is the formal Berezin transform of a star product with separation of variables $\tilde\ast$ on $M$ equipped with the opposite Poisson structure such that $$f \tilde\ast g = I^{-1}(Ig \ast If).$$ We call $\tilde\ast$ the dual of the star product $\ast$. The dual of $\tilde\ast$ is $\ast$.
Given a complex manifold $M$ which admits a pseudo-Kähler structure, we denote by $\Omega(M)$ the set of formal series $$\omega = \frac{1}{\nu}\, \omega_{-1} + \omega_0 + \nu \omega_1 + \ldots,$$ where $\omega_r, r \geq -1$, are closed (1,1)-forms on $M$ and $\omega_{-1}$ is nondegenerate. In particular, $(M, \omega_{-1})$ is a pseudo-Kähler manifold which has a Poisson structure corresponding to $\omega_{-1}$. It was proved in [@CMP1] that the nondegenerate star products with separation of variables on $M$ are bijectively parametrized by the elements of $\Omega(M)$. In [@BW] Fedosov’s geometric construction was modified in order to show that on every pseudo-Kähler manifold there exists a star product with separation of variables (of the Wick type). In [@N2] it was shown that every star product with separation of variables on a pseudo-Kähler manifold can be obtained via a generalized Fedosov’s construction.
The form $\omega$ parametrizing a nondegenerate star product with separation of variables $\ast$ is called its classifying form. The classifying form $\tilde\omega$ of the dual star product $\tilde\ast$ is such that $$\tilde\omega = -\frac{1}{\nu}\, \omega_{-1} + \tilde\omega_0 + \nu \tilde\omega_1 + \ldots.$$ The mapping $\omega \mapsto \tilde\omega$ is an involution on $\Omega(M)$. We call the form $$\label{E:phase}
\omega^{ph} = \frac{1}{2}\left(\omega - \tilde\omega\right) = \frac{1}{\nu}\, \omega_{-1} + \omega^{ph}_0 + \nu \omega^{ph}_1 + \ldots$$ the phase form of the star product $\ast$. Clearly, the phase form of the dual star product $\tilde\ast$ is $-\omega^{ph}$. Given a form $\omega \in \Omega(M)$, we will call the corresponding forms $\tilde\omega$ and $\omega^{ph}$ its dual and phase forms, respectively.
In this paper we will prove that the mapping $\omega \mapsto \omega^{ph}$ is a bijection of $\Omega(M)$ onto itself and thus the phase forms can be used as an alternative parametrization of the nondegenerate star products with separation of variables on $M$. This choice of parametrization is justified by the fact that the cohomology class of a star product with separation of variables is equal to the cohomology class of its phase form, which follows from results obtained in [@LMP1].
We will consider the action of a change of the formal parameter on a star product with separation of variables, its formal Berezin transform, classifying form, phase form, and canonical trace density. In particular, we will consider the action of proper involutions of the formal parameter such as $\nu \mapsto -\nu$ and describe the star products whose phase form is odd with respect to an involution.
Deformation quantizations with separation of variables
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In this section we will describe basic constructions related to star products with separation of variables on a pseudo-Kähler manifold obtained in [@CMP1], [@LMP1], and [@LMP2]. Fix a pseudo-Kähler manifold $M$ with a pseudo-Kähler form $\omega_{-1}$ and an element $$\omega = \frac{1}{\nu}\, \omega_{-1} + \omega_0 + \nu \omega_1 + \ldots$$ of $\Omega(M)$. Let $U$ be a contractible coordinate chart on $M$. Each form $\omega_r, r \geq -1,$ has a local potential $\Phi_r$ on $U$ so that $\omega_r = i{\partial}\bar{\partial}\Phi_r$. Thus, $$\Phi = \frac{1}{\nu}\, \Phi_{-1} + \Phi_0 + \nu \Phi_1 + \ldots$$ is a formal potential of $\omega$. The metric tensor $g_{k\bar l}$ of the form $\omega_{-1}$ is given on $U$ by the formula $$g_{k\bar l} = \frac{{\partial}^2 \Phi_{-1}}{{\partial}z^k {\partial}\bar z^l}.$$ Its inverse $g^{\bar lk}$ is a Poisson tensor of type (1,1). It was proved in [@CMP1] that there exists a unique globally defined star product with separation of variables $\ast_\omega$ on $(M, \omega_{-1})$ such that on each contractible coordinate chart $U$, $$L_{\frac{{\partial}\Phi}{{\partial}z^k}} = \frac{{\partial}\Phi}{{\partial}z^k} + \frac{{\partial}}{{\partial}z^k} \mbox{ and } R_{\frac{{\partial}\Phi}{{\partial}\bar z^l}} = \frac{{\partial}\Phi}{{\partial}\bar z^l} + \frac{{\partial}}{{\partial}\bar z^l}.$$ The mapping $\omega \mapsto \ast_\omega$ is a bijection of $\Omega(M)$ onto the set of all nondegenerate star products with separation of variables on $M$. The formal form $\omega$ is called the classifying form of the star product $\ast_\omega$. We drop the subscript $\omega$ in $\ast_\omega$ if it does not lead to confusion.
Let $(M,\omega_{-1})$ be a pseudo-Kähler manifold of complex dimension $m$ and $U \subset M$ be a contractible coordinate chart. Denote $\mathbf{g} = \det (g_{k\bar l})$ and fix a branch of $\log \mathbf{g}$ on $U$. The Ricci form of the metric $g_{k\bar l}$ is a closed global (1,1)-form on $M$ given locally by the formula $\rho = - i{\partial}\bar{\partial}\log\mathbf{g}$. We denote by $\varepsilon = [-\rho]$ the canonical class of $M$.
A local construction of the canonical trace density $\mu$ of a star product with separation of variables $\ast$ on $M$ was introduced in [@LMP2]. Below we give a slightly modified version of this construction. Fix an arbitrary formal potential $$\label{E:phi}
\Phi = \frac{1}{\nu}\, \Phi_{-1} + \Phi_0 + \nu \Phi_1 + \ldots$$ of the classifying form $\omega$ of the product $\ast$ on $U$. There exists a unique potential $\Psi$ of the dual form $\tilde\omega$ of the form $$\label{E:psi}
\Psi = - \frac{1}{\nu}\, \Phi_{-1} + (-\Phi_0 + \log \mathbf{g}) + \nu \Psi_1 + \ldots$$ satisfying the equation $$\label{E:dnuphi}
\frac{d\Phi}{d\nu} + I\frac{d\Psi}{d\nu} = \frac{m}{\nu},$$ where $I$ is the formal Berezin transform for the product $\ast$. The global function $\varkappa$ from (\[E:mu\]) is given by the formula $$\label{E:kappa}
\varkappa = \Phi + \Psi - \log \mathbf{g} = \nu(\Phi_1 + \Psi_1) + \nu^2(\Phi_2 + \Psi_2) + \ldots$$ on $U$. It follows from (\[E:kappa\]) that $$\label{E:omegatildeomega}
\omega + \tilde\omega = - \rho + i{\partial}\bar{\partial}\varkappa.$$ Since $\varkappa$ is global, the class of $i{\partial}\bar{\partial}\varkappa$ is trivial and $$[\omega] + [\tilde\omega] = \varepsilon.$$ It was proved in [@LMP1] that the class of the star product with separation of variables with the classifying form $\omega$ is $[\omega] - \varepsilon/2$, which is exactly the class of the corresponding phase form $\omega^{ph}$.
A star product with a given phase form
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In this section we will prove the existence and uniqueness of a nondegenerate star product with separation of variables on a pseudo-Kähler manifold $M$ whose phase form is a given arbitrary element of $\Omega(M)$.
Let $M$ be a pseudo-Kähler manifold of complex dimension $m$ and $\ast$ be a nondegenerate star product with separation of variables on $M$ with the classifying form $$\omega = \frac{1}{\nu}\, \omega_{-1} + \omega_0 + \nu \omega_1 + \ldots.$$
\[L:conomega\] For each $r \geq 1$, the bidifferential operator $C_r$ of the star product $\ast$ depends only on the forms $\omega_k$ with $-1 \leq k \leq r-2$.
Let $U \subset M$ be a contractible coordinate chart and $\Phi = \frac{1}{\nu}\Phi_{-1} + \Phi_0 + \ldots$ be a potential of $\omega$ on $U$. Fix $f \in C^\infty(U)$. It was proved in [@CMP1] that the formal differential operator $A = L_f$ on $U$ is completely determined by the following conditions: the operator $A$ does not contain antiholomorphic derivatives, it commutes with the operators $R_{{\partial}\Phi/{\partial}\bar z^l}$ for $1 \leq l \leq m,$ and satisfies $A1 = f$. In particular, $A = f + \nu A_1 + \nu^2 A_2 + \ldots$ and $A_r1 =0$ for all $r \geq 1$. For a function $g \in C^\infty(U), A_rg = C_r(f,g)$. The commutation condition written explicitly is as follows, $$\label{E:commut}
\left[f + \nu A_1 + \ldots, \frac{1}{\nu}\frac{{\partial}\Phi_{-1}}{{\partial}\bar z^l} + \left(\frac{{\partial}\Phi_0}{{\partial}\bar z^l} + \frac{{\partial}}{{\partial}\bar z^l}\right) + \nu \frac{{\partial}\Phi_1}{{\partial}\bar z^l} + \ldots \right] =0.$$ Observe that condition (\[E:commut\]) does not depend on the choice of a formal potential of the form $\omega$. Extracting the component of (\[E:commut\]) corresponding to $\nu^{r-1}$ we get $$\label{E:express}
\left[A_r, \frac{{\partial}\Phi_{-1}}{{\partial}\bar z^l}\right] + \left[A_{r-2}, \frac{{\partial}\Phi_0}{{\partial}\bar z^l} + \frac{{\partial}}{{\partial}\bar z^l}\right] + \ldots + \left[A_1, \frac{{\partial}\Phi_{r-2}}{{\partial}\bar z^l}\right] =0.$$ Using induction on $r$, we see from (\[E:express\]) that the commutator $$\label{E:commrs}
\left[A_r, \frac{{\partial}\Phi_{-1}}{{\partial}\bar z^l}\right]$$ is expressed in terms of the forms $\omega_k$ with $-1 \leq k \leq r-2$. It was shown in [@CMP1] that the knowledge of the commutators (\[E:commrs\]) for $1 \leq l \leq m$ and the condition $A_r1=0$ determine $A_r$ uniquely. It follows that the operator $A_r$, and therefore $C_r$, are expressed in terms of the forms $\omega_k$ with $-1 \leq k \leq r-2$.
Let $I = 1 + \nu I_1 + \ldots$ be the formal Berezin transform of a nondegenerate star product with separation of variables $\ast$ on $M$ with the classifying form $\omega$. According to Lemma \[L:conomega\], for each $r \geq 1$ the operator $I_r$ is expressed in terms of the forms $\omega_k$ with $-1 \leq k \leq r-2$. Set $\tilde I = I^{-1}$. The operator $\tilde I = 1 + \nu \tilde I_1 + \nu^2 \tilde I_2 + \ldots$ is the formal Berezin transform of the dual star product $\tilde\ast$. The following lemma is trivial.
\[L:inv\] For each $r \geq 1$, the operator $\tilde I_r$ is expressed in terms of the forms $\omega_k$ with $-1 \leq k \leq r-2$.
Assume that $$\omega^{ph} = \frac{1}{\nu} \omega_{-1} + \omega^{ph}_0 + \nu \omega^{ph}_1 + \ldots$$ is an arbitrary element of $\Omega(M)$. We want to construct a star product with separation of variables $\ast$ on $(M,\omega_{-1})$ whose phase form is $\omega^{ph}$ and show its uniqueness. We will construct inductively its classifying form $$\omega = \frac{1}{\nu} \omega_{-1} + \omega_0 + \nu \omega_1+ \ldots.$$ It follows from formulas (\[E:phi\]) and (\[E:psi\]) that $$\omega_0 = \omega_0^{ph} - \frac{1}{2}\, \rho.$$ Let $U \subset M$ be a contractible coordinate chart. We rewrite equation (\[E:dnuphi\]) on $U$ as follows, $$\label{E:commr}
\frac{d\Psi}{d\nu} = \frac{m}{\nu} - \tilde I\, \frac{d\Phi}{d\nu}.$$ Extracting the component of (\[E:commr\]) corresponding to $\nu^{r-1}$ for $r\geq 1$ we get the equation $$r \Psi_r = - r\Phi_r - \sum_{k = 1}^{r+1} (r-k) \tilde I_k \Phi_{r-k},$$ whence it follows that for $r \geq 1$, $$\label{E:omegar}
\omega_r = \omega_r^{ph} - \frac{i}{2r} {\partial}\bar {\partial}\sum_{k = 1}^{r+1} (r-k) \tilde I_k \Phi_{r-k}.$$ The sum on the right-hand side of (\[E:omegar\]) is a global (1,1)-form on $M$ expressed in terms of the forms $\omega_k$ with $-1 \leq k \leq r-1$. Therefore, the form $\omega$ can be inductively constructed from the phase form $\omega^{ph}$ and is uniquely defined. We have proved the following theorem.
\[T:existuniq\] Given a pseudo-Kähler manifold $M$, for any formal form $\omega^{ph} \in \Omega(M)$ there exists a unique nondegenerate star product with separation of variables on $M$ whose phase form is $\omega^{ph}$.
It is particularly easy to construct a classifying form $\omega$ with a given phase form $\omega^{ph}$ if the phase form is an invariant formal form on a homogeneous pseudo-Kähler manifold. A nondegenerate star product with separation of variables on a homogeneous pseudo-Kähler manifold is invariant if and only if its classifying form is invariant (see [@MBN]). If $\ast$ is an invariant star product with separation of variables on a homogeneous pseudo-Kähler manifold $M$, then its canonical trace density $\mu$ is invariant and therefore the function $\varkappa$ from (\[E:mu\]) is a formal constant. Now formula (\[E:omegatildeomega\]) implies that $$\omega + \tilde \omega = - \rho$$ and the corresponding phase form is $$\omega^{ph} = \omega + \frac{1}{2}\, \rho.$$ Vice versa, if $\omega^{ph}$ is an arbitrary invariant formal form from $\Omega(M)$, then $$\omega = \omega^{ph} - \frac{1}{2}\, \rho$$ is an invariant classifying form of an invariant star product with separation of variables on $M$ whose corresponding phase form is $\omega^{ph}$.
[*Example:*]{} The complex projective space ${\ensuremath{\mathbb{C}}}P^m$ equipped with the Fubini-Study form $\omega_{FS}$ is a homogeneous Kähler manifold under the action of the projective unitary group $PU(m+1)$. The Ricci form of the Fubini-Study metric is $(m+1)\omega_{FS}$. The invariant star product with the classifying form $$\omega = \frac{1}{\nu}\, \omega_{FS} - \frac{m+1}{2}\omega_{FS}$$ has the phase form $$\omega^{ph} = \frac{1}{\nu}\, \omega_{FS}.$$
Change of the formal parameter
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Let $\tau(\nu) = \tau_1 \nu + \tau_2 \nu^2 + \ldots$ be a formal series in $\nu$ with $\tau_r \in {\ensuremath{\mathbb{C}}}$ and $\tau_1 \neq 0$. The change of the formal parameter $\nu \mapsto \tau(\nu)$ defines, via a pullback, a ${\ensuremath{\mathbb{C}}}$-algebra automorphism $T = \tau^\ast$ of $C^\infty(M)[\nu^{-1},\nu]]$, $$(Tf)(\nu,x) = f(\tau(\nu), x),\ x \in M.$$ The action of $T$ extends to other formal geometric objects on $M$. We will be particularly interested in the involutive change of the formal parameter $\nu \mapsto -\nu$.
Given a star product $\ast$ on $(M, \{\cdot, \cdot\})$, the product $\ast_T$ defined by $$f \ast_T g = T( (T^{-1}f) \ast (T^{-1}g))$$ is a ${\ensuremath{\mathbb{C}}}[\nu^{-1},\nu]]$-bilinear star product on $(M, \tau_1\{\cdot, \cdot\})$. If the product $\ast$ is given by a formal bidifferential operator $C = \sum_{r =0}^\infty \nu^r C_r$, then the product $\ast_T$ is given by $$C_T = \sum_{r =0}^\infty (\tau(\nu))^r C_r.$$
\[L:it\] If $\ast$ is a (possibly degenerate) star product with separation of variables on a complex manifold $M$ with the formal Berezin transform $I$, then $\ast_T$ is also a star product with separation of variables whose formal Berezin transform $I_T$ is given by the formula $I_T = T I T^{-1}$.
If $a$ is a local holomorphic function on $M$, then so is $T^{-1}a$. We have $$a \ast_T f = T( (T^{-1}a) \ast (T^{-1}f)) = T( (T^{-1}a)(T^{-1}f)) = af.$$ Similarly, for a local antiholomorphic function $b$ we have $f \ast_T b = bf$. Therefore, $\ast_T$ is a star product with separation of variables. Now, $$\begin{aligned}
I_T(ab) = b \ast_T a = T( (T^{-1}b) \ast (T^{-1}a)) =\\
TI((T^{-1}a) (T^{-1}b)) = TIT^{-1}(ab),\end{aligned}$$ hence $I_T = TIT^{-1}$.
In the rest of the paper we will assume that $\ast$ is a nondegenerate star product with separation of variables on a pseudo-Kähler manifold $(M, \omega_{-1})$ of complex dimension $m$ with the classifying form $$\omega = \frac{1}{\nu}\omega_{-1} + \omega_0 + \ldots \in \Omega(M).$$ We fix a change of the formal variable $T = \tau^\ast$. The product $\ast_T$ is a star product with separation of variables on the pseudo-Kähler manifold $(M, (1/\tau_1)\omega_{-1})$.
\[L:omega\] The classifying form of the star product with separation of variables $\ast_T$ is $T\omega$.
Let $\Phi$ be a formal potential of $\omega$ on a contractible coordinate chart $U \subset M$. Given a function $f$ on $U$, we have $$\begin{aligned}
\frac{{\partial}(T\Phi)}{{\partial}z^k} \ast_T f =
T\left(\frac{{\partial}\Phi}{{\partial}z^k}\right) \ast_T f = T\left(\frac{{\partial}\Phi}{{\partial}z^k} \ast T^{-1}f\right) = \\
T\left(\frac{{\partial}\Phi}{{\partial}z^k} T^{-1}f + \frac{{\partial}}{{\partial}z^k} T^{-1}f \right) = \frac{{\partial}(T\Phi)}{{\partial}z^k} f + \frac{{\partial}f}{{\partial}z^k}.\end{aligned}$$ It follows that $T\Phi$ is a formal potential which determines the star product with separation of variables $\ast_T$ on $U$. Therefore, the classifying form of $\ast_T$ is $T\omega$.
Let $I$ be the formal Berezin transform and $\mu$ be the canonical trace density for the product $\ast$, and let $\tilde\omega$ be the dual of the form $\omega$. The canonical trace density of the star product $\ast_T$ is given by the formula $$\label{E:mut}
\mu_T = \frac{1}{m!}\left(\frac{1}{\tau_1 \nu}\omega_{-1}\right)^m e^{\varkappa_T},$$ where $\varkappa_T$ is a globally defined formal function on $M$ which vanishes at $\nu =0$. The expression $$\log\frac{\tau(\nu)}{\tau_1\nu} = \frac{\tau_2}{\tau_1}\nu + \ldots$$ gives a well defined formal series which also vanishes at $\nu =0$.
\[P:tildeomega\]
(a) The dual form of the form $T\omega$ is the form $T\tilde\omega$.
(b) The following formula holds, $$\varkappa_T = T\varkappa - m \log \frac{\tau(\nu)}{\tau_1 \nu}.$$
(c) The canonical trace density for the product $\ast_T$ is $$\label{E:tmu}
\mu_T = T\mu = \frac{1}{m!}\left(\frac{1}{\tau(\nu)}\, \omega_{-1}\right)^m e^{T\varkappa}.$$
Let $U \subset M$ be a contractible coordinate chart. Denote, as above, the metric tensor of $\omega_{-1}$ by $g_{k\bar l}$ and set $\mathbf{g} = \det (g_{k\bar l})$. The metric tensor for $(1/\tau_1)\omega_{-1}$ is $(1/\tau_1)g_{k\bar l}$ and $$\det \left(\frac{1}{\tau_1}g_{k\bar l}\right) = \left(\frac{1}{\tau_1}\right)^m \mathbf{g}.$$ Fix a branch of $\log \mathbf{g}$ on $U$ and a value of $\log \tau_1$, and set $$\log \left(\frac{1}{\tau_1}\right)^m \mathbf{g} = \log \mathbf{g} - m \log \tau_1.$$ Choose an arbitrary potential $\Phi$ of the form $\omega$ on $U$. Then, according to Lemma \[L:omega\], $T\Phi$ is a potential of the form $\omega_T$. We have $$T\Phi = \frac{1}{\tau_1 \nu}\Phi_{-1} + \left(-\frac{\tau_2}{\tau_1^2}\Phi_{-1}+\Phi_0\right) \pmod{\nu}.$$ There exists a unique potential $\Xi$ of the form dual to $T\omega$ such that $$\label{E:longxi}
\Xi = - \frac{1}{\tau_1\nu}\Phi_{-1} + \left(\frac{\tau_2}{\tau_1^2}\Phi_{-1}-\Phi_0 + \log \mathbf{g} - m \log \tau_1\right) \pmod{\nu},$$ and which satisfies the equation $$\label{E:itxi}
\frac{d}{d\nu}(T\Phi) +I_T \frac{d}{d\nu}\Xi = \frac{m}{\nu}.$$ The function $\varkappa_T$ in formula (\[E:mut\]) has the following local expression, $$\label{E:kappatau}
\varkappa_T = T\Phi + \Xi - \log \mathbf{g} + m \log \tau_1.$$ Applying $T$ to (\[E:dnuphi\]) and using the fact that $$T \frac{d}{d\nu}T^{-1} = \frac{1}{\tau'(\nu)}\frac{d}{d\nu},$$ we obtain that $$\label{E:overtau}
\frac{1}{\tau'(\nu)}\frac{d (T\Phi)}{d\nu} + \frac{1}{\tau'(\nu)} I_T \frac{d (T\Psi)}{d\nu} = \frac{m}{\tau(\nu)}.$$ Equation (\[E:overtau\]) is equivalent to the following one, $$\label{E:logfrac}
\frac{d}{d\nu} (T\Phi) + I_T \frac{d}{d\nu}\left(T\Psi - m\log\frac{\tau(\nu)}{\tau_1\nu}\right) = \frac{m}{\nu}.$$ We get from equations (\[E:itxi\]) and (\[E:logfrac\]) that $$\label{E:xitpsi}
\frac{d}{d\nu}\Xi = \frac{d}{d\nu}\left(T\Psi - m\log\frac{\tau(\nu)}{\tau_1\nu}\right).$$ Equation (\[E:psi\]) implies that $$\label{E:longtpsi}
T\Psi = - \frac{1}{\tau_1\nu}\Phi_{-1} + \left(\frac{\tau_2}{\tau_1^2}\Phi_{-1} - \Phi_0 + \log \mathbf{g}\right) \pmod{\nu}.$$ We obtain from Eqns. (\[E:longxi\], \[E:xitpsi\], \[E:longtpsi\]) that $$\label{E:xiviatpsi}
\Xi = T\Psi - m \log \tau_1 - m\log\frac{\tau(\nu)}{\tau_1\nu}.$$ Statement $(a)$ of the Proposition follows directly from formula (\[E:xiviatpsi\]). We obtain statement $(b)$ combining the formula $$T\varkappa = T\Phi + T\Psi - \log \mathbf{g}$$ with (\[E:kappatau\]) and (\[E:xiviatpsi\]). Statement $(b)$ readily implies $(c)$.
Let $\omega^{ph}$ denote the phase form of the product $\ast$.
The phase form of the star product $\ast_T$ is $T\omega^{ph}$.
Now assume that $\tau(\nu) = \tau_1\nu + \tau_2 \nu^2 + \ldots$ is a proper involution, i.e., $\tau \circ \tau = id$ and $\tau_1 = -1$, for example, $\tau(\nu) = - \nu$. For any $\omega \in \Omega(M)$, its odd part with respect to the proper involution $T = \tau^\ast$, $$\frac{1}{2}(\omega - T\omega),$$ also lies in $\Omega(M)$.
\[T:equiv\] Let $\ast$ be a nondegenerate star product with separation of variables on a pseudo-Kähler manifold $M$ with the classifying form $\omega \in \Omega(M)$, dual form $\tilde\omega$, phase form $\omega^{ph}$, and formal Berezin transform $I$, and let $T=\tau^\ast$ be a proper involution. Then the following conditions are equivalent:
(i) $T\omega^{ph} = - \omega^{ph}$;
(ii) $T\omega = \tilde \omega$;
(iii) $I_T = I^{-1}$.
$(i)\Rightarrow (ii)$. Assume that $\omega^{ph}$ is odd with respect to $T$, i.e., $T\omega^{ph} = - \omega^{ph}$. The phase form of the form $T\omega$ is $T\omega^{ph} = - \omega^{ph}$. The phase form of the dual form $\tilde\omega$ is also $-\omega^{ph}$. By Theorem \[T:existuniq\], $T\omega = \tilde\omega$. The implications $(ii)\Rightarrow (i)$ and $ (ii) \Leftrightarrow (iii)$ are straightforward.
Assume that $T = \tau^\ast$ is a proper involution and $\ast$ is a star product satisfying conditions $(i) - (iii)$ of Theorem \[T:equiv\].
\[P:kappat\] The canonical trace density $\mu_T$ of the star product $\ast_T$ is expressed through the canonical trace density $\mu$ of the product $\ast$ as follows, $$\mu_T = (-1)^m \mu.$$
Formula (\[E:mut\]) and the fact that $\tau_1 = -1$ imply that the statement of the Proposition is equivalent to the statement $$\label{E:ident}
\varkappa_T = \varkappa.$$ We will prove the formula $$\label{E:tkappaminus}
T\varkappa - \varkappa = m \log \frac{\tau(\nu)}{\tau_1 \nu}$$ which, according to part $(ii)$ of Proposition \[P:tildeomega\], is equivalent to (\[E:ident\]). Assume that $\omega \in \Omega(M)$ is such that $T\omega = \tilde\omega$. Let $\Phi$ and $\Psi$ be formal potentials of $\omega$ and $\tilde\omega$ on a contractible coordinate chart $U \subset M$ given by formulas (\[E:phi\]) and (\[E:psi\]), respectively. Formula (\[E:kappa\]) implies that $$\label{E:tphitpsi}
T\varkappa - \varkappa = T\Phi + T\Psi - \Phi - \Psi.$$ Since $I_T = I^{-1}$ by Theorem \[T:equiv\], equation (\[E:logfrac\]) can be rewritten as follows, $$\label{E:invlogfrac}
I\frac{d (T\Phi)}{d\nu} + \frac{d}{d\nu}\left(T\Psi - m\log\frac{\tau(\nu)}{\tau_1\nu}\right) = \frac{m}{\nu}.$$ Subtracting equation (\[E:dnuphi\]) from (\[E:invlogfrac\]) we obtain that $$\label{E:subtr}
I \frac{d}{d\nu}(T\Phi - \Psi) + \frac{d}{d\nu}\left(T\Psi - \Phi - m\log\frac{\tau(\nu)}{\tau_1\nu}\right) = 0.$$ Lemma \[L:omega\] implies that $T\Phi - \Psi = a + b$, where $a$ and $b$ are a formal holomorphic and antiholomorphic functions on $U$, respectively, whence $$I \frac{d}{d\nu}(T\Phi - \Psi) = \frac{d}{d\nu}(T\Phi - \Psi).$$ We get from (\[E:subtr\]) that $$\label{E:almost}
\frac{d}{d\nu}\left(T\Phi - \Psi + T\Psi - \Phi - m\log\frac{\tau(\nu)}{\tau_1\nu}\right) =0.$$ We see from equations (\[E:tphitpsi\]) and (\[E:almost\]) that $$\label{E:follows}
\frac{d}{d\nu}\left(T\varkappa - \varkappa - m\log\frac{\tau(\nu)}{\tau_1\nu}\right) =0.$$ Formula (\[E:tkappaminus\]) follows from (\[E:follows\]) and the fact that the formal series $T\varkappa, \varkappa$, and $\log(\tau(\nu)/\tau_1\nu)$ vanish at $\nu =0$.
If $\tau(\nu) = -\nu$, then the function $\varkappa$ is even in the formal parameter $\nu$.
The Corollary is an immediate consequence of formula (\[E:tkappaminus\]).
Given the formal Berezin transform $I$ corresponding to a classifying form $\omega = (1/\nu)\omega_{-1} + \ldots \in \Omega(M)$, the formal differential operator $X := \log I = \nu X_1 + \nu^2 X_2 + \ldots$ is a well defined global operator on $M$ with $X_1 = \Delta$ the Laplace-Beltrami operator for the pseudo-Kähler metric $\omega_{-1}$. Condition $(iii)$ of Theorem \[T:equiv\] is equivalent to the condition that $X$ is odd with respect to $T$, i.e., $TXT^{-1} = - X$. In [@CMP3] it was noticed that, for each $r \geq 1$, the order of the operators $X_{2r-1}$ and $X_{2r}$ is not greater than $2r$. This observation leads to the question whether the order of the operators $X_{2r}$ can be lowered further. It turns out that all operators $X_{2r}$ can simultaneously vanish. Namely, if the involution is $\tau(\nu) = -\nu$, one can see from Theorem \[T:equiv\] that all operators $X_{2r}$ vanish if and only if the phase form $\omega^{ph}$ of the form $\omega$ is odd in the formal parameter $\nu$.
[99]{} Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D.: Deformation theory and quantization. I. Deformations of symplectic structures. [*Ann. Physics*]{} [**111**]{} (1978), no. 1, 61 – 110. Bordemann, M., Waldmann, S.: A Fedosov star product of the Wick type for Kähler manifolds. [*Lett. Math. Phys.*]{} [**41**]{} (3) (1997), 243 – 253. Deligne, P.: Déformations de l’algèbre des fonctions d’une variété symplectique: Comparaison entre Fedosov et De Wilde, Lecomte. [*Sel. Math., New Ser.*]{} [**1**]{} (1995), 667– 697. Fedosov, B.: A simple geometrical construction of deformation quantization. [*J. Differential Geom.*]{} [**40**]{} (1994), no. 2, 213–238. Fedosov, B.: Deformation quantization and index theory. [*Mathematical Topics.*]{} Vol. 9. Akademie Verlag, Berlin (1996). Karabegov, A.: Deformation quantizations with separation of variables on a Kähler manifold. [*Commun. Math. Phys.*]{} [**180**]{} (1996), no. 3, 745–755. Karabegov A.V.: Cohomological classification of deformation quantizations with separation of variables. [*Lett. Math. Phys.*]{} [**43**]{} (1998), 347–357. Karabegov A.V.: On the canonical normalization of a trace density of deformation quantization. [*Lett. Math. Phys.*]{} [**45**]{} (1998), 217 – 228. Karabegov, A.: Formal symplectic groupoid of a deformation quantization. [*Commun. Math. Phys.*]{} [**258**]{} (2005), 223–256. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. [*Lett. Math. Phys.*]{} [**66**]{} (2003), 157 – 216. Müller-Bahns, M., Neumaier, N.: Invariant star products of Wick type: classification and quantum momentum mappings. [*Lett. Math. Phys.*]{} [**70**]{} (2004), 1 – 15. Nest, R., Tsygan, B.: Algebraic index theorem. [*Commun. Math. Phys.*]{} [**172**]{} (1995), 223–262. Nest, R., Tsygan, B.: Algebraic index theorem for families, [*Advances in Math.*]{} [**113**]{} (1995), 151–205. Neumaier, N.: Universality of Fedosov’s Construction for Star Products of Wick Type on Pseudo-Kähler Manifolds. [*Rep. Math. Phys.*]{} [**52**]{} (2003), 43 – 80.
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abstract: |
A design for a pump is suggested which is based on well known peristaltic pumps. In order to simply describe the operation of the proposed pump, an innovative interpretation of low Reynolds number swimmers is presented and thereafter a similar theoretical model would be suggested to quantify the behavior of the pumps. A coarse-grained molecular dynamic simulation is used to examine the theoretical predictions and measure the efficiency of the pump in nano scales. It is shown that this pump with a modest design is capable of being a good option for transport processes in nano scale.
*Keywords*: nano-pump, microfluidics, molecular dynamics.
author:
- 'F. Farahpour'
- 'M.R. Ejtehadi'
bibliography:
- 'biblio.bib'
title: Peristaltic pumps work in nano scales
---
Introduction
============
During the last decades there was an increasing interest in fabrication of systems in nano and micro scales and according to the importance of pumps as a part of microfluidic systems, so far great efforts have been done to fabricate pumps in miniature scales. During recent years several different kind of micro and nano pumps, based on different pumping methods and principles have been designed and constructed [@Laser2004; @Woias2005; @Tsai2007; @Iverson2008; @Chen2010; @Wang2012].\
Beside all of these efforts, fabrication and developing of valveless micropumps attracted more attention. Having no valve in fact means the elimination of some of mechanical parts of micropump which make the whole pump, ineffective in short time because of their unremitting wear and fatigue. One of the most important category of valveless micropumps, which in last years some different kind of it was manufactured and developed, is peristaltic pumps. Peristaltic and Pneumatic micropumps using piezoelectric disks as actuators, are one of the first developed micropumps [@Olsson1998]. Simply one can consider a peristaltic pump as a long tube in which contractile waves traveling along it, induce a fluid flow inside it [@Berg2003; @Laser2004]. Characteristics and efficiency of these simple pumps are vastly investigated in literature and is used for micro pumping in biology and industry [@Lee2012a; @Goldschmidtboing2005; @Chang2007; @Knight2004; @Jeong2007; @Huang2006; @Berg2003]. Simplicity of the structure of peristaltic and Pneumatic pumps is a great advantage which can be a base for their miniaturization.\
Fabrication of nanofluidic structures is more challenging than microfluidic systems. This is in principal due to difficulty of manufacturing of nano scale structures and significant changes in fluids behavior because of changing the size of the system in several orders of magnitude. Nevertheless, during last years some nanopumps have been designed and developed [@Wang2012; @Wang2007; @Xu2012a; @Lohrasebi2012; @Chen2010; @Tuzun1996; @Zhang2008].\
In 2003, Berg *etal.* suggested a two-stage discrete peristaltic micropump which seems to be the simplest kind of peristaltic pumps [@Berg2003]. This pump is only consisted of three connector tubes and two actuators which their specific periodic motion yields to the fluid flow in a distinctive direction. This pump needs no valve and in addition it is a bi-directional pump, which means that only by reversing the order of operation of two actuators, the flow direction will be reversed.\
One of the most important benefits of this pump which would be discussed in this paper is its capability to be miniaturized to nanometer scale. In addition to its structural simplicity, the actuation sequence in one full pumping cycle of this pump is non$-$reciprocal. One of the known challenges in micro and nano scales (low Reynolds regime) to create an effective directional motion or flow is that exertion of any force to the fluid in this regime could be completely neutralized by a time-reversed force [@Nelson] but it has been shown that a non reciprocal periodic motion generally can yield to an effective motion [@Shapere1989; @Nelson].\
In some feature, Operation of this pump is similar to non-reciprocal swimmers have already been introduced [@Purcell1977; @Najafi2004; @Avron2005]. In the next section of this paper this similarity would be described more clearly.\
In Berg’s suggested pump, actuator regions should be opened and closed completely but one can show that even in half-closed state, this pump can still generate a directional flow but its efficiency depends on the details of the procedure such as difference between actuators’ volume in open and close states and frequency of actuation. A simple analogous model for the Berg’s pump can be considered in which whole pump is modeled as a long tube which connects two reservoir tanks. Two regions of the tube, are flexible and act as actuators. In the next section this model would be discussed in detail.
Analogy between Berg pump and low Reynolds micro swimmers
=========================================================
For better understanding of operation of berg pump in nano scale, one can establish a similarity between Berg pump and low Reynolds micro swimmers like the swimmer introduced by Najafi and Golestanian (NG) or the one designed by Avron, Kenneth and Oaknin (AKO) which are the simplest non-reciprocal swimmers and their motion can be described just by two independent parameters. one can shows that their motion in phase$-$space, constructed by these parameters, forms a closed loop in a full cycle, instead of a repetitive back and forth motion. For example NG swimmer can be described by the length of its two arms. This model swimmer consists of three spheres with radius $R$ that are connected by two parallel rigid slender arms, with lengths $l_1$ and $l_2$. There are two internal engines on the middle sphere, which act as internal active elements responsible for making a nonreciprocal motion which is needed to propel the whole system [@Najafi2004]. One complete period of its motion, as can be seen in Fig. \[Golestanian-Avron\]-a, could be represented by a closed loop in $l_1-l_2$ space. It is shown that this swimmer by breaking the time-reversal symmetry as well as the translational symmetry can swim at low Reynolds number. Similarly, AKO swimmer, which has a non-reciprocal motion in $l-v$ space (Fig. \[Golestanian-Avron\]-b), is another example of such swimmers [@Avron2005].
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![Operation of two micro swimmers and their non-reciprocal loops. a) NG swimmer, with three fixed-volume spheres and two variable-length bonds. b) AKO swimmer, two variable-volume spheres and one variable-length bond. \[Golestanian-Avron\]](./images/Golestanian.eps "fig:"){width=".8\columnwidth"}
$(a)$
![Operation of two micro swimmers and their non-reciprocal loops. a) NG swimmer, with three fixed-volume spheres and two variable-length bonds. b) AKO swimmer, two variable-volume spheres and one variable-length bond. \[Golestanian-Avron\]](./images/Avron.eps "fig:"){width=".8\columnwidth"}
$(b)$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\
In an innovative look to NG swimmer, with a simple analysis one can realize that this swimmer, in fact, in a non-reciprocal cycle in $l_1-l_2$ space, changes its drag coefficient in such a way that forces can not be eliminated by each other in forward and backward motion. This changes in drag coefficient is exactly the origin of the breaking in time-reversal symmetry and hence the effective displacement of the swimmer. To show the simplicity of this analysis we assume that drag coefficient of two adjacent spheres, when they are far away from each other (expansion of the connector arm), is equal to ${\zeta }_1$ but when two beads are close to each other (contraction of the connector arm) the drag coefficient of their collection is ${\zeta }_2<2{\zeta }_1$, due to hydrodynamic interaction between them [@Nelson]. Meanwhile we consider that swimmer’s arms expand and contract by a uniform and constant velocity in whole cycle. The length of the arms in expanded and contracted state is $L$ and $L-e$, respectively. Periodic motion of this swimmer, as shown in Fig. \[Golestanian-Avron\]-a, is consisted of left-arm contraction, right-arm contraction, left-arm expansion and finally, right-arm expansion.\
By considering the stokes’ law for the motion in viscous environments, one can calculate the magnitude of displacement of the middle bead. In the $i$th stage of the motion, $i=1,\dots,4$, the velocity of the middle bead relative to the fluid, which is considered stationary, is $v_i$ and its displacement is $\Delta x_i$. Thus, for example, in the first stage, $$\begin{aligned}
\
{\zeta }_1\left(u+v_1\right)+{\zeta }_1v_1+{\zeta }_1v_1=0\\end{aligned}$$ is the governing equation of the swimmer’s motion and therefore $v_1=-\frac{u}{3}\ $and $\Delta x_1=-\frac{e}{3}$. But in the second stage, the drag coefficient of the constituents of the swimmer will change as described above and thus governing equation will change to a new form $$\begin{aligned}
\
{\zeta }_2v_2+{\zeta }_1\left(v_2-u\right)=0\\end{aligned}$$ and $v_2=\frac{{\zeta }_1}{{\zeta }_1+{\zeta }_2}u$ and $\Delta x_2=\frac{{\zeta }_1}{{\zeta }_1+{\zeta }_2}e$. Repeating these calculations in the next stages, gives us $\Delta x_3=\frac{{\zeta}_1}{{\zeta }_1+{\zeta }_2}e$ and $\Delta x_4=-\frac{e}{3}$, so in a complete cycle of motion, forward displacement of the swimmer is obtained by $$\begin{aligned}
\
\Delta x=\sum_i{\Delta x_i{\rm =}\frac{{\rm 2}}{{\rm 3}}{\rm
e(}\frac{{\rm 2}{\zeta }_{{\rm 1}}{\rm -}{\zeta }_{{\rm 2}}}{{\zeta
}_{{\rm 1}}{\rm +}{\zeta }_{{\rm 2}}}{\rm )}>0}.\end{aligned}$$\
In order to make an analogy between Berg pump and NG swimmer one can consider a simple model for the Berg pump in which whole pump is modeled as a long tube which connects two reservoir tanks. Two regions of the tube, are flexible and act as actuators (Fig.\[nanopump\]).
![Operation of Berg pump and its non-reciprocal loop in $r_1-r_2$ space. Flow direction and magnitude in each step is shown schematically with gray arrows.\[nanopump\]](./images/nanopump.eps){width=".9\columnwidth"}
\
For example one can assume three carbon nano tubes, which are connected together via two protein nano-pores, like OmpF channel, a biological pore found in the outer membrane of Escherichia coli, and the width of these nano-pores can be changed by a specific activation in a controlled manner. Another model which can be considered is a long tube that two specific proteins, with two activation modes (contracted and expanded), or thermosensitive particles, with hydrodynamic radius dependent on temperature, are fixed in two regions of it (e.g. attached to its interior wall). By applying an AC electromangnetic radiation or focusing laser beams on their positions, [@Senff1999] one may be able to activate the proteins in such a way that widening and narrowing of two actuator parts of the Berg’s pump is modeled. Such mechanisms of contraction and expansion of proteins in nanoscale is known today for some biologic gates, like OmpF, in selective transferring of ions.\
Two flexible actuator regions of the tube are indexed by $A_1$ and $A_2$ (Fig.\[nanopump\]). Three connector tubes are $T_1$, $T_2$ and $T_3$. The width of $A_1$ and $A_2$ regions decreases and increases in the order which is shown in Fig. \[nanopump\], hence an effective flow is generated. The characteristic parameters of this pump are the radius of $A_1$ and $A_2$ regions ($r_1$ and $r_2$) which their variation in $r_1-r_2$ space forms a nonreciprocal loop. For simplicity we consider that the changes in volume of actuators are made by varying their radius in a uniform way, and the radius of an actuator is equal throughout its length. The real deformation of actuator is certainly smoother, but this assumption has no effect in generality of the pump’s operation.\
In Berg pump, resistance of two flexible regions changes exactly in the same way of the drag coefficient in NG swimmer. The fluid flow inside a pipe in low Reynolds regime can be obtained from a simple relation, $Q=\frac{p}{Z}$ [@Nelson], in which $p$ is the pressure difference between two ends of the pipe and $Z=\frac{8L\eta }{\pi R^4}$ is the pipe resistance against fluid flow. The dependence of resistance on the forth power of pipe’s radius, gives us this ability to make an effective change in the resistance of the pipe just by a little change in its radius. We assume that narrowing a region of the pipe increases the pressure in that region from $p$ to $p+\Delta p$ and widening it decreases the pressure to $p-\Delta p$. The length of $A_1$ and $A_2$ regions and $T_1$, $T_2$ and $T_3$ connector tubes are $L$, initial radius of whole tube is $R$ and the radius of flexible parts in contracted state is $r$. In addition we suppose that pressure is uniform in a tube by passing a specific time after a local variation. There is a delay time between two consecutive stages, $\delta t$, which is essential for this last condition to be established and finally we define the resistance of the tubes in expanded and contracted form as $Z_e=\frac{8L\eta }{\pi R^4}$ and $Z_c=\frac{8L\eta }{\pi r^4}$, respectively.\
In the first stage of motion, pressure in $A_1$ region increases to $p+\Delta p$. Flow resistance for right and left part of the first actuator in this stage are $Z_r=\frac{8(3L)\eta}{\pi R^4}=3Z_e$ and $Z_l=\frac{8L\eta }{\pi R^4}=Z_e$, respectively. $Z_r$ is obtained by adding the resistance of two connector tubes ($T_2$ and $T_3$) with the right actuator ($A_2$) in expanded state but $Z_l$ is only resistance of one connector tube ($T_1$). So resistance against right direction flow in right part is greater and flow in right and left part will be $Q_{1r}=+\frac{\Delta p}{{3Z}_e}$ and $Q_{1l}=-\frac{\Delta p}{Z_e}$ (Gray arrows in Fig.\[nanopump\]).\
In second stage, pressure in $A_2$ actuator increases but this time $A_1$ is half-closed and therefore $Z_r=Z_e$ (resistance of $T_3$ connector tube in the right side of $A_2$ actuator) and $Z_l=2Z_e+Z_c$ (sum of resistance of two connector tubes ($T_1$ and $T_2$) and the left actuator ($A_1$) in contracted state), so for flows, one obtaines $Q_{2r}=+\frac{\Delta p}{Z_e}$ ** and $Q_{2l}=-\frac{\Delta p}{2Z_e+Z_c}$.\
In the third and forth stages, after widening the actuators sequentially the pressure decreases in respective actuator and fluid flows toward the operating actuator (Fig.\[nanopump\]). After some straightforward calculation one reaches to $Q_{3r}=-\frac{\Delta p}{2Z_e+Z_c}$ ,$Q_{3l}=+\frac{\Delta p}{Z_e}$ ,$Q_{4r}=-\frac{\Delta p}{Z_e}$ and $Q_{4l}=+\frac{\Delta p}{3Z_e}$. Finally, after summing the flow magnitude in all of the stages, one reaches to the total amount of flow in the whole cycle: $$\begin{aligned}
\
Q=\sum_i{\left(Q_{ir}+Q_{il}\right)=+2\Delta
p(\frac{1}{3Z_o}}-\frac{{\rm 1}}{2Z_o+Z_e}).\\end{aligned}$$\
According to the definition of $Z_e$ and $Z_c$ and by considering the fact that $r<R$, one can see that the expression in the parentheses is always positive and hence there is a fluid flow in right direction inside the pipe. In reverse order, when the second actuator is closed at first, a leftward flow is generated.\
To show the effective operation of this pump in nano scale, we constructed a simple model and investigated its general performance based on some relevant parameters of the system by a molecular dynamic simulation. To simplify the process of modeling and without any loss of generality, we modeled the pump with a long connecting pipe between input and output reservoir and the actuators by putting two fixed beads with variable volume in each actuator part. The volume of these beads increases and decreases according to the algorithm mentioned above (in order to make a contraction in each actuator, the corresponding bead should expand and vice versa) (Fig. \[tube-bead\]). Such a modeling has a great advantage which can show the duality between swimmers and pumps more distinctly. In other words one can imagine that this pump is a swimmer consisted of a rod with fix length that in two ends is connected to two beads with variable volume and the swimmer as a whole is fixed in middle point of the rod.
Simulations and results
=======================
![The simple model used in simulation for actuation of pump. \[tube-bead\]](./images/tube-bead.eps){width=".4\columnwidth"}
Nanometer pipe is modeled by a fixed carbon nanotube and the fluid filled it, as liquid argon. In Fig. \[snapshot\] a schema of simulation box included of two reservoirs, connecting pipe, liquid particles and two actuator beads are shown. Interaction potentials between liquid particles and wall, between liquid and two actuators and between liquid particles are Lennard-Jones potential: $$\begin{aligned}
\
V_{LJ}\left(r\right)=4{\epsilon}[{\left(\frac{\sigma}{r}\right)}^{12}-{\left(\frac{\sigma }{r}\right)}^6].\end{aligned}$$\
Lennard-Jones’ parameters are fixed as ${\sigma }_{c-Ar}=3.573\ \mathop{\rm A}\limits^\circ$, ${\epsilon}_{c-Ar}=0.2827\ kcal.{mol}^{-1}$, ${\sigma}_{Ar-Ar}=3.35\ \mathop{\rm A}\limits^\circ $ and ${\epsilon}_{Ar-Ar}=0.2862\ kcal.{mol}^{-1}$ [@Tuzun1996]. The simulation temperature and density are 80${K}$ and $1.4 g/cm^3$ respectively. From now on we use $Ar$ atom diameter as the unit of length, $\sigma_0$ and $Ar-Ar$ interaction energy as the unit of energy, ${\text{\usefont{OML}{cmr}{m}{n}\symbol{15}}}_0$ so time would be given in units of $\tau_0=\sigma_0\sqrt{m_0/{\text{\usefont{OML}{cmr}{m}{n}\symbol{15}}}_0}=1.96\times10^{-12}s$, where $m$ denotes the mass of an Argon atom. Unless explicitly stated, all the quantities in this paper are given in these reduced units. Before actuating the actuators, we let simulation run for 100000 steps to reach equilibrium. The expansion and contraction time of actuators are $5\tau_0$ and the delay time between two consecutive stages, $\delta t$, varies depending on the situation. Simulation box is periodic in all direction and so pressure of two reservoirs are the same.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![a) Schema of simulation box. b) A look through the tube before starting the stroke. liquid particles (blue) fill the tube and two actuator beads (red) are fixed. c) Right actuator is expanded (part $A1$ is narrowed). d) Both actuators are expanded. \[snapshot\]](./images/tube.png "fig:"){width=".48\columnwidth"} ![a) Schema of simulation box. b) A look through the tube before starting the stroke. liquid particles (blue) fill the tube and two actuator beads (red) are fixed. c) Right actuator is expanded (part $A1$ is narrowed). d) Both actuators are expanded. \[snapshot\]](./images/0b.png "fig:"){width=".48\columnwidth"}
$(a)$ $(b)$
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![a) Schema of simulation box. b) A look through the tube before starting the stroke. liquid particles (blue) fill the tube and two actuator beads (red) are fixed. c) Right actuator is expanded (part $A1$ is narrowed). d) Both actuators are expanded. \[snapshot\]](./images/1b.png "fig:"){width=".48\columnwidth"} ![a) Schema of simulation box. b) A look through the tube before starting the stroke. liquid particles (blue) fill the tube and two actuator beads (red) are fixed. c) Right actuator is expanded (part $A1$ is narrowed). d) Both actuators are expanded. \[snapshot\]](./images/2b.png "fig:"){width=".48\columnwidth"}
$(c)$ $(d)$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
\
Fig. \[profile\] shows a sample flow profile (number of fluid particles crossing a cross section of the pipe to the right) versus time. In these series of simulation $\delta t=10$, tube radius is $4$ and first and final radius of the actuator beads are $1$ and $4$, respectively. This flow profile interestingly follows the actuators’ operation and has distinctive rises (falls) at the beginning of each expansion (contraction) stage.
![Number of crossed particles from a cross section of pipe (N) versus time (t) in 10 cycles. Data are avaraged from 20 independent simulations. Dashed line is a linear fit to whole data and can be regarded as flow rate, $Q$. Here $Q=0.14/\tau_0$.\[profile\]](./images/profile.eps){width="1\columnwidth"}
In Fig. \[Q\] flow rate (slope of $N-t$ diagram), $Q$, versus $\delta t$, delay time between two consecutive stages, is plotted. Here, tube radius is $3$ and first and final radius of the actuator beads are $0.5$ and $2.5$. At small $\delta t$, fluid has not enough time to flow in true direction and at big $\delta t$, the delay is so long (much bigger than equilibrium time of the system) that fluid particles have enough time to reach their equilibrium state and rearrange again uniformly trough the simulation box and hence pump is not efficient in very small and big delay times. Such a dependence on delay time or frequency is generally observable in peristaltic pumps [@Lee2012a; @Goldschmidtboing2005].
![flow rate versus $\delta t$. Every point in this plot is obtained from 20 MD runs. \[Q\]](./images/Q.eps){width="1\columnwidth"}
\
Fig. \[act\_rad\] shows the effect of actuator radius on flow process. For four different value of final radius of actuator beads, $r$, flow profile is sketched versus time. Here $\delta t=5$, tube radius is $4$ and first radius of the actuator beads is $1$. As it can be seen the effective pump is the one with complete contraction and flow rate decreases very fast as final radius decreases. As it is expected the maximum flow rate corresponds to full clogging of the tube.
![Number of crossed particles from a cross section of the pipe (N) versus time (t) in 10 cycles for four different value of actuators final radius. Lines are linear fit to data. Each curve is average of 20 different MD runs. Inset shows the dependence of flow rate versus final radius of the actuators. Connecting lines are represented just to guid eyes. \[act\_rad\]](./images/act_rad.eps){width="1.0\columnwidth"}
\
Conclusion
==========
In this paper it is discussed that the simple designed peristaltic pump of Berg is able to generate an effective flow in a distinctive direction in nano scale. This can be shown by a simple theoretical model in which we consider that the change in the radius of tube yields to the change in flow resistance of two actuator parts of the tube in a non-reciprocal way. For the first time a modest interpretation of NG swimmer is proposed which has the essential features of the system to model its non-reciprocal behavior. It is shown that there is an analogy between NG swimmer and this pump, in both system non reciprocal change of qualitative parameters breaks the symmetry of space and different resistance in two parts of the system makes an effective relative motion between the fluid and the system. In our simple model, Berg pump is composed of a long tube connecting two fluid reservoirs. Two flexible parts, with two natural states, contracted and expanded, are fixed in two points along it. These two flexible parts act as actuators of the pump and by their periodic contraction and expansion, enforce the fluid to flow in a defined direction. In reality these actuators can be two proteins with two natural states, expanded and contracted, which by a specific active control, like emission of an electromagnetic wave, can switch between these two states. Biological gates such as OmpF are other candidates for doing these job. By considering difficulties of experimental design, only as a suggestion, one can consider a square pulse wave that is propagating from left of the setup (Fig. \[nanopump\]) and its amplitude oscillates between two value, one of them in the range that can makes the actuator protein to expand and the other forces it to contract. If the wave length of this wave is at least a bit larger than the distance between two actuator parts, process described in Fig. \[nanopump\] would be simulated.\
We simulated this nanoscale pump with a long narrow tube and two actuator beads and investigated its behavior under different frequency of actuator’s oscillation and tube radius. There is a frequency in which this pump have maximum fluid flow and this frequency depends on other parameters of the system such as tube radius.
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abstract: 'We present analytical and numerical studies of pinned superconducting states of open-ended Josephson ladder arrays, neglecting inductances but taking edge effects into account. Treating the edge effects perturbatively, we find analytical approximations for three of these superconducting states – the no-vortex, fully-frustrated and single-vortex states – as functions of the dc bias current $I$ and the frustration $f$. Bifurcation theory is used to derive formulas for the depinning currents and critical frustrations at which the superconducting states disappear or lose dynamical stability as $I$ and $f$ are varied. These results are combined to yield a zero-temperature stability diagram of the system with respect to $I$ and $f$. To highlight the effects of the edges, we compare this dynamical stability diagram to the thermodynamic phase diagram for the infinite system where edges have been neglected. We briefly indicate how to extend our methods to include self-inductances.'
address:
- 'Physics Department, Massachusetts Institute of Technology, Cambridge, MA 02139'
- 'Kimball Hall, Department of Theoretical and Applied Mechanics and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853'
- 'Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139'
author:
- 'Mauricio Barahona[^1]'
- 'Steven H. Strogatz'
- 'Terry P. Orlando'
date: 'To appear in Phys. Rev. B, Jan 1 1998'
title: Superconducting states and depinning transitions of Josephson ladders
---
Introduction
============
Arrays of Josephson junctions are of interest in several branches of physics [@scbooks]. They have many technological applications, including high-frequency emitters and detectors, parametric amplifiers, local oscillators, and voltage standards [@scbooks; @technology]. They also shed light on the structural [@kleiner] and pinning [@mpa1] properties of the high-T$_{\rm c}$ superconducting cuprates. At the same time, they provide model systems for the study of problems in both spatiotemporal nonlinear dynamics [@series; @shiseries; @shilong; @summary] and nonequilibrium statistical physics [@summary; @kardar0]. For instance, the depinning transitions and nonlinear wave propagation seen in Josephson arrays are analogous to those found in incommensurate systems, earthquake models, Type II superconductors, and charge-density waves.
From the standpoint of dynamical systems theory [@steve], Josephson arrays can be viewed as large collections of coupled nonlinear oscillators. Unfortunately, because of their nonlinearity and large number of degrees of freedom, these arrays are inherently difficult to analyze mathematically. A further complication is that there is an intrinsic physical coupling among junctions, due to fluxoid quantization, which is more awkward to handle than the nearest-neighbor interaction usually assumed in idealized models of coupled oscillators. And when the effects of self-fields and inductances are included, there is even less hope of making analytical progress.
Despite these obstacles, some encouraging advances have occurred recently in the mathematical analysis of Josephson arrays, especially for one-dimensional (1-D) systems where the junctions are connected in series [@series; @shiseries] or in parallel [@shilong]. The logical next step is to tackle two-dimensional (2-D) arrays. Much of the previous theoretical work on 2-D arrays has focused on numerical simulation of the current-voltage characteristics [@experIV], in an effort to link the rich spatiotemporal dynamics of 2-D arrays to the averaged quantities that are most readily measured experimentally. On the mathematical side, there are recent indications that 2-D arrays, like their 1-D counterparts, are also going to be tractable in some regimes [@analysis2d].
An ideal example to explore the crossover between 1-D and 2-D behavior is the Josephson ladder array (Figure \[fig:ladder\]). Following Kardar [@kardar1; @kardar2], several authors have studied various statistical properties of the frustrated ladder, including its ground state, the complicated landscape of solutions at zero temperature, the low-lying excitations, and the linear response regime [@benedict; @falo2; @denniston]. However, all of these authors restricted attention to ladders in the absence of a driving current. Only recently has the fully dynamical problem been addressed, through numerical simulations of the depinning transition [@stroudscreen] and vortex propagation [@strouddyn].
In this paper we use the tools of nonlinear dynamics to analyze the superconducting states of ladder arrays. (Other dynamical regimes will be discussed elsewhere [@mythesis; @usladder2].) Mathematically, the superconducting states correspond to fixed points of the governing circuit equations. We study the bifurcations of these fixed points with respect to variations in the applied dc bias current $I$ and the frustration $f$ introduced by an external magnetic field. As $I$ is increased from $0$ at fixed $f$, we find that the stable superconducting states are destroyed in saddle-node bifurcations at certain critical values of the current. Then the system depins from its original static configuration and evolves toward some other state. The new state might be another fixed point, or it might be a running solution, in which case a nonzero dc voltage appears across the array. Global depinning of the array occurs when the last stable fixed point is destroyed. One of the main results of our analysis is a set of approximate analytical formulas for the critical currents at which the fixed points are destroyed, as a function of $f$, for the three most important types of superconducting states: the no-vortex, fully-frustrated and single-vortex solutions.
Another important finding is that symmetry plays a crucial role in the dynamics of the ladder. As we will show below, much of the behavior of the ladder can be understood by focusing on states that are “up-down symmetric” – in other words, states where the phases of the horizontal junctions on the top and bottom of any given plaquette are equal in magnitude but opposite in sign at all times. All of the stable superconducting states possess this symmetry. Even when the parameters are chosen so that depinning occurs, the subsequent transients and long-term running solutions typically remain up-down symmetric. But there is at least one exception: when the single-vortex state is destabilized by lowering the frustration $f$ below some critical value $f_{\rm min}$, the system depins via a symmetry-breaking bifurcation. During the transient behavior, the up-down symmetry is lost temporarily, but is then recovered as the system expels flux from the array and evolves toward the no-vortex state.
These symmetry considerations establish an unexpected link between the study of Josephson arrays and some recent developments in nonlinear dynamics. In mathematical terms, the up-down symmetric states of a ladder array form an invariant manifold of the full state space. A symmetry-breaking bifurcation occurs when this manifold loses stability in a transverse direction. The same issue – the transverse stability of an invariant manifold – arises in the study of riddled basins, synchronized chaos, on-off intermittency, and blowout bifurcations [@transverse]. These connections suggest a promising line of future research on Josephson arrays, particularly with regard to their chaotic states.
We will also show that the dynamics of ladder arrays are very strongly influenced by edge effects. One might have supposed these effects to be negligible, especially in long ladders, since their influence on the superconducting solutions dies off exponentially fast away from the boundaries. Yet although the edges do indeed have a small effect on the [*form*]{} of the superconducting solutions, they have a large effect on the [*stability*]{} of those solutions. Much of this paper is devoted to investigating the effects of the edges, first on the superconducting states themselves, then on their stability, and finally on the entire phase diagram.
This paper is organized as follows. Section \[sec:system\] reviews the model equations for the ladder and discusses their symmetry properties. In Section \[sec:analytical\] we obtain analytical approximations for three numerically observed superconducting solutions: the no-vortex (nv), single-vortex (sv), and fully-frustrated (ff) solutions. In all cases, edge effects are taken into account via perturbation theory. Next, in Section \[sec:depinning\] we describe the dynamical simulations which reveal the depinning properties of the nv, sv, and ff configurations and relate them to the global depinning of the ladder. In Section \[sec:bifurc\] we establish the rigorous connection of the dynamical depinning with the stability of these three fixed points. For all of them, we characterize the bifurcations and study their stability diagrams, both in the presence and in the absence of edges. When possible, analytical approximations to the critical currents are obtained. We show that the depinning transitions correspond to saddle-node bifurcations that are edge-dominated for almost all values of the frustration. Moreover, we find that some of the superconducting states can be destabilized via a subcritical pitchfork bifurcation as the frustration is reduced; in physical terms, this is a symmetry-breaking bifurcation in which flux is expelled [*transversally*]{} from the ladder. In the final section we summarize our conclusions in two phase diagrams (for ladders with and without edges), and we relate our results to those found by previous authors. We also add two more technical appendices: Appendix A compares the single-vortex configuration in the ladder with the corresponding kink-like solution in 1-D parallel arrays; Appendix B briefly indicates how to extend our approach to include self-inductance effects.
The system {#sec:system}
==========
We study an open-ended Josephson ladder with $N$ square plaquettes, i.e., an array formed by two rows of $N+1$ weakly coupled superconducting islands (Figure \[fig:ladder\]). The array is driven by a perpendicular uniform dc current $I$, and a magnetic field is applied transverse to the plane of the device. Each weak link between islands constitutes a junction. Its state is described by the gauge-invariant phase difference $\phi_{j}$, arising from the macroscopic character of the quantum wavefunction of the superconductors.
Assuming zero temperature, negligible charging (quantum) effects, and identical junctions, the dynamics of each junction is given, in the three-channel RCSJ model [@scbooks], by the nonlinear differential equation $$\label{eq:pendulum}
I_{j}={\ddot \phi_{j}} + \beta_{c}^{-1/2}
{\dot \phi_{j}} + \sin \phi_{j} \equiv {\cal N}(\phi_{j}).$$ Here $\beta_{c}$ is the McCumber parameter [@scbooks]; $I_{j}$ is given in units of the critical current of each (identical) junction; derivatives are with respect to time normalized in units of $\omega_{J}^{-1}$, the inverse of the plasma frequency; and ${\cal N}$ is shorthand for the nonlinear differential operator defined by (\[eq:pendulum\]). Thus, each junction is formally equivalent to a damped driven pendulum [@steve].
The junctions are [*intrinsically*]{} coupled, even when inductances are neglected, through two physical restrictions: the quantization of the magnetic flux through each plaquette, and Kirchhoff’s current conservation law at each node. When all inductances are zero, i.e., self-fields are neglected, the flux quantization condition for the array in Figure \[fig:ladder\] becomes $$\label{eq:fluxoid}
\phi_{j}^{V}+\phi_{j}^{h}-\phi_{j+1}^{V}-\phi_{j}^{H}=2 \pi (n_{j}-f),
\hspace{.1in} j=1,\ldots,N$$ where $f$ is the [*external*]{} magnetic flux in units of the flux quantum $\Phi_{0}$. The set of integers $\{n_{j}\}$ indicate the presence ($n_{j}=\pm 1$) or absence ($n_{j}=0$) of topological vortices in each plaquette when the phases are restricted to the interval $[-\pi,\pi)$.
In addition, Kirchhoff’s current law yields $$\begin{aligned}
\label{eq:interiorequation1}
&I_{j-1}^{H} + I = I_{j}^{H} + I_{j}^{V},\;\;& j=2,\ldots,N\\
&I_{j-1}^{h} + I_{j}^{V} = I + I_{j}^{h},\;\;& j=2,\ldots,N
\label{eq:interiorequation2}\end{aligned}$$ at the interior nodes of the ladder. At the left edge, $$\begin{aligned}
\label{eq:edgeequation1}
&I = I_{1}^{H} + I_{1}^{V}, \\
&I_{1}^{V} = I + I_{1}^{h},
\label{eq:edgeequation2}\end{aligned}$$ while at the right edge, $$\begin{aligned}
\label{eq:edgeequation3}
&I_{N}^{H}+ I = I_{N+1}^{V}, \\
&I_{N+1}^{V} + I_{N}^{h} = I .
\label{eq:edgeequation4}\end{aligned}$$ In summary, equations (\[eq:pendulum\])–(\[eq:edgeequation4\]) define our model for the dynamics of the ladder array. A useful mechanical analog for the system is a frustrated lattice of coupled, damped, nonlinear pendula driven by a constant torque applied at the edges.
An important restriction on the currents immediately follows from the presence of the edges. Equations (\[eq:edgeequation1\]) and (\[eq:edgeequation2\]) imply that $I_{1}^{H} = -I_{1}^{h}$. Moving successively from the left edge to the interior of the ladder, Kirchhoff’s current law yields $$\label{eq:returnedges}
I_{j}^{H} = -I_{j}^{h},
\hspace{.2in} \forall \; j.$$
This condition (\[eq:returnedges\]) is automatically satisfied by any phase configuration whose evolution obeys the [*up-down symmetry*]{} $$\label{eq:phiupdown}
\phi^{H}_{j}(t)=-\phi^{h}_{j}(t), \hspace{.3in} \forall \; j, \,
\forall \; t$$ as can be seen from (\[eq:pendulum\]). Moreover, if the initial conditions satisfy $\phi^{H}_{j}(0)=-\phi^{h}_{j}(0)$ along with similar equalities on the first time-derivatives, the governing equations imply that those equalities will hold for [*all*]{} time. In geometrical terms, the set of all up-down symmetric states (\[eq:phiupdown\]) forms an invariant submanifold of the full phase space.
On the other hand, it is certainly possible to choose initial conditions that do not have this up-down symmetry. But our simulations indicate that for a wide range of parameters and initial conditions, arbitrary phase configurations rapidly evolve toward up-down symmetric states. In other words, the invariant manifold is typically attracting in the transverse directions — initial states that are off the manifold are soon drawn onto it. There are also exceptions to this rule: as we will see in Section \[sec:bifurc\], the single-vortex and fully-frustrated states can lose transverse stability as the frustration $f$ decreases. Nevertheless, a great deal of insight can be obtained by restricting attention to the submanifold of up-down symmetric states. Thus, for much of this paper we will assume that (\[eq:phiupdown\]) holds, and we will replace $\phi^{h}_{j}$ with $-\phi^{H}_{j}$ throughout the governing equations.
There is a simple condition for the transverse stability of an up-down symmetric fixed point. It can be shown [@mythesis] that such a fixed point is linearly stable to perturbations that are [*strictly normal*]{} to the manifold if and only if $$\label{eq:horizontalcond}
|\phi^{H}_{j}| < \pi /2,
\hspace{.2in} \forall \; j.$$ We have checked numerically that in the instances when the up-down symmetry is broken (and, thus, the system escapes the symmetric manifold), the subsequent evolution does in fact take place purely along the normal direction. Therefore, any up-down symmetric fixed point that is stable must satisfy this inequality (\[eq:horizontalcond\]). Hence (\[eq:horizontalcond\]) constitutes a [*necessary*]{} condition for stability. (It is not sufficient, however, because it only governs the transverse direction; it says nothing about the stability with respect to perturbations that preserve the symmetry. Further conditions would be needed to ensure stability in directions [*along*]{} the invariant manifold as will be shown below.)
In summary, the governing equations can be written compactly as ${\bf f(x) = 0}$, with ${\bf x}=(\phi^{V}_{1},\ldots,\phi^{V}_{N+1},\phi^{H}_{1},\ldots,
\phi^{H}_{N},\phi^{h}_{1},\ldots,\phi^{h}_{N})$, and with the components of ${\bf f(x)}$ defined by $$\begin{aligned}
\label{eq:eq1}
{\rm f}_{j}({\bf x})&=& I+{\cal N}(\phi^{H}_{j-1})-{\cal N}(\phi^{H}_{j})-
{\cal N}(\phi^{V}_{j}), \nonumber\\
&& \hspace*{1in}j=1, \ldots, N+1\\
{\rm f}_{N+1+j}({\bf x})&=&{\cal N}(\phi^{H}_{j})+{\cal N}(\phi^{h}_{j}),
\nonumber\\
&& \hspace*{1in}j=1, \ldots, N\\
\label{eq:eq2}
{\rm f}_{2N+1+j}({\bf x})&=&\phi_{j}^{V}+\phi_{j}^{h}-\phi_{j+1}^{V}
-\phi_{j}^{H}-2 \pi(n_{j}- f), \nonumber\\
&& \hspace*{1in}j=1,\ldots,N.
\label{eq:eq3}\end{aligned}$$ The dynamical evolution of the state $\{{\bf x}(t),{\dot {\bf x}}(t)\}$ of the system is obtained by numerically solving this system of coupled differential and algebraic equations. The dynamics also depend implicitly on the parameters $I, \beta_{c}$, $f$, and $N$.
In experiments, the most convenient way to probe the dynamics of the array is to measure its dc current-voltage ($IV$) characteristics. From the Josephson relations [@scbooks], the time-dependent voltage across each junction is directly proportional to ${\dot{\phi}_{j}}$, the time-derivative of its phase. Hence, the total dc voltage $V$ across the array in the vertical direction is proportional to the spatial [*and*]{} temporal average of all the phase derivatives. Although because of this averaging a great deal of dynamical information is lost about the spatiotemporal state of the system, the $IV$ curve still provides a useful (if somewhat coarse) indicator of changes in the underlying dynamics as the drive current $I$ is varied.
In the case of ladder arrays, the $IV$ curves display three regions associated with distinct dynamical behaviors [@strouddyn; @mythesis]. At low $I$, the system is [*superconducting*]{} ($V=0$) with pinned or slightly oscillating junctions (in the mechanical analog, the pendula are at rest or librating slightly). At a depinning current $I_{\rm dep}$, the array jumps to the [*flux-flow*]{} region, in which a finite voltage is produced by vortices of magnetic flux moving across the array. At still higher currents, the dynamics is characterized by [*whirling modes*]{} [@shilong] (in the mechanical analog, all the pendula rotate over the top at a nearly uniform angular velocity proportional to the applied torque). In this state, there is a linear [*ohmic*]{} dependence of $V$ on $I$. In the remainder of the article we focus on the superconducting states and the critical current $I_{\rm dep}$ at which depinning occurs.
Observed superconducting solutions and analytical approximations {#sec:analytical}
================================================================
We have performed dynamical simulations of the array in which the current is ramped up from zero with different initial conditions. The superconducting solutions observed in those simulations are always static states, i.e., fixed points of the system. (In principle, time-dependent solutions with high cancelling symmetry could also have zero total dc voltage, but we never see such states in our simulations.) More specifically, for any given $\beta_{c}$ and $N$, only three types of configurations appear in the numerics: [*no-vortex*]{} solutions (Figure \[fig:scnovortex\]) for $f$ smaller than $\sim 0.3$, and solutions of the [*single-vortex*]{} type (Figure \[fig:onevortex\]) and [*fully-frustrated*]{} type (Figure \[fig:schalfedge\]) for $f \rightarrow 1/2$. Far from the edges, the no-vortex (nv) state is characterized by identical phases for all junctions. The same applies to the single-vortex (sv) configuration far from both the edges and the center of the vortex. On the other hand, the fully-frustrated (ff) state has a spatial oscillatory pattern with a wavelength equal to two plaquettes. All of these states are modified by noticeable edge effects. Although there are many other static solutions of the system, our numerical simulations indicate that the no-vortex, single-vortex, and fully-frustrated states are the only ones needed to explain the depinning behavior of the array.
When dealing with [*fixed points*]{} with up-down symmetry (\[eq:phiupdown\]) the defining equations (\[eq:eq1\])–(\[eq:eq3\]) become $$\begin{aligned}
\label{eq:scequation1}
&I+\sin \phi^{H}_{j-1}=\sin \phi^{H}_{j}+\sin \phi^{V}_{j},\; \;
&j=1,\ldots,N+1 \\
&\phi^{V}_{j}-\phi^{V}_{j+1}-2 \phi^{H}_{j}= 2 \pi (n_{j}-f), \; \;
&j=1,\ldots,N
\label{eq:scequation2}\end{aligned}$$ where we have defined artificial phases $\phi^{H}_{0}=\phi^{H}_{N+1}=0$. Note that the McCumber parameter $\beta_{c}$ does not appear in the equations for the fixed points, and hence does not affect their existence. This is consistent with the numerically observed independence of the depinning behavior on $\beta_{c}$.
In the rest of this section we obtain analytical approximations for the nv, sv, and ff configurations mentioned above. We follow a common scheme for all of them. First, we obtain a no-edge approximation (denoted with a dagger $\dagger$) for the infinite ladder. Second, we introduce the effect of the edges perturbatively to obtain an edge-corrected approximation (identified by a double dagger $\ddagger$). The calculated configurations have been exhaustively compared with the results of numerical simulations with excellent agreement.
No-vortex solution
------------------
Figure \[fig:scnovortex\] (a)–(b) shows a plot of the no-vortex solution, as computed numerically, along with the analytical approximation described below. This state is characterized by the absence of topological vortices ($n_{j}=n_{j+1}=0, \; \forall j$) and, far from the edges, by the constancy of the phases.
As a first approximation, let $\{{\phi^{V}_{j}}^{\dagger},{\phi^{H}_{j}}^{\dagger} \}$ denote the phases of the no-vortex solution for the [*infinite*]{} ladder, i.e., in the absence of edge effects. To ease the notation, let ${\phi^{H}}^{\dagger}$ denote the common phase of the horizontal junctions (so ${\phi^{H}_{j}}^{\dagger}={\phi^{H}}^{\dagger}$ for all $j$), and define ${\phi^{V}}^{\dagger}$ similarly for the phases of the vertical junctions. The only physically acceptable solution of equations (\[eq:scequation1\])–(\[eq:scequation2\]) that also satisfies the stability condition (\[eq:horizontalcond\]) is: $$\label{eq:scsolnoed}
{\phi^{V}}^{\dagger}=\arcsin I, \hspace{.2in} {\phi^{H}}^{\dagger}=\pi f,$$ where $0 \leq f \leq 1/2$ and all the angles are restricted to the first quadrant.
This solution exists if and only if $I \leq 1$. A linear stability analysis shows that, for all $I < 1$, the solution is stable if ${\phi^{V}}^{\dagger} = \arcsin I \in [0,\pi/2)$. The other possible solutions with ${\phi^{V}}^{\dagger} = \pi - \arcsin I$ or ${\phi^{H}}^{\dagger} = \pi f -\pi$ are linearly unstable [@mythesis]. In summary, when the edges are completely neglected, the array behaves like a single junction: its only stable no-vortex solution of the observed form (\[eq:scsolnoed\]) disappears at $I=1$ through a saddle-node bifurcation. This existence criterion will be used in Section \[sec:bifurc\] when discussing the stability properties of the nv solution.
Figure \[fig:scnovortex\] shows that this infinite-ladder approximation works well near the center of the ladder, but breaks down close to the edges. We now take edge effects into account by considering an edge-corrected solution (denoted by $\ddagger$) $$\label{eq:scnvedge}
{\phi^{V}_{j}}^{\ddagger}={\phi^{V}}^{\dagger}+A_{j}, \hspace{.5in}
{\phi^{H}_{j}}^{\ddagger}={\phi^{H}}^{\dagger}-B_{j}.$$ where $\{A_{j}, B_{j}\}$ denote the corrections. From the fixed point equations (\[eq:scequation1\]), (\[eq:scequation2\]), the $\{A_{j},B_{j}\}$ must satisfy $$\begin{aligned}
\label{eq:corrections1}
&\hspace{-.5in} I+ \sin (\pi f - B_{j-1}) = \nonumber \\
& \hspace*{1in} \sin (\pi f- B_{j})+ \sin (\arcsin I + A_{j})\\
&A_{j}-A_{j+1}+2 B_{j}=0.
\label{eq:corrections2}\end{aligned}$$ The corrections $\{A_{j},B_{j}\}$ are expected to be small, except in a region very close to the edges. Thus, Equation (\[eq:corrections1\]) can be expanded to first order in $A_{j}$ and $B_{j}$ to obtain a second-order difference equation for $A_{j}$: $$\begin{aligned}
\label{eq:difference}
A_{j+1}-2 \alpha A_{j} + A_{j-1} =0\\
{\rm with}\;\; \alpha=1+\frac{\sqrt{1-I^{2}}}{\cos \pi f},
\label{eq:scnvalpha}\end{aligned}$$ from which the horizontal corrections are $$\label{eq:correctionhor}
B_{j}=\frac{A_{j+1}-A_{j}}{2}.$$
The general solution of (\[eq:difference\]) is $$\begin{aligned}
\label{eq:correctionvert}
A_{j}=P r^{j-(N+1)}+Q r^{1-j}\\
{\rm where} \; \; r=\alpha + \sqrt{\alpha^{2}-1} \equiv e^{1/\lambda}.
\label{eq:scnvr}\end{aligned}$$ Hence, the edges produce corrections that decay exponentially from both ends with a characteristic length $\lambda(I,f)$. This $\lambda=1/\ln r$ is a measure of how small perturbations decay inside of a region with the no-vortex superconducting solution. A similar result was recently obtained by Denniston and Tang [@denniston] using the transfer matrix method for the particular $I=0$ case.
To complete the solution, the constants $P, Q$ in (\[eq:correctionvert\]) have to be fixed from the boundary conditions $$\begin{aligned}
\label{eq:scboundcond1}
I = \sin(\pi f -B_{1}) + \sin(\arcsin I + A_{1}) \\
I + \sin(\pi f -B_{N}) = \sin(\arcsin I + A_{N+1}),
\label{eq:scboundcond2}\end{aligned}$$ which result from current conservation at nodes $1$ and $N+1$ respectively. Since $\{A_{j},B_{j}\}$ become largest at the edges, equations (\[eq:scboundcond1\])–(\[eq:scboundcond2\]) are solved numerically [*without*]{} linearization. When the array is long enough, such that $\lambda \ll N+1$, the effect of one edge on the other is negligible and the solution is further simplified as equations (\[eq:scboundcond1\])–(\[eq:scboundcond2\]) decouple. Then $Q$ and $P$ are obtained independently by solving $$\begin{aligned}
\label{eq:scnvleft}
I= \sin\left(\pi f + \frac{Q}{2}\left(1-\frac{1}{r}\right)\right)
+ \sin(\arcsin I + Q)\\
I+\sin\left(\pi f-\frac{P}{2}\left(1-\frac{1}{r}\right)\right) =
\sin(\arcsin I +P).
\label{eq:scnvright}\end{aligned}$$
Figure \[fig:scnovortex\] (a)–(b) shows that the above analytical solution agrees well with the results of simulations for long ($N=25$) arrays. The approximation accounts well for the effect of the open ends also for short ($N=7$) arrays (not shown). The exponential decay of the perturbation from the edges is also checked satisfactorily in Figure \[fig:scnovortex\](c). As expected on physical grounds, the edge effects become more important as both the field and the current are increased. Thus, the approximation is best when $f$ and $I$ are small, and worsens as $I \rightarrow 1$ and $f \rightarrow 1/2$ ($f=1/2$ is a singular limit, as seen from the vanishing denominator in (\[eq:scnvalpha\])). This establishes limits on the use of this approximation for the prediction of the depinning current at high values of the frustration.
Our analytical approximation also explains other features of the numerics. For instance, the corrections $A_{j}, B_{j}$ are spatially asymmetric with respect to the center of the array when $I>0$ — as seen in Figure \[fig:scnovortex\](a)–(b) by comparing the rightmost and leftmost phases. Note also that for $0< f \leq 1/2$ the largest vertical phase occurs at the right end of the ladder— in obvious connection with the preferred direction for flux-propagation in the array (${\bf -\hat x}$). Moreover, it can readily be shown that the change of the frustration from $f$ to $1-f$ has only one effect: the vertical phases for frustration $1-f$ are a mirror image, with respect to the center of the array, of the vertical phases with frustration $f$. This implies that the depinning current will be [*identical*]{} for both values of the frustration, as expected, but the direction of propagation is reversed [@mythesis].
Single-vortex solution
----------------------
An analytical approximation for the single-vortex configuration can be obtained in a similar fashion by realizing that the effect of a vortex located in cell $a$ of the array is similar to the edge effects in the no-vortex state. Note how, if the phases in Figure \[fig:onevortex\](a)–(b) were reduced to $[-\pi,\pi)$, the single-vortex configuration is composed of two halves, each of which is equivalent to a no-vortex superconducting solution when we move away from the edges and from the vortex center $a$.
Hence, the zeroth order single-vortex solution $\{{\phi^{V}_{j}}^{\dagger},{\phi^{H}_{j}}^{\dagger}\}$ is identical to the no-edge nv solution given in (\[eq:scsolnoed\]). And the edge and vortex-corrected approximation with a vortex distribution $n_{a}=1$ and $n_{j}=0, \;\; \forall j \neq a$ is given by $$\label{eq:onevortex}
%\phi^{V}_{j}=\arcsin I+ 2 \pi\; \Theta (j-a) + A_{j}, \;\;
{\phi^{V}_{j}}^{\ddagger}=\arcsin I+ A_{j}, \;\;
{\phi^{H}_{j}}^{\ddagger}= \pi f - B_{j},$$ where the corrections $\{A_{j}, B_{j}\}$ result now both from the presence of the edges and of the vortex in plaquette $a$.
Following the same steps as for the nv configuration, we obtain identical expressions (\[eq:corrections1\])–(\[eq:scnvr\]) for the corrections for [*each*]{} half of the array: $$\begin{aligned}
\label{eq:Aonevortex}
A_{j}&=&\left \{
\begin{array}{ll}
P r^{j-a}+Q r^{1-j},& j \leq a\\
P' r^{j-(N+1)}+Q' r^{a+1-j}, & j > a
\end{array} \right .\\
B_{j}&=&\frac{A_{j+1}-A_{j}}{2}, \hspace{1in} j \neq a
\label{eq:Bonevortex}\end{aligned}$$ where $r$ is, once more, given by (\[eq:scnvr\]). Therefore, the single-vortex solution is obtained by [*matching*]{} two edge-corrected no-vortex solutions. In fact, the vortex in cell $a$ effectively introduces two new “edges”, at $a$ and $a+1$, which also produce similar exponentially decaying corrections. The matching condition at $a$ and $a+1$ is given by the fluxoid quantization condition in the cell containing the topological vortex: $A_{a}-A_{a+1}+2B_{a}= 2 \pi$. Thus, $$\label{eq:Baonevortex}
B_{a}= \pi +\frac{P' r^{a-N}+Q'-P-Q r^{1-a}}{2}.$$
This completes the equations needed to determine the unknowns $P,Q,P',Q'$ in our solution (\[eq:Aonevortex\]). They can be calculated numerically [@mythesis], for given $I$ and $f$, from (\[eq:Baonevortex\]) and the boundary conditions from current conservation at nodes $1,a, a+1$ and $N+1$. The approximation is compared with numerical simulations in Figure \[fig:onevortex\] with excellent agreement, especially at small $f$.
We also note that although the ladder equations (\[eq:scequation1\])- (\[eq:scequation2\]) can be reduced approximately to a discrete sine-Gordon equation [@kardar1; @kardar2; @strouddyn], our analytical expression (\[eq:onevortex\]) is a better approximation than the much-used sine-Gordon kink, which is a good description in strictly 1-D parallel arrays [@parallelcontinuum]. A detailed comparison of both approaches is presented in Appendix \[sec:appsg\].
Fully-frustrated solution
-------------------------
The other relevant superconducting state is the fully-frustrated solution, which appears in simulations when $f \approx 1/2$ (Figure \[fig:schalfedge\]). To obtain an analytical approximation, we follow once more the same scheme as above: first, calculate a no-edge basic solution; then, introduce the edges perturbatively.
In this case, the basic solution is seen numerically to oscillate in space with a wavelength equal to two plaquettes. Thus, when edges are neglected (the infinite-ladder approximation), the phases can be approximated in general by $$\label{eq:halfsolution}
\left \{
\begin{array}{l}
{\phi^{V}_{j}}^{\dagger}=2 \pi \; [a + (-1)^{j} b] \\
{\phi^{H}_{j}}^{\dagger}=2 \pi \; [c + (-1)^{j} d]
\end{array}
\right.$$ where $a,b,c,d$ are to be determined from (\[eq:scequation1\])–(\[eq:scequation2\]) with $n_{j}=[1 \mp (-1)^{j}]/2$. First, substitution in Eq. (\[eq:scequation2\]) gives $c$ and $d$ $$\begin{aligned}
c&=&f/2-1/4\\
d&=&b \pm 1/4.\end{aligned}$$ Second, from Eq. (\[eq:scequation2\]) we obtain $$\begin{aligned}
&&\sin 2 \pi a \; \cos 2 \pi b = I \nonumber\\
\sin 2 \pi b \; &&\cos 2 \pi a- 2 \sin \pi f \; \cos 2 \pi b=0, \nonumber\end{aligned}$$ from which we then solve explicitly for $a$ and $b$ in terms of the parameters $f$ and $I$ $$\begin{aligned}
\label{eq:halfsolution2}
a&=&\frac{1}{2 \pi} \arcsin \sqrt{L/2}\\
b&=&\frac{1}{2 \pi} \arccos \sqrt{2 I^{2}/L}\end{aligned}$$ where $$\label{eq:Lhf}
L=(1+I^{2})\pm \sqrt{(1-I^{2})^{2}-16 I^{2} \sin^{2}\pi f}.$$ Figure \[fig:schalfedge\] (a)–(b) compares the analytical infinite-ladder fully-frustrated approximation (\[eq:halfsolution\])–(\[eq:Lhf\]) with numerical simulations. The agreement is good except near the ends, as expected.
The approximation above also yields an existence criterion for the no-edge fully-frustrated solution. In (\[eq:Lhf\]), the expression inside the square root must be non-negative; hence the infinite-ladder fully-frustrated solution does not exist if $$\label{eq:currhalf}
I > I_{\rm ff,th} = \sqrt{4 \sin^{2} \pi f +1} - 2 \sin \pi f,$$ where the subscript “ff” denotes fully-frustrated and “th” denotes a theoretical approximation of a bifurcation condition. We will use this condition (\[eq:currhalf\]) later when we discuss the depinning of the fully-frustrated solution.
This predicted form of the infinite-ladder fully-frustrated solution agrees with previous findings obtained for the special case when there is no driving current [@benedict]. For $I=0$, our solution (\[eq:halfsolution\])–(\[eq:Lhf\]) reduces to the stable configuration [@benednote] $${\phi^{V}_{j}}^{\dagger}=(-1)^{j} \arctan (2), \hspace{.2in}
{\phi^{H}_{j}}^{\dagger}=(-1)^{j+1} \arctan(1/2),$$ which coincides with the ground state for $f=1/2$ and $I=0$ calculated by Benedict [@benedict]. (To obtain this result from the expressions above, note that $2I^{2}/L \rightarrow (1+4 \sin^{2} \pi f)^{-1}$ as $I
\rightarrow 0$, for the solution corresponding to the minus sign in (\[eq:Lhf\]).) We also note that the infinite-ladder fully-frustrated solution exists [*for all*]{} $f$ when $I=0$, i.e., there is no critical magnetic field below which it ceases to exist, although it is energetically most favorable when $f \approx 1/2$.
The physical meaning of this solution is clear: it contains a topological vortex in every other cell, as seen from the alternating sequence of zeros and ones for the plaquette integers $\{n_{j}\}$ in the equations (\[eq:scequation1\])–(\[eq:scequation2\]).
In fact, although the solutions with $\{n_{\rm odd}=1,\; n_{\rm even}=0\}$ and $\{n_{\rm odd}=0,\; n_{\rm even}=1\}$ are degenerate in an infinite array, they are not so if the array is finite. However, we will show in Section \[sec:bifurc\] that the depinning of the fully-frustrated state is basically unaffected by the parity of the number of cells in the ladder.
As we did for the no-vortex solution, we now introduce corrections from the edges. The improved solution $\{{\phi^{V}_{j}}^{\ddagger},{\phi^{H}_{j}}^{\ddagger}\}$ is given by $$\left \{
\begin{array}{l}
{\phi^{V}_{2i-1}}^{\ddagger}={\phi^{V}_{\rm odd}}^{\dagger}+C_{i}\\
{\phi^{V}_{2i}}^{\ddagger}={\phi^{V}_{\rm even}}^{\dagger}+D_{i}
\end{array}
\right. ,
\left \{
\begin{array}{l}
{\phi^{H}_{2i-1}}^{\ddagger}={\phi^{H}_{\rm odd}}^{\dagger}-E_{i}\\
{\phi^{H}_{2i}}^{\ddagger}={\phi^{H}_{\rm even}}^{\dagger}-F_{i}
\end{array}
\right.$$ where $i=1,\ldots,\rm{ceil}(N/2)$ and there is an additional ${\phi^{V}_{N+1}}^{\ddagger}$ when $N$ is even and ${\phi^{H}_{N+1}}^{\ddagger}=0$ when $N$ is odd. The double cell is used to simplify the calculations, as suggested by the spatial periodicity of the infinite-ladder solution.
Again, far away from the ends the corrections are small and we linearize the governing equations (\[eq:scequation1\])–(\[eq:scequation2\]) around the basic solution (\[eq:halfsolution\]). Thus we obtain the following system of coupled difference equations: $$\begin{aligned}
&&C_{i}-D_{i}+2E_{i}=0\\
&&D_{i}-C_{i+1}+2F_{i}=0\\
-E_{i} \cos {\phi^{H}_{\rm odd}}^{\dagger}&&=
-F_{i} \cos {\phi^{H}_{\rm even}}^{\dagger}
+D_{i} \cos {\phi^{V}_{\rm even}}^{\dagger}\\
-F_{i} \cos {\phi^{H}_{\rm even}}^{\dagger}&&=
-E_{i+1} \cos {\phi^{H}_{\rm odd}}^{\dagger}
+C_{i+1} \cos {\phi^{V}_{\rm odd}}^{\dagger}.\end{aligned}$$ Eliminating $E_{i}, F_{i}$ and $D_{i}$ we get a second-order difference equation for $C_{i}$: $$C_{i+2}+2 \gamma C_{i+1} + C_{i}=0,$$ with $\gamma(I,f)$ given by $$\gamma=\frac{\sin^{2} \pi f + cb^{2} \cos 2 \pi f - 2 \left [
(\sin^{2} \pi f- sa^{2}) sb^{2} + ca^{2} \right
]}{\sin^{2}\pi f -cb^{2}}.$$ (Here $sa, sb, ca$ and $cb$ are shorthand for $\sin 2\pi a$, $\sin 2\pi b$, $\cos 2\pi a$ and $\cos 2\pi b$ respectively). This difference equation has the general solution $$\begin{aligned}
\label{eq:Cschalfedge}
&C_{i}= P r_{\rm ff}^{i} + Q r_{\rm ff}^{-i}\\
{\rm with} \;\; &r_{\rm ff}=-\gamma + \sqrt{ \gamma^{2}-1}
\equiv e^{1/\lambda_{\rm ff}}.
\label{eq:rschalfedge}\end{aligned}$$ And $r_{\rm ff}$ is related again to another characteristic penetration depth for the perturbations from the edges to die off, this time inside a region with the fully-frustrated solution. The coefficients $P$ and $Q$ have to be calculated numerically using the boundary conditions from nodes $1$ and $N+1$. The spatial dependence of $D_{i}$ is also of the same form $D_{i}= R r_{\rm ff}^{i} + S r_{\rm ff}^{-i}$. Note that in both equations, $i$ is the number that indexes the double cell.
Figure \[fig:schalfedge\] (c) illustrates the accuracy of these approximate formulas. Specifically, we plot the spatial dependence of the predicted deviations $$C_{i}^{\rm num}= \phi^{V}_{2i-1}- {\phi^{V}_{2i-1}}^{\dagger}, \hspace{.2in}
D_{i}^{\rm num}= \phi^{V}_{2i}- {\phi^{V}_{2i}}^{\dagger},$$ where ${\phi_{j}^{V}}$ is obtained from the numerical solution and ${\phi_{j}^{V}}^{\dagger}$ is the infinite-ladder approximation. The expected exponential decay close to the edges with characteristic length $\lambda_{\rm ff}=2/\ln r_{\rm ff}$ is verified in the figure.
Dynamical depinning transitions of the array {#sec:depinning}
============================================
In this section, we describe the depinning transitions as seen in dynamical simulations of the ladder array at zero temperature. In the following section, we will explain these dynamical results by relating them to the bifurcations of the no-vortex, single-vortex, and fully-frustrated superconducting solutions. The main goal is to give a rigorous mathematical explanation of the following observations: Dynamical simulations show that the array is superconducting at low values of the driving current $I$. As $I$ is increased, the array remains superconducting until a critical current $I_{\rm dep}(f)$ is reached, after which the array depins and develops a non-zero average voltage. This depinning current $I_{\rm dep}$ decreases monotonically as the frustration $f$ increases from $0$ to $1/2$.
These numerical observations are all at the averaged level of the $IV$ characteristics. They do not tell us anything about the detailed configuration of the individual junctions. In particular, there are several distinct superconducting states (e.g., the nv, sv, and ff states discussed in Section \[sec:analytical\], and states containing multiple vortices) but these are indistinguishable on the $IV$ curve. This ambiguity raises the question: what is the state of the ladder just before it depins?
We will show in Section \[sec:bifurc\] that for most values of $f$, the depinning of the ladder is caused by the destruction of the no-vortex state. Specifically, the global depinning current $I_{\rm dep}(f)$ can be predicted by calculating the current at which the no-vortex state is annihilated in a saddle-node bifurcation. The only exception occurs for values of $f$ close to $1/2$, where the depinning is due to saddle-node bifurcations of states of the fully-frustrated type. The noteworthy point here is that no other superconducting states play a role in the global depinning of the array.
However, the question arises as to how the depinning behavior would change when configurations with vortices are used as initial conditions (rather than the random or zero-phase initial conditions that we ordinarily use in our simulations of the $IV$ characteristics). To address this issue, we perform dynamical simulations from an initial condition with one $2 \pi$-step in the middle cell of the array: $$\phi^{H}_{j}(t=0)=0, \; \; \phi^{V}_{j}(t=0)=
2 \pi \; \Theta\left(j-{\rm ceil}\left(\frac{N+1}{2}\right)\right),$$ where $\Theta(x)$ is the Heaviside step function. This initial condition is [*not*]{} a solution of the system and, thus, under the dynamical equations (\[eq:eq1\])–(\[eq:eq3\]) it relaxes onto a true solution for the ladder. For most $\{f,I\}$, the [*single-vortex superconducting state*]{} (top panel of Figure \[fig:LAT\]) is reached, i.e., a static configuration with a topological vortex in cell $a$ of the array such that $n_{a}=1$ and $n_{j}=0, \;
\forall j \neq a, \; j=1,\ldots,N$. For some ranges of $f$ and $I$, this configuration is not dynamically stable and other solutions are found, as discussed in Section \[sec:bifurc\].
The numerical observations shown in Figure \[fig:LAT\] depict the dynamical behavior of the single-vortex state for most values of $f$, i.e. approximately $0.12 < f < 0.37$. They can be summarized as follows: For $f>f_{\rm min}$ and small driving current $I$, the system relaxes onto a static single-vortex solution with a vortex in the middle of the array. The solution is slightly distorted as the current is increased, until at $I=I_{\rm LAT}(f)$ the vortex moves from the center cell toward the left. (This current is analogous to the well-known Lobb-Abraham-Tinkham (LAT) depinning current for two-dimensional Josephson junction arrays [@LAT]). For currents very close to $I_{\rm LAT}$, the vortex moves slowly and gets trapped in another cell somewhere between the center and the edge. For somewhat larger $I > I_{\rm LAT}$, the vortex moves all the way to the left edge where it becomes pinned until, at a second critical current $I_{\rm left}$, it is expelled from the array and the no-vortex solution is recovered again. The no-vortex configuration then remains stable until, at $I_{\rm dep}$, global depinning of the array occurs. If instead of placing the vortex in the middle, we locate it closest to the [*right*]{} edge, it depins at a current $I_{\rm right}< I_{\rm LAT}$, and moves toward the center.
These observations can be clarified with the usual analogy [@terryvortex] of the vortex as a damped particle moving in a sinusoidal potential under the action of a $\bf{- \hat x}$ Lorentz-like force directly proportional to $I$. The maxima of the potential correspond to the vertical junctions and the minima are situated in the middle of the cells. Thus, the initial barrier which has to be overcome to begin the motion explains the critical $I_{\rm LAT}$. Moreover, the open boundaries can be thought to produce an exponentially-decaying envelope superimposed on the sinusoidal potential. Thus the motion of the vortex is easier when the vortex is close to the right edge and becomes increasingly difficult as the left boundary is approached. When $I_{\rm left}$ is reached, the vortex is able to overcome the edge barrier and is expelled from the array. Then the no-vortex configuration is recovered and no new vortex enters the ladder.
We mentioned before that the described behavior is observed for values of the frustration contained between two limiting values. First, there is a minimum frustration $f_{\rm min} \sim 0.12$ below which the single-vortex solution never ensues from this initial condition; in fact, the system settles on the no-vortex superconducting solution. Second, for $f$ larger than $\sim 0.37$, the vortex is [*not*]{} expelled from the array at the left edge before depinning. Depinning occurs in that case when vortices enter the array from the right edge. These observations will be clarified in Section \[sec:bifurc\].
We have also performed dynamical simulations for the multi-vortex case and reached similar conclusions for most values of $f$. In that case, the initial condition consists of $N f$ equally spaced $2 \pi$-steps which are then allowed to evolve dynamically. For $I=0$ and large enough $f$, the initial condition relaxes onto the expected solution with $N f$ vortices in the array. As the current is increased, these move towards the left end where they accumulate until they are expelled one by one at different currents. After this, the no-vortex solution is again recovered.
On the other hand, the picture changes when $f$ is close to $1/2$. There, solutions of the fully-frustrated type are obtained from the multi-vortex initial condition and there is no expulsion of vortices from the ladder. Instead, the array depins globally at the current where the ff solution ceases to exist, i.e. $I_{\rm dep}(f \rightarrow 1/2)
\approx I_{\rm ff,th}$.
We conclude that in the ladder, the depinning of one vortex (or a train of vortices) is [*not*]{} equivalent to the global edge-dominated depinning of the device. As we will confirm in Section \[sec:bifurc\], the no-vortex and fully-frustrated states are the relevant solutions for the depinning of the array; for moderate $f$, even if the initial conditions contain vortices, these are expelled from the array as the current is increased and, eventually, the no-vortex solution is recovered. For $f$ close to $1/2$, the system settles onto fully-frustrated solutions with distinct depinning properties.
We have also identified three other critical currents related to the single-vortex configuration: $I_{\rm right}$, at which a vortex at the right edge begins to move; $I_{\rm LAT}$, at which dynamical depinning occurs for a single vortex centered in the middle of the ladder; and $I_{\rm left}$, at which the vortex is expelled at the edge.
All of these dynamical observations are explained in detail in the next section where they are compared to their exact mathematical descriptions.
Bifurcation analysis of the depinning transitions {#sec:bifurc}
=================================================
In this section, we use bifurcation theory to obtain exact criteria for all the critical currents of the no-vortex, single-vortex, and fully-frustrated states. We have checked consistently that these bifurcations explain the dynamical depinning behavior of the array as described in Section \[sec:depinning\]. Furthermore, analytical simplifications to some of those criteria will be deduced from approximations of the exact depinning results.
The depinning of the ladder can be explained in dynamical terms as follows: The linear stability of the superconducting states of the ladder as a function of $I$ can be deduced from the Jacobian matrix ${\bf J_{\rm dyn}}$ of the dynamical system (\[eq:eq1\])–(\[eq:eq3\]) for a given value of $f$, and for a given fixed point—in particular, the no-vortex, single-vortex or fully-frustrated state. If all the eigenvalues have negative real parts, the fixed point is linearly stable. As we increase $I$, some of the eigenvalues move to the right in the complex plane, and the fixed point becomes less stable. The critical current for a given fixed point is defined by the condition that the maximum of the real parts of the eigenvalues becomes positive. Then, to predict the [*global*]{} depinning current, we compare the critical currents of the different superconducting states, and take the maximum of those. In other words, we predict that global depinning occurs when the “last” stable state bifurcates.
Recall that there are several scenarios [@steve] by which a stable fixed point can undergo such a bifurcation. First, in a zero-eigenvalue bifurcation, a single eigenvalue moves along the real axis, and passes from negative to positive at the bifurcation. There are three main subtypes of zero-eigenvalue bifurcation: saddle-node, transcritical, and pitchfork. In the saddle-node bifurcation, a stable fixed point collides with a saddle point, and both are annihilated. In contrast, in the transcritical and pitchfork bifurcations, the stable fixed point is not destroyed — it continues to exist but goes unstable. A second scenario is provided by the Hopf bifurcation which involves a pair of complex conjugate eigenvalues passing through the imaginary axis from the left half plane to the right half plane – again this bifurcation destabilizes the fixed point, but does not destroy it.
Since it can be shown that Hopf bifurcations are not possible in this system [@newpaper], we can simplify our calculations by using the [*static*]{} system (\[eq:scequation1\])–(\[eq:scequation2\]) to identify the location of the zero-eigenvalue bifurcations. Those bifurcation points are characterized by a change in the number of fixed points and, thus, from the implicit function theorem [@calculus], the Jacobian matrix ${\bf J}$ of the static system has zero determinant there. Hence, we use the superconducting (static) up-down symmetric system given by ${\bf f(x)= 0}$, $$\begin{aligned}
\label{eq:systemeq}
&{\rm f}_{i}({\bf x})= I + \sin {\rm x}_{N+i} -\sin {\rm x}_{N+1+i} -&
\sin {\rm x}_{i}, \nonumber\\
&&i=1,\ldots,N+1\\
&{\rm f}_{N+1+i}({\bf x})={\rm x}_{i}-{\rm x}_{i+1}-
2{\rm x}_{N+1+i}+&2\pi(f-n_{i}), \nonumber\\
&& i=1,\ldots,N\end{aligned}$$ with ${\bf x}=({\rm x}_{1},\ldots,{\rm x}_{2N+1})\equiv
(\phi^{V}_{1},\ldots,\phi^{V}_{N+1},\phi^{H}_{1},\ldots,\phi^{H}_{N})$. For a given $f$ and a given superconducting state, we compute the bifurcating fixed point $\bf x^{\star}$ and its associated critical current $I^{\star}(f)$ at which ${\rm det({\bf J})}=0$. To this end, we define an augmented algebraic system with the current $I$ as an extra variable, and the constraint on the determinant as an extra equation. Then $\bf x^{\star}$ and $I^{\star}$ are obtained by solving ${\bf F(X^{\star})= 0}$ where ${\bf X}=({\bf x},I)$ and $$\begin{aligned}
\label{eq:augmented1}
{\rm F}_{j}({\bf X})=&&{\rm f}_{j}, \hspace{.2in}\;\; j=1,\ldots,2N+1\\
{\rm F}_{2N+2}({\bf X})=&&{\rm det({\bf J})}.
%{\rm F}_{2N+2}({\bf X})=&&{\rm max}\{\Re[ {\rm eig}({\bf J})]\}.
\label{eq:augmented2}\end{aligned}$$ Figure \[fig:depinning\] shows that the dynamical depinning of the ladder is explained by $I^{\star}$ (zero-eigenvalue) bifurcations.
The rest of this section is devoted to analyzing the bifurcations of the no-vortex, single-vortex and fully-frustrated configurations. For the sake of clarity, we follow a parallel scheme for all of them and keep the notation consistent. For each of the states, we first calculate numerically the zero-eigenvalue bifurcations from (\[eq:augmented1\])–(\[eq:augmented2\]) (always denoted with a star $^{\star}$) and compare them with the depinning currents from [*dynamical*]{} simulations (always denoted with symbols in the figures). Then, when possible, we deduce analytical simplifications of these criteria in one of two ways: (a) by deducing that the instability is essentially caused by a bifurcation of the no-edge solution, or (b) by explaining the instability as edge-originated and, thus, localized at the boundaries. These theoretical analytical approximations are always denoted with the subscript “th”. Moreover, to emphasize the importance of the edges in the depinning transitions, we calculate the depinning of the no-edge solutions for all three configurations. Finally, note that no energy arguments are invoked in this discussion. Thermodynamic considerations are studied in detail in Section \[sec:conclusion\] where phase diagrams of these superconducting states for the no-edge and finite ladders are presented.
Bifurcation of the no-vortex solution
-------------------------------------
Figure \[fig:depinning\] shows that for most values of $f$, the depinning current $I_{\rm dep}(f)$ (obtained from dynamical simulations) coincides with the critical current $I_{\rm nv}^{\star}$ for the no-vortex state —calculated from the augmented system (\[eq:augmented1\])–(\[eq:augmented2\]). Indeed, we find that the bifurcating phase configuration ${\bf x}_{\rm nv}^{\star}$ matches the depinning configuration ${\bf x}_{\rm dep}$ observed in dynamical simulations. Hence, the bifurcation of the no-vortex state constitutes an exact criterion for the global depinning current, except for values of $f$ close to $1/2$, where the global depinning is caused by the destruction of the fully-frustrated solution, as explained below.
To gain intuition about how to derive analytical approximations for $I_{\rm dep}(f)$, it is helpful to characterize the depinning bifurcation more precisely. Our numerical computations indicate that the depinning of the no-vortex state is due to a saddle-node bifurcation. As $I$ approaches $I_{\rm nv}^{\star}$ from below, the stable no-vortex state approaches an unstable no-vortex state, and coalesces with it when $I = I_{\rm nv}^{\star}$, causing both states to disappear. Figure \[fig:sadnodnv\](a) shows the maximum [*dynamical*]{} eigenvalue for both the stable and unstable states – note that both of these eigenvalues are pure real, and they equal 0 at the critical current. As expected, this plot has the standard shape of a saddle-node bifurcation diagram [@steve]. Figure \[fig:sadnodnv\](b)–(c) plots the phase configuration for both states at $I=0$. They have similar spatial structure, except near the rightmost cell, where (in the language of the mechanical analog) the unstable state has an inverted pendulum.
Incidentally, Figure \[fig:sadnodnv\] also shows that both states satisfy the up-down symmetry $\phi^{H}_{j}=-\phi^{h}_{j}$ discussed in Section \[sec:system\]. Numerical simulations show that this symmetry continues to hold for all values of $I$ on both the stable and unstable branches. Thus, the global depinning bifurcation takes place entirely within the invariant manifold of up-down symmetric states – it is [*not*]{} a symmetry-breaking bifurcation.
### Analytical approximations for $I_{\rm nv}^{\star}(f)$
The conclusion that the depinning transition for most values of $f$ corresponds to a saddle-node bifurcation of the no-vortex superconducting state can be simplified further. We now obtain analytical approximations for $I_{\rm nv}^{\star}(f)$ using the approximate solutions calculated in Section \[sec:analytical\].
We recall that the bifurcation of the infinite-ladder no-vortex configuration does not explain the observed $f$-dependence of the finite-ladder nv depinning. As discussed in Section \[sec:analytical\], if the edges are neglected completely, the no-vortex solution (\[eq:scnvedge\])–(\[eq:correctionvert\]) is predicted to exist and be stable for all $I< 1$, independent of the frustration $f$, in analogy with the single junction. Thus, the depinning for the no-edge nv state occurs through a saddle-node bifurcation at $I^{\dagger}_{\rm nv,th}(f) =1, \;\; \forall f$.
To capture the observed $f$-dependence of the critical current $I_{\rm nv}^{\star}$, we need a more careful approximation. We now present two such approximations to $I_{\rm nv}^{\star}$ that clarify the physical picture of the transition.
The first strategy is to use the improved approximation (\[eq:scnvedge\]) for the no-vortex state (in which the edge effects are included perturbatively), and then write down a [*simplified augmented system*]{} ${\bf F^{\ddagger}(X^{\ddagger})=0}$ for this solution, similar to the expression (\[eq:augmented1\])–(\[eq:augmented2\]) for the full $(2N+2)$-dimensional system. We can then calculate the critical current $I_{\rm nv,red}$ for this reduced model, defined as the value of $I$ where the [*perturbative*]{} solution undergoes a saddle-node bifurcation. This renders the calculation analytically tractable since, for a given $f$, only three variables $\{P,Q,I\}$ suffice to describe the perturbative solution (\[eq:scnvedge\])—instead of $I$ and $2N+1$ phases for the full solution. The simplified augmented system is constituted by (\[eq:scnvleft\]), (\[eq:scnvright\]) together with the condition that the determinant of the $2 \times 2$ Jacobian matrix equals zero, corresponding to the zero-eigenvalue condition at a saddle-node bifurcation.
Furthermore, since equations (\[eq:scnvleft\]) and (\[eq:scnvright\]) are uncoupled when $N$ is not very small, even this three-dimensional system can be further reduced to a two-dimensional system with unknowns $P$ and $I$: $$\begin{aligned}
\label{eq:reduced1}
{\rm F}^{\ddagger}_{1}&=&I +
\sin \left ( \pi f- \frac{P}{2} \left(1-\frac{1}{r}\right)\right)-
\sin (\arcsin I + P)\\
{\rm F}^{\ddagger}_{2}&=&
\frac{\partial {\rm F}^{\ddagger}_{1}}{\partial P}
\label{eq:reduced2}
%=-\frac{r-1}{2r}
%\cos\left ( \pi f- \frac{P}{2} \left(1-\frac{1}{r}\right)\right)-
%\cos(\arcsin I +P)=0\end{aligned}$$ where $r=r(f,I)$ is given by (\[eq:scnvr\]). Note that this is a set of [*local*]{} equations referred to the rightmost end of the array. We numerically solve the $2 \times 2$ reduced system ${\bf F}^{\ddagger}(P_{\rm red},I_{\rm nv,red})={\bf 0}$ to obtain the approximate depinning current $I_{\rm nv,red}(f)$ and the value of the rightmost phase $P_{\rm red}$ at the bifurcation.
Figure \[fig:depanalyt\] shows that $I_{\rm nv,red}(f)$ predicts the exact $I^{\star}_{\rm nv}(f)$ reasonably well. As expected, the prediction gets worse as $f$ nears $1/2$ since the perturbative approximation of the no-vortex solution is less accurate in that limit.
Both the analyses of the eigenfunctions of the full $2N+2$ system (\[eq:augmented1\])–(\[eq:augmented2\]) and of the reduced system (\[eq:reduced1\])–(\[eq:reduced2\]) indicate that the global depinning of the ladder is caused by a local instability of the rightmost junction of the array. This is consistent with physical arguments which imply that after depinning occurs, vortices propagate in the array in the ${\bf - \hat x}$ direction under the effect of a magnetic Magnus (Lorentz-like) force.
The key role played by the rightmost junction suggests a second simplification, which we call a heuristic criterion for depinning. This criterion connects the global depinning of the ladder with the much simpler depinning transition in a single junction. Recall that when the phase of a single junction reaches $\pi/2$, its superconducting solution is destroyed in a saddle-node bifurcation [@steve]. Therefore, we intuitively propose that when the phase of the rightmost junction reaches $\pi/2$, the ladder depins. Replacing the no-vortex solution by its perturbative approximation (\[eq:scnvedge\]), we solve for the current $I_{\rm nv,th}$ by imposing $$\label{eq:heurcrit}
{\phi^{V}_{N+1}}^{\ddagger}= \pi/2$$ which implies $$\arcsin I_{\rm nv,th} + P(I_{\rm nv,th})=\pi/2.$$ Then, from (\[eq:scnvright\]), we obtain an implicit transcendental equation for $I_{\rm nv,th}(f)$: $$\label{eq:heuristic}
\arcsin(1-I_{\rm nv,th})+\frac{r-1}{2r}\arccos I_{\rm nv,th}=\pi f$$ with $r=r(I_{\rm nv,th})$ given by (\[eq:scnvr\]). This simple analytical prediction is shown to be in very good agreement with the exact results in Figure \[fig:depanalyt\].
The techniques described in this Section can be extended to include the effect of self-fields. In Appendix \[sec:appendixa\] we illustrate this approach and show how self-inductance modifies the approximate no-vortex solution and the corresponding depinning current.
Bifurcations of the single-vortex solution
------------------------------------------
We begin the study of the stability of the single-vortex (sv) configuration by considering the single-vortex far from the edges. The results obtained for the vortex in the center will be used subsequently to describe the effects of the edges on the sv state.
### Saddle-node bifurcation of the sv solution
As described in Section \[sec:depinning\], a vortex in the center of the array moves to the left over the potential barrier when the critical current $I_{\rm LAT}$ is reached. We show now that this depinning of the vortex corresponds to a saddle-node bifurcation of the single-vortex solution. To verify this, we restrict our attention to the single-vortex solutions that are centered in the middle of the ladder; then we look for the current $I^{\star}_{\rm sv}$ at which the determinant of the static Jacobian matrix ${\rm det({\bf J}|_{{\bf x}^{\star}_{\rm sv}})} =0$. As in (\[eq:augmented1\])–(\[eq:augmented2\]), we solve the augmented system ${\bf F}({\bf x^{\star}_{\rm sv}},I^{\star}_{\rm sv}) = {\bf 0}$ to find where the centered single-vortex state ceases to exist.
Figure \[fig:depinvortex\] shows the perfect agreement between the $I^{\star}_{\rm sv}(f)$ computed from the static augmented system and the $I_{\rm LAT}(f)$ obtained from simulations where a vortex is placed in the middle of the array and the current is increased until it moves, i.e., $I_{\rm LAT}=I^{\star}_{\rm sv}$.
Moreover, Figure \[fig:sadnodsv\] confirms that this depinning transition is indeed caused by a saddle-node bifurcation. An unstable single-vortex state collides with and annihilates the stable single-vortex state at the transition. Figure \[fig:sadnodsv\] shows that the two states have similar spatial structure —the difference is that the stable state has its vortex in the center of a cell (where the vortex sits in a potential well), while the unstable state has its vortex on a junction (poised on a potential hill).
Although conceptually similar, our $I_{\rm LAT}$ for the ladder is not equivalent to that calculated by Lobb-Abraham-Tinkham [@LAT]. Their current is estimated by a static calculation of the energy barrier $E_{b}$ in an infinitely extended two-dimensional array, while ours is the dynamic current at which the centered single-vortex state undergoes a saddle-node bifurcation in the quasi-one-dimensional ladder. Moreover, their static calculation does not include the effects of the field $f$ or the injected current $I$ on the solutions while, in our case, they are implicitly taken into account since the configurations—and therefore their stability—depend parametrically on $\{I,f\}$.
### Symmetry-breaking bifurcation of the sv solution
We noted above the numerical observation [@stroudscreen] that, when performing dynamical simulations, the static single-vortex solution is unstable below a [*critical field*]{} $f_{\rm min}(I)$. We show now that this is the result of a symmetry-breaking instability which is mathematically related to another zero-eigenvalue bifurcation. Therefore, once again, the dynamical $f_{\rm min}$ coincides with a static $f^{\star}_{\rm sv}$ calculated from the augmented system (\[eq:augmented1\])–(\[eq:augmented2\]) as the value of $f$ where the determinant of the static Jacobian matrix [**J**]{} is zero and, thus, a change in the number of fixed points is likely. Excellent agreement between $f_{\rm min}$ and $f^{\star}_{\rm sv}$ is shown in Figure \[fig:depinvortex\].
Figure \[fig:pitchforksv\] depicts detailed information about this bifurcation. Specifically, it shows three single-vortex states that co-exist for $f$ slightly greater than $f^{\star}_{\rm sv}$. These states appear very similar, but on close inspection, one notices that two of the states are asymmetric: $\phi^{H}_j \neq -\phi^{h}_j$ (this is especially clear for the central plaquette $j=a$.). As $f \rightarrow f^{\star}_{\rm sv}$ from above, these asymmetric states – which are unstable – simultaneously collide with the stable symmetric state, rendering it unstable. To visualize this transition in greater detail, Figure \[fig:pitchforksv\] plots the asymmetry for the central plaquette, $S \equiv \phi^{H}_a + \phi^{h}_a$ , as a function of the frustration $f$. The symmetric state exists both above and below the bifurcation, and satisfies $S=0$ throughout. The two unstable branches join the symmetric branch at $f=f^{\star}_{\rm sv}$. The scenario depicted in Figure \[fig:pitchforksv\], common in symmetric systems, is known as a subcritical pitchfork bifurcation [@steve].
From this, we conclude that there exists a region of the $(f,I)$ plane (Figure \[fig:depinvortex\]) where the single-vortex configuration exists but is always dynamically unstable. In this region, the vortex (magnetic flux) is expelled [*transversally*]{} from the array through transient modes which do [*not*]{} preserve the up-down symmetry of the horizontal phases.
We can also derive an analytical expression $f_{\rm sv,th}$ for the critical field $f^{\star}_{\rm sv}$. As given in (\[eq:horizontalcond\]), all up-down symmetric fixed points become unstable when the phase of any of the horizontal junctions is larger than $\pi/2$ in absolute value. Since this largest phase occurs at the central plaquette (as seen in Figure \[fig:pitchforksv\]), the following stability criterion ensues from the condition $\phi^{H}_{a} = -\pi /2$: $$\label{eq:stabhorizontal}
2 \pi f_{\rm sv,th} + [P(f_{\rm sv,th}) - Q'(f_{\rm sv,th})] = \pi,$$ where we have used (\[eq:onevortex\]) and (\[eq:Baonevortex\]) and we consider a long array such that the effect of the edges on the middle cell can be neglected. Here $P, Q'$ have to be calculated from the boundary conditions (current conservation) at nodes $a$ and $a+1$: $$\begin{aligned}
I+ \sin(\pi f - &&P (r-1)/(2r))= \nonumber \\
\sin( \arcsin I &&+ P) - \sin(\pi f - (Q'-P)/2) \nonumber\\
I - \sin(\pi f - &&(Q'-P)/2)= \nonumber \\
\sin( \arcsin I &&+ Q') +\sin (\pi f + Q'(r-1)/(2r))
\label{eq:boundhorizontal}\end{aligned}$$ with $r$ given by (\[eq:scnvr\]). Numerical solution of the critical condition (\[eq:stabhorizontal\]) yields the curve $f_{\rm sv,th}(I)$ which is almost indistinguishable from the curve for $f^{\star}_{\rm sv}$ from the full augmented system (Figure \[fig:depinvortex\]).
It is especially interesting to check the case $I=0$. Then $P=-Q'$ and the critical condition (\[eq:stabhorizontal\])–(\[eq:boundhorizontal\]) simplifies to $$\label{eq:crithorizontal}
\sin \left[\pi f_{\rm sv,th}^{o} + \left(f_{\rm sv,th}^{o}-\frac{1}{2}\right)
\frac{\pi (r-1)}{2r} \right]
= \cos (\pi f_{\rm sv,th}^{o}) -1,$$ which can be solved numerically to give $f_{\rm sv,th}^{o} \equiv f_{\rm sv,th}(I=0) = 0.1193$, in agreement with numerical findings from previous dynamical simulations [@stroudscreen].
### Single-vortex configurations at the edges
So far we have concentrated on a particular single-vortex state, namely one where the vortex occurs in the middle cell of the array. But there are many other single-vortex states, each differing from the previous one by displacing the vortex by one cell to the right or to the left. Each of those configurations becomes unstable through bifurcations similar to those discussed above for the case of the centered single-vortex state. For most values of $f$, as discussed in Section \[sec:depinning\], when the driving current $I$ is increased, a vortex in the ladder moves to the left (getting pinned in cells closer to the left edge as $I$ grows) until it is expelled from the array at the left boundary. Thus, in explaining the effect of the edges on the sv solution we are most interested in two critical currents: the critical current $I_{\rm right}$, at which a vortex at the right edge begins to move, and the critical current $I_{\rm left}$, at which the vortex is expelled at the left edge of the ladder.
To predict $I_{\rm right}$, we analyze the sv configuration with the vortex placed in the rightmost cell (the $000 \ldots 01$ configuration). The results of the analysis are similar to those for the vortex in the center. As shown in Figure \[fig:svedge\](a), this state can cease to exist through a saddle-node bifurcation ($I^{\star}_{\rm sv,right}$), or become unstable through a symmetry-breaking pitchfork bifurcation ($f^{\star}_{\rm sv,right}$). The agreement with the dynamical simulations is excellent.
The rigorous explanation of $I_{\rm left}$ turns out to be slightly more complicated. A careful examination of the numerics reveals that, depending on the value of $f$, the vortex can be expelled from the array in one of two ways: from the leftmost cell (cell number 1) as expected, or directly from cell number 2. Thus, we need to examine the dynamical stability of two sv states: the one with the vortex in cell 1 ($1000 \ldots 00$) and in cell 2 ($0100 \ldots 00$).
The dynamic stability of the $0100 \ldots 00$ configuration contains no new elements. The two observed bifurcations (a saddle-node $I^{\star}_{\rm sv,left2}$ and a subcritical pitchfork $f^{\star}_{\rm sv,left2}$) are similar to those explained above. The results are presented in Figure \[fig:svedge\](b) where the saddle-node bifurcation is seen to explain the dynamical $I_{\rm left}$ for $f<0.29$.
However, it is the stability analysis of the $1000 \ldots 00$ state that explains the expulsion of the vortex at the edge for $f>0.29$. As shown in Figure \[fig:svedge\](c), this configuration presents the usual saddle-node ($I^{\star}_{\rm sv,left1}$) bifurcation. The pitchfork ($f^{\star}_{\rm sv,left1}$) bifurcation is barely visible in the figure. There is also another saddle-node bifurcation at low $I$ and high $f$ which is irrelevant for the depinning considered here.
The results of this section are summarized in Figure \[fig:svedge\](d), which indicates the region where [*at least one*]{} single-vortex configuration in a ladder array [*with edges*]{} is dynamically stable. Figure \[fig:svedge\](d) is, in essence, the union of Figure \[fig:depinvortex\] with Figure \[fig:svedge\](a)–(c) and shows how the sv solutions either cease to exist through a saddle-node bifurcation when $I$ is increased for most values of the frustration $f$, or become unstable through a symmetry-breaking pitchfork bifurcation for small values of $f$.
Within this picture, the dynamical behavior in Figure \[fig:depinvortex\], where the vortex propagates along the ladder in the interval $I^{\star}_{\rm sv,right}(f) < I < I^{\star}_{\rm sv,left}(f)$, is the result of a succession of saddle-node bifurcations of single-vortex states situated in contiguous cells until the vortex is expelled at the left edge. This is in contrast with the symmetry-breaking exit of the vortex for $f < f^{\star}_{\rm sv}(I)$ where the flux is expelled in the transversal direction.
This description is valid for $f<0.37$. However, a subtle variation is observed beyond that point for the sv solutions. The comparison of Figure \[fig:svedge\](c) and Figure \[fig:depanalyt\] shows that the depinning of the sv and nv states is similar for high $f$, i.e. $I^{\star}_{\rm sv,left} \approx I^{\star}_{\rm nv}$, for $f>0.37$. This is due to the fact that at high $f$ the vortex is [*not*]{} expelled from the array before depinning. Instead, the sv configuration with the vortex at the left edge ceases to exist through a saddle-node bifurcation which, similarly to what happens to the nv state, is localized on the [*right*]{} edge of the ladder. The depinning transition of the single-vortex state at high f can be described in a similar fashion as the depinning of the no-vortex configuration: by the nucleation of vortices on the right edge of the ladder. Thus, the single-vortex configuration plays no role in the global depinning of the array: only the behavior of the nv and ff states has to be considered.
Bifurcations of the fully-frustrated state
------------------------------------------
The depinning of the ladder array as $f \rightarrow 1/2$ is determined by the stability of the fully-frustrated configuration. This can be readily seen by inspection of Figure \[fig:depinning\] which shows that at high $f$ the depinning of the no-vortex state occurs [*before*]{} the global depinning of the array.
To clarify the importance of these transitions we follow the same scheme as above. We first calculate the zero-eigenvalue bifurcations of this state and show that the depinning of the ladder at $f \simeq 1/2$ is indeed explained by a saddle-node bifurcation of the ff state. Then we obtain analytical approximations (“th”) to the exact bifurcations, using stability criteria for the infinite-ladder ff configuration.
In principle, the characterization of the ff bifurcations is more intricate than for the nv and sv configurations above since in the [*finite*]{} ladder several states of the ff type could play a role in the depinning. First, there exist different states in ladders with odd and even number of cells, as seen when the $\ldots 010101 \ldots$ alternating vortex pattern is fitted into a finite length. Second, there are many ff-states very close energetically with different dynamic stability structure. However, we will show that the landscape of relevant solutions is indeed clear, and depinning can be assigned to instabilities of [*one*]{} of those configurations.
Consider first an [*even*]{} ladder with ten plaquettes as an example. Of the several states of the ff type, three can be thought as relevant: (1a) $1010101010$, (1b) $0101010101$, and (1c) $1101010100$. We have analyzed the stability of these three states and conclude that only configuration (1a) is relevant for depinning of the even ladder. Both (1b) and (1c) cease to exist or become unstable at lower critical currents. This is physically reasonable since (1b) will tend to move one cell to the left under the action of a driving current to produce (1a), while (1c) ceases to exist through a low current saddle-node bifurcation caused by the expulsion of the vortex at the left boundary.
The case of the [*odd*]{} ladder with $N$ plaquettes has one further complication, namely that we cannot have exactly $N/2$ vortices in the array. Thus, the “pure” ff state (as calculated for the infinite case) is not possible under these topological constraints. However, states similar to the “pure” ff are those which contain $(N-1)/2$ and $(N+1)/2$ vortices. Take, for instance, a ladder with eleven plaquettes. We have then two groups of states: those with $6$ vortices, (2a) $10101010101$, (2b) $11010101010$, and (2c) $01010101011$; and those with $5$ vortices, (3a) $10101010100$, (3b) $01010101010$, and (3c) $00101010101$. The detailed analysis of these states shows that (3a) has the highest critical transition and is thus responsible for the depinning of the odd ladder.
There is a final important observation in our argument: the stability diagrams of the configurations which cause the depinning in odd and even ladders –(3a) and (1a) respectively– are [*indistinguishable*]{}. (This is also true for other odd-even related configurations like (2b) and (1c).) The conclusion is then clear: the depinning behavior of odd and even ladders at high $f$ is identical as it can be explained by the bifurcation of configurations (1a) and (3a) which are of the $101010 \ldots$ type.
The zero-eigenvalue bifurcations of these configurations are presented in Figure \[fig:depff\]. The bifurcations are of two types (similar to those obtained for the sv state): saddle-node ($I^{\star}_{\rm ff}$) and pitchfork ($f^{\star}_{\rm ff}$). Note also how the saddle-node bifurcation corresponds to two distinct dynamical instabilities: for $0.38< f <0.5$, the instability is spatially extended, while for $0.22<f<0.38$, the instability is localized at the [*right*]{} edge. Only the former (spatially extended) saddle-node bifurcation has any relevance for the global depinning of the array—as reflected by the agreement between $I_{\rm dep}$ and $I^{\star}_{\rm ff}$ for $f > 0.45$. (Of course, for $f < 0.45$, the depinning current and $I^{\star}_{\rm ff}$ no longer coincide, because the depinning is caused there by a saddle-node bifurcation of a different state, the no-vortex solution, as shown earlier.) The pitchfork bifurcation $f^{\star}_{\rm ff}$ is also spatially extended and, as observed for the single vortex, it implies the breaking of the up-down symmetry in the ff state. Thus, the flux is expelled in the transversal direction. We clarify this in the following by calculating some analytical approximations to these criteria.
### Analytical approximations to the ff bifurcations
The rigorous analysis above indicates that two of the bifurcating mechanisms imply spatially extended perturbations which are not localized at the edges. Hence, we turn to instabilities of the infinite-ladder fully-frustrated solution to obtain analytical approximations.
One of the criteria was already established in Section \[sec:analytical\] where we showed that the no-edge ff solution ceases to exist at a current $I_{\rm ff,th}(f)$ given by (\[eq:currhalf\]). Indeed, we find excellent numerical agreement between the analytical $I_{\rm ff,th}$ and the numerical $I^{\star}_{\rm ff}$ for $0.38< f < 0.5$ (Figure \[fig:depff\]). For instance, the predicted $I_{\rm ff,th}(f=1/2)= \sqrt 5 -2=0.236$ is very close to the dynamically computed $I_{\rm dep}(f=1/2)=0.238$.
To explain the observed pitchfork bifurcation, recall that the up-down symmetric manifold of solutions becomes unstable to normal perturbations when the absolute value of the horizontal phases is larger than $\pi/2$, as given in (\[eq:horizontalcond\]). Thus, from (\[eq:halfsolution\])–(\[eq:Lhf\]) and the critical condition $|{\phi^{H}}^{\dagger}|=\pi/2$ we obtain the following implicit equation for the instability boundary $f_{\rm ff,th}(I)$: $$I=\tan (\pi f_{\rm ff,th}) \sqrt{\cos^{2} (\pi f_{\rm ff,th}) -
4 \sin^{4} (\pi f_{\rm ff,th})},
\label{eq:ffftrans}$$ which is shown in Figure \[fig:depff\] and compared to the exact $f^{\star}_{\rm ff}$ with excellent agreement. As an example, it is easy to show analytically that the value of this critical field when $I=0$ is given by $$\label{eq:f_ff,th_zero}
f_{\rm ff,th}(I=0)=\frac{1}{\pi}
\arcsin \left[ \sqrt{\frac{17^{1/2}-1}{8}} \right ] = 0.2148.$$
In summary, we can explain in part the stability diagram of the ff state in the presence of edges with two bifurcations of the infinitely extended (no-edge) ff solution: the saddle-node bifurcation $I_{\rm ff,th}(f)$, and the subcritical pitchfork $f_{\rm ff,th}(f)$.
However, for the finite ladder, another saddle-node bifurcation is reached before $I_{\rm ff,th}$ for $0.22< f < 0.38$ as seen in Figure \[fig:depff\]. Numerical analysis shows that this bifurcation is local and it corresponds to an instability in the [*leftmost*]{} cell. It can be approximated heuristically by a criterion similar to (\[eq:heurcrit\]) for the nv state, i.e. for this interval of the frustration the ff state depins approximately when the leftmost junction becomes unstable.
Summary and discussion {#sec:conclusion}
======================
The first conclusion of our analysis is that for most values of $f$, the global depinning current of the array $I_{\rm dep}(f)$ coincides with the current $I_{\rm nv}^{\star}$ where the no-vortex state undergoes a saddle-node bifurcation (Figure \[fig:depinning\]). This bifurcation point can be well approximated by an analytical $I_{\rm nv,th}$, given in (\[eq:heuristic\]), derived from an instability criterion for the rightmost junction of the array. For values of $f$ close to $1/2$, however, the global depinning is caused by a saddle-node bifurcation of the fully-frustrated solution at a current $I_{\rm ff}^{\star}$. This bifurcation itself is well approximated by the global instability of the no-edge ff state at $I_{\rm ff,th}$, as given analytically in equation ([\[eq:currhalf\]]{}).
We have also shown that the depinning of the single vortex and its subsequent motion in the $\bf{- \hat x}$ direction (Figure \[fig:LAT\]) is the result of a cascade of saddle-node bifurcations of the single-vortex solution such that, for most values of $f$, the fluxoids are expelled from the ladder through its left edge. More surprisingly, for smaller $f$ the sv and ff configurations can also undergo another transition: a symmetry-breaking subcritical pitchfork bifurcation in which the up-down symmetry of the horizontal phases plays a crucial role. In this case, the fluxoids are expelled in the transversal ($\bf{- \hat y}$) direction through the horizontal junctions.
At a finer level of description, the approximations obtained in Section \[sec:analytical\] for the nv, ff and sv states all have a common feature: the corrections due to the existence of edges, or of topological vortices in the array, decay exponentially in space with a characteristic length dependent on $I$ and $f$, as seen in (\[eq:scnvr\]) for instance. Thus, the effect of the perturbations can usually be captured by a local analysis. This explains why, besides their independence from the purely dynamical parameter $\beta_{c}$, the depinning observables are largely independent of $N$, the length of the array.
We have summarized our results in a zero-temperature stability diagram (Figure \[fig:phasediagedge\]) where we present the different critical currents for the nv, sv, and ff superconducting solutions for the [*finite*]{} ladder (with edges). In short, the array ceases to be superconducting (depins globally) at $I_{\rm dep}$ when either the no-vortex, single-vortex or fully-frustrated solutions ceases to exist through saddle-node bifurcations.
The presence of vortices in the array does not change the observed depinning. If $f<f_{\rm min}(I) \sim 0.1$, the single-vortex solution is always unstable. For $f_{\rm min}<f<0.37$, a vortex in the array will depin at $I_{\rm LAT} < I_{\rm dep}$ and will be expelled at the left edge at $I_{\rm left} < I_{\rm dep}$. At that point the no-vortex solution is recovered. This behavior is the same for multivortex solutions with moderate $f$. For $f>0.37$, the single-vortex is not expelled at the left edge before depinning but its instability is almost identical to that of the no-vortex configuration since vortices enter from the right edge.
Note that no energy criteria have been invoked above. The calculation of energy boundaries for the relevant states remains open for further investigation.
To highlight the effect of the edges and to include energetic considerations we explore in more detail the phase diagram for the infinite ladder (no edges), which we present in Figure \[fig:phasediag\]. (This should be compared to the phase diagram of the ladder with edges in Figure \[fig:phasediagedge\].) When no edges are present, the dynamic bifurcation boundaries for the three states are given by $I^{\dagger}_{\rm nv,th}$, $I_{\rm ff,th}$, $f_{\rm ff,th}$, $I^{\star}_{\rm sv,center}$, and $f^{\star}_{\rm sv,center}$. These dynamic criteria do not contradict previous thermodynamic studies of the infinite ladder [@kardar1; @kardar2] with $I=0$ where the no-vortex solution was calculated to be energetically stable only for frustrations smaller than a thermodynamic critical field $f_{c1}$, above which the flux penetrates the ladder. We have extended these calculations for the [*driven*]{} infinite ladder and the three superconducting solutions addressed in this article. Defining the energy of a given configuration as $$E= - \sum_{all} \cos \phi_{j},$$ we have calculated the energy boundaries for the approximate no-edge no-vortex, fully-frustrated and single-vortex solutions: $f_{\rm nv-sv}(I)$, $f_{\rm nv-ff}(I)$ and $f_{\rm sv-ff}(I)$. The results of these [*thermodynamic*]{} calculations are also presented in Figure \[fig:phasediag\]. Note, for instance, that our calculated $f_{c1} = f_{\rm nv-sv}(I=0) \simeq 0.282$ agrees well with other estimates [@kardar2] $f_{c1}\simeq 2 \sqrt 2/\pi^{2}=0.287$.
Moreover, our approximate solutions produce some new analytical results. For example, a closed expression for the energy boundary between the no-vortex and fully-frustrated solutions $f_{\rm {nv-ff}}(I)$ can be obtained as: $$\begin{aligned}
\sqrt{1-I^2} - I \sqrt{2/L - 1} + 2 \cos (\pi f_{\rm {nv-ff}}) \nonumber \\
- 2 \sin(\pi f_{\rm {nv-ff}}) \sqrt{1-2 I^{2}/L}=0
\label{eq:nvhfbound}\end{aligned}$$ where $L$ is given by (\[eq:Lhf\]) with the negative sign. For the special case $I=0$, and using the limit $2I^{2}/L \rightarrow (1+4 \sin^{2} \pi f)^{-1}$, it can be shown analytically that this boundary crosses the $I=0$ axis at $f_{\rm{nv-ff}}(I=0)=1/3$.
We emphasize also that the regions in Figure \[fig:phasediag\] present distinct dynamical and thermodynamical stabilities. For instance, in some of them, the single-vortex solution is not the ground state of the system, although it is [*dynamically*]{} stable (metastable).
When comparing our results with those found in earlier work, one should carefully note the direction of injection of the driving current $I$. Previous analytical studies [@kardar2; @benedict] have focused on the $I=0$ case and considered the effect of a small [*parallel*]{} current in the ${\bf \hat x}$ direction. In contrast, here $I$ is injected in the [*perpendicular*]{} (${\bf{\hat y}}$) direction. The depinning depends on the direction of current injection, a factor to be taken into account when explaining recent numerical simulations of ladder arrays [@stroudscreen]. In those simulations, marked differences between the depinning current of a circular ladder with [*perpendicular*]{} injection, and of an open-ended ladder with [*parallel*]{} injection were reported.
Although we have not studied the ring ladder in this paper, it is closely connected to the infinite (no-edge) ladder. The solutions for the infinite ladder constitute a sub-manifold of the solutions for the ring ladder (i.e., the topological constraints are more strict on the infinite ladder than in the ring ladder). To understand this, note first that the governing equations (\[eq:eq1\])–(\[eq:eq3\]) are the same for both ring and infinite ladders. On the other hand, following the same reasoning given by the sequence of equations (\[eq:interiorequation1\])–(\[eq:returnedges\]), we conclude that the topological constraints in the ring ladder imply only that $$\label{eq:returnring}
I^{H}_{j}+I^{h}_{j}=C, \;\; \forall j,$$ where $C$ is a constant. Therefore, from (\[eq:returnedges\]), the infinite ladder is mathematically equivalent to the particular case of the ring with $C=0$, in which the concentric currents through the horizontal junctions in the outer $I^{H}$ and inner $I^{h}$ circles are equal and opposite.
Consequently, the results for the infinite ladder summarized in Figure \[fig:phasediag\] are also valid for the ring ladder if we restrict to the submanifold with $C=0$. In this case, the phase diagram has to be reinterpreted in terms of the new topological constraints. First, the depinning of the no-vortex solution is unchanged: $I^{\dagger}_{\rm nv,th}$ is still constant and equal to $1$. However, the two bifurcations of the single-vortex configurations have new dynamical meaning. If the sv state goes unstable through the saddle-node bifurcation $I^{\star}_{\rm sv} = I_{\rm LAT}$, flux cannot be expelled through the horizontal junctions and the vortex depins and moves along the ladder circularly. Thus, the ring ladder depins effectively at $I_{LAT}$. However, if $f<f^{\star}_{\rm sv}$, the flux can be expelled [*transversally*]{} from the ladder through the horizontal junctions and the no-vortex state is recovered. Then, the depinning occurs at $I^{\dagger}_{\rm nv,th}=1$. This is exactly the behavior reported from numerical simulations by Hwang [*et al.*]{} [@stroudscreen]. First, their isotropic $I_{c \perp}$ coincides with our calculated $I^{\star}_{\rm sv} = I_{\rm LAT}$. Second, they found a critical field $f_{c1 \perp}^{\ast}
\simeq 0.12$ below which the depinning current is $I_{c \perp}=1$ with exclusion of field inside the array. This corresponds to $f^{\star}_{\rm sv}$, the frustration below which the single-vortex configuration becomes dynamically unstable through a symmetry-breaking pitchfork bifurcation, which can be approximated analytically by $f_{\rm sv,th}(I)$, as given in (\[eq:stabhorizontal\]). (In the absence of driving, the analytical prediction (\[eq:crithorizontal\]) gives $f_{\rm sv,th}(I=0)=0.1193$, in perfect agreement with their numerical simulations [@stroudscreen].)
We note in passing that the same behavior should be expected for the fully-frustrated state in the ring ladder. The checkerboard pattern would begin to slide along the ring at $I_{\rm ff,th}$ producing a finite voltage. However, if the $f_{\rm ff,th}$ is crossed, the flux can be expelled transversally and the nv state would appear. These predictions would have to be checked numerically.
As a final remark, we also note that the second device considered by Hwang [*et al.*]{} [@stroudscreen]—an open-ended ladder with [*parallel*]{} current injection—cannot be compared directly with our ladder with [*perpendicular*]{} injection. However, the depinning current $I_{c}(f)$ follows a similar trend to our $I_{\rm dep}(f)$. In fact, the dependence of their $I_{c}(f)$ seems to be well explained with formulae calculated by Benedict [@benedict] for the same device by invoking a similar criterion: the onset of soft modes [@benedict2].
We thank Shinya Watanabe, Enrique Trías, and Herre van der Zant for consistently useful discussions, and Mehran Kardar for helpful guidance to the literature on ladder arrays. Research supported by a Spanish MEC-Fulbright predoctoral fellowship (MB), and by NSF grants DMS-9500948 (SHS) and DMR-9402020 (TPO).
Comparison of the single-vortex configuration in the ladder with the one-dimensional kink {#sec:appsg}
=========================================================================================
Although analogies between the ladder and strictly one-dimensional parallel arrays can be drawn, we show now how the single-vortex solution for the quasi-one-dimensional ladder is mathematically different from the kink-like vortex in 1-D parallel arrays.
Recall that the equations for the one-dimensional parallel array can be reduced to a discrete driven sine-Gordon equation [@shilong; @parallelcontinuum] if only self-inductances are considered. When $I=0$, the discrete single-vortex solution is well approximated by the kink solution of the undriven, time-independent, infinitely-extended, one-dimensional continuum sine-Gordon equation [@drazin]: $$\label{eq:kink}
\phi^{\rm SG}_{i}=4 \arctan\{\exp[(i-i_{0})/\lambda_{\rm SG}]\}.$$ Thus, the vortex corresponds to a $2 \pi$-jump centered at $i_{0}$ with a characteristic half-width $\lambda_{\rm SG}$. (Incidentally, it has also been shown [@parallelcontinuum] that by introducing an effective $\lambda_{\rm SG}^{\rm eff}$, this functional form is also valid when mutual inductances are included).
For our no-inductance ladder array, it is also possible to obtain an approximate sine-Gordon equation for the system. Although there is no explicit inductance in the problem, the coupling between the vertical junctions is provided by the horizontal junctions via the fluxoid quantization.
The approximate sine-Gordon equation has been most simply obtained [@kardar2; @strouddyn] by assuming that the horizontal phases are small: $\sin \phi^{H}_{j} \approx \phi^{H}_{j}$. Then, the [*zero-current*]{} time-independent equations for the ladder (\[eq:scequation1\])–(\[eq:scequation2\]) can be reduced to $\phi^{V}_{j+1}-2 \phi^{V}_{j} + \phi^{V}_{j-1} =
2 \sin \phi^{V}_{j}$. This gives, in the continuum limit, the time-independent sine-Gordon equation with no forcing and $\lambda_{\rm SG}=1/\sqrt{2}$, where the cell size is taken as length unit. However, a better linearization is suggested by the numerics if we take the phase change in the vertical junctions $(\phi^{V}_{j}-\phi^{V}_{j+1})/2 = \phi^{H}_{j} - \pi f \ll 1$ as the small parameter. In other words, one should linearize about $\phi^{H}_{j} = \pi f$, not $\phi^{H}_{j} = 0$. In that case, we obtain the following more accurate sine-Gordon equation: $\phi^{V}_{xx}-(2/\cos \pi f) \sin \phi^{V}=0$, with $\lambda_{\rm SG}^2=\cos(\pi f)/2$.
We argue now that the numerically observed single-vortex configuration is not as accurately approximated by the kink (\[eq:kink\]) as it is by our expression (\[eq:onevortex\])–(\[eq:Bonevortex\]). To compare them, we particularize (\[eq:onevortex\])–(\[eq:Bonevortex\]) for $I=0$ for a long array such that $1 \ll a \ll N+1$. Then, the vertical phases near the center of the vortex become $$\label{eq:vortexlad}
\phi^{V}_{j}= \left \{
\begin{array}{ll}
P \exp\{(j-a)/\lambda\}, & j\leq a\\
2 \pi -P \exp\{(a+1-j)/\lambda\}, & j > a
\end{array}
\right .$$ with $r$ and $\lambda$ given by (\[eq:scnvr\]) and $P$ by $$\sin P=\sin\left(\pi f -\frac{r-1}{2r}P\right)+ \sin(\pi f +P).$$ Note that in (\[eq:vortexlad\]) we have not reduced the phases to the interval $[-\pi, \pi)$, to facilitate the comparison with (\[eq:kink\]). Our solution resembles the sine-Gordon kink in that it describes a $2 \pi$-jump with odd symmetry with respect to . However, both the functional forms and the characteristic lengths are different. Figure \[fig:sinegordon\] shows that the numerics are better approximated over a wide range of $f$ by (\[eq:vortexlad\]) than by the sine-Gordon kink (\[eq:kink\]).
Given the relative inappropriateness of the sine-Gordon kink as a model of the vortex in the no-inductance ladder array, we conclude that even for the static case it is an oversimplification to reduce the ladder to a one-dimensional parallel array where the horizontal junctions are approximated by an effective inductance. Other dynamic phenomena observed in the ladder, e.g, the dynamical mechanism of retrapping from the whirling mode, reinforce this statement and will be discussed elsewhere [@usladder2].
No-vortex solution and depinning with self-inductance {#sec:appendixa}
=====================================================
Following on the concepts and notation in Sections \[sec:analytical\] and \[sec:depinning\], we briefly consider the ladder with self-inductance. This is a first approximation to explore the effect of self-fields on the no-vortex solution and, consequently, on the global depinning current of the array.
When self-inductances are included, the time-independent governing equations (\[eq:scequation1\])–(\[eq:scequation2\]) become $$\begin{aligned}
\label{eq:selfind1}
&I+\sin \phi^{H}_{j-1}=\sin \phi^{H}_{j}+\sin \phi^{V}_{j}\\
&\phi^{V}_{j}-\phi^{V}_{j+1}-2 \phi^{H}_{j}= 2 \pi (n_{j}-f)
- I_{j}^{m}/ \Lambda^2,
\label{eq:selfind2}\end{aligned}$$ where $I_{j}^{m}$ is the mesh current in plaquette $j$ and $\Lambda^2=L_{J}/L_{s}$ is the two-dimensional penetration depth and a measure of the discreteness of the array. Due to the geometrical constraints of the ladder, it is readily seen that $I_{j}^{m} = - I_{j}^{H}$ in this case.
For the no-vortex solution we still have $\phi_{j}^{V} \simeq \phi_{j+1}^{V} = {\phi^{V}}^{\ast}$ and $\phi_{j}^{H} \simeq \phi_{j+1}^{H}= {\phi^{H}}^{\ast}$ far from the edges. From (\[eq:selfind1\]) we then get $$\label{eq:selfindvertical}
{\phi^{V}}^{\ast}=\arcsin I,$$ as for the case with no inductance ($\Lambda = \infty$). However, the horizontal phase is different and has to be calculated from the following nonlinear equation: $$\label{eq:selfindhorizontal}
{\phi^{H}}^{\ast}+\frac{1}{2\Lambda^2}\;{\sin {\phi^{H}}^{\ast}} = \pi f.$$ This is a particular case of Kepler’s equation, studied in celestial mechanics [@celestial], which can be solved through the method of successive approximations. In our case, a good approximation to ${\phi^{H}}^{\ast}$ over the whole range of $f$ is given by the first iteration of that method as: $${\phi^{H}}^{\ast} \simeq \pi f - \frac{1}{2 \Lambda^2} \;
\sin \left(\frac{2 \Lambda^2 }{1+ 2 \Lambda^2} \; \pi f\right).$$
The procedure to account for the open boundaries is identical to that presented in Section \[sec:analytical\]. The results are also similar: the corrections decay exponentially from the edges as in (\[eq:correctionhor\])–(\[eq:scnvr\]). However, the characteristic length $\lambda_{s}=1/\ln r_{s}$ is now given by $r_{s}= \alpha_{s} + \sqrt{\alpha_{s}^{2}-1}$ where $$\label{eq:selfindalpha}
\alpha_{s}=1+\frac{\sqrt{1-I^{2}}}{\cos{\phi^{H}}^{\ast}} \;
\left ( 1 + \frac{1}{2 \Lambda^2} \cos{\phi^{H}}^{\ast}\right ).$$
To assess how the inductance affects the depinning current, we use (\[eq:selfindalpha\]) and the ${\phi^H}^{\ast}$ obtained from (\[eq:selfindhorizontal\]) with a heuristic criterion similar to (\[eq:heuristic\]), $$\label{eq:selfinddepin}
\arcsin (1-I_{\rm nv,th}) = {\phi^{H}}^{\ast} -
\frac{r_{s}-1}{2 r_{s}} \arccos I_{\rm nv,th},$$ to calculate $I_{\rm nv,th}(f,\Lambda^2)$. The results in Figure \[fig:selfinductance\] show that, for fixed $\Lambda$, $I_{\rm nv,th}(f)$ still decreases monotonically with $f$. However, for a given $f$, the value of the depinning current increases as $\Lambda$ diminishes. This is expected on physical grounds since an increase in the inductance $L_{s}$ implies larger self-fields which oppose the external applied magnetic field, thus decreasing the effective magnetic flux through the array.
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The first formula of Section VI should include a contribution from the driving current. In our choice of gauge, $$E= - \sum_{\rm{all}} \cos \phi_{j} - I \sum_{j=1}^{N+1} \phi^{V}_{j}.$$ Accordingly, Eq. (64) becomes $$\sqrt{1-I^2} + 2 \cos (\pi f_{\rm {NV-FF}})+I \arcsin(I)
- 2 \sin(\pi f_{\rm {NV-FF}}) (1-2 I^{2}/L)^{-1/2}
- I \arcsin(\sqrt{L/2})=0,$$ and $f_{\rm{NV-FF}}(I=0)=1/3$ remains unaltered. Figure 15 is very slightly modified as shown. The conclusions do not change.
[^1]: Present address: Ginzton Laboratory, Stanford University, Stanford, CA 94305.
|
---
abstract: 'This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the $N$-dimensional linear convection-diffusion equation with $N{\geqslant}1$ and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.'
author:
- Imene Aicha Djebour
- Takéo Takahashi
- Julie Valein
bibliography:
- 'biblio.bib'
title: Feedback stabilization of parabolic systems with input delay
---
[**Keywords:**]{} stabilizability, delay control, parabolic systems, finite-dimensional control
[**2010 Mathematics Subject Classification**]{} 93B52, 93D15, 35Q30, 76D05, 93C20.
Introduction
============
Time delay phenomena appear in many applications, for instance in biology, mechanics, automatic control or engineering and are inevitable due to the time-lag between the measurements and their exploitation. For instance in control problems, one need to take into account the analysis time or the computation time. In the context of stability problems for partial differential equations with delay, it is classical that a small delay in the feedback mechanism can destabilize a system (see for instance [@MR818942; @MR937679]). On the other hand, a delay term can also improve the performance of a system (see for instance [@abdallah1993delayed]). These features appear for hyperbolic systems and here our aim is to consider the stabilization problems for a large class of parabolic systems with a particular delay input.
More precisely this article is devoted to the feedback stabilization of the system $$\label{ijt000}
z'=Az+Bv+f, \quad z(0)=z^0,$$ where $A$ is the generator of an analytic semigroup $(e^{tA})_{t{\geqslant}0}$ on a Hilbert space $\mathbb{H}$, where $B : \mathbb{U} \to \mathcal{D}(A^*)'$ is a linear operator on a Hilbert space $\mathbb{U}$ and where $f$ is a given source satisfying an exponential decay at infinity. Our aim is to obtain a feedback control $v(t)$ that depends on the values of $z(s)$ for $s{\leqslant}t-\tau$, where $\tau>0$ is a positive constant corresponding to a delay. With such a feedback control, our aim is to obtain exponential stabilization of if we assume that it is the case without delay. A characterization of the exponential stabilization of in the case without delay is the well-known Fattorini-Hautus criterion (see [@Fattorini1966], [@Hautus] and [@BT2014]): $$\forall\varepsilon\in \mathcal{D}(A^*), \
\forall \lambda\in\mathbb{C},\
{\operatorname{Re}}\lambda{\geqslant}-\sigma\quad
A^*\varepsilon= \lambda \varepsilon \quad \text{and} \quad B^*\varepsilon=0
\quad\Longrightarrow\quad
\varepsilon = 0. \tag{$\text{UC}_{\sigma}$}\label{UCstab}$$ Here and in what follows, we denote by $A^* : \mathcal{D}(A^*) \to \mathbb{H}$ and by $B^* : \mathcal{D}(A^*) \to \mathbb{U}$ the adjoint operators of $A$ and $B$. This criterion is equivalent to the exponential stabilization of with a rate larger than $\sigma$ provided we assume the following hypotheses: $$\label{H1}\tag{Hyp1}
\begin{array}{c}
\text{The spectrum of $A$ consists of isolated eigenvalues $(\lambda_j)$ with
finite algebraic multiplicity $N_j$}\\
\text{and there is no finite cluster point in } \{\lambda \in \mathbb{C} \ ; \ {\operatorname{Re}}\lambda {\geqslant}-\sigma\},
\end{array}$$ $$\label{H2}\tag{Hyp2}
B\in \mathcal{L}(\mathbb{U},\mathbb{H}_{-\gamma}) \quad \text{for some } \gamma\in [0,1).$$ The spaces $\mathbb{H}_\alpha$ are defined as follows: we fix $\mu_0\in \rho(A)$, then $$\label{Halpha}
\mathbb{H}_\alpha:= \left\{
\begin{array}{ll}
\mathcal{D}((\mu_0-A)^{\alpha}) &\mbox{ if }\alpha{\geqslant}0\\
\mathcal{D}((\mu_0-A^*)^{-\alpha})' &\mbox{ if }\alpha< 0
\end{array}
\right.\quad \mbox{ and }\quad
\mathbb{H}_{\alpha}^*:= \left\{
\begin{array}{ll}
\mathcal{D}((\mu_0-A^*)^{\alpha}) &\mbox{ if }\alpha{\geqslant}0\\
\mathcal{D}((\mu_0-A)^{-\alpha})' &\mbox{ if }\alpha< 0.
\end{array}
\right.$$ To deal with the source $f$, we also assume the following hypothesis $$\label{H3}\tag{Hyp3}
\mathbb{H}_\alpha=[\mathbb{H},\mathcal{D}(A)]_{\alpha} \quad (\alpha \in [0,1]),$$ where $[\cdot,\cdot]_{\alpha}$ denotes the complex interpolation method. We assume that $$\label{ijt004}
f_\sigma:t\mapsto e^{\sigma t} f(t)\in L^2(0,\infty;\mathbb{H}_{-\gamma'}) \quad \gamma'<1/2.$$ We say that $f\in L_\sigma^2(0,\infty;\mathbb{H}_{-\gamma'})$ if $f_\sigma\in L^2(0,\infty;\mathbb{H}_{-\gamma'})$ and we write $$\left\|f\right\|_{L_\sigma^2(0,\infty;\mathbb{H}_{-\gamma'})}= \left\|f_\sigma\right\|_{L^2(0,\infty;\mathbb{H}_{-\gamma'})}.$$ The same definition can be extended to spaces of the kind $L^p_\sigma(0,\infty;\mathbb{X})$, $C_\sigma^0([0,\infty);\mathbb{X})$, $H^m_\sigma(0,\infty;\mathbb{X})$, with $\mathbb{X}$ a Banach space. Note that a sufficient condition for is that $A$ has compact resolvent. For all $\lambda_j$ eigenvalue of $A$, we define its geometric multiplicity $$\ell_j:=\dim \ker (A-\lambda_j{\operatorname{Id}})\in \mathbb{N}^*.$$ We also define the maximum of the geometric multiplicities of the unstable modes: $$\label{Nc}
N_+:=\max \{\ell_i, \ {\operatorname{Re}}\lambda_i{\geqslant}-\sigma\}.$$
Our main result is the following theorem:
\[thmain\] Let us consider $\sigma>0$ and let us assume , , and . Then there exist $K\in L^\infty_{\rm loc}(\mathbb{R}^2;\mathcal{L}(\mathbb{H}))$, $\zeta_k\in \mathcal{D}(A^*)$, $v_k\in B^*\left(\mathcal{D}(A^*)\right)$, $k=1,\ldots,N_+$, such that if $$\label{fb1}
v(t)=\mathds{1}_{[\tau,+\infty)}(t) \sum_{k=1}^{N_+} \left(z(t-\tau)+\int_0^{t-\tau} K(t-\tau,s)z(s) \ ds,\zeta_k\right)_{\mathbb{H}} v_k,$$ then for any $z^0 \in \mathbb{H}$, $f$ satisfying , the solution $z$ of satisfies $$\label{ijt008}
\|z(t)\|_{\mathbb{H}} {\leqslant}C e^{-\sigma t} \left(\|z^0\|_{\mathbb{H}} + \|f\|_{L^2_{\sigma}(0,\infty;\mathbb{H}_{-\gamma'})}\right)
\quad (t>0).$$ Assume moreover that $\gamma=0$, $\gamma'=0$ and that $z^0\in \mathbb{H}_{1/2}$. Then, $$z\in L^2_\sigma (0,\infty;\mathbb{H}_1)\cap C^0_\sigma ([0,\infty);\mathbb{H}_{1/2})\cap H^1_\sigma (0,\infty;\mathbb{H}),$$ and $$\label{ijt200}
\|z\|_{L^2_\sigma (0,\infty;\mathbb{H}_1)\cap C^0_\sigma ([0,\infty);\mathbb{H}_{1/2})\cap H^1_\sigma (0,\infty;\mathbb{H})}
{\leqslant}C\left(\|z^0\|_{\mathbb{H}_{1/2}} + \|f\|_{L^2_{\sigma}(0,\infty;\mathbb{H})}\right).$$
Here and in all what follows, $\mathds{1}_{\mathcal{O}}$ is the characteristic function of the set $\mathcal{O}$.
The above result shows that we can stabilize the above general class of linear parabolic systems with a finite number of controls and with a constant delay: the feedback control $v(t)$ at time $t$, given by , only depends on value of the state $z(s)$ for $s{\leqslant}t-\tau$. This result can be seen as a generalization of several recent results on the stabilization of parabolic systems with delay control, in particular [@MR3767485] where the authors constructed a feedback control for finite dimensional linear systems, and [@MR3936420] where the authors obtained a stabilizing feedback control of a one-dimensional reaction-diffusion equation with a boundary control subjected to a constant delay. Let us mention some ideas of their method that we adapt to prove our result: using that their operator is self-adjoint of compact resolvent they split the system into an unstable finite-dimensional part and a stable infinite-dimensional part. They are thus led to stabilize the finite-dimensional unstable system and to do this with a delay, they use the Artstein transformation and obtain an autonomous control system without delay satisfying the Kalman condition. Finally, by using an appropriate Lyapunov function, they prove that the feedback control designed in the finite-dimensional part actually stabilizes the whole system.
We can mention several articles in this direction: in [@lhachemi2019boundary], the authors consider the stabilization of a structurally damped Euler-Bernoulli beam. The corresponding system is parabolic but the main operator is no more self-adjoint. Then [@lhachemi2019feedback] generalizes the result of [@MR3936420] in the case where the main operator is a Riesz spectral operator with simple eigenvalues. In [@Lhachemi_2020] they manage to extend the result of [@MR3936420] to the case where the control contains some disturbances and where the delay can depend on time.
Here our aim is to extend the result of [@MR3936420] for a large class of parabolic systems, and in particular with the possibility to consider partial differential equations in dimension larger than one. We also precise the number of controls $N_+$ needed to stabilize the system by using the approach developed in [@MB1] in the case of the Navier-Stokes system or in [@BT2014], for general linear and nonlinear parabolic systems. We present two important examples, that is the reaction-diffusion equation and the Oseen system and we end this paper to show that within this framework, we can also handle some nonlinear parabolic systems such as the Navier-Stokes system.
The present paper is organized as follows. In \[sec\_proof\], the proof of \[thmain\] is given. As in [@MR3936420], it relies on the decomposition of the system into two parts: an unstable finite-dimensional part and an infinite-dimensional part. This decomposition is possible thanks to and [@Kato1966 Theorem 6.17, p.178]. Due to the presence of a constant delay, an equivalent autonomous control system is considered for the finite-dimensional part by means of the Artstein transformation. This system is exponentially stabilizable by using . Using the inverse of the Artstein transform, a stabilizing feedback control is designed in the finite-dimensional space that stabilizes exponentially the finite-dimensional unstable system (with delay control). Finally, we prove that the designed feedback stabilizes exponentially the complete system. Thereafter, we illustrate our results by some precise examples: the case of the feedback stabilization of the $N$-dimensional linear convection-diffusion equation with $N{\geqslant}1$ with delay boundary control in \[sec\_heat\], the case of the feedback stabilization of the Oseen system with delay distributed control in \[sec\_oseen\] and finally, a local feedback distributed stabilization of the Navier-Stokes system around a stationary state in \[sec\_navier\].
Proof of \[thmain\] {#sec_proof}
===================
Let us consider $\sigma>0$. We first decompose the spectrum of $A$ into the “unstable” modes and the “stable” modes: $$\label{ijt005}
\Sigma_+:= \{\lambda_j \ ; \ {\operatorname{Re}}\lambda_j {\geqslant}-\sigma\},
\quad
\Sigma_-:= \{\lambda_j \ ; \ {\operatorname{Re}}\lambda_j < -\sigma\}.$$ Using that $(e^{tA})_{t{\geqslant}0}$ is an analytic semigroup (see [@BDDM Theorem 2.11, p.112]) and , we see that $\Sigma_+$ is of finite cardinal.
Thus, we can introduce the projection operator (see [@Kato1966 Thm. 6.17, p.178]) defined by $$P_+:= \frac{1}{2\pi \imath}\int_{\Gamma_+} (\lambda -A)^{-1} \ d\lambda,$$ where $\Gamma_+$ is a contour enclosing $\Sigma_+$ but no other point of the spectrum of $A$. We can define $$\mathbb{H}_+:=P_+ \mathbb{H}, \quad \mathbb{H}_-:=({\operatorname{Id}}-P_+) \mathbb{H}.$$ From [@Kato1966 Thm. 6.17, p.178], we have $\mathbb{H}_+ \oplus \mathbb{H}_-=\mathbb{H}$ and if we set $$A_+:=A_{|\mathbb{H}_+} : \mathbb{H}_+ \to \mathbb{H}_+, \quad
A_-:=A_{|\mathbb{H}_-} : \mathcal{D}(A)\cap \mathbb{H}_- \to \mathbb{H}_-,$$ then the spectrum of $A_+$ (resp. $A_-$) is exactly $\Sigma_+$ (resp. $\Sigma_-$). By using the analyticity of $\left(e^{A t}\right)_{t{\geqslant}0}$, and , we deduce the existence of $\sigma_- >\sigma$ such that $$\label{ijt006}
\left\| e^{A_-t} \right\|_{\mathcal{L}(\mathbb{H}_-)} {\leqslant}Ce^{-\sigma_- t},
\quad
\left\| (\lambda_0-A)^{\gamma} e^{A_-t} \right\|_{\mathcal{L}(\mathbb{H}_-)} {\leqslant}C\frac{1}{t^\gamma} e^{-\sigma_- t}.$$
We can proceed similarly for $A^*$: we write $$P_+^*:= \frac{1}{2\pi \imath}\int_{\overline{\Gamma_+}} (\lambda -A^*)^{-1} \ d\lambda,$$ $$\mathbb{H}_+^*:=P_+^* \mathbb{H}, \quad \mathbb{H}_-^*:=({\operatorname{Id}}-P_+^*) \mathbb{H},$$ $$A_+^*:=A_{|\mathbb{H}_+^*} : \mathbb{H}_+^* \to \mathbb{H}_+^*, \quad
A_-^*:=A_{|\mathbb{H}_-^*} : \mathcal{D}(A^*)\cap \mathbb{H}_-^* \to \mathbb{H}_-^*.$$ Note that $P_+^*$ is the adjoint of $P_+$. In particular, we see that if $z\in \mathbb{H}_-$ and $\zeta\in \mathbb{H}_+^*$, then $$\label{oubli1}
\left(z,\zeta\right)_{\mathbb{H}}=\left(({\operatorname{Id}}-P_+)z,\zeta\right)_{\mathbb{H}}=\left(z,({\operatorname{Id}}-P_+^*)\zeta\right)_{\mathbb{H}}=0.$$
We also define $$\mathbb{U}_+:=B^* \mathbb{H}_+^*, \quad
\mathbb{U}_-:=B^* \mathbb{H}_-^*,$$ and $$p_+ : \mathbb{U} \to \mathbb{U}_+, \quad
p_- : \mathbb{U} \to \mathbb{U}_-, \quad
i_+ : \mathbb{U}_+ \to \mathbb{U}, \quad
i_- : \mathbb{U}_- \to \mathbb{U},$$ the orthogonal projections and the inclusion maps. Then we write $$B_+:=P_+ B i_+ \in \mathcal{L}(\mathbb{U}_+,\mathbb{H}_+),
\quad
B_-:=({\operatorname{Id}}-P_+) B i_- \in \mathcal{L}(\mathbb{U}_-,\mathbb{H}_-\cap \mathbb{H}_{-\gamma}).$$ We have that $$P_+B=B_+p_+
\quad
({\operatorname{Id}}-P_+)B=B_-p_-.$$ Indeed, for any $w\in \mathcal{D}(A)$ and $ v\in \mathbb{U}$ we can write $$\left\langle P_+Bv,w\right\rangle_{\mathbb{H}_{-1},\mathbb{H}_{1}}
=\left\langle v, B^*P^*_+ w \right\rangle_\mathbb{U}
=\left\langle p_+v, p_+B^*P^*_+ w \right\rangle_\mathbb{U}
=\left\langle P_+B i_+p_+v, w \right\rangle_{\mathbb{H}_{-1},\mathbb{H}_{1}}
=\left\langle B_+p_+v, w \right\rangle_{\mathbb{H}_{-1},\mathbb{H}_{1}},$$ and we can prove similarly the other relation.
From the above notation, we can split into the two equations (see [@BT2014; @RaymondThevenet2009]). $$\label{ijt001}
z_+'=A_+ z_+ + B_+ p_+ v + P_+f, \quad z_+(0)=P_+z^0,$$ $$\label{ijt002}
z_-'=A_- z_- + B_- p_- v + ({\operatorname{Id}}-P_+)f, \quad z_-(0)=({\operatorname{Id}}-P_+)z^0.$$ In order to study the stabilization of the finite-dimensional system , we use the Artstein transformation that allows us to pass from in the case of a delay input to an autonomous system. More precisely, we consider $$w(t):=z_+(t)+\int_t^{t+\tau} e^{(t-s)A_+} B_+ p_+ v(s) \ ds.$$ Then in what follows, we study the stabilization of the autonomous system satisfied by $w$ (\[L1\]). Since the corresponding feedback is expressed with $w$, we also consider the inverse of the Artstein transformation and more precisely show the existence of a kernel $K$ to write $w$ in terms of $z_+$ (\[L2\]).
\[L1\] Assume for $\sigma>0$. Then, there exist $C>0$ and $G\in\mathcal{L}(\mathbb{H}_+,\mathbb{U}_+)$, with ${\operatorname{rank}}G{\leqslant}N_+$ where $N_+$ is defined by , such that for any $f\in L_\sigma^2(0,\infty;\mathbb{H}_{-\gamma'})$ and $w^0 \in \mathbb{H}_+$, the solution of $$\label{eq12}
\left\{
\begin{array}{c}
w'=A_+w+e^{-\tau A_+}B_+p_+ Gw+P_+f,\\
w(0)=w^0 ,
\end{array}
\right.$$ satisfies $$\label{stab1}
\|w\|_{H^1_\sigma(0,\infty;\mathbb{H}_+)} {\leqslant}C\left(\|w^0\|_{\mathbb{H}_+} + \|P_+ f\|_{L^2_{\sigma}(0,\infty;\mathbb{H}_+)}\right).$$
Let us consider $\sigma_\star>\sigma.$ First we notice that $(A_+,e^{-\tau A_+}B_+p_+)$ satisfies the Fattorini-Hautus test: assume $$A_+^* \varepsilon =\overline{\lambda_j} \varepsilon, \quad
B_+^* e^{-\tau A_+^*} \varepsilon=0.$$ Then we deduce $$A^*\varepsilon =\overline{\lambda_j} \varepsilon, \quad
B_+^* e^{-\tau A_+^*} \varepsilon
=e^{-\tau \overline{\lambda_j} } B^* P_+^* \varepsilon
=e^{-\tau \overline{\lambda_j} } B^* \varepsilon=0.$$ Thus from , we deduce $\varepsilon=0$. We can thus use the standard result of Fattorini or Hautus (see also [@BT2014 Theorem 1.6]) to deduce that there exists $G\in\mathcal{L}(\mathbb{H}_+,\mathbb{U}_+)$, with ${\operatorname{rank}}G{\leqslant}N_+$ such that the solution of $$\label{eq12-2}
\left\{
\begin{array}{c}
w'=A_+w+e^{-\tau A_+}B_+ p_+ Gw,\\
w(0)=w^0 \in \mathbb{H}_+,
\end{array}
\right.$$ satisfies $$\label{stab1-2}
\left\|w(t)\right\|_{\mathbb{H}_+} {\leqslant}Ce^{-\sigma_\star t}\left\|w^0 \right\|_{\mathbb{H}_+} , \quad t{\geqslant}0.$$ Then we write the Duhamel formula for the solutions of $$w(t)=e^{A_\star t} w^0 +\int_0^t e^{A_\star(t-s)} P_+f(s) \ ds,$$ with $$A_\star=A_+ + e^{-\tau A_+}B_+ p_+ G,$$ to deduce .
\[L2\] There exists $K\in L^\infty_{\rm loc}(\mathbb{R}^2;\mathcal{L}(\mathbb{H}_+))$ such that $$\begin{gathered}
\label{k3}
K(t,s)= e^{(t-s-\tau)A_+}B_+p_+G\mathds{1}_{(\max\{t-\tau,0\},t)} (s) \\+ \int_{\max\{t-\tau,s\}}^te^{(t-\xi-\tau)A_+}B_+p_+GK(\xi,s) \ d\xi
\quad (t>0, s\in (0,t)).\end{gathered}$$
The proof relies on a fixed point argument. We set $$K_0(t):=e^{(t-\tau)A_+}B_+p_+G, \quad K_0 \in L^\infty(0,\tau;\mathcal{L}(\mathbb{H_+})),$$ so that writes $$K(t,s)=K_0(t-s)\mathds{1}_{(\max\{t-\tau,0\},t)} (s)
+ \int_{\max\{t-\tau,s\}}^t K_0(t-\xi) K(\xi,s) \ d\xi.$$
Let $T>0$, and let us define $$D_T=\{(t,s)\in\mathbb{R}^2,\quad t\in (0,T),\quad s\in (0,t)\} ,$$ and $$\Phi: L^\infty(D_T;\mathcal{L}(\mathbb{H_+}))\rightarrow L^\infty(D_T;\mathcal{L}(\mathbb{H_+})),$$ $$(\Phi K)(t,s)=\int_{\max\{t-\tau,s\}}^tK_0(t-\xi)K(\xi,s) \ d\xi \quad ((t,s)\in D_T).$$ The mapping $\Phi$ is well-defined, and is a linear and bounded operator of $L^\infty(D_T;\mathcal{L}(\mathbb{H_+}))$. Moreover, $$\left\| (\Phi K)(t,s) \right\|_{\mathcal{L}(\mathbb{H}_+)} {\leqslant}t \left\|K_0 \right\|_{L^\infty(0,\tau;\mathcal{L}(\mathbb{H_+}))}
\left\|K \right\|_{L^\infty(D_T;\mathcal{L}(\mathbb{H_+}))}.$$ This yields $$\left\| (\Phi^2 K)(t,s) \right\|_{\mathcal{L}(\mathbb{H}_+)} {\leqslant}\frac{t^2}{2} \left\|K_0 \right\|_{L^\infty(0,\tau;\mathcal{L}(\mathbb{H_+}))}^2
\left\|K \right\|_{L^\infty(D_T;\mathcal{L}(\mathbb{H_+}))},$$ and by induction $$\left\| (\Phi^n K)(t,s) \right\|_{\mathcal{L}(\mathbb{H}_+)} {\leqslant}\frac{t^n}{n!} \left\|K_0 \right\|_{L^\infty(0,\tau;\mathcal{L}(\mathbb{H_+}))}^n
\left\|K \right\|_{L^\infty(D_T;\mathcal{L}(\mathbb{H_+}))} \quad (n\in \mathbb{N}^*).$$ In particular, for $n$ large enough, $\Phi^n$ is a strict contraction and so is $$(\widetilde{\Phi} K)(t,s):= (\Phi K)(t,s) + K_0(t-s)\mathds{1}_{(\max\{t-\tau,0\},t)} (s).$$ Thus, $\widetilde \Phi$ admits a unique fixed point which is a solution of .
We are now in a position to prove the main result
We consider $G$ and $K(t,s)$ obtained in \[L1\] and \[L2\], and we set $$\label{fb12}
v(t)=\mathds{1}_{[\tau,+\infty)}(t)G\left[z_+(t-\tau)+\int_0^{t-\tau} K(t-\tau,s)z_+(s) \ ds \right].$$ Note that, we can write $G$ as $$G(\phi)=\sum_{k=1}^{N_+} \left(\phi,\zeta_k\right)_{\mathbb{H}} v_k, \quad (\phi\in \mathbb{H}_+)$$ with $\zeta_k\in \mathbb{H}_+^*$ and $v_k\in \mathbb{U}_+$, $k=1,\ldots,N_+$. We can take $\zeta_k\in \mathbb{H}_+^*$ due to a standard result in linear algebra: combining $\dim \mathbb{H}_+^*=\dim \mathcal{L}(\mathbb{H}_+,\mathbb{C})$ and , we can identify these two spaces. The interest to take $\zeta_k\in \mathbb{H}_+^*$ is that the above formula for $G$ can be applied to $\phi\in \mathbb{H}$ and extend $G$ as a linear bounded operator in $\mathbb{H}$ satisfying $G=0$ in $\mathbb{H}_-$ (see ). Extending also the family $K$ by $K(t,s)=0$ in $\mathbb{H}_-$, we see that can be written as .
Let us define $$\label{eqW}
w(t):=z_+(t)+\int_0^{t} K(t,s)z_+(s) \ ds,$$ so that can be written $$\label{eqZ+2}
\left\{
\begin{array}{cc}
z_+'(t)=A_+ z_+(t)+B_+p_+\mathds{1}_{[\tau,+\infty)}(t)G w(t-\tau)+P_+f(t) \quad t>0,\\
z_+(0)=P_+z^0.
\end{array}
\right.$$
Then we use , and the Fubini theorem to perform the following computation for $t>0$: $$\begin{gathered}
\label{calculs}
\int_{t}^{t+\tau}e^{(t-s)A_+}B_+p_+ Gw(s-\tau) \mathds{1}_{[\tau,+\infty)}(s) \ ds
=
\int_{\max \{t-\tau,0\}}^{t}e^{(t-s-\tau)A_+}B_+p_+ Gw(s) \ ds
\\
=\int_{\max \{t-\tau,0\}}^{t}e^{(t-s-\tau)A_+}B_+p_+ G\left[ z_+(s)+\int_0^{s} K(s,\xi)z_+(\xi) \ d\xi\right] \ ds
\\
=\int_{0}^{t}
\left[\mathds{1}_{(\max\{t-\tau,0\},t)} (s) e^{(t-s-\tau)A_+}B_+p_+ G
+\int_{\max \{t-\tau,s\}}^{t} e^{(t-\xi-\tau)A_+}B_+p_+ G K(\xi,s) \ d\xi
\right]z_+(s) \ ds
\\
=\int_{0}^{t} K(t,s)z_+(s) \ ds
=w(t)-z_+(t).\end{gathered}$$
Consequently, $$\label{tempo}
w(t)=z_+(t)+\int_{t}^{t+\tau}e^{(t-s)A_+}B_+p_+ Gw(s-\tau) \mathds{1}_{[\tau,+\infty)}(s) \ ds.$$ From , we deduce that $w$ is solution of with $w^0=z_+(0)$. Thus $w$ satisfies and from , $$\label{etoile}
\|z_+\|_{H^1_{\sigma}(0,\infty;\mathbb{H}_+)} {\leqslant}C\left(\|P_+ z^0\|_{\mathbb{H}_+} + \|P_+f\|_{L^2_{\sigma}(0,\infty;\mathbb{H}_+)}\right).$$ This yields in particular that if $z^0\in \mathbb{H}_{1/2}$ and if $f\in L^2_{\sigma}(0,\infty;\mathbb{H})$, then $$\label{stab1-strong}
\|z_+\|_{L^2_\sigma (0,\infty;\mathbb{H}_1)\cap C^0_\sigma ([0,\infty);\mathbb{H}_{1/2})\cap H^1_\sigma (0,\infty;\mathbb{H})}
{\leqslant}C\left(\|z^0\|_{\mathbb{H}_{1/2}} + \|f\|_{L^2_{\sigma}(0,\infty;\mathbb{H})}\right).$$
Then, we can consider the solution of : for $t{\geqslant}\tau$, $$\begin{gathered}
z_-(t)=e^{A_-t} ({\operatorname{Id}}-P_+)z^0
+\int_\tau^t (\lambda_0-A)^{\gamma} e^{A_-(t-s)} (\lambda_0-A)^{-\gamma} B_- p_- Gw(s-\tau) \ ds
\\
+\int_0^t e^{A_-(t-s)} ({\operatorname{Id}}-P_+)f(s) \ ds.\end{gathered}$$ Using and , we deduce that $$\begin{gathered}
\|z_-(t)\|_{\mathbb{H}} {\leqslant}Ce^{-\sigma_- t} \left\|z^0\right\|_{\mathbb{H}}
+C e^{-\sigma t} \int_\tau^t \frac{1}{(t-s)^\gamma} e^{-(\sigma_- -\sigma) (t-s)}\ ds \left(\|P_+ z^0\|_{\mathbb{H}_+} + \|P_+ f\|_{L^2_{\sigma}(0,\infty;\mathbb{H}_+)}\right)
\\
+C e^{-\sigma t} \int_0^t \frac{1}{(t-s)^{\gamma'}} e^{-(\sigma_- -\sigma) (t-s)} \left\| e^{\sigma s} f(s)\right\|_{\mathbb{H}_{-\gamma'}} \ ds.\end{gathered}$$ Using that $\sigma_->\sigma$, $\gamma<1$ and $\gamma'<1/2$, we deduce from the above estimate that $$\|z_-(t)\|_{\mathbb{H}_-} {\leqslant}C e^{-\sigma t} \left(\|z^0\|_{\mathbb{H}} + \|f\|_{L^2_{\sigma}(0,\infty;\mathbb{H}_{-\gamma'})}\right)
\quad (t>0).$$ Combing this with , we deduce .
Let us prove now . If $f\in L^2_{\sigma}(0,\infty;\mathbb{H})$, $B\in \mathcal{L}(\mathbb{U},\mathbb{H})$ and if $z^0\in \mathbb{H}_{1/2}$, then the first part remains unchanged, and we have and $$\label{ijt100}
\|v\|_{L^2_{\sigma}(0,\infty;\mathbb{U})} {\leqslant}C\left(\|z^0\|_{\mathbb{H}_+} + \|f\|_{L^2_{\sigma}(0,\infty;\mathbb{H}_+)}\right).$$ Consequently, $$\label{ijt101}
B_- p_- v + ({\operatorname{Id}}-P_+)f \in L^2_{\sigma}(0,\infty;\mathbb{H}_-),$$ and $$z_-(0)=z^0-P_+z^0 \in \mathbb{H}_{1/2}\cap \mathbb{H}_-.$$ Using that $A_-$ is the infinitesimal generator of an analytic semigroup of type smaller than $-\sigma$ (see, for instance [@BDDM Proposition 2.9, p.120]), then $$z_-\in L^2_\sigma (0,\infty;\mathbb{H}_1)\cap C^0_\sigma ([0,\infty);\mathbb{H}_{1/2})\cap H^1_\sigma (0,\infty;\mathbb{H}),$$ and from $$\begin{gathered}
\|z_-\|_{L^2_\sigma (0,\infty;\mathbb{H}_1)\cap C^0_\sigma ([0,\infty);\mathbb{H}_{1/2})\cap H^1_\sigma (0,\infty;\mathbb{H})}
{\leqslant}C\left(\|z^0-P_+z^0 \|_{\mathbb{H}_{1/2}} + \| B_- p_- v\|_{L^2_{\sigma}(0,\infty;\mathbb{H})} + \|({\operatorname{Id}}-P_+)f\|_{L^2_{\sigma}(0,\infty;\mathbb{H})}\right)\\
{\leqslant}C\left(\|z^0\|_{\mathbb{H}_{1/2}} + \|f\|_{L^2_{\sigma}(0,\infty;\mathbb{H})}\right).\end{gathered}$$ Combining this with , we deduce .
Feedback boundary stabilization of the convection-diffusion equation {#sec_heat}
====================================================================
Let $\Omega\subset \mathbb{R}^N$ ($N{\geqslant}1$) be a bounded domain of class $C^{1,1}$. In this section, we apply \[thmain\] for the stabilization of the convection-diffusion equation. Let us consider $\Gamma$ a non-empty open subset of $\partial \Omega$ and the control problem: $$\label{sys}
\left\{
\begin{array}{rl}
\partial_t z-\Delta z-b\cdot \nabla z-cz=0 & \text{in} \ (0,\infty)\times \Omega,\\
z=v& \text{on} \ (0,\infty)\times \Gamma,\\
z=0& \text{on} \ (0,\infty)\times (\partial \Omega\setminus\Gamma),\\
z(0,\cdot)=z^0 & \text{in}\ \Omega,
\end{array}
\right.$$ where $c,b,{\operatorname{div}}b\in L^\infty(\Omega)$. In order to write under the form , we introduce the following functional setting: $$\mathbb{H}=L^2(\Omega),\quad \mathbb{U}=L^{2}(\Gamma),$$ $$Az=\Delta z+b\cdot \nabla z+cz,\quad \mathcal{D}(A)=H^2(\Omega)\cap H^1_0(\Omega).$$ From standard results on this operator $A$ (see for example [@MR1625845 Theorem 5, p.305]), we see that holds true. To define the control operator $B$, we use a standard method (see, for instance [@TucsnakWeiss pp.341-343] or [@RaymondStokes]): we first consider the lifting operator $D_0 \in\mathcal{L}(L^{2}(\partial\Omega); L^2(\Omega))$ such that for any $v\in L^2(\partial\Omega)$, $w=D_0 v$ is the unique solution of the following system $$\left\{
\begin{array}{rl}
\lambda_0 w-\Delta w-b\cdot \nabla w-cw=0& \text{ in } \Omega,\\
w=v & \text{ on } \ \partial\Omega,
\end{array}
\right.$$ where $\lambda_0\in \rho(A)$. Then, we set $$B = (\lambda_0-A)D_0 : \mathbb{U}\longrightarrow (\mathcal{D}(A^*))',$$ where we have extended the operator $A$ as an operator from $L^2(\Omega)$ into $(\mathcal{D}(A^*))'$ and where we see $\mathbb{U}$ as a closed subspace of $L^2(\partial \Omega)$ (by extending by zero in $\partial \Omega\setminus\Gamma$ any $v\in \mathbb{U}$). Using standard results on elliptic equations, we have that $B$ satisfies for any $\gamma>3/4$.
Let us recall how we can see that with $A$ and $B$ defined as above writes as . We set $\widetilde z=z-w$, with $w=D_0v$. Then $\widetilde z$ satisfies the system $$\left\{\begin{array}{rl}
\partial_t \widetilde z-\Delta \widetilde z -b\cdot \nabla \widetilde z-c\widetilde z=-\partial_t w+\lambda_0 w
& \text{in} \ (0,\infty)\times \Omega,\\
\widetilde z=0 & \text{on} \ (0,\infty)\times \partial\Omega,\\
\widetilde z(0,\cdot)=\widetilde z^0:=z^0-w(0,\cdot)& \text{ in } \ \Omega.
\end{array}
\right.$$ Using the Duhamel formula, we have $$\widetilde z(t)=e^{tA}\widetilde z^0+\int_0^t e^{(t-s)A}(-\partial_t w(s)+\lambda_0 w(s)) \ ds.$$ By integrating by parts, we obtain $$z(t)=e^{tA}z^0+\int_0^t e^{(t-s)A}(\lambda_0-A) w(s)\ ds,$$ that is $$\left\{
\begin{array}{rl}
z'=Az+(\lambda_0-A) D_0 v,\\
z(0)=z^0.
\end{array}
\right.$$
To apply \[thmain\], we only need to check . We recall that $$\mathcal{D}(A^*)=H^2(\Omega)\cap H^1_0(\Omega),
\quad
A^*\varepsilon=\Delta \varepsilon-\overline b\cdot \nabla \varepsilon+(\overline{c-{\operatorname{div}}b})\varepsilon,$$ (see, for instance, [@TucsnakWeiss p.345]). Moreover, by classical results (see [@TucsnakWeiss Proposition 10.6.7]), we see that $$D_0^* :=-\frac{\partial}{\partial \nu} (\lambda_0-A^*)^{-1},$$ and thus $$B^*\varepsilon :=-\frac{\partial \varepsilon}{\partial \nu}_{|\Gamma}.$$ Thus if $\varepsilon$ satisfies $A^*\varepsilon= \lambda \varepsilon$ and $B^*\varepsilon=0$, then $$\left\{
\begin{array}{rl}
\lambda \varepsilon- \Delta \varepsilon+\overline b\cdot \nabla \varepsilon-(\overline{c-{\operatorname{div}}b})\varepsilon=0& \text{in}\ \Omega,\\
\varepsilon=0 & \text{on} \ \partial\Omega,\\
\frac{\partial \varepsilon}{\partial \nu}=0& \text{on} \ \Gamma.
\end{array}
\right.$$ From standard results on the unique continuation of the Laplace operator (see for instance [@MR0404822 Theorem 5.3.1, p.125]), we deduce that $\varepsilon=0$. Thus holds for any $\sigma$ and we deduce the following result by applying \[thmain\]:
\[stabheat\] Assume $\sigma>0$ and let us define $N_+$ by . Then there exist $K\in L^\infty_{\rm loc}(\mathbb{R}^2;\mathcal{L}(L^2(\Omega)))$, $\zeta_k\in H^2(\Omega)\cap H^1_0(\Omega)$, $v_k\in H^{1/2}(\Gamma)$, $k=1,\ldots,N_+$, such that the solution $z$ of with $$\label{fb1-heat}
v(t)=\mathds{1}_{[\tau,+\infty)}(t) \sum_{k=1}^{N_+} \left(\int_{\Omega} \left[z(t-\tau)+\int_0^{t-\tau} K(t-\tau,s)z(s) \ ds\right]\zeta_k \ dx\right) v_k,$$ and for $z^0 \in L^2(\Omega)$ satisfies $$\label{ijt008-heat}
\|z(t)\|_{L^2(\Omega)} {\leqslant}C e^{-\sigma t} \|z^0\|_{L^2(\Omega)} .$$
Feedback distributed stabilization of the Oseen system {#sec_oseen}
======================================================
Let $\Omega\subset \mathbb{R}^3$ be a bounded domain of class $C^{1,1}$. In this section, we apply \[thmain\] to the Oseen system: $$\label{oseen}
\left\{
\begin{array}{rl}
\partial_t z+(w^S\cdot \nabla)z+(z\cdot \nabla)w^S-\nu \Delta z+\nabla q=\mathds{1}_{\mathcal{O}} v & \text{in}\ (0,\infty)\times \Omega,\\
\nabla \cdot z=0& \text{in} \ (0,\infty)\times \Omega,\\
z=0 & \text{on} \ (0,\infty)\times \partial\Omega,\\
z(0,\cdot)=z^0 & \text{in} \ \Omega,
\end{array}
\right.$$ where $w^S\in [H^2(\Omega)]^3$ is a fixed (real) velocity and $v$ is the control that acts on the nonempty open subset $\mathcal{O}\subset \Omega$. We could also consider the boundary stabilization of the Oseen system by using the same method as in the above section but with some adaptations due the incompressibility condition and due to the pressure (see [@MB1] for more details).
Let us give the functional setting: $$\mathbb{H}=\{z\in [L^2(\Omega)]^3,\quad \nabla\cdot z=0 \text{ in }\Omega,\quad z\cdot n=0\text{ on }\partial\Omega\},
\quad
\mathbb{U}=[L^2(\mathcal{O})]^3.$$ We denote by $\mathbb{P}$ the orthogonal projection $\mathbb{P} : [L^2(\Omega)]^3\to \mathbb{H}$ and we define the Oseen operator: $$\mathcal{D}(A)= [H^2(\Omega)\cap H^1_0(\Omega)]^3 \cap \mathbb{H},
\quad
Az=\mathbb{P}\left(\nu\Delta z-(w^S\cdot \nabla)z-(z\cdot \nabla)w^S\right).$$ We recall (see, for instance [@MB1 Theorem 20]) that the operator $A$ is the infinitesimal generator of an analytic semigroup on $\mathbb{H}$ and has a compact resolvent. Moreover, $$\mathcal{D}(A^*)= [H^2(\Omega)\cap H^1_0(\Omega)]^3 \cap \mathbb{H},
\quad
A^* \varepsilon=\mathbb{P}\left(\nu\Delta \varepsilon+(w^S\cdot \nabla)\varepsilon-(\nabla w^S)^* \varepsilon\right).$$
We also define the control operator $B\in \mathcal{L}(\mathbb{U},\mathbb{H})$ by $$Bv= \mathbb{P} \left(\mathds{1}_{\mathcal{O}} v\right),$$ and we can check that $$B^* \varepsilon = \varepsilon_{|\mathcal{O}}.$$ In particular, we see that and hold true and if $\varepsilon$ satisfies $A^*\varepsilon= \lambda \varepsilon$ and $B^*\varepsilon=0$, then $$\left\{
\begin{array}{rl}
\lambda \varepsilon- \nu \Delta \varepsilon-(w^S\cdot \nabla)\varepsilon+(\nabla w^S)^* \varepsilon+\nabla \pi=0& \text{in}\ \Omega,\\
\nabla \cdot \varepsilon =0& \text{in}\ \Omega,\\
\varepsilon=0 & \text{on} \ \partial\Omega,\\
\varepsilon\equiv 0& \text{in} \ \mathcal{O}.
\end{array}
\right.$$ Then using [@FabreLebeau], we deduce that $\varepsilon=0$. Thus holds for any $\sigma$ and we deduce the following result by applying \[thmain\]:
\[staboseen\] Assume $\sigma>0$ and let us define $N_+$ by . Then there exist $K\in L^\infty_{\rm loc}(\mathbb{R}^2;\mathcal{L}(\mathbb{H}))$, $\zeta_k\in \mathcal{D}(A^*)$, $v_k\in [L^2(\mathcal{O})]^3$, $k=1,\ldots,N_+$, such that the solution $z$ of with $$\label{fb1-oseen}
v(t)=\mathds{1}_{[\tau,+\infty)}(t) \sum_{k=1}^{N_+} \left(\int_{\Omega} \left[z(t-\tau)+\int_0^{t-\tau} K(t-\tau,s)z(s) \ ds\right]\zeta_k \ dx\right) v_k,$$ and for $z^0 \in \mathbb{H}$ satisfies $$\|z(t)\|_{[L^2(\Omega)]^3} {\leqslant}C e^{-\sigma t} \|z^0\|_{[L^2(\Omega)]^3} .$$
Let us define $$\label{ijt105}
\mathbb{V}= [H^1_0(\Omega)]^3 \cap \mathbb{H},$$ then we have that $\mathbb{V}=\mathbb{H}_{1/2}$ (see again [@MB1 Theorem 20]). Thus applying \[thmain\], we have also the following result on $$\label{oseen-2}
\left\{
\begin{array}{rl}
\partial_t z+(w^S\cdot \nabla)z+(z\cdot \nabla)w^S-\nu \Delta z+\nabla q=\mathds{1}_{\mathcal{O}} v +f & \text{in}\ (0,\infty)\times \Omega,\\
\nabla \cdot z=0& \text{in} \ (0,\infty)\times \Omega,\\
z=0 & \text{on} \ (0,\infty)\times \partial\Omega,\\
z(0,\cdot)=z^0 & \text{in} \ \Omega.
\end{array}
\right.$$
\[staboseen-2\] Assume $\sigma>0$ and let us consider $v$ given by . Then for any $z^0 \in \mathbb{V}$ and for any $f\in L^2_\sigma(0,\infty;\mathbb{H})$ the solution of satisfies $$z\in L^2_\sigma (0,\infty;[H^2(\Omega)]^3)\cap C^0_\sigma ([0,\infty);[H^1(\Omega)]^3)\cap H^1_\sigma (0,\infty;[L^2(\Omega)]^3),$$ and $$\label{ijt210}
\|z\|_{L^2_\sigma (0,\infty;[H^2(\Omega)]^3)\cap C^0_\sigma ([0,\infty);[H^1(\Omega)]^3)\cap H^1_\sigma (0,\infty;[L^2(\Omega)]^3)}
{\leqslant}C\left(\|z^0\|_{[H^1(\Omega)]^3} + \|f\|_{L^2_{\sigma}(0,\infty;[L^2(\Omega)]^3)}\right).$$
Local feedback distributed stabilization of the Navier-Stokes system {#sec_navier}
====================================================================
We use the same notation as in the previous section. We consider the stabilization of the Navier-Stokes system with internal control: $$\label{navier}
\left\{
\begin{array}{rl}
\partial_t \widetilde z+(\widetilde z \cdot \nabla)\widetilde z-\nu \Delta \widetilde z+\nabla \widetilde q=\mathds{1}_{\mathcal{O}} v+f^S & \text{in}\ (0,\infty)\times \Omega,\\
\nabla \cdot \widetilde z=0& \text{in} \ (0,\infty)\times \Omega,\\
\widetilde z=b^S & \text{on} \ (0,\infty)\times \partial\Omega,\\
\widetilde z(0,\cdot)=\widetilde z^0 & \text{in} \ \Omega,
\end{array}
\right.$$ around the stationary state $$\label{navier-S}
\left\{
\begin{array}{rl}
(w^S \cdot \nabla) w^S-\nu \Delta w^S+\nabla r^S=f^S & \text{in}\ \Omega,\\
\nabla \cdot w^S=0& \text{in} \ \Omega,\\
w^S=b^S & \text{on} \ \partial\Omega.
\end{array}
\right.$$ We assume that $(w^S,r^S)$ is a solution of such that $w^S\in [H^2(\Omega)]^3$ as in the previous section. The functions $f^S\in [L^2(\Omega)]^3$ and $b^S\in [W^{3/2}(\partial \Omega)]^3$ are independent of time.
We define $$z=\widetilde z-w^S, \quad q=\widetilde q-r^S, \quad z^0=\widetilde z^0-w^S,$$ so that $$\label{navier-diff}
\left\{
\begin{array}{rl}
\partial_t z+(w^S\cdot \nabla)z+(z\cdot \nabla)w^S-\nu \Delta z+\nabla q=\mathds{1}_{\mathcal{O}} v -(z\cdot\nabla) z & \text{in}\ (0,\infty)\times \Omega,\\
\nabla \cdot z=0& \text{in} \ (0,\infty)\times \Omega,\\
z=0 & \text{on} \ (0,\infty)\times \partial\Omega,\\
z(0,\cdot)=z^0 & \text{in} \ \Omega.
\end{array}
\right.$$
Then we consider the following mapping $$\mathcal{Z} : L^2_\sigma(0,\infty;[L^2(\Omega)]^3) \to L^2_\sigma(0,\infty;[L^2(\Omega)]^3),
\quad
f\mapsto -(z\cdot\nabla) z,$$ where $z$ is the solution given in \[staboseen-2\], associated with $z^0\in \mathbb{V}$ and $f\in L^2_\sigma(0,\infty;\mathbb{H})$. Then by standard Sobolev embeddings, we find $$\label{nonlin}
\| z^1\cdot\nabla z^2\|_{L^2_\sigma(0,\infty;[L^2(\Omega)]^3)}
\\
{\leqslant}C \|z^1\|_{C^0_\sigma ([0,\infty);[H^1(\Omega)]^3)}\|z^2\|_{L^2_\sigma (0,\infty;[H^2(\Omega)]^3)}.$$ Thus $\mathcal{Z}$ is well-defined. Let us set $$R=\|z^0\|_{[H^1(\Omega)]^3},$$ and $$B_R = \left\{f\in L^2_\sigma(0,\infty;[L^2(\Omega)]^3) \ ; \ \|f\|_{L^2_\sigma(0,\infty;[L^2(\Omega)]^3)} {\leqslant}R \right\}.$$ Then from and , $$\|\mathcal{Z}(f)\|_{L^2_\sigma(0,\infty;[L^2(\Omega)]^3)} {\leqslant}4CR^2,$$ and $B_R$ is invariant by $\mathcal{Z}$ for $R$ small enough. Similarly, using and , for any $f^1,f^2\in B_R$, then $$\|\mathcal{Z}(f^1)-\mathcal{Z}(f^2)\|_{L^2_\sigma(0,\infty;[L^2(\Omega)]^3)} {\leqslant}2CR \|f^1-f^2\|_{L^2_\sigma(0,\infty;[L^2(\Omega)]^3)},$$ and thus $\mathcal{Z}$ is a strict contraction on $B_R$ for $R$ small enough. We thus deduce that $\mathcal{Z}$ admits a fixed point $f$ for $\|z^0\|_{[H^1(\Omega)]^3}$ small enough and we notice that the solution $z$ given in \[staboseen-2\], associated with $z^0\in \mathbb{V}$ and $f\in L^2_\sigma(0,\infty;[L^2(\Omega)]^3)$ is a solution of .
We have obtained the following local stabilization result for the Navier-Stokes system with internal control with delay:
\[thmNS\] Assume $\sigma>0$ and let us define $N_+$ by . Then there exist $K\in L^\infty_{\rm loc}(\mathbb{R}^2;\mathcal{L}(\mathbb{H}))$, $\zeta_k\in \mathcal{D}(A^*)$, $v_k\in [L^2(\mathcal{O})]^3$, $k=1,\ldots,N_+$ and $R>0$, such that for any $$\widetilde{z}^0\in [H^1(\Omega)]^3, \quad \nabla \cdot \widetilde{z}^0=0 \quad \text{in} \ \Omega,
\quad
\widetilde z^0=b^S \ \text{on} \ \partial\Omega,$$ and $$\|\widetilde{z}^0-w^S \|_{[H^1(\Omega)]^3}{\leqslant}R,$$ there exists a unique solution $z$ of with $$v(t)=\mathds{1}_{[\tau,+\infty)}(t) \sum_{k=1}^{N_+} \left(\int_{\Omega} \left[(\widetilde z-w^S)(t-\tau)+\int_0^{t-\tau} K(t-\tau,s)(\widetilde z-w^S)(s) \ ds\right]\zeta_k \ dx\right) v_k,$$ satisfying $$\widetilde z-w^S\in L^2_\sigma (0,\infty;[H^2(\Omega)]^3)\cap C^0_\sigma ([0,\infty);[H^1(\Omega)]^3)\cap H^1_\sigma (0,\infty;[L^2(\Omega)]^3).$$ Moreover we have the estimate $$\label{ijt213}
\|\widetilde z-w^S\|_{L^2_\sigma (0,\infty;[H^2(\Omega)]^3)\cap C^0_\sigma ([0,\infty);[H^1(\Omega)]^3)\cap H^1_\sigma (0,\infty;[L^2(\Omega)]^3)}
{\leqslant}C\|\widetilde z^0-w^S\|_{[H^1(\Omega)]^3}.$$
Acknowledgments. {#acknowledgments. .unnumbered}
================
The two first authors were partially supported by the ANR research project IFSMACS (ANR-15-CE40-0010). The third author was partially supported by the ANR research projects ISDEEC (ANR-16-CE40-0013) and ANR ODISSE (ANR-19-CE48-0004-01).
|
---
abstract: 'We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its symplectic fillings, and morever, fillings cannot contain certain symplectic surfaces. Applying these obstructions to canonical contact structures on links of normal surface singularities, we show that links of isolated singularities of surfaces in the complex 3-space are planar only in the case of $A_n$-singularities, and in general characterize completely planar links of normal surface singularities (in terms of their resolution graphs). We also establish non-planarity of tight contact structures on certain small Seifert fibered L-spaces and of contact structures compatible with open books given by a boundary multi-twist on a page of positive genus. Additionally, we prove that every finitely presented group is the fundamental group of a Leschetz fibration with planar fibers.'
address:
- 'CNRS, Laboratoire Jean Leray, Université de Nantes, Nantes, France'
- 'Department of Mathematics, Stony Brook University, Stony Brook, NY, U.S.A.'
author:
- Paolo Ghiggini
- Marco Golla
- Olga Plamenevskaya
bibliography:
- 'nonplanar.bib'
title: 'Obstructions to planarity of contact 3-manifolds'
---
Introduction and background
===========================
Since the groundbreaking work of Giroux [@Gi], open books have played a major role in 3-dimensional contact topology; certain properties of open books are related to questions of tightness and fillability. While a compatible open book decomposition is not unique, one can ask what the smallest possible genus of a page is.
In particular, contact manifolds that admit [*planar*]{} open book decompositions (i.e. with page of genus zero and possibly multiple boundary components) have a number of special properties. For example, Etnyre showed that any symplectic filling for a planar contact structure has a negative definite intersection form [@Etn]; it follows that any contact structure that arises as a perturbation of a taut foliation cannot be planar. This implies, by [@HRRR], that if $Y$ is a graph manifold which is not an L-space, then $Y$ admits a non-planar contact structure. (Recall that L-spaces, whose name derives from their Floer-homological similarity to lens spaces, are 3-manifolds with the simplest possible Heegaard Floer homology [@OSlens].) By contrast, all contact structures on lens spaces are planar [@Scho]; the same is known for some other L-spaces, although in general L-spaces can admit non-planar contact structures as well [@LS3]. (Note that overtwisted contact structures are always planar by [@Etn].)
In this paper we develop new obstructions, in terms of conditions on the intersection form of a Stein filling and presence of certain symplectic surfaces in weak fillings, and rule out planarity for a number of interesting contact structures. All contact manifolds in this paper are assumed closed and co-oriented. Before stating the general conditions, we interpret our obstructions for canonical contact structures on links of normal surface singularities. Our first result is for isolated singularities of hypersurfaces in ${\mathbb{C}}^3$, the second is for more general surfaces. (The definitions involved in the second statement are more technical, and we defer them to Section \[normal-links\].)
Consider a complex surface $\Sigma\subset {\mathbb{C}}^N$ with an isolated critical point at the origin. For a sufficiently small ${\varepsilon}>0$, the intersection $Y= \Sigma \cap S^{2N-1}_{{\varepsilon}}$ with the sphere $S^{2N-1}_{{\varepsilon}}= \{|z_1|^2+|z_2|^2+\dots +|z_N|^2 = {\varepsilon}\}$ is a smooth 3-manifold called the [*link of the singularity*]{}. The induced contact structure $\xi$ on $Y$ is the distribution of complex tangencies to $Y$, and is referred to as the [*canonical*]{} contact structure on the link. The contact manifold $(Y, \xi)$ is independent of the choice of ${\varepsilon}$, up to contactomorphism.
\[sing\] Let $(Y, \xi)$ be the link of an isolated singularity of a complex surface in ${\mathbb{C}}^3$ with its canonical contact structure. Then $\xi$ is planar if and only if the singularity is of type $A_n$.
\[sing2\] Let $(Y,\xi)$ be the link of a normal surface singularity with its canonical contact structure. Then $\xi$ is planar if and only if the singularity has a good resolution whose graph is a tree of spheres with no bad vertices. In particular, if $\xi$ is planar, then the singularity is rational.
The “if” direction of Theorem \[sing2\] was proven by Schönenberger [@Scho]. From a different perspective, contact structures on links of singularities whose graphs have no bad vertices were discussed by Némethi–Tosun [@NT], who showed that in this case the Milnor open books (associated to the Artin cycle $Z_{min}$) are planar. (In [@NT], the absence of bad vertices translates to the condition $Z_{min}=E$, where $E$ is the exceptional divisor of the resolution.)
As a corollary of the proof, we obtain the following; we say that a singularity is planar if the canonical contact structure on its link is planar.
\[deformation\] There can be no strong symplectic cobordism from a non-planar normal surface singularity to a planar one. In particular, a planar normal surface singularity cannot be deformed to one where the link is not planar.
This corollary goes in the direction of arguing that there can be no Weinstein cobordism from a non-planar contact structure to a planar one, or, more generally, that the support genus is non-increasing under symplectic cobordisms.
For general planar contact manifolds, we show that a contact structure given by a plumbing graph with a bad vertex cannot be planar if the vertices adjacent to the bad one have weight $-2$ or $-3$, as in conditions .
\[general\] A planar contact manifold [*cannot*]{} have a Stein filling $W$ with the following property: for some $k>0$, there exist homology classes $B_1, B_2, \dots B_k, X \in H_2(W)$ such that $$\label{bad-form}
\begin{array}{lll}
B_i \cdot X = 1, & & \quad i =1, \dots, k \\
B_i \cdot B_j = 0, & & \quad i \neq j \\
B_i \cdot B_i \in \{-2, -3\}, & & \quad i = 1, \dots, k\\
X \cdot X > -k. & &
\end{array}$$ In other words, the intersection graph of $W$ cannot have a configuration shown in Figure \[bad\].
$\xygraph{
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In particular, we have the following corollary for small Seifert fibered L-spaces:
\[seifert\] Tight contact structures on a Seifert fibered space $M(-2; r_1, r_2, r_3)$ are never planar if this manifold is an L-space and $r_1, r_2, r_3\geq \frac13$.
Here we use the notation $M(e_0; r_1, r_2, r_3)$ for small Seifert fibered spaces; $e_0 \in {\mathbb{Z}}$, $r_i \in (0,1)\cap {\mathbb{Q}}$, and the space is given by surgery diagram in Figure \[seifert-space\]. Contact structures on these spaces were extensively studied (see e.g. [@LS3] and references therein); classification of tight contact structures and some (non-)fillability results are known in many cases. Many contact structures turn out to be planar: every contact structure on $M(e_0; r_1, r_2, r_3)$ is planar if $e_0 \leq -3$ [@Scho], and the same is true for $M(e_0; r_1, r_2, r_3)$ for $e_0\geq -1$ whenever this manifold is an L-space [@LS3]. Corollary \[seifert\] contrasts these planarity results.
$e_0$ at 0 140 $-\frac1{r_1}$ at 55 90 $-\frac1{r_2}$ at 115 90 $-\frac1{r_3}$ at 175 90 ![The Seifert fibered space $M(e_0; r_1, r_2, r_3)$.[]{data-label="seifert-space"}](figures/seifert "fig:")
Our results stated above are special cases of a rather general examination of the homology of possible Stein fillings. The major tool comes from Wendl’s theorem, saying that any Stein filling of a planar contact manifold admits a Lefschetz fibration with the same planar fiber, and whose vanishing cycles can be obtained by a positive factorization of the monodromy of the planar open book [@We]. Note that Wendl’s theorem is extended to show that any minimal weak symplectic filling of a planar contact structure is deformation equivalent to a Lefschetz fibration with the same properties [@We2]. Given a Lefschetz fibration, we can compute the homology and intersection form of the filling from the factorization of monodromy; together with the examination of all possible fillings, this leads to proofs of Theorems \[sing\] and \[general\].
In the proof of Theorem \[sing2\], we consider the plumbing of [*symplectic*]{} surfaces and make use of the following statement, which we think is of independent interest.
\[positivegenus\] If a contact $3$-manifold $(Y,\xi)$ has a symplectic filling containing a symplectic surface of positive genus, it is not planar.
As an immediate corollary, we prove non-planarity of Boothby–Wang contact structures for $g>0$ and extend some results of Wand [@Wand Corollary 7.6]. Given two integers $g\ge 0$ and $b>0$, we denote by $\Sigma_{g,b}$ the surface of genus $g$ with $b$ holes, and by $\tau_{\partial}$ the boundary multi-twist, i.e. the product of right-handed Dehn twists along each boundary component. In fact, $Y_{g,b}$ is the total space of the circle bundle with Euler number $-b$ over a surface of genus $g$, and $\xi_{g,b}$ is the Boothby–Wang contact structure on $Y_{g,b}$; i.e. it is the (convex) boundary of the symplectic disk bundle over a surface of genus $g$, and Euler number $-b$; see, for instance, [@GayMark]. Since the 0-section of the disk bundle is symplectic, we have
\[c:prequantization\] Let $(Y_{g,b},\xi_{g,b})$ be the contact structure supported by the open book $(\Sigma_{g,b}, \tau_\partial)$; in other words, $\xi_{g,b}$ is the Boothby–Wang contact structure on the circle bundle $Y_{g,b}$. Then $\xi_{g,b}$ is planar if and only if $g = 0$.
It is useful to compare our obstructions to previous results. As noted above, Etnyre proved that any symplectic filling of a planar contact structure is negative definite, and that for a planar integral homology sphere, any symplectic filling must have a diagonal intersection form [@Etn]. This implies, for example, that the canonical contact structure on the Poincaré homology sphere (the link of the $E_8$-singularity) is not planar. In fact, one can observe that Etnyre’s proof yields a stronger statement: the intersection form of any symplectic filling of a planar *rational* homology sphere embeds in a diagonal lattice of some (possibly higher) rank. It follows that the canonical contact structures on the links of the $E_6$- and $E_7$-singularities cannot be planar, either. On the other hand, Etnyre’s result gives no information for the links of the $D_n$-singularities, as the corresponding intersection forms embed in the standard lattice.
Another obstruction to planarity, in terms of Heegaard Floer homology, was developed by Ozsváth–Stipsicz–Szabó [@OSS]. This obstruction is also trivial for the links of the $D_n$-singularities. More generally, the Heegaard Floer obstruction is always trivial for L-spaces. By contrast, our obstruction often gives non-trivial information in the case of L-spaces, see Corollary \[seifert\] above.
Using factorizations of mapping classes, Wand gave another obstruction to planarity [@Wand]. Wand’s results are closer in spirit to ours, as he also uses Wendl’s theorem and examines topology of fillings, however both the specific approach and the obstruction Wand obtains are different from ours. In particular, Wand shows that the sum of the Euler characteristic and signature is the same for all Stein fillings of a planar contact manifold. Then, if one is able to find two weak fillings $W_1$, $W_2$ for $(Y, \xi)$ such that $\chi(W_1)+\sigma(W_1) \neq \chi(W_2)+\sigma(W_2)$, it follows that $(Y, \xi)$ cannot be planar. (Wand also examines how certain relators in the mapping class group affect $\chi+\sigma$.) However, this obstruction fails to address the case when there is a unique filling; for example, it is known that the filling is unique for the links of the $D_n$-singularities [@OhtaOno], so Wand’s approach gives no obstruction. Wand’s obstruction is also trivial when the underlying contact 3-manifold is a rational homology sphere, and all its fillings are negative definite (this is true, in particular, for all L-spaces); indeed, for a negative definite Stein filling $W$ of a rational homology sphere we always have $\chi(W)+\sigma(W)=1$ since $b_3(W)=b_1(W)=0$. We are also able to answer a question of Wand in our Corollary \[c:prequantization\], proving non-planarity for a family of contact structures that cannot be handled by Wand’s means (see [@Wand Corollary 7.6] and subsequent discussion).
As a byproduct of our intersection form calculation, we also get the following corollaries. These were first proven by Oba [@Oba2 Lemma 3.1, Lemma 3.2] using the Heegaard Floer obstruction from [@OSS]. It is instructive to obtain these results more directly, from the basic topology of fillings.
\[blow-down\] Let $(W,\omega)$ be a weak symplectic filling of a planar contact manifold $(Y, \xi)$. If $B \in H_2(W)$ is a class of square $-1$, $B$ is represented by an embedded symplectic sphere that can be blown down.
\[IHS\] Let $Y$ be an integral homology sphere, equipped with a planar contact structure $\xi$. Then any minimal weak symplectic filling of $(Y,\xi)$ is an integral homology ball.
Non-trivial examples of fillings as in Corollary \[IHS\] do exist; a number of examples were constructed by Oba [@Oba]. More generally, we show that one can construct Stein fillings with prescribed fundamental groups.
\[p:ballgroups1\] Every finitely generated group is the fundamental group of a Lefschetz fibration with planar fibers.
A more precise version of this statement, yielding also examples for Corollary \[IHS\], is given in Proposition \[p:ballgroups\]. Note that Proposition \[p:ballgroups1\] is similar to a theorem of Amorós–Bogomolov–Katzarkov–Pantev [@ABKP] and to Gompf’s theorem [@Go]: Gompf showed that any finitely presented group is the fundamental group of a closed symplectic 4-manifold, and in [@ABKP], a closed symplectic 4-manifold with prescribed fundamental group is constructed as a symplectic Lefschetz fibration over a closed surface. Unlike [@Go; @ABKP], where no bounds are given for the genus of the fiber, we work with manifolds with boundary but restrict to Lefschetz fibrations with planar fibers.
### Organization: {#organization .unnumbered}
In Section \[compute\], we explain how to compute the intersection form and first Chern class of the filling constructed from a positive factorization of the monodromy of a planar open book, and prove Corollaries \[blow-down\] and \[IHS\], and Theorem \[positivegenus\]. In Section \[links\], we prove Theorem \[sing\] (after considering the key example of $D_4$). In Section \[therest\], we prove Theorem \[general\] and Corollary \[seifert\]. In Section \[normal-links\], we prove Theorem \[sing2\]. Finally, in Section \[pi1\] we discuss fundamental groups and prove Proposition \[p:ballgroups1\].
### Acknowledgements: {#acknowledgements .unnumbered}
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “Homology theories in low dimensional topology” when this work was undertaken. This work was supported by EPSRC grant no EP/K032208/1. The third author is grateful to András Stipsicz for a helpful conversation that led to Corollary \[IHS\]; the second author thanks Cagri Karakurt, András Némethi, Burak Ozbagci, Matteo Ruggiero, and Laura Starkston for interesting discussions and for giving many relevant references; we also thank József Bodnár for some motivating discussions. PG and MG are partially supported by the ANR project QUANTACT (ANR16-CE40-0017). MG was also supported by the Alice and Knut Wallenberg foundation and by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 674978). OP was partially supported by NSF grant DMS-1510091.
Computing topological invariants of a planar Lefschetz fibration {#compute}
================================================================
In this section, we explain how to compute the intersection form and the first Chern class for a Lefschetz fibration over a disk with planar fibers. The second homology classes of a Lefschetz fibration are given by certain linear combinations of the vanishing cycles, and both the intersection form and the evaluation of the first Chern class can be found directly in terms of the vanishing cycles. This is a consequence of fairly straightforward topological considerations, and we think that these facts, especially Proposition \[iform\], should be known to experts (see Remark \[Auroux-rmk\] below), but complete statements and the proofs seem to be absent from the literature.
Let $P$ be the planar page of an open book decomposition of $Y$; $P$ is the disk $D$ with a few holes. Let us assume that the monodromy $\phi$ of the open book is the product of positive Dehn twists about homologically non-trivial simple closed curves $\alpha_1,\dots,\alpha_m$ in $P$ for some $m$. Each curve $\alpha_i$ divides $P$ into two components, and we *orient* it as the boundary of the region $A_i$ disjoint from $\partial D$. With this orientation, $\alpha_i$ defines a class in $H_1(P)$; by abuse of notation, we also denote this class by $\alpha_i$. For convenience, we will also assume that the $\alpha_i$ are smoothly embedded and that they intersect transversely. This implies that the union of the $\alpha_i$ disconnects $P$ into finitely many connected components. Unless otherwise stated, homology is taken with coefficients in ${\mathbb{Z}}$.
Let $W$ be the total space of a Lefschetz fibration over a disk ${\mathcal{D}}$. If ${\mathcal{D}}' \subset {\mathcal{D}}$ is a small disk that contains no critical points, then $W$ is obtained from $P \times {\mathcal{D}}'$ by attaching 2-handles to copies of the vanishing cycles contained in the vertical boundary $P\times {\partial}{\mathcal{D}}'$ so that distinct handles are attached along knots contained in distinct fibers.
We first describe $H_2(W)$ and give a convenient way to visualize homology classes. We use the exact sequence of the pair $(W, P \times {\mathcal{D}}')$; since $P \times {\mathcal{D}}'$ retracts onto $P$, we can replace the former with the latter: $$0 \longrightarrow H_2(W) \stackrel{j_*}\longrightarrow H_2(W, P) \stackrel{{\partial}_*}\longrightarrow H_1(P) \longrightarrow H_1(W) \longrightarrow H_1(W,P) \longrightarrow 0.$$ The group $H_2(W, P)$ is freely generated by the cores of the attached 2-handles; we can identify these generators with the vanishing cycles. Next, $H_2(W)$ is isomorphic to ${{\operatorname}{im}}j_*$, which in turn equals $\operatorname{ker}{\partial}_*$. So $H_2(W)$ can be identified with null-homologous linear combinations of vanishing cycles (thought of as 1-chains in $P$).
Further, in $H_1(P)$ a linear combination $b_1\alpha_1 + b_2\alpha_2+\dots +b_m\alpha_m$ is null-homologous if and only if the total winding number at each hole of $P$ is zero. Notice that the curves correspond to distinct vanishing cycles, but their homology classes may coincide. In our setting, each $\alpha_i$ is a vanishing cycle, so it is a simple closed curve on the planar surface $P$; then, with the chosen orientation convention, each $\alpha_i$ has winding number $0$ or $1$ at each hole.
We represent the homology class corresponding to the linear combination $\alpha_{\mathbf{b}}= \sum b_i\alpha_i$ as follows. Consider the linear combination of $2$-chains $A_{\mathbf{b}}= \sum b_i A_i$ in $P$. While it is possible to compute the self-intersection already at this point, using transversality for singular chains, we find it more satisfactory to represent homology classes by embedded surfaces as follows.
Since $\alpha_{\mathbf{b}}= 0 \in H_1(P)$, the multiplicity of the 2-chain $A_{\mathbf{b}}$ has multiplicity 0 near each boundary component of $P$, its boundary is $\alpha_{\mathbf{b}}$. We construct a surface representing $[\alpha_{\mathbf{b}}]$ as follows. Let ${\mathcal{D}}''=\frac12 {\mathcal{D}}' \subset {\mathcal{D}}'$ be a smaller disk, and identify its boundary with $S^1\subset{\mathbb{C}}$. Let $|{\mathbf{b}}| = \sum
|b_i|$, and consider $|{\mathbf{b}}|$ fibers $P_1,\dots, P_{|{\mathbf{b}}|}$ of $P\times {\mathcal{D}}'$ over the points $\eta_j = \exp(2\pi ji/|{\mathbf{b}}|)$ in $\partial {\mathcal{D}}''$. Rewrite the sum $\sum b_i A_i$ as $\varepsilon_1A_{i_1} + \dots + \varepsilon_{|{\mathbf{b}}|}A_{i_{|{\mathbf{b}}|}}$, where each $\varepsilon_i$ is $\pm1$.
Look at a hole $h$ of $P$. We ignore all indices $i$ such that $\alpha_{i}$ has winding number 0 around $h$, since the corresponding 2-chain $A_i$ is disjoint from $h$. Since the winding number of $\alpha_{\mathbf{b}}$ around $h$ is 0, all other indices, considered with their multiplicity, can be paired up; more precisely, we can rewrite $A_{\mathbf{b}}= A_{j_1}-A_{k_1} + \dots + A_{j_n} - A_{j_n} + A'$, where $A'$ is a 2-chain disjoint from the hole $h$.
Using a standard innermost argument and connecting the paired-up 2-chains by tubes, we can actually tube away all intersections of $A_{\mathbf{b}}$ with $h\times \partial{\mathcal{D}}''$ by adding cylinders that are parallel to $\partial h \times \partial{\mathcal{D}}''$ in $P \times \partial{\mathcal{D}}''$. The result is an embedded surface in $P \times {\mathcal{D}}'$ whose boundary consists of a number of vanishing cycles. In $W$, vanishing cycles are null-homologous, so they can be capped off to make an embedded closed surface in $W$ representing the given homology class. See Figure \[surface\].
[0.4]{} $A_2$ at 120 120 $A_3$ at 450 120 $A_1$ at 450 450 ![Constructing an embedded surface representing a homology class; in this case, the linear combination is $\alpha_2-\alpha_1-\alpha_3$.[]{data-label="surface"}](figures/H1LF1 "fig:"){width="\textwidth"}
[0.5]{} $\eta_2$ at 160 60 $\eta_1$ at 270 110 $\eta_3$ at 415 90 ![Constructing an embedded surface representing a homology class; in this case, the linear combination is $\alpha_2-\alpha_1-\alpha_3$.[]{data-label="surface"}](figures/H1LF2 "fig:"){width="\textwidth"}
Now we can determine the intersection form of $W$, by computing the intersection of two classes. Given two classes $B$ and $B'$, we construct their representatives as above, starting with fibrations over disjoint small disks ${\mathcal{D}}'_1$ and ${\mathcal{D}}'_2$ away from the critical points. The parts of the surfaces contained in $P \times {\mathcal{D}}'_1$ resp. $P \times {\mathcal{D}}'_2$ are then disjoint, but intersections may appear after we cap off the vanishing cycles on the boundary of these surfaces. This is schematically depicted in Figure \[f:spiders\]. Intersections now come in two sorts: (i) the self-intersection of the cap (thimble) corresponding to the vanishing cycle $\alpha_i$, and (ii) the intersection of the caps corresponding to the distinct vanishing cycles $\alpha_i$ and $\alpha_j$. In case (i), this self-intersection equals, by construction, the framing of $\alpha_i$, relative to the page $P$ along which the corresponding $2$-handle is attached. In other words, each such cap contributes $(-1)$. In case (ii), the intersection of caps is given by the intersection of the curves $\alpha_i$ and $\alpha_j$ on the page $P$. Since $P$ is planar, we see that $\alpha_i \cdot \alpha_j =0$ for the simple closed curves $\alpha_i$ and $\alpha_j$; thus, case (ii) gives no contribution.
![The red and green figures are meant to represent two embedded surfaces constructed from two linear combinations of cycles as above. The tubes lie over the green and red circles. Intersections happen over the regular fibers (dotted) and over the vanishing cycles (crossed).[]{data-label="f:spiders"}](figures/spider.pdf)
Note that at this point of the proof we use planarity in an essential way; the formula would have additional terms for a higher-genus page (see Remark \[Auroux-rmk\] below).
To sum up, the intersection of two homology classes in a planar Lefschetz fibration can be found simply by counting the number of vanishing cycles shared by the linear combinations representing these classes. We have obtained the following proposition.
\[iform\] Consider two homology classes $B$, $B'$ in $H_2(W)$, corresponding to two null-homologous linear combinations $b_1\alpha_1 + \dots +b_m\alpha_m$ and $b'_1\alpha_1 + \dots +b'_m\alpha_m$ of the vanishing cycles. Then $$B \cdot B' = -(b_1b_1'+\dots+b_mb'_m).$$ In particular, $B\cdot B = -(b_1^2+\dots+b_m^2)$.
\[Auroux-rmk\] A version of Proposition \[iform\] holds in the case of a higher-genus fiber, with a similar proof, but there are extra terms given by the intersections $\alpha_i \cdot \alpha_j$ that can be non-trivial in general. This fact is mentioned, without proof, in the course of the proof of [@Aur Lemma 16]. We focused on the planar case sufficient for our purposes as the statement is simpler and the surface representatives are easier to visualize.
\[lessthan1\] Let $W$ be a Stein filling of a planar contact $3$-manifold, and $B \in H_2(W)$ a non-zero homology class. Then $B \cdot B \leq -2$.
We know that $B \cdot B < 0$. The class $B$ corresponds to a null-homologous linear combination $b_1\alpha_1 + \dots +b_m\alpha_m$, where not all of the coefficients $b_i$ vanish. Suppose that $B\cdot B = -(b_1^2+\dots+b_m^2) = -1$, this implies that all $b_i$ vanish except for one, say $b_1 = \pm1$. But this contradicts the fact that all vanishing cycles are homologically essential in $P$.
We immediately get Corollaries \[blow-down\] and \[IHS\].
Let $W$ be a weak symplectic filling of a planar contact manifold $(Y, \xi)$. Suppose that $B\cdot B=-1$ for a class $B \in H_2(W)$. By [@We2], if $W$ were minimal, $W$ would be deformation equivalent to a Stein filling (given by a Lefschetz fibration with planar fibers), and so the previous corollary would give a contradiction. Suppose now that $E$ is the homology class of an exceptional divisor in $W$; if $E = B$ the proof is complete. We claim now that, if $E\neq B$, then $E\cdot B = 0$. To this end, let $B\cdot E = x$, and look at the subspace of $H_2(W)$ generated by $E$ and $B$; the intersection form of $W$, restricted to this subspace, is $\big({\tiny\begin{array}{cc}-1&x \\ x&-1\end{array}}\big)$, and this matrix is negative definite if and only if $x=0$. It follows that $W$ can be blown down along a sphere in $E$, and that, in the blowdown, $B\cdot B = -1$. By induction, we can blow down to a minimal weak filling $W_0$; since this can be deformed to a Stein filling, Corollary \[lessthan1\] now gives a contradiction.
By results of Etnyre [@Etn], if $(Y, \xi)$ is an integral homology sphere that admits a planar open book decomposition, then the intersection form $Q$ of any minimal weak symplectic filling $W$ of $(Y, \xi)$ embeds in the negative definite diagonal lattice ${\mathbb{Z}}^N$ for some $N$. Since $Q$ is unimodular, $Q$ is in fact a direct summand of ${\mathbb{Z}}^N$, and in particular it is itself diagonalizable. Therefore, unless the filling is a rational homology ball, $H_2(W)$ must have a class with self-intersection $-1$, but this is not possible by the previous corollary. Finally, since $W$ can be deformed to be a Stein fillings, it has a handle decomposition with no 3-handles, and hence the inclusion of the boundary induces a surjection $H_1(Y)\to H_1(W)$; thus $W$ is an integral homology ball.
There are many examples of planar, fillable integral homology spheres that are not contactomorphic to the standard tight $S^3$; we discuss these in Section \[pi1\].
We now turn to the calculation of the first Chern class $c_1(J)$ for a compatible almost-complex structure on the Lefschetz fibration. Although planarity is crucial in the next proposition, much of the proof follows the lines of the well-known calculation of $c_1$ for Stein domains corresponding to Legendrian surgeries, [@Gompf-handle Proposition 2.3].
\[c1\] Let $(Y,\xi)$ be the contact structure associated to the planar open book $(P, \phi)$. Let $(W,\omega)$ be the symplectic filling of $(Y,\xi)$ associated to the factorization of $\phi$ into positive Dehn twists along the curves $\alpha_1,\dots,\alpha_m$, oriented coherently with the outer boundary of $P\subset D^2$. If $J$ is an almost-complex structure compatible with $\omega$, and $b_1\alpha_1 + \dots + b_m\alpha_m$ represents a class in $H_2(W)$ as above, then $$\langle c_1(J), [b_1\alpha_1 + \dots + b_m\alpha_m]\rangle = b_1 + \dots + b_m.$$
Note that something similar follows by work of Gay–Stipsicz [@GayStipsicz Corollary 2.3]; they observe that, up to deformation, $(W,\omega)$ embeds in the complement of a line in a blowup $X$ of $\mathbb{CP}^2$. Therefore $H_2(W)$ embeds in the lattice generated by the homology classes of (some of) the exceptional divisors of $X$; the first Chern class evaluates as 1 on each of these divisors, thus recovering an analogue of Proposition \[c1\]. However, there are examples of Stein 4-manifolds that admit such an embedding, but are nevertheless *not* planar; for instance, the following 4-manifold is realised as a subdomain in blowup of ${\mathbb{C}}^2$, as the corresponding embedding shows, but the planarity of its boundary is excluded by Theorem \[sing2\]. $$\xygraph{
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As before, the space $W$ is obtained from $P \times {\mathcal{D}}'$ by attaching 2-handles. The complex bundle $(TW, J)$ is trivial over $P \times {\mathcal{D}}'$, and $c_1(J)$ measures the obstruction to extending a trivialization over the 2-handles. We will argue that for each 2-handle, this obstruction is the same in the appropriate sense. We can assume that the $2$-handles are attached to fibers of $P\times {\mathcal{D}}'$ over points in a small arc in $\partial {\mathcal{D}}'$. Fix an embedding $P\subset {\mathbb{C}}$ and trivialize the complex bundle $T(P \times {\mathcal{D}}')= TP \times T{\mathcal{D}}'$ over the chosen fibers by a frame $(u, v)$, where $u$ is a constant vector field in $P \subset {\mathbb{C}}$ and $v$ is an inward normal to $\partial {\mathcal{D}}'$ in ${\mathcal{D}}' \subset {\mathbb{C}}$. This trivialization extends to a complex trivialization of $T(P \times {\mathcal{D}}')$ over the entire product $P \times {\mathcal{D}}'$. Each 2-handle $H_k$ can be identified with a fixed copy of $D^2 \times D^2 \subset i{\mathbb{R}}^2 \times {\mathbb{R}}^2$, and we can pick a complex trivialization of its tangent bundle that restricts to the circle $S^1\times 0 \subset H_k$ as the frame $(\tau, \nu)$, where $\tau$ is the tangent and $\nu$ the outward normal vector fields to $S^1 = \partial D^2 \subset i {\mathbb{R}}^2$. (Indeed, the frame $(\tau ,\nu)$ differs from the restriction of the product frame to $S^1$ by an element of $\pi_1(SU(2))$, and so $(\tau ,\nu)$ can be extended over the entire handle since $SU(2)$ is simply connected, cf [@Gompf-handle Proposition 2.3]). When we attach the handle by identifying $S^1 \times 0 \subset H_k$ with the vanishing cycle $\alpha_k$, $\nu$ is identified with $v$, and $\tau$ is identified with the tangent vector field to $\alpha_k$. Therefore, $\nu$ and $v$ together span a trivial complex line bundle, and $c_1(J)$ equals the first Chern class of the complex line bundle defined by $\tau$ and $\mu$. To evaluate the latter on the core of the handle $H_k$ (as a relative Chern class), we must look at the rotation number of $\mu$ relative to $\tau$ along the vanishing cycle $\alpha_k$ in the page $P$. Since $P$ is planar and $\alpha_k$ is a simple closed curve in $P$, it is clear that this rotation number equals $r=\pm 1$. (The sign depends on the orientation conventions). Note also that the value of $r$ is the same for all handles, since the tangent bundles over different pages are identified by our choice of trivialization, and different vanishing cycles in the same page $P \subset {\mathbb{C}}$ are identified via an isotopy in ${\mathbb{C}}$. It follows that the value of $c_1(J)$ on the homology class given by a linear combination $b_1\alpha_1 + \dots + b_m\alpha_m$ is given by $$\label{e:c1partial}
\langle c_1(J), [b_1\alpha_1 + \dots + b_m\alpha_m]\rangle = r(b_1 + \dots + b_m),$$ where $r=\pm 1$. To pin down the sign, we consider the lens space $L(3,1)$. The canonical contact structure $\xi_0$ on $L(3,1)$ is the Boothby–Wang structure associated to the disk bundle over $S^2$ with Euler number $-3$. As mentioned in the introduction, $\xi_0$ is supported by the open book on the 2-holed disk $P$, where the monodromy $\phi$ is the multi-twist along the boundary; more precisely, Gay–Mark [@GayMark] that this factorisation corresponds (up to deformation equivalence) to the symplectic disk bundle filling $(W,\omega)$ of $\xi_0$. As above, call $J$ an almost-complex structure compatible with $\omega$.
Note that homology class $S$ of the 0-section of the disk bundle is a generator, and it is represented by a a symplectic sphere; in particular, its homology class satisfies the adjunction formula, and the symplectic form integrates positively over it.
The linear combination $\alpha_1-\alpha_2-\alpha_3$, where $\alpha_1$ is parallel to the outer boundary, represents a generator of $H_2(W)$. Since the corresponding 2-chain is made by a part of the page (with positive orientation) and three vanishing cycles (where $\omega$ vanishes), the symplectic form integrates positively over this linear combination.
In particular, $\alpha_1-\alpha_2-\alpha_3$ represents the 0-section, with its symplectic orientation. Applying the adjunction formula and $$-r = \langle c_1(J), [\alpha_1-\alpha_2-\alpha_3] \rangle = \langle c_1(J), S \rangle = S\cdot S + \chi(S) = -1,$$ hence $r=1$, as claimed.
We now use Propositions \[iform\] and \[c1\] to prove Theorem \[positivegenus\].
Suppose that $(W,\omega')$ is a symplectic filling that contains a symplectic surface of genus $g > 0$, and call $A$ its homology class. Let $J'$ be an almost-complex structure compatible with $\omega'$. Without loss of generality, we may assume that $(W, \omega')$ is minimal; then Wendl’s theorem guarantees that there is a deformation from $(\omega',J')$ to $(\omega,J)$, such that $(W,\omega)$ is supported by a planar Lefschetz fibration, corresponding to a factorization of a planar monodromy $\phi$ of $\xi$ into positive Dehn twists along $\alpha_1,\dots,\alpha_m\subset P$.
Note that $\langle c_1(J),A\rangle = \langle c_1(J'),A\rangle$, since evaluation of $c_1(J)$ can only take discrete values. Representing $A$ as a linear combination $[b_1\alpha_1+\dots+b_m\alpha_m]$, from Proposition \[c1\] we obtain: $$\langle c_1(J),A\rangle = b_1 + \dots + b_m.$$ On the other hand, since $A$ is represented by an $\omega'$-symplectic surface of genus $g$, it satisfies the adjunction formula: $$\langle c_1(J'),A\rangle - A\cdot A = 2 - 2g \le 0.$$ Putting everything together: $$\sum (b_j+b_j^2) = b_1 + \dots + b_m + b_1^2+\dots + b_m^2 \le 0.$$ However, each of the summands on the left-hand side is non-negative, and evaluates to $0$ only if $b_j \in \{-1,0\}$ for each $j$. If $g>1$, we are already done. If $g=1$, the signs of all coefficients of $b_j$ agree, and therefore $b_1\alpha_1+\dots+b_m\alpha_m$ cannot represent a nontrivial linear combination of curves that is null-homologous in $P$, as desired.
The previous theorem rules out the presence of symplectic surfaces of genus $g>0$. Symplectic spheres can exist in a weak symplectic filling of a planar contact structure, and we will now describe their homology classes explicitly in terms of vanishing cycles of a Lefschetz fibration deformation equivalent to the given minimal symplectic filling.
Let us set up some notation and terminology first. We say that two curves $\alpha$ and $\alpha'$ in $P\subset D^2$ are *separated* if there is no hole in $P$ around which both $\alpha$ and $\alpha'$ have positive winding number. (Equivalently, this means that $\alpha$ and $\alpha'$ are homologous to $\beta = \partial D$ and $\beta' = \partial D'$ such that $D$ and $D'$ are disjoint.) We say that $\alpha$ *dominates* $\alpha'$, and we write $\alpha \succ \alpha'$, if there is no hole in $P$ around which the winding number of $\alpha'$ is larger than the winding number of $\alpha$. (Equivalently, this means that $\alpha$ and $\alpha'$ are homologous to $\beta = \partial D$ and $\beta' = \partial D'$ such that $D$ contains $D'$.) Note that $\succ$ is *not* a partial order on isotopy classes of curves, but rather it induces one on homology classes of embedded curves.
\[spheres-coefficients\] Suppose that $(W, \omega')$ is a minimal weak filling of a planar contact manifold $(Y, \xi)$, deformation equivalent to a Stein filling $(W, \omega)$ supported by a planar Lefschetz fibration with vanishing cycles $\alpha_1,\dots,\alpha_m\subset P$. If the linear combination $b_1\alpha_1+\dots+b_m\alpha_m$ of curves represents an embedded symplectic sphere in $(W, \omega')$, then all coefficients are either $0$ or $\pm1$, and there is exactly one coefficient $+1$. Without loss of generality, suppose that the sphere is represented by $\alpha_1-\alpha_2-\dots-\alpha_\ell$; then $\alpha_1 \succ \alpha_j$ for every $j=2,\dots,\ell$, and $\alpha_j$ and $\alpha_{j'}$ are separated for every $j\neq j'$ among $2,\dots,\ell$.
The proof is immediate once we write the adjunction formula as in the previous proof; indeed, the equation $$\sum (b_j+b_j^2) = 2$$ implies that all coefficients $b_j$ are either $0$ or $-1$, except for exactly one $j$, for which $b_j = -2$ or $b_j = 1$. However, the first case is excluded, since otherwise all coefficients would have the same sign.
The second part of the statement follows from elementary homological considerations in the page.
Links of hypersurface singularities {#links}
===================================
In this section, we turn our attention to links of isolated singularities of complex hypersurfaces in ${\mathbb{C}}^3$. Consider a complex hypersurface $\Sigma\subset {\mathbb{C}}^3$, given by an equation $F(z_1, z_2, z_3)=0$ with an isolated critical point at the origin, and let $(Y, \xi)$ be the link of the singularity with its canonical contact structure, so that $Y= \Sigma \cap \{|z_1|^2+|z_2|^2+|z_3|^2 = {\varepsilon}\}$. The manifold $(Y, \xi)$ is Stein fillable, with the standard filling given by the [*Milnor fiber*]{} $\{F(z_1, z_2, z_3)=\eta \} \cap D^6$ for small $\eta>0$; the Milnor fiber is the smoothing of $\Sigma \cap \{|z_1|^2+|z_2|^2+|z_3|^2 \le {\varepsilon}\}$.
We now consider an example, the link of the $D_4$-singularity. As a 3-manifold, this is described by the surgery diagram of Figure \[seifert-space\] where $e_0 = -1/r_1 = -1/r_2 = -1/r_3 = -2$; that is, it is the boundary of the plumbing associated to the graph $D_4$ (see Figure \[Dn\]). This example illustrates the main idea of our obstruction and will also be the key case of Theorem \[sing\].
\[d4\] The canonical contact structure on the link of the $D_4$-singularity is not planar.
Consider the Milnor fiber $W$ of the $D_4$-singularity. This is a Stein filling of the canonical contact structure on the link. The intersection form of $W$ is given by the $D_4$-graph (Figure \[Dn\]). We label its central vertex $X$, and the other vertices $A$, $B$, $C$.
For the sake of contradiction, suppose that the canonical contact structure on $D_4$ admits an open book with planar page $P$. As before, by Wendl’s theorem we know that $W$ admits the structure of a Lefschetz fibration whose fiber is the page $P$, and the vanishing cycles come from a positive factorization of the monodromy. The intersection form on $W$ can be computed as in Proposition \[iform\]; we now examine possibilities for vanishing classes that could produce $D_4$.
![Simple closed curves in $P$ are homologous if and only if they encircle the same holes. The curves shown are homologous but not homotopic in the three-holed disk.[]{data-label="homolo-curves"}](figures/homologous)
To begin, we need to have four classes with self-intersection $-2$. By Proposition \[iform\], each of these must be given by the difference of two curves, corresponding to two *distinct* vanishing cycles (which could, however, be isotopic as curves in $P$); moreover, it must be a null-homologous linear combination, so the two curves should be homologous. This means that the two curves must encircle the same holes of the disk. Note that the curves [*do not*]{} have to bound an annulus and [*do not*]{} have to be homotopic, see Figure \[homolo-curves\].
Let the class of the central vertex $X$ be given by the difference $\alpha-\beta$. Similarly, the class $A$ is given by two homologous curves, and since $A \cdot X = 1$, exactly one of these curves must coincide with $\alpha$ or $\beta$. We may assume that $A$ corresponds to the difference $\gamma-\alpha$ (where the vanishing cycle $\gamma$ is different from both $\alpha$ and $\beta$); note also that $\alpha$ and $\beta$ must be distinct. Similarly, both classes $B$ and $C$ must be given by pairs of vanishing cycles, so that exactly one of the curves in the difference representing each pair coincides with $\alpha$ or $\beta$. However, since $A \cdot B =A \cdot C = B \cdot C = 0$, no curves may be used in more than one pair, which is clearly not possible. Indeed, if $B$ is given by the vanishing cycles $\beta$ and $\delta$, $C$ can use neither $\alpha$ nor $\beta$.
We are now ready to prove Theorem \[sing\].
Since the $D_n$-graph contains $D_4$, the argument of Lemma \[d4\] applies to show that the link of the $D_n$-singularity is not planar for any $n>4$.
$
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!{(0,0) }*+{\bullet}="a1"
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!{(2.25,0) }*+{\dots}="dots"
!{(4.5,0) }*+{\bullet}="ak1"
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!{(-1.5,-1.1) }*+{-2}
"c"-"a1"
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"ak1"-"ak"
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}$
For links of all other surface singularities, the theorem follows from previously known results. Indeed, by [@Tjurina], the only surface singularities with negative definite Milnor fiber are the simple singularities $A_n$, $D_n$, $E_n$. Etnyre’s theorem says that every filling of a planar contact structure must be negative definite [@Etn], and since the Milnor fiber gives a Stein filling, it follows that only the links of A-D-E singularities can be planar. The case of $E_8$ is ruled out by [@Etn Theorem 1.2], as the corresponding link is an integral homology sphere with a non-standard intersection form. The cases of $E_6$ and $E_7$ are similarly ruled out using [@Etn Theorem 1.2]: although not stated explicitly in Etnyre’s paper, the same proof applies to show that for a planar rational homology sphere, the intersection form of any Stein filling must embed in a negative definite diagonal lattice. The links of $E_6$ and $E_7$ are rational homology spheres; the corresponding Milnor fibers, i.e. fillings given by the plumbing graphs, have intersection forms $E_6$ and $E_7$. Neither embeds into the standard lattice, thus the canonical structures on the links of $E_6$ and $E_7$ cannot be planar.
Alternatively, the cases of $E_6$ and $E_7$ follow from Lemma \[d4\], as the $E_6$- and $E_7$-graphs both contain the $D_4$-graph.
Finally, the links of the $A_n$-singularities are the lens spaces $L(n+1,n)$, and their canonical contact structures are easily seen to be planar [@Scho].
The homological obstruction {#therest}
===========================
We now prove Theorem \[general\]; the argument is very similar to the proof of Lemma \[d4\]. As in Section \[compute\], we use the notation $\alpha_{\mathbf{b}}$ to denote the linear combination $b_1\alpha_1+\dots+b_m\alpha_m$ of curves associated to the $m$-tuple ${\mathbf{b}}= (b_1,\dots,b_m)$. Moreover, given an $m$-tuple ${\mathbf{b}}$, we call the set $\{i \mid b_i \neq 0\}$ the *support* of ${\mathbf{b}}$. In other words, the support of ${\mathbf{b}}$ is the set of curves used by $\alpha_{\mathbf{b}}$. By extension, we also call the same set the support of the associated homology class $[\alpha_{\mathbf{b}}]$ (when this makes sense).
Suppose $W$ is the minimal filling of the contact structure $(Y,\xi)$, associated to the factorization of the monodromy $\phi: P\to P$, where $P$ is planar. Suppose $B_1, B_2 \in H_2(W)$ satisfy $$B_1\cdot B_1, B_2\cdot B_2 \in \{-2,-3\}, \quad B_1\cdot B_2 = 0.$$ Then, $B_1$ and $B_2$ have disjoint support.
The proof is split into three cases:
1. $B_1\cdot B_1 = B_2 \cdot B_2 = -2$,
2. $B_1\cdot B_1 = -2$, $B_2 \cdot B_2 = -3$, and
3. $B_1\cdot B_1 = B_2 \cdot B_2 = -3$.
(The case $B_1\cdot B_1 = -3$, $B_2 \cdot B_2 = -2$ is clearly symmetric to the second case, so we can omit it.)
To fix the notation, suppose that $W$ is associated to the factorisation of $\phi$ into Dehn twists along curves $\alpha_1,\dots,\alpha_m$. We recall that, if a class in a minimal weak filling of a planar contact structure has self-intersection $-2$, then it is represented by the difference of two homologous curves. Along the same lines, if a class as above has self-intersection $-3$, it corresponds to a linear combination $\pm(\alpha_i - \alpha_j - \alpha_k)$, where $[\alpha_i] = [\alpha_j] + [\alpha_k] \in H_1(P)$. In particular, there are holes in $P$ around which both $\alpha_i$ and $\alpha_j$ (respectively, $\alpha_k$) have both winding number 1.
1. Without loss of generality, suppose that $B_1 = [\alpha_1 - \alpha_2]$, where $\alpha_1$ and $\alpha_2$ are homologous. If the support of $B_2$ is not disjoint from that of $B_1$, then $B_2 = [\alpha_i-\alpha_j]$ where $i$ or $j$ is either 1 or 2, and $\alpha_i$ and $\alpha_j$ are homologous. One easily sees that neither combination works: for example, if $B_2=\alpha_3-\alpha_1$ with $\alpha_3\neq \alpha_1, \alpha_2$, then $B_1 \cdot B_2=1$, if $B_2=\alpha_2-\alpha_1$, then $B_1 \cdot B_2=2$, and other cases are similar.
2. As above, suppose $B_1 = \alpha_1 - \alpha_2$. If the support of $B_2$ is not disjoint from that of $B_1$, the only possibility is that $\pm B_2 = [\alpha_3 - \alpha_1 - \alpha_2]$, since $\alpha_1$ and $\alpha_2$ must appear with the same sign. But $\alpha_1$ and $\alpha_2$ have winding number 1 around the same holes, and so their sum cannot be homologous to a simple closed curve $\alpha_3$.
3. Without loss of generality, suppose that $B_1 = [\alpha_1 - \alpha_2 - \alpha_3]$. (Note that here we are using that the assumptions are unchanged if we change sign to either $B_1$ or $B_2$.) Suppose that the support of $B_2$ is not disjoint from that of $B_1$. Up to relabeling the indices and up to changing the sign of $B_2$, the only possibility is that $B_2 = [\alpha_4 - \alpha_1 - \alpha_2]$. But, again, as observed above, $\alpha_1$ and $\alpha_2$ have winding number 1 around *some* hole, and so their sum cannot be homologous to a simple closed curve $\alpha_4$.
Suppose we have a configuration of curves $B_1,\dots, B_k$, $X$ as in the statement. By the previous lemma, $B_1, \dots, B_k$ have pairwise disjoint supports. Since $X$ meets non-trivially each of $B_1,\dots,B_k$, its support must intersect at least the support of each of them, and in particular $X\cdot X \ge k$, thus leading to a contradiction.
For Seifert fibered L-spaces $M(-2; r_1, r_2, r_3)$, classification of tight contact structures was given in [@Gh]. Every tight contact structure on this space can be obtained by expanding the rational parameters $-\frac1{r_i}$ as continued fractions, $$-\frac1{r_i}=a_0^{i}-\cfrac{1}{a_1^{i}-\cfrac{1}{a_2^{i} - \cdots}}$$ and making a Legendrian surgery diagram where each $-\frac1{r_i}$-framed circle is replaced by a chain of Legendrian unknots with Thurston–Bennequin numbers given by the coefficients $a_0^{i}+1, a_1^{i}+1, \dots$. Since by assumption $r_1, r_2, r_3 \geq \frac13$, we have that $a_0^{1}, a_0^{2}, a_0^{3} \in \{-2, -3\}$. Thus, the corresponding plumbing graph for the Stein filling satisfies the hypotheses of Theorem \[general\], and therefore the contact structure is not planar.
We observe that Theorem \[general\] applies in many situations where the filling is not a plumbing of spheres.
$L_0$ at 88 23 $L_1$ at 32 86 $L_k$ at 144 86 ![The Legendrian surgery diagram for Example \[e:paolo\]. The Legendrian knot $L_i$ have $k-1 < \operatorname{tb}(L_0) < 0$, $\operatorname{tb}(L_1), \dots, \operatorname{tb}(L_{k}) \in \{-1, -2\}$.[]{data-label="f:paolo"}](figures/paolo "fig:")
\[e:paolo\] Consider the Legendrian surgery diagram of Figure \[f:paolo\], where $k > 2$ and the Legendrian tangles $L_0, \dots, L_k$ satisfy the following properties:
- the closure of $L_0$ is a Legendrian knot with $1-k < \operatorname{tb}(L_0) < 0$;
- for each $i > 0$ the closure of $L_i$ is a Legendrian link; the closure of the *arc* in the tangle is a Legendrian knot $L^0_i$ with $\operatorname{tb}(L^0_i)\in\{-1,-2\}$ for each $i$;
- at least one of the knots $L_0, L_1^0, \dots, L_k^0$ is nontrivial.
Then the corresponding Stein 4-manifold $(W,J)$ is not diffeomorphic to a plumbing of spheres, but it contains a configuration of homology classes that satisfy the assumptions of Theorem \[general\]. Therefore, the boundary $(Y,\xi)$ of $(W,J)$ is not a planar contact manifold.
Finally, we observe that the technique of the proof of Theorem \[general\] does indeed have a limit. Indeed, the configuration corresponding to the plumbing graph of the Seifert fibered space $M(-2;1/2,1/2,1/4)$ is unobstructed, as the following embedding (of intersection forms) exhibits: $$\xygraph{
!{<0cm,0cm>;<1cm,0cm>:<0cm,1cm>::}
!~-{@{-}@[|(2.5)]}
!{(0,0) }*+{\bullet}="a1"
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!{(-1.5,1.5) }*+{\bullet}="c"
!{(-1.5,-1.5) }*+{\bullet}="d"
!{(0,0.4) }*+{-2}
!{(1.5,0.4) }*+{-2}
!{(-1.5,1.9) }*+{-4}
!{(-1.5,-1.1) }*+{-2}
"c"-"a1"
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} \qquad = \qquad
\xygraph{
!{<0cm,0cm>;<1cm,0cm>:<0cm,1cm>::}
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!{(0,0) }*+{\bullet}="a1"
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!{(0.2,0.4) }*+{\phantom{}_{\alpha_2 - \alpha_3}}
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!{(-1.5,1.9) }*+{\phantom{}_{\alpha_1+\alpha_2-\alpha_5-\alpha_6}}
!{(-1.7,-1.1) }*+{\phantom{}_{\alpha_1-\alpha_2}}
"c"-"a1"
"a2"-"a1"
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}$$ Note that this embedding is in fact realised in the intersection form of the (unique) filling of $L(6,5)$ associated to the open book $(S^1\times I, \tau^6)$ (here $\tau$ is the right-handed Dehn twist along the core of the annulus); the underlying 4-manifold is diffeomorphic to a linear plumbing of five spheres of self-intersection $-2$ (represented by $\alpha_1-\alpha_2, \dots, \alpha_5-\alpha_6$).
Links of normal surface singularities {#normal-links}
=====================================
The goal of this section is to prove Theorem \[sing2\]. We will use terminology and results from Section \[compute\] without explicit mention; before proving the theorem, we recall a few facts and definitions concerning resolutions of surface singularities and plumbing graphs.
Recall that given a complex surface $X$ with an isolated singularity at $0$, we can consider its resolution $\pi:\tilde{X}\to X$. The resolution is [*good*]{} if the irreducible components of the exceptional divisor $\pi^{-1}(0)$ are smooth complex curves that intersect transversely at double points only. The (dual) resolution graph $\Gamma$ has vertices corresponding to irreducible components of the exceptional divisor; each vertex is labeled by the genus of the corresponding curve and its self-intersection (weight). The edges of $\Gamma$ record intersections of different irreducible components of the exceptional divisor. The 3-manifold $Y$ is then the boundary of the plumbing of disk bundles over surfaces according to $\Gamma$. A good resolution is not unique but plumbing graphs arising from different resolutions are related by a finite sequence of blow-ups/blow-downs of vertices corresponding to spheres with self-intersection $-1$. It is known that a surface singularity is normal if and only if its plumbing graph is negative-definite. (This property simultaneously holds or fails for plumbing graphs of all good resolutions.) For a graph with negative integer weights associated to its vertices, recall that a [*bad vertex*]{} is a vertex $v$ with weight $-w(v)$ such that $$0<w(v)< \text{valence of $v$.}$$ A normal surface singularity is known to be rational if its graph has no bad vertices, but the converse is not true. (We refer the reader to [@Nem5] for details of the definitions above and their topological significance.) Némethi [@NemL] proved that a normal surface singularity is rational if and only if its link is an L-space. Using this, Theorem \[sing2\] implies that if the canonical contact structure on a link of singularity is planar, then this link must be an L-space.
As mentioned in the introduction, it was proved by Schönenberger [@Scho] that canonical contact structures are planar for links of normal surface singularities whose plumbing graph has no bad vertices. More precisely, consider a plumbing graph where all vertices are spheres with weight less than $-1$. The corresponding surgery diagram admits a Legendrian realization, so that the plumbing graph gives a Stein filling for the contact manifold constructed from the Legendrian surgery diagram. Schönenberger showed that any contact structure obtained from a plumbing tree without bad vertices (as above) is planar [@Scho]. Note that Schönenberger considers an [*arbitrary*]{} Legendrian realizations of the plumbing graph, which is more general than the canonical contact structures in Theorem \[sing2\].
We begin with two lemmas; both are probably well-known, but for convenience we give their proofs.
\[l:minimalGamma\] Suppose the graph $\Gamma$ is minimal, i.e. it contains no vertices of weight $-1$ given by spheres. Then $(W, \omega')$ is minimal.
The assumption on $\Gamma$ translates as: for each vertex $E_i$, either the weight $w_i = E_i\cdot E_i$ satisfies $w_i < -1$, or $w_i = -1$ and $g(E_i) > 0$.
We claim that no homology class $E = \sum_i a_iE_i$ on which $\omega$ integrates positively can satisfy $E\cdot E = -1$ and $g(E) = 0$; in fact, such a class would also have to satisfy $c_1(E) = 1$, by adjunction.
We first claim that all the coefficients $a_i$ are all positive. Since any class with all negative coefficients cannot obviously be symplectic, it is enough to show that all coefficients must have the same sign. First, we observe that the class $E$ is indecomposable, i.e. if we write $E = E'+E''$, where $E'\cdot E'' = 0$, then either $E'=0$ or $E''=0$; in fact, since $\Gamma$ is negative definite, if $E$ decomposed as $E'+E''$, both $E'$ and $E''$ would have negative square, and $E\cdot E = E'\cdot E' + E''\cdot E'' \le -2$. This, in turn, implies that the support of $E$, i.e. the set of vertices for which $a_i \neq 0$, is connected. Suppose now that the coefficients do not all have the same sign; then there are two coefficients $a_i < 0 < a_j$ such that $E_i \cdot E_j = 1$. Write $|E|$ for the homology class $|E| = \sum_i |a_i|E_i$. We now observe that $$-1 = E\cdot E = \sum_i a_i^2 w_i + \sum_{i,j} a_ia_j E_i\cdot E_j < \sum_i a_i^2 w_i + \sum_{i,j} |a_i||a_j|E_i\cdot E_j = |E|\cdot|E| \le -1.$$
Now we know that $a_i \ge 0$ for each $i$. Adjunction for each vertex shows that $c_1(E_i) = 2 - 2g(E_i) + w_i$, and the latter quantity is never positive by assumption. Hence, $$1 = c_1(E) = \sum_i a_i c_1(E_i) \le 0.\qedhere$$
\[l:genus-curve-sing\] Let $(C,0)$ be a curve singularity in $({\mathbb{C}}^2,0)$. The Milnor fiber of $C$ has genus $0$ if and only if $(C,0)$ is either smooth or a double point.
Indeed, if $g$ is the genus of the Milnor fiber of $(C,0)$, $\mu$ its Milnor number, $r$ its number of branches, and $\delta$ its delta-invariant, then $g = 1 + \delta - r$; since the multiplicity of $(C,0)$ is at least $r$, then $\delta \geq r(r-1)/2$, $g$ is positive unless the singular point is smooth or an ordinary double point. (See, for instance, [@BrieskornKnorrer Pages 572–574] for more details.)
We now turn to the proof of Theorem \[sing2\]. The strategy is the following. Suppose we have a normal surface singularity, and let $\Gamma$ be the graph associated to its smallest good resolution. If $\Gamma$ is not minimal, then we blow down to the minimal graph; we will show that this either has a singular curve or a point of higher intersection (e.g. a tangency or a singular point with more than two branches). In either of the two cases, we can construct a divisor by smoothing (some of) the singularities, and this divisor will have positive genus. If, on the other hand, $\Gamma$ is minimal, we will apply Wendl’s theorem, and argue that there can be no vertices with higher genus, nor cycles in the graph, nor bad vertices (in a way similar to the proof of \[general\]).
The canonical contact structure of the link of a normal surface singularity $(X, 0) \subset {\mathbb{C}}^N$ has a symplectic filling given by a good resolution $\pi: \tilde{X} \to X$. Note that $\tilde{X}$ lives in a blowup of ${\mathbb{C}}^N$, hence it is Kähler, and in particular it has a symplectic form $\omega'$; the preimage $\pi^{-1}(0)$ is a complex divisor, and in particular it is symplectic. More precisely, $Y= X \cap S^{2N-1}_{{\varepsilon}}$ is filled by $W=\pi^{-1}(D^{2N}_{{\varepsilon}})$, with the (restriction of the) symplectic structure $\omega'$. The irreducible components of the exceptional divisor are then symplectic surfaces in $\tilde X$, so that $(Y, \xi)$ is the convex boundary of a plumbing of symplectic surfaces; as in the introduction, the plumbing is encoded by the resolution graph $\Gamma$.
We would like to use Wendl’s theorem and arguments with vanishing cycles as before, however the filling $(W, \omega')$ is not necessarily minimal, and we have to perform some blow-downs before a compatible Lefschetz fibration can be found. Reduction to the case of minimal fillings is done as follows. If $(W, \omega')$ is not minimal, i.e. it contains a symplectic sphere $E$ with $E\cdot E=-1$, we use Lemma \[l:minimalGamma\] to find a vertex of genus 0 and weight $-1$ in the graph $\Gamma$. Suppose now that the graph $\Gamma$ contains vertices of genus 0 and weight $-1$. We blow down the corresponding divisors until we get a minimal graph. The corresponding resolution may no longer be good; there may be singular curves among the components of the exceptional divisor or multiple intersection points. We can smooth out the singular curve by replacing the singular point (of the curve in a surface) by its Milnor fiber. Similarly, if there are intersection points of multiplicity greater than 2 or tangencies, we also smooth them out (as a reducible singularity). This process creates a divisor of positive genus, because, thanks to Lemma \[l:genus-curve-sing\], the Milnor fibre of a curve singularity is planar if and only if we have a smooth or a double point. As a result, we found a symplectic surface of positive genus in a symplectic filling of $(Y, \xi)$, so by Theorem \[positivegenus\], $(Y, \xi)$ cannot be planar in this case.
It remains to prove the statement of the theorem for the case where a good resolution $\tilde{X}$ is also minimal, i.e. $(W, \omega')$ contains no spheres of self-intersection $-1$. Obviously, in this case $\Gamma$ contains no vertices corresponding to spheres with weight $-1$. The minimal weak symplectic filling $(W, \omega')$ is deformation equivalent to a Lefschetz fibration, and we can use the results of Section \[compute\]. We need to prove the following three facts, for $(Y,\xi)$ planar:
- all surfaces in the plumbing have genus 0;
- there are no cycles in the graph;
- the graph has no bad vertices.
Condition (g) is guaranteed by Theorem \[positivegenus\], so we only need to prove conditions (c) and (b).
- Suppose that there is a cycle; namely, that there are classes $A_1,\dots, A_{c+1} = A_1$, such that $A_k \cdot A_{k+1} = 1$ for every $k = 1,\dots c$. The divisor $A_1 \cup \dots \cup A_c$ is represented by symplectic spheres with positive, transverse intersections; smoothing all intersections, we obtain a symplectic torus, which contradicts Theorem \[positivegenus\].
- The filling $(W, \omega')$ is deformation equivalent to a Lefschetz fibration with the planar fiber; write $\alpha_1, \dots \alpha_m$ for its vanishing cycles. The argument is now similar to the one in the proof of Theorem \[general\]. Suppose $B$ is a vertex of the graph with $B\cdot B = \ell$, connected to vertices $A_1,\dots, A_n$. Lemma \[spheres-coefficients\] describes the class of $B$; without loss of generality, assume that $B = \alpha_1 - \alpha_2 - \dots - \alpha_\ell$, so that $\alpha_1$ dominates the curves $\alpha_2, \dots, \alpha_{\ell}$.
Again by Lemma \[spheres-coefficients\], we know that there exist indices $i_k$ and collections of indices $J_k\not\ni i_k$ such that $A_k = [\alpha_{i_k} - \sum_{j\in J_k} \alpha_j]$, with $\alpha_{i_k} \succ \alpha_j$ for every $j \in J_k$. For convenience, let $J_B = \{2,\dots,\ell\}$. We can then compute: $$\begin{aligned}
0 &= A_k\cdot A_{k'} = \delta(i_k,J_{k'}) + \delta(i_{k'},J_k) - \delta(i_k,i_{k'}) - \#(J_k \cap J_{k'}), \label{e:AA}\\
1 &= B\cdot A_{k} = \delta(1,J_{k}) + \delta(i_{k},J_B) - \delta(1,i_{k}) - \#(J_B \cap J_{k}) \label{e:AB}\end{aligned}$$ where $\delta(i,J) = 1$ if $i\in J$, and is 0 otherwise, and $\delta(i,i') = 1$ if $i = i'$, and is 0 otherwise.
Let us focus on first. From it, we deduce that at least one among $1 \in J_{k}$ and $i_{k} \in J_B$ holds. Suppose that both hold simultaneously; then $\alpha_{1} \succ \alpha_{i_{k}}$ and $\alpha_{i_{k}} \succ \alpha_{1}$, which implies that the two curves $\alpha_{1}$ and $\alpha_{i_{k}}$ are homologous, and therefore that $B = [\alpha_1 - \alpha_{i_k}]$ and $A_k = [\alpha_{i_k} - \alpha_{1}]$, which clearly contradicts the assumption that $B\cdot A_k = 1$.
Next, we claim that $1 \in J_{k}$ can only hold for at most one of the classes $A_k = [\alpha_{i_k} - \sum_{j\in J_k} \alpha_j]$. Indeed, suppose that we have $1 \in J_{k}$ and $1 \in J_{k'}$ for two distinct classes $A_k$, $A_{k'}$. This implies that both leading terms $\alpha_{i_k}$ and $\alpha_{i_{k'}}$ dominate $\alpha_1$. Then we must have $\alpha_{i_k} \not\in J_{k'}$, because otherwise $\alpha_{i_k}$ would be separated from $\alpha_1$, and similarly $\alpha_{i_{k'}} \not\in J_{k}$. It follows that $A_k\cdot A_{k'} \leq -1$, a contradiction.
Finally, we want to show that $i_k \neq i_{k'}$ for every pair $k,k'$; to this end, we use . Suppose that there are two indices, $k, k'$, such that $i_k = i_{k'} = i$; this implies that $i\not\in J_k, J_{k'}$, and that $\delta(i_k, i_{k'}) = 1$. But then $0 = A_k \cdot A_{k'} \le -1$, clearly a contradiction.
Summarizing, we see that the set $J_B=\{2, \dots, \ell\}$ must contain all the leading elements $i_1, \dots i_n$ of the classes $A_1, \dots, A_n$, except possibly one. Since $i_1, \dots i_n$ are all distinct, it follows that that $n \le \ell$, i.e. that $B$ is not a bad vertex.
We conclude this section with the proof of Corollary \[deformation\].
Let $(Y,\xi)$ be a planar contact 3-manifold. The proof of Theorem \[sing2\] above shows that no filling of $(Y,\xi)$ can contain the exceptional divisor of a resolution of a non-planar singularity. If there were a strong symplectic cobordism $(W',\omega')$ from the link $(Y_s,\xi_s)$ of a non-planar normal surface singularity to $(Y,\xi)$, then one could glue the resolution of $(Y_s,\xi_s)$ to obtain a strong symplectic filling $(W,\omega)$ of $(Y,\xi)$ containing a forbidden configuration.
The second half of the statement is now straightforward, since a deformation from $(S,0)$ to $(S',0)$ gives rise to a Weinstein cobordism from the link of $(S',0)$ to the link of $(S,0)$.
Planar Lefschetz fibrations with prescribed fundamental group {#pi1}
=============================================================
We will now construct planar Lefschetz fibrations with prescribed fundamental group. Recall that the *deficiency* of a presentation $\langle x_1,\dots,x_m \mid r_1, \dots, r_n \rangle$ is $m -n$, and that the *deficiency* of a finitely presented group is the maximal deficiency over all its presentations. A group is [*perfect*]{} if its abelianization is trivial.
\[p:ballgroups\] Let $G$ be a finitely presented group. Then there exists a planar Lefschetz fibration on a $4$-manifold $W$ with fundamental group $G$. Moreover, if $G$ is perfect and has deficiency $0$, $W$ can be chosen to be an integral homology Stein $4$-ball. In this case, ${\partial}W$ is an integral homology $3$-sphere.
The family of perfect, finitely presented groups of deficiency 0 is quite rich: for instance, it contains fundamental groups of integral homology spheres. In fact, let $G = \pi_1(Y)$ be the fundamental group of an integral homology 3-sphere $Y$; $G$ is perfect since its abelianization is $H_1(Y) = 0$. Moreover, $G$ has non-positive deficiency, since it is perfect; it has non-negative deficiency since a genus-$g$ Heegaard decomposition of $Y$ gives a presentation of $G$ with $g$ generators and $g$ relators (which is, in particular, a finite presentation).
The proof of Proposition \[p:ballgroups\] easily follows from the following lemma.
\[l:goodpresentation\] Let $\langle x_1,\dots,x_m \mid r_1, \dots, r_{m-d} \rangle$ be a presentation of a group $G$ of deficiency $d$. Then there exists another presentation $\langle y_1,\dots,y_n \mid s_1, \dots, s_{n-d} \rangle$ of $G$ such that:
- each word $s_j$ is a positive word in $y_1, \dots, y_n$;
- each generator $y_i$ appears at most once in each word $s_1,\dots,s_{n-d}$;
- the cyclic order of the generators $y_i$ is preserved in each word $s_j$.
By Lemma \[l:goodpresentation\], $G$ has a presentation $\langle y_1,\dots,y_n \mid s_1, \dots, s_{n-d} \rangle$ with the properties (p), (r), and (c) as above. Consider the $n$-holed disk $P$, with fundamental group $\pi_1(P) = \langle y_1, \dots, y_n\rangle$. We assume that the generators $y_1, y_2, \dots, y_n$ are given by loops going around one hole each, as in Figure \[f:pi1gens\]. By the properties (p), (r), and (c), each word $s_j$ is represented by an embedded simple closed curve $\alpha_j$ in $P$. Indeed, since by (p) and (r) each generator enters in the word $s_j$ positively and at most once, we can take the curve $\alpha_j$ enclosing the corresponding holes, with a counterclockwise orientation. By (c), the cyclic order of the generators in the loop given by $\alpha_j$ is the same as in the word $s_j$; it follows that $\alpha_j$ represents $s_j$. See Figure \[f:allowedword\] for an example. Let $\phi$ be the product of positive Dehn twists along $\alpha_1,\dots,\alpha_{n-d}$. By construction, the associated Lefschetz fibration $W$ is a Stein domain whose fundamental group is precisely $G$.
If $G$ is perfect, $H_1(W)=G/G'$ vanishes; if, moreover, $G$ has deficiency 0, using a presentation with $d=0$ yields $H_2(W)=0$, since the classes $\alpha_1, \dots, \alpha_n$ are linearly independent in $H_1(P)$.
[0.4]{} $y_1$ at 91 277 $y_2$ at 123 120 $y_3$ at 273 80 $y_4$ at 350 227 $y_5$ at 228 329 ![Generators and a simple closed curve in the 5-holed disk.[]{data-label="f:prc"}](figures/pi1generators "fig:"){width="\textwidth"}
[0.4]{} $y_1$ at 103 283 $y_2$ at 113 125 $y_3$ at 273 80 $y_4$ at 350 215 $y_5$ at 241 332 ![Generators and a simple closed curve in the 5-holed disk.[]{data-label="f:prc"}](figures/allowedword "fig:"){width="\textwidth"}
Before proving the lemma, let us introduce the concept of badness for a presentation. We say that a word is *long* if its length is at least $3$, and *short* if it is of length $2$. Given a presentation $\mathcal{P} = \langle a_1,\dots,a_m \mid w_1, \dots, w_{m-d} \rangle$, we define its *badness* $b(\mathcal P)$ as follows. Let $b_-(\mathcal{P})$ be sum of the number of occurrences of $a_i^{-1}$ over all generators $a_i$; let also $b^i_+(\mathcal{P})$ be sum of the number of occurrences of $a_i$ in long words, and $b_+({\mathcal{P}}) = \sum_i\max\{b^i_+({\mathcal{P}})-1,0\}$ (that is, we are ignoring the first appearance of each $a_i$ in long words, if there is one, as well as all appearances of $a_i$ in short words). Let $b({\mathcal{P}}) = b_-({\mathcal{P}})+b_+({\mathcal{P}})$.
For instance, consider the presentation ${\mathcal{P}}= \langle a,b,c,d \mid bad, cab, ab{^{-1}}ac{^{-1}}\rangle$; then we have $b_-({\mathcal{P}}) = 2$, $b_+^a({\mathcal{P}}) = 4$, $b_+^b({\mathcal{P}}) = 2$, $b_+^c({\mathcal{P}}) =b_+^d({\mathcal{P}}) = 1$, and hence $b({\mathcal{P}}) = 6$.
A key feature of $b$ that will be used in the proof is that it is insensitive to the labelling of the generators, in the sense that it is invariant under permutation of the indices of generators.
Notice that a presentation of badness 0 satisfies properties (p), (r), and (c), up to reordering the generators. (The converse, however, is not true.) In fact, short words respect all cyclic orders, and, when the badness is 0, each generator appears in at most one long word; hence each long word can be used to define a compatible order on the corresponding subset of generators.
To prove the lemma, it is enough to show that, given a presentation $\mathcal{P}$ of deficiency $d$ and positive badness, we can always find another presentation $\mathcal{P'}$ for the same group with the same deficiency and with $b(\mathcal{P}') < b(\mathcal{P})$.
There are three cases to consider. Either the inverse of a generator appears, or a generator appears more than once somewhere in the presentation. Without loss of generality, assume that that this generator is $a_1$, and let $w$ be one of the culprit words.
In the first case, the presentation $\mathcal{P}'$ is obtained from $\mathcal{P}$ by adding a generator $a_{m+1}$ and the relation $a_1a_{m+1}$, so that $a_{m+1} = a_1{^{-1}}$; we then replace one occurrence of $a_1{^{-1}}$ in $w_1$ by $a_{m+1}$. We have replaced one occurrence of $a_1{^{-1}}$ with a new generator, so $b_-({\mathcal{P}}') = b_-({\mathcal{P}})-1$, and we created a positive short word, so $b_+({\mathcal{P}}') = b_+({\mathcal{P}})$.
In the second case, we add two generators $a_{m+1}, a_{m+2}$ and the relations $a_1a_{m+1}$, $a_{m+1}a_{m+2}$, so that $a_{m+2} = a_{m+1}{^{-1}}= a_1$; we then replace one occurrence of $a_1$ in $w$ by $a_{m+2}$. We have replaced one extra occurrence of $a_1$ with a new generator, and created two positive short words, so $b_-({\mathcal{P}}') = b_-({\mathcal{P}})$ and $b_+({\mathcal{P}}') = b_+({\mathcal{P}})-1$.
In either case, $b({\mathcal{P}}') = b({\mathcal{P}})-1$, and this concludes the proof of the lemma.
In fact, one can extract a bound on the Euler characteristic of the page $P$ in terms of the original presentation $\mathcal{P}$ for $G$: the algorithm above gives a page $P$ with $\chi(P) \ge 1-n-2b_+(\mathcal{P})-b_-(\mathcal{P})$.
|
---
abstract: |
This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center, while the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross-Pitaevskii equation (GPE) which combines the attractive potential $\sim r^{-2}$ and the quartic self-repulsive nonlinearity, induced by the Lee-Huang-Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity $\sim r^{-4/3}$ at $r\rightarrow 0$. Modes with embedded angular momentum exist too, and they have their stability region. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension $D$, with the density singularity $\sim r^{-2/(4-D)}$. Such states may be considered as a result of screening of a “bare" delta-functional attractive potential by the respective nonlinearity.
This paper was published in:
Condensed Matter **5**, 20 (2020); https://doi.org/10.3390/condmat5010020
author:
- 'Elad Shamriz, Zhaopin Chen, and Boris A. Malomed$^{1,2,3}$'
- Hidetsugu Sakaguchi
title: 'Singular Mean-Field States: a Brief Review of Recent Results'
---
The list of acronyms:
------ ------------------------------------------------------
2D two-dimensional
3D three-dimensional
BEC Bose-Einstein condensate
GPE Gross-Pitaevskii equation
GS ground state
LHY Lee-Huang-Yang (correction to the mean-field theory)
MF mean field
NLSE nonlinear Schrödinger equation
TF Thomas-Fermi (approximation)
VK Vakhitov-Kolokolov (stability criterion)
------ ------------------------------------------------------
Introduction
============
Singular states pulled to the center by attractive fields
---------------------------------------------------------
### Outline of the topic
A commonly adopted condition which distinguishes physically relevant states produced by various two- and three-dimensional (2D and 3D) models originating in quantum mechanics, studies of Bose-Einstein condensates (BECs), nonlinear optics, plasma physics, etc., is that the respective fields, such as wave functions in quantum mechanics and mean-field (MF) description of BEC, or local amplitudes of optical fields, must avoid singularities at $r\rightarrow 0$, where $r$ is the radial coordinate. A usual example of the relevance of this condition is provided by the quantum-mechanical Schrödinger equation with the attractive Coulomb potential, $\mathcal{U}(r)\sim -r^{-1}$: the stationary real wave function of states with integer azimuthal quantum number (alias vorticity) $l\geq 0$ has expansion $$u(r)=\left[ u_{0}+u_{1}r+\mathcal{O}\left( r^{2}\right) \right] r^{l}
\label{l}$$at $r\rightarrow 0,$ thus avoiding a singularity, despite the fact that the trapping potential is singular [@LL].
In quantum mechanics, a critical role is played by a more singular attractive potential, *viz*., $$\mathcal{U}(r)=-\left( U_{0}/2\right) r^{-2}, \label{U(r)}$$with $U_{0}>0$, which gives rise to the *quantum collapse*, alias “*fall onto the center*" [@LL]. This well-known phenomenon means nonexistence of the ground state (GS) in the 3D and 2D Schrödinger equations with potential (\[U(r)\]). In 3D, the collapse occurs when $U_{0}$ exceeds a finite critical value, $\left( U_{0}^{\mathrm{%
(3D)}}\right) _{\mathrm{cr}}$ (in the notation adopted below in Eq. ([3Dscaled]{}), $\left( U_{0}^{\mathrm{(3D)}}\right) _{\mathrm{cr}}=1/4$), while in two dimensions $\left( U_{0}^{\mathrm{(2D)}}\right) _{\mathrm{cr}%
}=0 $, i.e., the 2D collapse happens at any $U_{0}>0$.
In both 3D and 2D cases, potential (\[U(r)\]) may be realized as electrostatic pull of a particle (small molecule), carrying a permanent electric dipole moment, to a charge placed at the center, assuming that the local orientation of the dipole is fixed by the minimization of its energy in the central field [@HS1]. In addition to that, in the 2D case the same potential (\[U\]) may be realized as attraction of a magnetically polarizable atom to a thread carrying electric current (e.g., an electron beam) transversely to the system’s plane, or the attraction of an electrically polarizable atom to a uniformly charged transverse thread (other 2D settings in Bose-Einstein condensates (BECs) under the action of similar fields were considered in Refs. [@Austria1] and [@Austria2]).
A fundamental issue is stabilization of the 3D and 2D quantum-mechanical settings with pulling potential (\[U(r)\]) against the collapse, with the aim to create a missing GS. A solution was proposed in Refs. [@QTF1]-[@QTF3], which replaced the original quantum-mechanical problem by one based on a linear quantum-field theory. While such a model produces a GS, it does not answer a natural question: what an effective radius of the GS is for given parameters of the setting, such as $U_{0}$ in (\[U(r)\]) and the mass of the quantum particle, $m$. Actually, the field-theory solution defines the GS size as an arbitrary spatial scale, which varies as a parameter of the respective field-theory renormalization group. A related problem is the definition of self-adjoint Hamiltonians in the linear quantum theory including the interaction of a particle with singular potentials [Yafaev,Noja]{}.
Another solution was proposed in Ref. [@HS1], which replaced the 3D linear Schrödinger equation by a nonlinear Gross-Pitaevskii equation (GPE) [@GP] for a gas of particles attracted to the center by potential (\[U\]), with repulsive inter-particle collisions. In the framework of the mean-field (MF) approximation, the 3D GPE gives rise to the missing GS at all values of $U_{0}>\left( U_{0}^{\mathrm{(3D)}}\right) _{\mathrm{cr}}$. The radius of the GS is fully determined by model’s constants, i.e., $U_{0}$, $m$, the scattering length of the inter-particle collisions, and the number of particles, $N$. For typical values of the physical parameters, an estimate for the radius is a few microns. Beyond the framework of the MF, it was demonstrated that the many-body quantum theory, applied to the same setting, does not, strictly speaking, create GS, but the interplay of the attraction to the center and inter-particle repulsion gives rise to a metastable state. For sufficiently large $N$, the metastable state is nearly tantamount to GS, being separated from the collapse regime by a very tall potential barrier [@Gregory]. Further, the mean-field GS was also constructed in the 3D gas embedded in a strong uniform electric field, which reduces the symmetry of the effective pulling potential from spherical to cylindrical [@HS2], as well as in the two-component 3D gas [@HS3].
In 2D, the problem is more difficult, as the repulsive cubic term in GPE, which represents inter-atomic collisions in the MF approximation [@GP], is not strong enough to suppress the 2D quantum collapse and create the GS. The main issue is that, in 3D and 2D settings alike, the MF wave function, $%
\psi (r)$, produced by the respective GPE, has density $\left\vert \psi
(r)\right\vert ^{2}$ diverging $\sim r^{-2}$ at $r\rightarrow 0$. In terms of the integral norm,$$N=\left( 2\pi \right) ^{D-1}\int_{0}^{\infty }\left\vert \psi (r)\right\vert
^{2}r^{D-1}dr \label{N}$$($D=3$ or $2$ is the dimension), the density singularity $r^{-2}$ is integrable in 3D, but not in 2D, where it gives rise to a logarithmic divergence of the norm:$$N\sim \ln \left( r_{\mathrm{cutoff}}^{-1}\right) , \label{ln()}$$where $r_{\mathrm{cutoff}}$ is a cutoff (smallest) radius, which may be determined by the size of particles in the condensate. Actually, the cubic self-repulsion is *critical* in 2D, as any stronger nonlinear term is sufficient to stabilize the 2D setting. In 3D, the critical value of the repulsive-nonlinearity power, which also leads to the logarithmic divergence of $N$, is $7/3$; it is relevant to mention that the respective nonlinear term, $|\psi |^{4/3}\psi $, represents the effective repulsion in the density-functional model of the Fermi gas [@Fermi2]-[@Fermi3].
A solution for the 2D setting is offered by the quintic defocusing nonlinearity [@HS1], that may account for three-body repulsive collisions in the bosonic gas [@Abdullaev1; @Abdullaev2]. However, a difficulty in the physical realization of the quintic term is the fact that three-body collisions give rise to effective losses, kicking out particles from BEC to the thermal component of the gas [@loss1]-[@loss3].
In addition to original papers [@HS1; @HS2; @HS3] and [@Gregory], the above-mentioned results were summarized in short review [@CM].
### New results included in the review
Recently, much interest was drawn to quasi-2D and 3D self-trapped states in BEC in the form of “quantum droplets", filled by a nearly incompressible two-component condensate, which is considered as an ultradilute quantum fluid. This possibility was predicted in the framework of the 3D [@Petrov] and 2D [@Petrov-Astra; @Zin; @Santos] GPEs which include the Lee-Huang-Yang (LHY) corrections to the MF approximation [LHY]{}. They represent effects of quantum fluctuations around the MF states. The two-component structure of the condensate makes it possible to provide nearly complete cancellation between the inter-component MF attraction and intra-component repulsion (which, in turn, can be adjusted by means of the Feshbach-resonance technique [@Feshbach]) and thus create stable droplets through the balance of the relatively weak residual MF attraction and LHY-induced quartic self-repulsion. The quantum droplets of oblate (quasi-2D) [@Leticia1; @Leticia2] and fully 3D (isotropic) [Inguscio1,Inguscio2]{} shapes were created in a mixture of two different spin states of $^{39}$K atoms, as well as in a mixture of $^{41}$K and $^{87}$Rb atoms [@hetero]. Further, it was predicted that 2D [Raymond1,Raymond2,lattice]{} and 3D [@Barcelona] droplets with *embedded vorticity* have their stability regions too. The LHY effect has also opened the way to the creation of stable 3D droplets in single-component BEC with long-range interactions between atoms carrying magnetic dipole moments [@Pfau1]-[@Pfau5], although dipolar-condensate droplets with embedded vorticity are unstable [@Macri].
The LHY effect in the 3D model of BEC pulled to the center by potential (\[U(r)\]) was considered in Ref. [@CM], with a conclusion that the LHY term gives rise to a quantum phase transition at $U_{0}=2/9$ (note that it is close to but smaller than the above-mentioned critical value for the linear Schrödinger equation, $\left( U_{0}^{\mathrm{(3D)}}\right) _{%
\mathrm{cr}}=1/4$). The phase transition manifests itself in the change of the asymptotic form of the GS stationary wave function at $r\rightarrow 0$:$$u(r)\approx \left\{
\begin{array}{c}
\mathrm{const}\cdot r^{-\left( 1/2-\sqrt{1/4-U_{0}}\right) },~\mathrm{at}%
~U_{0}<2/9, \\
\left( U_{0}/2-1/9\right) ^{1/3}r^{-2/3},~\mathrm{at}~U_{0}>2/9.%
\end{array}%
\right. \label{transition}$$This phase transition may be categorized as one of the first kind, as power $%
\alpha $ in terms $r^{-\alpha }$ in Eq. (\[transition\]) undergoes a finite jump at $U_{0}=2/9$, from $1/3$ to $2/3$.
In Section 2 of the present review we summarize recent results which demonstrate that the stabilization of the GS in the quasi-2D bosonic gas pulled to the center by potential (\[U\]) may be provided by the LHY correction to the GPE [@we]. This possibility is essential because, as mentioned above, the alternative, in the form of the quintic self-repulsion, is problematic in the BEC setting. The underlying three-dimensional GPE, which includes the LHY quartic defocusing term, is$$i\hbar \frac{\partial \Psi }{\partial t}=-\frac{\hbar ^{2}}{2m}\nabla
^{2}\Psi +W(\mathbf{r})\Psi +\frac{4\pi \hbar ^{2}\delta a}{m}|\Psi
|^{2}\Psi +\frac{256\sqrt{2\pi }\hbar ^{2}}{3m}a^{5/2}|\Psi |^{3}\Psi ,
\label{3DGPE}$$where $\Psi $ stands for equal wave functions of two components of the BEC, $%
W(\mathbf{r})$ is the general trapping potential, $a>0$ is the scattering length of inter-particle collisions, $\delta a\gtrless 0$, with $\left\vert
\delta a\right\vert \ll a$, represents the above-mentioned small disbalance of the inter-component attraction and intra-component repulsion, and the last term in Eq. (\[3DGPE\]) is the LHY correction to the MF equation [Petrov]{}.
The reduction of Eq. (\[3DGPE\]) to the 2D form, with coordinates $\left(
x,y\right) $, under the action of tight confinement applied in the $z$ direction, was derived in Ref. [@Petrov-Astra], producing the GPE with a cubic term multiplied by an additional logarithmic factor, $$\left( \mathrm{nonlin}\right) _{\mathrm{2D}}\sim |\Psi |^{2}\ln \left( |\Psi
|^{2}/\Psi _{0}^{2}\right) \Psi . \label{ln}$$However, this limit implies extremely strong confinement in the $z$ direction, with the transverse size $a_{\perp }\ll \xi $, where the healing length is $\xi =\left( 32\sqrt{2}/3\pi \right) \left( a/\left\vert \delta
a\right\vert \right) ^{3/2}a\approx 5\left( a/\left\vert \delta a\right\vert
\right) ^{3/2}a$ [@Petrov]. For experimentally relevant parameters [Leticia1]{}-[@Inguscio2], an estimate is $\xi \simeq 30$ nm. On the other hand, a realistic size of the confinement length in the experiment is few $%
\mathrm{\mu }$m, implying relation $a_{\perp }\gg \xi $, opposite to the above-mentioned one necessary for the derivation of Eq. (\[ln\]). Therefore, it is relevant to reduce Eq. (\[3DGPE\]) to the 2D form, keeping the same nonlinearity as in Eq. (\[3DGPE\]).
To complete the derivation of the effective 2D equation, we first rescale three-dimensional Eq. (\[3DGPE\]), measuring the density, length, time, and the trapping potential in units of $\left( 36/25\right) n_{0}$, $\xi $, $%
\tau \equiv \left( m/\hbar \right) \xi ^{2}$, and $\hbar /\tau $, respectively:$$i\frac{\partial \Psi }{\partial t}=-\frac{1}{2}\nabla ^{2}\Psi +\sigma |\Psi
|^{2}\Psi +|\Psi |^{3}\Psi +W(\mathbf{r})\Psi , \label{3Dscaled}$$where $\sigma =\pm 1$ is the sign of $\delta a$, the potential is a sum of term (\[U(r)\]) and a transverse-confinement term, $\left( 1/2\right)
a_{\perp }^{-4}z^{2}$, with sufficiently small $a_{\perp }^{2}$. Then, the 3D $\rightarrow $ 2D reduction is performed by means of the usual substitution [@Luca; @Delgado], $\Psi \left( x,y,z,t\right) =\psi \left(
x,y,t\right) \exp \left( -z^{2}/2a_{\perp }^{2}\right) $, followed by averaging in the transverse direction, $z$. Additional rescaling, $\psi
\rightarrow \left( 2/\sqrt{5}\right) \psi $, $\left( x,y\right) \rightarrow
\left( \sqrt{5}/2\right) \left( x,y\right) $, and $t\rightarrow \left(
5/4\right) t$, casts the effective 2D equation, written in polar coordinates $\left( r,\theta \right) $, in the final form: $$i\frac{\partial \psi }{\partial t}=-\frac{1}{2}\left( \frac{\partial
^{2}\psi }{\partial r^{2}}+\frac{1}{r}\frac{\partial \psi }{\partial r}+%
\frac{1}{r^{2}}\frac{\partial ^{2}\psi }{\partial \theta ^{2}}\right) -\frac{%
U_{0}}{2r^{2}}\psi +\sigma \left\vert \psi \right\vert ^{2}\psi +|\psi
|^{3}\psi , \label{psi2d}$$which includes potential (\[U(r)\]).
In the framework of Eq. (\[psi2d\]), it is also relevant to consider the case of $\delta a=0$, which implies exact compensation of the inter-component attraction and intra-component repulsion. In this case, one should set $\sigma =0$ in Eq. (\[psi2d\]), keeping the nonlinearity which originates as the LHY correction to the MF field theory (cf. Ref. [LHY-only]{}).
Results for both GS and vortex states, produced by the analysis of the 2D equation (\[psi2d\]) [@we], are summarized, as a part of the present review, in Section 2. Essential conclusions are that all the GS solutions (with zero vorticity) are stable, while vortex modes are stable if $U_0$ exceeds a certain critical value, which depends on the vorticity.
Singular solitons: previously known results
-------------------------------------------
### Singular solitons in free space
The condition of the convergence of the integral norm is equally relevant for self-trapped states, which are predicted, as localized solutions, by models such as the nonlinear Schrödinger equation (NLSE) for a complex wave function, $\psi \left( x,y,z,t\right) $:$$i\psi _{t}+(1/2)\nabla ^{2}\psi -\sigma \left\vert \psi \right\vert ^{2\nu
}\psi =0, \label{NLSE}$$where $\sigma =1$ and $-1$ correspond, respectively, to the self-repulsive (defocusing) and attractive (focusing) signs of the nonlinearity. Typical physical realizations of the NLSE feature cubic ($\nu=1$) and quintic ($\nu
=2$) nonlinearities.
The commonly known bright-soliton solutions of the self-focusing ($\sigma
=-1 $) NLSE in 1D, with arbitrary real chemical potential $\mu <0$, are free of singularities:$$\psi =\exp (-i\mu t)\left( \frac{\sqrt{-\left( \nu +1\right) \mu }}{\mathrm{%
\cosh }\left( \sqrt{-2\mu }\nu x\right) }\right) ^{1/\nu }. \label{soliton}$$At $\nu \geq 2$, these 1D solutions are subject to instability against the *wave collapse*, i.e., catastrophic shrinkage of the self-focusing field [@Berge]. In the case of $\sigma =+1$ (self-defocusing), Eq. ([NLSE]{} gives rise to an exact solution in the form of a *singular soliton*,$$\psi =\exp \left( -i\mu t\right) \left( \frac{\sqrt{-\left( \nu +1\right)
\mu }}{\mathrm{\sinh }\left( \sqrt{-2\mu }\nu |x|\right) }\right) ^{1/\nu }.
\label{sing-sol}$$For $\nu \leq 2$, this singular solution is physically irrelevant because it gives rise to a divergent total norm, $N=\int_{-\infty }^{+\infty
}\left\vert \psi (x)\right\vert ^{2}dx$. In particular, in terms of the usual GPE, the divergence of the norm implies that the creation of BEC states in the form of singular solitons (\[sing-sol\]) would require an infinite number of atoms. In optics, where the 1D version of Eq. (\[NLSE\]), with $t$ replaced by the propagation distance, $z$, governs paraxial propagation of a stationary light beam in a planar nonlinear waveguide with transverse coordinate $x$, the divergence implies an infinite power of the beam.
On the other hand, at $\nu >2$ the norm of the singular solution ([sing-sol]{}) converges, the exact result being$$N_{\sinh }=\frac{\left( \nu +1\right) ^{1/\nu }}{\sqrt{2\pi }\nu }\Gamma
\left( \frac{1}{2}-\frac{1}{\nu }\right) \Gamma \left( \frac{1}{\nu }\right)
\left( -\mu \right) ^{-\left( 1/2-1/\nu \right) }, \label{Nsinh}$$where $\Gamma $ is the Euler’s Gamma-function. Note that, in the limit of $%
\mu \rightarrow -0$, solution (\[sing-sol\]) takes the form of$$\psi _{\mu =0}=\left( \frac{\sqrt{\nu +1}}{\sqrt{2}\nu |x|}\right) ^{1/\nu },
\label{mu=0}$$whose norm diverges at all values of $\nu ,$ in agreement with Eq. ([Nsinh]{}).
In the $D$-dimensional case, a known mathematical result is that Eq. ([NLSE]{}) with self-repulsion ($\sigma =1$) [@Veron] and attraction ($%
\sigma =-1$) [@Lions; @Gidas] gives rise to solutions with singular density at $r\rightarrow 0$:$$|\psi |^{2}\approx \left( \sigma \frac{1+\left( 2-D\right) \nu }{2\nu ^{2}}%
\right) ^{1/\nu }r^{-2/\nu }. \label{singular}$$At $D=1$ and $2$, the singular asymptotic form (\[singular\]) makes sense for $\sigma =1$ and all values of $\nu $, while it is irrelevant for $\sigma
=-1$. In particular, the singularity of exact solution (\[sing-sol\]) agrees with Eq. (\[singular\]), for $\sigma =1$ and $D=1$. The 3D version of Eq. (\[singular\]), $|\psi |^{2}\approx \left( \sigma \left( 1-\nu
\right) /\left( 2\nu ^{2}\right) \right) ^{1/\nu }r^{-2/\nu }$, with $\sigma
=1$ and $-1$, admits values, severally, $\nu <1$ and $\nu >1$, and it vanishes in the fundamentally important case of the cubic nonlinearity, $\nu
=1$. In this case, a more accurate consideration yields [@HS]$$\left\vert \psi _{\mathrm{3D}}^{\mathrm{(cubic)}}\right\vert ^{2}\approx
\frac{1}{4r^{2}\left\vert \ln \left( r_{0}/r\right) \right\vert },
\label{3D}$$where $r_{0}$ is an arbitrary radial scale (in terms of the asymptotic approximation), and appropriate regions are $r\ll r_{0}$ and $r\gg r_{0}$ in the cases of $\sigma =1$ and $-1$, respectively, i.e., $r_{0}$ should be chosen as a large or small scale in these two cases. The derivation of asymptotic expression (\[3D\]) is briefly presented below, see Eqs. ([V]{}) - (\[VVVV\]).
The singularity produced by Eq. (\[singular\]) is physically admissible, producing a convergent norm, for $\nu >1$ at $D=2$, and for $\nu >2/3$ at $%
D=3$. Thus, in the 3D case only the interval of $$2/3<\nu \leq 1 \label{2/3}$$is a physically relevant one for NLSE (\[NLSE\]) with the self-defocusing nonlinearity. If the norm converges, Eq. (\[NLSE\]) implies that it is related to the chemical potential by the following scaling formula:$$N\sim \left( -\mu \right) ^{(1/\nu )-(D/2)}. \label{scaling}$$It is worthy to note that, at $\nu >2/D$, relation (\[scaling\]) satisfies the *anti-Vakhitov-Kolokolov (anti-VK) criterion*, $dN/d\mu >0$, which is a necessary stability condition for a family of bound states maintained by any self-defocusing nonlinearity [@anti] (the VK criterion proper, $dN/d\mu <0$, is necessary for stability of soliton families created by self-focusing nonlinearity [@VK; @Berge]). In particular, the entire 3D existence interval (\[2/3\]) is compatible with the anti-VK condition, which suggests, in particular, that 3D solutions generated by the cubic self-repulsion may be stable, which is indeed true (see Section 3 below).
The 3D equation (\[NLSE\]) with $\nu >1$ and self-focusing sign of the nonlinearity produces the same scaling relation (\[scaling\]). However, it is expected that the respective solution families are unstable, as the relation *does not* satisfy the VK criterion.
### Solitons pinned to a singular potential
In the 1D setting, singular solitons may be regularized, leading to localized states with a finite norm, if a delta-functional attractive potential with strength $\varepsilon >0$ is added to Eq. (\[NLSE\]):$$i\psi _{t}+(1/2)\psi _{xx}-\left\vert \psi \right\vert ^{2\nu }\psi
=-\varepsilon \delta (x)\psi . \label{eps}$$An exact solution to Eq. (\[eps\]) is$$\psi =\exp \left( -i\mu t\right) \left( \frac{\sqrt{-\left( \nu +1\right)
\mu }}{\mathrm{\sinh }\left( \sqrt{-2\mu }\nu \left( |x|+\xi \right) \right)
}\right) ^{1/\nu }, \label{regularized}$$with shift $\xi $, which removes the singularity in solution ([regularized]{}), defined by relation$$\tanh \left( \sqrt{-2\mu }\nu \xi \right) =\sqrt{-2\mu }/\varepsilon .
\label{tanh}$$The finite amplitude of the regularized solution (\[regularized\]) is$$A\equiv \left\vert \psi (x=0)\right\vert =\left[ (1/2)\left( \nu +1\right)
\left( \varepsilon ^{2}+2\mu \right) \right] ^{1/\left( 2\nu \right) }.
\label{A}$$As it follows from Eq. (\[regularized\]), such modes, pinned to the delta-functional attractive potential, exist in a finite interval of negative values of the chemical potential:$$0<-\mu <\varepsilon ^{2}/2. \label{interval}$$Exact solutions for solitons pinned to the same potential, but produced by Eq. (\[eps\]) with the opposite (self-focusing) sign in front of the nonlinear term, were recently considered in Ref. [@Zhenya].
For the exact solution given by Eqs. (\[regularized\]) and (\[tanh\]), it is easy to calculate the norm in the case of the cubic nonlinearity, $\nu
=1$:$$N=2\left( \varepsilon -\sqrt{-2\mu }\right) . \label{Nsimple}$$Note that the $N(\mu )$ dependence given by Eq. (\[Nsimple\]) obviously satisfies the anti-VK criterion.
It is relevant to mention that the attractive delta-functional potential (generally, a complex one, which includes a local gain) may also produce stable dissipative solitons (in particular, exact ones) in the framework of the 1D complex Ginzburg-Landau equation, i.e., NLSE with complex coefficients in front of the second-derivative and nonlinear terms [HK1,HK2,Zezy,JOSAB]{}.
### New results included in the review
Recent work [@HS] has reported results for families of stable 1D, 2D, and 3D singular solitons produced by Eq. (\[NLSE\]) with, respectively, *septimal* (seventh-order, corresponding to $\nu =3$), quintic, and cubic self-repulsive nonlinearities. The results, which are directly relevant to the topic of the present review, are summarized in Section 3. In the same section we explain that all the relevant nonlinear terms, including the seemingly “exotic" septimal one, naturally occur in physical media (chiefly, in optics). An essential conclusion is similar to that presented in Section 2 for the singular states pulled to the attractive center: the repulsive nonlinearity helps to create stable singular GSs with finite norms. On the other hand, 2D states with embedded vorticity can be found in the model with the attractive sign of the quintic nonlinearity, but they are completely unstable.
Two-dimensional singular modes in the attractive potential, stabilized by the Lee-Huang-Yang (LHY) term
=======================================================================================================
Analytical approximations
--------------------------
### The asymptotic form of the solutions at $r\rightarrow 0$ and $%
r\rightarrow \infty $
Stationary solutions to Eq. (\[psi2d\]) with integer vorticity $%
l=0,1,2,... $, are looked for as$$\psi \left( r,t\right) =\exp \left( -i\mu t+il\theta \right) u(r),
\label{psichi2D}$$with real radial function obeying the equation$$\mu u=-\frac{1}{2}\left( \frac{d^{2}u}{dr^{2}}+\frac{1}{r}\frac{du}{dr}+%
\frac{U_{l}}{r^{2}}u\right) +\sigma u^{3}+u^{4}, \label{chi2D}$$$$U_{l}\equiv U_{0}-l^{2}. \label{Ul}$$At $r\rightarrow 0$, an asymptotic expansion of a relevant solution of Eq. (\[chi2D\]) is $$u=\left[ \frac{1}{2}\left( U_{l}+\frac{4}{9}\right) \right]
^{1/3}r^{-2/3}-\sigma \frac{9U_{l}+4}{27U_{l}+16}+O\left( r^{2/3}\right) ,
\label{r=0-2D}$$cf. Eq. (\[transition\]). Obviously, the density singularity corresponding to this asymptotic solution, $u^{2}\sim r^{-4/3}$, is weak enough to make the 2D integral norm (\[N\]) convergent at $r\rightarrow 0$.
Asymptotic expression (\[r=0-2D\]) suggests substitution $$u(r)\equiv r^{-2/3}\chi (r) \label{uchi}$$in Eq. (\[chi2D\]), from which one can derive an equation for the singularity-free radial functions $\chi (r)$: $$\mu \chi =-\frac{1}{2}\left[ \frac{d^{2}\chi }{dr^{2}}-\frac{1}{3r}\frac{%
d\chi }{dr}+\frac{\left( U_{l}+4/9\right) }{r^{2}}\chi \right] +\sigma \frac{%
\chi ^{3}}{r^{4/3}}+\frac{\chi ^{4}}{r^{2}}. \label{chi}$$Accordingly, the asymptotic form (\[chi\]) of the solution at $%
r\rightarrow 0$ is replaced by a singularity-free expansion,** **$$\chi =\left[ \frac{1}{2}\left( U_{l}+\frac{4}{9}\right) \right]
^{1/3}-\sigma \frac{9U_{l}+4}{27U_{l}+16}r^{2/3}+O\left( r^{4/3}\right) .
\label{r=0-2D_chi}$$
Usually, the presence of integer vorticity $l\geq 1$ implies that the amplitude vanishes at $r\rightarrow 0$ as $r^{l}$, which is necessary because the phase of the vortex field is not defined at $r=0$. However, the indefiniteness of the phase is also compatible with the amplitude diverging at $r\rightarrow 0$. In the linear equation, this is the divergence of the standard Neumann’s cylindrical function, $Y_{l}(r)\sim r^{-l}$, which makes the respective 2D state unnormalizable for all values $l\geq 1$. In the present case, Eq. (\[r=0-2D\]) demonstrates that the interplay of the attractive potential and quartic self-repulsion curtails the divergence to the level of $u\sim r^{-2/3}$, for all values of $l$, thus securing the normalizability of all the states under the consideration. This conclusion may be compared to what was found in Ref. [@HS1], where the quintic repulsive term produced another integrable singularity of the 2D density, with $u(r)\sim r^{-1/2}$.
The asymptotic form of the solution, given by Eq. (\[r=0-2D\_chi\]) is meaningful if it yields $\chi (r)>0$ \[otherwise, Eq. (\[chi2D\]) cannot be derived from Eq. (\[psi2d\])\], i.e., for $U_{l}>0$, as well as for $%
-4/9<U_{l}<0$. The latter interval implies, according to Eq. (\[Ul\]),$$l^{2}-4/9<U_{0}<l^{2}. \label{4/9}$$For the vortex states, with $l\geq 1$, condition (\[4/9\]) means, in any case, $U_{0}>0$. However for the GS with $l=0$, Eq. (\[4/9\]) admits an interval of *negative* values of $U_{0}$, namely, $$0<-U_{0}<4/9. \label{<4/9}$$While the existence of the bound state under the combined action of the repulsive potential, with $U_{0}<0$, and the defocusing quartic nonlinearity is a counter-intuitive finding, it is closely related to the above-mentioned fact, first reported in Ref. [@Veron], that the 2D equation with the self-defocusing nonlinearity acting in the free space (without any potential) admits singular solutions with asymptotic form (\[singular\]). If potential (\[U(r)\]) is added to the 2D version of Eq. (\[NLSE\]), expression (\[singular\]) is replaced by$$\left\vert \psi (r)\right\vert ^{2}\approx \left[ \frac{1}{2}\left( \frac{1}{%
\nu ^{2}}+U_{0}\right) \right] ^{1/\nu }r^{-2/\nu }, \label{p}$$which is tantamount to Eq. (\[r=0-2D\]) in the case of $\nu =3/2$.
In the limit of $r\rightarrow \infty $, the asymptotic form of the solution to Eq. (\[chi\]) is$$\chi (r)\approx \chi _{0}r^{1/6}\exp \left( -\sqrt{-2\mu }r\right) ,
\label{chi0}$$where $\chi _{0}$ is an arbitrary constant, and $\mu $ must be negative. Then, a coarse approximation for the global solution can be obtained as an interpolation bridging asymptotic expressions (\[r=0-2D\_chi\]) and ([chi0]{}):$$\chi _{\mathrm{interpol}}(r)=\left[ \frac{1}{2}\left( U_{l}+\frac{4}{9}%
\right) \right] ^{1/3}\exp \left( -\sqrt{-2\mu }r\right) . \label{inter}$$As mentioned above, families of localized states are usually characterized by dependences $N(\mu )$. In particular, calculating norm (\[N\]) for the approximate solution given by Eq. (\[inter\]), one obtains$$N_{\mathrm{interpol}}(\mu )=2\pi \int_{0}^{\infty }u_{\mathrm{interpol}%
}^{2}(r)rdr=\pi \Gamma \left( \frac{2}{3}\right) \frac{\left(
U_{l}+4/9\right) ^{2/3}}{\left( -4\mu \right) ^{1/3}}, \label{Ninter}$$where $\Gamma (2/3)\approx \allowbreak 1.\allowbreak 354$ is the value of the Gamma-function. Note that this dependence satisfies the anti-VK criterion.
### The Thomas-Fermi (TF) approximation
The TF approximation, which, strictly speaking, applies under condition $%
U_{0}\gg 1$ (irrespective of the value of $|\mu |$) amounts to dropping derivatives in Eq. (\[chi\]). In fact, the TF approximation may produce relevant results even when $U_{0}$ is not especially large, see below. It produces an explicit approximate solution in the case of $\sigma =0$ in Eq. (\[chi\]) (with the nonlinearity represented solely by the LHY term):$$\chi _{\mathrm{TF}}(r)=\left\{
\begin{array}{c}
\left[ \left( U_{l}+4/9\right) /2-|\mu |r^{2}\right] ^{1/3},~\mathrm{at}%
~~r<r_{0}\equiv \sqrt{\left( U_{l}+4/9\right) /\left( 2|\mu |\right) }, \\
0,~\mathrm{at}~~r>r_{0}~,%
\end{array}%
\right. \label{TF}$$for $\mu <0$. In the limit of $r\rightarrow 0$, Eq. (\[TF\]) yields the same exact value, $\chi (r=0)=\left[ \left( U_{l}+4/9\right) /2\right]
^{1/3} $, as given by Eq. (\[r=0-2D\]). On the other hand, an essential difference from the full solution is that the TF approximation predicts a finite radius $r_{0}$ of the GS, neglecting the exponentially decaying tail at $r\rightarrow \infty $, cf. Eq. (\[chi0\]). Further, the TF approximation (\[TF\]) makes it possible to calculate the corresponding $%
N(\mu )$ dependence for the GS family:$$N_{\mathrm{TF}}^{(\sigma =0)}(\mu )=2\pi \int_{0}^{r_{0}}\left[ r^{-2/3}\chi
_{\mathrm{TF}}(r)\right] ^{2}rdr=C\frac{U_{l}+4/9}{\left( -\mu \right) ^{1/3}%
}, \label{NTF}$$where a numerical constant is $C\equiv \pi \int_{0}^{1}\left(
x^{-2}-1\right) ^{2/3}xdx\approx 3.80$, cf. Eq. (\[Ninter\]).
Note that TF radius $r_{0}$ keeps the same value, as given by Eq. (\[TF\]), in the presence of the MF defocusing cubic term with $\sigma =1$ in Eq. (\[chi2D\]), although the shape of the GS is more complex than one given by Eq. (\[TF\]) for $\sigma =0$. In this case, the asymptotic limit of the respective $N_{\mathrm{TF}}^{(\sigma =1)}(\mu )$ dependence at $\mu
\rightarrow -\infty $ is the same as given by Eq. (\[NTF\]), while in the limit of $\mu \rightarrow -0$ it features a weaker singularity:$$N_{\mathrm{TF}}^{(\sigma =1)}(\mu )\approx \left( \pi /2\right) U_{l}\ln
\left( 1/|\mu |\right) .$$
Even in the case of the focusing sign of the MF term, corresponding to $%
\sigma =-1$ in Eq. (\[chi2D\]), the LHY-induced quartic nonlinearity is able to stabilize the condensate against the combined action of the MF self-attraction and pull to the center. In this case, the TF approximation, applied to Eq. (\[chi2D\]), cannot be easily resolved to predict $u_{%
\mathrm{TF}}(r)$, but it readily produces an inverse dependence, for $r$ as a function of $u$:$$r^{2}=\left( U_{l}/2\right) \left( -\mu -u^{2}+u^{3}\right) ^{-1}. \label{r}$$Then, looking for a maximum of expression (\[r\]), which is attained at $%
u_{\max }=2/3$, it is easy to find the corresponding size of the TF state: $$r_{0}^{(\sigma =-1)}=\frac{U_{l}}{2\left( |\mu |-4/27\right) }. \label{-1}$$Equation (\[-1\]) suggests that the GS exists, in the case of $\sigma =-1$, for $|\mu |$ exceeding a threshold value, $$|\mu |>\left( |\mu |\right) _{\mathrm{thr}}=4/27. \label{thr}$$According to Eq. (\[-1\]), the norm diverges at $\mu \rightarrow -4/27$ as$$N\approx 2\pi \left( r_{0}^{(\sigma =-1)}\right) ^{2}u_{\max }^{2}=\frac{%
2\pi }{9}\frac{U_{l}^{2}}{\left( |\mu |-4/27\right) ^{2}}. \label{NN}$$
The analytical predictions reported in this subsection are compared to their numerical counterparts in the following one.
Numerical results for the 2D modes stabilized by the LHY term
-------------------------------------------------------------
Stationary solutions of Eq. (\[chi\]) could be readily produced by means of the Newton’s iteration method. Then, stability of stationary solutions was identified by solution of the linearized eigenvalue problem for small perturbations, represented by terms $$\delta \psi \sim \exp \left( -i\mu t+\Lambda t\right) ,~\exp \left( -i\mu
t+\Lambda ^{\ast }t\right) \label{pert}$$added to the stationary states ($\ast $ stands for the complex conjugation), the stability condition being, as usual [@Yang], that all eigenvalues $%
\Lambda $ must be pure-imaginary. The so predicted (in)stability was verified by direct simulations of Eq. (\[psi2d\]). The results were produced for the model including the cubic term in Eq. (\[psi2d\]), with $%
\sigma =\pm 1$, as well as for the most fundamental case of $\sigma =0$, when the nonlinearity is represented solely by the LHY term.
A typical stable GS, obtained as a numerical solution of Eq. (\[chi\]) with $\sigma =0$, $U_{0}=10$, and $\mu =-1$, is displayed in Fig. \[fig1a\], together with its counterpart produced by the TF approximation (the interpolation-based approximation, given by Eq. (\[inter\]), is not displayed here, as it is not relevant for large values of $U_{0}$). It is seen that the TF approximation is very close to its numerical counterpart in the inner zone, $r<r_{0}$ (see Eq. (\[TF\])), while in the outer one, $%
r>r_{0}$, the TF approximation yields zero, being inaccurate, in this sense. Actually, the overall impact of the difference between the numerical and TF solutions in the outer zone is less significant due to the presence of factor $r^{-2/3}$ in expression (\[uchi\]) for the full solution, $u(r)$. As a result, in the case presented in Fig. \[fig1a\] the relative error of the norm of the TF solution, calculated as per Eq. (\[NTF\]), is $\approx
0.03$ .
The effect of the MF cubic term of either sign, repulsive ($\sigma =1$) or attractive ($\sigma =-1$), in comparison with the case of $\sigma =0$, on the GS is shown in Fig. \[fig2a\]. At $r=0$, all the three shapes converge to a common value, $\chi (r=0)\approx 1.20$, in agreement with Eq. ([r=0-2D\_chi]{}).
[![Numerically generated stable GS solutions of Eq. (\[chi\]) for $U_{0}=3,\protect\mu =-0.8$ , and $\protect\sigma =1$, $0$, and $-1$. The respective norms are $N(\protect\sigma =1)=11.7$, $N(\protect\sigma %
=0)=15.41$ and $N(\protect\sigma =-1)=24.62$.[]{data-label="fig2a"}](fig2a.pdf "fig:"){width="3.2in"}]{}
The counter-intuitive prediction of the existence of the GS in the presence of the *repulsive* potential in Eq. (\[chi\]), with $U_{0}$ belonging to interval (\[<4/9\]), was also confirmed by the numerical results. Figure \[fig3a\] displays numerically found GSs for $U_{0}=-0.40$, taken close to the limit value, $U_{0}=-4/9\approx \allowbreak -0.44$, for all the three values of the coefficient in front of the MF cubic term in Eq. ([chi]{}), *viz*., $\sigma =0$ and $\pm 1$. The same figure shows that the interpolating approximation for these solutions, provided by Eq. ([inter]{}), is quite accurate in this case (the TF approximation (\[TF\]) does not apply to $U_{0}<0$). It is seen too that the MF cubic term does not strongly affect the solution.
Dependences $N(\mu )$ for families of the GS solutions, obtained from Eq. (\[chi\]) without and with the repulsive or attractive MF cubic term ($%
\sigma =0$ and $\sigma =\pm 1$, respectively), are displayed in Figs. [fig4a]{}(a,b), for different values of strength $U_{0}$ of the potential, both $U_{0}>0$ and $U_{0}<0$. In panel (a), the $N(\mu )$ curves produced by the TF approximation as per Eq. (\[NTF\]) are compared to their numerical counterparts. The same panel demonstrates that the interpolating approximation is very accurate for $U_{0}=-0.4$ (while its accuracy is poor for $U_{0}>0$).
Panel (c) in Fig. \[fig4a\] confirms that, in the presence of the attractive cubic term ($\sigma =-1$), the GS exists at $|\mu |>\left( |\mu
|\right) _{\mathrm{thr}}$, as predicted by the TF approximation in Eq. ([thr]{}). Up to the accuracy of the numerical results, the threshold value is indeed $\left( |\mu |\right) _{\mathrm{thr}}=4/27$, as given by Eq. ([thr]{}). This finding is explained by the fact that the width of the GS diverges in the limit of $|\mu |\rightarrow \left( |\mu |\right) _{\mathrm{%
thr}}$, as seen in Eq. (\[-1\]), hence in this limit the derivatives become negligible in Eq. (\[chi2D\]), making the TF approximation asymptotically exact. In principle, $N(\mu )$ must steeply diverge at $|\mu
|\rightarrow 4/27$, according to Eq. (\[NN\]), but it is difficult to collect numerical data very close to the threshold, as the GS is extremely broad in this limit.
The computation of eigenvalues for small perturbations, as well as direct simulations, demonstrate full stability of the GS solutions for both positive and negative values of $U_{0}$ (at which the GS exists), at all $%
\mu <0$. In particular, Fig. \[fig3a\](b) demonstrates the stability in the counter-intuitive case of the repulsive potential, with $U_{0}=-0.4$. The stability is not affected either by the TF cubic term, holding for $%
\sigma =0$ and $\pm 1$.
Lastly, an analytical consideration, verified by numerical data [@we], has demonstrated that vortex modes with $l=1$ (see Eq. (\[psichi2D\])), are stable at $U_0>14/9$, and unstable at $U_0<14/9$. For the vortices with $l=2$, the critical value of the pulling-potential strength, below which they are unstable, is much larger, *viz*., $U_0=77/9$. Simulations of the evolution of unstable vortex modes demonstrate that the vortical pivot drifts in the outside direction, along an unwinding spiral trajectory. Eventually, the pivot is ousted to periphery, and the vortex mode transforms into a stable GS with zero vorticity. In this case, the norm of the residual GS is essentially smaller than the original value, due to intensive emission of small-amplitude waves in the course of the transient evolution.
Singular solitons in one, two, and three dimensions
===================================================
This section summarizes original results which were recently reported in Ref. [@HS]. In the 2D case, the results are related to those presented in the previous section for the model with the external potential.
As mentioned in the Introduction, relevant self-defocusing nonlinearities which are necessary to support singular self-trapped states in 3D, 2D, and 1D settings, are represented by the cubic, quintic, or septimal terms, that correspond, respectively, to $\nu =1$, $2$, and $3$ in Eq. (\[NLSE\]). Realizations of the cubic and quintic nonlinearities of either sign (focusing or defocusing) in diverse physical media are well known [Michinel,Anger,review,Brazil]{}. In particular, such nonlinear terms with controllable strengths (including the “exotic" septimal term), can be realized in optical waveguides filled by suspensions of metallic nanoparticles, control parameters being the density and size of the particles [@Cid2; @Cid1; @Cid3].
Results outlined below provide not only solutions in analytical and numerical forms, but also an interpretation of the physical meaning of the singular solitons.
Analytical results
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### The one-dimensional model with the septimal nonlinearity
Singular solitons created by the 1D version of Eq. (\[NLSE\]) with the seventh-order defocusing ($\sigma =1$) nonlinearity, $\nu =3$, are looked for as $$\psi \left( x,t\right) =\exp \left( -i\mu t\right) \mathrm{U}(x), \label{uU}$$with real function $\mathrm{U}(x)$ satisfying equation$$\frac{1}{2}\frac{d^{2}\mathrm{U}}{dx^{2}}=-\mu \mathrm{U}+\mathrm{U}^{7}.
\label{U}$$In the application to the planar optical waveguide, $t$ is not time, but the propagation distance (often denoted $z$), $x$ is the transverse coordinate, and $-\mu $ is the propagation constant.
The exact solution of Eq. (\[U\]) is given by Eq. (\[sing-sol\]):$$\mathrm{U}(x)=\left( \frac{2\sqrt{-\mu }}{\mathrm{\sinh }\left( 3\sqrt{-2\mu
}|x|\right) }\right) ^{1/3}. \label{Uexact}$$The asymptotic form of solutions to Eq. (\[U\]) at $x\rightarrow 0$ does not depend on $\mu $,
$$\mathrm{U}(x)\approx \left( 2/9\right) ^{1/6}|x|^{-1/3}. \label{-1/3}$$
Note that expression (\[-1/3\]) is an exact solution of Eq. (\[U\]) with $\mu =0$, but its integral norm diverges at $|x|\rightarrow \infty $. For $%
\mu <0$, the exponentially decaying asymptotic form of Eq. (\[Uexact\]) at $|x|\rightarrow \infty $ is $$\mathrm{U}(x)\approx \left( 4\sqrt{-\mu }\right) ^{1/3}\exp \left( -\sqrt{%
-2\mu }|x|\right) . \label{tail}$$
Lastly, the norm of the 1D soliton family is given by Eq. (\[Nsinh\]),$$N=\frac{2^{2/3}}{3\sqrt{2\pi }}\Gamma \left( \frac{1}{6}\right) \Gamma
\left( \frac{1}{3}\right) \left( -\mu \right) ^{-1/6}\approx \allowbreak
3.1478\left( -\mu \right) ^{-1/6}, \label{N1D}$$with the numerical coefficient which is accidentally close to $\pi $. This $%
N(\mu )$ dependence satisfies the anti-VK criterion, which, as mentioned above, is necessary for the stability of localized modes supported by repulsive nonlinearities [@anti].
### Physical interpretation of the 1D singular soliton: screening of a “bare" $\protect\delta $-functional potential
Although the existence of the stable singular solitons under the action of the septimal self-repulsive nonlinearity is firmly established by the above analysis, this result may seem counter-intuitive, as it is commonly believed that localized modes may only be supported by self-attraction. The purport of the result may be understood by comparing Eq. (\[U\]) to a modified equation,$$\frac{1}{2}\frac{d^{2}\mathrm{U}}{dx^{2}}=-\mu \mathrm{U}+\mathrm{U}%
^{7}-\varepsilon \delta (x)\mathrm{U}, \label{eps2}$$which includes an attractive delta-functional potential with strength $%
\varepsilon >0$, cf. Eq. (\[eps\]). An exact solution to Eq. (\[eps2\]) is produced by Eqs. (\[regularized\]), (\[tanh\]), and (\[A\]):$$\mathrm{U}(x)=\left( \frac{2\sqrt{-\mu }}{\mathrm{\sinh }\left( 3\sqrt{-2\mu
}\left( |x|+\xi \right) \right) }\right) ^{1/3}, \label{U nu=3}$$$$\xi =\frac{1}{6\sqrt{-2\mu }}\ln \left( \frac{\varepsilon +\sqrt{-2\mu }}{%
\varepsilon -\sqrt{-2\mu }}\right) \approx \frac{1}{3\varepsilon }.
\label{xi nu=3}$$$$A=\left[ 2\left( \varepsilon ^{2}+2\mu \right) \right] ^{1/6} \label{A nu=3}$$(recall $A\equiv \left\vert \mathrm{U}(x=0)\right\vert $ is the amplitude of the mode pinned to the attractive delta-functional potential). The approximate value of the offset in Eq. (\[xi nu=3\]) corresponds to the limit of $\varepsilon \gg \sqrt{-2\mu }$.
This solution may be considered as a regularized version of its singular counterpart (\[-1/3\]), which obviously converges to the singular state for $\varepsilon \rightarrow \infty $. It is also relevant to calculate the value of the Hamiltonian, corresponding to Eq. (\[eps2\]),$$H_{\varepsilon }=\int_{-\infty }^{+\infty }\left[ \frac{1}{2}\left( \frac{d%
\mathrm{U}}{dx}\right) ^{2}+\frac{1}{4}|\mathrm{U}|^{8}\right]
dx-\varepsilon |\mathrm{U}(x=0)|^{2}. \label{H}$$In the limit of $\varepsilon \gg \sqrt{-2\mu }$, it is $$H_{\varepsilon }\approx -\left( 1/5\right) \left( 3\varepsilon \right)
^{5/3}, \label{Heps}$$the negative sign suggesting that solution may be the GS.
A physical interpretation of the ability of the self-repulsive 1D model to create singular solitons is offered by the fact that, according to Eq. ([xi nu=3]{}), the strength (“bare charge") $\varepsilon $ of the delta-functional attractive potential diverges in the limit of $\xi
\rightarrow 0$, which brings one back to the underlying singular solution given by Eqs. (\[-1/3\])-(\[tail\]). This observation implies that an *infinitely large* “bare attractive charge", embedded in the self-defocusing septimal medium, is *completely screened* by the nonlinearity, which builds the singular soliton with the convergent norm, for that purpose. This mechanism roughly resembles the renormalization procedure in quantum electrodynamics, where an infinite *bare charge* of the electron cancels with other diverging factors, making it possible to produce finite observable predictions.
### The two-dimensional model with the quintic nonlinearity
In the 2D setting, it is relevant to consider Eq. (\[NLSE\]) with the quintic self-defocusing term: $$i\psi _{t}=-\frac{1}{2}\nabla ^{2}\psi +|\psi |^{4}\psi . \label{u2D}$$In terms of optics, Eq. (\[u2D\]) models the paraxial propagation of light in a bulk waveguide, with $t$ being the propagation distance (usually denoted $z$).
In polar coordinates $\left( r,\theta \right) $, solutions of Eq. (\[u2D\]) with integer vorticity, $l=0,1,2,...$, is looked for as $$\psi =\exp \left( -i\mu t+il\theta \right) \mathrm{U}(r), \label{M}$$with real amplitude function $\mathrm{U}(r)$ satisfying the radial equation, cf. Eq. (\[U\]):$$\frac{1}{2}\left( \frac{d^{2}\mathrm{U}}{dr^{2}}+\frac{1}{r}\frac{d\mathrm{U}%
}{dr}-\frac{l^{2}}{r^{2}}\mathrm{U}\right) =-\mu \mathrm{U}+\mathrm{U}^{5}.
\label{U2D}$$For 2D singular solitons with $l=0$, Eq. (\[U2D\]) produces the following expansion at $r\rightarrow 0$: $$\mathrm{U}_{\mathrm{2D}}(r)\approx 2^{-3/4}r^{-1/2}-2^{1/4}\mu r^{3/2},
\label{2Drep}$$with the respective 2D integral norm converging at $r\rightarrow 0$. For $%
\mu =0$, $$\mathrm{U}_{\mathrm{2D}}^{(\mu =0)}(r)=2^{-3/4}r^{-1/2} \label{u0}$$is an exact solution of Eq. (\[U2D\]), but its norm diverges at $%
r\rightarrow \infty $. The asymptotic form of the solution at $r\rightarrow
\infty $ is found from the linearized version of Eq. (\[U2D\]),$$\mathrm{U}_{\mathrm{2D}}(r)\approx \frac{C}{\sqrt{r}}\left( 1-\frac{1}{8%
\sqrt{-2\mu }r}\right) \exp \left( -\sqrt{-2\mu }r\right) \label{2Dexp}$$\[cf. Eq. (\[tail\])\] where $C$ is a constant, and the second term in the parenthesis its a correction to the lowest approximation.
For vortex states given by Eq. (\[M\]) with $l\geq 1$, a singular solution with convergent norm is obtained with the *opposite* (self-focusing) sign in front of the quintic term in Eq. (\[u2D\]), the asymptotic approximation at $r\rightarrow 0$ being $$\mathrm{U}_{\mathrm{2D}}^{(l)}(r)\approx \left[ \frac{1}{2}\left( l^{2}-%
\frac{1}{4}\right) \right] ^{1/4}r^{-1/2}, \label{2Dattr}$$cf. Eq. (\[2Drep\]). For $\mu =0$, Eq. (\[2Dattr\]) represents an exact solution of Eq. (\[U2D\]) with the opposite (self-focusing) sign in front of the quintic term, but its integral norm diverges at $r\rightarrow \infty $.
An exact scaling relation for the norm of the solutions, with $l=0$ and $%
l\geq 1$ alike, follows from Eq. (\[u2D\]) with either sign of the quintic term: $$N_{\mathrm{2D}}(\mu )=\mathrm{const}/\sqrt{-\mu } \label{N2D}$$(for $l=0$, a numerically found value is $\mathrm{const}\approx 21.2$). This dependence satisfies the anti-VK criterion, $dN/d\mu >0$ hence the respective GS solutions with $l=0$, maintained by the defocusing quintic nonlinearity, may be (and indeed are) stable, but it contradicts the VK condition per se, $dN/d\mu <0$, hence the vortex states, which exist in the case of self-focusing, are definitely unstable, as corroborated by numerical simulations [@HS].
### Interpretation of the 2D singular soliton: screening of a ring-shaped attractive potential
Similar to what is outlined above for the 1D model, it is possible to introduce a version of Eq. (\[u2D\]) with a delta-functional potential; however, it is concentrated on a ring of a small radius, $\rho $, instead of the single point. The so modified equation (\[U2D\]) with $l=0$ is$$\frac{1}{2}\left( \frac{d^{2}\mathrm{U}}{dr^{2}}+\frac{1}{r}\frac{d\mathrm{U}%
}{dr}\right) =-\mu \mathrm{U}+\mathrm{U}^{5}-\varepsilon _{\mathrm{2D}%
}\delta (r-\rho )\mathrm{U}=0, \label{2Deps}$$cf. Eq. (\[eps2\]). At $r>\rho $ the solution of Eq. (\[2Deps\]) is sought for in a form similar to that given by Eq. (\[2Drep\]), while inside the ring it is taken as $\mathrm{U}=\mathrm{const}\approx
2^{-3/4}\rho ^{-1/2}$. Eventually, straightforward manipulations, which take into regard the jump of $d\mathrm{U}/dr$ at $r=\rho $, yield a relation between the strength of the delta-functional potential concentrated on the ring and the ring’s radius: $$\varepsilon _{\mathrm{2D}}=1/\left( 4\rho \right) , \label{eps2D}$$cf. Eq. (\[xi nu=3\]). Then, in the limit of $\rho \rightarrow 0$, an effective “charge" of the 2D attractive potential is$$Q_{\mathrm{2D}}=2\pi \rho \cdot \varepsilon _{\mathrm{2D}}=\frac{\pi }{2}.
\label{Q2D}$$Thus, the 2D singular soliton may be construed as a solution providing the screening of the finite “charge" by the defocusing quintic nonlinearity.
### Effects of additional nonlinear terms on 1D and 2D singular solitons
Even if the septimal or quintic term represents the dominant nonlinearity in the underlying NLSE, the lower-order terms, i.e., cubic and/or quintic ones (with respective coefficients $g_{3}$ and $g_{5}$), should be included in the realistic model of the light propagation [@Cid2; @Cid1; @Cid3]. In particular, the accordingly amended septimal 1D equation (\[U\]) is replaced by $$\frac{1}{2}\frac{d^{2}\mathrm{U}}{dx^{2}}=-\mu \mathrm{U}+g_{3}\mathrm{U}%
^{3}+g_{5}\mathrm{U}^{5}+\mathrm{U}^{7}. \label{1D amended}$$The additional terms produce negligible corrections to the asymptotic singular form (\[-1/3\]) at $x\rightarrow 0$ [@HS]:$$\mathrm{\delta U}_{\mathrm{1D}}(x)\approx -\left( \frac{32}{81}\right) ^{1/6}%
\frac{g_{5}}{5}|x|^{1/3}-\frac{3g_{3}}{7\sqrt{2}}|x|. \label{delta1D}$$Further, if, in the 2D setting, the cubic term (the same as in Eq. (\[1D amended\])) is added to Eq. (\[U2D\]), it also produces a negligible correction to the singular asymptotic form given by Eq. (\[2Drep\]) [HS]{}: $\mathrm{\delta U}_{\mathrm{2D}}(r)\approx -2^{-5/4}g_{3}r^{1/2}$, cf. Eq. (\[delta1D\]).
### The 3D model with the cubic nonlinearity
In the 3D case, the relevant equation is the NLSE with the usual cubic term: $$i\psi _{t}=-\frac{1}{2}\nabla ^{2}\psi +\sigma |\psi |^{2}\psi . \label{u3D}$$It has a plethora of physical realizations for both the defocusing ($\sigma
=1$) and focusing ($\sigma =-1$) signs of the nonlinearity [@review], including the GPE for BEC with, respectively, repulsive or attractive interactions between atoms [@GP]. For isotropic stationary states, $\psi
=\exp (-i\mu t)\mathrm{U}(r)$, where $r$ is the 3D radial coordinate, the real radial functions obeys the commonly known equation,$$\frac{1}{2}\left( \frac{d^{2}\mathrm{U}}{dr^{2}}+\frac{2}{r}\frac{d\mathrm{U}%
}{dr}\right) -\sigma \mathrm{U}^{3}=-\mu \mathrm{U}. \label{radial}$$For the asymptotic consideration of singular solutions at $r\rightarrow 0$, the term on the right-hand side of Eq. (\[radial\]) is negligible. Dropping this term and looking for solutions in the form of$$\mathrm{U}(r)=r^{-1}\mathrm{V}\left( \tau \equiv -\ln (r/r_{0})\right) ,
\label{V}$$where $r_{0}$ is an arbitrary radial scale, one can transform Eq. ([radial]{}) into the following form (which is an exact transformation for $\mu
=0)$:$$\frac{d^{2}\mathrm{V}}{d\tau ^{2}}=-\frac{d\mathrm{V}}{d\tau }+2\sigma
\mathrm{V}^{3}. \label{VV}$$Formally, Eq. (\[VV\]) is tantamount to the equation of motion of a mechanical unit-mass particle with coordinate $\mathrm{V}$ and time $\tau $ in the normal ($\sigma =-1$) or inverted ($\sigma =1$) quartic potential, $W(%
\mathrm{V})=-\left( \sigma /2\right) \mathrm{V}^{4}$, under the action of the friction force with coefficient $1$.
For $\sigma =-1$ (the focusing nonlinearity), an appropriate asymptotic solution to Eq. (\[VV\]) is one dominated by the balance of the potential and friction forces at $\tau \rightarrow +\infty $, which is relevant for $%
r\ll r_{0}$, according to the definition of $\tau $ in Eq. (\[V\]): $$\mathrm{V}\approx (1/2)\tau ^{-1/2}. \label{VVV}$$For $\sigma =1$ (defocusing), the appropriate solution to Eq. (\[VV\]) is also determined by the balance of the potential force and friction, the relevant region being $\tau \rightarrow -\infty $, i.e., $r\gg r_{0}$:$$\mathrm{V}\approx (1/2)\left( -\tau \right) ^{-1/2}. \label{VVVV}$$Both solutions (\[VVV\]) and (\[VVVV\]) eventually translate into the asymptotic expression for the 3D density given above by Eq. (\[3D\]).
In the opposite limit of $r\rightarrow \infty $, a straightforward consideration yields an asymptotic expression for the solution in the form of$$\mathrm{U}_{\mathrm{3D}}(r)\approx Cr^{-1}\exp \left( -\sqrt{-2\mu }r\right)
, \label{3Dexp}$$with constant $C$, cf. Eq. (\[2Dexp\]).
An exact scaling relation between the 3D norm and chemical potential, as it follows from the cubic NLSE (\[u3D\]), is the same as its counterpart ([N2D]{}) in the 2D model with the quintic nonlinearity:$$N_{\mathrm{3D}}(\mu )\equiv 4\pi \int_{0}^{\infty }\mathrm{U}^{2}(r)r^{2}dr=%
\mathrm{const}\cdot \left( -\mu \right) ^{-1/2}. \label{N3D}$$This relation satisfies the anti-VK criterion, hence the family of the 3D singular solitons may be (and indeed is) stable in the case of the defocusing cubic nonlinearity, $\sigma =1$.
### Interpretation of the 3D singular solitons: screening of an attractive spherical potential
Similar to what is presented above for the 2D model, one can augment the 3D model by the attractive delta-functional potential concentrated on a sphere of radius $\rho $, the respective stationary equation being (cf. Eq. ([2Deps]{}))$$\frac{1}{2}\left( \frac{d^{2}\mathrm{U}}{dr^{2}}+\frac{2}{r}\frac{d\mathrm{U}%
}{dr}\right) =-\mu \mathrm{U}+\mathrm{U}^{3}-\varepsilon _{\mathrm{3D}%
}\delta (r-\rho )\mathrm{U}, \label{3Deps}$$where it is assumed that, although $\rho $ is small, it must be larger than $%
r_{0}$ in Eq. (\[V\]). A stationary solution to Eq. (\[3Deps\]) is sought for in the form given by Eqs. (\[V\]) and (\[VVVV\]) at $r>\rho $, and as $\mathrm{U}=\mathrm{const}\approx \left( 2\rho \sqrt{\ln \left(
\rho /r_{0}\right) }\right) ^{-1}~~$at$~~r<\rho $. Then, calculations similar to those outlined above in the 2D setting lead to a relation between $\rho $ and the strength of the attractive potential, $\varepsilon _{\mathrm{%
3D}}=1/\left( 2\rho \right) $, cf. Eq. (\[eps2D\]). In the limit of $\rho
\rightarrow 0$ (which is imposed along with $r_{0}\rightarrow 0$, so as to keep condition $\rho >r_{0}$ valid), the respective charge" of the 3D attractive potential is $Q_{\mathrm{3D}}=4\pi \rho
^{2}\cdot \varepsilon _{\mathrm{3D}}=2\pi \rho \rightarrow 0$, cf. Eq. ([Q2D]{}) Thus, one may realize the 3D singular soliton as a state which provides the screening of the vanishingly small “charge".
Numerical results for the 1D, 2D, and 3D singular solitons
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The numerical scheme for producing singular solitons as solutions of the NLSEs with the self-repulsive nonlinearity must be adjusted to the fact that, in the analytical form, the solutions take infinite values at the origin. In Ref. [@HS], a finite-difference scheme was used for this purpose. It was defined on a grid with spacing $\Delta $, constructed so that closest to the origin were points with coordinates $$\left( x,y,z\right) =\left( \pm \Delta /2,\pm \Delta /2,\pm \Delta /2\right)
\label{Delta}$$in the 3D case, and similarly in 1D and 2D. At these points, boundary conditions with large but finite values of $|\psi |$ were fixed according to the asymptotically exact analytical expressions (\[-1/3\]), (\[2Drep\]), and ( \[3D\]).
A typical shape of the 1D singular soliton, produced by exact solution ([Uexact]{}) with $\mu =-1$, is plotted in Fig. \[fig1b\] with stepsize $%
\Delta x=10^{-5}$. In agreement with the fact that the entire family of the 1D singular solitons satisfies the anti-VK criterion, numerical results corroborate the full stability of the family.
![The shape of the singular 1D soliton, as given by Eq. ([Uexact]{}) with $\protect\mu =-1$.[]{data-label="fig1b"}](fig1b.pdf){height="5.5cm"}
In the 2D setting, numerical solution of the quintic equation (\[u2D\]) has corroborated the existence and full stability of the singular solitons with zero vorticity ($l=0$). As an illustration, Fig. \[fig3b\] displays a global view of the stable 2D soliton produced by direct simulations of Eq. (\[u2D\]), starting from the initial conditions taken as per exact solution (\[u0\]) corresponding to $\mu =0$ (the formal divergence of its integral norm at $r\rightarrow \infty $ is restricted by the finite size of the integration domain). In particular, he numerical simulations confirm the stability of the 2D singular solitons against azimuthal perturbations which attempt to break the axial symmetry of the solitons.
![A stable 2D singular soliton produced by simulation of the full 2D equation (\[u2D\]), starting from input (\[u0\]). []{data-label="fig3b"}](fig3b.pdf){height="7.0cm"}
In the 3D setting, numerical solution of Eq. (\[u3D\]) produces a family of 3D singular solitons. Comparison of these solutions with the analytical prediction, given by Eqs. (\[V\]) and (\[VVVV\]) for small $r$, is not straightforward, as the analytical expression contains indefinite parameter $%
r_{0}$. Therefore, an example, displayed in Fig. \[fig4b\](a) presents the comparison of the numerical solution to analytical profile $\mathrm{const}%
\cdot r^{-1}$ at small $r$, with the constant selected as the best-fit parameter ($\mathrm{const}=0.001$ for the case of $\mu =-1$, displayed in Fig. \[fig4b\](a)).
Further, one can try to use the asymptotic form of the 3D solution at large $%
r$, given by Eq. (\[3Dexp\]), with $C$ set equal to the best-fit value of $%
\mathrm{const}$ selected at $r\rightarrow 0$, as a global analytical approximation. Figure \[fig4b\](a) demonstrates that such an approximation is extremely close to its numerical counterpart at all values of $r$.
Finally, direct simulations of the evolution of the 3D singular solitons confirm stability of the soliton family, see an example in Fig. \[fig4b\](b).
![(a) The comparison, on the double-logarithmic scale, of the numerically found stationary solution of Eq. (\[u3D\]) for the 3D singular soliton with $\protect\mu =-1$ (the continuous line) with profile $%
0.001\cdot r^{-1}$ relevant at small $r$ (the long-dashed line), and the global approximation provided by Eq. (\[3Dexp\]) with $C=0.001$ (the short-dashed line), see explanation in the text. The difference between the latter analytical approximation and the numerical solution is barely visible. (b) The radial cross section of the 3D singular soliton, produced at $t=2$ by direct simulations of Eq. (\[u3D\]), with the input taken from panel (a). The result confirms stability of the 3D singular soliton. In this figure, labels for $|\protect\psi |$ and $\mathrm{U}_{%
\mathrm{3D}}$ are replaced by $|u|$.[]{data-label="fig4b"}](fig4b.pdf){height="7.0cm"}
Conclusion
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The aim of this paper is to produce a brief review of findings recently reported in works [@HS] and [@we], that demonstrate the existence of several types of singular bound states in 3D, 2D, and 1D models with self-repulsive nonlinearities. These states feature a singular structure of the density at $r\rightarrow 0$, while the total norm converges, making the states physically relevant solutions. One model combines the attractive potential $\sim -1/r^{2}$ in the 2D space and the LHY (quartic) self-repulsive term. The respective GPE (Gross-Pitaevskii equation) is a model of BEC composed of particles carrying permanent electric dipole moments, which are pulled to the central charge. Previously, it was found, in the framework of the MF and many-body quantum settings alike [HS1,Gregory]{}, that the quantum collapse, driven by this attractive potential, can be effectively suppressed in the 3D space by the usual cubic nonlinearity (which represents, as usual, repulsive interactions of colliding particles), while the same cubic nonlinearity cannot stabilize the 2D condensate. A brief review of those results was presented in Ref. [CM]{}.
The new result, reported in Ref. [@we] and summarized in Section 2 of the present review, is that the quartic self-repulsive term is sufficient to suppress the 2D quantum collapse and restore the missing GS (ground state), which is a completely stable one. States with embedded angular momentum (vorticity) are constructed too. They are stable if the strength of the attractive potential exceeds a certain threshold value, which depends on the vorticity. An essential peculiarity is that stable 2D GS modes exist, counter-intuitively, even when the attractive central potential is replaced by the repulsive one, with a sufficiently small strength, as defined by Eq. (\[<4/9\]). The latter feature is closely related to another topic relevant to the consideration of singular states, which was elaborated in Ref. [@HS] and is summarized in Section 3 of the review. It deals with NLSEs (nonlinear Schrödinger equations) in 1D, 2D, and 3D free space, which contain the septimal, quintic, and cubic repulsive terms, respectively. These equations demonstrate the existence of stable singular solitons, that realize the model’s GS in all the cases, with the 1D solitons found in an exact analytical form. The physical interpretation of these counter-intuitive findings is provided by the possibility to construe the singular solitons as a result of screening, by the respective self-repulsive nonlinearity, of a delta-functional attractive potential, whose integral strength is divergent in 1D, finite in 2D, and vanishingly small in 3D.
A common feature of both models considered in this paper is that, on the contrary to the stable zero-vorticity GSs, two-dimensional vortex states are completely unstable. Therefore, a challenging problem for further work is search for physically relevant modifications of the 2D models which may allow the existence of stable vortices. A still more challenging objective is to construct 3D states with embedded vorticity and investigate their stability. Another subject for further work may be singular dissipative solitons in complex Ginzburg-Landau equations [@HS]. Quite challenging may also be extension of the analysis for singular modes (if any) in models of Fermi gases based on density-functional equations.
**Acknowledgment**The work on topics relevant to the mini-review was supported, in part, by the Israel Science Foundation through Grant No. 1286/17.
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---
abstract: '[Many link prediction algorithms require the computation of a similarity metric on each vertex pair, which is quadratic in the number of vertices and infeasible for large networks. We develop a class of link prediction algorithms based on a spectral embedding and the $k$ closest pairs algorithm that are scalable to very large networks. We compare the prediction accuracy and runtime of these methods to existing algorithms on several large link prediction tasks. Our methods achieve comparable accuracy to standard algorithms but are significantly faster.]{} [link prediction; graph embedding; commute time; resistance distance; closest pairs.]{}'
author:
- |
[Benjamin Pachev, Benjamin Webb]{}$^*$\
*Department of Mathematics, Brigham Young University, Provo, Utah*\
$^*$
title: Fast link prediction for large networks using spectral embedding
---
Introduction
============
The study of networks has become increasingly relevant in our understanding of the technological, natural, and social sciences. This is owing to the fact that many important systems in these areas can be described in terms of networks [@networks_an_introduction], where vertices represent the system’s individual components, e.g. computer routers, neurons, individuals, etc. and where edges represent interactions or relationships between these components.
An essential feature of the large majority of these networks is that they have a dynamic topology, i.e. a structure of interactions that evolves over time [@adaptive_networks]. The structure of social networks, for instance, change over time as relationships are formed and dissolved. In information networks such as the WWW the network’s structure changes as information is created, updated, and linked.
Although understanding the mechanisms that govern this structural evolution is fundamental to network science, these mechanisms are still poorly understood. Consequently, predicting a network’s eventual structure, function, or whether the network is likely to fail at some point are all currently out of reach for even simple networks.
In an attempt to determine which processes cause changes in a network’s structure we are lead to the following link prediction problem: Given a network, which of the *links*, i.e. edges between existing vertices, are likely to form in the near future. Here we adopt the standard convention that links are to be predicted solely on the basis of the network’s current topology (see, for instance, [@link_prediction_social_networks]).
Importantly, the link prediction problem can be used to study more than just which edges will appear in a network. It can also be used to predict which of the non-network edges are, in fact, in the network but currently undetected. Similarly, it can be used to detect which of the current network edges have been falsely determined to be a part of the network.
This notion of link prediction is of central importance in numerous applications. Companies such as Facebook, Twitter, and Google need to know the current state and efficiently predict the future structure of the networks they use to accurately sort and organize data [@quercia2012tweetlda]. Biologists need to know whether biochemical reactions are caused by specific sets of enzymes to infer causality and so on [@barzel2013network].
The barrier in determining whether network links truly exist in these and other settings, is that testing and discovering interactions in a network requires significant experimental effort in the laboratory or in the field [@clauset2008hierarchical]. Similarly, determining experimentally when and where a new link will form may also be impractical, especially if the precise mechanism for link formation is unknown. For these reasons it is important to develop models for link prediction.
At present, there is an ever increasing number of proposed methods for predicting network links [@srinivas2016link]. Not surprisingly, certain methods more accurately predict the formation of links in certain networks when compared with others. Additionally, each of these methods has a runtime that scales differently with the size of the network. In our experiments, we discover that a number of link predictors have a runtime that is so high that it effectively prohibits their use on moderately large networks.
Here we propose a class of link predicting algorithms that scale to large networks. This method, which we refer to as the *approximate resistance distance predictor*, integrates a spectral embedding of the network with a known algorithm for efficiently finding the $k$ closest pairs of points in Euclidean space. The spectral embedding aspect of the algorithm is derived as a low-rank approximation of the effective resistance between network vertices, as in [@fouss2007random]. The $k$ closest pairs component of the algorithm is taken from [@lenhof1992k] and can be used to predict links based on this embedding.
Here we compare the prediction accuracy and runtime of this method against several well-known algorithms on a number of coauthorship networks and a social network consisting of a small subset of Facebook users. We find that our method is achieves the best accuracy on some networks and scales to networks that many other link predictors cannot.
The paper is structured as follows. In Section \[sec:linkpred\] we describe the link prediction problem and outline a number of standard link prediction algorithms. In Section \[sec:spectral\] we introduce the method of resistance distance embedding and prove that it is optimal as a low rank approximation of effective resistance (see Proposition \[prop:resistance\_distance\_approx\]). In Section \[sec:setup\] we describe the experimental setup. Section \[sec:results\] numerical results comparing the performance of the resistance distance embedding algorithm to other algorithms are given. Section VI concludes with some closing remarks including a number of open questions for future work.
The Link Prediction Problem {#sec:linkpred}
===========================
The link prediction problem can be stated as follows. Given a connected graph $G = (V, E)$, and $k$, the number of predicted nonadjacent links, we seek $k$ pairs of vertices which are most likely to become connected. While the choice of $k$ depends on the application, we adopt the convention that $1 \leq k \leq |E|$.
The general paradigm for link prediction is to compute a similarity metric $score(x,y)$ on each vertex pair $(x,y)$. The predicted links are then the $k$ $(x,y) \in V \times V-E$ for which $score(x,y)$ is maximal. By contructing a matrix from the scores, we obtain a *graph kernel*. We can also go in the other direction. Any real $n \times n$ matrix, where $n = |V|$, defines a score function on pairs of vertices, and can be used for link prediction.
We now give a sampling of existing link prediction algorithms.
Local Methods
-------------
A *local method* for link prediction is an algorithm that uses vertex neighborhoods to compute similarity.
**Common Neighbors:** Common neighbors simply assigns $$\label{eq:common}score(x,y) = |\Gamma(x)\cap\Gamma(y)|,$$ where $\Gamma(x)$ is the neighbor set for $x \in V$.
**Jaccard’s Coefficient:** Jaccard’s coefficient is a normalized version of common neighbors that takes into account the total number of neighbors for both vertices. It is given by $$\label{eq:jaccard}score(x,y) = \frac{|\Gamma(x)\cap\Gamma(y)|}{|\Gamma(x)\cup\Gamma(y)|}.$$
**Preferential Attachment:** Preferential attachement is based on the idea that highly connected nodes are more likely to form links, an observed pattern in coathourship networks [@newman2001clustering]. This leads to $$\label{eq:prefattach}score(x,y) = |\Gamma(x)||\Gamma(y)|.$$
**Adamic-Adar:** $$\label{eq:adamic}score(x,y) = \sum_{z \in \Gamma(x)\cap\Gamma(y)}\frac{1}{log|\Gamma(z)|}$$
**Resource Allocation:** $$\label{eq:resource_alloc}score(x,y) = \sum_{z \in \Gamma(x)\cap\Gamma(y)}\frac{1}{|\Gamma(z)|}$$
Path-based Methods
------------------
*Path-based methods* consider all or a subset of the paths between two vertices to compute similarity. Unlike local similarity measures, they can capture global information about the network.
**Shortest Path:** This link predictor defines $score(x,y)$ as the negated length of the shortest path from $x$ to $y$.
**Katz:** The Katz metric counts all paths between two nodes, and discounts the longer paths exponentially. Define $path^{\ell}_{x,y}$ to be the set of all paths of length $\ell$ from $x$ to $y$. Then given a weight $0 < \beta < 1$,
$$\label{eq:katz}score(x,y) = \sum_{\ell=1}^{\infty}\beta^{\ell}|path^{\ell}_{x,y}|$$
A closed form for the associated graph kernel is given by $(I-\beta A)^{-1} - I$ = $\sum_{\ell=1}^{\infty}(\beta A)^{\ell}$, where $A$ is the adjacency matrix of $G$.
Random walks
------------
A *random walk* on G starts at some node x and iteratively moves to new nodes with uniform probability. There are a multitude of link predictors based on random walks. These are some of the fundamental ones.
**Hitting and Commute Time:** The *hitting time* $H_{x,y}$ is the expected number of steps required to reach $y$ in a random walk starting at $x$. Commute time is defined as $C_{x,y} = H_{x,y} + H_{y,x}$. Negated hitting time can be used as a link predictor, but the hitting time is assymetric in general, so we use instead the negated commute time, which is symmetric.
The commute time and its variants will be discussed further in Section \[sec:spectral\].
**Rooted Page Rank:** A problem with hitting and commute time is that random walks can become lost exploring distant portions of the graph. Rooted Page Rank deals with this problem by introducing random resets. Given a root node $x$, we consider a random walk starting at $x$. At each step, with probability $\alpha$ the walk returns back to $x$. With probability $1-\alpha$ the walk proceeds to a random neighbor. Given a root node $x$, for each other node $y$, $score(x,y)$ is defined as the stationary probability of $y$ under the random walk rooted at $x$. The corresponding graph kernel is given by $(1-\alpha)(I-\alpha D^{-1}A)^{-1}$, where $D$ is the degree matrix and $A$ is the adjacency matrix.
Scaling Link Predictors to Large Networks
-----------------------------------------
Many link predictors, such as Katz, require the computation of a matrix inverse. This is heinously expensive for large networks, as it is cubic in the number of vertices. One way to circumvent such problems is via a low-rank approximation of the score matrix. We investigate such a low-rank approximation for the commute-time or resistance distance kernel in the next section.
Even the simpler local predictors such as common neighbors or preferential attachment face difficulties at scale. This is because for sufficiently large networks, it is not possible to compute scores for each pair of vertices and then find the maximal ones. Instead, efficient search techniques must be employed to search only a small subset of the potential links in order to find those of maximal score. In Section \[sec:spectral\] we will demonstrate how a class of graph embedding based predictors can efficiently find the $k$ links of maximal score.
Spectral Embedding {#sec:spectral}
==================
We begin by deriving the *approximate resistance distance link predictor* as a best low-rank approximation to commute time and show how to evaluate its link prediction scores with a spectral embedding. We then show that this link predictor is part of a family of graph embedding based link predictors that use the $k$ closest pairs algorithm to efficiently find the links of maximal score. Finally, we discuss efficient ways to compute the spectral embedding upon which the approximate resistance distance predictor relies.
Approximating Commute Time
--------------------------
Let $L$ = $D-A$ be the Laplacian matrix of a graph $G=(V,E)$, and let $n$ = $|V|$. Let $L^{\dagger}$ be the Moore-Penrose inverse of L. Then the commute time is given by $$\label{eq:commute_time}C_{x,y} = |E|(L^{\dagger}_{x,x} + L^{\dagger}_{y,y} - 2L^{\dagger}_{x,y}),$$ where the quantity $r_{x,y} = (L^{\dagger}_{x,x} + L^{\dagger}_{y,y} - 2L^{\dagger}_{x,y})$ is known as the *effective resistance* or the *resistance distance* [@fouss2007random]. Since resistance distance differs from commute-time by a (network-dependant) constant scaling factor, they can be used interchangeably for link prediction.
For many networks, $G$ is too large to compute $L^{\dagger}$ exactly, so an approximation must be used. A natural choice is a best rank-$d$ approximation to $L^{\dagger}$ for some fixed dimension $d$. The resulting approximation of the resistance distances is closely related to distances between points in Euclidean space.
\[prop:resistance\_distance\_approx\]Let $d$ be a positive integer and let $G=(V,E)$ be a connected, undirected graph. Then $\exists$ a best rank-$d$ approximation $S$ of $L^{\dagger}$, and a map $f:V\rightarrow R^{d}$ so that $\forall $ $x,y \in V$, $S_{x,x} + S_{y,y} - 2S_{x,y} = {\left\lVert f(x)-f(y) \right\rVert}_2^2$. We call this map the *resistance distance embedding*.
${\noindent\textbf{Proof.}\quad}$ For a connected graph, the Laplacian matrix is positive semidefinite, with eigenvalues $0=\lambda_1 < \lambda_2 \leq \dots \leq \lambda_n$ and corresponding eigenvectors $v_{1}, v_{2}, v_{3}, \dots, v_{n}$. Then we have the spectral decompositions $$L = \sum_{i=2}^{n}\lambda_{i}v_{i}v_{i}^T$$ and $$L^{\dagger} = \sum_{i=2}^{n}\frac{1}{\lambda_{i}}v_{i}v_{i}^T.$$ Hence, $S = \sum_{i=2}^{d+1}\frac{1}{\lambda_{i}}v_{i}v_{i}^T$ is a best rank-$d$ approximation to $L^{\dagger}$ in the 2-norm. Then note $$S_{x,x} + S_{y,y} - 2S_{x,y} = (e_{x}-e_y)^{T}S(e_{x}-e_y)$$ $$= \sum_{i=2}^{d+1}\frac{1}{\lambda_{i}}(e_{x}-e_y)^{T}v_{i}v_{i}^T(e_{x}-e_y)$$ $$= \sum_{i=2}^{d+1}\frac{1}{\lambda_{i}}(v_{i,x}-v_{i,y})^2$$ $$= {\left\lVert f(x)-f(y) \right\rVert}_2^2$$ where $$\label{eq:res_embedding}f(x) = [\frac{v_{2,x}}{\sqrt{\lambda_2}}, \frac{v_{3,x}}{\sqrt{\lambda_3}}, \dots , \frac{v_{d+1,x}}{\sqrt{\lambda_{d+1}}}]^{T} \in R^{d}$$
We define the *approximate resistance distance link predictor* of dimension $d$ by setting $$\label{eq:approx_predictor}score(x,y) = -(S_{x,x} + S_{y,y} - 2S_{x,y}) = -{\left\lVert f(x)-f(y) \right\rVert}_2^2,$$ where $S$ and $f$ are defined as in Proposition \[prop:resistance\_distance\_approx\].
In the next section, we will see that the approximate resistance distance link predictor is part of a class of link predictors that avoid brute-force search when predicting links.
Link Prediction with Graph Embeddings
-------------------------------------
The resistance distance embedding is a special case of a *graph embedding*, which is a map $f$ from $V$ to $R^d$, $d$ a positive integer. We can use graph embeddings to create link predictors. A natural choice is to set $score(x,y)$ = $-{\left\lVert f(x)-f(y) \right\rVert}_2$, (so maximizing score corresponds to minimizing distance). We refer to this score function as the *Euclidean score*.
If $f$ is the resistance distance embedding, then link prediction with the Euclidean score is equivalent to the approximate resistance distance predictor. Recall that the approximate resistance distance score function is $-{\left\lVert f(x)-f(y) \right\rVert}_2^2$. The $k$ predicted links of maximal score correspond to the $k$ nonadjacent pairs of vertices $(x,y)$ for which $-{\left\lVert f(x)-f(y) \right\rVert}_2^2$ is maximal. These are precisely the $k$ links for which ${\left\lVert f(x)-f(y) \right\rVert}_2$ is minimal and are predicted with the Euclidean score.
Link prediction with the Euclidean score is related to the $k$ *closest pairs problem*. The closest pairs problem is as follows. Given a set of vectors $\{x_1,x_2, \dots, x_n\} \subset R^d$ we seek the k unordered pairs $(x_i, x_j), i \ne j$ of minimal distance (here we use the Euclidean norm but any $L_{p}$ norm can be used, $1 \leq p \leq \infty$). There is an algorithm to solve this problem in $$\label{eq:k_closest}O(d(n\,log\,n + k\,log\,n\,\,log(\frac{n^2}{k})))$$ [@lenhof1992k].
We can think of the link prediction problem as the closest pairs problem applied to the set of vectors $\{f(y), y \in V\}$, with the additional constraint that the best pairs must correspond to non-edges in $G$. The extra constraint can be handled by finding the $|E|+k$ closest pairs, then selecting the best $k$ which are non-edges. As there can be no more than $|E|$ edges, this approach is sure to work. We then have the worst-case complexity bound of $$\label{eq:link_pred_complexity}O(d(n\,log\,n + (|E|+k)log\,n\,\,log(\frac{n^2}{|E|+k}))).$$
Recalling that we require $1 \leq k \leq |E|$, and assuming that $G$ is connected so $|E| \geq n-1$, this complexity bound can be simplified to $$\label{eq:link_pred_simple_complexity}O(d\,|E|\,log^2n).$$ For large, sparse networks, $|E| << n^2$, and this is a tremendous speedup over the $O(n^2)$ brute-force approach.
**Cosine Similarity Score:** Another link prediction score function that can be derived from a graph embedding is the cosine similarity score, defined by $$\label{eq:cosine_similarity}score(x,y)=\frac{<f(x),f(y)>}{{\left\lVert f(x) \right\rVert}{\left\lVert f(y) \right\rVert}}.$$
If the cosine similarity score is used, the link prediction problem can still be solved without brute-force search. It is equivalent to the link prediction problem with Euclidean score on a modified graph embedding. The modified embedding is obtained from the original by normalizing the embedding vectors as follows.
\[prop:cosine\_score\] Given a graph embedding $f:V\rightarrow R^d$, the link prediction problem using $$score(x,y)=\frac{<f(x),(f(y))>}{{\left\lVert f(x) \right\rVert}{\left\lVert f(y) \right\rVert}} = \cos\theta$$ is equivalent to the link prediction problem with the Euclidean score function on the modified embedding given by $g(y) = \frac{f(y)}{{\left\lVert f(y) \right\rVert}}$.
${\noindent\textbf{Proof.}\quad}$ Let $x,y \in V$. Note $$<g(x), g(y)> = \cos\theta = score(x,y).$$ We have $${\left\lVert g(x)-g(y) \right\rVert}_{2}^2 = {\left\lVert g(x) \right\rVert}_{2}^2 + {\left\lVert g(y) \right\rVert}_{2}^2 - 2<g(x), g(y)>$$ $$= 2 - 2\cos\theta = 2-2score(x,y).$$ This shows that minimizing Euclidean distance for the modified embedding is the same as maximizing cosine similarity score on the original, so link prediction with Euclidean score on the modified embedding is equivalent to link prediction with the cosine similarity score on the original.
This section introduced a class of link predictors that avoid a brute-force search when predicting links. These link predictors rely on a precomputed graph embedding. The graph embedding needs to be efficiently computable in order for the overall prediction algorithm to be fast. We are concerned with link predictors that rely on the resistance distance embedding. Consequently, rapid computation of this particular graph embedding is the subject of the next section.
Computing the Resistance Distance Embedding
-------------------------------------------
Computing the resistance distance embedding of dimension $d$ requires finding the smallest $d$ nonzero eigenvalues and associated eigenvectors of the Laplacian matrix $L$. Fortunately, specialized, efficient algorithms exist for this problem which exploit the positive semi-definiteness and sparsity of $L$. These include TRACEMIN-Fiedler [@manguoglu2010tracemin] and a multilevel solver MC73\_FIEDLER [@hu2003hsl]. TRACEMIN-Fiedler is simpler to implement, and is also parallelizable, so we use it in our experiments.
Experimental Setup {#sec:setup}
==================
In this section we compare the performance of our link prediction algorithm to others on several large social networks. In a social network, nodes correspond to persons or entities. Edges correspond to an interaction between nodes, such as coauthouring a paper or becoming linked on a social media website.
The Networks
------------
**Arxiv High Energy Physics Theory (hep-th):** This network is a coauthorship network obtained from the Konect network collection. [@konect:leskovec107; @konect:2016:ca-cit-HepTh].
**Arxiv High Enery Physics Phenomenology (hep-ph):** This is another coauthorship network from the Konect network collection [@konect:leskovec107; @konect:2016:ca-cit-HepPh].
**Facebook Friendship (facebook):** This social network consists of a small subset of facebook users, where edges represent friendships [@konect:2016:facebook-wosn-links; @viswanath09].
**Arxiv Condensed Matter Physics (cond-mat):** This dataset was obtained from Mark Newman’s website [@newman2001structure], and is also a coathourship network. Unlike the other datasets, the edges are not timestamped.
Creating Training Graphs
------------------------
In order to perform link prediction, we partition edges into a training set and a test set. Edges in the training set occur before those in the test set and are used to construct a training graph. We run link prediction algorithms on the training graph to predict the contents of the test set. In most cases, edges have timestamps, and we can choose a cutoff time to partition the edges.
For one network (cond-mat) the edges are not timestamped. However, there are two versions of the cond-mat network available. One contains all collaborations up to 2003. The second is an updated network with all collaborataions up to 2005. We use the first network as the training graph. The test set consists of all edges in the second network for which both nodes are in the earlier network.
Choosing the cutoff between the training and test edges is somewhat arbitrary. If too few edges are used for training, link predictors will struggle. If too few are left for testing, then results may be statistically insignificant. See Table \[table:train\_net\_stats\] for a comparison of the training networks and original networks.
-------------------- -------- ----------- ----------------
Network Nodes Edges Average Degree
\[0.5ex\] cond-mat 15,803 60,989 7.7187
cond-mat train 13,861 44,619 6.4381
facebook 63,731 817,035 12.8201
facebook train 59,416 731,929 24.6374
hep-ph 28,093 3,148,447 112.0723
hep-ph train 26,738 2,114,734 158.1819
hep-th 22,908 2,444,798 106.7225
hep-th train 21,178 1,787,157 168.7749
\[1ex\]
-------------------- -------- ----------- ----------------
: Training network statistics[]{data-label="table:train_net_stats"}
Our spectral embedding based link prediction algorithms require a connected graph. To solve this problem, we reduce each training graph to its largest connected component. For each network we consider, the largest component contains the vast majority of the vertices.
The Predictors
--------------
We perform experiments with two spectral embedding based predictors. Each uses the resistance distance embedding of dimension $d$, with $d$ a parameter to be varied. The first uses the Euclidean score function and is equivalent to the approximate resistance distance predictor of dimension $d$. The second uses the cosine similarity score. We refer to these link predictors as spec\_euclid and spec\_cosine respectively (spec for spectral). In tables, the dimension of the embedding is indicated by a number after the predictor name. For example, spec\_euclid8 refers to the spec\_euclid predictor using an 8-dimensional resistance distance embedding.
The other link prediction algorithms used in our experiments are preferential attachment, common neighbors, Adamic Adar, Rooted Page Rank and Katz (with $\beta=.01$). Some networks are too large for certain algorithms to handle, so not every algorithm is run on each network. For example, the facebook training graph has 59,416 nodes. Computing the Katz score on this graph requires finding the inverse of a 59,416$\times$59,416 matrix, and is very expensive in time and space, so we do not use the Katz algorithm for the facebook graph.
All experiments were performed on the same 4 core machine. The common neighbors, preferential attachment, and Adamic Adar algorithms were implemented in Python and were not parallelized. Our spectral link predictors, Katz, and Rooted Page Rank use the Python library Numpy to parallelize linear algebra operations. All code that was used in the experments in this paper can be found at the git repository bitbucket.org/thorfax/spectral\_research.
For each network, we fix the number of links to be predicted. With the exception of hep-th, this number is equal to 10% of the maximum possible number of correct predictions (i.e the number of new links in the test set). For the hep-th network we discovered that the spec\_euclid and spec\_cosine predictors achieve nearly perfect accuracy when predicting 1000 links. As this is not the case for any other network we considered, we report this unusual phenomenon.
For all of the networks we consider, the probability of randomly predicting a correct link is very low. Most of the algorithms we consider do much better than the random baseline, but have low raw accuracy since there are few new links compared to the number of possible links. See Table \[table:train\_prob\_info\] for a summary of the number of links predicted and baseline probability of randomly predicting a correct link.
-------------------- -------- --------------------- --
Network Links Random Accuracy (%)
\[0.5ex\] cond-mat 1190 0.012
facebook 7858 0.004
hep-ph 101466 0.286
reduced hep-ph 1988 0.661
hep-th 1000 0.296
reduced hep-th 135 0.084
\[1ex\]
-------------------- -------- --------------------- --
: Link Prediction Task Setup[]{data-label="table:train_prob_info"}
Results {#sec:results}
=======
On the cond-mat and facebook networks, both the spec\_euclid and spec\_cosine predictors performed worse than the simple common neighbors predictor. In addition to the full networks, we also compared predictor accuracy on reduced versions of the hep-th and hep-ph networks, because the full networks are too large for methods like Katz, common neighbors, and Rooted Page Rank to complete in a reasonable amount of time. On our reduced version of the hep-th network, our embedding-based predictors did better than common neighbors but not as well as the Rooted Page Rank predictor. On the reduced hep-ph network, the spec\_euclid predictor performed significantly better than all other competitors, including our other embedding-based predictor, spec\_cosine.
As Table \[table:cond-mat\] shows, the best predictors for the cond-mat network were Katz and common neighbors. Note that for both spec\_euclid and spec\_cosine, the accuracy increases with the dimension of the embedding.
----------------- ------------- ---------- --
Predictor Correct (%) Time (s)
\[0.5ex\] katz 5.97 62.96
commonNeighbors 5.97 1.55
prefattach 1.93 0.35
spec\_euclid1 1.51 2.99
spec\_cosine1 0.25 3.35
spec\_euclid2 1.51 3.65
spec\_cosine2 1.18 3.80
spec\_euclid4 1.76 10.54
spec\_cosine4 1.34 10.89
spec\_euclid8 1.68 11.41
spec\_cosine8 1.34 10.73
spec\_euclid16 1.68 29.91
spec\_cosine16 1.43 32.31
\[1ex\]
----------------- ------------- ---------- --
: Performance of link predictors on the cond-mat network[]{data-label="table:cond-mat"}
As previously mentioned, the facebook graph was too large to run the Katz predictor on it in a reasonable amount of time. As with the cond-mat network, the simple common neighbors predictor performs best.
--------------------------- ------------- ---------- --
Predictor Correct (%) Time (s)
\[0.5ex\] commonNeighbors 5.29 151.76
prefattach 0.41 7.00
spec\_euclid1 0.42 9.86
spec\_cosine1 0.00 47.20
spec\_euclid2 0.50 11.52
spec\_cosine2 0.42 12.96
spec\_euclid4 1.40 24.05
spec\_cosine4 1.02 25.96
spec\_euclid8 1.95 61.21
spec\_cosine8 2.58 62.27
\[1ex\]
--------------------------- ------------- ---------- --
: Performance of link predictors on the facebook network[]{data-label="table:facebook"}
Our spectral embedding link predictors performed significantly better on the hep-th and hep-ph networks, as Table \[table:hep-th\] and Table \[table:hep-ph\] show.
The common neighbors algorithm did not scale to the hep-th and hep-ph networks, unlike the facebook network. Although the facebook network had more nodes, it has a lower average node degree and fewer distance two pairs. The common neighbors algorithm computes intersections of neighbor sets for each distance two pair. Because the average node degree is higher for the hep-th and hep-ph networks, these intersections are more expensive to compute and there are more distance two pairs for which intersections must be computed.
---------------------- ------------- ---------- --
Predictor Correct (%) Time (s)
\[0.5ex\] prefattach 0.00 9.47
spec\_euclid1 94.50 10.88
spec\_cosine1 1.50 17.17
spec\_euclid2 98.70 17.88
spec\_cosine2 99.60 15.97
spec\_cosine4 100.00 15.67
spec\_euclid4 99.90 18.53
spec\_euclid8 100.00 29.81
spec\_cosine8 100.00 29.55
spec\_euclid16 99.90 96.47
spec\_cosine16 99.90 100.93
\[1ex\]
---------------------- ------------- ---------- --
: Performance of link predictors on the hep-th network[]{data-label="table:hep-th"}
---------------------- ------------- ---------- --
Predictor Correct (%) Time (s)
\[0.5ex\] prefattach 0.00 16.85
spec\_euclid1 3.93 18.25
spec\_cosine1 0.14 32.37
spec\_euclid2 9.16 21.98
spec\_cosine2 3.44 23.46
spec\_euclid4 19.25 27.04
spec\_cosine4 13.65 26.09
spec\_euclid8 22.90 46.69
spec\_cosine8 21.12 49.62
spec\_euclid16 24.62 148.51
spec\_cosine16 23.97 135.83
\[1ex\]
---------------------- ------------- ---------- --
: Performance of link predictors on the hep-ph network[]{data-label="table:hep-ph"}
In order to compare the performance of our spectral predictors to other predictors on the hep-ph and hep-th network data, we conducted another experiment using downsampled versions of these networks. To downsample, we used only the top 10% highest degree nodes. Our spectral predictors performed the best on the reduced hep-ph network (see Table \[table:hep-ph small\]), while the Rooted Page Rank algorithm was best for the reduced hep-th network (see Table \[table:hep-th small\]).
---------------------- ------------- ---------- --
Predictor Correct (%) Time (s)
\[0.5ex\] prefattach 0.00 1.52
katz 1.16 2.75
commonNeighbors 9.36 23.96
pageRank 11.87 2.94
adamicAdar 8.85 754.40
spec\_euclid1 1.81 2.80
spec\_cosine1 0.10 6.21
spec\_euclid2 4.73 3.73
spec\_cosine2 2.57 6.86
spec\_euclid4 13.13 5.80
spec\_cosine4 11.12 9.01
spec\_euclid8 16.40 7.39
spec\_cosine8 9.31 7.44
spec\_euclid16 14.13 17.32
spec\_cosine16 4.93 14.90
\[1ex\]
---------------------- ------------- ---------- --
: Performance of link predictors on the reduced hep-ph network[]{data-label="table:hep-ph small"}
---------------------- ------------- ---------- --
Predictor Correct (%) Time (s)
\[0.5ex\] prefattach 0.00 1.28
katz 0.00 1.61
commonNeighbors 2.22 16.70
pageRank 11.11 1.97
adamicAdar 2.22 788.11
spec\_euclid1 0.00 2.02
spec\_cosine1 0.00 3.77
spec\_euclid2 0.74 4.45
spec\_cosine2 0.00 4.61
spec\_euclid4 0.74 6.84
spec\_cosine4 0.00 6.25
spec\_euclid8 2.22 10.59
spec\_cosine8 1.48 11.02
spec\_euclid16 8.89 26.82
spec\_cosine16 5.93 24.92
\[1ex\]
---------------------- ------------- ---------- --
: Performance of link predictors on the reduced hep-th network[]{data-label="table:hep-th small"}
Conclusion {#sec:conclusion}
==========
We present a link prediction framework that can scale to very large networks by avoiding the quadratic costs inherent in methods that exhaustively search all candidate pairs of nonadjacent nodes. We investigated the performance of a set of predictors based on this framework and the spectrum and eigenvectors of the graph’s Laplacian matrix. These methods achieved high levels of accuracy on certain real-world link prediction tasks, and scaled well to networks with tens of thousands of nodes and millions of edges.
We emphasize that there are many other possible graph embeddings to invesitigate. Virtually all the runtime of our spectral link predictors is spent computing the resistance distance embedding. The $k$ closest pairs component of our algorithm is very fast in practice, with nearly linear temporal complexity in the number of edges Replacing the resistance distance embedding with one that is cheaper to compute could potentially produce link predictors that can scale to much larger networks than the ones we consider in this paper.
Our approximate resistance distance link predictor was derived as a low-rank approximation of resistance distance, an established link prediction score that is expensive to compute. Many other well-known predictors are expensive to compute, such as Katz and Rooted Page Rank. There is much room to explore low-rank approximations of such predictors and investigate whether they can be converted into accurate, scalable, graph embedding based, link predictors of the form we considered.\
\
**Funding**\
This work was supported by the Defense Threat Reduction Agency \[grant number HDTRA1-15-1-0049\].
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Magneto–transport measurements in the vicinity of quantum Hall (QH) transitions provided over the recent years an extensive variety of data [@phas-tran; @chi; @dhi; @dual; @hlldual; @reslaw; @qhi], which stimulated a considerable confusion. The present paper suggests a way to settle the apparent disagreement between different experimental results..
The traditional point of view asserts that transport properties near the transitions between QH plateaux, and from a QH liquid to the insulator, should reflect the proximity to a second order quantum phase transition. Correspondingly, the d.c. resistivity tensor $\rho_{ij}$ at a given filling fraction $\nu$ and temperature $T$ should be described by a universal function $f(X)$ of a single parameter: \_[ij]{}=\_[ijc]{}f([T\^]{}) , \[rscale\] where $\Delta\nu=\nu-\nu_c$ (the deviation from the critical filling $\nu_c$), and $\rho_{ijc}$, $\kappa$ are universal. Here $\kappa$ is a combination of critical exponents, $\kappa=1/zx$, where $z$ is the dynamical exponent and $x$ characterizes the divergance of the correlation length near criticality: $\xi\sim|\Delta\nu|^{-x}$. Theoretical studies have predicted $x=7/3$ [@bodo]; experimentally, a number of groups have obtained data in remarkable consistency with the scaling ansatz, at different (integer as well as fractional) QH transitions [@yscl]. These results are further supported by data showing scaling at finite frequency [@fscl] and current [@cscl], which moreover confirm the theoretical prediction $x=7/3$ and yield $z\approx 1$ (indicating the relevance of Coulomb interactions [@phas-tran] near the quantum critical point).
The validity of the single parameter scaling form Eq. (\[rscale\]) has been, however, recently challenged [@reslaw]. Motivated by an earlier work [@dual], which identified a surprisingly robust (duality) symmetry relating the current–voltage curves $I(V)$ in the QH liquid phase ($\Delta\nu>0$ [@fn1]) to $V(I)$ in the insulator at $-\Delta\nu$, the parameter $\Delta\nu$ was marked as a relevant scaling variable [@dualt]. The Ohmic resistivity plotted as a function of $\Delta\nu$ is indeed fitted (in a wide range of parameters) by the formula \_[xx]{}=[he\^2]{} , \[rlaw\] However, counter to the expectation, $\nu_0(T)$ does not scale as $T^\kappa$, but rather exhibits a linear dependence on $T$: \_0(T)=T+ . \[nu0\] The resistivity law Eqs. (\[rlaw\]), (\[nu0\]) holds in various different samples, as well as in different transitions, including plateau–to–plateau transitions [@hlldual] (with an appropriate definition of the analog of $\rho_{xx}$). The parameters $\alpha$ and $\beta$ are sample–dependent, and define a “saturation temperature” $T_s=\beta/\alpha$ which ranges between $~0.05K$ and $\sim 0.5K$. Moreover, even if one ignores this saturation, attributing it to incomplete cooling of the carriers, a linear scaling of $\nu_0(T)$ with $T$ is inconsistent with any sensible theory for the quantum critical behavior. It should be noted, that a partial set of curves obeying Eqs. (\[rlaw\]), (\[nu0\]) (corresponding to a restricted range of parameters) can be collapsed on a “traditional” scaling curve [@shahar], with an exponent $0<\kappa<1$. This observation raises a serious doubt concerning the interpretation of the data in earlier experimental works: the distinction between a pure power law $T^\kappa$ with $\kappa\sim0.4-0.5$, and an alternative function which interpolated between $\sim T$ and a constant, may turn out to be rather difficult. Nevertheless, a recent experimental result [@CZ] indicates a cross–over from one behavior to another, which will be discussed later in this paper in more detail.
Another set of experimental observations that appear to be inconclusive involve the Hall resistance in the high magnetic field insulator. While part of the data [@chi] support the theoretical prediction of a “Hall insulator phase” [@KLZ], in which $\rho_{xx}$ diverges in the limit $T\rightarrow 0$ yet $\rho_{xy}$ behaves as in a classical conductor (linearly dependent on the magnetic field $B$), other data [@dhi] exhibit a tendency to divergence of $\rho_{xy}$ as well. Yet a third class of experimental data [@dual; @qhi; @qhirus] have established the existance of a “quantized Hall insulator” (QHI) behavior in the insulator close to a fundamental QH state ($1/k$ with $k$ an odd integer): in this regime, $\rho_{xy}$ is not only finite as asserted in [@KLZ], but moreover maintains the quantized plateau value $kh/e^2$. This phenomenon is a specific manifestation of the “semi–circle law” [@ruzin], which however extends beyond the range of validity expected from that theory. In particular, the observation of a QHI behavior in the non–linear response regime [@dual] indicates a surprising robustness of the phenomena, and has provided support to the idea that it is intimately related to the validity of duality symmetry [@dualt].
In a previous work [@SA], Shimshoni and Auerbach proposed a transport mechanism consistent with the above described QHI phenomenon. The mechanism involves hopping across the junction between edge–states surrounding nearest neighbors in a random network of $1/k$-QH liquid puddles, carried out by quantum tunneling assisted by the temperature and current bias. Then, neglecting the quantum interference between different junctions in the network, it is proven that the Hall resistance is quantized at the value dictated by the QH liquid, irrespective of the details of the longitudinal resistance associated with the hopping processes in the junctions. Note that this network model can be extended to the case where the liquid puddles do not consist of a single type (a situation that is likely to be applicable in an insulating regime close to more than one fundumental QH state), in which case $\rho_{xy}\sim B$ can be established (similarly to data in Ref. [@chi]). It is later shown [@SAK], that within this model one also obtains an activated–like behavior similar to Eq. (\[rlaw\]) (though with $\nu_0(T)$ interpolating between $\sim T$ and a constant in a way different from the linear expression Eq. (\[nu0\])). The linear dependence on $\Delta\nu$ in this model is attributed to the relation between area–fraction of the liquid and the barrier heights.
Underlying the above described model for the transport, there is an essential assumption of a finite [*dephasing length*]{} $L_\phi$, beyond which quantum interfernce terms are suppressed. As long as the size of a QH liquid puddle, around which electronic edge–states are extended, is larger than $L_\phi$ – the [*classical*]{} random resistor network model is justified. In this sense, the transport regime dominated by nearest–neighbor hopping processes is “classical”. This is even though the resistance associated with a single junction in the network (and given by a Landauer formula, where the two neighboring puddles are regarded as macroscopic reservoirs), is possibly dictated by quantum tunneling through the barrier.
The classical model of Ref. [@SA] is not obviously a unique scenario which yields a quantized Hall resistance away from the strict QH phase. To test this, in a recent work [@PA] Pryadko and Auerbach have examined the effect of quantum interference in the network on $\rho_{xy}$, and found a deviation from the quantized value. In the case where $L_\phi$ is much larger than a puddle size, $\rho_{xy}$ vs. $B$ indicates an exponential divergence towards the insulating regime. A similar trend is indicated in other recent numerical data as well [@SW]. This suggests that (counter to the naive intuition!) a classical scenario is actually a [*necessary condition*]{} for supporting a QHI behavior; it also follows, that the range of parameters in which it is observed should coincide with the range of validity of Eqs. (\[rlaw\]), (\[nu0\]).
A cross–over from the classical to a quantum transport regime is expected, when the size of the QH puddles becomes smaller than $L_\phi$. Then, the nearest neighbor localized puddle is not necessarily the optimal destination of a hopping electron. Typically, randomly distributed electronic states that are close in energy are not close in real space. As a consequence, the dominant transport mechanism is variable range hopping (VRH) [@vrh]. In this regime, a typical hop occurs between localized states separated by a distance $R_h(T)$, which minimizes the exponential suppression of the hopping probability due to the difference in both energy and real space. Assuming a Coulomb gap in the density of states [@ES], this hopping length is given by R\_h(T)\~(e\^2k\_B T)\^[1/2]{} ; \[hopl\] here $\xi$ is the localization length, and $\epsilon$ the dielectric constant. The resulting expression for the longitudinal resistance in the insulator is \_[xx]{}\~\_0 , T\_0=[e\^2k\_B]{} . \[rhop\] (Similarly, in the QH phase Eq. (\[rhop\]) holds for $1/\rho_{xx}$; to avoid confusion, in the rest of the paper the expressions for $\rho_{xx}$ correspond to the insulator.)
To convert Eq. (\[rhop\]) into a dependence on the filling fraction, note that the localization length $\xi$ (which by definition describes a typical cluster over which an electronic state is extended), coincides with the correlation length which tends to diverge near the transition [@PS]: =\_0([||\_c]{})\^[-x]{} \[xidiv\] (where $\xi_0$ is the value of $\xi$ deep in the localized phase). The critical behavior Eq. (\[xidiv\]) is valid for $\Delta\nu\ll \nu_c$, so that $\xi\gg \xi_0$. On the other hand, the mechanism of VRH dominates the transport as long as $\xi$ is finite and [*smaller*]{} than $L_\phi$. Provided the latter is a few orders of magnitude larger than $\xi_0$, there is a range of parameters where Eq. (\[rhop\]) holds [*in coincidence*]{} with Eq. (\[xidiv\]) [@crhop]. As a result, one obtains \_[xx]{}\~\_0 , C . \[rqlaw\]
In the regime where Eq. (\[rqlaw\]) is applicable, the experimental data exhibit three prominent features: (a) the scaling form Eq. (\[rscale\]) is recovered with $\kappa=1/x\approx 0.43$ (given that indeed $x=7/3$), and $f(X)\sim e^{(CX^x)^{1/2}}$; (b) at a given $\Delta\nu$, \_[xx]{}(T)\~T\^[-1/2]{} ; \[rvsT\] and (c) isotherms plotted as a function of $\nu$ are of the form \_[xx]{}()\~||\^[1.15]{} . \[rvsnu\] Comparing Eq. (\[rvsnu\]) with the empirical resistivity law (\[rlaw\]), we observe that [*by mere coincidence*]{} (which stems from the specific value of the exponent $x$), the two functional forms are practically indistinguishable. Similarly, it is hard to distinguish the temperature dependence Eq. (\[rvsT\]) from the fit to $T^{-\kappa}$ employed in Ref. [@CZ]; it is suggested that the VRH scenario is, in fact, a more appropriate basis for interpretation of the data in the low $T$ regime.
As mentioned above, the VRH scenario is consistent in the regime where the transport is quantum coherent, namely for $\xi<L_\phi$. Hence, the quantum regime terminate once $\xi$ approaches $L_\phi$ due to either increase of temperature, or the divergence of $\xi$ sufficiently close to $\nu_c$. To estimate the boundary of the corresponding region in parameter space, an explicite expression for $L_\phi$ is needed. It turns out [@vrh] that in the VRH regime, the length scale which plays the role of a dephasing length is the hopping length $R_h(T)$ (Eq. (\[hopl\])). This implies that a cross–over to a “classical” transport regim occurs at $T\sim T_0$ (where $T_0$ is defined in Eq. (\[rhop\])). Note that this criterion is consistent with the observation, that for $T>T_0$ the longitudinal resistivity no longer indicates the exponential divergence characteristic of strong localization. Employing Eq. (\[xidiv\]) we conclude that for a fixed $\nu$, a cross–over to the classical regime occurs at a temperature $T_x$, where T\_x\~T\_0= [e\^2k\_B\_0]{}([||\_c]{})\^x . \[Tx\] Alternatively, for a fixed $T$, the cross–over occurs at $|\Delta\nu|_x$, where \~([\_0 k\_BTe\^2]{})\^[1/x]{} . \[dnux\] As argued in Ref. [@PS], the latter expression defines the width of the peaks in $\sigma_{xx}$ near QH transitions. However, it should be emphasized that (at a fixed $T$) critical scaling of the data is expected to hold [*out side*]{} this width, while $|\Delta\nu|<|\Delta\nu|_x$ corresponds to a classical transport regime.
I next show that the data which clearly manifest the activated–like behavior resistivity law (\[rlaw\]), (\[nu0\]) (Ref. [@reslaw]) and a QHI behavior (Refs. [@qhi; @qhirus]), mostly correspond to the classical regime by the criterion suggested above. A quantitative estimate of $|\Delta\nu|_x$ from Eq. (\[dnux\]) is possible provided the “bare” localization length $\xi_0$ is known. Unfortunately, this parameter can not be extracted independently from the available information about the samples. However, the fact that the integer QH effect is observed indicates that the single–electron states are localized over a length scale at least as large as the magnetic length $l=(\hbar c/eB)^{1/2}$. Hence, the insertion $\xi_0\sim l$ provides a [*minimal*]{} estimate of $|\Delta\nu|_x$ for a given $T$. In Ref. [@reslaw], close to the critical field in the InGaAs/InP sample ($B_c=2.14\, T$, corresponding to $\nu_c=0.562$ and carrier density $n=3\times 10^{10}\, cm^{-2}$), one gets $l\approx 170\AA$. The implied lower bound on the width of the classical regime is $(\Delta\nu)_x/\nu_c\sim \pm 0.2$ for the highest temperature isotherm ($T=2.21\, K$), and $(\Delta\nu)_x/\nu_c\sim\pm 0.1$ for $T=0.3\, K$. Comparing with the data, it turns out that the range of $\nu$’s where $\log\rho_{xx}$ vs. $\nu$ is strictly linear is not much larger than this lower bound. A more conclusive statement can be made regarding the quantized Hall resistance data of Hilke [*et al.*]{} in Ref [@qhi]: there, $l\approx 130\AA$ which implies that at the lowest displayed temperature ($T\approx 0.3\, K$), the classical regime extends at least within $\Delta B/B_c\sim 0.09$. Indeed, this estimate implies an upper field $B_u=B_c\times 1.09$ which roughly coincides with the field at which the $\rho_{xy}$ data terminate (due to insufficient accuracy of the measurement). Note that the range of observed QHI increases with $T$ or with an increased current bias, as long as a plateau in the QH phase is preserved. Beyond a certain $T$, the quantization in the insulator is destroyed at the same time with the entire plateau, due to excitations to higher Landau levels.
To further test the central arguments of this paper, one should examine experimental data that extend over a wide enough range of temperatures below and above the cross-over point $T_x$. The classical and quantum regimes are then clearly distinct in terms of the $T$-dependence of $\rho_{xx}$ for a given $\nu$. The functional dependence on $\nu$ is, however, nearly identical: $\log\rho_{xx}$ is expected to be approximately linear in $\nu$ in a wide range of parameters extending over both regimes. This is possibly a major source of confusion in the literature. In particular, in Ref. [@CZ] a cross–over in the $T$-dependence is clearly observed, however the single cross–over point identified there ($T_x\approx 0.1\, K$) is an average over a range of filling fractions. Similarly, it is possible that in Ref. [@reslaw] as well, the outskirts of the range of $\Delta\nu$ indicating $\log\rho_{xx}\sim\nu$ extend into the quantum regime. The formula Eq. (\[nu0\]) is then not necessarily the only possible fit of the slope.
To summarize, I propose an interpretation of the extensive set of data close to QH transitions which distinguishes two conduction regimes. The classical regime is established closer to the critical point and at relatively high $T$. It is dominated by hopping between nearest neighbor hopping between conducting QH puddles, whose typical size is larger than the dephasing length $L_\phi$. Hence, the transport coefficients do not depend on $L_\phi$, but rather on the details of the narrow junctions separating the puddles. When mapped to a QH liquid–to–insulator transition, the characteristic behavior of the resistivity tensor in this regime is an activated–like $\rho_{xx}$, and quantization of $\rho_{xy}$ in the insulator. The quantum regime is established at lower $T$ and farther from the critical point, where the transport is dominated by VRH. The limit of validity of VRH (corresponding to $\xi\sim R_h(T)$, where $R_h(T)$ is identified with $L_\phi$), provides an estimate of the boundary between the regimes (Eqs. (\[Tx\]), (\[dnux\])). The classical and quantum regimes are very hard to distinguish by $\nu$–dependence of $\log\rho_{xx}$ ($\sim\Delta\nu$ vs. $\sim(\Delta\nu)^{1.15}$, respectively). It is predicted that the cross–over between them should be more clearly indicated by a change in the $T$–dependence, accompanied by a deviation of $\rho_{xy}$ in the insulator from a quantized plateau.
I thank A. Auerbach, S. Girvin, M. Hilke, D. Huse, S. Murphy, D. Shahar, U. Sivan and S. Sondhi for useful conversations, and P. Coleridge for informing me of his data prior to publication. This work was supported by grant no. 96–00294 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel.
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|
---
abstract: 'Ground-state properties of a few attractively interacting ultra-cold atoms of different mass confined in a one-dimensional harmonic trap are studied. The analysis is performed in terms of the noise correlation, which captures the two-particle correlations induced by the mutual interactions. Depending on the mass ratio between the components’ atoms, the inter-particle correlations change their properties significantly from a strong pair-like correlation to an almost uncorrelated phase. This change is accompanied by a simultaneous change in the structure of the many-body ground state. A crucial role of the quantum statistics is emphasized by comparing properties of the Fermi-Fermi mixture with a corresponding Fermi-Bose system.'
author:
- Daniel
- Tomasz
bibliography:
- 'bibtex.bib'
title: |
Intercomponent correlations\
in attractive one-dimensional mass-imbalanced few-body mixtures
---
Introduction
============
Recent years have brought many examples that the systems consisting of a few ultra-cold atoms can be prepared and well-controlled experimentally with extreme precision [@wenz2013fewToMany; @zurn2012fermionization; @serwane2011deterministic; @Murmann2015AntiferroSpinChain; @Kaufman2015Entangling; @Andrea2018Imaging; @zurn2013Pairing]. It became possible to measure (as functions of mutual interactions) not only single-particle properties of the system but also higher multi-particle correlations. The latter are fundamentally important since they directly reflect different non-classical multi-particle properties of the system being direct manifestations of indistinguishability and entanglement forced by interactions. It is quite obvious that these higher correlations cannot be neglected if one needs to characterize an obtained quantum state appropriately. As shown recently, the two-body position and momentum correlation functions can be measured and they are indeed a very powerful tool to characterize quantum states [@bergschneider2018correlations]. In principle, one can have experimental access also to higher-order correlations between particles. For example, it can be done by using atomic microscopes which allow one to measure positions of all particles at the same time [@bakr2010probing; @sherson2010single; @OmranBloch2015PauliBlocking; @Cheuk2015FermionicMicroscope; @Cheuk2016MottMicroscope; @Cheuk2016Correlations; @ParsonsPRL2015; @EdgePRA2015ImagingFermions; @haller2015single; @ParsonsPRL2015]. All this means that on an experimental level, the ultra-cold physics starts to explore much more complicated features of many-body systems than simple single-particle densities.
Theoretical studies of one-dimensional few-body mixtures are very rich in the literature. In a great majority, due to the experimental motivation from the Heidelberg group [@serwane2011deterministic; @zurn2013Pairing], these considerations concern two-component mixtures of repelling particles with equal mass (see for example [@brouzos2013two; @SowinskiGrass2013FewInteracting; @Gharashi2013UpperBranchCorrelations; @GarciaMarch2014Localization; @fogarty2018fast; @Garcia-MarchPRA2014; @Valiente2014Multi; @volosniev2015hyperspherical; @koscik2012quantum]). Accordingly less attention is given for attractive systems for which some precursors of the Cooper-like pairing were observed [@zurn2013Pairing], theoretically explained [@zinner2013comparing; @DAmico2015Pairing; @Sowinski2015Pairing] and explored [@conduit2013fflo; @rammelmuller2016ground; @mckenney2016ground; @BjerlinReimann2016Higgs].
Recent years have brought tremendous progress in experimental studies of fermionic (Li-K, Dy-K) [@Wille6Li40K; @tiecke2010Feshbach6Li40K; @cetina2016ultrafast; @Grimm2018DyK] as well as bosonic-fermionic (Li-Na, Rb-K, Cs-Li, Li-K) [@Hadzibabic2002LiNa; @Esslinger2006RbK; @Bloch2009RbK; @Wu2011KKLi; @Chin2014CsLi; @Grimm2018KLi] mixtures of large number of ultra-cold atoms of different mass. Although for such mixtures the few-body regime has not been achieved yet, first theoretical predictions show that such systems may have essentially different properties [@JasonHo2013PhaseSeparation; @loft2014variational; @Pecak2016Separation; @Pecak2017Ansatz; @harshman2017masses; @mistakidis2018repulsive] than systems with equal mass atoms.
In this general context, the question of properties of mass-imbalanced few-body mixtures in the attractive regime seems to be very relevant and important. In the following, we perform the first step in this direction and we analyze a destructive effect of a mass difference on inter-component pairing correlations emerging in a strong attractive regime. We identify, describe, and quantify these highly non-classical correlations as functions of mass ratio between particles forming opposite components and their number. We also emphasize the role of the quantum statistics in this destructive process.
The work is organized as follows. In Sec. \[sec:model\] we introduce the theoretical model of the few-body ultra-cold system studied and we briefly discuss a numerical method of treatment used. Next in Sec. \[sec:noise\], we refresh the concept of noise correlation and we introduce a natural measure quantifying an amount of inter-component correlations in the system. In Sec. \[sec:twoatoms\] we analyze the simplest situation of two atoms for repulsive and attractive interactions and different masses. Importantly, in Sec. \[sec:manybody\] we broadly discuss inter-component correlations induced by attractions for different strengths of interactions and different mass ratios. For completeness, in Sec. \[sec:boseFermi\] we examine the consequences of the quantum statistics by studying Bose-Fermi mixtures. Finally, in Sec. \[sec:conclusion\] we conclude.
The Model {#sec:model}
=========
In the following we consider a two-component mixture of ultra-cold fermions of masses $m_\downarrow$ and $m_\uparrow$ confined in a one-dimensional harmonic trap of a frequency $\omega$. We assume that the particles belonging to different species interact dominantly in the $s$-wave channel and we model this interaction with the $\delta$-like potential. In contrast, for fermions of the same kind (for which the $s$-wave channel is closed due to the Pauli exclusion principle) mutual interactions are negligible and we ignore them. Under these assumptions the many-body Hamiltonian of the system reads: $$\begin{gathered}
\label{eq:ham}
\hat{\cal H} = \sum_{\sigma\in\{\uparrow,\downarrow\}} \int\!\!\mathrm{d}x\,\hat{\Psi}^\dag_\sigma(x)H_\sigma\hat{\Psi}_\sigma(x) \\
+ g\int\!\! \mathrm{d}x\,\hat{\Psi}^\dag_\uparrow(x)\hat{\Psi}^\dag_\downarrow(x)\hat{\Psi}_\downarrow(x)\hat{\Psi}_\uparrow(x),\end{gathered}$$ where the single-particle Hamiltonians $H_\sigma$ are given by:
\[1PHam\] $$\begin{aligned}
H_\downarrow &= - \frac{\hbar^2}{2m_\downarrow} \frac{\mathrm{d}^2}{\mathrm{d} x^2} + \frac{m_\downarrow\omega^2}{2} x^2, \\
H_\uparrow &= - \frac{\hbar^2}{2m_\uparrow} \frac{\mathrm{d}^2} {\mathrm{d} x^2} + \frac{m_\uparrow\omega^2}{2} x^2.\end{aligned}$$
Note, that for simplicity the frequencies $\omega$ are equal for both components and that implies only one energy scale $\hbar\omega$ in the system. The field operator $\hat\Psi_\sigma(x)$ annihilates a particle of type $\sigma$ at given point $x$. The quantum statistics is reflected in the natural anti-commutation relations, $\{\hat\Psi_\sigma(x),\hat\Psi^\dagger_\sigma(x')\} = \delta(x-x')$ and $\{\hat\Psi_\sigma(x),\hat\Psi_\sigma(x')\} = 0$. Note that particles of different types are fundamentally distinguishable. Therefore any appropriate relations between fields $\hat\Psi_\uparrow(x)$ and $\hat\Psi_\downarrow(x)$ are equivalent to the commutation relations $[\hat\Psi_\uparrow(x),\hat\Psi^\dagger_\downarrow(x')] = [\hat\Psi_\uparrow(x),\hat\Psi_\downarrow(x')]=0$. Evidently, the Hamiltonian does commute with the particle number operators in a given component $\hat{N}_\sigma=\int\mathrm{d}x\hat{\Psi}_\sigma^\dagger(x)\hat{\Psi}_\sigma(x)$. Therefore, the properties of the system can be examined independently in the subspaces of given $N_\uparrow$ and $N_\downarrow$. To make a whole analysis as clear as possible, in this work we focus on balanced systems, [*i.e.*]{}, the systems with equal number of particles in both components, $N_\uparrow=N_\downarrow$. The effective one-dimensional interaction strength $g$ between fermions from opposite components can be derived from the full three-dimensional theory of scattering by integrating out the perpendicular degrees of motion [@Olshanii1998]. In the following we express all quantities in the natural harmonic oscillator units with respect to the $\downarrow$ component, [*i.e.*]{}, energies are measured in $\hbar\omega$, positions in $\sqrt{\hbar/(m_\downarrow\omega)}$, the interaction strength $g$ in units of $(\hbar^3\omega/m_\downarrow)^{1/2}$, [*etc.*]{} For convenience we also denote the mass ratio of atoms of different species as $\mu=m_\uparrow/m_\downarrow$.
By expanding the field operators $\hat{\Psi}_\sigma(x) = \sum_i \phi_{i\sigma}(x) \hat{a}_{i\sigma}$ in the eigenbasis $\{\phi_{i\sigma}(x)\}$ of the appropriate single-particle Hamiltonians we rewrite the Hamiltonian to the following form: $$\label{eq:hamSQ}
\hat{{\cal H}} = \sum_\sigma \sum_i E_i \hat{a}_{i\sigma}^\dag \hat{a}_{i\sigma} +
g \sum_{ijkl} U_{ijkl} \hat{a}_{i\downarrow}^\dag \hat{a}_{j\uparrow}^\dag \hat{a}_{k\uparrow}^{\phantom{\dag}} \hat{a}_{l\downarrow}^{\phantom{\dag}},$$ where $E_i$ are the single-particle eigenenergies and the interaction coefficients $U_{ijkl}$ read: $$\label{eq:uijkl}
U_{ijkl} = \int \mathrm{d}x \phi^*_{i\downarrow}(x) \phi^*_{j\uparrow}(x) \phi_{k\uparrow}(x) \phi_{l\downarrow}(x).$$
To find numerically the ground-state $|G_0\rangle$ of an interacting system, we calculate the matrix elements of the Hamiltonian in the Fock space spanned by the many-body noninteracting Fock states $\{|F_j\rangle\}$ constructed from the lowest single-particle orbitals and we diagonalize the matrix obtained via the Arnoldi method [@ARPACK1998Sorensen]. In this way we find the decomposition of the many-body ground-state in this basis, $|G_0\rangle = \sum_j \alpha_j|F_j\rangle$. The size of the cropped Fock space is carefully selected in such a way that the final results are almost insensitive for further extension of the Fock space. It is worth to note that the many-body eigenenergies obtained via the exact diagonalization method converge slowly with the size of the Hilbert space. However, in the case of the many-body wave functions the convergence, understood as a numerical stabilization of coefficients in the decomposition of the many-body ground-state into the Fock basis, is much faster (for details see for example [@Pecak2017com] and Appendix \[AppendixConv\]). Since in the following article we focus on the two-body correlations, therefore a poor convergence of the ground-state energy does not reduce the credibility of the results. We checked that in the cases studied it is sufficient to use the first ten single-particle states in the decomposition of the field operator $\hat{\Psi}_\sigma(x)$ to obtain reliable results. The corresponding sizes of the many-body Hilbert space is presented in Tab. \[tab\].
----------------------------- -------- ---------
$N_\downarrow=N_\uparrow=2$ 2 025 2 475
$N_\downarrow=N_\uparrow=3$ 14 400 26 400
$N_\downarrow=N_\uparrow=4$ 44 100 150 150
$N_\downarrow=N_\uparrow=5$ 63 504 504 504
----------------------------- -------- ---------
: \[tab\] The size of the cropped many-body Hilbert space for different number of particles occupying no more than ten first single-particle orbitals $\phi_{i\sigma}(x)$, $i\in\{0,\ldots 9\}$. Note, that for Bose-Fermi mixtures the size of the Hilbert space is significantly larger.
Noise correlation {#sec:noise}
=================
The simplest observable that characterizes an interacting few-body system is the single-particle density profile being the diagonal part of the single-particle density matrix: $$\label{eq:SPx}
\rho_\sigma^{(1)}(x) = \langle G_0 | \hat{\Psi}^\dag_\sigma (x) \hat{\Psi}_\sigma (x) | G_0 \rangle.$$ It can be simply understood as the probability density of finding a single particle from the component $\sigma$ at position $x$. Similarly, the probability density of finding a single particle with the momentum $p$ reads: $$\label{eq:SPp}
\tau_\sigma^{(1)}(p) = \langle G_0 | \hat{\Psi}^\dag_\sigma (p) \hat{\Psi}_\sigma (p) | G_0 \rangle.$$ Note that in the latter definition the field operator is expressed in the momentum domain. These two single-particle quantities are the simplest (apart from the energy of the state) measurable observables which characterize the many-body quantum system. However, they do not possess any information about correlations between simultaneously measured particles belonging to opposite components. These features are captured by the two-body correlations which are encoded complementarily in the two-particle densities in position and momentum domains:
\[eq:TP\] $$\label{eq:TPx}
\rho^{(2)}(x;y) = \langle G_0 | \hat{\Psi}^\dag_\downarrow(x) \hat{\Psi}^\dag_\uparrow(y) \hat{\Psi}_\uparrow(y) \hat{\Psi}_\downarrow(x) | G_0 \rangle,$$ $$\label{eq:TPp}
\tau^{(2)}(p;k) = \langle G_0 | \hat{\Psi}^\dag_\downarrow(p) \hat{\Psi}^\dag_\uparrow(k) \hat{\Psi}_\uparrow(k) \hat{\Psi}_\downarrow(p) | G_0 \rangle.$$
In the noninteracting case ($g=0$) the many-body wave function of the ground state is a simple product of two antisymmetric wave functions, one for each component. Consequently, two-particle densities are the products of the corresponding single-particle densities, $\rho^{(2)}(x;y) = \rho^{(1)}_\downarrow(x) \rho^{(1)}_\uparrow(y)$ and $\tau^{(2)}(p;k) = \tau^{(1)}_\downarrow(p) \tau^{(1)}_\uparrow(k)$. When the interactions are turned on ($g\neq 0$), these relations do not hold anymore since inter-component correlations emerge in the system. It turns out that these additional correlations forced by interactions are well captured by the so-called noise correlations introduced in [@lukin2004noisCorr; @Altman20081D; @Altman2009lowDim] and exploited recently in the context of repulsive few-body systems [@brandt2017two; @brandt2018HOM]. These quantities are defined as the following:
\[eq:Noise\] $$\label{eq:NoiseX}
{\cal G}_\rho(x;y) = \rho^{(2)}(x;y) - \rho^{(1)}_\downarrow(x) \rho^{(1)}_\uparrow(y),$$ $$\label{eq:NoiseP}
{\cal G}_\tau(p;k) = \tau^{(2)}(p;k) - \tau^{(1)}_\downarrow(p) \tau^{(1)}_\uparrow(k)$$
and they just show the differences between the exact two-particle densities and one predicted by the single-particle picture. It is worth noting that the noise correlation can be measured experimentally in the position as well as in the momentum domain [@folling2005spatial; @bergschneider2018correlations].
As explained above, the noise correlations are appropriate quantifiers of inter-component correlations. However, having two different noise correlations for two different experimental parameters it is very hard to select one having higher correlations. Therefore, it is very convenient to introduce some geometric distance between an actual two-particle density profile and that obtained as a product of single-particle ones. Fortunately, single- and two-particle density profiles have all mathematical properties of density distributions. Therefore, the natural metric in their space exists. The Frobenius distance (known also as Hilbert-Schmidt norm) [@deza2009encyclopedia; @rachev2013methods] can be extracted directly from the noise correlations:
\[eq:measure\] $$\begin{aligned}
||{\cal G}_\rho|| &= \left(\int\!\mathrm{d}x\,\mathrm{d}y\, |{\cal G}_\rho(x;y)|^2\right)^{1/2}, \\
||{\cal G}_\tau|| &= \left(\int\!\mathrm{d}p\,\mathrm{d}k\, |{\cal G}_\tau(p;k)|^2\right)^{1/2}.\end{aligned}$$
It is quite obvious that the distance vanishes for the noninteracting system and it grows when an average magnitude of the inter-component correlations increases. In the following, we will quantify correlations mainly in the language of this quantity.
Two-atom system {#sec:twoatoms}
===============
![The noise correlations ${\cal G}_\rho$ and ${\cal G}_\tau$ for a system of two particles ($N_\uparrow=N_\downarrow=1$). (a) The equal mass case ($\mu=1$) for strong inter-particle repulsions (upper row) and attractions (bottom row). (b) Strongly mass-imbalanced case ($\mu=4$) for strong inter-particle attractions. Note, that independently of the mass ratio the noise correlations demonstrate similar correlations (anti-correlations) in the position (momentum) space. See the main text for explanation. In all plots positions and momenta are measured in natural units of the harmonic oscillator, $\sqrt{\hbar/m_\downarrow\omega}$ and $\sqrt{\hbar m_\downarrow\omega}$, respectively. []{data-label="fig1"}](fig1.png){width="\columnwidth"}
Before we analyze inter-component correlations for a larger number of attractively interacting particles let us start from the simplest situation of two particles of equal mass ($\mu=1$) for which the exact analytical expression for the ground-state wave function and its energy is known [@Busch1998]. In the context of the noise correlation, the ground state of a few repelling fermions ($g>0$) was considered recently in [@brandt2017two], where the interactions were modeled by the Gaussian-shaped inter-particle potential. The width of the Gaussian was much smaller than the natural harmonic oscillator length, hence we reproduce the results by using the pure $\delta$ potential. As seen in the upper row of Fig. \[fig1\]a, for strong repulsions ($g=5$) the noise correlation ${\cal G}_\rho$ become negative on the diagonal. As noticed in [@brandt2017two], it is a direct manifestation of the fact that due to repulsions it is almost not possible to find two particles in the same position. Importantly, this effect cannot be captured by a simple product of single-particle densities. For the same interaction, also some non-trivial behavior of the noise correlation in the momentum domain ${\cal G}_\tau$ is present (right panel in Fig. \[fig1\]a).
The situation changes qualitatively for the attractive scenario ($g=-5$). In this case, the probability of finding two particles at the same position is highly enhanced when compared to quite poor predictions of single-particle distributions. The most prominent difference between repulsive and attractive systems is however visible in the momentum domain. For an attractive system, one finds a very strong anti-correlation between interacting particles signified as a high positive value of the noise correlation along the line $p=-k$ (see the bottom right plot in Fig. \[fig1\]a). This means that the probability of finding two particles having exactly opposite momenta is significantly larger than that predicted by the single-particle picture. Importantly, in the case of the two particles studied, the situation does not change significantly when different masses of particles are considered. As it is seen in Fig. \[fig1\]b, even for a large mass ratio ($\mu=4$), the strong correlation in positions and anticorrelation in momenta are present in the system.
Many-body system {#sec:manybody}
================
![The noise correlations ${\cal G}_\rho$ and ${\cal G}_\tau$ for a strongly attractive system ($g=-5$) of $N_\uparrow=N_\downarrow=4$ particles and different mass ratio $\mu=1,2,3,4$. Note that while the strong correlations in the position domain survive when the mass imbalance in the system is introduced, the anti-correlation in the momentum domain is strongly suppressed and reduced. Position and momentum are measured in the natural units of the harmonic oscillator, $\sqrt{\hbar/m_\downarrow\omega}$ and $\sqrt{\hbar m_\downarrow\omega}$, respectively.[]{data-label="fig2"}](fig2.png){width="\columnwidth"}
The two-particle system described above is trivial from the quantum statistics point of view. Therefore in this section, we focus on attractive systems with a larger number of particles ($N_\uparrow=N_\downarrow=4$). First, we calculate the noise correlation for the balanced system of equal-mass particles (upper row in Fig. \[fig2\]). As it is seen, the inter-component correlations for the attractive scenario ($g=-5$) qualitatively resemble main features observed in the two-body scenario – strong correlations in positions and anti-correlations in momenta are clearly visible. It should be noted, however, that the noise distribution in the momentum domain is much flatter along the line $p=-k$ than the corresponding one obtained for a smaller number of particles. This effect is forced by an inherent indistinguishability of fermions and it can be viewed as one of the indicators of the Cooper-like pairing in the system [@Sowinski2015Pairing].
The situation changes when some factor lifting the balance in the system is present. In principle, in our case, there are two distinct mechanisms leading to the imbalance. The first originates in a direct difference of the number of particles in each component. The second is forced by different masses of the atoms forming opposite components ($\mu\neq 1$). In the case of harmonic confinement there exists a quite important difference between these two scenarios. It is clearly visible in the noninteracting limit. When a difference in numbers of particles is considered, contributions to the total energy of the system of both components are different (they have different Fermi energies). In contrast, the Fermi energy is insensitive to any change in the mass of particles since in the case of harmonic confinement the single-particle energies $E_i$ do not depend on mass (due to the same frequency $\omega$). All this suggests that these two different mechanisms may have a different impact on the properties of the system. In this work, we focus only on the imbalance forced by the mass difference ($\mu\neq 1$) assuming always a balance in the particle number $N_\uparrow=N_\downarrow$.
As explained in previous works [@Pecak2016Separation; @Pecak2016Transition], with varying $\mu$ the single-particle harmonic orbitals change their shape and they become different for different components. Although the single-particle energies remain unchanged, the mutual repulsions force the system to excite lighter particles. As a consequence, for the repulsion ($g>0$) strong enough the separation of the density profiles emerge. A similar effect of the phase separation driven by the mass imbalance was also studied in the case of homogeneous systems and non-harmonic confinements [@conduit2011itinerant; @JasonHo2013PhaseSeparation; @Fratini2014ZeroTemp].
In the case of attractive interactions ($g<0$), the situation also changes when compared to the balanced system $\mu=1$. It is clearly visible when the noise correlations are considered (Fig. \[fig2\]). Although in the position domain the main effect caused by $\mu\neq 1$ is quite trivial (the distribution of the heavier component is just narrower), in the momentum domain the change is significant. As it is seen in Fig. \[fig2\], the strong anti-correlations along the line $p=-k$ are smeared and for a large enough mass ratio $\mu$ any traces of correlated pairs almost vanish.
![(a) The total Frobenius distance $||{\cal G}_\tau||$ as a function of the interaction strength $g$ for the system of $N_\uparrow=N_\downarrow=4$ particles and different mass ratio $\mu$. In the inset we show the slope $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}\mu$ calculated in the region of vanishing interactions $g$. (b) Contributions ${\cal C}_1$ (solid thin), ${\cal C}_2$ (solid thick), and ${\cal C}_3$ (dashed) of different sectors of the many-body Hilbert space to the ground state of the system as a function of mass ratio $\mu$ for weak and strong interaction $g$. Note that around $\mu\approx 2$ (the exact value depends on $g$) two sectors are strongly suppressed and the excitations are dominated only by one type of states with $k=1$. The interactions are measured in the natural units of the harmonic oscillator, $(\hbar^3\omega/m_\downarrow)^{1/2}$. []{data-label="fig3"}](fig3a.pdf "fig:"){width="\columnwidth"} ![(a) The total Frobenius distance $||{\cal G}_\tau||$ as a function of the interaction strength $g$ for the system of $N_\uparrow=N_\downarrow=4$ particles and different mass ratio $\mu$. In the inset we show the slope $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}\mu$ calculated in the region of vanishing interactions $g$. (b) Contributions ${\cal C}_1$ (solid thin), ${\cal C}_2$ (solid thick), and ${\cal C}_3$ (dashed) of different sectors of the many-body Hilbert space to the ground state of the system as a function of mass ratio $\mu$ for weak and strong interaction $g$. Note that around $\mu\approx 2$ (the exact value depends on $g$) two sectors are strongly suppressed and the excitations are dominated only by one type of states with $k=1$. The interactions are measured in the natural units of the harmonic oscillator, $(\hbar^3\omega/m_\downarrow)^{1/2}$. []{data-label="fig3"}](fig3b.pdf "fig:"){width="\columnwidth"}
To show quantitatively how the inter-component correlations change with varying parameters of the system, first, we focus on the Frobenius distance $||{\cal G}_\tau||$ as a function of the attraction $g$ for the fixed mass ratio $\mu$ (Fig. \[fig3\]). From this figure, one can easily deduce the general behavior of the system. First, it is clearly visible that independently of the mass ratio $\mu$ the inter-component correlations grow with an amplitude of interactions. This fact is in full accordance with our intuition – stronger inter-component forces lead to stronger correlations between particles. One also notices that for fixed interactions and increasing mass ratio $\mu$ correlations measured by $||{\cal G}_\tau||$ decreases, [*i.e.*]{}, particles become less correlated. We can quantify this behavior more precisely in a few different ways. The simplest one is by calculating the derivative $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}g$ close to the perturbative regime ($-0.5\lesssim g<0$) where a linear growth of correlations is visible. As shown in the inset of Fig. \[fig3\], in this range of interactions, the slope of the derivative (the second derivative of $||{\cal G}_\tau||$) evidently depends on mass and around $\mu\gtrsim 2$ the rate $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}g$ becomes almost independent of $\mu$. To find the origin of this surprising change of the slope, we performed a direct numerical inspection of the many-body ground state. We find that in the case studied, around $\mu\approx 2$, a specific change of the ground-state structure is clearly visible. It can be viewed by performing a specific decomposition of the many-body ground state. Generally, the ground-state of the system can be written as a superposition of Fock states belonging to four disconnected sectors of the many-body Hilbert space: $$|G_0\rangle = \alpha_0|F_0\rangle + \sum_{k=1}^{3}\sum_j \alpha^{(k)}_j |F^{(k)}_j\rangle.$$ The first sector contains only the ground state of the noninteracting system $|F_0\rangle$. Three other sectors spanned by vectors $\{|F^{(k)}_j\rangle\}$ have the following properties. For $k=1$, the states $|F^{(1)}_j\rangle$ are products of the noninteracting ground state of $\uparrow$ (heavier) particles and different excited states of $\downarrow$ (lighter) ones. Conversely, for $k=2$, the states $|F^{(2)}_j\rangle$ are products of different excited states of $\uparrow$ (heavier) particles and the noninteracting ground state of $\downarrow$ (lighter) particles. Finally, for $k=3$ the states $|F^{(3)}_j\rangle$ are built only from the excited states in both components. Having this decomposition, one can calculate contributions from different sectors to the interacting ground state of the system. These contributions are quantified by four numbers ${\cal C}_k = \sum_j|\alpha_j^{(k)}|^2$ and ${\cal C}_0=|\alpha_0|^2$. Inter-component correlations are encoded in excitations of the system and therefore they are directly reflected in non-vanishing values of ${\cal C}_k$ with $k=1,2,3$. In Fig. \[fig3\]b we plot these quantities as functions of the mass ratio $\mu$ for $g=-1$ and $g=-5$. As it is seen, for equal mass system $\mu=1$, all sectors of the system’s excitations contribute to building the correlations. However, when the mass ratio increases, one of the sectors ($k=1$) starts to dominate. At the same time the other sectors are strongly suppressed and from around $\mu_c\approx 2$ (the exact value depends on interaction strength) the many-body ground state can be written almost perfectly as a superposition of Fock states having all heavy fermions located in the lowest harmonic oscillator orbitals. It means that for $\mu>\mu_c$ the inter-component correlations are much less sensitive to any further variations of the mass ratio and they come only from variations of the internal structure of the lighter component.
The transition in the ground-state structure at around $\mu_c$ can be also visualized by plotting the distance $||{\cal G}_\tau||$ and its derivative $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}\mu$ as functions of the mass ratio $\mu$ for different numbers of particles and different interactions (see Fig. \[fig4\]). As it is seen, for some particular value of the mass ratio $\mu_c\approx 2$ the derivative has a clearly visible minimum. Although the critical value $\mu_c$ depends on system parameters (interaction $g$, number of particles), the mechanism is always the same — for mass ratio larger than $\mu_c\approx 2$ the exact form of the ground-state is significantly simplified and it manifests a very high probability of finding all heavy particles in their noninteracting ground state.
To lower the energy of the attractive system it is preferred that the two kinds of particles have the same spatial distributions — then the interaction integrals $U_{ijkl}$ are the largest. In principle, spatial distributions can be adjusted by exciting particles to higher single-particles orbitals. Although the excitation cost is the same for both components, due to the different length scales for the components, adjusting the density profile of heavier particles requires much more excitations. Therefore it is energetically favorable to excite light particles keeping the heavy component almost in the noninteracting ground state. This phenomenological explanation is in full accordance with our numerical many-body calculations described above. Obviously, it cannot predict the exact value of the critical mass ratio $\mu_c$ which is surprisingly small. Let us also note that this argumentation is also in full agreement with the mechanism of the spatial separation induced by repulsions in mass-imbalanced systems described in [@Pecak2016Separation].
To make the analysis as complete as possible, we also discuss the inter-component correlations in terms of the von Neumann entropy which has been successfully used for the bosonic system also in the context of the mass imbalance [@garcia2016entanglement]. In contrast to the noise correlation which is based on the two-particle reduced density matrix, the von Neumann entropy is calculated from the reduced density matrix of the whole component. Therefore, it quantifies the total amount of correlations between components. It is defined straightforwardly as $${\cal S} = -\mathrm{Tr}\left(\hat\rho_\uparrow\ln\hat\rho_\uparrow\right),$$ where $\hat\rho_\uparrow=\mathrm{Tr}_\downarrow\left(|G_0\rangle\langle G_0|\right)$ is the reduced density matrix of a chosen component calculated by tracing out the remaining component’s degrees of freedom. As it is shown in the bottom row of Fig. \[fig4\], the behavior of the von Neumann entropy is in full agreement with predictions based on the noise correlation. With increasing mass ratio $\mu$ the entropy rapidly decreases and after crossing some $\mu_c$ it slowly saturates on a small non-zero value. When comparing the von Neumann entropy $\cal S$ to the Frobenius distance $||{\cal G}_\tau||$ we see that these both quantities behave in a very similar way. It may suggest that all inter-component correlations are encoded mostly in the two-particle ones which probably dominate in the attractively interacting system.
![The Frobenius distance $||{\cal G}_\tau||$ (upper row), its derivative $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}\mu$ (middle row), and the von Neumann entropy $\cal S$ (bottom row) as functions of the mass ratio $\mu$ for different number of particles and different interactions. Note the clearly visible minima of derivatives at which the distances $||{\cal G}_\tau||$ and the entropies $\cal S$ significantly change their behaviour. It is related to the change in the structure of the many-body ground state of the system. See the main text for details. []{data-label="fig4"}](fig4a.pdf){width="\columnwidth"}
Role of the Quantum Statistics {#sec:boseFermi}
==============================
![ The noise correlation ${\cal G}_\rho$ and ${\cal G}_\tau$ calculated for the strongly attractive Bose-Fermi system ($g=-5$) described by the Hamiltonian in the regime of strong repulsions between bosons [($g'=8$)]{} for $N_B=4$ bosons and $N_F=4$ fermions. While the inter-component correlations in the position domain are very similar to those obtained for fermionic mixtures, the anti-correlations in the momentum domain predicted previously are significantly destroyed even in the equal mass case ($\mu=1$). Position and momentum are measured in natural units of the harmonic oscillator, $\sqrt{\hbar/m_\downarrow\omega}$ and $\sqrt{\hbar m_\downarrow\omega}$, respectively. []{data-label="fig5"}](fig5.png){width="\columnwidth"}
Finally, let us also discuss the role of the quantum statistics in forming inter-component correlations in the system studied. This analysis can be done systematically by changing one of the fermionic components to the bosonic one with the same number of particles and masses. In such a case the system is described by the modified Hamiltonian of the form: $$\begin{gathered}
\label{eq:hamBF}
\hat{\cal H} = \int\!\!\mathrm{d}x\,\left[\hat{\Psi}^\dag(x)H_F\hat{\Psi}(x)+\hat{\Phi}^\dag(x)H_B\hat{\Phi}(x)\right] \\
+ g\int\!\! \mathrm{d}x\,\hat{\Psi}^\dag(x)\hat{\Phi}^\dag(x)\hat{\Phi}(x)\hat{\Psi}(x) \\
+ g'\int\!\! \mathrm{d}x\,\hat{\Phi}^\dag(x)\hat{\Phi}^\dag(x)\hat{\Phi}(x)\hat{\Phi}(x).\end{gathered}$$ Here, the single-particle Hamiltonians $H_F$ and $H_B$ are given by equations , respectively (we replace spin-$\uparrow$ fermions described by the field $\hat\Psi_\uparrow(x)$ by interacting spinless bosons described by the field $\hat\Phi(x)$). The bosonic field operator $\hat{\Phi}(x)$ obeys the standard commutation relations $[\hat\Phi(x),\hat\Phi^\dagger(x')]=\delta(x-x')$ and $[\hat\Phi(x),\hat\Phi(x')]=0$.
The additional term in the Hamiltonian which is proportional to $g'$ describes mutual interactions between bosons. To mimic the fermionic nature of bosons, in the following one assumes that $g'$ tends to infinite repulsions, [*i.e.*]{}, according to the Bose-Fermi mapping there exists a one-to-one correspondence between bosonic and fermionic wave functions in the component with replaced statistics [@Girardeau1960]. In consequence, the spatial densities of infinitely repelling bosons and noninteracting fermions are exactly the same. Note, however, that there is a significant difference when density distributions of momenta are compared. This theoretical prediction was recently observed experimentally for two distinguishable fermions [@zurn2012fermionization] as well as for bosons confined in elongated traps [@paredes2004tonks; @kinoshita2004observation]. Of course, it is not possible to set $g'\rightarrow\infty$ in the numerical approach used. However, we checked that in the case of four bosons setting $g'=8$ appropriately mimics very strong repulsions for bosons and we use this value as the benchmark of infinite repulsions (see Appendix \[appendixBosons\] for details).
In the limit of infinite repulsions between bosons ($g'\rightarrow\infty$), the only difference between the two systems studied (modeled by Hamiltonians and ) lies in the symmetry of the many-body wave function under exchange of two heavy particles. Despite this fact, the inter-component correlations (in the momentum domain) forced by attractive forces have significantly different properties. As it is seen in Fig. \[fig5\], the noise correlations in the position domain for Bose-Fermi and Fermi-Fermi mixtures are very similar independently of the mass ratio $\mu$. This observation is a direct manifestation of the mapping mentioned above. However, in the momentum domain, the correlations described by the noise ${\cal G}_\tau$ are completely different. Even for the equal mass case $\mu=1$ the evident anti-correlation of momenta, previously clearly visible for fermions, is smeared and destroyed. This observation strongly suggests that the fermionic statistics present simultaneously in both components is crucial in building strong pairing (anti-correlations in momenta) in the system.
Conclusion {#sec:conclusion}
==========
To conclude, in this paper we discussed the properties of a two-component mixture of a few ultra-cold atoms in a one-dimensional harmonic trap with attractive mutual interactions. We focus on the inter-component correlations in terms of the noise correlation which effectively filters out single-particle features of the system from the two-body densities. In this way, we show that inter-component correlations strongly depend on the mass ratio between the atoms forming individual components. When the mass ratio is above $\mu_c\approx2$, the many-body ground state of the system undergoes a specific transition of its structure and it can be viewed as an almost perfect product of the noninteracting ground state of the heavier component and some well-defined state of the lighter particles. In consequence, inter-component correlations are strongly suppressed and are almost insensitive to the strength of attractive mutual interactions. Our numerical calculations predict a surprisingly small value of the critical mass (much below the ratio for a K-Li mixture, $\mu=40/6$, and Dy-K mixture, $\mu=161/40$) at which the transition occurs. It is much smaller than for an impurity problem in the bosonic system studied recently [@garcia2016entanglement].
In addition, by studying two-component Bose-Fermi mixtures, we show that the quantum statistics play a crucial role in forming inter-component correlations. In this kind of a system, the anti-correlation of particles is strongly disturbed even in the system of equal mass components.
Since the noise correlation can be measured in nowadays experiments [@bergschneider2018correlations], our results may shed some light on incoming experiments with attractively interacting few-body systems. Our analysis is quite general and it might be also important for building our understanding of different condensed matter problems related to the ’few’ to ’many’ crossover or unconventional superconductivity which originates in a pairing of different-mass fermions [@HFsystems1984; @matsuda2007fulde; @kinnunen2018fulde]. For the same-mass fermions, the Fermi surfaces of both components match each other. Whenever the Fermi surfaces do not match perfectly, a non-zero net momentum of pairs together with unconventional correlations may appear in the system. A similar mechanism can occur for equal-mass systems with different trapping frequencies.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Remigiusz Augusiak for valuable comments and Jacek Dobrzyniecki for a thorough reading of the manuscript. This work was supported by the (Polish) National Science Center Grants No. 2016/21/N/ST2/03315 (DP) and 2016/22/E/ST2/00555 (TS). Numerical calculations were partially carried out in the Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw (ICM) under the computational grant No. G75-6.
Numerical convergence
=====================
\[AppendixConv\]
![ The successive fidelity ${\cal F}_K$ of the many-body ground state of $N_\uparrow=N_\downarrow=4$ equal-mass fermions obtained by the exact diagonalization for different values of the cut-off $K$ and two interaction strength (a) $g=-1$ and (b) $g=-5$. For the cut-off large enough the fidelity stabilizes (horizontal dashed line). Note that for better visibility, we use a nonlinear scaling on the horizontal axis. []{data-label="Fig6"}](fig6.pdf){width="\columnwidth"}
![Dependence of different quantities on the cut-off $K$. (a) The Frobenius distance $||{\cal G}_\tau||$ calculated for $N_\uparrow=N_\downarrow=4$ and two different mass ratios $\mu=1$ and $\mu=6$ (compare with Fig. \[fig3\]a). (b) A derivative of the Frobenius distance $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}\mu$ calculated for $N_\uparrow=N_\downarrow=5$ and $g=-5$ (compare with the right middle plot in Fig. \[fig4\]).[]{data-label="Fig8"}](fig7a.pdf "fig:"){width="\columnwidth"} ![Dependence of different quantities on the cut-off $K$. (a) The Frobenius distance $||{\cal G}_\tau||$ calculated for $N_\uparrow=N_\downarrow=4$ and two different mass ratios $\mu=1$ and $\mu=6$ (compare with Fig. \[fig3\]a). (b) A derivative of the Frobenius distance $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}\mu$ calculated for $N_\uparrow=N_\downarrow=5$ and $g=-5$ (compare with the right middle plot in Fig. \[fig4\]).[]{data-label="Fig8"}](fig7b.pdf "fig:"){width="\columnwidth"}
The convergence of the numerical method is ascertained by checking the successive fidelity of the many-body ground-state [@Pecak2017com]. The many-body Fock basis $\{|F_j\rangle\}$ is built from the lowest $K$ single-particle orbitals $\phi_{i\sigma}(x)$, $i\in\{0,\ldots,K-1\}$. After numerical diagonalization of the many-body Hamiltonian one obtains the many-body ground state $|G^{\{K\}}_0\rangle$ and its energy $E^{\{K\}}_0$. By performing calculations for successive values of the cut-off $K$ we calculate the successive fidelity defined as $${\cal F}_K = |\langle G^{\{K-1\}}_0|G^{\{K\}}_0\rangle|.$$ In Fig. \[Fig6\] we plot the difference $1-{\cal F}_K$ as a function of the cut-off $K$ for the system of $N_\uparrow=N_\downarrow=4$ fermions ($\mu=1$) and two different interaction strengths $g=-1$ and $g=-5$ (Fig. \[Fig6\]a and Fig. \[Fig6\]b, respectively). For convenience, we also display the corresponding sizes of the cropped many-body Hilbert space. We assume that the ground-state is found with sufficient accuracy if the changes of the fidelity are stabilized with the increasing cut-off (horizontal lines in Fig. \[Fig6\]). Note, that we use a nonlinear scaling on the horizontal axis, [*i.e.*]{}, the fidelity changes very slowly (in the stabilization region) with an increasing dimension of the Hilbert space. In these cases no significant changes of single- and two-particle densities with increasing cut-off are visible. The differences are also not significant when other quantities discussed are considered. In Fig. \[Fig8\]a we show the Frobenius distance $||{\cal G}_\tau||$ for $\mu=1$ and $\mu=6$ calculated with different cut-off $K$. As it is seen, in the range of attractions considered ($|g|\lessapprox 5$) the final result is well converged for $K\ge 10$. The situation is less obvious in the case of a derivative of the Frobenius distance $\mathrm{d}||{\cal G}_\tau||/\mathrm{d}\mu$ (middle row in Fig. \[fig4\]) which is much more sensitive to any changes of the Fock basis (see example for $N_\uparrow=N_\downarrow=5$ and $g=-5$ case in Fig. \[Fig8\]b). However, all the curves with $K\ge 10$ unambiguously support the observation that the ground state of the system undergoes a specific transition at around $\mu_c\approx 2$.
Fermionization limit
====================
\[appendixBosons\]
![ Comparison of the spatial properties of $N_F=4$ noninteracting fermions and the same number of repelling bosons for different interaction strengths $g'$. (a) The single-particle density distributions for noninteracting fermions and bosons with different repulsions. (b) The two-particle density distributions of corresponding four-particle systems. Note that in the case of $g'=8$ the single- and two-particle distributions are very close to the corresponding distributions of noninteracting fermions. []{data-label="Fig7"}](fig8a.pdf "fig:"){width="\columnwidth"} ![ Comparison of the spatial properties of $N_F=4$ noninteracting fermions and the same number of repelling bosons for different interaction strengths $g'$. (a) The single-particle density distributions for noninteracting fermions and bosons with different repulsions. (b) The two-particle density distributions of corresponding four-particle systems. Note that in the case of $g'=8$ the single- and two-particle distributions are very close to the corresponding distributions of noninteracting fermions. []{data-label="Fig7"}](fig8b.pdf "fig:"){width="\columnwidth"}
In the case of mixed statistics mixtures (bosons and fermions) discussed in Sec. \[sec:boseFermi\], we assumed that the infinite repulsion limit between bosons ($g'\rightarrow\infty$) is appropriately mimicked by the finite value of interactions $g'=8$. To clarify this assumption we compare the ground-state spatial properties of $N_F=4$ noninteracting fermions (having the same spatial properties as infinitely repelling bosons) to $N_B=4$ bosons with different mutual interactions, $g'\in\{0, 1, 8\}$. Assuming that the ground state of $N_B$ interacting bosons ($N_F$ fermions) is $|G_B\rangle$ ($|G_F\rangle$) one defines the single-particle density as
\[rhoBF\] $$\begin{aligned}
\rho^{(1)}_B(x)&=\langle G_B|\hat\Phi^\dagger(x)\hat\Phi(x)|G_B\rangle, \\
\rho^{(1)}_F(x) &= \langle G_F|\hat\Psi^\dagger(x)\hat\Psi(x)|G_F\rangle\end{aligned}$$
for bosons and fermions, respectively. These distributions are shown in Fig. \[Fig7\]a. Similarly, the two-particle density distributions for the same systems of bosons and fermions are shown in Fig. \[Fig7\]b and are defined as
\[rho2BF\] $$\begin{aligned}
\rho^{(2)}_B(x;y)&=\langle G_B|\hat\Phi^\dagger(x)\Phi^\dagger(y)\hat\Phi(y)\hat\Phi(x)|G_B\rangle, \\
\rho^{(2)}_F(x;y) &= \langle G_F|\hat\Psi^\dagger(x)\Psi^\dagger(y)\hat\Psi(y)\hat\Psi(x)|G_F\rangle,\end{aligned}$$
respectively. It is clearly seen that the distributions of noninteracting fermions and noninteracting or weakly interacting bosons are essentially different. However, for strong repulsions the bosonic system undergoes fermionization and for $g'=8$ its spatial distributions become very close to corresponding distributions of the noninteracting fermionic system. This observation supports our assumption that the system of $N_B=4$ bosons interacting with the strength $g'=8$ can be safely treated as a benchmark of infinite repulsion.
|
---
abstract: 'Cities are different around the world, but does this fact have any relation to culture? The idea that urban form embodies idiosyncrasies related to cultural identities captures the imagination of many in urban studies, but it is an assumption yet to be carefully examined. Approaching spatial configurations in the built environment as a proxy of urban culture, this paper searches for differences potentially consistent with specific regional cultures or cultures of planning in urban development. It does so focusing on the elementary components shaping cities: buildings and how they are aggregated in cellular complexes of built form. Exploring Shannon’s work, we introduce an entropy measure to analyse the probability distribution of cellular arrangements in built form systems. We apply it to downtown areas of 45 cities from different regions of the world as a similarity measure to compare and cluster cities potentially consistent with specific spatial cultures. Findings suggest a classification scheme that sheds further light on what we call the ‘cultural hypothesis’: the possibility that different cultures and regions find different ways of ordering space.'
author:
- |
Vinicius M. Netto\
Associate Professor\
Department of Urbanism\
Universidade Federal Fluminense (UFF)\
Visiting Scholar\
Center for Urban Science and Progress (CUSP NYU)\
`[email protected]`\
Edgardo Brigatti\
Associate Professor\
Institute of Physics\
Universidade Federal do Rio de Janeiro (UFRJ)\
`[email protected]`\
Caio Cacholas\
Researcher\
Graduate Programme of Architectural and Urban Studies (PPGAU UFF)\
bibliography:
- 'refs\_01.bib'
title: |
From form to information:\
Analysing built environments in different spatial cultures
---
=1
Introduction
============
Cities of different cultural types and different scales embody different spatial identities... human societies order their spatial milieu in order to construct a spatial culture, that is, a distinctive way of ordering space.\
Hillier [@hillier1989architecture p.5-6]
The idea that urban form embodies idiosyncrasies that express cultural identities seems to be a frequent assumption in urban studies. It has to do with the contextual role of custom and institutional settings, from regional idiosyncrasies assimilated to traditional ways of building to the dichotomies of planned and unplanned cities, shaped through top-down agencies or as chance-grown arrangements [@kostof1991city]. However, can local cultures actually leave traces in urban space? Despite its persistence in the urban imagination, the problem of how built environments might embody specific cultural identities seems yet to be fully addressed in urban morphology. To begin with, there is an “evident lack of a quantitatively rigorous, comprehensive and systematic framework for the analysis of urban form” [@venerandi2017form]. In this sense, historically- and culturally-informed quantitative methods are essential for uncovering forms and patterns resulting from city organization processes [@boeing2019urban]. In this paper, we look closely into that assumption, and address the question of whether cities find distinct regional characteristics as material forms and cultural milieu, or take on physically specific forms under certain cultural conditions [@hillier1989architecture]. This implies examining the existence of contextualised ways of shaping cities – and features that might transcend context. We shall do so approaching the spatial configurations of the built environment as a proxy of urban culture, looking into the very constituents of urban form. Differently from emphases on street networks [@hillier1993natural; @porta2006network], our approach focuses on the elementary components shaping the tangible spaces of cities: buildings and how they are aggregated in complexes of built form. It also means taking into account a feature that seems to differentiate cities from non-urban settlements: the systems of built forms arranged in urban blocks. Closely related to systems of streets and open spaces, the urban block has become emblematic, uniquely defining the form of cities in urban societies emerged in regions and cultures seemingly with no contact with one other [@netto2017social]. We will look into 45 cities around the world and measure their spatial configurations to assess differences and similarities between them. In order to do so, we shall lay down an approach based on Shannon’s [@shannon1948mathematical] measure of information and entropy. We will argue that Shannon’s measure is particularly suited for the task of capturing amounts of information related to randomness and order in configurations of built form. Our approach takes the following steps:
- Inquire into built form as ‘spatial culture’.
- Propose a measure of configuration of built form based on Shannon’s entropy.
- Apply this measure to examine cities of different regions of the world.
- Finally, use the results as a similarity measure to compare and cluster the studied cities, as ‘information signatures’ potentially associated with specific regions or spatial cultures.
Does culture leave traces in urban space?
=========================================
‘Culture’ was famously described by Raymond Williams [@williams1995sociology] as one of the most complex words in the English language, an elusive phenomenon notoriously difficult to conceptualise and frequently challenged as an explanatory category [@mitchell2000cultural; @gregory2011dictionary]. We use the term not to refer to an ‘independent entity’ with explanatory force [@duncan1980superorganic] but as an ongoing process involving the practices, works and products of human activity situated in time and place. Therefore, ‘culture’ is embedded in material contexts and social frameworks, and relates to institutions and institutionalised behaviours, values, meanings and orientations, and capacities for self-regulation [@bennett1998culture]. Such processual notion of culture also as a field of action [@hastrup2004action] takes into account forms of self-organisation and coordination between agencies in material production. In this sense, we wish to explore urban form as an expression of cultural systems as inherently material processes, aware of potential contingencies that must be considered in empirical analysis. One of the under-examined assumptions about the connections between society and urban form is that the latter may somehow express cultural identities that constitute the former. Conzen [@conzen1960alnwick] was a pioneer in studying patterns of change in urban form in relation to changes in the economic, social, and technological milieu, proposing a cyclical nature of the development of urban form [@rashid2017geometry]. Going a step further, Aldo Rossi [@rossi1982architecture] argued that the material form of the city is intrinsic to its sociological and cultural reality. Later on, Hillier [@hillier1989architecture] addressed the possibility of cities of different cultural types embodying different spatial identities. His analytical approach allowed him to claim that human societies order their built environment to construct a ‘spatial culture’, a ‘distinctive way of ordering space’. Cities take on different forms in different cultural conditions in non-contingent ways, as spatial arrangements shape the field of encounters that animate different social cultures. Physical space is systematically ordered to reproduce culturally-specific patterns of social behaviour based on co-presence as a principle for ordering social relations.
Recent discussions have enriched this construct by approaching a spatial culture as “a fundamentally performative and temporal process” [@griffiths2016spatial p.xxiv] Focusing on “questions of cultural specificity in the formation of space”, these works assess how culture affects spatial formation, and the possibility of “encoding and transmitting social and cultural information” in urban space [@koch2016spatial]. Contingencies are added by the possibility that “different cultures invest differently in space, be it in regards to what is manifested, or to what extent society is manifested through built form” [@koch2016spatial] (p.i). Difficulties are also of an epistemological nature, since the space-culture relation may be indivisible analytically [@palaiologou2016reclaiming], and therefore hard to be scrutinised. In a careful opposition to some of these views, the late urban historian Spiro Kostof [@kostof1991city] was suspicious of the belief that buildings and city-forms fully embody recognisable idiosyncrasies enough to be medium of cultural expression. Even though his works relate processes like ‘reading’, ‘encoding’ and ‘information’ to culture, he claimed that a same urban form does not express an invariable human content. Despite this position, the quest for underlying explanations for systematic differences in urban form has led to the idea that physical patterns encapsulate an extra-physical reality, as different social and cultural agencies are seen to shape physical space. These agencies can range from tradition and custom, material requirements of interaction, associations with socially shared symbols or principles of societal organisation. For instance, cities with irregular physical patterns are thought to be the result of development left entirely to individuals, as bottom-up processes leading to the random ways of the unplanned city. In turn, top-down processes triggered by governing bodies would be able to guide the organisation of urban land and built form, leading to uniformly ordered cities \[Castagnoli in [@kostof1991city], p.43)\]. Some studies looked into spatial features, logics or organising principles in comparative studies of cities consistent with distinct regions. To be sure, most of these works deal with street networks rather than built form systems. Medeiro’s [@medeiros2013urbis] topological analysis of betweenness centrality and depth in the street networks of 164 cities in different parts of the world identified regional differences. For instance, American and Canadian cities appear prominently with the highest levels of accessibility, as opposed to Brazilian cities in South America, the most spatially segregated. Louf and Barthelemy [@louf2014typology] searched for the ‘fingerprints’ of cities analysing the distribution of blocks extracted from street networks of 131 city centres. Their classification scheme is based on information about the area and a simplified proxy for the shape of blocks. The method identifies that nearly two thirds of American cities in their sample are structurally different from European cities. Rashid [@rashid2017geometry] carried on a comparative study of urban morphology in 104 cities in six continents. Using uni-variate statistics of data for 44 spatial measures of street configurations and basic geometric measures of built form, like block perimeters and areas, he found limited differences between downtown areas in developed and developing countries. Furthermore, disaggregated measures of geometry of urban layouts have little power to describe actual form (say, of blocks) and do not grasp information encoded in relations between components of built form. In turn, our configurational approach focuses on buildings as the elementary components shaping cities, and how they are aggregated in combinations and complexes. By looking into frequencies of cellular arrangements representing buildings in selected cities, we wish to understand *if* and *to what extent* their configurations can be seen as particular cultural features, regardless of whether these features are intentionally embodied in urban space. Recognising that urban structures are different around the word, and approaching spatial configuration as a proxy of urban culture, we attempt to measure such configurations to assess their differences and similarities. For that, we shall explore Shannon’s view of ‘information’ and ‘entropy’ to investigate whether spatial cultures entail ‘distinctive ways of ordering space’, as Hillier suggests.
Information and entropy in physical spatial systems
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A number of works have explored information and entropy measures in relation to urban systems, beginning with Wilson’s [@wilson2011entropy] pioneering study of utility-maximising systems in 1970. The entropy-maximising paradigm was frequently used to derive model formulations for spatial interactions and urban distributions, microeconomic behaviour and input-output analysis [@batten1981entropy]. Batty developed a number of studies on entropy in spatial aggregations and interaction since the early 1970s [@batty1972entropy; @batty1974spatial; @batty1976entropy]. More recently, Batty et al. [@batty2014entropy] proposed a measure of complexity based on Shannon information able to grasp the complexity of cities as they vary in scale, size and spatial distribution of population, dealing with spatial entropy related to the distribution of information, and with information density related to city size. Other approaches used modifications of Shannon entropy and information-theoretical metrics as methods to capture, quantify and group similar two-dimensional spatial patterns in landscape ecology, including efforts towards a universal classification of configuration types in a linear sequence according to increasing values [@claramunt2012towards; @altieri2018new; @nowosad2019information]. Entropy measures have also been applied to purely urban morphological problems, namely in street network analysis. Gudmundsson and Mohajeri [@gudmundsson2013entropy] developed a method based on Shannon’s entropy to measure angular variation between streets, applied to 41 British cities. Boeing [@boeing2019urban] applied Gudmundsson and Mohajeri’s method to analyse 100 cities around the world focusing on street networks downloaded from Open Street Maps (OSM). However interesting as morphological approaches, these applications do not seek to uncover spatial information patterns, focusing instead on entropy as a measure of variation in street angles and lengths. The entropy measure of the distribution of crossing angles does not necessary capture the global degree of order/disorder of street networks. Even if they do, street orientation does not describe spatial configurations in a relational sense. These considerations are reflected by the fact that the resulting values of the measure applied are concentrated in the extremes, suggesting that it is not so sensitive. Furthermore, entropy measures applied to street orientation are not a comprehensive morphological approach since they not take into account entropy in built form. In other words, they ignore discrepancies between levels of order in street networks and built form systems. Cities can be physically disordered even if their street networks are perfectly ordered. As Kostof [@kostof1991city] put it, “\[s\]treets that read as straight and uniform on the city plan may be compromised by the capricious behavior of the bordering masses” (p.44). In short, we can have low entropy in street orientation, yet highly disordered morphological structures (figure \[fig:Fig 1\_built form variation\_h.png\]).
![A same street network can support very different built form systems. []{data-label="fig:Fig 1_built form variation_h.png"}](Fig1_built_form_variation_h.png){width="100.00000%"}
More comprehensively, Haken and Portugali [@haken2003face; @haken2014information] focused on how the built environment actually embodies information. They explore Shannon information quantitatively in connection with Haken’s synergetic qualitative approach to semantic information, in order to empirically assess how basic cellular arrangements and categorisations of building facades convey different amounts of information. Finally, other approaches to spatial information have adopted different measures of entropy, distribution of spatial co-occurrences, or information density to assess the amount of redundancy and grouping related to cognitive efforts to extract task relevant information from the built environment [@woodruff1998constant; @rosenholtz2007measuring]. In turn, our approach will explore Shannon entropy to measure levels of randomness and disorder in physical space, namely in cellular arrangements of built form. We shall look into the possibility that consistent differences between cities can be perceived at this scale, and that cultures and regions find different ways of ordering such configurations, which may be captured by this measure.
Analysing built form systems
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Our first procedure involves a reduction of urban form to two-dimensional arrangements based on building footprints. Since Giambattista Nolli’s 1748 Map of Rome, the figure/ground diagrams have become a classic methodological resource in urban studies, showing built/unbuilt distinctions. For instance, Nineteenth century scholar Camillo Sitte [@sitte1979art] represented public buildings and the ordinary fabric of the city exploring such diagrams. More recently, Rowe and Koetter [@rowe1983collage] have described the theoretical significance of the figure-ground map or ‘Nolli map’ [@rashid2017geometry; @hwang2005heuristic; @verstegen2013giambattista]. The figure-ground diagram provides a spatial data-driven method to analyse and study the urban form and circulation networks that structure human activities and social relations [@boeing2019urban]. Our second procedure looks into different cellular arrangements of built form and attempts to characterize their configurations. We do so analysing the probability distribution of built form configurations, by estimating the Shannon entropy [@shannon1948mathematical] of Nolli maps of different cities of the world. Of course, this has to do with the level of randomness in the cellular arrangements of built form in cities. By analysing cellular arrangements, we capture the structures of urban blocks in relation to the open spaces of streets and public squares. Indeed, the layout of the environment encodes more information than two-dimensional configurations can express. However, we opted for an analytic approach able to sufficiently describe differences in built form – hence the reduction of 3-dimensions (3D) urban form to 2-dimensions (2D) cellular aggregations (figure \[fig:3D\_2D\_config2.png\]).
![Reduction from 3D to 2D configurations of built form (Manhattan, NYC). []{data-label="fig:3D_2D_config2.png"}](3D_2D_config2.png){width="80.00000%"}
We characterise the spatial information encoded in two-dimensional configurations of buildings in the following terms. As mentioned, information will be quantified measuring Shannon entropy, operationally estimated by looking at the sequence of bits 1 and 0 representing built form cells and open space cells within sections of cities. Theoretically, this corresponds to measuring the Shannon entropy of a 2D symbolic sequence of 1 and 0. In this context, information finds a precise meaning: the entropy of the sequence, a measure of the surprise a source that produces the sequence causes in the observer [@shannon1948mathematical]. Physical arrangements characterised by higher levels of randomness, uncertainty or unpredictability are associated with high entropy. In contrast, the presence of regularities and patterns in urban structures corresponds to lower entropy, which means a higher predictability.
The next step involves the preparation of our set of empirical cases, and the conversion of city maps into Nolli maps. We selected cities for their importance in their region or country. Selection also had to take into account the availability of information on built form. Many cities, particularly in Latin America, Africa and Asia, have incomplete information regarding building footprints, i.e. their precise location, position and form.
For methodological reasons, we selected areas within these cities for the application of our measure. This selection procedure follows two critical considerations. The first and most important one observes that it is interesting to decouple the analysis of urban structures between small-scale, detailed and denser urban areas, and large-scale regional and peripheral urban areas. In fact, the two areas are different, and for this reason, they can be naturally described using different methodologies. The first small-scale urban area is defined by specific features such as buildings and urban blocks, which introduce typical characteristic scales. This means that there are some well-defined scales related to the distance above which configurations loose their correlations. These characteristic scales define sub-systems characterised by typical local patterns (urban blocks, individual buildings and possible neighborhoods). Here, human action is the principal vector defining shapes and patterns which generally appear in a stratified form, like the ones we see in older and traditional central areas. In turn, large-scale regional and peripheral urban areas are likely to include sparse occupation, frequently with a scale-free character. This means that the characteristics of their patterns are independent of the scale we fix for analysing them. Looking at different scales, the underlying structure remains the same. In these regions, physical features linked to topography, geographical formations and barriers (e.g.. water bodies, mountains, and valleys), along with the presence of very large infrastructures (e.g. highways) might play relevant roles in the definition of the spatial patterns. In this work, we will focus only on small-scale areas with dense urban form.
The second consideration takes into account that our method is well fitted for estimating entropy for dense and continuous urban areas. Fixing the density of built form cells allows us to obtain results independent from this parameter. The high continuity and homogeneity of built form allows us to use a specific extrapolation technique that will prove useful for estimating the entropy of our 2D symbolic sequences. For these reasons, the selection of sections was based on the identification of dense areas, with a high spatial continuity in the fabric of built form. We will consider occupation rates close to 50%, which means avoiding large empty areas or rarefied patterns of urbanisation.
We prepared our sample extracting building footprints in sections of cities from the public map repository Google Maps API. We tested trade-offs between resolution and availability of data for distinct scales. We chose geographic areas of $9,000,000$ m$^2$, which were considered sufficient for representing the general spatial characteristics of dense urban areas regarding the configuration of buildings, urban blocks and open spaces of 45 cities around the world (figure \[fig:Fig3\_new2.png\]). Built form maps of the selected cities were then prepared and exported in high resolution, filtering layers and converting entities representing buildings into solid raster cells. Images underwent a re-sizing process for $1000^2$ cells and were converted to a monochrome system and then into a matrix of size $1000\times 1000$ cells with binary numerical values (figure \[fig:Figure\_4\_45 cities.png\]).\
![Location of 45 cities in our sample. Colours show amounts of Shannon entropy found by our method in cellular configurations of built form, from blue (low entropy) to red (high entropy). []{data-label="fig:Fig3_new2.png"}](Fig3_new2.png){width="100.00000%"}
![Nolli maps with building footprint distributions in downtown areas of 45 analysed cities (9,000,000 m$^2$ windows, 1,000,000 cells), extracted from Google Maps. These sections are used to compute Shannon entropy. Rotation in grids and built form systems does not affect results. []{data-label="fig:Figure_4_45 cities.png"}](Figure_4_45_cities.png){width="70.00000%"}
Estimation of the Shannon entropy of the considered 2D cellular arrangements uses a method commonly applied for estimating the entropy of sequences of symbols encoded in one-dimensional strings [@schurmann1996entropy]. For 1D data sets, the method consists of defining the block entropy of order $n$ through
$$H_n=-\sum_k p_n(k) \log_{2}[p_n(k)],
\label{1entropy}$$
where blocks are string segments of size $n$, and the sum runs over all the $k$ possible $n$-blocks. Equation (\[1entropy\]) corresponds to the Shannon entropy of the probability distribution $p_n(k)$. The Shannon entropy of the considered system (the whole 1D string) [@schurmann1996entropy; @lesne2009entropy], which we indicate with $h$, is obtained from the following limit:
$$h=\lim_{n \to \infty} H_n/n,
\label{hentropy}$$
which measures the average amount of randomness per symbol that persists after all correlations and constraints are taken into account. The above limit exists for all spatial-translation invariant systems, as demonstrated in [@cover1991elements]. More details about this method can be found in [@schurmann1996entropy; @lesne2009entropy].
This approach can be generalized to sequences of symbols in two dimensions, which correspond to our situation. We have to define the $n$-blocks for a two-dimensional matrix [@feldman2003structural]. The most intuitive idea is to consider a block of size $n$ as a square which contains $n$ cells. To obtain the sequence of $H_{n}$ also for $n$ values that do not correspond to squares, we considered blocks that interpolate perfect squares, as described in figure \[fig:blocks+20cells2.png\]. Note that there is no unique natural way to scan a 2D matrix. We tested our approach for different reasonable forms of constructing the blocks, and the use of different paths does not seem to significantly influence the estimation of $H_n$ for the considered data set.
![ Areas in Rio de Janeiro and Manhattan, NYC (left). Examples of blocks with nine cells shown in red are amplified on the right. Configurations of the type (a) show great variation, like those found in Rio, while the type (b) shows regular arrangements frequently found in Manhattan. Blocks are constructed following the fixed path represented on the bottom right. Numbers indicate the order in which cells are added to blocks. The first block of size 1 corresponds to cell 1. Neighbouring cells are added in the corresponding order. Nolli maps are scanned with this set of different cell blocks.[]{data-label="fig:blocks+20cells2.png"}](blocks+20cells2.png){width="80.00000%"}
Equation \[hentropy\] gives precisely the entropy for a theoretical infinite set of data. In real situations, where the data set is finite, our method estimates the probabilities of distinct arrangements of cells within blocks up to a certain size $n$, counting their frequencies. For example, for $H_1$, it is sufficient to have knowledge of the symbol distribution $p_1(2)$, which is approximated by the frequency of 0 and 1 present in the data set. It is important to note that it is common to find in the literature of image processing, urban studies and ecological landscapes approaches that perform some entropy based analysis measuring our $H_1$ or, at best, our $H_2$. Unfortunately, sometimes these quantities are wrongly referred to as the Shannon entropy of the system, which, in contrast, is our $h$.
If our data were a purely random set, $h$ would coincide with $H_1$, and $p_1(2)$ would give a full account of the spatial configuration. This is obviously not true for urban situations, where evident structures and strong long-range correlations are present. In this case, estimating entropy is a difficult task, as taking correlations into account means computing $H_n$ for a large $n$. In fact, the estimation of $h$ is good when the spatial range of correlations is smaller than the maximum size of the block entropy we are able to compute. This estimation can be rendered difficult because of the exponential increase in the number of distinct cells arrangements in blocks with $n$. When working with two symbols, as in our case, the estimation of $H_n$ becomes not reasonable when $2^{n}\approx N$, where $N$ is the number of elements in our data set. Thus, in our case, this condition is verified for $n\approx 20$. Even if this is a rough evaluation, it reasonably fixes the maximum size of the blocks that can be investigated with sufficient statistical quality [@lesne2009entropy]. The limit taken in equation \[hentropy\] can be empirically obtained fitting the set of $H_n/n$ points with an appropriate function and then taking its limit for $n \to \infty$.
We found heuristically that, for all examined cases, the following ansatz provides an excellent fit: $$H_n/n \approx a+b/n^c, \qquad b,c>0.
\label{fitting}$$ The fitted value of $a$ gives a reasonable extrapolation of the Shannon Entropy $h$.\
Considering our database of 45 cities from North America, Europe, Asia, Oceania, Africa and South America, our goal is to develop a classification scheme based on the similarities and differences between the entropy levels of the sampled cities. In this sense, the next step consists in performing a proximity network analysis based on the measured entropy values, with the aim of identifying the presence of communities or clusters of cities sharing similar entropy levels. In short, entropy estimation will allow us to order our pool of cities and define a classification scheme. This scheme may help us find similarities possibly consistent with same spatial cultures or world regions. Once we obtained the entropy $h$ for all considered cities, we can quantify the levels of similarity defining a distance between cities $i$ and $j$ based on the values of $h$: $d_{ij}=|h_i-h_j|$. We created a matrix of distances for the analysed cities and then defined a network where cities are nodes, and edges (links between nodes $i$ and $j$) are present only if the value of $d_{ij}$ is smaller than a fixed threshold value. The detection of clusters displayed by this network is a straightforward task considering the relatively small size of our data set.
We further developed the cluster analysis applying a method for constructing a dendrogram representation of the distance matrix. We used the unweighted pair group method with arithmetic mean (UPGMA). This method constructs a dendrogram that reflects the structure present in the similarity matrix, building a hierarchy of clusters. The algorithm used in the analysis is part of the module Bio.Phylo in the Biopython package [@biopython]. When this approach incorporates a reliable dating of entities, it can be used to identify cultural phylogenies, like in the work of Barbrook et al. in the phylogenetic analysis of written texts [@barbrook1998phylogeny; @howe2001manuscript]. This is an interesting exception. In general, like in our case, cultural objects are related in an involved form between themselves, and dating is a major challenge for long standing living entities, whether they are cities or languages [@cavalli1997genes; @cavalli2001genes; @benedetto2002language].
Results: proximity networks and hierarchical clustering
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The use of the empirical functions of equation \[fitting\] provides an excellent fit for all the considered cities. The values of the parameters $c$ are contained in the interval $[0.37,0.68]$. These values are consistent with the entropy convergence found in written texts, where $c$ ranges from 0.4 to 0.6 [@ebeling1991entropy; @ebeling1997prediction; @ebeling1994entropy], and with a result for a Beethoven sonata where an exponent 0.75 was found [@anishchenko1994power]. These results seem typical of language-like systems, where the presence of long-range order is characterised by a slowly decaying contribution to the asymptotics of the entropy for large $n$. Despite the relative slow convergence, the fine quality of the fits allows a good extrapolation of the Shannon Entropy $h$. As an example, the results for the estimation of $H_n$/$n$ and the corresponding fitting procedure for the city of Los Angeles are displayed in figure \[fig:ranking\_2.png\]. Results for the estimation of entropy $h$ for the sampled cities can also be seen on a horizontal axis in figure \[fig:ranking\_2.png\], showing how this measure introduces a clear sorting among our data.
![**Left**: An example of the estimated values of $H_n/n$ for the city of Los Angeles. The continuous line represents the best fitting of our data using the function of equation \[fitting\]. All the analysed cities present a very similar behaviour. **Right:** Estimated values of $h$ for the 45 cities under analysis.[]{data-label="fig:ranking_2.png"}](Fitting1.png "fig:"){width="40.00000%"} ![**Left**: An example of the estimated values of $H_n/n$ for the city of Los Angeles. The continuous line represents the best fitting of our data using the function of equation \[fitting\]. All the analysed cities present a very similar behaviour. **Right:** Estimated values of $h$ for the 45 cities under analysis.[]{data-label="fig:ranking_2.png"}](ranking_2.png "fig:"){width="59.00000%"}
The similarity networks were constructed fixing the threshold value to $0.018$, which corresponds to the 90% confidence interval of the extrapolated values of $h$. We chose to implement the clustering analysis in increasing subsets of our pool of cities, starting within a same region. This way, it was easier to extract and visualise potential patterns or clusters of cities sharing similar entropy levels.
We started by looking into European cities (figure \[fig:treesEu\]). Selected cities in Europe cluster in two main groups in the proximity network and corresponding dendrogram. The first one includes predominantly cities in Northern Europe, along with Barcelona and Madrid, which present lower levels of entropy. The second one includes mostly cities in Southern Europe, with higher entropy levels. The clustering displayed by the proximity network shows how Madrid and Amsterdam lie at the connection between both communities.
![ **Left:** Proximity network of the considered European cities based on the value of $h$. The edge lengths are not proportional to the levels of proximity between entropy values. **Right:** Dendrogram constructed using the UPGMA method applied to the distance matrix obtained in terms of the values of $h$. The important information displayed by the tree is its topology. The tree features essentially two main morphological groups corresponding to cities in Northern and Southern Europe. []{data-label="fig:treesEu"}](Net_Eu20.png "fig:"){width="50.00000%"} ![ **Left:** Proximity network of the considered European cities based on the value of $h$. The edge lengths are not proportional to the levels of proximity between entropy values. **Right:** Dendrogram constructed using the UPGMA method applied to the distance matrix obtained in terms of the values of $h$. The important information displayed by the tree is its topology. The tree features essentially two main morphological groups corresponding to cities in Northern and Southern Europe. []{data-label="fig:treesEu"}](Tree_Eu20.png "fig:"){width="40.00000%"}
Next, we analysed the cities of Europe and the Americas along with Lagos in Africa (figure \[fig:tresEuAm\]). We can distinguish different clusters in the proximity network. While Brazilian cities São Paulo, Rio de Janeiro and Fortaleza remain as isolated clusters, other Latin American cities Mexico City, Ecatepec and Lagos in Africa form a small cluster joint to a large cluster dominated by cities of Southern Europe. Another major cluster aggregates cities of Northern Europe and Canada, along with US city San Francisco, Spanish cities Barcelona and Madrid, and South American cities Buenos Aires and Santiago de Chile. A smaller connected cluster is formed by major cities in the United States, whereas Chicago stands as an outlier. The dendrogram further clarifies these relations: Mexico City and Ecatepec along with Lagos share a common branch with the cluster comprised of Brazilian cities. Major US cities Chicago, Los Angeles, New York and Washington are placed in related branches, close to other North American cities (except San Francisco). Birmingham, Santiago and Buenos Aires relate to a same branch, as cities with the lowest entropy levels in their respective regions. There is also a branch relating Northern European cities, Madrid and Barcelona, and a major cluster dominated by Southern European cities.
![Proximity network and dendrogram of the analysed European, North and South American cities, and Lagos in Africa. []{data-label="fig:tresEuAm"}](Net_EuUSa36_fix.png "fig:"){width="59.00000%"} ![Proximity network and dendrogram of the analysed European, North and South American cities, and Lagos in Africa. []{data-label="fig:tresEuAm"}](Tree_EuUsa36.png "fig:"){width="40.00000%"}
The concluding analysis joins together all the considered cities, adding the Asian and Oceanian data. The number of clusters in the proximity network is similar to the previous analysis, with the addition of a new one with the most ordered cities, Beijing and Chicago. Apart from this fact, the community structure seems unchanged. Other Asian cities distribute themselves among pre-existing clusters: most Asian cities join either the cluster dominated by Southern European cities or the cluster with most Latin American cities. Furthermore, the network shows an interesting connectivity, from the most ordered cities Beijing and Chicago to major North American cities, then to a mixed cluster formed by cities from different regions sharing relatively low entropy levels, connected through Shanghai to a large cluster dominated by Southern European cities. This cluster in turn connects to the highest entropy groups, from Mexico City to Tokyo and Brazilian cities. The complete dendrogram can be seen in figure \[fig:tresEuAmAS\].
![Proximity network and dendrogram of North American, European, Asian, Oceanian, African and South American cities under analysis.[]{data-label="fig:tresEuAmAS"}](Net_EuUsaAs45_fix.png "fig:"){width="58.00000%"} ![Proximity network and dendrogram of North American, European, Asian, Oceanian, African and South American cities under analysis.[]{data-label="fig:tresEuAmAS"}](Tree_EuUsaAs45.png "fig:"){width="40.00000%"}
Discussion
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What can the proximity networks and hierarchical clusters based on entropy values tell us about the *cultural hypothesis?* That means examining the possibility that similarities in the ways of ordering built form can be explained either by (i) regional proximity, as cities from a geographically defined culture and identity (e.g. ‘American cities’, ‘Italian cities’, ‘Islamic cities’); or by (ii) similarities in the form of producing patterns historically shaped by tradition in self-organised, bottom-up processes or by top-down agencies of self-regulation, allowing us to find elements in common even between different regions. Of course, our analysis brings no value judgement in the sense of pointing out a certain level of entropy as desirable. We may start by interpreting these differences in the light of the *‘planned versus unplanned’ dichotomy* so persistent in the urban imagination [@kostof1991city]. Beginning with the analysis of European cities, we found a subtle difference in levels of entropy between Northern and Southern cities. Northern European (i.e. Anglo-Saxon, Germanic and Russian) cities in our sample displayed in general lower levels of entropy – from Birmingham (0.209) and Munich (0.225) to Amsterdam (0.254) and Vienna (0.263) – than Southern (i.e. Latin European) cities, from Rome (0.260) to Paris (0.286) and Marseille (0.292), with the exception of Spanish cities Barcelona (0.227) and Madrid (0.240). While the analysed area of Madrid is composed as a patchwork, its parts are mostly regular in themselves. In turn, a large part of contemporary Barcelona was notoriously built according to Ildefonso Cerdà’s 1859 *Eixample* orthogonal plan. These features echo the Spanish tradition of regular grids deployed in colonized regions in Latin America, coupled with a strict alignment of buildings frontal facades, and contribute to set them apart from other Southern European cities. Potential common traces in Northern cities include grids usually composed like a patchwork of partially regular areas (e.g. London, Munich, Amsterdam). This development pattern is frequently related to prior rural ownership and property boundaries. Regular grid sections relate to resources like land survey and delimitation based on measurement, prior to subdivision into building plots [@kostof1991city]. These cities also display considerable consistency in the building type adopted, leading to regularity in urban block surfaces. Even though Munich’s historical core shows curved urban blocks, frontal and back facades are predominantly aligned. Geometric variation in the position of rear facades may be intense, combined with frequently sinuous urban blocks (e.g. Moscow, Vienna, Brussels). In turn, frequent curves in streets and block systems may follow medieval footpaths of previous open fields and rural field divisions related to landscape features (e.g. historic cores of Milan, Lisbon and Athens). Practical modes of plot division and building seem to closely relate to topography (e.g. Lisbon) and watercourses (e.g. Nice, Toulouse and Zaragoza). In these areas, buildings can be frequently strung along topographic lines and watercourses. Despite such irregular features, there is considerable consistency in the position of frontal facades aligned along streets and open spaces.
To be sure, *bottom-up processes of cellular aggregation* take morphogenetic paths involving randomness [@hillier1984social; @batty2013], trial and error [@alexander1964notes; @alexander2003new], and path dependence [@netto2017fabric]. These are processes where location decisions may influence the direction of subsequent decisions. If an urban system shows positive feedback from a particular configuration, an increasing proportion of that choice increases the probability of another building being added in a similar way to the system, favouring the dominant pattern. This means that the built form system can phase-lock in a specific, path-dependent configuration. Geometric consistencies resulting from trial and error processes and urban advantages triggered by increasing densities and decreasing internal distances [@netto2017social] can be reproduced as traditional modes of building. This process may eventually lead to institutionalised rules, like those prescribing particular building types, facade alignments or uniform setbacks even along originally unplanned street networks.
When we take the 45 cities into account, we notice three main branches in the hierarchical clusters (figure \[fig:tresEuAmAS\], at a branch length around 0.075). A first cluster clearly emerges with cities with the lowest levels of entropy in the sample lower entropy cases. It is further divided into three initial branches. Beijing ($h=$0.111) and Chicago (0.116) have the lowest levels of entropy, and are in a branch of their own. Beijing is an exception in the Asian context, which generally has higher entropy values, from Shanghai (0.243) and Kyoto (0.206) to Tokyo (0.380). Beijing is probably the most strictly planned city in China. Planning was implemented rigorously along cardinal directions (East, West, South, North) following a tradition traced back to early Ming dynasty (1368-1644 AD), in turn based on ‘regulations of construction’ from the Fifth century BC, as expressions of both regal power and social order [@wainwright2016]. Buildings and urban blocks frequently display regular forms and aligned facades. In turn, Chicago epitomises the US tradition of planning cities based on orthogonal grids – and it does so with great regularity in urban blocks and building surfaces. Other branches bifurcate into a group with major American cities New York (0.174), Washington (0.167) and Los Angeles (0.162), and configurations with the lowest entropy levels from other world regions, like Kyoto, Melbourne and Birmingham, along with other US/Canadian cities Montreal (0.190), Toronto (0.202) and Philadelphia (0.208). Interestingly, Buenos Aires (0.198) and Santiago (0.209) cluster here, quite apart from other cities in Latin America, with high entropy levels. This somewhat surprising result *runs counter the first aspect of the cultural hypothesis:* the similarity in entropy levels for cities within a same culture or region. This might have to do with the evolution of these cities in comparison to others in the Latin American region. Cities founded in the Sixteenth century by Spanish colonizers in the Americas were often created in a rigid orthogonal pattern, following the 1573 [*Ordenanzas de Poblaciones*]{}, the first code of urbanism of the early modern period in the West. This was the case for Santiago and particularly Buenos Aires, with its plain topography [@salcedo1996urbanismo]. These areas became the historical and economic core of these cities, with high density and compact patterns of built form. As these cities expanded, patchworks were added around the core’s regular structure, adding entropy to the mix. Nevertheless, the levels of order in those central configurations are felt in the analysis, bringing them to closer to cities with higher levels of order in built form, like Toronto and Philadelphia.
The second cluster highlights the highest entropy group in the sample, comprised of Brazilian cities Rio de Janeiro and São Paulo, in Latin America ($h=$ 0.391 and 0.382, respectively), followed closely by Tokyo (0.380) and another Brazilian city, Fortaleza (0.347). The third major cluster divides into communities from high to middle entropy. This cluster further bifurcates into a group with slightly lower entropy levels, Mexico City (0.303) and Ecatepec (0.320) in Mexico, Istanbul (0.322) in Turkey, and Lagos in Africa (0.315). Another bifurcating branch between the opposing clusters is comprised of Southern European cities Marseilles, Porto, Toulouse, Athens and Paris, along with Brussels. A final large branch of middle to lower entropy cities bifurcates into cities from diverse regions, like Manila (0.274) in the Philippines, Milan (0.277), Rome (0.260), Lisbon (0.268), Zaragoza (0.260) and Nice (0.262) in Southern Europe, and Amsterdam and Vienna in Northern Europe; and into more diverse groups with lower entropy cities, like Shanghai and Madrid, Moscow (0.231) and San Francisco (0.233), Sydney (0.224) and Munich, London (0.223) and Barcelona. These distinct clusters show that *we cannot associate particular levels of entropy exclusively with particular regions, a first possibility of verifying the cultural hypothesis*. We have to ask ourselves what in different regions could have triggered similar entropy levels. The idea of a planned-unplanned dichotomy suggests that we should look into the actual evolution and planning conditions existing (or not) in these different cities, many of them having faced considerable growth in the twentieth century. We checked the existence of modern planning rules that act specifically upon built form, namely: (1) Land parcelling: how land is divided into urban plots, and whether there are rules guiding the shape and regularity of plots. (2) The layout of urban blocks and streets: what are the rules for layouts – say, whether they impose orthogonal systems or ‘planned picturesque’ systems like intentionally curved and varied block shapes and street networks. (3) Regulations on building design and location: whether there are rules that specify the position of buildings in plots (e.g. frontal and lateral setbacks), and in relation to neighbouring buildings. We examined the legislation in emblematic cases in Turkey, Nigeria, China, Brazil, Mexico, United States, England and The Netherlands. We found something that goes counter the planned-unplanned account of ordered and disordered cities: cities which have top-down planning may also exhibit high built form entropy. They do have rules and government agencies that regulate building and urbanisation.
But how can high entropy in built form be somehow influenced by top-down rules? We found that cities from different regions – namely, Brazil, Nigeria, Mexico and Turkey – may have certain aspects of planning in common, which allow great variation in built form to come into being. For instance, these cities share emphases on parcel-based, piecemeal developments. New urbanised areas are mostly exempt from requirements to keep connections to neighbouring areas, including street continuity and grid alignment. Another crucial instance here is how individual buildings can be positioned in their plots. Some regulations may enforce frontal and lateral setbacks, and define rules like increasing setbacks as buildings grow taller. *Simple local rules focused exclusively on individual buildings* rather than coordinated construction among nearest neighbours *lead to a high level of fragmentation* in built form.
Going a step further, whole areas in these cities are urbanised and built by people’s own hands in informal settlements, hence apart from planning regulations. This is especially the case throughout the Twentieth century, when cities in developing countries experienced fast growth. We are likely to find high entropy mostly associated with variation in the shape of urban blocks (related to angular variation in surrounding streets) in those settlements. In short, parcel-based, piecemeal developments, patchworks of diverse blocks and street networks, and fragmented built form are key features of highly entropic urban landscapes.
![Urban sections (500x500m) with similar entropy levels, different spatial configurations: Toronto ($h=$ 0.202), Buenos Aires (0.198) and Melbourne (0.207); and Istanbul ($h=$ 0.322), Ecatepec (0.320) and Lagos (0.315).[]{data-label="Figure_10.png"}](Figure_10.png){width="80.00000%"}
All this shows that cities from distinct regions may share similar entropy levels as far as built form is concerned. Their typical combinations of cells might be different, and they might share neither geographical proximity nor common historical roots, but they still can contain similar levels of disorder, as captured by our measure (figure 10). This suggests certain common traits between different regional cultures shaping how built form is ordered. That said, even though regions do not have entropy values necessarily different from others, *individual regions do seem to converge around certain values.* This interesting pattern emerges once we visually distribute a classification of the 45 cities according to increasing entropy values on a global map (figure \[fig:Fig3\_new2.png\]). Some regions show higher levels of regularity and predictability in built form systems than others. We suggest that our measure seems to capture spatial information potentially related to *different emphases on order and coordination latent in different planning cultures*, the second aspect of the cultural hypothesis seen above. How do these findings on regional differences compare with previous studies, based on different spatial entities and methods? We have seen that Medeiros’s [@medeiros2013urbis] analysis of street networks based on betweenness centrality and topological depth identified clusters of US/Canadian cities with the highest levels of accessibility, in contrast with Brazilian cities in South America, followed by European cities. Largely echoing Medeiros’s findings, Boeing [@boeing2019urban] explored angular orientation entropy and grid order indicators to identify US/Canadian cities with the lowest orientation entropy. European cities also exhibit higher orientation entropy than Latin American cities. Louf and Barthelemy’s [@louf2014typology] classification based on block areas also identifies differences between most American cities and European cities.
US/Canadian cities display low entropy in our analysis as well, but our results on European cities differ from those studies. Consistency in built form in European cities brings entropy to lower levels. Our approach was also able to identify differences between Northern and Southern European cities. In their turn, São Paulo and Rome exhibit the highest entropy levels in Boeing’s study. In our approach, despite the varied shapes in its block system, Rome’s consistency around aligned buildings lowers its entropy to a level far from São Paulo. Interestingly, these different approaches converge about Brazilian cities: they exhibit the lowest average betweenness centrality, and highest orientation entropy and built form entropy in these different samples – probably due to an extraordinary variation in the position of buildings in plots, coupled with fragmented grid patchworks. Nevertheless, the differences between findings are clearly related to differences between the morphologies of street networks and built form systems: the fact that a same street network can support endlessly different configurations of buildings. Levels of order in street networks do not necessary cause low entropy in built form.
Conclusion
==========
In this paper, we developed an approach to spatial information based on Shannon entropy. The approach was designed to (1) measure the entropy characterising levels of order and disorder in cellular configurations present in 45 cities around the world. (2) we applied the method to investigate the hypothesis of ‘spatial cultures’ as ways of ordering urban form. Put another way, we verified whether the entropy measure could accurately grasp features and differences in built form systems; then we looked for traces of ‘information signatures’ potentially consistent with specific regions or cultures. This method is intended as a step towards a more precise understanding of *spatial cultures as emergent patterns* – i.e. how typical configurations of built form emerge from local rules of aggregation active at the scale of cellular configurations. Of course, any search for ‘information signatures’ of spatial cultures embodied in the tangible spatiality of cities faces certain risks: (a) Different cultures or regions may not have distinct ways of ordering space. In other words, there could be no ‘spatial cultures’ related to regions or even enough differences between cities to be associated with a particular culture. (b) Spatial cultures may well have specific information signatures, but these may not be encoded at the scale of local cellular configurations of built form systems. (c) In case the possibilities above were wrong, a measure of spatial information based on Shannon’s entropy in cellular configurations may not be precise enough to capture information signatures or even qualitative differences in configurations.
In the research process, our method allowed us to find the distribution and clustering of cities around certain values of built form entropy. We would like to conclude our work discussing such entropy values and clusters in connection with characteristics of these cities, including aspects particularly related to what we called ‘cultural hypothesis’: the idea that similarities in the ways of ordering built form can be explained by (i) regional proximity, as geographically defined cultures and identities, or by (ii) common features in urban morphogenesis shared by distinct regions, say rules in similar ‘planning cultures’. That meant looking for reasons of non-contingent similarities and differences between cities.
The usual association of bottom-up processes of spatial production in disordered, unplanned cities, as opposed to top-down processes of spatial production in ordered, planned cities, suggested that we should look specifically into planning rules guiding built form. We found that the ‘planned/unplanned’ dichotomy in urban studies may have been valid in pre-modern periods of certain urban cultures, but it seems of limited explanatory power once we consider modern and contemporary planning. Cities with top-down planning may also have high built form entropy. A key difference lies in the kind of rules applied and how they deal with buildings. Cities with high entropy in different regions in our sample seem to have in common rules that focus mostly on individual buildings, allowing great variation in how they are placed in plots and blocks, including increasing lateral and frontal setbacks as buildings grow taller. This focus may happen for specific periods of their histories – long enough to shape the evolution and morphology of large portions of these cities. This is the case of planning in countries like Brazil, Mexico, Nigeria and Turkey, especially when the analysed cities faced fast growth in the Twentieth century. In short, we found that *simple local rules centred on individual buildings rather than coordinated construction lead to high fragmentation in ensembles of built form*. These rules are frequently coupled with piecemeal developments and grid patchworks, including informal settlements, shaping a visible fragmentation of urban landscapes.\
Our analysis brings other findings. First, proximity networks and hierarchical clusters show similarities in cities from different regions (e.g. high entropy cities including São Paulo, Tokyo, Istanbul and Lagos), with close entropy values even if they have geometrically distinct arrangements (figure 10). This suggests that the measure does not necessarily generate specific values as exclusive ‘information signatures’ for each region, a first possibility of verifying the cultural hypothesis. Second, despite that fact, the measure seems to capture something of the ‘planning culture’ of these regions. We found higher frequencies of certain regular arrangements in cities with top-down planning coupled with a strong focus on rules for coordinated modular construction, each building adjusting and aligning to those around, taking into account systemic consequences of ensembles of built form. The high frequency of certain arrangements can also be found in cases of bottom-up processes of cellular aggregation potentially involving path dependence, i.e. built form systems locked into specific configurations, reproduced in traditional modes of building – patterns that can be eventually institutionalised into formal planning rules. This seems to be the case especially in the passage from pre-modern to modern urbanisation of European cities. On the other hand, we found plenty of variation in cellular aggregations in urban cultures that allow the construction of buildings in uncoordinated actions between individual developers. This clearly leads to less regularity and higher unpredictability in what surrounding built forms will be like as cities grow. Summing up, in both top-down and bottom-up form-making processes, local rules guiding how to position buildings in relation to others seem to trigger bifurcated developments as the built form system evolves in size and complexity, leading either into greater consistency or into greater fragmentation. But that is not the whole story, of course. We may find many possibilities in between those archetypal paths, or combinations of them in different parts of cities, like patchworks, or intermingled in layers of ordered and disordered aggregations – say, the iconic case of Manhattan, based on the regularity of a gridiron street layout, and planning rules that made room for enormous variation in built form.
Third, although regions do not necessarily have exclusive values of built form entropy, *individual regions do seem to converge around certain values.* Our results show certain consistencies, grouping cities from a same region (e.g. Brazilian cities, American cities). To use Hillier’s words [@hillier1989architecture], this echoes the idea that societies create their own spatial cultures – their distinctive ways of ordering space and shaping cities. Such finding needs to be further examined through a larger sample of cities and comparisons with other approaches, along the lines we explored above. Of course, deep historical conditions and local contingencies are likely be at play, and must be carefully taken into account.
Finally, differences between results obtained from street network-based measures and our measure of entropy shed light on the *potential dissociation between the morphology of streets and the morphology of built form systems* in every city: the endless combinatorial possibilities of configurations of buildings, missing from street network approaches, add complexity to urban phenomena and suggest the need for a renewed interest in built form systems.\
Our sample is not a random set, which would be impossible due to the lack of information on building footprints in many cities and countries. Methodologically, at the present stage, our approach takes account of the spatial information latent in the arrangements of cells capturing relations of proximity, but eventually missing some correlations at large distances. On the one hand, cellular growth shapes larger structures as fundamental features of cities – a subject explored in other works [@haken2003face; @hillier2012genetic; @battyetal1989; @batty2013]. On the other hand, humans have a clear hierarchical reading [@alexander1964city], and structures at larger scales seem to have more weight than structures at smaller scales to differentiate objects. Further development of this research will look into broader spatial structures in cities by introducing measures of statistical complexity. In addition, we wish to expand this approach to other forms of physical information, such as three-dimensional differences between buildings, physical cues and landmarks [@couclelisetal1987; @stern1999environmental].
Even though there is no value judgement in our work or claims of particular levels of entropy as desirable, different levels of built form entropy may well trigger different cognitive and practical responses from people. Higher degrees of entropy may be associated with spatial and visual surprises in navigation. Surprises can be considered desirable by some, as famously suggested by Camillo Sitte [@sitte1979art] and explored by Gordon Cullen’s [@cullen1995concise] concept of ‘serial vision’. Notwithstanding, empirical studies in spatial cognition and neuroscience have shown that certain regularities and alignment effects (say, between paths or objects like buildings, or triggered by cardinal directions) improve our judgement of relative direction in navigation and our capacity to determine the position of objects in a surrounding area, affecting intelligibility and our memory of the built environment [@mcnamara2003egocentric; @frankenstein2012map; @ekstrom2018human]. Effects of urban form on cognition are a hot research topic and could benefit from explorations into entropy and regularity in physical space as informational features in navigation [@haken2003face; @haken2014information; @hillier1996space]. Furthermore, human knowledge of spatial properties and patterns goes beyond physical information and can integrate configurational, visual and semantic aspects of an urban environment [@mulligann2011analyzing; @haken2014information; @netto2018cities]. More work is needed to understand how physical information is associated with non-physical information, and is enacted by social agents making decisions and cooperating in cities.
Acknowledgements
================
We would like to thank the following researchers, architects and urban designers for their support in the analysis of planning regulations in a number of cities around the world: Cynthia Adeokun (Lagos and London), Ilgi Toprak (Istanbul), Mayra Gamboa Gonzáles, Juan Ángel Demerutis Arenas and Claudia Ortiz-Chao (Mexico City and Ecatepec), Tatiana Rivera Pabón (Buenos Aires and Santiago), Akkie Van Ness (Oslo and Amsterdam) and Chaogui Kang (Chinese cities). We also thank Lilian Laranja for discussions on culture and built form entropy. Any errors in interpretation are the authors’ responsibility.
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abstract: 'We use images of high spatial, spectral and temporal resolution, obtained using both ground- and space-based instrumentation, to investigate the coupling between wave phenomena observed at numerous heights in the solar atmosphere. Analysis of $4170$[Å]{} continuum images reveals small-scale umbral intensity enhancements, with diameters $\sim$$0{\,}.{\!\!}{\arcsec}6$, lasting in excess of $30$ minutes. Intensity oscillations of $\approx$$3$ minutes are observed to encompass these photospheric structures, with power at least three orders-of-magnitude higher than the surrounding umbra. Simultaneous chromospheric velocity and intensity time series reveal an $87\pm8\degr$ out-of-phase behavior, implying the presence of standing modes created as a result of partial wave reflection at the transition region boundary. We find a maximum wave guide inclination angle of $\approx$$40\degr$ between photospheric and chromospheric heights, combined with a radial expansion factor of $<$$76$%. An average blue-shifted Doppler velocity of $\approx$$1.5$ kms$^{-1}$, in addition to a time lag between photospheric and chromospheric oscillatory phenomena, confirms the presence of upwardly-propagating slow-mode waves in the lower solar atmosphere. Propagating oscillations in EUV intensity are detected in simultaneous coronal fan structures, with a periodicity of $172\pm17$ s and a propagation velocity of $45\pm7$ kms$^{-1}$. Numerical simulations reveal that the damping of the magneto-acoustic wave trains is dominated by thermal conduction. The coronal fans are seen to anchor into the photosphere in locations where large-amplitude umbral dot oscillations manifest. Derived kinetic temperature and emission measure time-series display prominent out-of-phase characteristics, and when combined with the previously established sub-sonic wave speeds, we conclude that the observed EUV waves are the coronal counterparts of the upwardly-propagating magneto-acoustic slow-modes detected in the lower solar atmosphere. Thus, for the first time, we reveal how the propagation of $3$ minute magneto-acoustic waves in solar coronal structures is a direct result of amplitude enhancements occurring in photospheric umbral dots.'
author:
- |
D. B. Jess$^{1}$, I. De Moortel$^{2}$, M. Mathioudakis$^{1}$, D. J. Christian$^{3}$,\
K. P. Reardon$^{4,1}$, P. H. Keys$^{1}$, F. P. Keenan$^{1}$
title: 'The Source of Three-minute Magneto-acoustic Oscillations in Coronal Fans'
---
Introduction
============
Early white-light eclipse photographs of the Sun revealed elongated, faint columns of enhanced density stretching far out into the corona [@van50; @Sai65]. These structures, now commonly referred to as coronal plumes, can be viewed over a wide range of wavelengths, in particular the extreme ultraviolet [EUV; @Boh75; @Ahm77]. Plumes are just one type of EUV feature that are seen to exist in the solar corona. Other examples include coronal loop and fan structures, which are observed to outline the coronal magnetic-field topology, and demonstrate a wide range of oscillatory behaviour [[e.g.,]{} @Asc99; @Asc02; @Nak99; @Jes08c; @Ofm08; @Ver08; @Van08; @Bal11].
One of the first studies which uncovered propagating wave phenomena in coronal structures was that by @Def98. These authors utilised the Extreme-ultraviolet Imaging Telescope on board the Solar and Heliospheric Observatory spacecraft to identify quasi-periodic perturbations in the brightness of $171$ [Å]{} images. @DeM00 undertook a similar study using the higher spatial resolution Transition Region and Coronal Explorer [TRACE; @Han99], and concluded that these oscillations were signatures of slow magneto-acoustic waves, which propagate upwards along the coronal waveguides with velocities of $70$–$165$ kms$^{-1}$ and periods in the range $180$–$420$ s. Energy estimates for these motions exhibit an incredibly wide range of values, typically $10^{2}$–$10^{5}$ ergscm$^{-2}$s$^{-1}$ [@Def98; @DeM00].
Since magnetic fields play an essential role in plume/fan/coronal-loop formation and structuring, they are often modelled using magnetohydrodynamic (MHD) equations [@Del97]. Utilising non-linear, two-dimensional MHD simulations, @Ofm99 were able to replicate previous observational results, and concluded that outward-propagating slow magneto-acoustic waves may be able to contribute significantly to the heating of the lower corona through compressive dissipation. Furthermore, theoretical modelling has suggested that the propagation characteristics of a magneto-acoustic intensity perturbation depends on a number of factors, including the dissipation of the wave energy [@Kli04; @DeM12]. Numerical simulations indicate that thermal conduction may be the dominant damping mechanism behind the dissipation of magneto-acoustic wave energy in the solar corona [@Ofm02; @DeM03; @DeM04; @Men04].
{width="\textwidth"}
Through examination of coronal structures in the close proximity of active regions, @Flu01 and @Mar06 were able to reveal how structures situated above sunspot regions displayed intensity oscillations with a period of the order of three minutes, while oscillations in “non-sunspot” structures demonstrated much longer periodicities [@DeM02]. The authors concluded that the most likely explanation for the observed longitudinal waves revolves around a driver directly exciting the magnetic footpoints. This scenario requires the magneto-acoustic wave trains to be able to propagate from the lower solar atmosphere, through the transition region, and into the corona. Utilising numerical simulations of the Sun’s lower atmosphere, @Kho06 have revealed how longitudinal oscillations, generated with a periodicity of $\sim$$3$ minutes in sunspot umbrae, can readily propagate upwards from the photosphere and into the chromosphere.
Previously, three minute umbral oscillations have been notoriously difficult to detect at photospheric heights. @Bal87 were unable to detect any photospheric signatures using the Locarno solar station at the G[ö]{}ttingen Observatory, while @Bru85 suggested that they may get lost in the noise, as a result of their very low amplitude. Indeed, @Nag07 utilised Hinode/SOT image sequences to show how oscillatory power, at all frequencies, is significantly reduced in sunspot umbrae. More recently, @Kob08 [@Kob11] have not only detected photospheric three minute oscillations, but the authors also claim that the location of maximum chromospheric power also corresponds to a co-spatial decrease in power of the photospheric oscillations. However, the spatial resolution obtained by @Kob08 [@Kob11] was on the order of 1, so precise diagnostics of the exact umbral structures displaying three minute periodicities was impossible.
In this paper, we utilise ground- and space-based instrumentation, with high spatial, temporal and spectral resolution, to investigate the origin of $3$ minute magneto-acoustic waves observed in EUV images of coronal fan structures. We employ a multi-wavelength approach to study the photospheric counterpart of these coronal phenomena, and analyse the resulting wave propagation characteristics from the photosphere, through the chromosphere, and out into the corona.
Observations & Data Preparation
===============================
Ground-based Data
-----------------
The observational data presented here are part of a sequence obtained during $13$:$32$ – $14$:$04$ UT on $2011$ July $13$, with the Dunn Solar Telescope (DST) at Sacramento Peak, New Mexico. We employed the Rapid Oscillations in the Solar Atmosphere [ROSA; @Jes10b] camera system to image a $69$ $\times$ $69$ region encompassing active region NOAA $11250$, positioned at heliocentric co-ordinates ($-146$, $-486$), or S$26.3$E$10.1$ in the conventional north-south-east-west co-ordinate system. A spatial sampling of $0{\,}.{\!\!}{\arcsec}069$ per pixel was used for the ROSA cameras, to match the telescope’s diffraction-limited resolution in the blue continuum to that of the CCD. This results in images obtained at longer wavelengths being slightly oversampled. However, this was deemed desirable to keep the dimensions of the field-of-view the same for all ROSA cameras.
A recent addition to the DST’s imaging capabilities is a new high quantum efficiency device, the Hydrogen-Alpha Rapid Dynamics camera (HARDcam). This camera, an iXon X3 DU-887-BV[^1] model manufactured by Andor Technology, consists of a back-illuminated, $512$$\times$$512$ pixel$^{2}$ electron-multiplying CCD, with a quantum efficiency exceeding 95[%]{} at $6500$ [Å]{}. As a result, this camera is best suited to imaging in the red portion of the optical spectrum. The triggering and readout architectures are identical to the first-generation ROSA instrumentation, allowing HARDcam to be seamlessly integrated with the existing setup. To optimise the exploitation of its high quantum efficiency, HARDcam was employed behind a $0.25$ [Å]{} H$\alpha$ core filter, incorporating a spatial sampling of $0{\,}.{\!\!}{\arcsec}138$ per pixel, providing a field-of-view size ($71$ $\times$ $71$) comparable to existing ROSA image sequences.
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---------------------- --------------- ------------- ---------------
Filter Used Exposure Time Frames per Reconstructed
(ms) Second Cadence (s)
Continuum ($4170$ Å) $7$ $30.3$ $1.06$
H$\alpha$ core $35$ $27.9$ $1.26$
Ca ($8542.12$ Å) $250$ $^{a}$$3.0$ $^{b}$$43.4$
---------------------- --------------- ------------- ---------------
: ROSA/HARDcam/IBIS filter and cadence overview. \[table1\]
\
$^{a}$: Average frames per second including Fabry-Perot tuning time\
$^{b}$: Cadence of a complete Ca profile scan\
In addition to ROSA and HARDcam observations, the Interferometric BIdimensional Spectrometer [IBIS; @Cav06] was used to simultaneously sample the Ca absorption profile at $8542.12$ [Å]{}. IBIS employed a spatial sampling of $0{\,}.{\!\!}{\arcsec}097$ per pixel, which allowed ROSA’s near square field-of-view to be contained within the circular aperture provided by IBIS. Thirteen discreet wavelength steps, with ten exposures per step to assist image reconstruction, were used to provide a complete scan cadence of $43.4$ s. A whitelight camera, synchronised with the IBIS feed, was utilised to assist the processing of narrowband images. Full details of the observations presented here, including filters and exposure times used, can be found in Table \[table1\], while sample images can be viewed in Figure \[fig1\].
During the observations, high-order adaptive optics [@Rim04] were used to correct wavefront deformations in real-time. The acquired images were further improved through speckle reconstruction algorithms [@Wog08], utilizing $32 \rightarrow 1$ restorations for the G-band and 4170[Å]{} continuum images. The remaining HARDcam and IBIS images were processed with $35 \rightarrow 1$ and $10 \rightarrow 1$ restorations, respectively. Post-reconstruction cadences are displayed in the fourth column of Table \[table1\]. A full image-reconstructed IBIS scan through the Ca absorption line includes a blueshift correction, required due to the use of classical etalon mountings [@Cau08]. Atmospheric seeing conditions remained excellent throughout the time series. However, to insure accurate co-alignment in all bandpasses, broadband time series were Fourier co-registered and de-stretched using a $40 \times 40$ grid, equating to a $\approx$$1{\,}.{\!\!}{\arcsec}7$ separation between spatial samples [@Jes07; @Jes08a]. Narrowband images, including those from IBIS, were corrected by applying destretching vectors established from simultaneous broadband reference images [@Rea08; @Jes10a; @Rea11].
{width="\textwidth"}
Space-based Data
----------------
The Atmospheric Imaging Assembly [AIA; @Lem11] onboard the Solar Dynamics Observatory [SDO; @Pes11] was utilised to provide simultaneous EUV images of active region NOAA $11250$. This instrument images the entire solar disk in 10 different channels, incorporating a two-pixel spatial resolution of $1{\,}.{\!\!}{\arcsec}2$, and a cadence of $12$ s. Here, we selected $7$ EUV datasets spanning $13$:$30$ – $14$:$05$ UT on $2011$ July $13$, consisting of $175$ images in each of the $94$[Å]{}, $131$[Å]{}, $171$[Å]{}, $193$[Å]{}, $211$[Å]{}, $304$[Å]{}, and $335$[Å]{} channels. In addition, one contextual $4500$[Å]{} continuum image, acquired at $14$:$00$ UT, was obtained for the purposes of co-aligning AIA datasets with images of the lower solar atmosphere.
The EUV bandpasses were specifically chosen to cover a wide range of transition region and coronal temperatures, spanning approximately $50{\,}000$ K – $7$ MK, under non-flaring conditions. Transition region imaging is covered by the He -dominated $304$[Å]{} bandpass, with a typical formation temperature of $\sim$$50{\,}000$ K. The selected coronal channels, consisting of the $94$[Å]{}, $131$[Å]{}, $171$[Å]{}, $193$[Å]{}, $211$[Å]{}, and $335$[Å]{} bandpasses, demonstrate typical effective temperatures of approximately $7.0$ MK, $0.4$ MK, $0.7$ MK, $1.6$ MK, $2.0$ MK, and $2.8$ MK respectively [@Odw10; @Bro11]. Thus, the AIA datasets, in conjunction with our ground-based observations of the lower solar atmosphere, provide us with the ideal opportunity to investigate the coupling of lower atmospheric (photosphere and chromosphere) phenomena with their multi-million degree coronal counterparts at unprecedented spatial and temporal resolutions.
The AIA data were processed using the standard routine, and include the removal of energetic particle hits, in addition to the co-registration of images from different wavelengths on to a common plate scale. Subsequently, $200$ $\times$ $200$ sub-fields were extracted from the processed data, with a central pointing close to that of the ground-based image sequences. A sub-field image, including an outline of the field-of-view obtained with ground-based observations, is shown in the upper-left panel of Figure \[fig2\]. Using the AIA $4500$[Å]{} context image to define absolute solar co-ordinates, our ground-based observations were subjected to cross-correlation techniques to provide sub-pixel co-alignment accuracy between the imaging sequences. To do this, the plate scales of our ground-based observations were first degraded to match that of the AIA image[^2]. Next, squared mean absolute deviations were calculated between the datasets, with the ground-based images subsequently shifted to best align with the AIA reference image (see the lower panels of Figure \[fig2\] for interlaced examples). Following co-alignment, the maximum x- and y-displacements are both less than one tenth of an AIA pixel, or $0{\,}.{\!\!}{\arcsec}06$ ($\approx$$45$ km).
Data Analysis and Discussion
============================
Using a combination of ground- and space-based datasets, we were able to analyse image sequences from the photosphere through to the corona. Due to the wide range of atmospheric heights covered, this section is divided into sub-sections in which we discuss different regions of the solar atmosphere.
Photosphere
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To investigate oscillatory phenomena occurring in the sunspot umbra, we first masked out all other areas of the field-of-view. This was deemed desirable so that intrinsically faint umbral structures would become much more apparent in scaled intensity images. To isolate the umbra, we first had to create a binary map, whereby umbral pixels were given a value of ‘$1$’, and non-umbral pixels assigned a value of ‘$0$’. This form of binary map was created by first averaging the $4170$[Å]{} continuum images over the entire $32$ minute duration of the dataset. Next, the umbral pixels were defined as those with an intensity below $45$% of the median granulation intensity, and subsequently assigned a value of ‘$1$’. This accurately defined the perimeter of the dark umbra. However, some small-scale structures, which existed inside the umbra ([e.g.,]{} umbral dots; UDs), exhibited higher relative intensity values. Therefore, to include all structures which existed inside the umbral boundary, all pixels which lay inside the outer perimeter were assigned a value of ‘$1$’, while all other pixels were given a value of ‘$0$’. The time-averaged $4170$[Å]{} continuum image, including the outline of the umbral binary map, is shown in the left panel of Figure \[fig3\].
![[*[Upper:]{}*]{} Intensity time series of the four umbral dots labelled UD1 (red line), UD2 (blue line), UD3 (green line), and UD4 (pink line) in Figure \[fig3\], following temporal filtering using a $2$ – $4$ minute bandpass filter. The black line displays the intensity of the umbral region away from UD structures, devoid of long-lived or transient brightenings, and scaled (by a factor of $5$) for clarity. [*[Lower:]{}*]{} The relative phase angles between the $3$ minute umbral dot and background umbral oscillations, plotted as a function of time. The red, blue, green, and pink lines represent umbral dots UD1, UD2, UD3, and UD4, respectively, while a black dashed line highlights a phase angle of $0$$\degr$.[]{data-label="fig4"}](fig04_lowres.eps){width="\columnwidth"}
Examination of the time-averaged umbral intensity map (middle panel of Figure \[fig3\]) revealed a collection of near-circular brightenings. These brightenings are consistent with the signatures of UDs, whereby they typically exhibit diameters between $0{\,}.{\!\!}{\arcsec}2$ – $0{\,}.{\!\!}{\arcsec}8$, and intensities $1.3$ – $1.5$ times brighter than the background umbral value [@Den98; @Tri02; @Sob05]. Due to their sharp appearance in a time-averaged intensity image, it hints at the persistent nature of such structures in the same spatial location, either through long lifetimes [[e.g.,]{} $30$ minutes or more; @Sob97a; @Sob97b], or by the reappearance of new UDs in the same spatial location. Analysing simultaneous G-band and continuum filtergrams, @Rim08 found that UDs, when observed in the G-band, displayed dark fine-structuring towards their core. Contrarily, this author found that these dark features are not as readily visible in the continuum, and concluded this may be due to the reduced resolution of the longer wavelength and longer exposure images. However, our continuum images were obtained at a shorter wavelength than the G-band ($4170$[Å]{} versus $4305$[Å]{}), in addition to being captured with a much shorter exposure time ($7$ ms versus $20$ ms). Thus, we cannot attribute a lack of dark fine-scale structuring in the UDs observed in the $4170$[Å]{} continuum to either reduced spatial resolution, or exposure-time related smearing of the images. @Jes12 recently determined the contribution functions of the G-band and $4170$[Å]{} continuum filter bandpases used at the DST, and concluded that the blue continuum is formed at a height of $\sim$$25$ km, while the G-band is formed approximately $100$ km higher. This atmospheric height separation between the two bandpasses, in addition to the appearance of diatomic CH molecules in the G-band images, may contribute to the presence of dark cores observed in G-band UDs.
Since we are concerned with the larger macro-scale fluctuations in umbral intensity, rather than the very small-scale structuring ([e.g.,]{} dark features within UDs) which may be close to, if not overlap with the telescope’s diffraction limit, our photospheric umbral time series was created purely from consecutive $4170$[Å]{} continuum images. These images were multiplied by the umbral binary map, producing a dataset which could be analysed with Fourier techniques. Following the methodology of @Jes07 [@Jes07b], the wavelet analysis routines of @Tor98 were applied to the $4170$[Å]{} continuum umbral time series to search for the presence of oscillatory behaviour. Considerable oscillatory power was present throughout the sunspot umbra, with dominant peaks in the Fourier spectrum at $3$ and $5$ minutes. This is consistent with the generalised $p$-mode spectrum of sunspot oscillations [@Lit82; @Lit84; @Lit86; @Bog06]. However, when locations of high oscillatory power were examined, it became evident that the UDs displayed significant enhancements of wave amplitude, at periodicities of approximately $3$ minutes. The middle panel of Figure \[fig3\] shows the spatial locations where oscillatory power at approximately $3$ minutes is in excess of $1000$ times the background umbral value. To test whether these long-lived umbral brightenings oscillate in phase with the surrounding umbra, temporally filtered time series of these structures were first created by passing each spatially-averaged UD intensity through a $2$ – $4$ minute bandpass filter. The resulting lightcurves, dominated by power corresponding to the $3$ minute $p$-mode oscillations, are displayed in the upper panel of Figure \[fig4\], along with a filtered time series representing a region of the quiet umbra devoid of any transient brightenings. It is clear that not only do the peaks and troughs of the UD oscillations appear similar in time, but these signatures also closely follow the oscillations originating from within the background umbra.
To quantify any small-scale differences in the oscillating UD time series, a phase difference analysis with respect to the background umbra was performed. The lower panel of Figure \[fig4\] reveals the phase difference between each of the four UD time series, and an isolated region of the background umbra ([i.e.]{} the solid black line in the upper panel of Figure \[fig4\]). It is clear that a preference exists for the UDs to oscillate in phase with the background umbra. Maximum deviations in the phase angle reach approximately $\pm$$40$$\degr$, although these may be due to slight drifts in oscillation period between the respective time series. For example, the phase angle related to UD1 (red line in the lower panel of Figure \[fig4\]) drifts a total of $\approx$$80$$\degr$ over a $700$ s time duration, providing an average shift of $\approx$$0.11$$\degr$s$^{-1}$. This drift can be explained by a period discrepancy of only $20$ s between the UD and the background umbra, something which is well within the normal range of $p$-mode frequencies [@Tho85]. Thus, we interpret the umbra as a single oscillating “drum skin”, which not only induces wave motion in its inherent UDs, but also causes these structures to oscillate in phase with one another.
While @Cho09 and @Sta11 have shown that $3$ minute magneto-acoustic power is suppressed in sunspot umbrae as a result of local absorption and emissivity reduction, their spatial resolution was insufficient to allow studies of the smallest sunspot features ([e.g.,]{} umbral dots). Recently, @She09 have shown that on small spatial scales, the curvature and strength of magnetic field lines have substantial influence on the efficiency of magneto-acoustic wave propagation. Indeed, these authors suggest that wave power can actually be amplified under certain atmospheric conditions. Following on from this, our observational data suggests that UDs are able to enhance the background $p$-mode power by at least three orders-of-magnitude (see [e.g.,]{} middle panel of Figure \[fig3\]). This may be a result of the magnetic field lines acting as efficient conduits for magneto-acoustic wave propagation [[e.g.,]{} @Sin92; @She06; @Kho08; @Erd10].
Chromosphere
------------
Chromospheric information comes from the H$\alpha$ core imaging dataset, in addition to the Ca $8542$[Å]{} spectral imaging scans taken with IBIS. Furthermore, Doppler shifts of the Ca profile minimum allow a series of two-dimensional velocity maps to be generated [@Jes10a]. The Ca core image shown in the right panel of Figure \[fig1\] is a true intensity map, created by establishing the line-profile minimum at each pixel. By displaying Doppler-compensated line-centre intensities, rather than rest-wavelength intensities, brightness variations throughout the image are more indicative of the source function than of the velocities present in the line-forming region [@Lee10].
In a process identical to that applied to the photospheric image sequence, the H$\alpha$ and Doppler-compensated Ca data sets were multiplied by the umbral binary map, and subsequently analysed using wavelet techniques. As for the $4170$[Å]{} continuum time series, considerable oscillatory power was present throughout the entire sunspot umbra, with dominant peaks in the Fourier spectrum at $3$ and $5$ minutes. However, remaining consistent with @Tho85, chromospheric power at periodicities of $\sim$$5$ minutes was much reduced when compared to those found in the photospheric umbra. To investigate whether oscillatory power detected in the chromosphere can be related to similar phenomena found in the photosphere, the locations of high oscillatory power were examined. The right panel of Figure \[fig3\] outlines the spatial locations where power, in both intensity and velocity signals at approximately $3$ minutes, is in excess of $1000$ times the background umbral value. Similar to the locations of high photospheric power displayed in the middle panel of Figure \[fig3\], several distinct groups manifest in a horseshoe shape in the south-west quadrant of the sunspot umbra.
The locations of significant chromospheric power have two distinct differences when compared to their photospheric counterparts. First, their spatial positions are slightly offset from those displayed in the middle panel of Figure \[fig3\]. Secondly, the spatial sizes of high chromospheric power are substantially larger than those found in the photosphere. These effects can be explained by the geometry of the magnetic field lines which stretch outwards from the solar surface. For example, high photospheric power encompassing UD2 (middle panel of Figure \[fig3\]) has a local maximum at the heliocentric co-ordinate ($-133{\,}.{\!\!}{\arcsec}7$, $-483{\,}.{\!\!}{\arcsec}7$), while the chromospheric local maximum is at ($-133{\,}.{\!\!}{\arcsec}1$, $-482{\,}.{\!\!}{\arcsec}2$). Thus, the offset between these two maxima is $\approx$$1{\,}.{\!\!}{\arcsec}6$, or $\approx$$1100$ km. A separation between the $4170$[Å]{} continuum and the H$\alpha$/Ca core formation heights of $\sim$$1800$ km [@Ver81; @Jes12] requires an inclination angle of $\approx$$40{\degr}$ to support the assumption that the oscillatory signals are from a continuation of the same solar structure [[e.g.,]{} @Cen06]. This is the largest offset present in our observations, and forms our upper limit of the magnetic field inclination angle. @Rem09 have recently shown that the inclination angles (to the vertical) of magnetic flux tubes within sunspot umbrae can be quite large ($>$$45{\degr}$), with these structures often becoming horizontal near the penumbral boundary. Furthermore, @Mar09 utilized stereoscopic observations of propagating slow-mode waves in coronal structures to infer an absolute inclination angle $\sim$$40{\degr}$. Thus, an inclination angle of $<$$40{\degr}$ supports the interpretation that oscillatory behaviour detected at photospheric and chromospheric layers are directly related by the magnetic field lines which extend upwards from the solar surface.
![[*[Upper:]{}*]{} An unfiltered time series, created by spatially averaging Ca intensity (solid line) and velocity (dashed line) signals, along with H$\alpha$ core intensities (red line), originating from within the contours outlining ‘UD2’ in the right panel of Figure \[fig3\]. [*[Middle:]{}*]{} The same time series from the upper panel, following temporal filtering through a $2$ – $4$ minute bandpass filter. A $-90\degr$ phase delay between velocity and intensity signals indicates the presence of oscillations which are magneto-acoustic in nature, while a close agreement between H$\alpha$ and Ca intensities is a result of their similar formation heights. [*[Lower:]{}*]{} The relative phase angles between the photospheric and chromospheric umbral dot oscillations, plotted as a function of time. The black, red, green, and blue lines represent umbral dots UD1, UD2, UD3, and UD4, respectively, while a black dashed line highlights a phase angle of $0$$\degr$. []{data-label="fig5"}](fig05_lowres.eps){width="\columnwidth"}
An increase in the area of the oscillating regions can be associated with a physical expansion of the magnetic flux tubes as a function of atmospheric height. Through examination of intense magnetic field concentrations ($>$$1000$ G), @Jes09 were able to show that the radial expansion of magnetic flux tubes between photospheric and chromospheric heights can be as large as a factor of $2$. Assuming these magnetic flux tubes demonstrate an expanding cylindrical geometry, doubling the radial dimension will result in an area increase of $400$%. Continuing with the previous example, the region of high oscillatory power encompassing UD2 covers $326$ pixels ($0.8$ Mm$^{2}$ at $2500$ km$^{2}$pixel$^{-1}$) and $256$ pixels ($2.5$ Mm$^{2}$ at $10{\,}000$ km$^{2}$pixel$^{-1}$) at photospheric and chromospheric heights, respectively. As a result, an expansion of only $310$% is observed, meaning our observations are well within the limits of previous expanding magnetic flux tube models [[e.g.,]{} @Mei07; @Rud08; @She10; @Kar11; @Fed11].
The specific mode of oscillation can be determined through investigation of the coupling between intensity and velocity signals. Here, we adopt the $V-I$ convention [@Deu89; @Fle89], which shows the delay of maximum intensity ($I$) with respect to maximum blueshift velocity ($V$). Thus, a wave which has a velocity signal trailing its intensity signal by $1/4$ of a period will have a $V-I$ phase angle of $-90\degr$. Due to the Ca absorption line being sensitive to temperature fluctuations [@Bee69], we follow the common practice to adopt the intensity ($I$) as a proxy for the local temperature. @Mei77 has shown that when waves propagate along moderately inclined flux tubes, the phase lag between fluctuations in velocity and temperature are the same as those found in purely acoustic waves which have been modified by gravity. Therefore, in a fully adiabatic scenario, theory predicts a $V-I$ phase angle approaching $0\degr$ for running acoustic waves, which can increase to $-90\degr$ when aspects of wave reflection create standing acoustic modes [@Hof96; @Al98; @Nig99]. Under isothermal conditions, the phase lag can further increase up to $-180\degr$ [@Mih81; @Mih82]. Examination of the upper panel in Figure \[fig5\] reveals how the unfiltered velocity time series (dashed line) trails the co-spatial intensity lightcurve (solid line) by approximately $1/4$ of a wave period. This effect becomes even more pronounced when the same time series are passed through a $2$ – $4$ minute bandpass filter (middle panel of Figure \[fig5\]). Chromospheric waves detected within the contours of the right panel of Figure \[fig3\] have a spatially and temporally averaged phase angle of $-87\pm8\degr$, suggesting these waves are magneto-acoustic in nature, with characteristics consistent with standing acoustic modes [@Deu74; @Cra78]. The generation of a chromospheric standing wave may be the result of a portion of the acoustic wave energy being reflected back at the transition region boundary [@Sch92; @Nak04; @Fed11].
The spatial and temporal averaging of velocity signals found in the locations of high chromospheric power results in a net blueshift velocity of $\approx$$1.5$ kms$^{-1}$ (upper panel of Figure \[fig5\]). While a $V-I$ phase angle of $-87\pm8\degr$ implies the presence of wave reflection, a net blueshift velocity may also indicate that a portion of the magneto-acoustic wave is propagating in the upward direction. To examine the propagation characteristics of the magneto-acoustic waves associated with the UDs, spatially-averaged H${\alpha}$ intensity time-series were constructed for each of the regions contoured in the right panel of Figure [\[fig3\]]{}, and subsequently passed through a $2$ – $4$ minute bandpass filter to isolate oscillatory phenomena around 3 minutes. A phase difference analysis was performed between these chromospheric H${\alpha}$ lightcurves and their corresponding photospheric $4170$[Å]{} continuum counterparts created in Section [\[photosphere\]]{}, and displayed in the upper panel of Figure [\[fig4\]]{}. A preference for negative phase angles is shown in the lower panel of Figure [\[fig5\]]{}, which indicates the oscillatory signals are first observed in the photospheric $4170$[Å]{} continuum, before later being detected in the chromospheric H${\alpha}$ time-series. By averaging the derived phase angles over all UDs, an average phase lag of $-43\pm12\degr$ is found, implying the presence of upwardly propagating waves. Using the dominant periodicity of $\approx$$3$ minutes, a relative phase angle of $-43\degr$ equates to a time lag of $\approx$$22$ s. This is consistent with the photosphere-to-chromosphere time-lag measurements detailed in @Kob11. However, due to a height separation of $\sim$$1800$ km between the $4170$[Å]{} continuum and H$\alpha$ core image sequences [@Ver81; @Jes12], a time lag of $\approx$$22$ s requires a phase speed exceeding $80$ kms$^{-1}$. This is clearly unfeasible since a velocity this large will be supersonic, and violates our interpretation that the observed waves are magneto-acoustic slow modes. However, because the wave trains exist at the start of our observing sequence, and continue to be observed as the time-series finishes, a factor of $n2\pi$ ($n360\degr$), where $n$ is an integer, may be absent from our derived phase angle. By considering $n=1$, the phase lag increases to $\approx$$-403\degr$, with the resulting phase speed reducing considerably to $\approx$$8$ kms$^{-1}$. Only by observing the start and/or end of a propagating wave train can the exact time lag (and associated phase speed) be conclusively determined. Nevertheless, the detected waves are best described as upwardly-propagating magneto-acoustic modes, which travel along magnetic flux tubes inclined to the vertical by less than $\approx$$40\degr$. Such an inclination angle may explain why @Kob08 [@Kob11] were unable to correlate (on a pixel-by-pixel basis) high chromospheric oscillatory power to that occurring in the underlying photosphere.
Our H$\alpha$ dataset has a much higher cadence than the Ca core image sequence ($1.26$ s instead of $43.4$ s). Thus, examination of the H$\alpha$ intensity time series can substantially reduce the associated errors when deriving the periodicities of the propagating magneto-acoustic waves. As can be seen in Figure \[fig5\], the H$\alpha$ data displays prominent intensity oscillations, most of which last for the entire duration of the time series. A spatial and temporal averaging over the regions encompassed by the red contours in the right panel of Figure \[fig3\] yields a peak chromospheric periodicity of $168\pm7$ s, consistent with the generalised $p$-mode spectrum of chromospheric sunspot oscillations [@OSh01; @OSh02; @Ban02]. The close resemblance between H$\alpha$ and Ca intensity time series suggests their formation heights are remarkably similar. This is also apparent by the similarities present in simultaneous snapshots through their respective filters ([e.g.,]{} Figure \[fig1\]).
Transition Region and Corona
----------------------------
Due to the reduced spatial resolution of AIA images ($2$ pixel resolution $\approx$$1{\,}.{\!\!}{\arcsec}2$ $\approx$$870$ km), when compared with our simultaneous datasets of the lower solar atmosphere ($2$ pixel resolution $\approx$$0{\,}.{\!\!}{\arcsec}138$ $\approx$$100$ km), it is imperative to concentrate on the larger-scale structures which will be apparent in all image sequences. Following inspection of the co-aligned images displayed in the lower panels of Figure \[fig2\], it is clear that a number of fan structures rise out of the photospheric sunspot umbra, and into the corona. These structures are particularly visible as intensity enhancements in the coronal AIA images, where the $131$[Å]{}, $171$[Å]{}, $193$[Å]{}, $211$[Å]{}, and $335$[Å]{} bandpasses show similar structuring extending outwards in the south-west direction. However, these fan structures are mostly absent in the transition region $304$[Å]{} images.
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To investigate whether an absence of the fan brightenings in the $304$[Å]{} images is a consequence of the structures lying outside of the filter’s temperature response curve, differential emission measure (DEM) techniques were employed [[e.g.,]{} @Han12]. Utilising the six coronal EUV channels on AIA, we were able to construct emission measure ($EM$) and temperature ($T_{e}$) estimates of active region NOAA $11250$, including its immediate vicinity. Following the methodology presented by @Asc11, each batch of near-simultaneous EUV exposures allowed us to construct a time series of $EM$ and $T_{e}$ variables, thus allowing the evolutionary changes in each sequence to be studied. Importantly, the temperature of the coronal fan structures currently under investigation is in the range of $0.5$ – $1.2$ MK (right panel of Figure \[fig6\]). Their relatively cool temperature, when compared to the multi-million degree values found at the centre of the active region, probably manifest as a result of open magnetic field lines which cannot trap heated plasma. An alternative explanation could revolve around their connection with quasi-separatrix layers, which may be subject to a peculiar heating function [@Sch10]. While the $304$[Å]{} bandpass has two distinct temperature response functions covering approximately $50{\,}000$ K and $1.5$ MK, the fact that active region NOAA $11250$ was positioned close to solar disk centre suggests that the resulting images should be dominated by the He emission formed at $\sim$$50{\,}000$ K, with contributions from the $1.5$ MK Si $202.22$[Å]{} emission line minimal [@Odw10]. This helps to explain why features near one million degrees ([e.g.,]{} the fan structures extending outwards from the underlying sunspot), are not readily apparent in the transition region data.
Examination of time-lapse movies of coronal EUV images revealed clear and distinctly periodic outflows along the coronal fans. To quantify the associated periodicities and flow velocities, a series of one-dimensional slits were placed along the motion path in each of the coronal EUV channels. The resulting time-distance cuts reveal numerous propagating wavefronts, indicated by straight, diagonal trends in the bottom panel of Figure \[fig7\]. The AIA EUV images, in the temperature range of $0.4$ – $2.8$ MK (incorporating the 131[Å]{}, 171[Å]{}, 193[Å]{}, 211[Å]{}, and 335[Å]{} bandpasses), display a dominant periodicity of $172\pm17$ s, and a propagation velocity of $45\pm7$ kms$^{-1}$. Red and blue contours in the lower panel of Figure \[fig7\] outline the intensity signals present in the higher temperature $211$[Å]{} and $335$[Å]{} bandpasses, respectively. It is clear that not only do the periodicities of the wave fronts closely resemble one another in both space and time, but the intensity gradients (and hence wave speeds) are also similar. Propagating wave fronts are observed co-spatially, and simultaneously in all but the $94$[Å]{} AIA bandpass. The signal-to-noise of the $94$[Å]{} channel is too low to extract a time series of sufficient quality for analysis. This may be compounded by the fact that the fan structures demonstrate a temperature much lower than the $94$[Å]{} channel’s peak response.
A periodicity of $\approx$$172$ s and a propagation velocity $\approx$$45$ kms$^{-1}$ are consistent with @Kin03, who utilised the $171$[Å]{} and $195$[Å]{} filters onboard the TRACE satellite to reveal how the propagation velocity of such waves will not exceed the local sound speed, even with a large inclination angle of the wave guides away from the observer’s line-of-sight. The typical sound speed associated with the dominant Fe emission from the $171$[Å]{} bandpass is $\approx$$150$ kms$^{-1}$ [@DeM06], which requires the fans currently under investigation to have an inclination angle exceeding $70\degr$ before the observed wave motion would become supersonic. An inclination angle this large is highly unrealistic due to the location of the active region on the solar disk, in addition to previous surveys of coronal magnetic geometries [[e.g.,]{} @Asc02; @Asc08; @Asc09]. Thus, we can conclude that the observed wave phenomena is best described as propagating slow-mode waves.
![An AIA $171$[Å]{} image (top), co-spatial with our field-of-view of the lower solar atmosphere. Dashed white lines outline a coronal fan structure, originating from within the underlying sunspot umbra, where propagating wave phenomena are readily apparent. The perimeter of the sunspot umbra is highlighted by a solid red line, demonstrating how the coronal fan is anchored into regions of high chromospheric oscillatory power (see e.g., the right panel in Figure \[fig3\]). A sample time-distance diagram of this fan is displayed in the lower panel, where $0$ km represents the sunspot umbra. Red and blue contours outline $211$[Å]{} and $335$[Å]{} intensities, respectively, which are $60$[%]{} above the local quiescent background. A black dashed line represents the propagation of a typical wave front, where the gradient provides the wave speed, typically $45\pm7$ kms$^{-1}$. Wave phenomena which are co-spatial, co-temporal, and propagating with the same wave speed are readily apparent over a range of coronal temperatures.[]{data-label="fig7"}](fig07_lowres.eps){width="\columnwidth"}
Since propagating wave fronts are detected in many of the AIA EUV bandpasses, emission measure and temperature maps were subsequently investigated for the presence of similar signatures. As the emission measure maps created here are the sum of (squared) electron densities along a given line-of-sight, the total mass distribution, regardless of the local temperature, can be studied as a function of time [@Asc11]. Furthermore, the derived pixel-by-pixel temperatures correspond to the peak of the DEM distribution for a given line-of-sight, thus allowing the entire temperature range of the coronal plasma to be easily studied [@AscBoe11]. Applying the same techniques used on the EUV image sequences, time-distance diagrams of the $EM$ and $T_{e}$ variables derived for the fan structures reveal identical periodicities and propagation velocities as found in the EUV images (middle and upper panels of Figure \[fig8\]). The emission measure is usually defined as [@Kin82; @Doy85], $$EM (T_{e}) = \int n_{e}^{2}~dh \ ,$$ where $n_{e}$ is the electron density and $h$ is an emitting depth along the observer’s line-of-sight. Since the compressive phase of a magneto-acoustic wave mode will cause the emitting volume to decrease, thus reducing the overall emission measure, one would expect the $EM$ time series to oscillate in phase with the corresponding EUV intensities. However, as the plasma becomes compressed, the associated kinetic temperature will increase, thus causing the $T_{e}$ signal to oscillate out-of-phase with the $EM$ and EUV intensity signals. Indeed, perturbations in the EUV intensity, depicted in Figure \[fig7\], are found to occur in phase with the derived emission measure fluctuations. Furthermore, the lower panel in Figure \[fig8\] reveals how oscillations in $T_{e}$ are found to be $180\degr$ out-of-phase with respect to those detected in the $EM$ signal. Quantitatively, the detected fluctuations in the $EM$ and $T_{e}$ values are typically in the range $21.50\pm0.03$ cm$^{-5}$K$^{-1}$ and $0.55\pm0.03$ MK, respectively. This strengthens our interpretation that the observed wave phenomena are best described as magneto-acoustic slow modes.
![Time-distance diagrams of the emission measure (in units of cm$^{-5}$K$^{-1}$; top) and temperature (in units of log($T_{e}$); middle), derived for the fan structure where propagating wave phenomena in EUV intensity images was detected. The dashed white lines highlight the spatial position along the slice ($2500$ km from the underlying umbra) where $EM$ and $T_{e}$ time series were created. These values are displayed in the lower panel, where the solid line represents the emission measure, and the dashed line traces the temperature. Both time series are displayed in their native units, as used in the top and middle panels. A clear anti-correlation between the $EM$ and $T_{e}$ time series is readily apparent.[]{data-label="fig8"}](fig08_lowres.eps){width="\columnwidth"}
Numerical Simulations
---------------------
{width="12cm"}
To model our observational findings, we performed a series of 1D numerical simulations using the Lagrangian-Remap code [; @Arb01], in which thermal conduction and optically thin radiation have been included [@Owe09]. The $1$D model setup is constructed to closely match the observed AIA $171$[Å]{} emission at the lowest coronal part of the fan structure. In particular, the density profile (upper-left panel of Figure \[fig9\]) along the fan includes gravitational stratification, where the initial ($z=0$) value is determined through equalisation of the forward-modelled and observed $171$[Å]{} intensities, as can be seen in the upper-right panel of Figure \[fig9\]. Here, the solid line corresponds to the spatial variation in $171$[Å]{} emission, obtained through forward-modelling of the temperature and density profiles inferred from the observed fan structure [where the forward modelling is undertaken using a code developed by @DeM08]. A model temperature profile is constructed which utilises the minimum and maximum values of the (isothermal) temperatures extracted directly from the AIA emission. The background magnetic field is taken to be constant. From the upper-right panel of Figure \[fig9\], it is clear that the theoretical model is in close agreement with the observed $171$[Å]{} emission during the first $15$ Mm along the fan. After this, the model emission declines more steeply than the observed $171$[Å]{} emission. However, we are primarily concerned with the behaviour within the first $15$ Mm, and therefore can neglect larger distances where the emission becomes faint, and the observations become more affected by detector noise.
A slow magneto-acoustic wave is driven into the domain at the lower boundary, through periodic perturbations of the field-aligned velocity component. A snapshot of the velocity perturbations at $t=30$ minutes, using a driving period of $172$ s to match that of the observations, is shown in the lower-left panel of Figure \[fig9\]. A slow wave will propagate along the field at the characteristic tube speed, which is of the order of the local sound speed. Incorporating a temperature $\approx$$1$ MK, and a background magnetic field strength $\approx$$10$ G, the tube speed is of the order of $122$ kms$^{-1}$. Combined with a driving period of $172$ s, we expect a wavelength of the order of 22 Mm, which can be verified in the lower panels of Figure \[fig9\]. After an initial increase, due to the rapidly decreasing background density, the velocity amplitudes are strongly damped through a combination of thermal conduction, optically thin radiation, and compressive viscosity. The corresponding density perturbations (i.e., the density profile at $t=0$ has been subtracted from the density at $t=30$ minutes) are shown in the lower-right panel of Figure \[fig9\]. Such strong damping is predominantly caused by thermal conduction, as was previously uncovered by @DeM03 [@DeM04] and @Owe09. Only a small fraction of the overall damping can be attributed to optically thin radiation and compressive viscosity.
To achieve a more meaningful comparison with the perturbations observed by the AIA instrument, we utilised the model density and energy (temperature) distributions along the loop, to forward-model the emission for different AIA channels, at a multitude of time steps. We find that the modelled $171$[Å]{} emission is significantly more intense over the first $15$ Mm of the coronal fan structure, when compared to the $193$[Å]{} and $211$[Å]{} intensities. Although these AIA channels contain some emission lines at cooler temperatures, the $193$[Å]{} and $211$[Å]{} bandpasses mainly respond to hotter coronal temperatures [i.e., above $1.5$ MK; @Odw10; @Bro11], and hence only begin to display elevated intensities when the density reduces and the local temperature increases beyond $1$ MK. This is consistent with the AIA observations, whereby the cooler $171$[Å]{} emission dominates the first $15$ Mm from the underlying sunspot. When forward-modelling intensity oscillations, it is important to remember that a variation in amplitude of the intensity perturbations is a combination of the change in amplitude of the model density and temperature fluctuations, as well as the individual response functions of the AIA channels to different temperatures.
![Time-averaged global wavelet power spectra, for the observed (top) and simulated (bottom) AIA time series, plotted as a function of distance along the coronal fan. The red, green, and blue lines highlight the oscillatory power contained within the $171$[Å]{}, $193$[Å]{}, and $211$[Å]{} bandpasses, respectively, with each power spectrum normalised for clarity. Dashed green lines displayed in the observed spectra (top) relate to the power associated with the hot ($\sim$$1.6$ MK) and cool (${\lesssim}$$1$ MK) spectral components of the $193$[Å]{} channel, with the cool component closely following the $171$[Å]{} trend, and the hot component tending more towards the $211$[Å]{} spectrum. The damping of magneto-acoustic wave power, at hotter temperatures first, is an indication that thermal conduction is the dominant damping mechanism. []{data-label="fig10"}](fig10_lowres.eps){width="\columnwidth"}
Finally, we utilise the global wavelet power, at each spatial position along the coronal fan, to compare the amplitude decay rates found in the different AIA bandpasses in a more quantitative way. To do this, a wavelet transform was performed at each position along the fan, with the temporally-averaged global wavelet power subsequently calculated. The red, green, and blue lines in Figure \[fig10\] represent the time-averaged oscillatory power around 3 minutes, for the AIA $171$[Å]{}, $193$[Å]{}, and $211$[Å]{} observed and forward-modelled bandpasses, respectively. Note that each of these have been normalised to their individual maxima for display purposes. Of particular note is the observed $193$[Å]{} power spectrum. Here, the solid green line represents the oscillatory power registered through the entire AIA bandpass, while the dashed green lines relate to the power associated with the hot ($\sim$$1.6$ MK) and cool (${\lesssim}$$1$ MK) spectral components of this channel. These components have been separated using the methods detailed in @Kid12, with the resulting cool emission peaking alongside the $171$[Å]{} power at $\approx$$4.5$ Mm, and the hot emission displaying a peak between the $171$[Å]{} and $211$[Å]{} bandpasses ($\approx$$3$ Mm). In reality, the global wavelet power of the $193$[Å]{} and $211$[Å]{} emission is substantially lower than that of the $171$[Å]{} emission. However, the time-averaged power of the $193$[Å]{} and $211$[Å]{} modelled intensities are very similar, as is the case for the observations. For these wavelengths, the simulated global wavelet power reaches a minimum around $10$ Mm, while the $171$[Å]{} bandpass reaches its minimum power slightly further along the fan structure, at about $15$ Mm. These results agree qualitatively with the observed time-averaged power, where the $193$[Å]{} and $211$[Å]{} channels near-simultaneously reach a minimum at about $9$ Mm along the coronal fan, with the $171$[Å]{} power diminishing completely by $15$ Mm. This reiterates the importance of thermal conduction in the damping of coronal slow-mode waves. The increase in time-averaged global wavelet power, after $\approx$$10$ Mm in both the simulated and observed $193$[Å]{} and $211$[Å]{} time series, is due to the fact that, despite the decreasing amplitudes of the model density and temperature, the percentage intensity perturbations in these channels are actually increasing. This is a result of the increased sensitivity of the AIA $193$[Å]{} and $211$[Å]{} bandpasses to the $>$$1$ MK temperatures which have been reached by this distance along the fan structure. The increase is not as apparent in the observed $193$[Å]{} and $211$[Å]{} time-averaged wavelet power spectra, which may be due to the decreased signal-to-noise levels found in these channels, especially when compared to the $171$[Å]{} bandpass.
Overview and Concluding Remarks
===============================
Here we present high-cadence observations of the solar atmosphere, obtained using the latest ground- and space-based facilities. Prominent oscillatory behaviour is detected throughout the optical and EUV image sequences, with remarkable similarities found between the detected wave modes. First, a number of UD structures in the photospheric umbra are found to display intensity oscillations with a $\approx$$3$ minute periodicity. These oscillations exhibit considerable power, with regions encompassing the UDs displaying more than three orders-of-magnitude stronger power than the background umbra (Figure \[fig3\]). Next, chromospheric intensity [*[and]{}*]{} velocity measurements were analysed for the presence of co-spatial and co-temporal oscillations. Such phenomena were detected, both on larger spatial scales, and with small central offsets with respect to the underlying photospheric oscillations. By considering the extension of magnetic flux tubes from the solar surface out into the upper solar atmosphere, a geometric expansion of only $76$% in the radial direction and an inclination angle $<$$40\degr$ allows the observed oscillations to be interpreted as originating from within the same magnetic flux tube.
Following examination of the phase lag between chromospheric velocity and intensity components, a $V-I$ phase angle of $-87\pm8\degr$ was derived, allowing these waves to be described as a magneto-acoustic mode, with characteristics consistent with standing acoustic modes (middle panel of Figure \[fig5\]). The generation of a chromospheric standing wave may be the result of partial wave reflection at the transition region boundary. An average blueshift velocity of $\approx$$1.5$ kms$^{-1}$ was found in the locations where high chromospheric oscillatory power was present. This is a sub-sonic velocity, and coupled with a time lag between photospheric and chromospheric oscillatory phenomena, strengthens our interpretation that the observed oscillations are upwardly-propagating magneto-acoustic waves, which originate in UD structures located in the photospheric umbra.
A prominent fan structure is present in the simultaneous coronal EUV images, namely those from the AIA $131$[Å]{}, $171$[Å]{}, $193$[Å]{}, $211$[Å]{}, and $335$[Å]{} bandpasses. This fan is not readily apparent in either the transition-region dominated $304$[Å]{} emission, or in the higher temperature ($\sim$$7.0$ MK) $94$[Å]{} bandpass. Using DEM techniques, we constrained the temperature of the coronal fan to $0.5$ – $1.2$ MK, thus placing it outside the temperature range of both the $94$[Å]{} and $304$[Å]{} filtergrams (Figure \[fig6\]). Time-distance techniques were employed on the EUV imaging data where the fan structure was readily apparent, allowing the characteristics of propagating wave phenomena to be uncovered. Most coronal channels, regardless of their absolute temperature sensitivity, revealed outwardly propagating wave fronts with an average periodicity and velocity of $172\pm17$ s and $45\pm7$ kms$^{-1}$, respectively (lower panel of Figure \[fig7\]). The out-of-phase nature between the derived temperature and emission measure signals indicates the presence of a compressive wave mode (Figure \[fig8\]). This, coupled with a sub-sonic wave speed ($\approx$$45$ kms$^{-1}$), highlights the fact that these coronal phenomena are best described as upwardly-propagating magneto-acoustic slow mode waves.
Employing numerical simulations, we were able to accurately simulate the behaviour of the coronal EUV emission. Utilising input parameters derived directly from the AIA observations, forward-modelling techniques allowed us to evolve velocity, density, and emissivity values forward in time, creating a time series which could be directly compared to the AIA observations. Crucially, our simulations revealed that thermal conduction is the primary damping mechanism behind the dissipation of magneto-acoustic slow-mode waves in the corona. Other mechanisms, including optically thin radiation and compressive viscosity, play a secondary role in the damping of these oscillations.
The fan structure observed in the AIA images, which displays signatures of propagating magneto-acoustic waves, appears to have anchor points in the south-west quadrant of the photospheric sunspot umbra (lower panels of Figure \[fig2\]). These locations are also consistent with the presence of large-amplitude wave phenomena detected in simultaneous photospheric and chromospheric image sequences. The co-temporal and co-spatial relationship between these upwardly-propagating magneto-acoustic wave modes, detected throughout the entire solar atmosphere, suggests such coronal phenomena may be driven by UD oscillations occurring inside the sunspot umbra. With this conclusion, it appears that photospheric structures which are on the order of $0{\,}.{\!\!}{\arcsec}5$ ($360$ km) in diameter, can have a strong influence on coronal structures not only several thousand km above their position, but on structures which have expanded into the local plasma to diameters often exceeding $10$ ($7000$ km).
DBJ thanks the Science and Technology Facilities Council (STFC) for a Post-Doctoral Fellowship. IDM acknowledges support from a Royal Society University Research Fellowship. PHK is grateful to the Northern Ireland Department of Education and Learning for a PhD studentship. DJC thanks the CSUN College of Science for start-up funding related to this project. Solar Physics research at QUB is supported by STFC. The ROSA project is supported by The European Office of Aerospace Research & Development. HARDcam observations were made possible by a Royal Society Research Grant (2009). We are grateful for the use of SDO/AIA images, which were obtained courtesy of NASA/SDO and the AIA, EVE, and HMI science teams.
[*Facilities:*]{} & .
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[^1]: Full specifications available at http://www.andor.com
[^2]: Data analysis was performed on full-resolution ([i.e.]{} non-degraded) image sequences
|
---
author:
- 'Ehsan Emamjomeh-Zadeh[^1]'
- 'David Kempe[^2]'
bibliography:
- 'names.bib'
- 'conferences.bib'
- 'references.bib'
title: '[A General Framework for Robust Interactive Learning]{}'
---
[^1]: Department of Computer Science, University of Southern California, [email protected]
[^2]: Department of Computer Science, University of Southern California, [email protected]
|
---
abstract: 'Organize songs, albums, and artists in groups with shared similarity could be done with the help of genre labels. In this paper, we present a novel approach for automatic classifying musical genre in Brazilian music using only the song lyrics. This kind of classification remains a challenge in the field of Natural Language Processing. We construct a dataset of $138,368$ Brazilian song lyrics distributed in 14 genres. We apply SVM, Random Forest and a Bidirectional Long Short-Term Memory (BLSTM) network combined with different word embeddings techniques to address this classification task. Our experiments show that the BLSTM method outperforms the other models with an F1-score average of $0.48$. Some genres like *gospel*, *funk-carioca* and *sertanejo*, which obtained $0.89$, $0.70$ and $0.69$ of F1-score, respectively, can be defined as the most distinct and easy to classify in the Brazilian musical genres context.'
author:
- |
Raul de Araújo Lima\
Department of Informatics, PUC-Rio\
Rua Marques de Sao Vicente 225\
Rio de Janeiro, RJ, 22420-030, Brazil\
`[email protected]`\
Rômulo César C. de Sousa\
Department of Informatics, PUC-Rio\
Rua Marques de Sao Vicente 225\
Rio de Janeiro, RJ, 22420-030, Brazil\
`[email protected]`\
Simone Diniz Junqueira Barbosa\
Department of Informatics, PUC-Rio\
Rua Marques de Sao Vicente 225\
Rio de Janeiro, RJ, 22420-030, Brazil\
`[email protected]`\
Hélio Cortês Vieira Lopes\
Department of Informatics, PUC-Rio\
Rua Marques de Sao Vicente 225\
Rio de Janeiro, RJ, 22420-030, Brazil\
`[email protected]`\
bibliography:
- 'references.bib'
title: 'Brazilian Lyrics-Based Music Genre Classification Using a BLSTM Network'
---
Introduction
============
Music is part of the day-to-day life of a huge number of people, and many works try to understand the best way to classify, recommend, and identify similarities between songs. Among the tasks that involve music classification, genre classification has been studied widely in recent years [@ying2012genre] since musical genres are the main top-level descriptors used by music dealers and librarians to organize their music collections [@scaringella2006automatic].
Automatic music genre classification based only on the lyrics is considered a challenging task in the field of Natural Language Processing (NLP). Music genres remain a poorly defined concept, and boundaries between genres still remain fuzzy, which makes the automatic classification problem a nontrivial task [@scaringella2006automatic].
Traditional approaches in text classification have applied algorithms such as Support Vector Machine (SVM) and Naïve Bayes, combined with handcraft features (POS and chunk tags) and word count-based representations, like bag-of-words. More recently, the usage of Deep Learning methods such as Recurrent Neural Networks (RNNs) and Convolutional Neural Networks (CNNs) has produced great results in text classification tasks.
Some works like [@laurier2008multimodal], [@hu2009lyric] [@hu2010lyrics] focus on classification of mood or sentiment of music based on its lyrics or audio content. Other works, like [@scaringella2006automatic], and [@tsaptsinos2017lyrics], on the other hand, try to automatically classify the music genre; and the work [@fell2014lyrics] tries to classify, besides the music genre, the best and the worst songs, and determine the approximate publication time of a song.
In this work, we collected a set of about 130 thousand Brazilian songs distributed in 14 genres. We use a Bidirectional Long Short-Term Memory (BLSTM) network to make a lyrics-based music genre classification. We did not apply an elaborate set of handcraft textual features, instead, we represent the lyrics songs with a pre-trained word embeddings model, obtaining an F1 average score of $0.48$. Our experiments and results show some real aspects that exist among the Brazilian music genres and also show the usefulness of the dataset we have built for future works.
This paper is organized as follows. In the next section, we cite and comment on some related works. Section \[sec:methods\] describes our experiments from data collection to the proposed model, presenting some important concepts. Our experimental results are presented in Section \[sec:results\], and Section \[sec:conclusion\] presents our concluding remarks and future work.
Related Works {#sec:related_works}
=============
Several works have been carried out to add textual information to genre and mood classification. Fell and Sporleder [@fell2014lyrics] used several handcraft features, such as vocabulary, style, semantics, orientation towards the world, and song structure to obtain performance gains on three different classification tasks: detecting genre, distinguishing the best and the worst songs, and determining the approximate publication time of a song. The experiments in genre classification focused on eight genres: Blues, Rap, Metal, Folk, R&B, Reggae, Country, and Religious. Only lyrics in English were included and they used an SVM with the default settings for the classification.
Ying *et al.* [@ying2012genre] used Part-of-Speech (POS) features extracted from lyrics and combined them with three different machine learning techniques – k-Nearest-Neighbor, Naïve Bayes, and Support Vector Machines – to classify a collection of 600 English songs by the genre and mood.
Zaanen and Kanters [@van2010automatic] used the term frequency and inverse document frequency statistical metrics as features to solve music mood classification, obtaining an accuracy of more than 70%.
In recent years, deep learning techniques have also been applied to music genre classification. This kind of approach typically does not rely on handcraft features or external data. In [@tsaptsinos2017lyrics], the authors used a hierarchical attention network to perform the task in a large dataset of nearly half a million song lyrics, obtaining an accuracy of more than 45%. Some papers such as [@kumar2018genre] used word embedding techniques to represent words from the lyrics and then classify them by the genre using a 3-layer Deep Learning model.
Methods {#sec:methods}
=======
In this chapter we present all the major steps we have taken, from obtaining the dataset to the proposed approach to address the automatic music genre classification problem.
Data Acquisition
----------------
In order to obtain a large number of Brazilian music lyrics, we created a crawler to navigate into the *Vagalume*[^1] website, extracting, for each musical genre, all the songs by all the listed authors. The implementation of a crawler was necessary because, although the *Vagalume* site provides an API, it is only for consultation and does not allow obtaining large amounts of data. The crawler was implemented using Scrapy[^2], an open-source and collaborative Python library to extract data from websites.
From the Vagalume’s music web page, we collect the song title and lyrics, and the artist name. The genre was collected from the page of styles[^3], which lists all the musical genres and, for each one, all the artists. We selected only 14 genres that we consider as representative Brazilian music, shown in Table \[tab:dataset\_info\]. Figure \[fig:vagalume\_music\] presents an example of the Vagalume’s music Web page with the song *“Como é grande o meu amor por você”*[^4], of the Brazilian singer Roberto Carlos. Green boxes indicate information about music that can be extracted directly from the web page. From this information, the language in which the lyrics are available can be obtained by looking at the icon indicating the flag of Brazil preceded by the *“Original”* word.
![An example of a Vagalume’s song web page[]{data-label="fig:vagalume_music"}](img/vagalume_music.png){width="0.8\linewidth"}
After extracting data, we obtained a set of $138,368$ songs distributed across 14 genres. Table \[tab:dataset\_info\] presents the number of songs and artists by genre. In order to use the data to learn how to automatically classify genre, we split the dataset into tree partitions: training ($96,857$ samples), validation ($27,673$ samples), and test ($13,838$ samples). The total dataset and splits are available for download[^5].
**Genre** **\#songs** **\#artists**
-------------- ------------- ---------------
Gospel 33,344 800
Sertanejo 27,417 543
MPB 16,785 282
Forró 11,861 191
Pagode 8,199 174
Rock 8,188 396
Samba 6,221 111
Pop 4,629 338
Axé 4,592 63
Funk-carioca 4,557 279
Infantil 4,550 70
Velha-guarda 3,179 24
Bossa-nova 3,105 38
Jovem-guarda 1,741 18
: The number of songs and artists by genre[]{data-label="tab:dataset_info"}
Word Embeddings
---------------
Word embeddings is a technique to represent words as real vectors, so that these vectors maintain some semantic aspects of the real words. Basically, vectors are computed by calculating probabilities of the context of words, with the intuition that semantically similar words have similar contexts, and must therefore have similar vectors.
Word2Vec[^6], by Mikolov *et al.* [@mikolov2013distributed], is one of the first and most widely used algorithms to make word embeddings. It has two architectures to compute word vectors: Continuous Bag-Of-Words (CBOW) and Skip-gram. CBOW gets a context as input and predicts the current word, while Skip-gram gets the current word as input and predicts its context.
In this work, we use the Python Word2Vec implementation provided by the Gensim[^7] library. The Portuguese pre-trained word embeddings created by [@hartmann2017portuguese] and available for download[^8] was used to represent words as vectors. We only used models of dimension 300 and, for Word2Vec, Wang2Vec, and FastText, skip-gram architectured models.
Bidirectional Long Short-Term Memory
------------------------------------
Long Short-Term Memory (LSTM) is a specification of Recurrent Neural Network (RNN) that was proposed by Hochreiter and Schmidhuber [@hochreiter1997long]. This kind of network is widely used to solve classification of sequential data and is designed to capture time dynamics through graph cycles. Figure \[fig:lstm\_unit\] presents an LSTM unity, which receives an input from the previous unit, processes it, and passes it to the next unit.
The following equations are used to update $C_t$ and $h_t$ values. $$f_t = \sigma(W_f h_{t-1} + U_f x_t + b_f)$$ $$i_t = \sigma(W_i h_{t-1} + U_i x_t + b_i)$$ $$\widetilde{C}_t = tanh(W_C h_{t-1} + U_C x_t + b_C)$$ $$C_t = f_t \times C_{t-1} + i_t \times \widetilde{C}_t$$ $$o_t = \sigma(W_o h_{t-1} + U_o x_t + b_o)$$ $$h_t = o_t \times tanh(C_t)$$ where $W_f$, $W_i$, $W_C$, $W_o$ are the weight matrices for $h_{t-1}$ input; $U_f$, $U_i$, $U_C$, $U_o$ are the weight matrices for $x_t$ input; and $b_f$, $b_i$, $b_C$, $b_o$ are the bias vectors.
![The Long Short-Term Memory unit.[]{data-label="fig:lstm_unit"}](img/LSTM_unit.pdf){width=".7\textwidth"}
Basically, a Bidirectional LSTM network consists of using two LSTM networks: a forward LSTM and a backward LSTM. The intuition behind it is that, in some types of problems, past and future information captured by forward and backward LSTM layers are useful to predict the current data.
Proposed Approach
-----------------
Our proposed approach consists of three main steps. Firstly, we concatenate the title of the song with its lyrics, put all words in lower case and then we clean up the text by removing line breaks, multiple spaces, and some punctuation (,!.?). Secondly, we represent the text as a vector provided by a pre-trained word embeddings model. For classical learning algorithms like SVM and Random Forest, we generate, for each song, a vectorial representation by calculating the average of the vectors of each word in the song lyrics that can be can be expressed by the equation below: $$vector(L) = \frac{1}{n} \sum_{w \in L} vector(w)$$ where $L$ is the song lyrics, $w$ is a word in $L$, and $n$ is the number of words in $L$. If a word does not have a vector representation in the word embeddings model, it is not considered in the equation. For the BLSTM algorithm, the representation was made in the format of a matrix, as shown in Figure \[fig:blstm\], where each line is a vector representation of a word in the lyrics. In the third step, we use as features the generated representation for the genre classification tasks using SVM, Random Forests, and BLSTM.
![Our BLSTM model architecture[]{data-label="fig:blstm"}](img/blstm.pdf){width="60.00000%"}
Experimental Results {#sec:results}
====================
In this section, we describe our experiments. We used the Linear SVM and Random Forest Scikit-learn[^9] implementations and Keras[^10] on top of TensorFlow[^11] for the BLSTM implementation. In this study, we did not focus on finding the best combination of parameters for the algorithms, so that for SVM we used the default parameters, and for Random Forest we used a number of 100 trees. Our BLSTM model was trained using 4 epochs, with Adam optimizer, and 256 as the size of the hidden layer.
As we can see in Table \[tab:results\], our BLSTM approach outperforms the other models with an F1-score average of $0.48$. In addition, we can note that the use of Wang2Vec pre-trained word embeddings made it possible to obtain better F1-score results in BLSTM, which is not necessarily noticed in other cases, since for SVM and Random Forest, Glove and FastText, respectively, were the techniques that obtained better F1-scores.
**System** **Model** **Precision** **Recall** **F1-score**
------------ ----------- --------------- ------------ --------------
Word2Vec 0.290 0.143 0.120
Wang2Vec 0.297 0.144 0.121
FastText 0.278 0.144 0.120
Glove 0.296 0.144 **0.124**
Word2Vec 0.388 0.197 0.212
Wang2Vec 0.386 0.197 0.207
FastText 0.404 0.203 **0.215**
Glove 0.394 0.199 0.210
Word2Vec 0.492 0.454 0.465
Wang2Vec **0.515** **0.460** **0.477**
FastText 0.417 0.350 0.360
Glove 0.492 0.460 0.470
: Classification results for each classifier and word embeddings model combination[]{data-label="tab:results"}
Table \[tab:results\_BLSTM\] shows the BLSTM classification results for each genre. We can see that the genres *gospel*, *funk-carioca* and *sertanejo* have a greater distinction in relation to the other genres, since they were better classified by the model. In particular, *funk-carioca* obtained a good classification result although it did not have a large number of collected song lyrics.
In gospel song lyrics, we can identify some typical words, such as *“Deus”* (God) , *“Senhor”* (Lord), and *“Jesus”* (Jesus); in *funk-carioca*, songs have the words *“bonde”* (tram), *“chão”* (floor) and *“baile”* (dance ball), all used as slang; in *sertanejo*, some of the most common words are *“amor”* (love), *“coração”* (heart) and *“saudade”* (longing). The occurrence of these typical words could contribute to the higher performance of F1-scores in these genres.
**Genre** **F1-score**
-------------- --------------
Gospel 0.89
Funk-carioca 0.70
Sertanejo 0.69
Forró 0.53
Axé 0.49
MPB 0.49
Pagode 0.48
Infantil 0.47
Rock 0.46
Velha-guarda 0.38
Samba 0.35
Bossa-nova 0.31
Pop 0.26
Jovem-guarda 0.19
*Average* *0.481*
: Detailed result of BLSTM[]{data-label="tab:results_BLSTM"}
The *bossa-nova* and *jovem-guarda* genres, which have few instances in the dataset, are among the most difficult ones to classify using the model. The *pop* genre, by contrast, has a small distribution between the number of songs and the number of artists, and could not be well classified by our model. This may indicate that our model was unable to identify a pattern due to the low number of songs per artist, or that the song lyrics of this genre cover several subjects that are confused with other genres.
![Normalized confusion matrix[]{data-label="fig:cm"}](img/cm.png){width="70.00000%"}
Figure \[fig:cm\] shows the confusion matrix of the results produced by our BLSTM model. We can notice that many instances of class *forró* are often confused with class *sertanejo*. Indeed, these two genres are very close. Both *Forró* and *sertanejo* have as theme the cultural and daily aspects of the Northeast region of Brazil. Instances of class *infantil* are often confused with class *gospel*: in *infantil* we have music for children for both entertainment and education. In some of the songs, songwriters try to address religious education, which could explain the confusion between those genres. The *MPB* (Brazilian Popular Music) genre was the most confused of all, which may indicate that song lyrics of this genre cover a wide range of subjects that intersect with other genres.
Conclusion and Future Works {#sec:conclusion}
===========================
In this work we constructed a dataset of $138,368$ Brazilian song lyrics distributed in 14 genres. We applied SVM, Random Forest, and a Bidirectional Long Short-Term Memory (BLSTM) network combined with different word embeddings techniques to address the automatic genre classification task based only on the song lyrics. We compared the results between the different combinations of classifiers and word embedding techniques, concluding that our BLSTM combined with the Wang2Vec pre-trained model obtained the best F1-score classification result. Beside the dataset construction and the comparison of tools, this work also evidences the lack of an absolute superiority between the different techniques of word embeddings, since their use and efficiency in this specific task showed to be very closely related to the classification technique.
As future work, it is possible to explore the dataset to identify genre or artist similarities, generating visualizations that may or may not confirm aspects pre-conceived by the consumers of Brazilian music. It is also possible to perform classification tasks by artists of a specific genre.
[^1]: https://www.vagalume.com.br/
[^2]: https://scrapy.org/
[^3]: <https://www.vagalume.com.br/browse/style/>
[^4]: <https://www.vagalume.com.br/roberto-carlos/como-e-grande-o-meu-amor-por-voce-letras.html>
[^5]: <https://drive.google.com/open?id=1b681ChByK737CpASYImFB4GfuqPdrvBN>
[^6]: <https://code.google.com/archive/p/word2vec/>
[^7]: https://radimrehurek.com/gensim/index.html
[^8]: <http://nilc.icmc.usp.br/embeddings>
[^9]: <https://scikit-learn.org/>
[^10]: <https://keras.io/>
[^11]: <https://www.tensorflow.org/>
|
---
abstract: 'We extend and improve the results in [@DK16]: showing that weak solutions to full elliptic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form equations. We extend the weak type-(1, 1) estimates in [@DK16] and [@Es94] up to the boundary and derive a Harnack inequality for non-negative adjoint solutions to non-divergence form elliptic equations, when the leading coefficients have Dini mean oscillation.'
address:
- 'Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, United States of America'
- 'UPV/EHU, Dpto. Matemáticas, Barrio Sarriena s/n 48940 Leioa, Spain'
- 'Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 03722, Republic of Korea'
author:
- Hongjie Dong
- Luis Escauriaza
- Seick Kim
title: 'On $C^1$, $C^2$, and weak type-$(1,1)$ estimates for linear elliptic operators: Part II'
---
[^1]
[^2]
[^3]
Introduction and main results
=============================
Let $\Omega \subset {\mathbb{R}}^n$ be a bounded domain. We consider a second-order elliptic operator $L$ in divergence form $$\label{master-d}
L u= \sum_{i,j=1}^n D_i(a^{ij}(x) D_j u + b^i(x) u) + \sum_{i=1}^n c^i(x) D_iu + d(x)u,$$ where the coefficients $\mathbf{A}=(a^{ij})_{i,j=1}^n$, ${\boldsymbol{b}}=(b^1,\ldots, b^n)$, ${\boldsymbol{c}}=(c^1,\ldots, c^n)$, and $d$ are measurable functions defined on $\overline{\Omega}$. We assume that the principal coefficients $\mathbf{A}=(a^{ij})$ are defined on ${\mathbb{R}}^n$ and satisfy the uniform ellipticity condition $$\label{ellipticity-d}
\lambda {\lvert\xi\rvert}^2 \le \sum_{i,j=1}^n a^{ij}(x) \xi^i \xi^j,\quad \forall \xi=(\xi^1,\ldots, \xi^n) \in {\mathbb{R}}^n,\quad\forall x \in {\mathbb{R}}^n$$ and the uniform boundedness condition $$\label{bdd-d}
\sum_{i,j=1}^n \,{\lverta^{ij}(x)\rvert}^2 \le \Lambda^2,\quad \forall x \in {\mathbb{R}}^n$$ for some positive constants $\lambda$ and $\Lambda$.
We say that a nonnegative measurable function $\omega: (0,a) \to {\mathbb{R}}$ is a Dini function provided that there are constants $c_1, c_2 >0$ such that $$\label{eq10.31tu}
c_1 \omega(t) \le \omega(s) \le c_2 \omega(t)$$ whenever $\tfrac12 t \le s \le t$ and $0<t<a$ and that $$\int_0^t \frac{\omega(s)}s \,ds <+\infty,\quad \forall t \in (0,a).$$ For $x\in {\mathbb{R}}^n$ and $r>0$, we denote by $B(x,r)$ the Euclidean ball with radius $r$ centered at $x$, and denote $$\Omega(x,r):=\Omega \cap B(x,r).$$ For a locally integrable function $g$ on $\Omega$, we shall say that $g$ is uniformly Dini continuous (in $\Omega$) if the function $\varrho_g: {\mathbb{R}}_+ \to {\mathbb{R}}$ defined by $$\varrho_g(r):=\sup_{x \in \Omega} \,\sup_{y, y' \in \Omega(x,r)} {\lvertg(y)-g(y')\rvert}$$ is a Dini function, while we shall say that $g$ is of *Dini mean oscillation* (in $\Omega$) if the function $\omega_g: {\mathbb{R}}_+ \to {\mathbb{R}}$ defined by $$\omega_g(r):=\sup_{x\in \overline{\Omega}} \fint_{\Omega(x,r)} \,{\lvertg(y)-\bar {g}_{\Omega(x,r)}\rvert}\,dy \quad \left(\;\bar g_{\Omega(x,r)} :=\fint_{\Omega(x,r)} g\;\right)$$ is a Dini function. We point out that the condition is satisfied by $\rho_g$ and also by $\omega_g$; see [@Y.Li2016]. Moreover, it should be clear that if $g$ is uniformly Dini continuous, then it is of Dini mean oscillation and $\omega_g(r) \le \varrho_g(r)$. It is worthwhile to note that if $\Omega$ is such that for any $x\in \overline{\Omega}$, $$\label{cond_a}
{\lvert\Omega(x,r)\rvert} \ge A_0 r^n,\quad 0< \forall r\le {\operatorname{diam}}\Omega\quad (\text{$A_0$ is a positive constant})$$ and if $g$ is of Dini mean oscillation, then $g$ is uniformly continuous with a modulus of continuity determined by $\omega_g$.
In a recent paper [@Y.Li2016], Yanyan Li raised a question whether weak solutions of $$\sum_{i,j=1}^n D_i(a^{ij}(x) D_j u)=0$$ are continuously differentiable when $\mathbf{A}=(a^{ij})$ are of Dini mean oscillation.[^4] In [@DK16], the first and third named authors showed that the answer to his question is positive. This paper is a sequel to [@DK16] and extends and improves results presented there. More precisely, we show that weak solutions to with zero Dirichlet boundary conditions are continuously differentiable up to boundary provided that the leading coefficients $\mathbf{A}$ and ${\boldsymbol{b}}$ are of Dini mean oscillation, lower order coefficients ${\boldsymbol{c}}$ and $d$ belong to $L^q$ with $q>n$, and $\partial \Omega$ has $C^{1,Dini}$ boundary. We prove a similar result when the operator is in non-divergence form. In [@Es94], the second named author investigated (interior) weak type-$(1,1)$ estimates for solutions of $$\sum_{i,j=1}^n a^{ij}(x) D_{ij} u=f\;\mbox{ in }\; B(0,1),\quad u=0\;\mbox{ on }\;\partial B(0,1),$$ and showed that if $\mathbf{A}=(a^{ij})$ belong to the class of functions with vanishing mean oscillations (VMO), then the $D^2 u$ satisfies weak type-$(1,1)$ estimates with respect to $W\,dx$. Here $W$ is a nonnegative solution to the adjoint equation, which is a good Muckenhoupt weight as $\log W$ was proved to be in VMO, so that the associated measure $W\,dx$ is better adjusted to the equation than $dx$. Moreover, it is also shown in [@Es94] that the standard weak type-$(1,1)$ estimates (i.e., the estimate with $W=1$) do not hold even if $\mathbf{A}$ is uniformly continuous. In this paper, we prove that if $\mathbf{A}$ is of Dini mean oscillation, then the standard weak type-$(1,1)$ estimates hold up to the boundary. We also show that in this case, the weight $W$ mentioned above satisfies a Harnack type inequality.
Now, we state the main results more precisely.
Let $\Omega \subset {\mathbb{R}}^n$ be open and bounded, $k=1,2,\cdots$. We say $\partial \Omega$ is $C^{k,Dini}$ if for each point $x_0 \in \partial\Omega$, there exist $r>0$ independent of $x_0$ and a $C^{k, Dini}$ function (i.e., $C^k$ function whose $k$th derivatives are uniformly Dini continuous) $\gamma: {\mathbb{R}}^{n-1} \to {\mathbb{R}}$ such that (upon relabeling and reorienting the coordinates axes if necessary) in a new coordinate system $(x',x^n)=(x^1,\ldots,x^{n-1},x^n)$, $x_0$ becomes the origin and $$\Omega \cap B(0, r)={\{ x \in B(0, r) : x^n > \gamma(x^1, \ldots, x^{n-1})\}},\quad \gamma(0')=0.$$
\[cond-n\] The coefficients $\mathbf{A}=(a^{ij})_{i,j=1}^n$ and ${\boldsymbol{b}}=(b^1,\ldots, b^n)$ are of Dini mean oscillation in $\Omega$ and ${\boldsymbol{c}} =(c^1,\ldots, c^n)$, $d \in L^{q}(\Omega)$ with $q>n$.
\[thm-main-d\] Let $\Omega$ have $C^{1, Dini}$ boundary and the coefficients of $L$ in satisfy the conditions , , and Condition \[cond-n\]. Let $u \in W^{1,2}_{0}(\Omega)$ be the weak solution of $$Lu={\operatorname{div}}{\boldsymbol{g}} + f\;\mbox{ in } \; \Omega,\quad u=0\;\mbox{ on }\;\partial\Omega,$$ where ${\boldsymbol{g}}=(g^1,\ldots, g^n)$ are of Dini mean oscillation in $\Omega$ and $f \in L^q(\Omega)$ with $q>n$. Then, we have $u\in C^1(\overline{\Omega})$.
The proof of Theorem \[thm-main-d\] is given Section \[sec2\], where an upper bound for the modulus of continuity of $Du$ can be found.
We also consider elliptic operators ${\mathcal{L}}$ in non-divergence form $$\label{master-nd}
{\mathcal{L}}u= \sum_{i,j=1}^n a^{ij}(x) D_{ij} u+ \sum_{i=1}^n b^i(x) D_i u + c(x)u,$$ where the coefficients $\mathbf{A}$ are assumed to be symmetric, i.e. $a^{ij}=a^{ji}$, and satisfy the uniform ellipticity and boundedness condition $$\label{ellipticity-nd}
\lambda {\lvert\xi\rvert}^2 \le \sum_{i,j=1}^n a^{ij}(x) \xi^i \xi^j \le \Lambda {\lvert\xi\rvert}^2,\quad \forall \xi=(\xi^1,\ldots, \xi^n) \in {\mathbb{R}}^n,\quad \forall x \in {\mathbb{R}}^n$$ for some constants $0< \lambda \le \Lambda$.
\[cond-nd\] The coefficients $\mathbf{A}=(a^{ij})_{i,j=1}^n$, ${\boldsymbol{b}}=(b^1,\ldots, b^n)$, and $c$ are of Dini mean oscillation in $\Omega$.
\[thm-main-nd\] Let $\Omega$ have $C^{2, Dini}$ boundary and the coefficients of ${\mathcal{L}}$ in satisfy the condition and Condition \[cond-nd\]. Let $u \in W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega)$ be a strong solution of $${\mathcal{L}}u= g\;\mbox{ in } \; \Omega,\quad u=0\;\mbox{ on }\;\partial\Omega,$$ where $g$ is of Dini mean oscillation in $\Omega$. Then, we have $u\in C^2(\overline{\Omega})$.
The proof of Theorem \[thm-main-nd\] is also given Section \[sec2\]. The formal adjoint operator of the non-divergence operator $\mathcal L$ is defined by $$\mathcal L^\ast u = D_{ij}(a^{ij}(x) u)-D_i(b^i(x) u)+c(x)u.$$ We also deal with the boundary value problem $$\label{master-adj}
\mathcal L^\ast u={\operatorname{div}}^2 \mathbf g + f\; \text{ in }\; \Omega,\quad u= \psi +\frac{\mathbf{g}\nu\cdot \nu}{\mathbf{A}\nu\cdot \nu} \; \text{ on }\; \partial\Omega,$$ where $\mathbf g = (g^{kl})_{k,l=1}^n$ is a symmetric matrix and ${\operatorname{div}}^2\mathbf g=\sum_{k,l=1}^n D_{kl}g^{kl}$. At first, the appearance of the term $\frac{\mathbf{g}\nu\cdot \nu}{\mathbf{A}\nu\cdot \nu}$ as a part of boundary value may look strange, but it helps to make $\mathbf{g}$ to disappear from the boundary integral in the identity , which formally defines a “weak” adjoint solution to ; see [@EM2016 Definition 2] for more details.
Let $\Omega \subset {\mathbb{R}}^n$ be a bounded $C^{1,1}$ domain with unit exterior normal vector $\nu$. Assume that $\mathbf{g} \in L^p(\Omega)$, $f \in L^p(\Omega)$, and $\psi \in L^p(\partial\Omega)$, where $1<p<\infty$. We say that $u \in L^p(\Omega)$ is an adjoint solution to if $u$ satisfies $$\label{eq13.53pde}
\int_{\Omega} u\,\mathcal L v\,dx=\int_{\Omega} {\operatorname{tr}}(\mathbf{g}\,D^2 v)\, dx + \int_{\Omega} f v \,dx + \int_{\partial\Omega} \psi\, \mathbf{A} Dv \cdot \nu\, dS_x,$$ for any $v \in W^{2,p'}(\Omega)\cap W^{1,p'}_0(\Omega)$, where $\frac{1}{p}+\frac{1}{p'}=1$. By a local adjoint solution of , we mean a function $u$ in $L^p_{loc}(\Omega)$ that verifies when $v$ is in $W^{2,p'}_0(\Omega)$.
\[cond-adj\] The coefficients $\mathbf{A}=(a^{ij})_{i,j=1}^n$ are of Dini mean oscillation over an open set containing $\overline\Omega$ and ${\boldsymbol{b}}=(b^1,\ldots, b^n) \in L^q(\Omega)$, $c \in L^{\frac{q}{2}}(\Omega)$, for some $q>n$.
\[thm-main-adj\] Let $\Omega$ have a $C^{1,1}$ boundary, the coefficients of ${\mathcal{L}}$ in satisfy the condition and Condition \[cond-adj\]. Let $u \in L^2(\Omega)$ be an adjoint solution of the problem $$\mathcal L^\ast u={\operatorname{div}}^2 \mathbf{g}+f\; \text{ in }\; \Omega,\quad u= \psi +\frac{\mathbf{g}\nu\cdot \nu}{\mathbf{A}\nu\cdot \nu} \; \text{ on }\; \partial\Omega,$$ where $\mathbf{g}$ is of Dini mean oscillation in $\Omega$, $f\in L^s(\Omega)$ with $s>\frac{n}{2}$, and $\psi \in C(\partial\Omega)$. Then, $u\in C(\overline \Omega)$.
The proof of Theorem \[thm-main-adj\] is also given in Section \[sec2\]. We note that in Theorem \[thm-main-adj\], we assume $\mathbf{A}$ and $\mathbf{g}$ are of Dini mean oscillation, and thus $\frac{\mathbf{g}\nu\cdot \nu}{\mathbf{A}\nu\cdot \nu}$ becomes a uniformly continuous function in $\partial \Omega$. Therefore, the boundary data in the above theorem include all continuous functions defined on $\partial\Omega$. See [@S1973] for previous results on interior $C^\alpha$-regularity, $0<\alpha <1$, for solutions to with $\mathbf{g}=0$.
In section \[sec3\], we provide an improvement of the weak type-$(1,1)$ estimates given in [@DK16]. In particular, they are shown to hold up to the boundary, while in the non-divergence case, the weak type-$(1,1)$ estimate is shown to hold without imposing further conditions on the principal coefficients $\mathbf{A}$ other than being of Dini mean oscillation over an open set containing $\overline\Omega$.
\[thm-weak11-d\] Let $\Omega$ have a $C^{1,Dini}$ boundary and the coefficients $\mathbf{A}=(a^{ij})$ satisfy the conditions , , and the following: $$\label{eq15.08m}
\exists c>0 \;\mbox{such that }\;\omega_{\mathbf{A}}(r) \le c ( \ln r )^{-2}, \quad\forall r\in(0,\tfrac12).$$ Assume that $\partial \Omega$ is locally represented as a graph of $C^1$ function $\gamma$ satisfying $${\lvertD \gamma(x) - D\gamma(y)\rvert} \le c(\ln {\lvertx-y\rvert})^{-2},\quad {\lvertx-y\rvert}<\tfrac12.$$ For ${\boldsymbol{f}} =(f^1,\ldots, f^n) \in L^2(\Omega)$, let $u \in W^{1,2}_0(\Omega)$ be a unique weak solution to $$\sum_{i,j=1}^n D_i(a^{ij} D_j u)= {\operatorname{div}}{\boldsymbol{f}}\;\mbox{ in }\; \Omega,\quad
u=0 \;\mbox{ on } \; \partial \Omega.$$ Then for any $t>0$, we have $${\left\lvert{\{x \in \Omega : {\lvertDu(x)\rvert} > t\}}\right\rvert} \le \frac{C}{t} \int_{\Omega} {\lvert{\boldsymbol{f}}\rvert},$$ where $C=C(n, \lambda, \Lambda, \omega_{\mathbf{A}}, \Omega)$.
A similar result can be proved for the adjoint problem $$\sum_{i,j=1}^n D_{ij}(a^{ij} u) = {\operatorname{div}}^2 \mathbf{g}\;\mbox{ in }\; \Omega,\quad
u=\frac{\mathbf{g}\nu\cdot\nu}{\mathbf{A}\nu\cdot\nu} \;\mbox{ on } \; \partial \Omega.$$ The statement and its proof are similar to those of Theorem \[thm-weak11-d\] and omitted.
\[thm-weak11-nd\] Let $\Omega$ have a $C^{1,1}$ boundary, the coefficients $\mathbf{A}=(a^{ij})$ have Dini mean oscillations over an open set containing $\overline\Omega$ and satisfy the condition . For $f \in L^2(\Omega)$, let $u \in W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega)$ be the unique solution to $$\label{E: nodivergenete}
\sum_{i,j=1}^n a^{ij} D_{ij} u= f\;\mbox{ in }\; \Omega,\quad
u=0 \;\mbox{ on } \; \partial \Omega.$$ Then for any $t>0$, we have $$\label{eq15.08wh}
{\left\lvert{\{x \in \Omega : {\lvertD^2u(x)\rvert} > t\}}\right\rvert} \le \frac{C}{t} \int_{\Omega}\, {\lvertf\rvert}\,dx,$$ where $C=C(n, \lambda, \Lambda, \omega_{\mathbf A}, \Omega)$.
We recall in Remark \[R: 1\] the previously known interior weak type-$(1,1)$ properties for solutions to and sketch out how to extend them up to the boundary, when the leading coefficients matrix $\mathbf{A}$ is only in VMO over an open set containing $\overline\Omega$. We also explain why Theorem \[thm-weak11-nd\] is optimal for its comparison with counterexamples in [@Es94 §3].
The paper is organized as follows. In Section \[sec2\] we provide some preliminary lemmas and propositions and the proofs of Theorems \[thm-main-d\], \[thm-main-nd\], and \[thm-main-adj\]. Section \[sec3\] is devoted to the proof of Theorems \[thm-weak11-d\] and \[thm-weak11-nd\]. Section \[appendinx1\] is an appendix where we outline for the reader’s convenience a complete proof of Lemma \[lem01-stein\], which is standard, and a proof of Lemma \[lem8.06\], where a Harnack type inequality for nonnegative adjoint solutions are presented. Lemma \[lem01-stein\] is used in the proofs of Theorems \[thm-weak11-d\], \[thm-weak11-nd\] and of Lemmas \[lem02-weak11\], \[lem-weak11-nd\], and \[lem-weak11-adj\].
Finally, a few remarks are in order. Theorems \[thm-main-d\], \[thm-main-nd\], and \[thm-weak11-d\] are easily extended to elliptic systems since their proofs do not use any scalar structure. The same is true for Theorem \[thm-main-adj\] if we keep $\psi \equiv 0$ there. In Theorem \[thm-main-d\] (resp. Theorem \[thm-main-nd\]), instead of assuming zero Dirichlet data, we may assume $u=\psi$ on $\partial\Omega$ with $\psi \in C^{1,Dini}(\overline{\Omega})$ (resp. $\psi \in C^{2,Dini}(\overline{\Omega})$). Finally, the conditions on lower order terms in Theorems \[thm-main-d\], and \[thm-main-adj\] can be relaxed a little. For example, in Theorem \[thm-main-d\] we may assume that ${\boldsymbol{c}}$, $d$, and $f$ belong to suitable Morrey-Campanato spaces instead of $L^p$ spaces.
Proof of Theorems \[thm-main-d\], \[thm-main-nd\], and \[thm-main-adj\] {#sec2}
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We write $x=(x^1,\ldots, x^n)=(x', x^n)$. Hereafter, we shall denote $$B^{+}(0,r)=B(0,r) \cap {\{x^n >0\}}\quad\text{and}\quad T(0,r)=B(0,r) \cap {\{x^n=0\}}.$$ We fix a smooth (convex) domain ${\mathcal{D}}$ satisfying $B^{+}(0,\frac12) \subset {\mathcal{D}}\subset B^{+}(0,1)$ so that $\partial {\mathcal{D}}$ contains a flat portion $T(0, \frac12)$. For $\bar x \in \partial {\mathbb{R}}^n_{+}={\{x^n=0\}}$, we then set $$B^{+}(\bar x,r)=B^{+}(0,r)+ \bar x,\quad T(\bar x,r)=T(0,r)+\bar x,\quad\text{and}\quad {\mathcal{D}}(\bar x,r)= r {\mathcal{D}}+ \bar x.$$ Throughout the rest of paper, we adopt the usual summation convention over repeated indices. Also, for nonnegative (variable) quantities $A$ and $B$, the relation $A \lesssim B$ should be understood that there is some constant $c>0$ such that $A \le c B$. We write $A \approx B$ if $A \lesssim B$ and $B \lesssim A$.
Preliminary lemmas
------------------
\[lem00\] Let $\Omega$ satisfy the condition . If $f$ is uniformly Dini continuous and $g$ is of Dini mean oscillation in $\Omega$, then $fg$ is of Dini mean oscillation in $\Omega$.
For any $x\in \overline\Omega$ and $r>0$, we have $$\begin{aligned}
\fint_{\Omega(x,r)} \,{\left\lvertfg-\overline{fg}_{\Omega(x,r)}\right\rvert}
&\le \fint_{\Omega(x,r)} \,{\left\lvertfg-f\,\bar{g}_{\Omega(x,r)}\right\rvert}
+\fint_{\Omega(x,r)} \,{\left\lvertf\,\bar{g}_{\Omega(x,r)}-\overline{fg}_{\Omega(x,r)}\right\rvert}\\
&\le \sup_{\Omega(x,r)} f \cdot \omega_g(r) + \varrho_f(r) \cdot \fint_{\Omega(x,r)} {\lvertg\rvert},\end{aligned}$$ where we used $$\sup_{\Omega(x,r)}\, {\left\lvertf\,\bar{g}_{\Omega(x,r)}-\overline{fg}_{\Omega(x,r)}\right\rvert}
\le \varrho_f(r) \cdot \fint_{\Omega(x,r)} {\lvertg\rvert}.$$ Therefore, we get $$\label{eq11.39w}
\omega_{fg}(r) \le {\lVertf\rVert}_{L^\infty(\Omega)}\, \omega_g(r)+{\lVertg\rVert}_{L^\infty(\Omega)}\,\varrho_f(r)$$ and thus $\omega_{fg}$ is a Dini function.
\[lem02-weak11\] Let $\bar{\mathbf{A}}=(\bar a^{ij})$ be a constant matrix satisfying and . For ${\boldsymbol{f}} \in L^2({\mathcal{D}})$ let $u \in W^{1,2}_0({\mathcal{D}})$ be a unique weak solution to $$\sum_{i,j=1}^n D_i (\bar a^{ij} D_j u)= {\operatorname{div}}{\boldsymbol{f}}\;\mbox{ in }\; {\mathcal{D}};\quad
u=0 \;\mbox{ on } \; \partial {\mathcal{D}}.$$ Then for any $t>0$, we have $${\left\lvert{\{x \in {\mathcal{D}}: {\lvertDu(x)\rvert} > t\}}\right\rvert} \le \frac{C}{t} \int_{{\mathcal{D}}}\, {\lvert{\boldsymbol{f}}\rvert},$$ where $C=C(n, \lambda, \Lambda, {\mathcal{D}})$.
See proof of [@DK16 Lemma 2.2] and Lemma \[lem01-stein\].
\[lem-weak11-nd\] Let $\bar{\mathbf{A}}=(\bar a^{ij})$ be a constant symmetric matrix satisfying . For $f \in L^2({\mathcal{D}})$ let $u \in W^{2,2}({\mathcal{D}})\cap W^{1,2}_0({\mathcal{D}})$ be a unique solution to $$\sum_{i,j=1}^n \bar a^{ij} D_{ij} u= f\;\mbox{ in }\; {\mathcal{D}};\quad
u=0 \;\mbox{ on } \; \partial {\mathcal{D}}.$$ Then for any $t>0$, we have $${\left\lvert{\{x \in {\mathcal{D}}: {\lvertD^2 u(x)\rvert} > t\}}\right\rvert} \le \frac{C}{t} \int_{{\mathcal{D}}}\, {\lvertf\rvert},$$ where $C=C(n, \lambda, \Lambda, {\mathcal{D}})$.
See proof of [@DK16 Lemma 2.20] and Lemma \[lem01-stein\].
\[lem-weak11-adj\] Let $\bar{\mathbf{A}}=(\bar a^{ij})$ be a constant symmetric matrix satisfying . For $\mathbf g \in L^2({\mathcal{D}})$ let $u \in L^{2}({\mathcal{D}})$ be a unique adjoint solution to $$\sum_{i,j=1}^n D_{ij} (\bar a^{ij} u)={\operatorname{div}}^2 \mathbf g\;\mbox{ in }\; {\mathcal{D}};\quad
u=\frac{\mathbf g\nu\cdot\nu}{\bar{\mathbf A}\nu\cdot\nu} \;\mbox{ on } \; \partial {\mathcal{D}}.$$ Then for any $t>0$, we have $${\left\lvert{\{x \in {\mathcal{D}}: {\lvertu(x)\rvert} > t\}}\right\rvert} \le \frac{C}{t} \int_{{\mathcal{D}}}\, {\lvertg\rvert},$$ where $C=C(n, \lambda, \Lambda, {\mathcal{D}})$.
See proof of [@DK16 Lemma 2.23] and Lemma \[lem01-stein\].
We finish this subsection by a Lipschitz estimate for the following equation, which will be used in the proof of Theorem \[thm-main-adj\]: $$\label{eq1.59}
D_{ij}(\bar{a}^{ij} u)={\operatorname{div}}^2 \bar{\mathbf g}\;\text{ in }\; B^+(0,2),\quad
u=\frac{\bar{\mathbf g}\nu\cdot\nu}{\bar{\mathbf A}\nu\cdot\nu}\;\text{ on }\;T(0,2),$$ where $\bar{\mathbf A}=(\bar{a}^{ij})$ and $\bar{\mathbf g}=(\bar{g}^{ij})$ are constant symmetric matrices.
\[lem-bdry-Lip\] Let us denote $ B^+_r:=B^+(0,r)$. Suppose that $u\in L^2(B^+_2)$ satisfies . Then for any $p\in (0,1)$ and $c\in {\mathbb{R}}$, we have $$\label{eq2.02}
{\lVertDu\rVert}_{L^\infty(B_{1}^+) }\le C {\lVertu-c\rVert}_{L^p(B_2^+)},$$ where $C=C(n, \lambda, \Lambda, p)$.
First we notice that $u$ is smooth in $B^+(0,2) \cup T(0,2)$ and satisfies $$\bar a_{ij}D_{ij} u=0\;\text{ in }\; B^+(0,2),\quad u=\text{constant}\;\text{ on }\;T(0,2).$$ Obviously, $u-c$ enjoys the same properties for any $c\in {\mathbb{R}}$. Thus, without loss of generality, we may assume that $c=0$. By a linear transformation and a covering argument, we may further assume that $\bar a_{ij}=\delta_{ij}$, i.e. $\bar{\mathbf A}=\mathbf{I}$. The problem is thus reduced to $$\label{eq1.59b}
\Delta u=0\;\text{ in }\; B^+(0,2),\quad u=\text{constant}\;\text{ on }\;T(0,2).$$ By differentiating in the tangential direction $x_k$ for $k=1,2, \ldots, n-1$, we see that $v_k=D_{k}u$ satisfies $$\Delta v_k=0\;\text{ in }\; B^+(0,2),\quad v_k=0\;\text{ on }\;T(0,2).$$ By classical estimates for harmonic functions, we thus have $$\label{eq2.09}
{\lVertD_{k} u\rVert}_{L^\infty(B_{1}^+)}+{\lVertDD_{k} u\rVert}_{L^\infty(B_{1}^+)} \le C {\lVertD_{k} u\rVert}_{L^2(B_{3/2}^+)},\quad k=1,2, \ldots, n-1.$$ Next, from the equation, we find that $D_{nn}u= -\sum_{k=1}^{n-1} D_{kk} u=0$ on $T(0,2)$. Therefore, the normal derivative $v_n=D_n u$ satisfies $$\Delta v_n=0\;\text{ in }\; B^+(0,2),\quad D_n v_n=0\;\text{ on }\; T(0,2).$$ Again, by classical estimates for harmonic functions, we have $$\label{eq2.09b}
{\lVertD_{n}u\rVert}_{L^\infty(B_{1}^+)}+{\lVertDD_{n}u\rVert}_{L^\infty(B_{1}^+)} \le C {\lVertD_{n}u\rVert}_{L^2(B_{3/2}^+)}.$$ Combining and yields $$\label{eq2.13}
{\lVertDu\rVert}_{L^\infty(B_{1}^+)}+ {\lVertD^2 u\rVert}_{L^\infty(B_{1}^+)} \le C{\lVertDu\rVert}_{L^2(B_{3/2}^+)}.$$ Now, follows from , the interpolation inequality (proved by using the standard mollification technique): $${\lVertDu\rVert}_{L^2(B_{3/2}^+)}
\le C{\lVertDu\rVert}_{L^\infty(B_{3/2}^+)}
\le \varepsilon {\lVertD^2 u\rVert}_{L^\infty(B_{3/2}^+)}
+C(\varepsilon,n,p)\, {\lVertu\rVert}_{L^p(B_{3/2}^+)},$$ and a standard iteration argument.
Proof of Theorem \[thm-main-d\]
-------------------------------
First, we develop an interior $C^1$ estimate.
\[prop2.5p\] For any $\Omega' \subset\subset \Omega$, we have $u \in C^1(\overline \Omega{}')$.
Since $a^{ij}$ are uniformly continuous over $\overline\Omega$ and $\partial\Omega$ is $C^1$, by moving the lower-order terms to the right-hand side of the equation, we can show that $u \in W^{1, p}(\Omega)$, for any $1<p<\infty$. Indeed, we can rewrite the equation as $$D_i(a^{ij} D_j u)= {\operatorname{div}}({\boldsymbol{g}}- u {\boldsymbol{b}}+\nabla v),\ \text{in}\ \Omega,\quad u=0,\ \text{on}\ \partial\Omega,$$ where $v$ is the Newtonian potential of $(f- c^i D_i u - du)\chi_{\Omega}$. By the Calderón-Zygmund theory (see e.g. [@GT Theorem 9.9]) and Hölder’s inequality, we have $${\lVertD^2 v\rVert}_{L^{s}({\mathbb{R}}^n)} \le C \left( {\lVertf\rVert}_{L^q(\Omega)} + {\lVertc\rVert}_{L^q(\Omega)} {\lVert\nabla u\rVert}_{L^2(\Omega)}+ {\lVertd\rVert}_{L^q(\Omega)}{\lVertu\rVert}_{L^2(\Omega)} \right),$$ where $\frac{1}{s}=\frac{1}{q}+\frac{1}{2}$. Let us set $${\boldsymbol{g}}':={\boldsymbol{g}} - u {\boldsymbol{b}}+ \nabla v$$ and note that ${\boldsymbol{g}}' \in L^{r}(\Omega)$, where $\frac{1}{r}=\frac{1}{q}+\frac{1}{2}-\frac{1}{n}$, and $${\lVert{\boldsymbol{g}}'\rVert}_{L^{r}(\Omega)} \le C \left( {\lVert{\boldsymbol{g}}\rVert}_{L^\infty(\Omega)} + {\lVertf\rVert}_{L^q(\Omega)} \right) + C \left( {\lVert{\boldsymbol{b}}\rVert}_{L^\infty(\Omega)} + {\lVertc\rVert}_{L^q(\Omega)}+ {\lVertd\rVert}_{L^q(\Omega)}\right) {\lVertu\rVert}_{W^{1,2}(\Omega)}.$$ On the other hand, by the energy estimates (see e.g. [@LU68 §3.4]), we get $${\lVertu\rVert}_{W^{1,2}(\Omega)} \le C {\lVertf\rVert}_{L^q(\Omega)}+ C{\lVert{\boldsymbol{g}}\rVert}_{L^\infty(\Omega)}+ C {\lVertu\rVert}_{L^1(\Omega)}.$$ Then we apply the global $W^{1,p}(\Omega)$ theory (see, e.g. [@AuscherQafsaoui02 Theorem 1]) to get $${\lVertu\rVert}_{W^{1,r}(\Omega)} \le C {\lVertf\rVert}_{L^q(\Omega)}+ C{\lVert{\boldsymbol{g}}\rVert}_{L^\infty(\Omega)} + C {\lVertu\rVert}_{L^1(\Omega)},$$ where $C$ is a constant depending only on $n$, $\lambda$, $\Lambda$, $q$, $\Omega$, $\partial \Omega$, $\omega_{\mathbf{A}}$, ${\lVert{\boldsymbol{c}}\rVert}_{L^q(\Omega)}$, ${\lVertd\rVert}_{L^q(\Omega)}$, and ${\lVert{\boldsymbol{b}}\rVert}_{L^\infty(\Omega)}$. Then, by standard bootstrapping argument, we have $u \in W^{1, p}(\Omega)$, for any $1<p<\infty$, and $${\lVertu\rVert}_{W^{1,p}(\Omega)} \le C {\lVertf\rVert}_{L^q(\Omega)}+ C{\lVert{\boldsymbol{g}}\rVert}_{L^\infty(\Omega)} + C {\lVertu\rVert}_{L^1(\Omega)}$$ as claimed with $C$ depending additionally on $p$.
By Morrey’s inequality, we have $u \in C^{0,\mu}(\Omega)$ for any $0<\mu<1$ and $${\lVertu\rVert}_{C^{0,\mu}(\Omega)} \le C {\lVertf\rVert}_{L^q(\Omega)}+ C{\lVert{\boldsymbol{g}}\rVert}_{L^\infty(\Omega)} + C {\lVertu\rVert}_{L^1(\Omega)}.$$ Also, note that $f - c^i D_iu -du \in L^{p}(\Omega)$ for $n<p <q$ and $${\lVertf-c^i D_iu -du\rVert}_{L^{p}(\Omega)} \le C \left( {\lVertf\rVert}_{L^q(\Omega)}+ {\lVert{\boldsymbol{g}}\rVert}_{L^\infty(\Omega)} + {\lVertu\rVert}_{L^1(\Omega)} \right) \left( 1+ {\lVert{\boldsymbol{c}}\rVert}_{L^{q}(\Omega)}+{\lVertd\rVert}_{L^{q}(\Omega)} \right).$$ Recall that $v$ is the Newtonian potential of $f- c^i D_i u - du$. By [@GT Theorem 9.9] and Morrey’s inequality, we find $\nabla v \in C^{0,\delta}({\mathbb{R}}^n)$ with $\delta=1-\frac{n}{p}$ and $$[\nabla v]_{C^{0,\delta}({\mathbb{R}}^n)} \le C \left( {\lVertf\rVert}_{L^q(\Omega)}+ {\lVert{\boldsymbol{g}}\rVert}_{L^\infty(\Omega)}+ {\lVertu\rVert}_{L^1(\Omega)} \right) \left( 1+ {\lVert{\boldsymbol{c}}\rVert}_{L^{q}(\Omega)}+{\lVertd\rVert}_{L^{q}(\Omega)} \right).$$ Therefore, by Lemma \[lem00\], we see that ${\boldsymbol{g}}'$ is of Dini mean oscillation in $\Omega$.
In summary, we see that $u$ is a weak solution of $${\operatorname{div}}(\mathbf{A}\nabla u)= {\operatorname{div}}{\boldsymbol{g}}' \;\text{ in }\;\Omega,$$ where ${\boldsymbol{g}}'$ is of Dini mean oscillation and $\omega_{{\boldsymbol{g}}'}$ is completely determined by the given data (namely $n$, $\lambda$, $\Lambda$, $\Omega$, $\omega_{\mathbf{A}}$, $p$, $q$, ${\lVertf\rVert}_{L^q(\Omega)}$, ${\lVert{\boldsymbol{c}}\rVert}_{L^q(\Omega)}$, ${\lVertd\rVert}_{L^q(\Omega)}$, $\omega_b$, ${\lVert{\boldsymbol{b}}\rVert}_{L^\infty(\Omega)}$, $\omega_{{\boldsymbol{g}}}$, and ${\lVert{\boldsymbol{g}}\rVert}_{L^\infty(\Omega)}$) and ${\lVertu\rVert}_{L^1(\Omega)}$. By [@DK16 Theorem 1.5], we thus find that $u \in C^1(\overline \Omega{}')$ and ${\lVertu\rVert}_{C^1(\overline \Omega{}')}$ is bounded by a constant $C$ depending only on the above mentioned given data, ${\lVertu\rVert}_{L^1(\Omega)}$, and $\Omega'$.
Next, we turn to $C^1$ estimate near the boundary. We shall write $B^+_r=B^+(0,r)$. Let ${\boldsymbol{g}}'$ and ${\boldsymbol{g}}''$ be as given in the proof of Proposition \[prop2.5p\]. Under a volume preserving mapping of flattening boundary $$y={\boldsymbol{\Phi}}(x)=(\Phi^1(x),\ldots, \Phi^n(x)), \quad (\det D{\boldsymbol{\Phi}} =1)$$ let $\tilde u(y)=u(x)$, which satisfies $$D_i(\tilde a^{ij} D_j \tilde u)= {\operatorname{div}}\tilde{{\boldsymbol{g}}}'$$ and $$\begin{gathered}
\tilde a^{ij}(y)= D_l\Phi^i D_k\Phi^j a^{kl}(x), \quad \tilde {{\boldsymbol{g}}}'(y)= D {\boldsymbol{\Phi}}{^{\mkern-1.5mu\mathsf{T}}}{\boldsymbol{g}}'(x).\end{gathered}$$ By Lemma \[lem00\], we see that the coefficients $\tilde a^{ij}$ and the data $\tilde {{\boldsymbol{g}}}'$ are still of Dini mean oscillation. Therefore, we are reduced to prove the following.
\[prop01\] If $u \in W^{1,2}(B^+_4)$ is a weak solution of $$D_i(a^{ij} D_j u)= {\operatorname{div}}{\boldsymbol{g}}\;\mbox{ in }\;B^+_4$$ satisfying $u=0$ on $T(0,4)$, then $u \in C^1(\overline B{}^+_1)$.
The rest of this subsection is devoted to the proof of Proposition \[prop01\]. The proof of Proposition \[prop01\] is in the spirit of Campanato’s method [@Campanato] as presented in a modern textbook [@GM2012]. We shall derive an a priori estimate of the modulus of continuity of $Du$ by assuming that $u$ is in $C^1(\overline B{}^+_3)$. The general case follows from a standard approximation argument.
Fix any $p \in (0,1)$. For $x \in B^+_4$ and $r>0$, we define $$\label{eq12.14fo}
\phi(x,r):=\inf_{{\boldsymbol{q}} \in {\mathbb{R}}^n} \left( \fint_{B(x,r) \cap B^{+}_4} \,{\lvertDu - {\boldsymbol{q}}\rvert}^p \right)^{\frac1p}$$ and choose a vector ${\boldsymbol{q}}_{x,r}\in {\mathbb{R}}^n$ satisfying $$\label{eq0955w}
\phi(x,r) = \left( \fint_{B(x,r) \cap B^{+}_4} \,{\lvertDu - {\boldsymbol{q}}_{x,r}\rvert}^p \right)^{\frac1p}.$$ Also, for $\bar x \in T(0,4)$ and $r>0$, we introduce an auxiliary quantity $$\label{eq12.14fop}
\varphi(\bar x, r):=\inf_{q\in {\mathbb{R}}}\, \left( \fint_{B^{+}(\bar x,r) \cap B^{+}_4} \,{\lvertDu - q {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p}\quad ({\boldsymbol{e}}_n=(0,\ldots,0,1) \in {\mathbb{R}}^n)$$ and fix a number $\bar q_{\bar x, r} \in {\mathbb{R}}$ satisfying $$\label{eq0955wp}
\varphi(\bar x, r) = \left( \fint_{B^{+}(\bar x,r) \cap B^{+}_4} \,{\lvertDu - \bar q_{\bar x, r}\, {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p}.$$ We present a series of lemmas (and their proofs) that will provide key estimates for the proof of Proposition \[prop01\].
\[lem-01\] Let $\beta \in (0,1)$. For any $\bar x \in T(0,3)$ and $0<\rho \le r \le \frac12$, we have $$\label{eq1225f}
\varphi(\bar x, \rho) \le C\left(\frac{\rho}{r}\right)^{\beta}\,r^{-n} {\lVertDu\rVert}_{L^1(B^{+}(\bar x, r))}+C {\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,2r))}\,\tilde\omega_{\mathbf{A}}(2\rho) + C \tilde \omega_{{\boldsymbol{g}}}(2\rho),$$ where $C=C(n, \lambda, \Lambda, p, \beta)$ are constants and $\tilde\omega_\bullet(t)$ is a Dini function derived from $\omega_\bullet(t)$.
Note that we have $B^{+}(\bar x, 2r) \subset B^+_4$ and $$\label{eq12.14f}
\varphi(\bar x, r) \le \left(\fint_{B^{+}(\bar x,r)} {\lvertDu\rvert}^p\right)^{\frac1p} \lesssim r^{-n} {\lVertDu\rVert}_{L^1(B^{+}(\bar x, r))}.$$ We decompose $u=v+w$, where $w \in W^{1,2}_0({\mathcal{D}}(\bar x,2r))$ is the solution of the problem $${\operatorname{div}}(\bar{\mathbf{A}} \nabla w) = -{\operatorname{div}}((\mathbf{A}- \bar{\mathbf{A}}) \nabla u) +{\operatorname{div}}({\boldsymbol{g}}-\bar{{\boldsymbol{g}}})\;\mbox{ in }\;{\mathcal{D}}(\bar x,2r);\quad
w=0 \;\mbox{ on }\;\partial {\mathcal{D}}(\bar x,2r).$$ Here and below, we use the simplified notation $$\bar{\mathbf{A}}=\bar{\mathbf{A}}_{B^{+}(\bar x,2r)},\quad \bar{{\boldsymbol{g}}} = \bar{{\boldsymbol{g}}}_{B^{+}(\bar x,2r)}.$$ By Lemma \[lem02-weak11\] with scaling, for any $t>0$, we have $${\lvert{\{x\in B^{+}(\bar x,r): {\lvertDw(x)\rvert} > t\}}\rvert}
\lesssim \frac{1}{t}\left({\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,2r))} \int_{B^{+}(\bar x,2r)} {\lvert\mathbf{A}-\bar{\mathbf{A}}\rvert} + \int_{B^{+}(\bar x,2r)} {\lvert{\boldsymbol{g}} -\bar {{\boldsymbol{g}}}\rvert} \,\right),$$ where we used $B^{+}(\bar x,r) \subset {\mathcal{D}}(\bar x,2r) \subset B^{+}(\bar x,2r)$. Then, we have (cf. [@DK16 (2.11)]) $$\label{eq15.50a}
\left(\fint_{B^{+}(\bar x,r)} {\lvertDw\rvert}^p \right)^{\frac1p} \lesssim \omega_{\mathbf{A}}(2r) \,{\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,2r))} + \omega_{{\boldsymbol{g}}}(2r).$$ On the other hand, $v=u-w$ satisfies $$\label{eq08.48th}
{\operatorname{div}}(\bar{\mathbf{A}} \nabla v) =0 \;\mbox{ in }\; B^{+}(\bar x,r);\quad v=0 \;\mbox{ on }\; T(\bar x,r).$$ Note that the same is satisfied by $D_j v$ for $j=1, \ldots, n-1$. By standard boundary estimates for elliptic equations (or systems) with constant coefficients, we have $${\lVertD D_{j} v\rVert}_{L^\infty(B^{+}(\bar x, \frac12 r))} \lesssim r^{-1} \left(\fint_{B^{+}(\bar x,r)} {\lvertD_jv\rvert}^p\,\right)^{\frac1p} \lesssim r^{-1} \left(\fint_{B^{+}(\bar x,r)} {\lvertD_{x'}v\rvert}^p\,\right)^{\frac1p},$$ where ${\lvertD_{x'} v\rvert}^2 := \sum_{j=1}^{n-1} (D_j v)^2$. Since $$D_{nn} v = -\frac{1}{\bar a^{nn}} \sum_{(i,j)\neq (n,n)}\bar a^{ij} D_{ij}v,$$ we obtain $$\label{eq9.33th}
{\lVertD^2 v\rVert}_{L^\infty(B^{+}(\bar x, \frac12 r))}\lesssim r^{-1} \left(\fint_{B^{+}(\bar x, r)} {\lvertD_{x'}v\rvert}^p\,\right)^{\frac1p}.$$ Therefore, we have $$\label{int-reg}
{\lVertD^2 v\rVert}_{L^\infty(B^{+}(\bar x, \frac12 r))}\lesssim r^{-1} \left(\fint_{B^{+}(\bar x,r)} {\lvertDv -q{\boldsymbol{e}}_n\rvert}^p\,\right)^{\frac1p}, \quad \forall q \in {\mathbb{R}}.$$ Let $0< \kappa < \frac12$ to be a number to be fixed later. Note that we have $$\left(\fint_{B^{+}(\bar x,\kappa r)} {\lvertD_nv - \overline{D_n v}_{B^{+}(\bar x,\kappa r)}\rvert}^p \right)^{\frac1p} \le 2\kappa r {\lVertD^2 v\rVert}_{L^\infty(B(\bar x, \frac12 r))},$$ while, for $j=1, \ldots, n-1$, we have $$\left(\fint_{B^{+}(\bar x,\kappa r)} {\lvertD_j v\rvert}^p \right)^{\frac1p} = \left(\fint_{B^{+}(\bar x,\kappa r)} {\lvertD_j v - D_jv(\bar x)\rvert}^p \right)^{\frac1p} \le 2\kappa r {\lVertD^2 v\rVert}_{L^\infty(B(\bar x, \frac12 r))}.$$ Hence, by we obtain $$\label{eq15.50b}
\left(\fint_{B^{+}(\bar x,\kappa r)} {\lvertDv - \overline{D_n v}_{B^{+}(\bar x,\kappa r)} \, {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p}
\le C_0 \kappa \left(\fint_{B^{+}(\bar x, r)} {\lvertDv -q {\boldsymbol{e}}_n\rvert}^p \, \right)^{\frac1p},\quad \forall q \in {\mathbb{R}},$$ where $C_0$ is an absolute constant determined only by $n$, $\lambda$, $\Lambda$, and $p$. By using the decomposition $u=v+w$, we obtain from that $$\begin{aligned}
&\left(\fint_{B^{+}(\bar x, \kappa r)} {\lvertDu - \overline{D_n v}_{B^{+}(\bar x,\kappa r)}\, {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p} \\
&\qquad\qquad \le 2^{\frac{1-p}{p}} \left(\fint_{B^{+}(\bar x, \kappa r)} {\lvertDv - \overline{D_n v}_{B^{+}(\bar x,\kappa r)}\, {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p}+ C\left(\fint_{B^{+}(\bar x, \kappa r)} {\lvertDw\rvert}^p \right)^{\frac1p}\\
&\qquad \qquad \le 4^{\frac{1-p}p}C_0\kappa \left(\fint_{B^{+}(\bar x, r)} {\lvertDu - q {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p} +C (\kappa^{-\frac{n}{p}}+1) \left(\fint_{B^{+}(\bar x, r)} {\lvertDw\rvert}^p \right)^{\frac1p}.\end{aligned}$$ Since $q\in {\mathbb{R}}$ is arbitrary, by using , we thus obtain $$\varphi(\bar x,\kappa r) \le 4^{\frac{1-p}{p}} C_0 \kappa \, \varphi(\bar x,r)+ C (\kappa^{-\frac{n}{p}}+1) \left( \omega_{\mathbf{A}}(2r) \,{\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,2r))} + \omega_{{\boldsymbol{g}}}(2r) \right).$$ For any given $\beta \in (0,1)$, let $\kappa \in (0,\frac12)$ be sufficiently small so that $4^{\frac{1-p}{p}} C_0 \le \kappa^{\beta-1}$. Then, we obtain $$\varphi(\bar x,\kappa r) \le \kappa^{\beta} \varphi(\bar x,r)+ C \left( \omega_{\mathbf{A}}(2r) \,{\lVertDu\rVert}_{L^\infty(B^{+}(\bar x, 2r))} + \omega_{{\boldsymbol{g}}}(2r) \right).$$ Note that $\kappa^\beta <1$. By iterating, for $j=1,2,\ldots$, we get $$\varphi(\bar x,\kappa^j r) \le \kappa^{j \beta} \varphi(\bar x,r)
+C \left({\lVertDu\rVert}_{L^\infty(B^{+}(\bar x, 2r))} \sum_{i=1}^{j} \kappa^{(i-1)\beta} \omega_{\mathbf{A}}(2\kappa^{j-i} r) + \sum_{i=1}^{j} \kappa^{(i-1)\beta} \omega_{{\boldsymbol{g}}}(2\kappa^{j-i} r) \right).$$ Therefore, we have $$\label{eq22.25f}
\varphi(\bar x,\kappa^j r) \le \kappa^{j \beta} \varphi(\bar x,r) +C {\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,2r))}\,\tilde \omega_{\mathbf{A}}(2\kappa^{j} r) + C \tilde \omega_{{\boldsymbol{g}}}(2\kappa^{j} r),$$ where we set $$\label{eq14.27w}
\tilde \omega_{\bullet}(t)= \sum_{i=1}^{\infty} \kappa^{i \beta} \left(\omega_{\bullet}(\kappa^{-i}t)\, [\kappa^{-i}t \le 1] + \omega_{\bullet}(1)\, [\kappa^{-i}t >1] \right).$$ Here, we used Iverson bracket notation; i.e., $[P] = 1$ if $P$ is true and $[P] = 0$ otherwise. We recall that $\tilde \omega_{\bullet}(t)$ is a Dini function; see [@Dong2012 Lemma 1].
Now, for any $\rho$ satisfying $0<\rho \le r$, we take $j$ to be the integer satisfying $\kappa^{j+1} < \rho/r \le \kappa^j$. Then, by $$\label{eq0947f}
\varphi(\bar x, \rho) \le C \left(\frac{\rho}{r}\right)^{\beta} \varphi(\bar x, \kappa^{-j} \rho) +C {\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,2r))}\,\tilde \omega_{\mathbf{A}}(2\rho) + C \tilde \omega_{{\boldsymbol{g}}}(2\rho).$$ Therefore, we get from and .
\[lem-02\] Let $\beta \in (0,1)$. For any $x \in B^+_3$ and $0<\rho\le r \le \frac14$, we have $$\label{eq10.39tu}
\phi(x, \rho) \le C\left(\frac{\rho}{r}\right)^{\beta}\, r^{-n} {\lVertDu\rVert}_{L^1(B(x ,3r)\cap B^+_4)}+C{\lVertDu\rVert}_{L^\infty(B(x,5r)\cap B^+_4)}\, \hat\omega_{\mathbf{A}}(\rho) + C \hat\omega_g(\rho),$$ where $C=C(n, \lambda, \Lambda,p, \beta)$ are constants and $\hat\omega_\bullet(t)$ is a Dini function derived from $\omega_\bullet(t)$.
In this proof we shall denote $$\bar x= (x^1, \ldots, x^{n-1}, 0).$$ There are three possibilities.
i. $\rho \le r \le x^n$: We utilize an interior $C^1$ estimate developed in [@DK16] as follows. Since $B(x, r) \subset B^+_4$, we observe that $\phi(x,\rho)$ is identical to that introduced in the proof of [@DK16 Theorem 1.5]. We recall that it satisfies $$\phi(x,\kappa r) \le 2^{\frac2p-1}C_0 \kappa \, \phi(x, r)+ C (\kappa^{-\frac{d}{p}}+1) \left( \omega_A(r) \,{\lVertDu\rVert}_{L^\infty(B(x, r))} + \omega_g(r) \right).$$ Therefore, similar to , we get $$\label{eq22.25fo}
\phi(x, \kappa^j r) \le \kappa^{j \beta} \phi(x, r) +C {\lVertDu\rVert}_{L^\infty(B(x, r))}\,\tilde \omega_A(\kappa^{j} r) + C \tilde \omega_g(\kappa^{j} r),$$ where $\tilde\omega_\bullet$ is as defined in and we take $\kappa$ in and to be identical. Then we get an inequality similar to , namely, for $0<\rho \le r \le \frac14$ $$\label{eq0946fo}
\phi(x, \rho) \le C \left(\frac{\rho}r\right)^{\beta}\, \phi(x, \kappa^{-j} \rho) +C {\lVertDu\rVert}_{L^\infty(B(x,r))}\,\tilde \omega_{\mathbf{A}}(\rho) + C \tilde \omega_{{\boldsymbol{g}}}(\rho),$$ where $j$ is the integer satisfying $\kappa^{j+1} < \rho/r \le \kappa^j$. By , we have $$\phi(x, \kappa^{-j} \rho) \le \left(\fint_{B(x, \kappa^{-j}\rho)} {\lvertDu\rvert}^p\right)^{\frac1p} \le \kappa^{-n} \fint_{B(x,r)} {\lvertDu\rvert}\,,$$ and thus, we obtain $$\label{eq0947fo}
\phi(x, \rho) \le C \left(\frac{\rho}r\right)^{\beta}\, r^{-n}{\lVertDu\rVert}_{L^1(B(x,r))} +C {\lVertDu\rVert}_{L^\infty(B(x,r))}\,\tilde \omega_{\mathbf{A}}(\rho) + C \tilde \omega_{{\boldsymbol{g}}}(\rho).$$
ii. $x^n \le \rho \le r$: Since $B(x, \rho) \cap B^{+}_4 \subset B^{+}(\bar x, 2\rho) \subset B^+_4$, we have $$\begin{aligned}
\nonumber
\phi(x, \rho) &=\left( \fint_{B(x, \rho) \cap B^{+}(0,4)} \,{\lvertDu - {\boldsymbol{q}}_{x, \rho}\rvert}^p \right)^{\frac1p} \le \left( \fint_{B(x, \rho) \cap B^{+}(0,4)} \,{\lvertDu - \bar q_{\bar x, 2\rho}\, {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p} \\
\label{eq1448fr}
&\le 2^{\frac{n}p} \left( \fint_{B^{+}(\bar x, 2\rho)} \,{\lvertDu - \bar q_{\bar x, 2\rho}\,{\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p} = 2^{\frac{n}{p}} \varphi(\bar x, 2\rho).\end{aligned}$$ Therefore, by Lemma \[lem-01\], and using ${\lvertx-\bar x\rvert}=x^n \le r$, we obtain $$\begin{aligned}
\nonumber
\phi(x, \rho) &\le C \left(\frac{2\rho}{2r}\right)^\beta r^{-n}{\lVertDu\rVert}_{L^1(B^{+}(\bar x, 2r))}+ C{\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,4r))}\, \tilde \omega_{\mathbf{A}}(4\rho) + C \tilde \omega_{{\boldsymbol{g}}}(4\rho)\\
\label{eq0919tu}
&\le C \left(\frac{\rho}{r}\right)^\beta r^{-n}{\lVertDu\rVert}_{L^1(B(x, 3r)\cap B^+_4)}+ C{\lVertDu\rVert}_{L^\infty(B(x,5r) \cap B^+_4)}\, \tilde\omega_{\mathbf{A}}(4\rho) + C \tilde\omega_{{\boldsymbol{g}}}(4\rho).\end{aligned}$$
iii. $\rho \le x^n \le r$: Take $R=x^n$ and let $j$ be the integer satisfying $\kappa^{j+1} < \rho/R \le \kappa^j$. Since $B(x, \kappa^{-j} \rho) \subset B(x,R) \subset B^+_4$ and $B(x,R) \subset B^{+}(\bar x, 2R)$, we have $$\begin{aligned}
\nonumber
\phi(x, \kappa^{-j} \rho) &= \left( \fint_{B(x, \kappa^{-j} \rho)} \,{\lvertDu - {\boldsymbol{q}}_{x, \kappa^{-j} \rho}\rvert}^p \right)^{\frac1p} \le \left( \fint_{B(x, \kappa^{-j} \rho)} \,{\lvertDu - \bar q_{\bar x, 2R}\, {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p} \\
\label{eq20.09fr}
&\le 2^{\frac{n-1}p} \kappa^{-\frac{n}{p}}\left( \fint_{B^{+}(\bar x, 2R)} \,{\lvertDu - \bar q_{\bar x, 2R}\, {\boldsymbol{e}}_n\rvert}^p \right)^{\frac1p} = 2^{\frac{n-1}p} \kappa^{-\frac{n}{p}} \varphi(\bar x, 2R).\end{aligned}$$ Therefore, by and Lemma \[lem-01\], we get $$\begin{aligned}
\phi(x, \rho) &\le C\left(\frac{\rho}{R}\right)^{\beta} \varphi(\bar x, 2R) +C {\lVertDu\rVert}_{L^\infty(B(x,R))}\,\tilde \omega_{\mathbf{A}}(\rho) + C \tilde \omega_{{\boldsymbol{g}}}(\rho) \\
&\le C\left(\frac{\rho}{R}\right)^{\beta} \left\{\, \left(\frac{2R}{2r}\right)^{\beta}\,r^{-n} {\lVertDu\rVert}_{L^1(B^{+}(\bar x, 2r))}+ {\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,4r))}\,\tilde\omega_{\mathbf{A}}(4R) +\tilde\omega_{{\boldsymbol{g}}}(4R) \right\}\\
&\qquad \qquad+C {\lVertDu\rVert}_{L^\infty(B^{+}(\bar x,r))}\,\tilde \omega_{\mathbf{A}}(\rho) + C \tilde \omega_{{\boldsymbol{g}}}(\rho).\end{aligned}$$ Therefore, by setting $$\omega^{\sharp}_{\bullet}(t):= \sup_{s \in [t,1]} \,\left(\frac{t}{s}\right)^\beta\, \tilde\omega_{\bullet}(s) \quad (0<t\le 1)$$ and using ${\lvertx-\bar x\rvert}=x^n \le r$, we obtain $$\begin{aligned}
\nonumber
\phi(x, \rho) \le C\left(\frac{\rho}r\right)^{\beta}\,& r^{-n}{\lVertDu\rVert}_{L^1(B(x ,3r)\cap B^+_4)}+C{\lVertDu\rVert}_{L^\infty(B(x,5r)\cap B^+_4)}\, \omega^\sharp_{\mathbf{A}}(4\rho) + C \omega_g^\sharp(4\rho)\\
\label{eq14.24m}
&+C{\lVertDu\rVert}_{L^\infty(B(x,2r) \cap B^+_4)}\, \tilde \omega_{\mathbf{A}}(\rho) + C \tilde \omega_{{\boldsymbol{g}}}(\rho).\end{aligned}$$
We have covered all three possible cases and obtained bounds for $\phi(x, \rho)$, namely, , , and . Therefore, if we set $\hat\omega_{\bullet}(t)$ as $$\hat\omega_{\bullet}(t):= \tilde \omega_{\bullet}(t)+\tilde\omega_{\bullet}(4t)+\omega_{\bullet}^{\sharp}(4t),$$ then follows. To complete the proof, we only need to show that $\omega^{\sharp}_{\bullet}(t)$ is a Dini function. By [@Dong2012 Lemma 1], it is enough to show $$\label{eq15.39m}
\omega^{\sharp}_{\bullet}(t) \lesssim \sum_{j=0}^{\infty} \frac{1}{2^{j \beta}} \left(\tilde \omega_{\bullet}(2^j t)\, [ 2^j t \le 1] + \tilde\omega_{\bullet}(1)\, [2^j t >1] \right).$$ To see this, we first recall $\tilde \omega_{\bullet}$ satisfies the property . See Remark \[rmk2.33\] below. Since for any $s \in [t, 1]$, there is an integer $j$ be an integer such that $2^{j-1}t \le s < 2^j t$ and follows from the definition of $\omega^\sharp_\bullet(t)$.
\[rmk2.33\] It can be easily seen that $\tilde \omega_\bullet$ satisfies the condition ; see [@DK16]. From the construction of $\hat \omega_\bullet$ in the above proof, it is routine to verify that $\hat \omega_\bullet$ satisfies the property as well.
\[lem-03\] We have $$\label{eq10.23m}
{\lVertDu\rVert}_{L^\infty(B^+_2)} \le C {\lVertDu\rVert}_{L^1(B^+_4)} + C\int_0^{1} \frac{\hat \omega_{{\boldsymbol{g}}}(t)}t \,dt,$$ where $C>0$ is a constant depending only on $n$, $\lambda$, $\Lambda$, $p$, and $\omega_{\mathbf{A}}$.
For $x \in B^+_3$ and $0<r \le \frac14$, let ${\{{\boldsymbol{q}}_{x,2^{-k}r}\}}_{k=0}^\infty$ be a sequence of vectors in ${\mathbb{R}}^n$ as given in . Since we have $${\lvert{\boldsymbol{q}}_{x,r} - {\boldsymbol{q}}_{x, \frac12 r}\rvert}^p \le
{\lvertDu(y)-{\boldsymbol{q}}_{x,r}\rvert}^p + {\lvertDu(y) - {\boldsymbol{q}}_{x, \frac12 r}\rvert}^p,$$ by taking average over $y \in B(x,\frac12 r) \cap B^{+}_4$ and then taking $p$th root, we obtain $$\label{eq1426fr}
{\lvert{\boldsymbol{q}}_{x,r} -{\boldsymbol{q}}_{x,\frac12 r}\rvert} \lesssim \phi(x,r) + \phi(x, \tfrac12 r).$$ Then, by iterating, we get $$\label{eq1408th}
{\lvert{\boldsymbol{q}}_{x,2^{-k} r} - {\boldsymbol{q}}_{x, r}\rvert} \lesssim \sum_{j=0}^k \phi(x, 2^{-j} r).$$ Note that implies $$\lim_{k \to \infty} \phi(x, 2^{-k}r)=0,$$ and thus, by the assumption that $u \in C^1(\bar B^{+}_3)$, we find $$\lim_{k \to \infty} {\boldsymbol{q}}_{x, 2^{-k}r}=Du(x).$$ Therefore, by taking $k\to \infty$ in , using and Remark \[rmk2.33\], we get $${\lvertDu(x)-{\boldsymbol{q}}_{x,r}\rvert} \lesssim r^{-n}{\lVertDu\rVert}_{L^1(B(x ,3r)\cap B^+_4)}+
{\lVertDu\rVert}_{L^\infty(B(x,5r)\cap B^+_4)} \int_0^r \frac{\hat \omega_{\mathbf{A}}(t)}t \,dt+\int_0^r \frac{\hat\omega_{{\boldsymbol{g}}}(t)}t \,dt.$$ By averaging the obvious inequality $${\lvert{\boldsymbol{q}}_{x, r}\rvert}^p \le {\lvertDu(y) -{\boldsymbol{q}}_{x, r}\rvert}^p + {\lvertDu(y)\rvert}^p$$ over $y \in B(x, r)\cap B^+_4$ and taking $p$th root, we get $${\lvert{\boldsymbol{q}}_{x, r}\rvert} \lesssim \phi(x, r) + \left(\fint_{B(x,r)\cap B^+_4} {\lvertDu\rvert}^p\right)^{\frac1p}.$$ Combining these together and using $$\phi(x,r) \lesssim r^{-n} {\lVertDu\rVert}_{L^1(B(x,r)\cap B^+_4)},$$ we obtain $${\lvertDu(x)\rvert} \lesssim r^{-n} {\lVertDu\rVert}_{L^1(B(x,3r)\cap B^+_4)} + {\lVertDu\rVert}_{L^\infty(B(x,5r)\cap B^+_4)} \int_0^r \frac{\hat\omega_{\mathbf{A}}(t)}t \,dt+\int_0^r \frac{\hat\omega_{{\boldsymbol{g}}}(t)}t \,dt.$$ Now, taking supremum for $x\in B(x_0, r)\cap B^+_4$, where $x_0 \in B^+_3$ and $r\le \frac14$, we have $$\begin{gathered}
{\lVertDu\rVert}_{L^\infty(B(x_0, r)\cap B^+_4)} \le
C r^{-n} {\lVertDu\rVert}_{L^1(B(x_0, 4r)\cap B^+_4)}\\
+C {\lVertDu\rVert}_{L^\infty(B(x_0, 6r)\cap B^+_4)} \int_0^r \frac{\hat\omega_{\mathbf{A}}(t)}t \,dt + C \int_0^r \frac{\hat\omega_{{\boldsymbol{g}}}(t)}t \,dt .\end{gathered}$$ We fix $r_0 <\frac14$ such that for any $0<r\le r_0$, $$C \int_0^{r} \frac{\hat\omega_{\mathbf{A}}(t)}t \,dt \le \frac1{3^n}.$$ Then, we have for any $x_0 \in B^+_3$ and $0<r\le r_0$ that $${\lVertDu\rVert}_{L^\infty(B(x_0, r)\cap B^+_4)} \le
3^{-n}{\lVertDu\rVert}_{L^\infty(B(x_0,6r) \cap B^+_4)} + C r^{-n} {\lVertDu\rVert}_{L^1(B(x_0,4r) \cap B^+_4)} + C \int_0^r \frac{\hat\omega_{{\boldsymbol{g}}}(t)}t \,dt.$$ For $k=1,2,\ldots$, denote $r_k=3-2^{1-k}$. Note that $r_{k+1}-r_k=2^{-k}$ for $k\ge 1$ and $r_1=2$. For $x_0\in B^+_{r_k}$ and $r\le 2^{-k-3}$, we have $B(x_0, 6r) \cap B^+_4 \subset B^{+}_{r_{k+1}}$. We take $k_0$ sufficiently large such that $2^{-k_0-3}\le r_0$. It then follows that for any $k\ge k_0$, $${\lVertDu\rVert}_{L^\infty(B^+_{r_k})} \le C 2^{kn} {\lVertDu\rVert}_{L^1(B^+_4)} + C \int_0^{1} \frac{\hat\omega_{{\boldsymbol{g}}}(t)}t \,dt+3^{-n} {\lVertDu\rVert}_{L^\infty(B^+_{r_{k+1}})}.$$ By multiplying the above by $3^{-kn}$ and then summing over $k \ge k_0$, we reach $$\sum_{k=k_0}^\infty 3^{-kn}{\lVertDu\rVert}_{L^\infty(B^+_{r_k})} \le C {\lVertDu\rVert}_{L^1(B^+_4)} + C \int_0^{1} \frac{\hat\omega_{{\boldsymbol{g}}}(t)}t \,dt
+\sum_{k=k_0}^\infty 3^{-(k+1)n} {\lVertDu\rVert}_{L^\infty(B^+_{r_{k+1}})}.$$ Since we assume that $u\in C^1(\overline B{}^+_3)$, the summations on both sides are convergent and we obtain .
With $\tilde\omega_\bullet(t)$ and $\hat\omega_\bullet(t)$ as in Lemmas \[lem-01\] and \[lem-02\], we define $$\label{eq10.46tu}
\omega^*_\bullet(t):= \hat\omega_\bullet(t)+ \int_0^t \frac{\tilde \omega_\bullet(s)}s \,ds +\tilde \omega_\bullet(4t)+ \int_0^t \frac{\tilde \omega_\bullet(4s)}s \,ds.$$
\[lem-04\] Let $\beta \in (0,1)$. For any $x \in B^+_3$ and $0< r \le \frac15$, we have $$\label{eq11.44fr}
{\lvertDu(x)-{\boldsymbol{q}}_{x,r}\rvert} \le Cr^{\beta}\,{\lVertDu\rVert}_{L^1(B(x,\frac35) \cap B^+_4)} + C{\lVertDu\rVert}_{L^\infty(B(x,1)\cap B^+_4)}\, \omega^*_{\mathbf{A}}(r) + C\omega_{{\boldsymbol{g}}}^*(r),$$ where $C=C(n, \lambda, \Lambda,p, \beta)$ are constants and $\omega^*(t)$ is defined as in .
As in the proof of Lemma \[lem-02\], we denote $\bar x= (x^1, \ldots, x^{n-1}, 0)$ and $\kappa<\frac12$ be the same constant as and . Let ${\{{\boldsymbol{q}}_{x,\kappa^i r}\}}_{i=0}^\infty \in {\mathbb{R}}^n$ and ${\{\bar q_{\bar x, 2\kappa^i r}\}}_{i=0}^\infty \in {\mathbb{R}}$ be sequences that are chosen accordingly as in and . By using $$\lim_{i \to \infty} {\boldsymbol{q}}_{x,\kappa^i r}=Du(x)$$ and a computation similar to , we get $${\lvertDu(x) -{\boldsymbol{q}}_{x,r}\rvert} \le \sum_{i=0}^{\infty} \,{\lvert{\boldsymbol{q}}_{x, \kappa^i r}-{\boldsymbol{q}}_{x,\kappa^{i+1}r}\rvert}
\lesssim \sum_{i=0}^\infty \phi(x, \kappa^i r).$$ Therefore, it suffices to bound $\sum_{i=0}^\infty \phi(x, \kappa^i r)$ by the right-hand side of .
In the case when $r\le x^n$, by and [@DK16 Lemma 2.7] we have $$\sum_{i=0}^\infty \phi(x, \kappa^i r) \lesssim \phi(x,r)+ {\lVertDu\rVert}_{L^\infty(B(x,\frac15) \cap B^+_4)} \int_0^{r} \frac{\tilde \omega_{\mathbf{A}}(t)}t \,dt +\int_0^{r} \frac{\tilde \omega_{{\boldsymbol{g}}}(t)}t \,dt.$$ By Lemma \[lem-02\], we get $$\phi(x, r) \lesssim r^{\beta}\,{\lVertDu\rVert}_{L^1(B(x,\frac35) \cap B^+_4)} + {\lVertDu\rVert}_{L^\infty(B(x,1)\cap B^+_4)}\, \hat\omega_{\mathbf{A}}(r) + \hat\omega_{{\boldsymbol{g}}}(r),$$ By combining the above two inequalities, we obtain .
In the case when $r>x^n$, let $i_0$ be the integer such that $\kappa^{i_0+1}r \le x^n < \kappa^{i_0} r$. By and [@DK16 Lemma 2.7], we have $$\sum_{i=i_0+1}^\infty \phi(x, \kappa^i r) \lesssim \phi(x, \kappa^{i_0+1} r) + {\lVertDu\rVert}_{L^\infty(B(x,r) \cap B^+_4)} \int_0^{r} \frac{\tilde \omega_{\mathbf{A}}(t)}t \,dt +\int_0^{r} \frac{\tilde \omega_{{\boldsymbol{g}}}(t)}t \,dt.$$ By a computation similar to , we have $$\phi(x, \kappa^{i_0+1} r) \le 2^{\frac{n-1}p} \kappa^{-\frac{n}{p}} \varphi(\bar x, 2 \kappa^{i_0} r)$$ and by , for $0 \le i \le i_0$, we have $$\phi(x, \kappa^i r) \le 2^{\frac{n}p} \varphi(\bar x, 2\kappa^i r).$$ Hence, we have $$\sum_{i=0}^\infty \phi(x, \kappa^i r) \lesssim \sum_{i=0}^{i_0} \varphi(\bar x, 2\kappa^i r) + {\lVertDu\rVert}_{L^\infty(B(x,r) \cap B^+_4)} \int_0^{r} \frac{\tilde \omega_{\mathbf{A}}(t)}t \,dt +\int_0^{r} \frac{\tilde \omega_{{\boldsymbol{g}}}(t)}t \,dt.$$ On the other hand, by and assumption ${\lvertx-\bar x\rvert}=x^n <r \le \frac15$, we have $$\sum_{i=0}^\infty \varphi(\bar x, 2\kappa^i r) \lesssim \varphi(\bar x, 2r)+ {\lVertDu\rVert}_{L^\infty(B(x, 1) \cap B^+_4)} \int_0^{r} \frac{\tilde \omega_{\mathbf{A}}(4t)}t \,dt +\int_0^r \frac{\tilde \omega_{{\boldsymbol{g}}}(4t)}t \,dt.$$ By Lemma \[lem-01\] and using ${\lvertx-\bar x\rvert} \le \frac15$, we find $$\varphi(\bar x, 2r) \lesssim r^{\beta}\, {\lVertDu\rVert}_{L^1(B(x,\frac35)\cap B^{+}_4)}+ {\lVertDu\rVert}_{L^\infty(B(x,1)\cap B^+_4)}\,\tilde \omega_{\mathbf{A}}(4r) + \tilde \omega_{{\boldsymbol{g}}}(4r).$$ Combining these together, we get as well.
Now, we are ready to show that $u \in C^1(\bar B^+_1)$. For $x, y \in B^+_1$, we have $${\lvertDu(x)-Du(y)\rvert} \le {\lvertDu(x)- {\boldsymbol{q}}_{x,r}\rvert}\ + {\lvert{\boldsymbol{q}}_{x,r}-{\boldsymbol{q}}_{y,r}\rvert}+{\lvertDu(y)-{\boldsymbol{q}}_{y,r}\rvert}.$$ In the case when ${\lvertx-y\rvert} <\frac15$, set $r={\lvertx-y\rvert}$ and apply Lemma \[lem-04\] to get $${\lvertDu(x)- {\boldsymbol{q}}_{x,r}\rvert} +{\lvertDu(y)-{\boldsymbol{q}}_{y,r}\rvert}
\lesssim r^{\beta}\,{\lVertDu\rVert}_{L^1(B^+_2)} + {\lVertDu\rVert}_{L^\infty(B^+_2)}\, \omega^*_{\mathbf{A}}(r) + \omega_{{\boldsymbol{g}}}^*(r).$$ Take the average over $z \in B(x,r)\cap B(y,r) \cap B^+_4$ in the inequality $${\lvert{\boldsymbol{q}}_{x,r}-{\boldsymbol{q}}_{y,r}\rvert}^p \le {\lvertDu(z)- {\boldsymbol{q}}_{x,r}\rvert}^p +{\lvertDu(z)-{\boldsymbol{q}}_{y,r}\rvert}^p$$ and take the $p$th root and apply Lemma \[lem-02\] to get $${\lvert{\boldsymbol{q}}_{x,r}-{\boldsymbol{q}}_{y,r}\rvert} \lesssim \phi(x,r) + \phi(y,r) \lesssim r^\beta\, {\lVertDu\rVert}_{L^1(B^+_2)} + {\lVertDu\rVert}_{L^\infty(B^+_2)}\, \hat\omega_{\mathbf{A}}(r) + \hat\omega_{{\boldsymbol{g}}}(r).$$ Combining these together and using Lemma \[lem-03\], we obtain (note $\hat\omega_\bullet (t) \le \omega^*_\bullet(t)$) $$\begin{gathered}
\label{eq08.33st}
{\lvertDu(x)-Du(y)\rvert} \lesssim {\lVertDu\rVert}_{L^1(B^+_2)}\,{\lvertx-y\rvert}^\beta \\
+\left( {\lVertDu\rVert}_{L^1(B^+_4)} + \int_0^1 \frac{\hat \omega_g(t)}t\,dt \right) \omega^*_{\mathbf{A}}({\lvertx-y\rvert})+ \omega^*_{{\boldsymbol{g}}}({\lvertx-y\rvert}).\end{gathered}$$ In case when ${\lvertx-y\rvert} \ge \frac15$, we use ${\lvertDu(x) - Du(y)\rvert} \le 2{\lVertDu\rVert}_{L^\infty(B^+_1)}$, apply Lemma \[lem-03\], and still obtain . This completes the proof of Proposition \[prop01\] and that of Theorem \[thm-main-d\].
We note that the modulus of continuity estimate is sharper than the corresponding interior estimate in [@DK16]. In particular, if $\mathbf A$ and ${\boldsymbol{g}}$ are Hölder continuous with exponent $\alpha \in (0,1)$, then by taking $\beta \in (\alpha, 1)$ in , one can verify that $Du$ is Hölder continuous with the same exponent $\alpha$, recovering the classical Schauder estimates. This fact was not clear in [@DK16].
Proof of Theorem \[thm-main-nd\]
--------------------------------
The idea of proof is essentially the same as that of Theorem \[thm-main-d\]. We first establish interior $C^2$ estimates.
\[prop2.45p\] For any $p\in (1,\infty)$, we have $u\in W^{2,p}(\Omega)$. Moreover, for any $\Omega' \subset\subset \Omega$, we have $u \in C^2(\overline \Omega{}')$.
By the $W^{2,p}$ theory, we have $u \in W^{2, p}(\Omega)$ for any $1<p<\infty$ and $${\lVertu\rVert}_{W^{2,p}(\Omega)} \le C {\lVertg\rVert}_{L^\infty(\Omega)} + C {\lVertu\rVert}_{L^1(\Omega)},$$ where $C$ is a constant depending only on $n$, $\lambda$, $\Lambda$, $p$, $\Omega$, $\partial\Omega$, and the coefficients of ${\mathcal{L}}$; see, for instance, [@Kr08 Theorem 11.2.3]. Therefore, by the Morrey-Sobolev embedding, $u \in C^{1,\mu}(\Omega)$ for any $0<\mu<1$ and $${\lVertu\rVert}_{C^{1,\mu}(\Omega)} \le C{\lVertg\rVert}_{L^\infty(\Omega)}+C{\lVertu\rVert}_{L^1(\Omega)}.$$ In particular, we have $$\varrho_{Du}(t)+ \varrho_u(t) \le C \left( {\lVertg\rVert}_{L^\infty(\Omega)} + {\lVertu\rVert}_{L^1(\Omega)}\right) t^{\mu}.$$ We rewrite the equation as $$a^{ij} D_{ij} u= g-b^i D_i u-cu =: g'.$$ Then $g'$ is of Dini mean oscillation by Lemma \[lem00\]. Moreover, by , we have $$\omega_{g'}(t) \le \omega_g(t) + C\left( {\lVertg\rVert}_{L^\infty(\Omega)} + {\lVertu\rVert}_{L^1(\Omega)}\right) \left\{\omega_b(t)+ \omega_c(t)+ \left({\lVert{\boldsymbol{b}}\rVert}_{L^\infty(\Omega)} + {\lVertc\rVert}_{L^\infty(\Omega)}\right)t^\mu \right\}.$$ Therefore, $\omega_{g'}$ is a Dini function that is completely determined by the given data (namely $n$, $\lambda$, $\Lambda$, $\Omega$, $\omega_{\mathbf{A}}$, $p$, $\omega_b$, ${\lVertb\rVert}_{L^\infty(\Omega)}$, $\omega_c$, $\omega_g$, and ${\lVertg\rVert}_{L^\infty(\Omega)}$) and ${\lVertu\rVert}_{L^1(\Omega)}$. By [@DK16 Theorem 1.6], we thus find that $u \in C^2(\overline \Omega{}')$ and ${\lVertu\rVert}_{C^2(\overline \Omega{}')}$ is bounded by a constant $C$ depending only on the above mentioned given data, ${\lVertu\rVert}_{L^1(\Omega)}$, and $\Omega'$.
Next, we turn to $C^2$ estimate near the boundary. Let $g'$ be as given in the proof of Proposition \[prop2.45p\]. Under a mapping of flattening boundary $$y={\boldsymbol{\Phi}}(x)=(\Phi^1(x),\ldots, \Phi^n(x)),$$ let $\tilde u(y)=u(x)$, which satisfies $$\tilde a^{ij} D_{ij} \tilde u=\tilde g'-\tilde b^iD_i\tilde u=:\tilde h,$$ where $$\tilde a^{ij}(y)= D_l\Phi^i D_k\Phi^j a^{kl}(x),\quad \tilde b^i(y)= D_{kl}\Phi^i a^{kl}(x),\quad \tilde g'(y)=g'(x).$$ By Lemmas \[lem00\], we see that the coefficients $\tilde a^{ij}$ and the data $\tilde h$ are of Dini mean oscillation. As before, we are thus reduced to prove the following.
\[prop01nd\] If $u \in W^{2,2}(B^+_4)$ is a strong solution of $$a^{ij} D_{ij} u= g\;\mbox{ in }\;B^+_4$$ satisfying $u=0$ on $T(0,4)$, then $u \in C^2(\overline B{}^+_1)$.
The rest of this subsection is devoted to the proof of Proposition \[prop01nd\]. As in the proof of Proposition \[prop01\], we shall derive an a priori estimate of the modulus of continuity of $D^2u$ by assuming that $u$ is in $C^2(\overline B{}^+_3)$.
Let $\mathbb S(n)$ be the set of all symmetric $n\times n$ matrices and let $$\mathbb S_0(n)={\{ \mathbf q= (q^{ij}) \in \mathbb S(n) : q^{ij}=0\;\text{ for }\; i,j =1, 2, \ldots, n-1\}}.$$ Fix any $p \in (0,1)$. Similar to and , for $x\in B^+_4$ and $r>0$, we define $$\phi(x,r):=\inf_{\mathbf q \in \mathbb{S}(n)} \left( \fint_{B(x,r) \cap B^{+}_4} \,{\lvertD^2u - \mathbf q\rvert}^p \right)^{\frac1p}$$ and fix a matrix $\mathbf q_{x,r}\in \mathbb S(n)$ satisfying $$\phi(x,r) = \left( \fint_{B(x,r) \cap B^{+}_4} \,{\lvertD^2u - \mathbf q_{x,r}\rvert}^p \right)^{\frac1p}.$$ Also, similar to , for $\bar x \in T(0,4)$ and $r>0$, we introduce an auxiliary quantity $$\varphi(\bar x, r):=\inf_{\mathbf q \in \mathbb S_0(n)}\left( \fint_{B^{+}(\bar x,r)} {\lvertD^2u -\mathbf q\rvert}^p \right)^{\frac1p}.$$
The following lemma is in parallel with Lemma \[lem-01\].
\[lem-01nd\] Let $\beta \in (0,1)$. For any $\bar x \in T(0,3)$ and $0<\rho \le r \le \frac12$, we have $$\varphi(\bar x, \rho) \le C\left(\frac{\rho}{r}\right)^{\beta}\,r^{-n} {\lVertD^2u\rVert}_{L^1(B^{+}(\bar x, r))}+C {\lVertD^2u\rVert}_{L^\infty(B^{+}(\bar x,2r))}\,\tilde\omega_{\mathbf{A}}(2\rho) + C \tilde\omega_g(2\rho),$$ where $C=C(n, \lambda, \Lambda, p, \beta)$ are constants and $\tilde\omega_\bullet(t)$ is a Dini function derived from $\omega_\bullet(t)$.
For $\bar x \in T(0,3)$ and $0<r \le \frac12$, we decompose $u=v+w$, where $w \in W^{2,2}({\mathcal{D}}(\bar x, 2r)) \cap W^{1,2}_0({\mathcal{D}}(\bar x, 2r))$ is a unique solution of the problem $$\bar a^{ij} D_{ij} w = -{\operatorname{tr}}((\mathbf{A}- \bar{\mathbf{A}}) D^2 u) + g-\bar g \;\mbox{ in }\; {\mathcal{D}}(\bar x, 2r);\quad
w=0 \;\mbox{ on }\;\partial {\mathcal{D}}(\bar x, 2r).$$ By Lemma \[lem-weak11-nd\] with scaling, we have for any $\alpha>0$, $${\lvert{\{x\in B^{+}(\bar x, r): {\lvertD^2w(x)\rvert} > \alpha\}}\rvert} \lesssim \frac{1}{\alpha}\left(\,{\lVertD^2 u\rVert}_{L^\infty(B^{+}(\bar x, 2r))} \int_{B^{+}(\bar x, 2r)} {\lvert\mathbf{A}-\bar{\mathbf{A}}\rvert} + \int_{B^{+}(\bar x, 2r)} {\lvertg -\bar g\rvert}\,\right).$$ Therefore, we have $$\left(\fint_{B^{+}(\bar x, r)} {\lvertD^2w\rvert}^p \right)^{\frac1p} \lesssim \omega_{\mathbf{A}}(2r) \,{\lVertD^2 u\rVert}_{L^\infty(B^{+}(\bar x, 2r))} + \omega_g(2r).$$ Since $v=u-w$ satisfies $$\bar a^{ij} D_{ij} v= D_i (\bar a^{ij} D_j v)= \bar g\;\mbox{ in }\;B^{+}(\bar x, r);\quad v=0\;\mbox{ on }\;T(\bar x, r),$$ we see that $D_k v$ satisfies for $k=1, \ldots, n-1$. Therefore, by , we have $${\lVertD^2D_k v\rVert}_{L^\infty(B^{+}(\bar x, \frac12 r))}\lesssim r^{-1} \left(\fint_{B^{+}(\bar x, r)} {\lvertD_{x'}^2 v\rvert}^p\,\right)^{\frac1p},$$ where ${\lvertD_{x'}^2 v\rvert}^2 = \sum_{k, l=1}^{n-1} (D_{kl} v)^2$. Since $\bar a^{ij} D_{ij} (D_n v)= 0$, we find $$D_{nnn} v=D_{nn} D_nv = -\frac{1}{\bar a^{nn}} \sum_{(i,j)\neq (n,n)}\bar a^{ij} D_{ijn}v,$$ and thus, we obtain $${\lVertD^3 v\rVert}_{L^\infty(B^{+}(\bar x, \frac12 r))}\lesssim r^{-1} \left(\fint_{B^{+}(\bar x, r)} {\lvertD_{x'}^2 v\rvert}^p\,\right)^{\frac1p}.$$ Therefore, similar to , we have $${\lVertD^3 v\rVert}_{L^\infty(B^{+}(\bar x, \frac12 r))}\lesssim r^{-1} \left(\fint_{B^{+}(\bar x,r)} {\lvertD^2v -\mathbf q\rvert}^p\,\right)^{\frac1p},\quad \forall \mathbf q \in \mathbb S_0(n).$$ Note that for $i, j=1, \ldots, n-1$ and $0<\kappa < \frac12$, we have $$\left(\fint_{B^{+}(\bar x,\kappa r)} {\lvertD_{ij} v\rvert}^p \right)^{\frac1p} = \left(\fint_{B^{+}(\bar x,\kappa r)} {\lvertD_{ij} v - D_{ij}v(\bar x)\rvert}^p \right)^{\frac1p} \le 2\kappa r {\lVertD^3 v\rVert}_{L^\infty(B(\bar x, \frac12 r))}.$$ Therefore, if we take $\bar {\mathbf q}_{\bar x, \kappa r} \in \mathbb S_0(n)$ whose $(i,n)$ entry is $\overline{D_{in}v}_{B^+(\bar x, \kappa r)}$ for $i=1,\ldots, n$, then similar to , we have $$\left(\fint_{B^{+}(\bar x,\kappa r)} {\lvertD^2v - \bar {\mathbf q}_{\bar x, \kappa r}\rvert}^p \right)^{\frac1p}
\le C_0 \kappa \left(\fint_{B^{+}(\bar x, r)} {\lvertD^2v -\mathbf q\rvert}^p\, \right)^{\frac1p}, \quad \forall \mathbf q \in \mathbb S_0(n).$$ By the same argument that led to , we find that there is $\kappa \in (0, \frac12)$ such that $$\varphi(\bar x, \kappa^j r) \le \kappa^{j \beta} \varphi(\bar x, r) +C {\lVertD^2 u\rVert}_{L^\infty(B^{+}(\bar x, 2r))}\,\tilde \omega_{\mathbf{A}}(2\kappa^{j} r) + C \tilde \omega_g(2\kappa^{j} r),$$ where $\tilde\omega_\bullet(t)$ is the same as in . The rest of proof is the same as that of Lemma \[lem-01\].
By modifying the proof of Lemmas \[lem-02\], \[lem-03\], and \[lem-04\] in a straightforward way, we obtain the following lemmas.
Let $\beta \in (0,1)$. For any $x \in B^+_3$ and $0<\rho\le r \le \frac14$, we have $$\phi(x, \rho) \le C\left(\frac{\rho}{r}\right)^{\beta}\, r^{-n} {\lVertD^2u\rVert}_{L^1(B(x ,3r)\cap B^+_4)}+C{\lVertD^2u\rVert}_{L^\infty(B(x,5r)\cap B^+_4)}\, \hat\omega_{\mathbf{A}}(\rho) + C \hat\omega_g(\rho),$$ where $C=C(n, \lambda, \Lambda,p, \beta)$ are constants and $\hat\omega_\bullet(t)$ is a Dini function derived from $\omega_\bullet(t)$.
We have $${\lVertD^2 u\rVert}_{L^\infty(B^+_2)} \le C {\lVertD^2 u\rVert}_{L^1(B^+_4)} + C\int_0^{1} \frac{\hat\omega_g(t)}t \,dt,$$ where $C>0$ is a constant depending only on $n$, $\lambda$, $\Lambda$, $p$, and $\omega_{\mathbf{A}}$.
Let $\beta \in (0,1)$. For any $x \in B^+_3$ and $0< r \le \frac15$, we have $${\lvertD^2u(x)-\mathbf q_{x,r}\rvert} \le Cr^{\beta}\,{\lVertD^2u\rVert}_{L^1(B(x,\frac35) \cap B^+_4)} + C{\lVertD^2u\rVert}_{L^\infty(B(x,1)\cap B^+_4)}\, \omega^*_{\mathbf{A}}(r) + C\omega_g^*(r),$$ where $C=C(n, \lambda, \Lambda,p, \beta)$ are constants and $\omega^*(t)$ is defined as in .
With the above lemmas at hand, we obtain, similar to , the following estimates for $x, y \in B^+_1$: $$\begin{gathered}
{\lvertD^2 u(x)-D^2 u(y)\rvert} \le C{\lVertD^2u\rVert}_{L^1(B^+_2)}\,{\lvertx-y\rvert}^\beta \\
+C\left( {\lVertD^2 u\rVert}_{L^1(B^+_4)} + \int_0^1 \frac{\hat\omega_g(t)}t\,dt \right) \omega^*_{\mathbf{A}}({\lvertx-y\rvert})+ C\omega^*_g({\lvertx-y\rvert}),\end{gathered}$$ where $\beta =\beta(n,\lambda, \Lambda, p)$ and $C=C(n, \lambda, \Lambda, p, \omega_{\mathbf{A}})$, and $\omega^*_\bullet(t)$ is defined as in . $\beta \in (0,1)$ is any given number, $C=C(n, \lambda, \Lambda, p, \omega_{\mathbf{A}}, \beta)$, and $\omega^*_\bullet(t)$ is defined as in . We have shown that $u \in C^2(\bar B^+_1)$ as desired. This completes the proof of Proposition \[prop01nd\] and that of Theorem \[thm-main-nd\].
Instead of the condition of Dini mean oscillations (in the $L^1$ sense), we may also consider coefficients and data with Dini mean oscillations in the $L^p$ sense with some $p\in (1,\infty)$, i.e., the function $\omega_{g, p}: {\mathbb{R}}_+ \to {\mathbb{R}}$ defined by $$\omega_{g,p}(r):=\sup_{x\in \overline{\Omega}} \left(\fint_{\Omega(x,r)} \,{\lvertg(y)-\bar {g}_{\Omega(x,r)}\rvert}^p\,dy \right)^{\frac 1 p}$$ is a Dini function. In this case, by modifying the proofs below and using the $L^p$ estimates instead of the weak type-$(1,1)$ estimates, we can show that in the non-divergence case $D^2 u$ has Dini mean oscillations in the $L^p$ sense with the same $p$. Similar results hold for divergence form equations, and the adjoint problem of non-divergence form equations (with the boundary data $\psi=0$).
Proof of Theorem \[thm-main-adj\]
---------------------------------
As before, we adopt the usual summation convention over repeated indices. We first establish the following interior $C^0$ estimates.
\[prop2.45p-adj\] For any $p\in (1,\infty)$, we have $u\in L^p(\Omega)$. Moreover, for any $\Omega' \subset\subset \Omega$, we have $u \in C(\overline \Omega{}')$.
Let $w$ be a unique $W^{1,2}_0(\Omega)$ solution of $$\label{eq18.16hgm}
\Delta w= D_i(b^i u)-cu +f\;\text{ in }\;\Omega,\quad w=0 \;\text{ on }\;\partial\Omega.$$ Since $u \in L^2(\Omega)$, we have $w \in L^{r}(\Omega)$, where $\frac1r=\frac12+\frac1q-\frac1n<\frac12$. Then, by setting $\mathbf{g}'=\mathbf{g}+w \mathbf{I} \in L^r(\Omega)$, we see that $u$ becomes an adjoint solution of $$\label{eq15.55kaist}
D_{ij}(a^{ij} u)={\operatorname{div}}^2 \mathbf{g}'\;\text{ in }\;\Omega,\quad
u= \psi +\frac{\mathbf{g}'\nu\cdot \nu}{\mathbf{A}\nu\cdot \nu} \; \text{ on }\; \partial\Omega.$$ Therefore, by [@EM2016 Lemma 2], we see that $u \in L^{r}(\Omega)$. By bootstrapping, i.e., feeding $u\in L^r(\Omega)$ back to , we find that $u \in L^p(\Omega)$ for any $p\in (1,\infty)$ with ${\lVertu\rVert}_{L^p(\Omega)}$ controlled by the given data. Then, by the Morrey-Sobolev embedding, we have $w \in C^{0,\mu}(\Omega)$ for some $\mu>0$ with ${\lVertw\rVert}_{C^{0,\mu}(\Omega)}$ controlled by the given data, and $$\varrho_{w}(t) \le C\left( {\lVert\mathbf{g}\rVert}_{L^\infty(\Omega)} + {\lVertf\rVert}_{L^{s}(\Omega)}+ {\lVert\psi\rVert}_{L^\infty(\partial\Omega)}+ {\lVertu\rVert}_{L^1(\Omega)}\right) t^\mu.$$ Therefore, $\omega_{\mathbf g'}$ is a Dini function that is completely determined by the given data. By [@DK16 Theorem 1.10], we thus find that $u \in C(\overline \Omega{}')$ and ${\lVertu\rVert}_{C(\overline \Omega{}')}$ is bounded by a constant $C$ depending only on the given data, ${\lVertu\rVert}_{L^1(\Omega)}$, and $\Omega'$.
Next, we turn to continuity estimate near the boundary. The following lemma shows that it is enough to consider the case when $\psi=0$ in .
\[lem-cont-bdry-adj\] The adjoint problem $$D_{ij}(a^{ij} v)=0\;\text{ in }\;\Omega,\quad v=\psi\;\text{ on }\;\partial\Omega$$ has a unique solution $v \in C(\overline\Omega)$.
The proof of the above lemma is deferred to Section \[sec3\], where we introduce normalized adjoint solutions. Let $\mathbf{g}'$ be as in the proof of Proposition \[prop2.45p-adj\]. Under a volume preserving mapping of flattening boundary $$y={\boldsymbol{\Phi}}(x)=(\Phi^1(x),\ldots, \Phi^n(x))$$ as before, let $\tilde u(y)=u(x)$, which satisfies $$D_{ij}(\tilde a^{ij} \tilde u)={\operatorname{div}}^2 \tilde {\mathbf g}' +D_i(\tilde b^i \tilde u),$$ where $$\begin{gathered}
\tilde a^{ij}(y)= D_l\Phi^i D_k\Phi^j a^{kl}(x),\quad \tilde b_i(y)=D_{kl}\Phi^i a^{kl}(x),\quad
\tilde{\mathbf{g}}'(y)= D \mathbf{\Phi}{^{\mkern-1.5mu\mathsf{T}}}\mathbf{g}' D\mathbf{\Phi} (x).\end{gathered}$$ We may assume without loss of generality that ${\boldsymbol{\Phi}}$ is a $C^{1,1}$-diffeomorphism on ${\mathbb{R}}^n$. If we set $\tilde w$ to be a solution to $$\Delta \tilde w=D_i(\tilde b^i \tilde u)\;\text{ in }\;{\boldsymbol{\Phi}}(\Omega), \quad \tilde w=0\; \text{ on }\; {\boldsymbol{\Phi}}(\partial\Omega),$$ then $\tilde u$ satisfies $$D_{ij}(\tilde a^{ij} \tilde u)={\operatorname{div}}^2 (\tilde {\mathbf{g}}'+\tilde w \mathbf{I})\;\text{ in }\;{\boldsymbol{\Phi}}(\Omega),\quad \tilde u=\frac{(\mathbf{\tilde{g}}'+\tilde w \mathbf{I})\nu\cdot\nu}{\mathbf A\nu\cdot\nu}\;\text{ on }\;{\boldsymbol{\Phi}}(\partial\Omega).$$ By Lemma \[lem00\] and the proof of Proposition \[prop2.45p-adj\], we see that the coefficients $\tilde a^{ij}$ and the data $\tilde{\mathbf g}'+\tilde w \mathbf{I}$ are of Dini mean oscillation. As before, we are thus reduced to prove the following.
\[prop01adj\] If $u \in L^{2}(B^+_4)$ is an adjoint solution satisfying $$D_{ij}(a^{ij} u)= {\operatorname{div}}^2 \mathbf{g}\;\mbox{ in }\;B^+_4,\quad u=\frac{\mathbf{g}\nu\cdot \nu}{\mathbf{A}\nu\cdot\nu}\; \text{ on }\;T(0,4),$$ then $u \in C(\overline B{}^+_1)$.
The rest of this subsection is devoted to the proof of Proposition \[prop01adj\]. As in the proof of Propositions \[prop01\] and \[prop01nd\], we shall derive an a priori estimate of the modulus of continuity of $u$ by assuming that $u$ is in $C(\overline B{}^+_3)$.
Fix any $p \in (0,1)$. Similar to and , for $x\in B^+_4$ and $r>0$, we define $$\phi(x,r):=\inf_{q \in {\mathbb{R}}} \left( \fint_{B(x,r) \cap B^{+}_4} \,{\lvertu - q\rvert}^p \right)^{\frac1p}$$ and fix a number $q_{x,r}\in {\mathbb{R}}$ satisfying $$\phi(x,r) = \left( \fint_{B(x,r) \cap B^{+}_4} \, {\lvertu - q_{x,r}\rvert}^p \right)^{\frac1p}.$$
The following lemma is in parallel with Lemmas \[lem-01\] and \[lem-01nd\].
Let $\beta \in (0,1)$. For any $\bar x \in T(0,3)$ and $0<\rho \le r \le \frac12$, we have $$\phi(\bar x, \rho) \le C\left(\frac{\rho}{r}\right)^{\beta}\,r^{-n} {\lVertu\rVert}_{L^1(B^{+}(\bar x, r))}+C {\lVertu\rVert}_{L^\infty(B^{+}(\bar x,2r))}\,\tilde\omega_{\mathbf{A}}(2\rho) + C \tilde\omega_{\mathbf{g}}(2\rho),$$ where $C=C(n, \lambda, \Lambda, p, \beta)$ are constants and $\tilde\omega_\bullet(t)$ is a Dini function derived from $\omega_\bullet(t)$.
For $\bar x \in T(0,3)$ and $0<r \le \frac12$, we decompose $u=v+w$, where $w \in L^2(B(\bar x, r)) $ is a unique adjoint solution of the problem $$\begin{aligned}
D_{ij}(\bar a^{ij} w) &= -{\operatorname{div}}^2 ((\mathbf{A}- \bar{\mathbf{A}}) u) + {\operatorname{div}}^2(\mathbf{g}-\bar{\mathbf{g}}) \;\mbox{ in }\; {\mathcal{D}}(\bar x, 2r),\\
w&=\frac{\left(\mathbf{g}-\bar{\mathbf{g}}-(\mathbf{A}- \bar{\mathbf{A}}) u\right)\nu\cdot\nu}{\bar{\mathbf{A}}\nu\cdot\nu} \;\mbox{ on }\;\partial {\mathcal{D}}(\bar x, 2r).\end{aligned}$$ By Lemma \[lem-weak11-adj\] with scaling, we have for any $\alpha>0$, $${\lvert{\{x\in B^{+}(\bar x, r): {\lvertw(x)\rvert} > \alpha\}}\rvert} \lesssim \frac{1}{\alpha}\left(\,{\lVert u\rVert}_{L^\infty(B^{+}(\bar x, 2r))} \int_{B^{+}(\bar x, 2r)} {\lvert\mathbf{A}-\bar{\mathbf{A}}\rvert} + \int_{B^{+}(\bar x, 2r)} {\lvert\mathbf{g}-\bar{\mathbf{g}}\rvert}\,\right).$$ Therefore, we have $$\left(\fint_{B^{+}(\bar x, r)} {\lvertw\rvert}^p \right)^{\frac1p} \lesssim \omega_{\mathbf{A}}(2r) \,{\lVertu\rVert}_{L^\infty(B^{+}(\bar x, 2r))} + \omega_{\mathbf{g}}(2r).$$ Since $v=u-w$ satisfies $$D_{ij}(\bar a^{ij} v)={\operatorname{div}}^2\bar{\mathbf{g}}\;\mbox{ in }\;B^{+}(\bar x, r),\quad v=\frac{\bar{\mathbf{g}}\nu\cdot\nu}{\bar{\mathbf{A}}\nu\cdot\nu}\;\mbox{ on }\;T(\bar x, r),$$ by Lemma \[lem-bdry-Lip\] with scaling, we have $${\lVertD v\rVert}_{L^\infty(B^{+}(\bar x, \frac12 r))}\lesssim r^{-1} \left(\fint_{B^{+}(\bar x,r)} {\lvertv -q\rvert}^p\,\right)^{\frac1p},\quad \forall q \in {\mathbb{R}}.$$ Thus similar to , we have $$\left(\fint_{B^{+}(\bar x,\kappa r)} {\lvertv - \bar {v}_{\bar x, \kappa r}\rvert}^p \right)^{\frac1p}
\le C_0 \kappa \left(\fint_{B^{+}(\bar x, r)} {\lvertv -q\rvert}^p\, \right)^{\frac1p}, \quad \forall q \in {\mathbb{R}}.$$ By the same argument that led to , we find that there is $\kappa \in (0, \frac12)$ such that $$\phi(\bar x, \kappa^j r) \le\kappa^{j \beta} \phi(\bar x, r) +C {\lVertu\rVert}_{L^\infty(B^{+}(\bar x, 2r))}\,\tilde \omega_{\mathbf{A}}(2\kappa^{j} r) + C \tilde \omega_{\mathbf{g}}(2\kappa^{j} r),$$ where $\tilde\omega_\bullet(t)$ is the same as in . The rest of proof is the same as that of Lemma \[lem-01\].
By modifying the proof of Lemmas \[lem-02\], \[lem-03\], and \[lem-04\] in a straightforward way, we obtain the following lemmas.
Let $\beta \in (0,1)$. For any $x \in B^+_3$ and $0<\rho\le r \le \frac14$, we have $$\phi(x, \rho) \le C\left(\frac{\rho}{r}\right)^{\beta}\, r^{-n} {\lVertu\rVert}_{L^1(B(x ,3r)\cap B^+_4)}+C{\lVertu\rVert}_{L^\infty(B(x,5r)\cap B^+_4)}\, \hat\omega_{\mathbf{A}}(\rho) + C \hat\omega_{\mathbf{g}}(\rho),$$ where $C=C(n, \lambda, \Lambda,p, \beta)$ are constants and $\hat\omega_\bullet(t)$ is a Dini function derived from $\omega_\bullet(t)$.
We have $${\lVertu\rVert}_{L^\infty(B^+_2)} \le C {\lVertu\rVert}_{L^1(B^+_4)} + C\int_0^{1} \frac{\hat\omega_{\mathbf{g}}(t)}t \,dt,$$ where $C>0$ is a constant depending only on $n$, $\lambda$, $\Lambda$, $p$, and $\omega_{\mathbf{A}}$.
Let $\beta \in (0,1)$. For any $x \in B^+_3$ and $0< r \le \frac15$, we have $${\lvertu(x)-q_{x,r}\rvert} \le Cr^{\beta}\,{\lVertu\rVert}_{L^1(B(x,\frac35) \cap B^+_4)} + C{\lVertu\rVert}_{L^\infty(B(x,1)\cap B^+_4)}\, \omega^*_{\mathbf{A}}(r) + C\omega_{\mathbf{g}}^*(r),$$ where $C=C(n, \lambda, \Lambda,p, \beta)$ are constants and $\omega^*_\bullet(t)$ is defined as in .
With the above lemmas at hand, we obtain, similar to , the following estimates for $x, y \in B^+_1$: $$\begin{gathered}
{\lvertu(x)-u(y)\rvert} \le C{\lVertu\rVert}_{L^1(B^+_2)}\,{\lvertx-y\rvert}^\beta \\
+C\left( {\lVertu\rVert}_{L^1(B^+_4)} + \int_0^1 \frac{\hat\omega_{\mathbf{g}}(t)}t\,dt \right) \omega^*_{\mathbf{A}}({\lvertx-y\rvert})+ C\omega^*_{\mathbf{g}}({\lvertx-y\rvert}),\end{gathered}$$ where $\beta \in (0,1)$ is any given number, $C=C(n, \lambda, \Lambda, p, \omega_{\mathbf{A}}, \beta)$, and $\omega^*_\bullet(t)$ is defined as in . We have shown that $u \in C(\bar B^+_1)$ as desired. This completes the proof of Proposition \[prop01adj\] and that of Theorem \[thm-main-adj\].
Weak type-$(1,1)$ estimates {#sec3}
============================
Proof of Theorem \[thm-weak11-d\]
---------------------------------
We modify the proof of [@DK16 Theorem 3.2]. Since the map $T: {\boldsymbol{f}} \mapsto Du$ is a bounded linear operator in $L^2(\Omega)$, it suffices to show that $T$ satisfies the hypothesis of Lemma \[lem01-stein\].
Set $c=8$. For fixed $x_0 \in \overline{\Omega}$ and $0<r< \frac12 {\operatorname{diam}}\Omega$, let ${\boldsymbol{b}} \in L^2(\Omega)$ be supported in $\Omega(x_0, r)$ and satisfy $\int_\Omega {\boldsymbol{b}}=0$. Let $u \in W^{1,2}_0(\Omega)$ be the unique weak solution of $$\sum_{i,j=1}^n D_{i}(a^{ij} D_j u)= {\operatorname{div}}{\boldsymbol{b}}\;\mbox{ in }\; \Omega;\quad
u=0 \;\mbox{ on } \; \partial \Omega.$$ For any $R\ge 8r$ such that $\Omega\setminus B(x_0, R) \neq \emptyset$ and ${\boldsymbol{g}} \in C^\infty_c(\Omega(x_0, 2R)\setminus B(x_0,R))$, let $v \in W^{1,2}_0(\Omega)$ be a weak solution of an adjoint problem $$\sum_{i,j=1}^n D_i(a^{ji} D_j v)= {\operatorname{div}}{\boldsymbol{g}}\;\mbox{ in }\; \Omega;\quad
v=0 \;\mbox{ on } \; \partial \Omega.$$ Then, we have the identity $$\label{eq9.24}
\int_{\Omega} Du \cdot {\boldsymbol{g}} = \int_{\Omega} {\boldsymbol{b}} \cdot Dv = \int_{\Omega(x_0 ,r)} {\boldsymbol{b}} \cdot \left(Dv-\overline{Dv}_{\Omega(x_0 ,r)} \right).$$ Since ${\boldsymbol{g}}=0$ in $\Omega(x_0, R)$, by flattening the boundary and using a similar argument that led to , we get $${\lvertDv(x)-Dv(y)\rvert} \lesssim\left(\left(\frac{{\lvertx-y\rvert}}{R}\right)^\beta + \omega_{\mathbf{A}}^*({\lvertx-y\rvert}) \right) R^{-\frac{d}2} {\lVertDv\rVert}_{L^2(\Omega(x_0, \frac14 R))}$$ for $x, y \in \Omega(x_0, \frac18 R)$. Since $r\le R/8$, we thus have $$\begin{aligned}
{\lVertDv -\overline{Dv}_{\Omega(x_0, r)}\rVert}_{L^\infty(\Omega(x_0, r))} &\lesssim R^{-\frac{d}2} {\lVertDv\rVert}_{L^2(\Omega)} \left( r^\beta R^{-\beta} + \sup_{0<t\le r\omega^*_{\mathbf{A}}(t)}\right)\label{eq15.51w}.\end{aligned}$$ Using and , it is routine to check that $$\tilde \omega_{\mathbf{A}}(r)\lesssim ( \ln r )^{-2},\quad \hat \omega_{\mathbf{A}}(r)\lesssim ( \ln r )^{-2},\quad \omega_{\mathbf{A}}^*(r)\lesssim ( \ln (4/r) )^{-1}, \quad\forall r\in(0,\tfrac12).$$ See, for instance, [@DK16 Lemma 3.4]. By the construction of $v$, we have $${\lVertDv\rVert}_{L^2(\Omega)} \lesssim {\lVert{\boldsymbol{g}}\rVert}_{L^2(\Omega)}={\lVert{\boldsymbol{g}}\rVert}_{L^2(\Omega(x_0,2R)\setminus B(x_0,R))}.$$ Therefore, we have by and that $$\begin{gathered}
{\left\lvert\int_{\Omega(x_0,2R)\setminus B(x_0,R)} Du \cdot {\boldsymbol{g}}\,\right\rvert}\\
\lesssim {\lVert{\boldsymbol{b}}\rVert}_{L^1(\Omega(x_0, r))}\left(r^\beta R^{-\beta} +\{\ln (4/r)\}^{-1}\right) R^{-\frac{d}2} {\lVert{\boldsymbol{g}}\rVert}_{L^2(\Omega(x_0,2R)\setminus B(x_0,R))}.\end{gathered}$$ and thus, by duality and Hölder’s inequality, we get $${\lVertDu\rVert}_{L^1(\Omega(x_0,2R)\setminus B(x_0,R))} \lesssim \left(r^\beta R^{-\beta} +\{\ln (4/r)\}^{-1}\right)\,{\lVert{\boldsymbol{b}}\rVert}_{L^1(\Omega(x_0, r))}.$$ Let $N$ be the smallest positive integer such that $\Omega \subset B(x_0, 2^{N +3}r)$. Clearly, $N \lesssim \ln(1/r)$. By taking $R=2^3r,2^4r,\ldots,2^{N+2} r$, we have $$\int_{\Omega \setminus B(x_0, 8r)} {\lvertD u\rvert} \lesssim \sum_{k=1}^N \left(2^{-\beta k} +\{\ln (4/r)\}^{-1}\right)\ {\lVert{\boldsymbol{b}}\rVert}_{L^1(B(x_0, r))} \sim \int_{B(x_0, r)} {\lvert{\boldsymbol{b}}\rvert},$$ and thus we are done.
Proof of Theorem \[thm-weak11-nd\]
----------------------------------
For simplicity of argument, we may assume that $\Omega$ is contained in $B_5=B(0,5)$ and $\mathbf{A}$ has Dini mean oscillation on $B_{10}=B(0,10)$.
Let $W$ be be the adjoint solution to the problem $$D_{ij}(a^{ij} W) = 0 \;\mbox{ in }\;B_{10},\quad W=1 \;\mbox{ on }\;\partial B_{10}.$$ It is known that $W$ is a nonnegative Muckenhoupt weight in the reverse Hölder class $\mathcal B_{\frac{n}{n-1}}(B_{10})$, with constants which depend only on $n$, $\lambda$, and $\Lambda$; i.e., $$\label{doubling}
W(B(x_0, 2r)) \lesssim W(B(x_0,r)), \quad \left( W(B(x_0,r)) := \int_{B(x_0,r)} W \,dx\right),$$ $$\label{E: Muckenhoupt}
\left(\fint_{B(x_0,r)} W^{\frac{n}{n-1}}\, dx\right)^{\frac{n-1}{n}} \lesssim \fint_{B(x_0,r)}W\, dx,$$ whenever $B(x_0, 2r) \subset B_{10}$; see [@FS1984]. Also, $W(B_{10})\approx 1$; see [@Es94; @FGMS1988].
\[L:1\] For $x_0 \in B_1$ and $0<r \le 1$, we have $$\label{eq10.53f}
\sup_{B(x_0, r)} W \le C \fint_{B(x_0, r)} W,$$ where $C$ depends only on $n, \lambda, \Lambda$, and $\omega_{\mathbf{A}}$.
In the proof of [@DK16 Theorem 1.10], it is shown that for $x,y \in B(x_0,r)$ $${\lvertW(x)-W(y)\rvert} \lesssim r^{-n} {\lVertW\rVert}_{L^1(B(x_0, 4r))}\left( r^{-\beta} {\lvertx-y\rvert}^\beta + \int_0^{{\lvertx-y\rvert}} \frac{\tilde \omega_{\mathbf{A}}(t)}{t}\,dt \right),$$ where $\tilde \omega_{\mathbf{A}}$ is as in . Therefore, we have $$W(x) \lesssim r^{-n} {\lVertW\rVert}_{L^1(B(x_0, 4r))}\left( r^{-\beta} {\lvertx-y\rvert}^\beta + \int_0^{{\lvertx-y\rvert}} \frac{\tilde \omega_{\mathbf{A}}(t)}{t}\,dt \right) + W(y).$$ By averaging over $y \in B(x_0, r)$, $$W(x) \lesssim \left(1 + \int_0^{2r} \frac{\tilde \omega_{\mathbf{A}}(t)}{t}\,dt \right) \fint_{B(x_0, 4r)} W+ \fint_{B(x_0, r)} W.$$ By using the doubling property of $W$, we get .
We say that $\tilde v$ is a normalized adjoint solution (for the operator $a^{ij}D_{ij}$) in an open subset $\Omega'$ of $\Omega$ if $\tilde{v}$ is a continuous function defined in $\Omega'$ such that $\tilde{v} W$ is an adjoint solution in $\Omega'$, i.e., $D_{ij}(a^{ij}\tilde{v} W)=0$ in $\Omega'$.
We record the following property of normalized adjoint solutions $\tilde{v}$ on $B(x_0,r)\subset \Omega$: There are constants depending only on $\lambda, \Lambda$, and $n$ such that the following holds: $$\label{11.11thr}
{\lVert\tilde v\rVert}_{L^\infty(B(x_0,\frac r2))} \le \frac{C}{W(B(x_0,r))} \int_{B(x_0,r)} {\lvert\tilde v\rvert} W\,dx,$$ $$[\tilde v]_{C^{\alpha}(B(x_0,r))} \le Cr^{-\alpha} {\lVert\tilde v\rVert}_{L^\infty(B(x_0,2r))}.$$ See [@Ba84; @Es94]. There is also a boundary version of the above estimates. Namely, if $\tilde v$ is a normalized adjoint solution in $\Omega(x_0, r)$ with $x_0\in \overline{\Omega}$ and $\tilde v=0$ on $B(x_0,r)\cap \partial\Omega$, then $$\label{eq14.16tu}
{\lVert\tilde v\rVert}_{L^\infty(\Omega(x_0,\frac r2))} \le \frac{C}{W(B(x_0, r))} \int_{\Omega(x_0,r)} {\lvert\tilde v\rvert} W$$ and $$\label{eq14.17tu}
[\tilde v]_{C^{\alpha}(B(x_0,r))} \le Cr^{-\alpha} {\lVert\tilde v\rVert}_{L^\infty(\Omega(x_0,2r))}.$$ We note that the constants $C$ and $0<\alpha\le 1$ in the above estimates depend only on $n$, $\lambda$, and $\Lambda$; see [@Ba84; @FGMS1988].
We are now ready to prove the theorem. We shall make first the qualitative assumption that the coefficients $\mathbf{A}=(a^{ij})$ are smooth. However, the constant $C$ that appears in will not depend on the extra smoothness of the coefficients. Let ${\{Q_l\}}$ be a collection of disjoint cubes as those used in the proof of Lemma \[lem01-stein\] so that we have $$\label{eq12.10frd}
t<\fint_{Q_l}\, {\lvertf\rvert} \le A_1 t$$ and ${\lvertf(x)\rvert} \le t$ for a.e. $x \in \Omega \setminus \textstyle \bigcup_l Q_l$. We decompose $f=g+b$, with $b=\sum_l b_l$, such that $$g= \tilde m_l(f):=\frac{1}{W(Q_l)} \int_{Q_l} f W \;\text{ on }\;Q_l,$$ $g=f$ on $\Omega\setminus \bigcup_l Q_l$, and set $$b_l= \chi_{Q_l}\left(f - \tilde m_l(f) \right).$$ It is obvious that $$\label{eq11.16f}
\int_{\Omega} b_l W=\int_{Q_l} b_l W=0.$$ By we find (a ball $B(x_0, r)$ can be easily replaced by a “cube”) $${\left\lvert\tilde m_l(f)\right\rvert} \le \frac{1}{W(Q_l)}\, {\lVertW\rVert}_{L^\infty(Q_l)} \int_{Q_l} {\lvertf\rvert}\ \lesssim \fint_{Q_l} {\lvertf\rvert}$$ and thus, we have $$\label{eq11.18f}
\int_{Q_l} {\lvertb_l\rvert} \,dx \le \int_{Q_l} {\lvertf\rvert} + {\lvertQ_l\rvert}\,\tilde m_l(f) \lesssim \int_{Q_l} {\lvertf\rvert} \lesssim t {\lvertQ_l\rvert}.$$ Also, we find that (c.f. in the Appendix) $${\lvertg(x)\rvert} \lesssim t,\quad\text{for a.e. }x \in \Omega.$$ Now, we write $u=v+w$, where $v \in W^{1,2}_0(B_1)\cap W^{2,2}(B_1)$ is a unique solution to $$a^{ij}D_{ij} v= g\;\mbox{ in }\; \Omega,\quad
v=0 \;\mbox{ on } \; \partial \Omega.$$ By the standard elliptic theory, we have $${\lVertD^2 v\rVert}_{L^2(\Omega)} \lesssim {\lVertg\rVert}_{L^2(\Omega)}$$ and thus, we have (c.f. in the Appendix) $$\begin{aligned}
\nonumber
{\left\lvert{\{x \in \Omega : {\lvertD^2 v(x)\rvert} > \tfrac12 t\}}\right\rvert} &\lesssim \frac{1}{t^2} \int_{\Omega} {\lvertD^2 v\rvert}^2 \lesssim \frac{1}{t^2} \int_{\Omega} {\lvertg\rvert}^2\ \\
\label{eq10.31m}
&\lesssim \frac{1}{t} \int_{\Omega \setminus \bigcup_l Q_l} {\lvertf\rvert} + \sum_l \,{\lvertQ_l\rvert} \lesssim \frac1t \int_{\Omega} {\lvertf\rvert}.\end{aligned}$$ For each $l=1,2, \cdots$, let $w_l \in W^{1,2}_0(\Omega)\cap W^{2,2}(\Omega)$ be the unique solution to $$\label{eq14.28frd}
a^{ij}D_{ij} w_l= b_l\;\mbox{ in }\; \Omega,\quad
w_l=0 \;\mbox{ on } \; \partial \Omega.$$ Take $c=8$. We associate each $Q_l$ with a ball $B_l=B(x_l, r_l)$ as in the proof of Lemma \[lem01-stein\], and denote $B_l^*=B(x_l, 8r_l)$. Since $w =\sum_l w_l$, we have $$\int_{\Omega \setminus \bigcup_l B_l^*} \, {\lvertD^2 w\rvert} \le \sum_l \int_{\Omega \setminus B_l^*}\, {\lvertD^2 w_l\rvert}.$$ We claim that $$\label{eq14.15f}
\int_{\Omega \setminus B_l^*} \, {\lvertD^2 w_l\rvert} \lesssim \int_{Q_l} \, {\lvertb_l\rvert}.$$ Take the claim for now. Then, by and , we get $$\int_{\Omega \setminus \bigcup_l B_l^*} \,{\lvertD^2 w\rvert} \lesssim \sum_l \int_{Q_l} \, {\lvertb_l\rvert} \lesssim t \sum_l \,{\lvertQ_l\rvert} \lesssim \int_{\Omega}\, {\lvertf\rvert},$$ which shows that $${\left\lvert{\{x \in \Omega: {\lvertD^2 w(x)\rvert} > \tfrac12 t\}} \setminus \textstyle\bigcup_l Q_l^*\right\rvert} \lesssim \frac1t \int_{\Omega}\, {\lvertf\rvert}.$$ However, $${\left\lvert\textstyle \bigcup_j B_l^*\right\rvert} \le 8^n \sum_l\, {\lvertB_l\rvert} \lesssim \sum_l \,{\lvertQ_l\rvert} \lesssim \frac1t \int_{\Omega}\, {\lvertf\rvert}.$$ Together then, the last two estimates imply $${\left\lvert{\{x \in \Omega: {\lvertD^2 w(x)\rvert} > \tfrac12 t\}}\right\rvert} \lesssim \frac1t \int_{\Omega} \, {\lvertf\rvert},$$ which combined with gives the theorem since $u=v+w$.
We now prove the claim . To do this, we follow the same line of proof of [@DK16 Lemma 2.20]. Recall that $w_l \in W^{1,2}_0(\Omega)\cap W^{2,2}(\Omega)$ satisfies with $b_l$ supported in $Q_l\subset B_l\cap \Omega = \Omega(x_l, r_l)$. For any $R\ge 8r_l$ such that $\Omega \setminus B(x_l, R) \neq \emptyset$ and smooth functions $\mathbf{h}=(h^{km})_{k,m=1}^n$ with a compact support in $\Omega(x_l, 2R)\setminus \overline B(x_l,R)$, let $v_l \in L^2(\Omega)$ be a unique adjoint solution of $$D_{ij}(a^{ij} v_l)= {\operatorname{div}}^2 \mathbf{h}\;\mbox{ in }\; \Omega;\quad
v_l=0 \;\mbox{ on } \; \partial \Omega,$$ and let $$\tilde v_l:= v_l/W.$$ Then, we have the identity $$\label{eq22.44f}
\int_{\Omega} {\operatorname{tr}}(D^2 w_l \, \mathbf{h}) = \int_{\Omega} v_l b_l = \int_{\Omega} \tilde v_l W b_l = \int_{Q_l} \left(\tilde v_l-(\tilde v_l)_{x_l,r_l} \right) W b_l,$$ where we set $$(\tilde v_l)_{x_l,r_l}:=\fint_{\Omega(x_l,r_l)}\, \tilde v_l$$ and used . Since $\mathbf{h}=0$ in $\Omega(x_l,R)$, we find that $\tilde v_l$ is a normalized adjoint solution in $\Omega(x_l, R)$. Thus, by and , for $x \in Q_l$, we have $$\begin{aligned}
{\left\lvert\tilde v_l(x) - (\tilde v_l)_{x_l,r_l}\right\rvert} & \le \fint_{\Omega(x_l,r_l)} {\lvert\tilde v_l(x) - \tilde v_l(y)\rvert}\, dy \lesssim \left(\frac{r_l}{R}\right)^\alpha {\lVert\tilde v_l\rVert}_{L^\infty(\Omega(x_l, \frac14 R))}\\
&\lesssim \left(\frac{r_l}{R}\right)^\alpha \frac{1}{W(B(x_l, \frac12 R))} \int_{\Omega(x_l, R)} \,{\lvert\tilde v_l\rvert} W
= \left(\frac{r_l}{R}\right)^\alpha \frac{1}{W(B(x_l, \frac12 R))} \int_{\Omega(x_l, R)}\, {\lvertv_l\rvert}.\end{aligned}$$ Therefore, by , , and the doubling property of $W$, we have $$\begin{aligned}
{\left\lvert\int_{\Omega(x_l,2R)\setminus B(x_l,R)} {\operatorname{tr}}(D^2 w_l \, \mathbf{h}) \right\rvert}
&\lesssim \left(\frac{r_l}{R}\right)^\alpha \frac{1}{W(B(x_l,\frac12 R))} {\lVertv_l\rVert}_{L^1(\Omega(x_l, R))} \,{\lVertW\rVert}_{L^\infty(B(x_l, \frac14 R))}\, {\lVertb_l\rVert}_{L^1(Q_l)}\\
&\lesssim \left(\frac{r_l}{R}\right)^\alpha \frac{W(B(x_l, \frac14 R))}{W(B(x_l, \frac12 R))\, {\lvertB(x_l, \frac14 R)\rvert}}\, {\lVertv_l\rVert}_{L^1(\Omega(x_l, R))}\,{\lVertb_l\rVert}_{L^1(Q_l)}\\
&\lesssim r_l^\alpha R^{-\alpha -\frac{n}2}\,{\lVertb_l\rVert}_{L^1(Q_l)} \,{\lVertv_l\rVert}_{L^2(\Omega)}\\
&\lesssim r_l^\alpha R^{-\alpha-\frac{n}2}\,{\lVertb_l\rVert}_{L^1(Q_l)} \,{\lVert\mathbf h\rVert}_{L^2(\Omega(x_l,2R)\setminus B(x_l,R))},\end{aligned}$$ where we used Hölder’s inequality and the estimate $${\lVertv_l\rVert}_{L^2(\Omega)} \lesssim {\lVert\mathbf h\rVert}_{L^2(\Omega)}$$ in the last two inequalities. Therefore, by duality and Hölder’s inequality, we get $${\lVertD^2 w_l\rVert}_{L^1(\Omega(x_l,2R)\setminus B(x_l,R))} \lesssim r_l^\alpha R^{-\alpha} \,{\lVertb_l\rVert}_{L^1(Q_l)}.$$ Now let $N$ be the smallest positive integer such that $\Omega\subset B(x_l, 2^{N} 8r_l)$. By taking $R=2^3r_l,2^4r_l,\ldots,2^{N+2}r_l$ in the above, we have $$\int_{\Omega \setminus B(x_l, 8r_l)} {\lvertD^2 w_l\rvert} \lesssim \sum_{k=1}^{N} 2^{-\alpha k} {\lVertb_l\rVert}_{L^1(Q_l)} \simeq \int_{Q_l} {\lvertb_l\rvert}$$ as desired.
Finally, we shall show how to get rid of extra smoothness assumption on the coefficients $\mathbf{A}=(a^{ij})$. Let $\mathbf{A}_k=(a^{ij}_k)$ and $f_k$ are smooth functions such that $a^{ij}_k$ converges uniformly to $a^{ij}$ in $\Omega$ and $f_k$ converges to $f$ in $L^2(\Omega)$ as $k \to \infty$. We may further assume that $\mathbf{A}_k$ satisfies condition and that $\omega_{\mathbf{A}_k} \le \omega_{\mathbf{A}}$. Let $u_k \in W^{1,2}_0(\Omega)\cap W^{2,2}(\Omega)$ satisfy $$a^{ij}_k D_{ij} u_k= f_k\;\mbox{ in }\; \Omega,\quad
u_k=0 \;\mbox{ on } \; \partial \Omega.$$ Then, by , we have an estimate $$\label{eq15.59wh}
{\left\lvert{\{x \in \Omega : {\lvertD^2u_k(x)\rvert} > t\}}\right\rvert} \le \frac{C}{t} \int_{\Omega}\, {\lvertf_k\rvert}\,dx,$$ which is uniform for all $k=1,2,\cdots$. Note that $$a^{ij}_k D_{ij} (u-u_k) = f-f_k + (a^{ij}_k-a^{ij}) D_{ij} u$$ so that, by the $W^{2,2}$ theory, we have $${\lVertD^2 u - D^2 u_k\rVert}_{L^2(\Omega)} \le C \left( {\lVertf-f_k\rVert}_{L^2(\Omega)}+ {\lVert(a^{ij}_k-a^{ij})D_{ij}u\rVert}_{L^2(\Omega)} \right).$$ In particular, we find that $D^2 u_k \to D^2 u$ in $L^2(\Omega)$. Therefore, by taking limit $k\to \infty$ in , we get .
\[R: 1\] It is shown in [@Es94] that when the coefficients of the elliptic operator $\mathcal L$ in non-divergence form are VMO functions in $B_{10}$, i.e., $$\label{E:20}
\lim_{r\to 0 +} \sup_{x\in B_{10}}\fint_{Q(x,r)} {\left\lvert\mathbf{A}-\bar{\mathbf{A}}_{Q(x,r)}\right\rvert} =0,$$ where $Q(x,r)$ denotes a cube in ${\mathbb{R}}^n$ centered at $x$ with edges of length $r$ and sides parallel to the coordinate axis, the Muckenhoupt property of the weight $W$ improves because then, $\log W$ is in $\text{VMO}(B_5)$ (see [@Es94 Theorem 1.2]), i.e., $$\label{E: estaenvmo}
\lim_{r\to 0+} \sup_{x\in B_5}\fint_{Q(x,r)} {\left\lvert\log{W}-\overline{\log{W}}_{Q(x,r)}\right\rvert} =0,$$ while the weight $W$ is shown to be unbounded or to vanish inside $B_{10}$ for some of these operators; see [@Es94 §3]. In fact, when $\mathbf{A}$ has Dini mean oscillation in $B_{10}$, the ideas behind [@DK16 Theorem 1.10] and [@EM2016 Lemma 3] imply that nonnegative adjoint solutions $W$ verify the following Harnack inequality: there is $C=C(\lambda, \Lambda, n, \omega_\mathbf{A})$ such that $$\sup_{B(x,r)}W\le C\inf_{B(x,r)}W,$$ when $B(x,2r)\subset B_8$. See Lemma \[lem8.06\] in the Appendix. This local Harnack inequality fails when $\mathbf{A}$ is continuous on $B_{10}$; see [@Es94 §3].
It is shown in that under the same hypothesis, the solution $u$ to verifies an interior weak type-$(1,1)$ property with respect to the weight $W\,dx$, when $f\in C_c^\infty(\Omega)$. Also, [@Es94 §3] provides counterexamples of operators $\mathcal L$ in non-divergence form with continuous coefficients for which the weak type-$(1,1)$ fails in the interior of $\Omega$. Notice that the coefficient matrices there just fail to be Dini continuous or have Dini mean oscillations.
Finally, the methods in this paper, , – and are easily seen to help to extend up to the boundary the interior weak type-$(1,1)$ property with respect to the weight $W\,dx$ in . In particular, under the weaker VMO condition one can show that there is $C=C(n, \lambda, \Lambda, \Omega,\mathbf{A})$ such that for any $t>0$, $$W\left(\{x \in \Omega : {\lvertD^2u(x)\rvert} > t\}\right) \le \frac{C}{t} \int_{\Omega}\, {\lvertf\rvert} W\,dx,$$ when $u$ is the solution to , $\Omega\subset B_5$, and $f\in C^\infty(\overline{\Omega})$. Observe there are operators with uniformly continuous coefficients such that the adjoint solution $W$ is unbounded above or below or it is not a local $A_1$ Muckenhoupt weight (See [@Ba84b] and [@Es94 §3]).
Proof of Lemma \[lem-cont-bdry-adj\]
------------------------------------
As in the proof of Theorem \[thm-weak11-nd\], let us assume that $\Omega$ is contained in $B_5=B(0,5)$ and $\mathbf{A}$ has Dini mean oscillation on $B_{10}$. Let $W$ be as given in the proof of Theorem \[thm-weak11-nd\]. By [@DK16 Theorem 1.10], we find that $W$ is uniformly continuous in $B_5$ with its modulus of continuity determined by $n$, $\lambda$, $\Lambda$, and $\omega_{\mathbf A}$. Also, Lemma \[lem8.06\] in the Appendix implies that $W$ is bounded from above and below in $B_5$ with its lower and upper bounds depending only on $n$, $\lambda$, $\Lambda$, and $\omega_{\mathbf A}$.
Therefore, by [@Es00 Theorem 2.8], there is a unique normalized adjoint solution $\tilde v$ that satisfies $$D_{ij}(a^{ij}\tilde v W)=0\; \text{ in }\;\Omega,\quad \tilde v = \frac{\psi}{W}\; \text{ on }\;\partial\Omega.$$ Moreover, $\tilde v \in C(\overline\Omega)$ with a modulus of continuity controlled by $n$, $\lambda$, $\Lambda$, the Lipschitz character of $\Omega$, and the modulus of continuity of $\frac{\psi}{W}$. The latter in turn is controlled by $n$, $\lambda$, $\Lambda$, $\omega_{\mathbf A}$, and $\psi$. It is clear that $v=\tilde v W$ satisfies all the desired properties.
Appendix {#appendinx1}
========
The following lemma is a slight generalization of [@DK16 Lemma 2.1]. For the completeness, we present a proof here.
\[lem01-stein\] Let $\Omega\subset {\mathbb{R}}^n$ be a bounded domain satisfying the condition and let $T$ be a bounded linear operator from $L^2(\Omega)$ to $L^2(\Omega)$. Let $\mu \in (0,1)$ be a constant. Suppose that for any $x_0 \in \Omega$ and $0<r< \mu {\operatorname{diam}}\Omega$, we have $$\label{eq14.58w}
\int_{\Omega \setminus B(x_0, cr)} {\lvertTb\rvert} \le C \int_{B(x_0, r)\cap \Omega} {\lvertb\rvert}$$ whenever $b \in L^2(\Omega)$ is supported in $B(x_0, r)\cap \Omega$, $\int_{\Omega} b =0$, and $c>1$ and $C>0$ are constants. Then for $f \in L^2(\Omega)$ and any $t>0$, we have $$\label{eq13.38w}
{\left\lvert{\{x \in \Omega : {\lvertTf(x)\rvert} > t\}}\right\rvert} \le \frac{C'}{t} \int_{\Omega} {\lvertf\rvert},$$ where $C'=C'(n, c , C, \mu, \Omega, A_0)$ is a constant.
To begin with, we note that $\Omega$ equipped with the standard Euclidean metric and the Lebesgue measure (restricted to $\Omega$) is a space of homogeneous type. By [@Ch90 Theorem 11], there exists a collection of open subsets (called “cubes”) $${\{Q^k_\alpha \subset \Omega : k \in {\mathbb{Z}}, \; \alpha \in I_k\}},$$ with $I_k$ at most countable set and constants $\delta \in (0,1)$, $a_0>0$ and $C_1<\infty$ such that
i) ${\left\lvert\Omega \setminus \bigcup_\alpha Q^k_\alpha\right\rvert} =0\quad \forall k$.
ii) If $\ell \ge k$ then either $Q^\ell_\beta \subset Q^k_\alpha$ or $Q^\ell_\beta \cap Q^k_\alpha=\emptyset$.
iii) For each $(k,\alpha)$ and each $\ell < k$ there is a unique $\beta$ such that $Q^k_\alpha \subset Q^\ell_\beta$.
iv) ${\operatorname{diam}}Q^k_\alpha \le C_1 \delta^k$.
v) Each $Q^k_\alpha$ contains some ball $B(z^k_\alpha, a_0 \delta^k) \cap \Omega$.
From the above, we can infer the following.
(a) There is constant $A_1\ge 1$ such that if $Q^{k-1}_\beta$ is the parent of $Q^k_\alpha$ (resp. if $B^k_\alpha$ is the Euclidean ball in ${\mathbb{R}}^n$ centered at $z^k_\alpha$ with radius $r={\operatorname{diam}}Q^k_\alpha$), then we have $$\label{p1}
{\lvertQ^{k-1}_\beta\rvert} \le A_1 {\lvertQ^k_\alpha\rvert}\quad (\text{resp. }\;{\lvertB^{k}_\alpha\rvert} \le A_1 {\lvertQ^k_\alpha\rvert}\,).$$
(b) The Lebesgue differentiation theorem is available for the chain of cubes shrinking to a point because the maximal function defined as $$M(f)(x)=
\begin{cases}
\sup_{x\in Q^k_\alpha}\fint_{Q^k_\alpha}{\lvertf\rvert}\, dx,\ &\text{when}\ x\in\bigcap_k\bigcup_{\alpha\in I_k}Q^k_\alpha,\\
0,\ &\text{otherwise},
\end{cases}$$ is of weak type-$(1,1)$ over $\Omega$.
By i) – v) above and , choose $k_0\in {\mathbb{Z}}$ with $\theta:=\inf_{\alpha\in I_{k_0}}|Q^{k_0}_\alpha|>0$. To get when $$\frac 1t\int_{\Omega} {\lvertf\rvert}\ dx>\theta,$$ it suffices to choose $C'\ge \theta^{-1}{\lvert\Omega\rvert}$. Otherwise, $$\fint_{Q^{k_0}_\alpha}{\lvertf\rvert}\ dx\le t,\ \text{for all}\ \alpha\in I_{k_0}.$$ Let then ${\{Q_l\}}$ denote the set of cubes chosen as follows. For $k=k_0+1$ and $\alpha \in I_k$, the cube $Q=Q^k_\alpha$ satisfies either $\fint_{Q}\, {\lvertf\rvert} \le t$ or $\fint_{Q}\, {\lvertf\rvert} > t$. In the second case, we select $Q=Q^k_\alpha$ as one of the cubes in ${\{Q_l\}}$. Note that in this case, we have by $$t<\fint_{Q} {\lvertf\rvert}\,dx \le A_1 t.$$ In the first case, we subdivide $Q=Q^k_\alpha$ further into subcubes $Q'=Q^{k+1}_\beta$, and repeat the process until (if ever) we are forced into the second case. By observation (b), we find that ${\lvertf(x)\rvert} \le t$ for a.e. $x \in \Omega \setminus \bigcup_l Q_l$.
We decompose $f=g+b$, with $b=\sum_l b_l$, such that $$g= m_l(f):=\fint_{Q_l} f \,dx\;\text{ on }\;Q_l,$$ $g=f$ on $\Omega \setminus \bigcup_l Q_l$, and set $$b_l= \chi_{Q_l}\left(f - m_l(f) \right).$$ It is obvious that $\int_{Q_l} b_l \,dx =0$ and we have $$\label{a11.18f}
\int_{Q_l} {\lvertb_l\rvert} \,dx \le \int_{Q_l} {\lvertf\rvert}\,dx + {\lvertQ_l\rvert} m_l(f) \le 2 \int_{Q_l} {\lvertf\rvert} \,dx \le 2A_1 t {\lvertQ_l\rvert}.$$ Also, we see that $$\label{a11.17f}
{\lvertg(x)\rvert} \le A_1 t \quad\text{for a.e. }x \in \Omega.$$ Indeed, for a.e. $x \in \Omega \setminus \bigcup_l Q_l$, we have ${\lvertg(x)\rvert} ={\lvertf(x)\rvert} \le t$ and ${\lvertg(x)\rvert} \le A_1 t$ on $Q_l$. By Chebyshev’s inequality and the $L^2$ boundedness of $T$, we have $$\begin{aligned}
\nonumber
{\left\lvert{\{x \in \Omega : {\lvertTg(x)\rvert} > \tfrac12 t\}}\right\rvert} &\lesssim \frac{1}{t^2} \int_{\Omega}\, {\lvertTg\rvert}^2\,dx \lesssim \frac{1}{t^2} \int_{\Omega}\, {\lvertg\rvert}^2\,dx \\
\label{a10.31m}
&\lesssim \frac{1}{t} \int_{\Omega \setminus \bigcup_l Q_l} {\lvertf\rvert}\,dx + \sum_l \,{\lvertQ_l\rvert} \lesssim \frac1t \int_{\Omega} {\lvertf\rvert}\,dx,\end{aligned}$$ where we used and the property that $$\label{a14.46m}
\sum_l \,{\lvertQ_l\rvert} \le \frac{1}{t} \int_{\Omega} {\lvertf\rvert}\,dx.$$ We associate each $Q_l=Q^k_\alpha$ with a Euclidean ball $B_l=B(x_l, r_l)$, where $x_l=z^k_\alpha\in \Omega$ and $r_l={\operatorname{diam}}Q^k_\alpha$. Let us denote $B_l^*=B(x_l, cr_l)$. Since $Tb =\sum_l Tb_l$, we have $$\int_{\Omega \setminus \bigcup_l B_l^*} {\lvertTb\rvert}\,dx \le \sum_l \int_{\Omega \setminus B_l^*} {\lvertT b_l\rvert}\,dx.$$ By the hypothesis together with and , we get $$\int_{\Omega \setminus \bigcup_{l} B_l^*} {\lvertTb\rvert}\,dx \le C \sum_l \int_{Q_l} {\lvertb_l\rvert}\,dx \lesssim t \sum_l {\lvertQ_l\rvert} \lesssim \int_{\Omega} {\lvertf\rvert}\,dx,$$ which via Chebyshev’s inequality shows that $${\left\lvert{\{x \in \Omega: {\lvertTb(x)\rvert} > \tfrac12 t\}} \setminus \textstyle\bigcup_l B_l^*\,\right\rvert} \lesssim \frac1t \int_{\Omega} {\lvertf\rvert}\,dx.$$ Also, by , we have $${\left\lvert\textstyle \bigcup_l B_l^*\right\rvert} \le c^n \sum_l\, {\lvertB_l\rvert}\ \lesssim \sum_l \,{\lvertQ_l\rvert} \lesssim \frac1t \int_{\Omega} {\lvertf\rvert}\,dx.$$ Together then, the last two estimates imply $${\left\lvert{\{x \in \Omega: {\lvertTb(x)\rvert} > \tfrac12 t\}}\right\rvert} \lesssim \frac1t \int_\Omega {\lvertf\rvert}\,dx,$$ which combined with gives since $Tf=Tg+Tb$.
Finally we prove the following Harnack type inequality for nonnegative adjoint solutions.
\[lem8.06\] Assume the coefficients $\mathbf{A}=(a^{ij})$ are of Dini mean oscillation and satisfy the condition . Let $w\in L^2(B_4)$ be a nonnegative solution to $D_{ij}(a^{ij} w)=0$ in $B_4=B(0,4)$ and ${\lVertw\rVert}_{L^1(B_3)}=1$. Then we have $$c\le \inf_{B_1}\,w,\quad \sup_{B_1}\,w \le C,$$ where $c$ and $C$ are positive constants depending only on $n$, $\lambda$, $\Lambda$, and $\omega_{\mathbf{A}}$.
The upper bound follows with the same type of reasoning as in the proof of Lemma \[L:1\], because from [@DK16 (2.25)], we have $$\label{eq2.45}
{\lvertw(x)-w(y_0)\rvert} \le C \left(\left(\frac{{\lvertx-y_0\rvert}}{R}\right)^\beta + \int_0^{{\lvertx-y_0\rvert}} \frac{\tilde\omega_{\mathbf{A}}(t)}t \,dt \right) R^{-d} {\lVertw\rVert}_{L^1(B(y_0,R))}$$ for $x \in B(y_0, \frac12 R)$, $y_0\in B_1$ and $R\in (0,1]$. Here $\beta>0$ is an absolute constant and $\tilde \omega_{\mathbf{A}}$ is defined as in .
We prove the lower bound by contradiction. Suppose the claim is not true. Then we can find a sequence of coefficients ${\{\mathbf{A}_k\}}$ satisfying $$\sup_k \omega_{\mathbf{A}_k}(t) \le \omega(t)$$ for some Dini function $\omega$ and a sequence of nonnegative solutions ${\{w_k\}}$ to $$D_{ij}(a^{ij}_k w_k)=0\quad \text{in}\,\, B_4$$ such that $${\lVertw_k\rVert}_{L^1(B_3)}=1\quad\text{and}\quad w_k(x_k) \le 1/k$$ for some $x_k\in B_1$. After passing to a subsequence, we may assume that $x_k\to y_0 \in \bar B_1$. By [@DK16 Theorem 1.10], ${\{w_k\}}$ is uniformly bounded and equicontinuous in $\bar B_2$. Of course, ${\{\mathbf{A}_k\}}$ is also uniformly bounded and equicontinuous in $\bar B_2$. Therefore, by the Arzelà–Ascoli theorem, they have subsequences, still denoted by ${\{w_k\}}$ and ${\{\mathbf{A}_k\}}$, which converge to $w$ and $\mathbf{A}$ uniformly in $\bar B_2$, with the same moduli of continuity. It is easily seen that $w$ is a nonnegative solution of $$D_{ij}(a^{ij}w)=0 \quad \text{in}\quad B_2$$ and $w(y_0)=0$. By the doubling property of $w$ (see [@FS1984]), ${\lVertw_k\rVert}_{L^1(B_2)}$ is bounded from below and above uniformly. It then follow from the uniform convergence that ${\lVertw\rVert}_{L^1(B_2)}$ is also bounded from below and above.
Let $\kappa\in (0,1/2)$ be a small constant to be specified later. From , for any $R\in (0,1]$, we have $$\fint_{B(y_0, \kappa R)}w\le N\left(\kappa^\beta+\int_0^{\kappa R}
\frac{\tilde\omega_{\mathbf{A}}(t)}t \,dt \right) \fint_{B(y_0,R)}w,$$ where $N$ is independent of $\kappa$. We then fix $\kappa$ sufficiently small such that $2N\kappa^\beta\le \kappa^{\beta/2}$. Then for any small $R$ such that $$\int_0^{\kappa R}\frac{\tilde\omega_{\mathbf{A}}(t)}t \,dt\le \kappa^{\beta},$$ we obtain $$\fint_{B(y_0, \kappa R)}w\le \kappa^{\beta/2} \fint_{B(y_0, R)}w.$$ By iteration, we deduce $\fint_{B(y_0, r)} w\le Nr^{\beta/2}$. This, however, contradicts with [@Es00 Theorem 1.5], which reads that for any $\varepsilon>0$, it holds that $\fint_{B(y_0, r)}w \gtrsim r^{\varepsilon}$ for all $r\in (0,1)$.
Part of this work was done at the time when the second author was attending the Harmonic Analysis Program held at M.S.R.I. from January to May 2017. He would like to thank the members of the Institute and the organizers of the program for their hospitality.
[m]{}
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[^1]: H. Dong was partially supported by the NSF under agreement DMS-1056737 and DMS-1600593.
[^2]: L. Escauriaza is supported by grants MTM2014-53145-P and IT641-13 (GIC12/96).
[^3]: S. Kim is partially supported by NRF Grant No. NRF-2016R1D1A1B03931680.
[^4]: In fact, the condition on $\mathbf{A}$ imposed by Yanyan Li was slightly stronger.
|
---
author:
- |
Xutan Peng^$\dagger$^, Chenghua Lin^$\dagger$^, Mark Stevenson^$\dagger$^, Chen li^$\ddagger$^\
^$\dagger$^NLP Group, Department of Computer Science, The University of Sheffield\
^$\ddagger$^School of Computer Science and Engineering, Beihang University\
[{x.peng, c.lin, mark.stevenson}@sheffield.ac.uk]{},\
[[email protected]]{}
bibliography:
- 'analogy2linearity.bib'
title: |
Revisiting the linearity in cross-lingual embedding mappings:\
from a perspective of word analogies
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|
---
abstract: 'The notion of *entangling power* of unitary matrices was introduced by Zanardi, Zalka and Faoro \[PRA, 62, 030301\]. We study the entangling power of permutations, given in terms of a combinatorial formula. We show that the permutation matrices with zero entangling power are, up to local unitaries, the identity and the swap. We construct the permutations with the minimum nonzero entangling power for every dimension. With the use of orthogonal latin squares, we construct the permutations with the maximum entangling power for every dimension. Moreover, we show that the value obtained is maximum over all unitaries of the same dimension, with possible exception for $36$. Our result enables us to construct generic examples of $4$-qudits maximally entangled states for all dimensions except for $2$ and $6$. We numerically classify, according to their entangling power, the permutation matrices of dimension $4$ and $9$, and we give some estimates for higher dimensions.'
author:
- Lieven Clarisse
- Sibasish Ghosh
- Simone Severini
- Anthony Sudbery
title: Entangling Power of Permutations
---
Introduction
============
The notion of entangling power of a quantum evolution was introduced by Zanardi, Zalka and Faoro [@z] (see also [@z0; @z1; @z2]). Let $\mathcal{H}_{A}$, $\mathcal{H}_{B}$ and $\mathcal{H}=\mathcal{H}_{A}\otimes
\mathcal{H}_{B}$ be Hilbert spaces where $\dim \mathcal{H}_{A}=\dim \mathcal{H}_{B}=d$. The entangling power of a unitary $U\in \mathcal{U(}\mathcal{H})\cong U(d^{2})$ is the average amount of entanglement produced by $U$ acting on a given (uncorrelated) distribution of product states. As the pure entanglement measure we use the linear entropy $S_{L}(\cdot )$ of the reduced density matrix. For $|\psi \rangle \in \mathcal{H}$, let [@note1]$$S_{L}(|\psi \rangle ):=\displaystyle{\frac{d}{d-1}(1-\operatorname{Tr}\rho ^{2})},\quad \mbox{where}\quad \rho =\operatorname{Tr}_{B}|\psi \rangle \langle \psi |.$$The *entangling power* of $U$ is defined as [@z] $$\epsilon (U):=\int_{\langle \psi _{1}|\psi _{1}\rangle =1}\int_{\langle \psi
_{2}|\psi _{2}\rangle =1}S_{L}(U|\psi _{1}\rangle |\psi _{2}\rangle )d\psi
_{1}d\psi _{2}, \label{epu}$$where $d\psi _{1}$ and $d\psi _{2}$ are normalized probability measures on unit spheres. As the linear entropy is a concave function of the reduced density matrix, it is an entanglement monotone [@vidal], and therefore a legitimate pure state entanglement measure.
Consider now the scenario where Alice and Bob share an unknown product state and want to use a unitary operator to create a state which is as highly entangled as possible. This means that Alice and Bob are looking for unitaries with the maximum entangling power. Zanardi [@z0] observed that there are permutation matrices with the maximum entangling power over all unitaries in $\mathcal{U(}\mathcal{H})$ when $d$ is odd or $d=4n$, but the case $d=4n+2$ was left open. In this paper we extend this result and study the entangling power $\epsilon (P)$ of a permutation matrix $P$.
The paper is organized as follows. In Section II we give a combinatorial expression for $\epsilon (P)$ (Theorem 2). In Section III we determine the non-entangling permutations (Theorem \[ent\]). In Section IV we construct permutations with the maximum entangling power over all unitaries in $\mathcal{U(}\mathcal{H})$ for every $d\neq 6$. Moreover, we construct permutations with the minimum entangling power for all $d$ (Theorem \[mib\]). As a corollary, we give an upper bound to the number of permutations with different entangling power. In Section V, we give a complete numerical classification of permutation matrices of dimension $4$ and $9$ according to their entangling power. For higher dimensions, we report numerical estimates. Finally, in Section VI we draw conclusions and propose some open problems.
Entangling power of permutations
================================
For our purposes we find it useful to rewrite Equation in another more concrete form as described by Zanardi [@z0] (see Equation in Lemma , below). The new expression for $\epsilon
(U)$ is based on a correspondence between quantum operations on bipartite systems and certain quantum states which we are now going to recall.
Let $|\psi \rangle =\sum_{i,j}X_{ij}|ij\rangle $ be a state in $\mathcal{H}$ and let $X$ be the corresponding operator on $\mathcal{H}$, so that $X_{ij}=d\langle i|X|j\rangle $. The singular values of $X$ are equal to the Schmidt coefficients of $|\psi \rangle $. The state $|\psi \rangle $ can be written in terms of this operator as $|\psi \rangle =X\otimes I|\psi
_{+}\rangle $, where $|\psi _{+}\rangle =\frac{1}{\sqrt{d}}\sum_{i=1}^{d}|ii\rangle $ is a maximally entangled state and $I$ is the identity operator. This relation establishes a bijection between pure states in $\mathcal{H}\cong \mathcal{H}_{A}\otimes \mathcal{H}_{B}$ and operators $X $ acting on $\mathcal{H}_{A}\cong \mathcal{H}_{B}$.
Now, following Cirac *et al.* [@c], suppose $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are themeselves bipartite: $\mathcal{H}_{A}=\mathcal{H}_{1}\otimes \mathcal{H}_{2}$ and $\mathcal{H}_{B}=\mathcal{H}_{3}\otimes
\mathcal{H}_{4}$. We can repartition $\mathcal{H}\cong \mathcal{H}_{A}\otimes \mathcal{H}_{B}$ and regard it as the tensor product of $\mathcal{H}_{1}\otimes \mathcal{H}_{3}$ and $\mathcal{H}_{2}\otimes \mathcal{H}_{4}$. Applying the above construction to an operator $X_{13}$ yields a state $$|X\rangle _{A|B}=|X\rangle _{12|34}:=\frac{1}{N}(X_{13}\otimes I_{24})|\Psi
_{+}\rangle _{13|24}, \label{vecfrommat}$$where $$|\Psi _{+}\rangle _{13|24}=\frac{1}{d}\sum_{i,j=1}^{d}|ij\rangle
_{13}\otimes |ij\rangle _{24},$$and $N$ is an appropriate normalization factor. This has an important physical interpretation: the operator $X_{13}$, acting on the Hilbert space $\mathcal{H}_{A1}\otimes \mathcal{H}_{B3}$, corresponds to a quantum operation. If this quantum operation is a tensor product of two quantum operations, one acting on the system $1$ and another on the system $3$, then $|X\rangle _{12|34}$ is a product state with respect to the systems $A$ and $B$. In fact, $X_{13}\otimes I_{24}$ and $|X\rangle _{12|34}$ have the same Schmidt decomposition if $X_{13}$ is a unitary operator [@z0]. It turns out that if both parties $A$ and $B$ share an entangled state $|X\rangle
_{12|34}$ then the non-local quantum operation $X_{13}\otimes I_{24}$ can be implemented with certainty by making use of local operations and classical communication only (see Equation 6 of [@c]). With this in mind, the entangling power of a unitary $U$ is related to the entanglement of $|U\rangle $ as defined in Equation . We use the linear entropy as an entanglement measure. It might clarify matters to give the following two extremal examples:
- (Identity) If $U=I$ then the corresponding state is given by $$| I\rangle _{12|34}=\frac{1}{d}\sum_{i,j=1}^{d}|ii\rangle _{12}\otimes |
jj\rangle _{34},$$which is separable with respect to the split $A|B=12|34$. In this case we have $S_{L}(|I\rangle )=0$.
- (Swap) If $U=S=\sum_{ij}^{d}|ij\rangle \langle ji|$ then the corresponding state is given by $$| S\rangle _{12|34}=|\Psi _{+}\rangle _{A|B}=\frac{1}{d}\sum_{i,j=1}^{d}|ij\rangle _{12}\otimes |ij\rangle _{34},$$which is a maximally entangled vector. In this case we have $S_{L}(|S\rangle
)=1$.
The swap corresponds to the parties interchanging their systems, and might therefore be regarded as the most non-local operation possible. Nevertheless, the swap operation is non-entangling (that is, it never creates entanglement). This discrepancy between *non-local* and *non-entangling* means that we cannot *just* use the entanglement of $|
U\rangle $ as a measure for the entangling power of the matrix $U$. However with a small modification one can still express $\epsilon (U)$ in terms of the linear entropy, as is done in the following lemma.
\[zana\] The entangling power of a unitary $U\in \mathcal{U(}\mathcal{H}) $ is given by $$\epsilon (U)=\frac{d}{d+1}[S_{L}(|U\rangle )+S_{L}(|US\rangle
)-S_{L}(|S\rangle )], \label{newe}$$where $0\leq \epsilon (U)\leq \frac{d}{d+1}$ [@note2].
A *permutation* of $[n]=\{1,2,...,n\}$ is a bijection from $[n]$ to itself. Every permutation $p$ of $[n]$ induces an $n\times n$ matrix $P=(p_{ij})$, called a *permutation matrix*, such that $p_{ij}=1$ if $p(i)=j$ and $p_{ij}=0$ otherwise. Equivalently a permutation on $[n]$ induces a linear map of an $n$-dimensional Hilbert space which permutes a given basis of the space (a *permutation operator* on $\mathcal{H}$). If $n=d^{2}$ we can replace $[n]$ by $[d]\times \lbrack d]$ and write $p(i,j)=(k_{ij},l_{ij})$; thus a permutation of $[d^{2}]$ is represented by a pair of $d\times d$ matrices $K=(k_{ij})$ and $L=(l_{ij})$. The corresponding permutation operator permutes the elements $|i\rangle |
j\rangle $ of a product basis of $\mathcal{H}$:$$P(|i\rangle |j\rangle )=|k_{ij}\rangle |l_{ij}\rangle .$$
Let $P=\sum_{ij}|k_{ij}l_{ij}\rangle \langle ij|$ be a permutation matrix in $\mathcal{U}(\mathcal{H})$. The entangling power of $P$ is given by$$\epsilon (P)=\frac{d^{4}+d^{2}-Q_{P}-Q_{PS}}{d(d-1)(d+1)^{2}},$$with $$Q_{P}=\sum_{i,j,m,n=1}^{d}a_{ijm}a_{ijn}b_{imn}b_{jmn}, \label{defQ}$$where $$\begin{aligned}
a_{ijm} &=&\langle l_{im}|l_{jm}\rangle =a_{jim}, \\
b_{imn} &=&\langle k_{im}|k_{in}\rangle =b_{inm}.\end{aligned}$$The quantity $Q_{PS}$ is the corresponding expression for the permutation $PS$.
> From Lemma \[zana\], $$\epsilon (P)=\frac{d^{4}+d^{2}-d^{4}[\text{Tr}\rho _{PS}^{2}+\text{Tr}\rho
_{P}^{2}]}{d(d-1)(d+1)^{2}},$$where $\rho _{P}$ and $\rho _{PS}$ are the reduced density matrices of the states $|P\rangle \langle P|$ and $|PS\rangle \langle PS|$, respectively. We can express $d^{4}$Tr$\rho _{P}^{2}$ in terms of the matrices $(k_{ij})$ and $(l_{ij})$. Applying the formula in Equation \[vecfrommat\], we find the state corresponding to $P$ as $$| P\rangle _{12|34}=\frac{1}{d}\sum_{i,m=1}^{d}|k_{im}i\rangle _{12}\otimes
| l_{im}m\rangle _{34}.$$This leads to $$\rho _{P}=\text{Tr}_{B}|P\rangle \langle P|=\frac{1}{d^{2}}\sum_{i,j,m=1}^{d}a_{ijm}|k_{im}i\rangle \langle k_{jm}j|.$$By squaring and taking the trace, we obtain $$d^{4}\text{Tr}\rho _{P}^{2}=\sum_{i,j,m,n=1}^{d}a_{ijm}a_{ijn}b_{imn}b_{jmn}.$$The same analysis applies to $Q_{PS}$.
QTP[ite]{} The problem of classifying bipartite permutation operators according to their entangling power now reduces to finding what different values can be taken by $Q_{P}+Q_{PS}$. Observe that the coefficients $a_{ijn} $ and $b_{imn} $ (which are either $0$ or $1$), and the combination $$r_{ijmn}=a_{ijm}a_{ijn}b_{imn}b_{jmn},$$which occurs in , have the following interpretation:
- $a_{ijn} = 1$ if and only if $(i,n)$ and $(j,n)$, which are in the same column of the square $[d]\times[d]$, are taken by $P$ to elements in the same column;
- $b_{imn} = 1$ if and only if $(i,m)$ and $(i,n)$, which are in the same row of the square, are taken by $P$ to elements in the same row;
- $r_{ijmn}=1$ if and only if $(i,m),(i,n),(j,m)$ and $(j,n)$, which form a rectangle in $[d]\times \lbrack d]$, are taken by $P$ to a rectangle with the same orientation.
We will denote the rectangle $(i,m),(i,n),(j,m),(j,n)$ (formed by the intersection of rows $i$ and $j$ and columns $m$ and $n$) as $R_{ijmn}$. We note that if this rectangle is non-degenerate, that is $i\neq j$ and $m\neq
n $, then it contributes either $0$ or $4$ to $Q_{P}$, since $$r_{ijmn}=r_{jimn}=r_{ijnm}=r_{jinm}=0\text{ or }1;$$if $i=j$ or $m=n$ but not both, $R_{ijmn}$ contributes $0$ or $2$; while if $i=j$ and $m=n$, then $R_{ijmn}$ contributes $1$.
QTP[ite]{} QTP[ite]{}
Non-entangling permutations
===========================
QTP[ite]{} Two permutation matrices $P,Q\in \mathcal{U(H)}$ are said to be *locally unitarily connected* (for short, *LU-connected*) if there are unitaries $V$ acting on $\mathcal{H}_{A}$ and $W$ on $\mathcal{H}_{B}$ such that $(V\otimes W)P=Q$. Then $V$ and $W$ are actually permutation operators. Note that if two permutations are LU-connected then they have the same entangling power. The set of non-entangling permutation matrices is denoted by $E^{0}$. The following result seems to be known to specialists (see, *e.g.*, [@rez]), but we provide a proof for completeness
\[ent\]Let $P\in\mathcal{U(H)}$ be a permutation matrix. Then $P\in
E^{0} $ if and only if one of the following two conditions is satisfied:
1. $P$ is LU-connected to $I$;
2. $P$ is LU-connected to $S$.
The permutation matrix $P$ is non-entangling if $P(|\varphi \rangle |\chi
\rangle )$ is a product state for all $|\varphi \rangle \in \mathcal{H}_{A}$ and $|\chi \rangle \in \mathcal{H}_{B}$. Consider two basis elements $|
i\rangle |j_{1}\rangle $ and $|i\rangle |j_{2}\rangle $, and suppose $$\begin{aligned}
P(|i\rangle |j_{1}\rangle )& =|i_{1}^{\prime }\rangle |j_{1}^{\prime
}\rangle , \\
P(|i\rangle |j_{2}\rangle )& =|i_{2}^{\prime }\rangle |j_{2}^{\prime
}\rangle .\end{aligned}$$Then either $i_{1}^{\prime }=i_{2}^{\prime }$ or $j_{1}^{\prime
}=j_{2}^{\prime }$, otherwise $P(|i\rangle (|j_{1}\rangle +|j_{2}\rangle ))$ would be entangled. Suppose $i_{1}^{\prime }=i_{2}^{\prime }=i^{\prime }$. Then for all $j_{3}$ we must have $P(|i\rangle |j_{3}\rangle )=|i^{\prime
}\rangle |j_{3}^{\prime }\rangle $, for if not $P(|i\rangle |j_{3}\rangle
)=|i^{\prime \prime }\rangle |j_{1}^{\prime }\rangle $ and then $P(|i\rangle
(|j_{2}\rangle +|j_{3}\rangle ))$ is entangled. So there is a permutation $p_{B}$ such that $P(|i\rangle |j\rangle )=|i^{\prime }\rangle |
p_{B}(j)\rangle $. Thus as a permutation of the square lattice $\{(i,j):1\leq i,j\leq d\}$, $P$ takes the row $(i,\cdot )$ to a row. Consider an element $(i_{2},j_{1})$ in the same column as $(i,j_{1})$. The permutation $p$ cannot take $(i_{2},j_{1})$ to an element in the row $(i^{\prime },\cdot )$, because that is already full. So $p$ must take $(i_{2},j_{1})$ to an element in the column $(\cdot ,j_{1}^{\prime })$. Thus $p$ takes the column $(\cdot ,j_{1})$ to a column. It then follows that $p$ takes every row to a row and every column to a column, that is $P((|i\rangle
| j\rangle )=|p_{A}(i)\rangle |p_{B}(j)\rangle _{2}$ for some permutation $p_{A}$.
In the second case, $j_{1}^{\prime }=j_{2}^{\prime }$, the permutation $p$ takes two elements in a row to elements in a column, and we can similarly show that it takes every row to a column and every column to a row, that is $P(|i\rangle |j\rangle )=|p_{A}(j)\rangle |p_{B}(i)\rangle $.
QTP[ite]{} The probability of sampling a permutation matrix in $E^{0}$ over all permutations of $[d^{2}]$ goes to $0$ as $d\rightarrow \infty $. In fact, by Theorem \[ent\], the number of elements in $E^{0}$ is $2(d!)^{2}$, and therefore the probability is $$\begin{tabular}{lll}
$\dfrac{2(d!)^{2}}{d^{2}!}\rightarrow 0$ & as & $d\rightarrow \infty $.\end{tabular}$$
Entangling permutations
=======================
Maximum
-------
QTP[ite]{} In this section we construct and count the permutations with the maximum entangling power $d/(d+1)$ that can be attained by any unitary operator on $\mathcal{H}_{A}\otimes \mathcal{H}_{B}$. We will make use of latin squares. Recall that a *latin square* of *side* $d$ is a $d\times d$ matrix with entries from the set $[d]=\{1,\ldots ,d\}$ such that every row and column is a permutation of $\{1,\ldots ,d\}$, and two $d\times
d$ latin squares $(k_{ij})$ and $(l_{ij})$ are *orthogonal* if $(k_{ij},l_{ij})$ is a permutation of $[d]\times \lbrack d]$. Equivalently, two $d\times d$ latin squares $(k_{ij})$ and $(l_{ij})$ are orthogonal if the ordered pairs $(k_{ij},l_{ij})$ are distinct for all $i$ and $j$. Euler believed that there are no orthogonal latin squares of side $4n+2$. Only in 1960 was it shown by Bose, Shrikhande and Parker that, except for side $6$, Euler’s conjecture was false [@bsp]. The study of latin squares is an important area of combinatorics with connections to design theory, projective geometries, graph theory, *etc.* [@crc; @lm; @ls]. Orthogonal latin squares have been recently applied in quantum information theory [@ww; @hhh; @kg].
\[max\]Let $P$ be a permutation operator on $\mathcal{H}_{A}\otimes
\mathcal{H}_{B}$ defined by $P(|i\rangle |j\rangle )=|k_{ij}\rangle |
l_{ij}\rangle $. Then the entangling power of $P$ equals the maximum value $\epsilon (P)=d/(d+1)$ over $\mathcal{U(H)}$ if and only if the matrices $(k_{ij})$ and $(l_{ij})$ are orthogonal latin squares.
By Theorem 2, the entangling power of $P$ is maximised when $Q_{P}+Q_{PS}$ is minimised. Now $Q_{P}$ is equal to the number of rectangles in $[d]\times
\lbrack d]$ which are taken to rectangles by $P$, with the horizontal lines remaining horizontal and the vertical lines remaining vertical. This is at least $d^{2}$, since every rectangle consisting of a single point must be taken to a rectangle. It is precisely $d^{2}$ if and only if no nonzero horizontal line is taken to a horizontal line and no nonzero vertical line is taken to a vertical line, i.e. if $$k_{im}\neq k_{in}\quad \text{ whenever }\quad m\neq n$$and $$l_{im}\neq l_{jm}\quad \text{ whenever }\quad i\neq j.$$On the other hand, $Q_{PS}$ is equal to the number of rectangles in $[d]\times \lbrack d]$ which are taken to rectangles by $P$, but with the horizontal lines becoming vertical and the vertical lines becoming horizontal. This will be precisely $d^{2}$ if and only if no nonzero vertical line is taken to a horizontal line and vice versa, that is if and only if $$k_{im}\neq k_{jm}\quad \text{ whenever }\quad i\neq j$$and $$l_{im}\neq l_{in}\quad \text{ whenever }\quad m\neq n.$$Together, these are the conditions for the matrices $(k_{ij})$ and $(l_{ij})$ to be latin squares. Since $(k_{ij},l_{ij})$ form a permutation of $(i,j)$, the two latin squares are orthogonal.
**Corollary** \[mib1\] *For every* $d\neq 2,6$* there is a permutation matrix* $P\in \mathcal{U(H)}$* such that* $\epsilon (P)$* is maximum over* $\mathcal{U(H)}$.
It follows from Theorem \[max\] together with the fact that there are two orthogonal latin squares for every $d\neq 2,6$ (see, *e.g.*, [@ls]).
As an example of a permutation matrix satisfying Theorem 4, consider$$R=\left[
\begin{array}{ccc|ccc|ccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right] .$$For the permutation matrix $R$ we have $\epsilon (R)=\frac{3}{4}$ which is the maximum over all unitaries in $\mathcal{U}(\mathcal{H})=(\mathcal{H}_{A}\otimes \mathcal{H}_{B})$ where $\dim \mathcal{H}_{A}=\dim \mathcal{H}_{B}=3$.
By looking at $d^{2}\times d^{2}$ permutation matrices as made up of $d^{2}$ blocks, we can state an alternative version of Theorem \[max\].
\[fourway\]Let $P\in \mathcal{U(H)}$ be a permutation matrix. Then $\epsilon (P)$ is maximum over $\mathcal{U(H)}$ if and only if $P$ satisfies the following conditions:
1. Every block contains one and only one nonzero element;
2. All blocks are different;
3. Nonzero elements in the same block-row are in different sub-columns;
4. Nonzero elements in the same block-column are in different sub-rows.
By Theorem 2, the quantity $Q_{P}$ is maximum if and only if $a_{ijm}=b_{imn}=1$ for all $1\leq i,j,n,m\leq d$. In this case $Q_{P}=d^{4}$ (for example, when$P=I$). On the other hand $Q_{P}$ is minimum if and only if $a_{ijn}=\delta _{ij}$ and $b_{imn}=\delta _{mn}$, and in this case the sum reduces to $$\begin{tabular}{lll}
$Q_{P}=\sum_{i,n=1}^{d}a_{iin}b_{inn}=d^{2}$ & & (for example, when $P=S$).\end{tabular}$$To obtain the maximum entangling power we need to find the permutation matrix that minimizes $Q_{P}+Q_{PS}$. We have seen that the maximum entangling power is $d/(d+1)$, and in [@z0] is shown that this value can be obtained if and only if $$S_{L}(|U\rangle )=S_{L}(|US\rangle )=S_{L}(|S\rangle )$$or, equivalently, $$Q_{P}=Q_{PS}=Q_{S}=d^{2}.$$It is then easy to observe that the conditions $1$ and $2$ express the minimality of $Q_{P}$ and the conditions $3$ and $4$ express the minimality of $Q_{PS}$.
Theorem \[fourway\] implies that in a $d^{2}\times d^{2}$ permutation matrix $P$ attaining the maximum value $d/(d+1)$, every block contains one and only one nonzero entry (as in the above permutation matrix $R$). It is then possible to represent $P$ by a $d\times d$ array $\widetilde{P}=(\widetilde{p}_{ij})$. The cell $\widetilde{p}_{ij}$ specifies the coordinates of the nonzero entry in the $ij$-th block of $P$. For the above permutation matrix $R$, we have $$\widetilde{R}=\begin{tabular}{|ccc|}
\hline
$11$ & $23$ & $32$ \\
$22$ & $31$ & $13$ \\
$33$ & $12$ & $21$ \\ \hline
\end{tabular}\text{ }.$$Note that the $ij$-th cell of $\widetilde{R}$ is of the form $(k_{ij},l_{ij}) $ where $K=(k_{ij})$ and $L=(l_{ij})$ are the orthogonal latin squares$$\begin{tabular}{lll}
$K=\begin{tabular}{|lll|}
\hline
$1$ & $2$ & $3$ \\
$2$ & $3$ & $1$ \\
$3$ & $1$ & $2$ \\ \hline
\end{tabular}$ & and & $L=\begin{tabular}{|lll|}
\hline
$1$ & $3$ & $2$ \\
$2$ & $1$ & $3$ \\
$3$ & $2$ & $1$ \\ \hline
\end{tabular}\text{ }.$\end{tabular}$$It follows from Theorem \[fourway\] that a permutation matrix $P$ has maximal entangling power if and only if $\widetilde{P}$ is obtained by *superimposing* two orthogonal latin squares.
Direct calculations give the following result.
The following statements are true:
1. For $d=2$ the matrix $P=CNOT$ attains the value $\epsilon (P)=\frac{4}{9}$ which is maximum over all unitaries in $\mathcal{U(}\mathcal{H})$.
2. For $d=6$ the value $\epsilon (P)=\frac{628}{735}$ is maximum over all permutations $P\in \mathcal{U(}\mathcal{H})$ and the maximizing $P$ is associated to $$\widetilde{P}=\begin{tabular}{|llllll|}
\hline
$11$ & $22$ & $33$ & $44$ & $55$ & $66$ \\
$23$ & $14$ & $45$ & $36$ & $61$ & $52$ \\
$32$ & $41$ & $64$ & $53$ & $16$ & $25$ \\
$46$ & $35$ & $51$ & $62$ & $24$ & $13$ \\
$54$ & $63$ & $26$ & $15$ & $42$ & $31$ \\
$65$ & $56$ & $12$ & $21$ & $33$ & $44$ \\ \hline
\end{tabular}\text{ }. \label{ptil}$$
*Part 1.* It has been shown by Rezakhani [@rez] that for $d=2$, the entangling power of any unitary $U\in U(4)$ is given by $$\begin{aligned}
\epsilon (U)& =\frac{1}{3}-\frac{1}{9}\times \{\cos (4c_{1})\cos (4c_{2})+
\notag \\
& \cos (4c_{1})\cos (4c_{3})+\cos (4c_{2})\cos (4c_{3}))\},\end{aligned}$$where $c_{1},c_{2},c_{3}\in \mathbb{R}$ and $|c_{3}|\leq c_{2}\leq c_{1}\leq
\pi /4$. It is easy to show that $\epsilon (U)$ takes its maximum value $4/9$ when either $c_{1}=c_{2}=\pi /4,c_{3}=0$ or $c_{1}=\pi /4,c_{2}=c_{3}=0$. Then, every permutation matrix which is LU-connected to any of the four matrices attains this maximum value (see Table I, given in Section V):$$\begin{tabular}{rr}
$CNOT=\left(
\begin{array}{llll}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0\end{array}\right) ,$ & $S\cdot CNOT,$ \\
& \\
$M=\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0\end{array}\right) ,$ & $S\cdot M.$\end{tabular}$$
*Part 2.* The permutation associated to $\widetilde{P}$ arises from the latin squares $$\begin{tabular}{|llllll|}
\hline
$1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\
$2$ & $1$ & $4$ & $3$ & $6$ & $5$ \\
$3$ & $4$ & $6$ & $5$ & $1$ & $2$ \\
$4$ & $3$ & $5$ & $6$ & $2$ & $1$ \\
$5$ & $6$ & $2$ & $1$ & $4$ & $3$ \\
$6$ & $5$ & $1$ & $2$ & $3$ & $4$ \\ \hline
\end{tabular}$$and$$\begin{tabular}{|llllll|}
\hline
$1$ & $2$ & $3$ & $4$ & $5$ & $6$ \\
$3$ & $4$ & $5$ & $6$ & $1$ & $2$ \\
$2$ & $1$ & $4$ & $3$ & $6$ & $5$ \\
$6$ & $5$ & $1$ & $2$ & $4$ & $3$ \\
$4$ & $3$ & $6$ & $5$ & $2$ & $1$ \\
$5$ & $6$ & $2$ & $1$ & $3$ & $4$ \\ \hline
\end{tabular}\text{ },$$which are *‘as close to being orthogonal as possible’* [@h]. For $P$, we have that $Q_{P}=40$ and $Q_{PS}=36$. Since there do not exist two orthogonal latin squares of side $6$, these are the smallest values obtainable, as it is explained by the following argument. The array$$\widehat{P}=\begin{tabular}{|llllll|}
\hline
$11$ & $22$ & $3\mathbf{3}$ & $4\mathbf{4}$ & $55$ & $66$ \\
$24$ & $13$ & $46$ & $35$ & $62$ & $51$ \\
$56$ & $65$ & $12$ & $21$ & $43$ & $34$ \\
$63$ & $54$ & $25$ & $16$ & $31$ & $42$ \\
$45$ & $36$ & $61$ & $52$ & $14$ & $23$ \\
$32$ & $41$ & $5\mathbf{3}$ & $6\mathbf{4}$ & $26$ & $15$ \\ \hline
\end{tabular}$$represents the action of $P$ on the set $[6]\times \lbrack 6]$. The $ij$-th cell of $\widehat{P}$ is $kl$ if in $P$ the contribution of the term $|kl\rangle \langle ij|$ is nonzero (for example, the $22$-th cell of $\widehat{P}$ is $13$ because $|13\rangle \langle 22|$ is nonzero in $P$). We have printed in boldface the symbols that occur twice in a row or in a column. Observe that, given any permutation matrix $P\in U(36)$, since every element of the set $[6]\times \lbrack 6]$ has to occur exactly once in $\widehat{P}$, it is never possible to have only one pair of equal symbols in the same column, without this being the case for also another column. Hence, the minimum value attainable by $Q_{P}$ would be $2$ steps ahead of $36$, that is $40$.
We have established a bijection between pairs of orthogonal latin squares of side $d$ and $d^{2}\times d^{2}$ permutations with maximal entangling power. The number of pairs of orthogonal latin squares of side $d$ is known only for small $d$ (see A072377, [@sloan]). It is $36$ for $d=3$ and $3456$ for $d=4$. Note that these values apply to unordered pairs of orthogonal latin squares. This means that the number of $d^{2}\times d^{2}$ permutations with the maximum entangling power is twice the number of pairs of orthogonal latin squares of side $d$. General methods for constructing pairs of orthogonal latin squares are presented in [@gras; @ls; @l].
For a unitary $U$ that reaches the upper bound $\epsilon (U)=d/(d+1)$, we have $S_{L}(|U\rangle )=S_{L}(|US\rangle )=1$. In other words, $|U\rangle $ is maximally entangled with respect to the bipartite splits $12|34$ and $23|14$. It turns out that this is also maximally entangled with respect to the split $13|24$. The easiest way to see this is by looking at Equation [vecfrommat]{} defining $|U\rangle $. The state $|U\rangle _{13|24}$ is obtained by two parties $A=13$ and $B=24$ sharing a maximally entangled state, by applying the unitary $U$ on Alice’s system. Since this is a local reversible transformation, the entanglement is preserved, hence the state $|U\rangle _{13|24}$ is maximally entangled. Thus a unitary $U$ is maximally entangling if and only $|U\rangle $ is maximally entangled with respect to the three possible bipartite splits [@Zalka]. Note that $|U\rangle $ is also maximally entangled in all the four splits $1|234$, $2|134$, $3|124$ and $4|123$. Then $|U\rangle $ is indeed a maximally entangled state. The construction of maximally entangling unitaries thus provides us with a canonical way of constructing such maximally entangled $4$-qudits for all dimensions except $d=2$ and $d=6$. It has been known for some time that this is not possible for $d=2$, *i.e.* $4$-qubits (see [@tony]), leaving only the case $d=6$ open.
Minimum
-------
In this section we construct the permutations with the minimum nonzero entangling power that can be attained by permutation operators on $\mathcal{H}_{A}\otimes \mathcal{H}_{B}$. Two permutation matrices $P,Q\in \mathcal{U(H)}$ are said to be in the same *entangling class* if $\epsilon
(P)=\epsilon (Q)$.
\[mib\] Let $P\in \mathcal{U(H)}$ be a permutation matrix. Then $\epsilon (P)$ is nonzero but minimum over all permutations in $\mathcal{U(H)}
$ if$$\widehat{P}=\begin{tabular}{|lllll|}
\hline
$11$ & $12$ & $\cdots $ & $\cdots $ & $1d$ \\
$21$ & $22$ & $\ldots $ & $\ldots $ & $2d$ \\
$\vdots $ & $\vdots $ & & & $\vdots $ \\
$(d-1)1$ & $(d-1)2$ & $\ldots $ & $\ldots $ & $(d-1)d$ \\
$d1$ & $d2$ & $\ldots $ & $dd$ & $d(d-1)$ \\ \hline
\end{tabular}\ .$$In such a case $$\epsilon (P)=\frac{8(d-1)}{d(d+1)^{2}}. \label{minepsilon}$$
Since $P$ is the permutation matrix *closest* to $I$, it is clear that $\epsilon (P)$ is miminum. To find its value, we compare $Q_{P}$ with $Q_{I}$ where I denotes the identity permutation. The rectangles $R_{ijmn}$ which contribute to $Q_{I}$ but not to $Q_{P}$ are those containing $(d-1,d)$ or $(d,d)$ or both, except for the degenerate rectangles contained in the bottom line of the square (i.e. those with $i=j=d$). There are $(d-1)(d-2)$ non-degenerate rectangles containing $(d-1,d)$ but not $(d,d-1)$, each contributing $4$ to $Q_{I}$, and $d-1$ degenerate vertical rectangles, with two equal vertices at $(d,d-1)$ and two equal vertices at $(i,d-1)$ where $i\neq d$. Each of these contributes 2 to $Q_{I}$, so the total contribution from rectangles containing $(d,d-1)$ but not $(d,d)$ is $4(d-1)(d-2)+2(d-1)$. There is an equal contribution from rectangles containing $(d,d)$ but not $(d,d-1)$. Finally, there are $d-1$ non-degenerate rectangles containing both $(d,d-1)$ and $(d,d)$, each contributing 4 to $Q_{I}$. The total contribution to $Q_{I}$ which is not included in $Q_{P}$ is $8(d-1)^{2}$. Since $Q_{I}=d^{4}$, we have $$Q_{P}=d^{4}-8(d-1)^{2}.$$On the other hand, $Q_{PS}=d^{2}$ since no nonzero horizontal line is taken to a vertical line by $P$. Hence, by Theorem 2, $\epsilon (P)$ is given by .
**Corollary** *An upper bound to the number of different entangling classes of permutations is given by* $$B=2+\frac{1}{2}(d^{4}-d^{2}-8(d-1)^{2}). \label{boun}$$
We can write $Q_{P}=\sum_{i,j,m,n=1}^{d}r_{ijmn}$. If $i\neq j$ and $m\neq n$, the contribution of the rectangle $R_{ijmn}$ is $0$ or $4$, which is even. If either $i=j$ or $m=n$, the contribution of $R_{ijmn}$ is $0$ or $2$, which is again even. If $i=j$ and $m=n$, the contribution of $R_{ijmn}$ is $1 $, and in total we have $d^{2}$ such terms. It follows that $Q_{P}$ is even if and only if $d$ is even. The same analysis applies to $Q_{PS}$. Then $Q_{P}+Q_{PS}$ is even for all $d$.
Now we have seen that the zero entangling power corresponds to $Q=d^{4}+d^{2} $, and the maximum to $Q=2d^{2}$, so that $\frac{1}{2}(d^{4}-d^{2}+2)$ is an upper bound to the number of classes, where the coefficient $1/2$ comes from the fact that two consecutives values of $Q_{P}$ ($Q_{PS}$) differ by a multiple of $2$. We can tighten this bound and obtain the value $(B-2)$ by making use of the fact from Theorem \[mib\] that the value of $Q_{P}+Q_{PS} $ is $d^{4}-8(d-1)^{2}+d^{2}$ when $\epsilon (P)$ is nonzero but minimum. The first term $2$ on the RHS of Equation \[boun\] occurs because of the two classes corresponding to zero and the maximum entangling power.
Numerical results
=================
In this section we report some numerical results. We are interested in counting the number of different entangling classes for different dimensions. We are also interested in the average entangling power over all permutations of a given dimension. The results are given in the following tables:
--------------------------------- ---------------------
Entangling Power $\epsilon (P)$ Number of elements
in entangling class
0 8
4/9 16
--------------------------------- ---------------------
: Classes of permutations with different entangling power and the number of elements in each class for $d=2$.
--------------------------------- ---------------------
Entangling Power $\epsilon (P)$ Number of elements
in entangling class
0 72
1/3 2592
3/8 864
5/12 1296
182/375 10368
23/48 20736
1/2 27432
25/48 36288
13/24 44064
9/16 101376
7/12 44712
29/48 46656
5/8 22464
2/3 3888
3/4 72
--------------------------------- ---------------------
: Classes of permutations with different entangling power and the number of elements in each class for $d=3$.
Dimension $d$ Number of classes Average entangling power
--------------- ------------------- -----------------------------
$2$ $2$ $\frac{8}{27}\approx 0.29$
$3$ $15$ $\frac{31}{56}\approx 0.55$
$4$ $\geq 65$ $0.67\pm 0.01$
$5$ $\geq 190$ $0.74\pm 0.01$
: Number of classes of permutations with different entangling power and the average entangling power as a function of the dimension $d$.
Conclusion and open problems
============================
In this paper we have studied the entangling power of permutations. We have shown that the permutation matrices with zero entangling power are, up to local unitaries, the identity and the swap. For all dimensions, we have constructed the permutations with (nonzero) minimum entangling power. With the use of orthogonal latin squares, we have constructed the permutations with the maximum entangling power for every dimension. Moreover, we have shown that this value is maximum over all unitaries of the same dimension, with a possible exception for $36$. Our result enabled us to construct generic examples of $4$-qudits maximally entangled states for all dimensions except for $2$ and $6$. We have reported numerical results about a complete classification of permutation matrices of dimension $4$ and $9$ according to their entangling power.
We conclude with a list of open problems:
- Describe a general classification of the permutation matrices according to their entangling power.
- Give a formula for the average entangling power over all permutation matrices of a given dimension.
- For $d=6$, does there exist $U\in \mathcal{U(}\mathcal{H})$ such that $\epsilon (U)>\frac{628}{735}$?
- The formula in Equation (\[newe\]) only works for the linear entropy. It would be desirable to have a similar simple formula of the entangling power of a unitary in terms of the von Neumann entropy.
- Study the entangling power of permutation matrices in relation to multipartite systems. In this context, it is conceivable that the permutation matrices with maximum entangling power are related to sets of mutually orthogonal latin squares.
*Acknowledgements.* We would like to thank S. L. Braunstein for helpful comments, and Ch. Zalka for pointing out the connection between maximally entangling unitaries and $4$-qudit states that are maximally entangled with respect to all bipartite splits. LC is supported by a WW Smith Scholarship. SG and SS are supported by EPSRC grants GR/87406 and GR/S56252, respectively.
[99]{} In refs. [@z] and [@z0], $S_{L}(|\psi \rangle )$ is taken as $S_{L}(|\psi \rangle )=1-$Tr$\rho ^{2}$, where $\rho =$ Tr$_{B}(|\psi \rangle \langle \psi |)$. In order to have $S_{L}(|\psi \rangle
)\in \lbrack 0,1]$, we have taken $S_{L}(|\psi \rangle )=\frac{d}{d-1}(1-$Tr$\rho ^{2})$.
In ref. [@z], the coefficient in front of the term $[S_{L}(|U\rangle )+S_{L}(|US\rangle )-S_{L}(|S\rangle )]$ is $\left( \frac{d-1}{d+1}\right) \frac{1}{S_{L}(|S\rangle )}$, which in our case is $\frac{d}{d+1}$, as we used a modified definition of $S_{L}(|\psi \rangle )$.
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---
abstract: 'The number of absorption and Raman optical modes for each Wyckoff orbit in the high-temperature tetragonal (I4/mmm) parent lattice of the Ruddlesden-Popper compounds Ca$_3$X$_2$O$_7$, with X=Mn or X=Ti is given. We analyze the effect of sequential perturbations which lower the symmetry to Cmcm and Cmc2$_1$ and finally include magnetic ordering. We determine the power law behavior (within mean field theory) for the cross section for photon absorption and Raman scattering of modes which appear as the symmetry is successively lowered. In the Cmc2$_1$ phase we give a symmetry analysis to discuss the magnon-phonon coupling which in other systems gives rise to “electromagnons." From our results we suggest several experiments to clarify the phase diagram and other properties of these systems.'
author:
- 'A. B. Harris'
title: 'Phonons and their Coupling to Magnons in $n=2$ Ruddlesden-Popper Compounds.'
---
INTRODUCTION
============
The Ruddlesden-Popper (RP) compounds\[\] are those of the type A$_{n+1}$B$_n$ C$_{3n+1}$, where the valences of the ions are A $=+2$, B $=+4$, and C $=-2$ (usually C is oxygen). At high temperature the crystal structure for many of these compounds is tetragonal, but the structure can distort in several ways as the temperature passes through structural phase transitions. In the lowest symmetry phase, these systems are usually ferroelectric and can also be magnetically ordered, although, in contrast to many magnetoelectrics,\[\] these two degrees of freedom do not appear at the same temperature. Nevertheless the RP systems do have many interesting couplings between their magnetic and dielectric properties. For example, their spontaneous polarization can be rotated by the application of a magnetic field and the weak ferromagnetic moment can be rotated by the application of an electric field.\[\] The symmetry of the RP’s is very similar to that of the Aurivillius compound SrBi$_2$Ta$_2$O$_9$ analyzed in detail by Perez-Mato [*et al.*]{}\[\] and many of our results should also apply to it. Although there have been several first principles calculations of the properties of CMO and/or CTO,\[\] the calculations of CF are most relevant to the present paper.
![\[FIG1\] (Color online) The tetragonal phase of CMO or CTO (space group I4/mmm). The oxygen ions are the blue spheres, the Ca ions are the open green circles, and the Mn or Ti ions (not shown) are at the centers of the octahedra of oxygen ions. There are two molecules per conventional unit cell.](FIG1.eps){width="5.0"}
Here we will be interested in the two RP compounds Ca$_3$Mn$_2$O$_7$ (CMO) and Ca$_3$Ti$_2$O$_7$ (CTO), which consist of bilayers of oxygen octahedra enclosing a Mn or Ti ion. It is believed that the high-temperature phase is tetragonal (I4/mmm)\[\] and the low-temperature phase is orthorhombic (Cmc2$_1$)\[\] and in Fig. \[GSG\] we show the three possible paths that connect these structures.\[\] First principles calculations\[\] indicate that the most likely scenario involves the appearance, as the temperature is lowered, of the intermediate state Cmcm at a temperature we denote $T_>$. In fact, the Cmcm phase has been observed in the isostructural compounds LaCa$_2$Mn$_2$O$_7$\[\] at and below room temperature and Bi$_{0.44}$Ca$_{2.56}$Mn$_2$O$_7$\[\] at room teperature. Accordingly, we will assume that this scenario applies to CMO and CTO although for these systems no intermediate phase has been observed up to now. However, recent measurements on ceramic CMO find a clear pyroelectric signal consistent with the onset of ferroelectric order close to $T_>=280$K.\[\] This transition is identified as the temperature at which Cmc2$_1$ appears. Since this ferroelectric transition seems to be a continuous and well-developed one and since a direct continuous transition between I4/mmm and Cmc2$_1$ is inconsistent with Landau theory,\[\] the seemingly inescapable conclusion is that the phase for $T$ slightly greater than $T_>$ is [*not*]{} I4/mmm, but is some phase which does not allow a spontaneous polarization. Thus the phase at temperature just above $T_>$ may be the Cmcm phase. In this scenario the Cmc2$_1$ phase would appear at a lower temperature $T_<$ (which experimental data implies is quite close to $T_>$). The purpose of the present paper is to discuss the symmetry of the zone center phonons and the Raman scattering cross section as they are modified by the successive lowering of symmetry as the temperature is reduced through the structural phase transitions at $T_>$ and $T_<$. Such a symmetry analysis can be done using standard group theory methods. Here we will supplement that analysis with an analysis of the phonon cross sections from which we determine their power law dependence on the order parameters (OP’s) which characterize the distortions from tetragonal symmetry to the lower symmetry structures. This enables us to predict the temperature dependence of the newly allowed cross sections which appear at the structural phase transitions. We then give a similar analysis of the magnon-phonon interaction. As a result of this coupling, the magnon absorption cross section (which usually results from a magnetic dipole allowed transition) is enhanced by now being electric-dipole allowed. This enhancement has led some authors\[\] to call such magnons “electromagnons."
Briefly, this paper is organized as follows. In Sec. II we give a symmetry analysis of the zone center phonons in the tetragonal (I4/mmm) phase and in Secs. III and IV we give similar analyses for the orthorhombic Cmcm and Cmc2$_1$ phases. We discuss the photon absorption and Raman scattering of these modes. For modes which only appear as a consequence of a phase transition we also determine the power-law dependence of their optical cross sections on the emergent OP’s. In Sec. V we discuss the coupling of these modes to magnetic excitations (magnons). Our results are summarized in Sec. VI.
![\[GSG\] The group-subgroup structure of CMO and CTO. We indicate the irreducible representations (irreps) by which the tetragonal phase is distorted to reach one of the intermediate phases. The numbering of the space groups follows Ref. . The superscripts on the irreps give the parity with respect to spatial inversion. The irrep $\Gamma_5^-$ is at zero wave vector and the $X$ irreps are at the wave vectors ${\bf q}=(1/2,\pm 1/2,0)$. In the final step to reach the A2$_1$am (equivalent to Cmc2$_1$) structure the other two irreps are introduced simultaneously,\[\] so that Cmc2$_1$ can be regarded as the result of simultaneous condensation of the three irreps.](FIG2.eps){width="8.6"}
SYMMETRY ANALYSIS FOR THE TETRAGONAL STRUCTURE
==============================================
The occupied sites in space group I4/mmm for the $n=2$ RP system are given in the first three columns of Table \[basis\], below. The character table for the little group of ${\bf q}=0$ for the I4/mmm tetragonal structure is given in Table \[T4\]. For the one dimensional irreducible representations (irreps) the characters are the matrices. For the two dimensional irreps and for the irreps for the $X$ points at ${\bf q}=(1/2,\pm 1/2,0)$ (in rlu’s) the matrices are given in Table \[IRREP\].\[\]
[|| c || c || c | c | c | c | c || c | c | c | c | c ||]{} Number & Operator & $\Gamma_1^+$ & $\Gamma_2^+$ & $\Gamma_3^+$ & $\Gamma_4^+$ & $\Gamma_5^+$ & $\Gamma_1^-$ & $\Gamma_2^-$ & $\Gamma_3^-$ & $\Gamma_4^-$ & $\Gamma_5^-$ \
$1$& $(xyz)$ & $1$ & $1$ & $1$ & $1$ & $2$ & $1$ & $1$ & $1$ & $1$ & $2$\
$37$,$40$ & $\ \ (yxz),(\overline y , \overline x, z)\ \ $ & $1$ & $-1$ & $-1$ & $1$ & $0$ & $-1$ & $1$ & $1$ & $-1$ & $0$\
$4$ & $(\overline x \overline yz)$ & $1$ & $1$ & $1$ & $1$ & $-2$ & $1$ & $1$ & $1$ & $1$ & $-2$\
$3,2$ & $(\overline xy \overline z), (x \overline y \overline z )$ & $1$ & $1$ & $-1$ & $-1$ & $0$ & $1$ & $1$ & $-1$ & $-1$ & $0$\
$39,38$ & $(\overline y x \overline z),(y \overline x \overline z)$ & $1$ & $-1$ & $1$ & $-1$ & $0$ & $-1$ & $1$ & $-1$ & $1$ & $0$\
$25$ & $(\overline x \overline y \overline z)$ & $1$ & $1$ & $1$ & $1$ & $2$ & $-1$ & $-1$ & $-1$ & $-1$ & $-2$\
$13,16$ & $(\overline y \overline x \overline z),(yx \overline z)$ & $1$ & $-1$ & $-1$ & $1$ & $0$ & $1$ & $-1$ & $-1$ & $1$ & $0$\
$28$ & $(xy \overline z)$ & $1$ & $1$ & $1$ & $1$ & $-2$ & $-1$ & $-1$ & $-1$ & $-1$ & $2$\
$27,26$ & $(x\overline yz),(\overline x y z)$ & $1$ & $1$ & $-1$ & $-1$ & $0$ & $-1$ & $-1$ & $1$ & $1$ & $0$\
$15,14$ & $(y\overline xz),(\overline y x z)$ & $1$ & $-1$ & $1$ & $-1$ & $0$ & $1$ & $-1$ & $1$ & $-1$ & $0$\
${\bf P}$ && $a(\hat i \hat i + \hat j \hat j)$ & $c(\hat i \hat i $ & & $d(\hat i \hat j $ & $e(\hat i \hat k + \hat k \hat i)$ & & & $g \hat k$ & & $h \hat i$\
or ${\bf R}$ & & $+b \hat k \hat k$ & $-\hat j \hat j )$ & & $+\hat j \hat i)$ & $e(\hat j \hat k + \hat k \hat j)$ & & & & & $h \hat j$\
\[h!\]
[|| c | c c c ||]{} ${\cal O}=$ & ${\cal R}_4$ & $m_d$ & $m_z$\
${\bf r}' = $ & $(\overline y, x,z)$ & $(y,x,z)$ & $(x,y, \overline z)$ \
${\bf M}^{({\bf X}_1^+)}({\cal O})=$ & $- \sigmav_x$ & $1$ & ${\bf 1}$\
${\bf M}^{({\bf X}_2^+)}({\cal O})=$ & $ \sigmav_x$ & $-1$ & ${\bf 1}$\
${\bf M}^{({\bf X}_3^+)}({\cal O})=$ & $-i\sigmav_y$ & $\sigmav_z$ & $-{\bf 1}$\
${\bf M}^{({\bf X}_4^+)}({\cal O})=$ & $i \sigmav_y$ & $-\sigmav_z$ & $-{\bf 1}$\
${\bf M}^{({\bf X}_1^-)}({\cal O})=$ & $ - \sigmav_x$ & $-{\bf 1}$ & $-{\bf 1}$\
${\bf M}^{({\bf X}_2^-)}({\cal O})=$ & $ \sigmav_x$ & ${\bf 1}$ & $-{\bf 1}$\
${\bf M}^{({\bf X}_3^-)}({\cal O})=$ & $-i\sigmav_y$ & $-\sigmav_z$ &${\bf 1}$\
${\bf M}^{({\bf X}_4^-)}({\cal O})=$ & $i \sigmav_y$ & $\sigmav_z$ &${\bf 1}$\
${\bf M}^{(\Gamma_5^-)}({\cal O})=$ & $i \sigmav_y$ & $\sigmav_x$ & ${\bf 1}$\
${\bf M}^{(\Gamma_5^+)}({\cal O})=$ & $i \sigmav_y$ & $\sigmav_x$ & $-{\bf 1}$\
\[h!\]
[|| c | c | c || c c c | c c c ||c c c | c c c ||c c c | c c c ||]{} Site & $n$ & ${\bf r}$ & & & & & &\
\
e & 1& $(0,0,\rho +1/2) $ & $u$ & 0 & 0 & 0 & $u$ & 0 & $a$ & $-a$ & 0 & $-a$ & $-a$ & 0 & 0 & 0 & 0 & 0 & 0 & 0\
e & 2& $(0,0,- \rho +1/2)$ &$u$&0&0 &0&$u$&0 & $a$ & $-a$ & 0 & $-a$ & $-a$ & 0 & 0 & 0 & 0 & 0 & 0 & 0\
b & 3& $(0,0,1/2) $ & $v$ & 0 & 0 & 0 & $v$ & 0 & $b$ & $-b$ & 0 & $-b$ & $-b$ & 0 & 0 & 0 & 0 & 0 & 0 & 0\
\
e & 4& $(0,0,\xi) $ & $w$ & 0 & 0 & 0 & $w$ & 0 & $c$ & $-c$ & 0 & $-c$ & $-c$ & 0 & 0 & 0 & 0 & 0 & 0 & 0\
e & 5& $(0,0,-\xi) $ & $w$ & 0 & 0 & 0 &$w$ & 0 & $c$ & $-c$ & 0 & $-c$ & $-c$ & 0 & 0 & 0 & 0 & 0 & 0 & 0\
\
a & 6 & $(0,0,0) $ & $x$ & 0 & 0 & 0 & $x$ & 0 & $d$ & $-d$ & 0 & $-d$ & $-d$ & 0 & 0 & 0 & 0 & 0 & 0 & 0\
e & 7 & $(0,0,\chi) $ & $y$ & 0 & 0 & 0 & $y$ & 0 & $e$ & $-e$ & 0 & $-e$ & $-e$ & 0 & 0 & 0 & 0 & 0 & 0 & 0\
e & 8 & $(0,0,-\chi) $ & $y$ & 0 & 0 & 0 &$y$ & 0 & $e$ & $-e$ & 0 & $-e$ & $-e$ & 0 & 0 & 0 & 0 & 0 & 0 & 0\
g & 9 & $(0,1/2,\tau)$ & $z_1$ & 0 & 0 & 0 & $z_2$ & $0$ & $0$ & $0$ & $f$ & $0$ & $0$ & $f$ & $-h$ & $-g$ & 0 & $-h$ & $g$ & 0\
g & 10 & $(0,1/2,-\tau)$ & $z_1$ &0 &0 & 0 & $z_2$ &0 & $0$ & $0$ & $-f$ & $0$ & $0$ & $-f$ & $-h$ & $-g$ & 0 & $-h$ & $g$ & 0 \
g & 11 & $(1/2,0,\tau)$ & $z_2$ &0 &0 &0 & $z_1$ &0 & $0$ & $0$ & $-f$ & $0$ & $0$ & $f$ & $g$ & $h$ & 0 & $-g$ & $h$ & 0\
g & 12 & $(1/2,0,-\tau)$ & $z_2$ &0 &0 &0 & $z_1$ &0 & $0$ & $0$ & $f$ & $0$ & $0$ & $-f$ & $g$ & $h$ & 0 & $-g$ & $h$ & 0\
We now discuss briefly the introduction of order parameters (OP’s). Starting from the high-temperature tetragonal, I4/mmm, structure, distortions of the irreps $\Gamma_5^-$, $X_3^-$ and $X_2^+$ are condensed as the temperature is lowered to eventually reach the low-symmetry low-temperature Cmc2$_1$ structure. A tabulation of these symmetry-adapted distortion vectors, $\Psi_n^{(X)}$, for irrep $X$ is given in Table \[basis\]. In the dominant component of the distortion of irrep $X_2^+$ the octahedra are rotated about the crystal $c$ axis like interlocking gears. The rotations in the upper ($z=\tau$) and lower ($z=-\tau$) planes of a bilayer are equal, so in the notation of Refs. and these rotations are both denoted $(0,0,\Theta)$. The dominant component of the distortion of irrep $X_3^-$ consists of 1) a rotation about the tetragonal \[1,0,0\] with the rotation alternating in sign as one moves along the \[1,0,0\] direction and 2) a rotation about the tetragonal \[0,1,0\] direction with the rotation alternating in sign as one moves along the \[0,1,0\] direction. Since this distortion is an even function of $z$, we identify this distortion as $(\Phi,\Phi,0)$ in the upper layer and $(-\Phi, -\Phi,0)$ in the lower layer.
If we assume that the distortions are small, then the crystal structure in any of the phases can be approximated as a distortion from the tetragonal structure which is written as linear combination of the distortions of the three irreps listed in Table \[basis\]: $$\begin{aligned}
\Phi &=& \sum_{m,X} Q_m^{(X)} \Psi_m^{(X)} \ , \nonumber\end{aligned}$$ where $\Psi_m^{(X)}$ is the 12 component vector of irrep $X$ as in Table \[basis\]. Since the notation to distinguish the various irreps which we will encounter is somehwat complicated, we summarize the notation in Table \[NOTE\]. If we regard the basis vectors $\Psi_m^{(X)}$ as fixed, then application of an operator ${\cal O}$ in the symmetry group of the tetragonal space group to $\Phi$ will induce a transformation of the OP’s as \[\] $$\begin{aligned}
{\cal O} Q_m^{(X)} &=& \sum_k M^{(X)}_{k,m}({\cal O}) Q_k^{(X)} \ ,
\nonumber \end{aligned}$$ where the representation matrices ${\bf M}^{(X)} ( {\cal O} )$ for the all the relevant operators ${\cal O}$ can be constructed from those given for the generators of the irrep $X$ given in Table \[IRREP\]. This equation defines the symmetry of the OP’s.
[|c|c c c c c|]{} For & Full & & Abbrev & & Abrev’\
$m=1,2$ & $Q_m^{(\Gamma_5^-)}$ & $\equiv$ & $Q_5^-(m)$ & $\equiv$ & $Q_5(m)$\
$m=1,2$ & $Q_m^{(\Gamma_5^+)}$ & $\equiv$ & $Q_5^+(m)$ & &\
$p<5$ & $Q_1^{(\Gamma_p^-)}$ & $\equiv$ & $Q_p^-(0)$ & &\
$p<5$ & $Q_1^{(\Gamma_p^+)}$ & $\equiv$ & $Q_p^+(0)$ & &\
& $[Q_1^{(\Gamma_5^-)}+Q_2^{(\Gamma_5^-)}]\sqrt 2$ & $\equiv$ & $Z^-$ & $\equiv$ & $Z$\
& $[Q_1^{(\Gamma_5^-)}-Q_2^{(\Gamma_5^-)}]\sqrt 2$ & $\equiv$ & $Y^-$ & $\equiv$ & $Y$\
& $[Q_1^{(\Gamma_5^+)}+Q_2^{(\Gamma_5^+)}]/\sqrt 2$ & $\equiv$ & $Z^+$ & &\
& $[Q_1^{(\Gamma_5^+)}-Q_2^{(\Gamma_5^+)}]/\sqrt 2$ & $\equiv$ & $Y^+$ & &\
& $Q_1^{(\Gamma_3^-)}$ & $\equiv$ & $Q_3^-(0)$ & $\equiv$ & $X$\
$m=1,2$ & $Q_m^{(X_2^+)}$ & $\equiv$ & $Q_2^+({\bf q}_m)$ & $\equiv$ & $Q_2({\bf q}_m)$\
$m=1,2$ & $Q_m^{(X_3^-)}$ & $\equiv$ & $Q_3^-({\bf q}_m)$ & $\equiv$ & $Q_3({\bf q}_m)$\
$m=1,2$, $p \not=2$ & $Q_m^{(X_p^+)}$ & $\equiv$ & $Q_p^+({\bf q}_m)$ & &\
$m=1,2$, $p \not=3$ & $Q_m^{(X_3^-)}$ & $\equiv$ & $Q_3^-({\bf q}_m)$ & &\
As explained in Ref. these distortions form two families, one of which involves wave vector ${\bf q}_1 \equiv (1/2,1/2,0)$ and the other of which involves wave vector ${\bf q}_2 \equiv (1/2,-1/2,0)$.\[\] The wave vectors $(1/2,\pm 1/2,0)$ correspond to a doubling of the size of the primitive unit cell. The first family is described by the three OP’s $Q_3^-({\bf q}_1)$, $Q_2^+({\bf q}_1)$, and $Z \propto Q_1^{(\Gamma_5^-)} +Q_2^{(\Gamma_5^-)}$, and the second by the OP’s $Q_3^-({\bf q}_2)$, $Q_2^+({\bf q}_2)$, and $Y\propto Q_1^{(\Gamma_5^-)} - Q_2^{(\Gamma_5^-)}$. In the absence of any external fields, when the sample orders, one expects an equal population of domains of the two families. However, it is possible to select a single family by invoking the coupling\[\] $$\begin{aligned}
V &=& a \epsilon_{xy} [Q_k({\bf q}_1)^2 - Q_k({\bf q}_2)^2] \ ,
\label{STRESSEQ} \end{aligned}$$ where $\epsilon_{xy}$ is a shear component of strain, and $k=2$ or 3. One can utilize this interaction by cooling the sample through $T_>$ in the presences of a shear stress conjugate to $\epsilon_{xy}$. This will preferentially select ${\bf q}_1$ or ${\bf q}_2$, depending on the sign of the stress and the sign of the constant $a$. It is possible that the selection of the wave vector can also be accomplished by cooling in an electric or magnetic field.\[\]
Our aim is describe the evolution of the cross section for a) zero wave vector photon absorption and b) Raman scattering in terms of the various OP’s.
The single photon absorption process results from the electromagnetic coupling $V_{\rm E} = - {\bf p} \cdot {\bf E}$, where ${\bf E}$ is the electric field of the incident electromagnetic wave and ${\bf p}$ is the dipole moment operator of the system. The symmetry indicated in Tables \[T4\] and \[IRREP\] indicates that in the tetragonal phase one has $$\begin{aligned}
p_x & = & a_x Q_5(1) \ , \ \
p_y = a_y Q_5(2) \ , \ \
p_z = a_z Q_3^-(0) \ ,
\nonumber \end{aligned}$$ with $a_y=a_x$. Note: in reality we ought to index the modes by an additional index to distinguish between different modes of the same symmetry. Thus, for instance, $$\begin{aligned}
p_x & = & \sum_{k=1}^{n(\Gamma_5^-)} a_{k,x} Q_5(1)_k \ ,
\label{EQPX} \end{aligned}$$ where $n(\Gamma_5^-)$ is the number of occurrences of $\Gamma_5^-$, which, in this case, is \[see Eq. (\[REP1EQ\]), below\] $n(\Gamma_5^-)=7$. For simplicity we will usually not display this index explicitly, but in all our results sums over such an index (or indices) are implied. We discuss the use of Eq. (\[EQPX\]) in Appendix \[APPA\]. The absorption cross section is proportional to the square of the scalar product of the polarization vector of the incident photon and the dipole moment vector for the irrep in question.\[\] (For the two dimensional irrep one has to sum the squares of the two scalar products.)
Likewise, the Raman scattering process results from the electromagnetic coupling $V_{\rm E} = - \sum_{\beta \gamma} \alpha_{\beta \gamma} E_\beta E_\gamma$, where $\alphav$ is the polarizability tensor, which depends on the phonon displacements as $$\begin{aligned}
\alpha_{\beta \gamma} &=& \alpha_{\beta \gamma}^{(0)} +
\sum_\tau \frac{\partial \alpha_{\beta \gamma}} {\partial Q_\tau} Q_\tau \ .
\nonumber \end{aligned}$$ We consider the cross section (or intensity) for a process in which the incident photon has an electric field vector in the $\beta$ direction and the scattered photon has an electric field vector in the $\gamma$ direction (we call this the $\beta$-$\gamma$ polarization) and a phonon $Q_\tau$ is created (or destroyed). The cross section for this process is proportional to the square of the tensor product of the Raman tensor $\partial \alpha_{\beta \gamma} / \partial Q_\tau$ with the polarizations of the incoming and outgoing photons.\[\] (For the two dimensional irreps one sums these squares over the two states of the irreps.) For both absorption and scattering we implicitly assume that the photon wave length is very large in comparison to the size of the unit cell.
Now we determine which modes (at zero wave vector) are allowed in the tetragonal lattice. Using standard methods we find that the reducible representations $\Gamma$ corresponding to the space of zero wave vector displacements of the Wyckoff orbits a, e, and g, are decomposed into their irreps as $$\begin{aligned}
\Gamma (a) &=& \Gamma_5^- + \Gamma_3^- \ , \nonumber \\
\Gamma (e) &=& \Gamma_5^- + \Gamma_3^- + \Gamma_5^+ + \Gamma_1^+ \ ,
\nonumber \\
\Gamma(g) &=& 2 \Gamma_5^- + \Gamma_4^- + \Gamma_3^- + 2 \Gamma_5^+
+ \Gamma_2^+ + \Gamma_1^+ \ ,
\label{TETIRREP} \end{aligned}$$ so that the reducible representation $\Gamma$ corresponding to the space of the vector displacements of all 12 ions in the primitive unit cell has the decomposition into irreps as $$\begin{aligned}
\Gamma &=& 2 \Gamma (a) + 3 \Gamma (e) + \Gamma(g) =
7 \Gamma_5^- + \Gamma_4^- \nonumber \\ &&
+ 6 \Gamma_3^-
+ 5 \Gamma_5^+ + 4 \Gamma_1^+ + \Gamma_2^+ \ .
\label{REP1EQ} \end{aligned}$$ The allowed absorption modes \[$x$ and $y$ from $\Gamma_5^-(0)$ and $z$ from $\Gamma_3^-(0)$\] are listed in Table \[T9a\]. Thus we find that in the tetragonal phase there are 20 absorption modes, 6 from $\Gamma_3^-$ which have $z$ polarization and 7 pairs of doubly degenerate modes (one with $x$ polarization and one with $y$ polarization) coming from $\Gamma_5^-$. Note that of these 20 absorption modes three are acoustic (zero frequency) and are not observed in IR experiments. There are also 15 Raman active modes, five nondegenerate modes coming from irreps $\Gamma_1^+$ and $\Gamma_2^+$ and 5 pairs of doubly degenerate modes coming from $\Gamma_5^+$. Because the lowest symmetry sites of the space group I4/mmm are not occupied in CMO or CTO, it happens that there are no modes for which the Raman polarization $xy$ is nonzero. Indeed, this result can be used as a signature of the tetragonal phase of CTO or CMO.
\[h!\]
[||c || c | c | c || c | c | c | c | c ||]{} & &\
Mode$_t$ & $Q_5^- (1)$ & $Q_5^- (2)$ & $Q_3^-(0)$& $Q_5^+(1)$ & $Q_5^+(2)$ & $Q_4^+(0)$ & $Q_2^+(0)$ & $Q_1^+ (0)$ \
Pol.$_t$ & $x$ & $y$ & $z$ & $zx$ & $zy$ & $xy$ & $x^2-y^2$ & $x^2+y^2,z^2$\
Mode$_O$ & $Y$ & $Z$ & $X$ & $Y^+$ & $Z^+$ & & &\
Pol.$_O$ & $Y$ & $Z$ & $X$ & $XY$ & $XZ$ & $(Y^2-Z^2)$ & $YZ$ & $X^2,(Y^2+Z^2)$\
RP(a) & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0\
RP(e) & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1\
RP(g) & 2 & 2 & 1 & 2 & 2 & 0 & 1 & 1\
TOT$^{\rm a}$ & 7 & 7 & 6 & 5 & 5 & 0 & 1 & 4\
a\) Here and below the absorption modes include an acoustic mode for each polarization.
Orthorhombic Space Group Cmcm (\# 63)
=====================================
Analysis based on Cmcm symmetry
-------------------------------
We now consider the modifications in the optical spectrum when we condense irrep $X_3^-$ at wave vector ${\bf q}_1 = (1/2,1/2,0)$ to go into the space group Cmcm. The transformation from tetragonal to orthorhombic coordinates is illustrated in Fig. \[TO\] and is given by $$\begin{aligned}
X_O &=& z_t , \hspace{0.4 in}
Y_O = \frac{x_t-y_t}{2} - \frac{1}{4} , \hspace{0.4 in}
\nonumber \\ Z_O &=& \frac{x_t+y_t}{2} - \frac{1}{4} ,
\label{TRANSEQ} \end{aligned}$$ where here and below we use lower case letters for tetragonal coordinates and capitals for orthorhombic coordinates. The O unit cell is twice as large as the t unit cell. That is why $x_t \pm y_t$ is divided by 2 rather than by $\sqrt 2$. In Table \[T7\] we give the character table for the little group of ${\bf q}=0$ for Cmcm as well as the characters for the reducible representation associated with the vector displacement over each of the Wyckoff orbits.
![\[TO\] (Color online) Tetragonal (large black arrows) and orthorhombic (small red arrows) axes.](FIG3.eps){width="5.0"}
[|| c || c || c | c | c | c || c | c | c | c ||c|c|c||]{} Class & Operator & $\Gamma_1^+$ & $\Gamma_2^+$ & $\Gamma_3^+$ & $\Gamma_4^+$ & $\Gamma_1^-$ & $\Gamma_2^-$ & $\Gamma_3^-$ & $\Gamma_4^-$ & a & e & g \
$E$& $(X,Y,Z)$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ & 6 & 12 & 24\
$2_X$ & $\ \ (X,\overline Y, \overline Z)\ \ $ & $1$ & $-1$ & $1$ & $-1$ & $1$ & $-1$ & $1$ & $-1$ & 0 & 0 & $-8$\
$m_Z$ & $(X,Y, \frac{1}{2} -Z)$ & $1$ & $-1$ & $-1$ & $1$ & $1$ & $-1$ & $-1$ & $1$ & 2 & 4 & 0\
$m_Y$ & $(X, \overline Y, \frac{1}{2} + Z)$ & $1$ & $1$ & $-1$ & $-1$ & $1$ & $1$ & $-1$ & $-1$ & 0 & 0 & 0\
${\cal I}$ & $(\overline X, \overline Y, \overline Z)$ & $1$ & $1$ & $1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & 0 & 0 & 0\
$m_X$ & $(\overline X, Y, Z)$ & $1$ & $-1$ & $1$ & $-1$ & $-1$ & $1$ & $-1$ & $1$ & 2 & 0 & 0\
$2_Z$ & $(\overline X, \overline Y, \frac{1}{2} + Z)$ & $1$ & $-1$ & $-1$ & $1$ & $-1$ & $1$ & $1$ & $-1$ & 0 & 0 & 0\
$2_Y$ & $(\overline X, Y, \frac{1}{2} - Z)$ & $1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & $1$ & $1$ & $-2$ & 0 & 0\
& 1 & $XZ$ & $YZ$ & $XY$ & $X$ & $Z$ & $XYZ$ & $Y$ & & &\
To find out how many Cmcm optically active modes there are, we need to convert the tetragonal coordinates into the orthorhombic coordinates but we do not need to explicitly include the distortion that makes the orthorhombic structure. From the parent tetragonal orbit a we have the orthorhombic orbit $$\begin{aligned}
{\bf r}_1 &=& (0,0,0)_t = (0, - \frac{1}{4} , - \frac{1}{4})_o \ , \nonumber \\
{\bf r}_2 &=& (1,0,0)_t = (0, \frac{1}{4}, \frac{1}{4})_o \ , \ \
\nonumber \end{aligned}$$ where the subscript t means tetragonal and the subscript o means Cmcm. We find that $$\begin{aligned}
\Gamma (a_{Cmcm}) &=& \Gamma_1^+ + \Gamma_3^+ + \Gamma_4^+
+ \Gamma_1^- + \Gamma_2^- + \Gamma_4^- \ .
\label{AEQ} \end{aligned}$$
For the parent tetragonal orbit e we have $$\begin{aligned}
{\bf r}_1 &=& (0,0,z)_t = (z, - \frac{1}{4}, - \frac{1}{4})_o \ , \nonumber \\
{\bf r}_2 &=& (1,0,z)_t = (z, \frac{1}{4}, \frac{1}{4})_o \ , \nonumber \\
{\bf r}_3 &=& (0,0,-z)_t = (-z, - \frac{1}{4}, -\frac{1}{4})_o \ , \nonumber \\
{\bf r}_4 &=& (1,0,-z)_t = (-z, \frac{1}{4}, \frac{1}{4})_o \ , \nonumber\end{aligned}$$ and we find that $$\begin{aligned}
\Gamma (e_{Cmcm}) &=& 2 \Gamma_1^+ + \Gamma_2^+ + \Gamma_3^+ + 2\Gamma_4^+
+ 2\Gamma_1^- + \Gamma_2^- \nonumber \\ && \ + \Gamma_3^- + 2 \Gamma_4^- \ .
\label{EEQ} \end{aligned}$$
Similarly, for the parent tetragonal orbit g we find that $$\begin{aligned}
\Gamma (g_{Cmcm}) &=& 2 \Gamma_1^+ + 4\Gamma_2^+ + 2\Gamma_3^+ + 4\Gamma_4^+
+ 2\Gamma_1^- \nonumber \\ && \ + 4\Gamma_2^- + 2\Gamma_3^- + 4 \Gamma_4^- \ .
\label{GEQ} \end{aligned}$$ Thus the decomposition of the reducible representation for the displacements of all ions in the unit cell is $$\begin{aligned}
\Gamma &=& 2 \Gamma(a_{Cmcm}) + 3 \Gamma(e_{Cmcm}) + \Gamma(g_{Cmcm})
\nonumber \\ &=& 10 \Gamma_1^+ +7 \Gamma_2^+ +7 \Gamma_3^+ +12 \Gamma_4^+
+10 \Gamma_1^- \nonumber \\ && \
+ 9 \Gamma_2^- +5 \Gamma_3^- +12 \Gamma_4^- \ . \nonumber\end{aligned}$$ So in the Cmcm phase we have 10 modes with $X_O$ polarization, 12 modes with $Y_O$ polarization, and 9 modes with $Z_O$ polarization, as given in Table \[T9\]. (The degeneracies of the tetragonal structure are removed in the Cmcm structure.) Results for the Raman-active modes are also given in Table \[T9\].
\[h!\]
[||c || c | c | c || c | c | c | c ||]{} & &\
IRREP & $\Gamma_1^-$ & $\Gamma_2^- $ & $\Gamma_4^-$ & $\Gamma_1^+$ & $\Gamma_2^+$ & $\Gamma_3^+$ & $\Gamma_4^+$ \
Polarization & $X_O$ & $Z_O$ & $Y_O$ & $X_O^2$, $Y_O^2,Z_O^2$ & $X_OZ_O$ & $Y_OZ_O$ & $X_OY_O$\
RP(a) & 1 & 1 & 1 & 1 & 0 & 1 & 1\
RP(e) & 2 & 1 & 2 & 2 & 1 & 1 & 2\
RP(g) & 2 & 4 & 4 & 2 & 4 & 2 & 4\
TOT & 10 & 9 & 12 & 10 & 7 & 7 & 12\
TET & 6 & 7 & 7 & 4 & 5 & 1 & 5\
Analysis based on lowering of tetragonal symmetry
-------------------------------------------------
We now discuss the phonon modes by analyzing the perturbative effect of distortions from the tetragonal parent structure. Compared to the previous analysis, this analysis provides information on how the intensity of the new Cmcm modes depends on the amplitude of the orthorhombic distortion from tetragonal symmetry. Let us suppose that $Q_3^-({\bf q}_1)$ has condensed due to the phase transition at temperature $T=T_>$ into the Cmcm phase. (Near this phase transition one can choose modes so that only a single mode of this symmetry is involved. As the temperature is lowered below $T_>$ other modes of this symmetry will progressively be introduced.) To see how the existence of the new order parameter can modify modes of the tetragonal system consider the Hamiltonian ${\cal H}_5$ for $Q_{5,n}^-$, $$\begin{aligned}
{\cal H}_5 &=& \frac{1}{2} \kappa_{5} \Biggl( Q_5(1)^2 + Q_5(2)^2
\Biggr) + \frac{1}{2} \kappa_{3-} [Q_{3}^-({\bf q}_1)]^2 \nonumber \\ && \
+ \sum_n b_n Q_5(1) Q_5(2) \langle Q_3^-({\bf q}_1) \rangle^n \ ,
\label{EQHAM} \end{aligned}$$ where the $\kappa$’s are stiffnesses and we put angled brackets around $Q_3({\bf q}_1)$ to indicate that these higher than quadratic terms give rise to effective quadratic terms written here when $Q_3^-({\bf q}_1)$ (whose value minimizes the free energy) is regarded as the constant $\langle Q_3^- ({\bf q}_1)\rangle$. To make the above Hamiltonian invariant under all the operations of the parent tetragonal system, we would have to include analogous terms involving ${\cal R}_4 Q_3^-({\bf q}_1) = Q_3^-({\bf q}_2)$, where ${\bf q}_2 = (1/2,-1/2,0)$. However, since we are considering a scenario in which $Q_3^-({\bf q}_2)$ is zero, we do not need to write these terms here. Since $b_1=0$ due to wave vector conservation, Eq. (\[EQHAM\]) is $$\begin{aligned}
{\cal H}_5 &=& \frac{1}{2} \Biggl( \kappa_5 - b_2
\langle Q_3^-({\bf q}_1) \rangle^2 \Biggr) \frac{[Q_5(1) -
Q_5(2)]^2} {2} \nonumber \\ && \ + \frac{1}{2} \Biggl( \kappa_5 + b_2
\langle Q_3^-({\bf q}_1) \rangle^2 \Biggr) \frac{[Q_5(1) +
Q_5(2)]^2} {2} \nonumber \\ & \equiv &
\frac{1}{2}\kappa_Y Y^2 + \frac{1}{2}\kappa_Z Z^2 \ ,
\label{H5EQ}\end{aligned}$$ where $Y= [Q_5(1) - Q_5(2)]/ \sqrt 2$, and $Z = [Q_5(1) + Q_5(2)]/ \sqrt 2$. This notation (of Table \[NOTE\]) is motivated by the labelling of the axes in the orthorhombic phase (see Eq. (\[TRANSEQ\]) and Fig. \[TO\]). We therefore see that the presence of a nonzero value of $\langle Q_3^-({\bf q}_1)\rangle$ lifts the degeneracy in energy between the two $Q_5^-$ modes and leads to the proper linear combinations $Y$ and $Z$ as the new normal modes of the Cmcm structure. The analogous splitting occurs in the same way for the $Q_5^+$ modes and leads to the redefinition of normal modes as $Y^+ = [Q_5^+(1) - Q_5^+(2)]/\sqrt 2$ and $Z^+ = [Q_5^+(1) + Q_5^+(2)]/\sqrt 2$.
### Induced Absorption Modes
We next investigate the mixing of modes due to the nonzero value of $\langle Q_3^-({\bf q}_1)\rangle$. The Hamiltonian we consider for the mixing of absorptive modes is ${\cal H}_3 + {\cal H}_5 +V_5$, where $$\begin{aligned}
{\cal H}_3 &=& \frac{1}{2} \kappa_3 \sum_k Q_3^- ({\bf q}_k)^2 \ ,
\nonumber \end{aligned}$$ $$\begin{aligned}
V_5 &=& \frac{1}{2} \sum_n \Biggl( \kappa_{U_n} U_n^2
+ \kappa_{V_n} V_n^2 + \kappa_{W_n} W_n^2 \Biggr) \nonumber \\
&& \ + \sum_n \langle Q_3^-({\bf q}_1) \rangle^n
\Biggl( a^{(n)} U_n Z + b^{(n)} V_n Y
\nonumber \\ && \ + c^{(n)} W_n Q_3^-(0) \Biggr) \ ,
\label{VVEQ} \end{aligned}$$ where $U_n$, $V_n$, $W_n$ are single phonon operators (to be determined) which, as we will see, become optically active by virtue of their coupling to the optically active $Z$, $Y$, or $Q_3^-(0)$ modes by the interaction $V_5$ at order $\langle Q_3^-({\bf q}_1) \rangle^n$. To identify these newly active modes, we analyze the symmetry of the coupling terms in Eq. (\[VVEQ\]). Note that wave vector conservation indicates that terms in Eq. (\[VVEQ\]) with $n$ even involve newly active modes at zero wave vector and those with $n$ odd involve newly active zero wave vector modes which, before the distortion, were at wave vector ${\bf q}_1$ in the tetragonal Brillouin zone. The character table for the ${\bf X}$ wave vectors was given in Table \[IRREP\].
To see more specifically what the Hamiltonian of Eqs. (\[H5EQ\]) and (\[VVEQ\]) implies, consider only the terms involving $U_n$ and $Z$: $$\begin{aligned}
{\cal H}_X &=& \frac{1}{2}\kappa_Z Z^2 + \frac{1}{2}
\sum_n \kappa_{U_n} U_n^2 \nonumber \\ && \ + \sum_n
a^{(n)} \langle Q_3^-({\bf q}_1) \rangle^n U_n Z \ .
\label{EQ31} \end{aligned}$$ In carrying out perturbation theory relative to the tetragonal phase we use the correct linear combinations $Z$ and $Y$. At the moment we ignore the perturbation in the energy due to this interaction which was addressed by Eq. (\[H5EQ\]). Our interest here lies in the mixing of modes. When the quadratic Hamiltonian of Eq. (\[EQ31\]) is diagonalized, the new modes ${\tilde U}_n$ perturbed through their coupling to $Z$ will now be $$\begin{aligned}
\tilde U_n &=& U_n - \frac{ a^{(n)} \langle Q_3^-({\bf q}_1)\rangle^n
Z} {\kappa_{U_n} - \kappa_Z} \ .
\label{MIX} \end{aligned}$$ By analogy we will also have new modes $$\begin{aligned}
\tilde V_n &=& V_n - \frac{ b^{(n)} \langle Q_3^-({\bf q}_1)\rangle^n
Y} {\kappa_{V_n} - \kappa_Y} \nonumber\end{aligned}$$ $$\begin{aligned}
\tilde W_n &=& W_n - \frac{ c^{(n)} \langle Q_3^-({\bf q}_1)\rangle^n
Q_3^-(0)} {\kappa_{W_n} - \kappa_{3}} \ . \nonumber \end{aligned}$$ What this means is that the modes $U_n$, $V_n$, and $W_n$ (which we will identify in a moment) are optically active because they are linear combinations with the optically active modes $Z_k$, $Y_k$, and $Q_3(0)_k$, respectively, whose polarizations they inherit. For quantitative work (see Appendix A), it is necessary to keep track of the coupling to the different optically active modes, $Z_k$, $Y_k$, and $Q_3^-(0)_k$. For simplicity, since we are mainly interested in which modes are newly induced, we need not keep the sum over $k$.
Near $T_>$, $\langle Q_3({\bf q}_1) \rangle$ will be proportional to $(T_>-T)^\beta$, where $\beta=1/2$ in mean field theory. Thus the temperature dependence of terms with different powers of this variable will be distinguishably different. Analogous terms for the scenario when the wave vector ${\bf q}_2$ has condensed can be obtained by applying the operator ${\cal R}_4$ to ${\cal H}_3 + {\cal H}_5+V_5$. Now we study how symmetry restricts the $U$’s, $V$’s and $W$’s of Eq. (\[VVEQ\]). Since ${\cal R}_4$ is not a relevant symmetry operation, we require ${\cal H}_3 + {\cal H}_5 +V_5$ to be invariant under ${\cal I}$, $m_d$, and $m_z$. If $n$ is odd (even), then $U_n$, $V_n$, and $W_n$ are at wave vector ${\bf q}_1$ (wave vector zero). To be invariant under inversion, $U_n$, $V_n$, and $W_n$ have to have parity $(-1)^{n+1}$ under inversion. Also $U_n$ and $V_n$ must be even under $m_z \equiv (xy\overline z)$ and $W_n$ must be odd under $(xy\overline z)$. Finally we have to consider the effect of $m_d=(yxz)$. From Tables \[T4\] and \[IRREP\] we find that $$\begin{aligned}
m_d Y &=& - Y \ , \ \
m_d Z = Z \ , \nonumber \\
m_d Q_3^-(0) &=& Q_3^-(0) \ , \ \
m_d Q_3^-({\bf q}_1) = - Q_3^-({\bf q}_1) \ . \nonumber\end{aligned}$$ For $V_5$ to be invariant under $m_d$, we therefore must have $$\begin{aligned}
m_d U_n & = & (-)^{n} U_n \ , \ \ \ \
m_d V_n = (-)^{n+1} V_n \ , \nonumber \\
m_d W_n &=& (-)^{n} W_n \ .
\label{EQMDV} \end{aligned}$$
Note that all the above constraints depend on $n$ only through $(-1)^n$, so that to find the leading temperature dependences just below $T_>$, we only need to consider two distinct cases, $n=1$ and $n=2$. First consider $n=1$. Then $$\begin{aligned}
{\cal I} U_1 ({\bf q}_1) &=& U_1({\bf q}_1) \ , \ \ \ \
m_z U_1({\bf q}_1) = U_1({\bf q}_1) \ , \nonumber \\
m_d U_1({\bf q}_1) &=& -U_1({\bf q}_1) \ , \ \ \ \
{\cal I} V_1({\bf q}_1) = V_1({\bf q}_1) \ , \nonumber \\
m_z V_1({\bf q}_1) &=& V_1({\bf q}_1) \ , \ \ \ \
m_d V_1({\bf q}_1) = V_1({\bf q}_1) \ , \nonumber \\
{\cal I} W_1 ({\bf q}_1) &=& W_1({\bf q}_1) \ , \ \ \ \
m_z W_1({\bf q}_1) = - W_1({\bf q}_1) \ , \nonumber \\
m_d W_1({\bf q}_1) &=& - W_1({\bf q}_1) \ , \nonumber\end{aligned}$$ so that $$\begin{aligned}
U_1 &=& \alpha_1Q_2^+({\bf q}_1) \ , \ \
V_1 = \beta_1 Q_1^+({\bf q}_1) \ , \ \
W_1 = \gamma_1 Q_4^+({\bf q}_1) \ , \nonumber\end{aligned}$$ where $\alpha_1$, $\beta_1$, and $\gamma_1$ are constants whose values are not fixed by symmetry. The meaning of $U_1$ is that because of the allowed coupling between $Z$ and [*any*]{} operator having the symmetry of $Q_2^+({\bf q}_1)$ this coupling will render all such modes optically active with polarization $Z$ inherited from coupling to $Z$. This coupling can be analoyzed for each Wykoff orbit, as we do in Table \[T14\]. To summarize: the coupling at first order in $\langle Q_3^-({\bf q}_1) \rangle$, which we denote $V_c^{(1)}$ \[and later the superscript 2 indicates second order in $\langle Q_3^-({\bf q}_1) \rangle$\] is $$\begin{aligned}
V_c^{(1)} &=& \langle Q_3({\bf q}_1) \rangle \Biggl[
\sum_{k=1}^{n(X_2^+)]} a^{(1)}_k Z Q_2^+({\bf q}_1)_k \nonumber \\ && \ +
\sum_{k=1}^{n(X_1^+)]} b^{(1)}_k Y Q_1^+({\bf q}_1)_k \nonumber \\
&& \ + \sum_{k=1}^{n(X_4^+)]} c^{(1)}_k Q_3^-(0) Q_4^+({\bf q}_1)_k
\Biggr] \ .
\label{VC1EQ} \end{aligned}$$ The fact that there are more than one occurrence of $Q_3({\bf q}_1)$ affects the intensity but not the number or symmetry of new modes, $Q_2^+({\bf q}_1)_k$, $Q_1^+({\bf q}_1)_k$, and $Q_4^+({\bf q}_1)_k$. A more complete treatment of this coupling is given in Appendix A.
We now obtain the $n[\Gamma ({\bf q}_1)]$ required in Eq. (\[VC1EQ\]). Using standard methods we find for wave vector ${\bf q}_1$ that the reducible representations of the displacement vectors over each orbit have the decomposition $$\begin{aligned}
\Gamma [a({\bf q}_1)] &=& X_2^-({\bf q}_1) + X_3^-({\bf q}_1) + X_4^-({\bf q}_1)
\ , \nonumber \\
\Gamma [e({\bf q}_1)] &=& X_2^-({\bf q}_1) + X_3^-({\bf q}_1) + X_4^-({\bf q}_1)
\nonumber \\ && \ + X_1^+({\bf q}_1) + X_3^+({\bf q}_1) + X_4^+({\bf q}_1)
\nonumber \\
\Gamma [g({\bf q}_1)] &=& 2X_1^-({\bf q}_1) + 2X_2^-({\bf q}_1)
+ X_3^-({\bf q}_1) \nonumber \\ && \ + X_4^-({\bf q}_1)
+ 2X_1^+({\bf q}_1) + 2X_2^+({\bf q}_1) \nonumber \\ && + X_3^+({\bf q}_1)
+ X_4^+({\bf q}_1) \ ,
\label{REP2EQ} \end{aligned}$$ Therefore, for example, $n[X_4^+({\bf q}_1)]$ assumes the values 0, 1, 1 for orbits a, e, and g, respectively, leading to the results in the second column of Table \[T14\].
\[h!\]
[||c || c | c | c | c || c | c | c | c | c ||]{} & &\
Mode & $Q_4^+({\bf q}_1)$ & $Q_2^+({\bf q}_1) $ & $Q_1^+ ({\bf q}_1)$ & $Q_2^-(0)$ & $Q_1^-({\bf q}_1)$ & $Q_2^-({\bf q}_1)$ & $Q_3^-({\bf q}_1)$ & $Q_4^-({\bf q}_1)$ & $Q_3^+(0)$ \
Pol. & $X$ & $Z$ & $Y$ & $X$ & $XZ$ & $XY$ & $X^2$, $Y^2$, $Z^2$ & $YZ$ & $Y^2-Z^2$\
Order & $\sigma$ & $\sigma$ & $\sigma$ & $\sigma^2$ & $\sigma$ & $\sigma$ & $\sigma$ & $\sigma$ &$\sigma^2$\
RP(a) & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0\
RP(e) & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0\
RP(g) & 1 & 2 & 2 & 0 & 2 & 2 & 1 & 1 & 0\
TOT & 4 & 2 & 5 & 0 & 2 & 7 & 6 & 6 & 0\
We also show figures of the induced absorption modes in Fig. \[M1\]. The easiest way to generate a mode of a given symmetry ($X_k^+$ or $X_k^-$) is to consider each Wyckoff orbit in turn. For a given orbit assign one site an arbitrary displacement vector $(u_x, u_y, u_z)$, then generate the displacements of the other sites using the characters of Table \[IRREP\]. In doing this one will find that one or more of the initial displacements can not be nonzero when one includes the restriction that the wave function have the given wave vector, in this case ${\bf q}_1$.
![\[M1\] (Color online) Modes of irrep $X_2^+$, left and $X_1^+$, center, and $X_4^+$, right, all at tetragonal wave vector ${\bf q}_1$. These modes become absorption-active in the Cmcm phase. The arrows represent displacements in the $x$-$y$ plane and $+$ and $-$ are the signs of displacements along the $z$-axis. Here the thick-lined squares delineate the equatorial oxygens (at $z= \pm \tau$) and the uncircled arrows and $\pm$’s (if any) in the center of the square indicate the displacement of the top apical e-site oxygens in the upper layer of the bilayer at $z = \chi$ and the circled arrows or $\pm$’s (if any) refer to the bottom apical oxygens in the lower layer of the bilayer at $z=-\chi$. All the e-site ions move along the same axis as the apical oxygens, but with independent amplitudes. There are two modes of $X_2^+$ symmetry corresponding to independently choosing the amplitudes of the large black and small red arrows. If the large black arrow is dominant, this mode is the $(0,0,\Theta)$ rotation about the tetragonal axis. There are five $X_1^+$ modes because the displacements of the apical oxygen, the other two e-site ions (not shown), and the two components of the equatorial oxygens are all independent parameters. There are four $X_4^+$ modes because the tilting, the displacements of the apical oxygens and of the two other e-site ions are independent parameters. (Some of the modes, such as $Q_1^+({\bf q}_1)$ and $Q_4^+({\bf q}_4)$, which grossly distort the octahedra will have relatively high frequencies.) Without the admixtures of order $\langle Q_3^-({\bf q}_1)\rangle$ all these modes are of the wrong symmetry to be optically active. However, the admixing, [*e. g.*]{} Eq. (\[VC1EQ\]) in the Cmcm phase makes these modes optically active.](FIG4.eps){width="8.6"}
Now we turn to the case $n=2$, for which $$\begin{aligned}
{\cal I} U_2 (0) &=& - U_2(0) \ , \ \
m_z U_2(0) = U_2(0) \ , \nonumber \\
m_d U_2(0) &=& U_2(0) \ , \ \
{\cal I} V_2(0) = - V_2(0) \ , \nonumber \\
m_z V_2(0) &=& V_2(0) \ , \ \ \ \
m_d V_2(0) = - V_2(0) \ , \nonumber \\
{\cal I} W_2 (0) &=& - W_2(0) \ , \ \ \ \
m_z W_2(0) = - W_2(0) \ , \nonumber \\
m_d W_2(0) &=& W_2(0) \ . \nonumber\end{aligned}$$ The solutions for $U_2(0)$ and $V_2(0)$ do not produce new modes but instead add to the intensity of already existing modes. New modes arise from $W_2(0)=Q_2^-(0)$, so that the relevant second order coupling perturbation is $$\begin{aligned}
V_c^{(2)} &=& \langle Q_3^-({\bf q}_1) \rangle^2 \sum_{k=1}^{n[\Gamma_2^-(0)]}
c^{(2)}_k Q_3^-(0) Q_2^-(0)_k \ , \nonumber\end{aligned}$$ where ${n[\Gamma_2^-(0)]}$ is given in Eq. (\[REP1EQ\]). We summarize our results in Table \[T14\]. Note that all the newly induced modes have absorption intensity proportional to $\langle Q_3^-({\bf q}_1)\rangle^2$ only because the low symmetry sites of I4/mmm (which allow the OP $Q_2^-(0)$ to appear) are not occupied for CMO or CTO: $n[\Gamma_2^-(0)]=0$ for CMO or CTO. But an argument equivalent to what we have given here would be needed to establish this result.
### *Induced Raman Modes*
Here we consider the admixing of the Raman active modes (see Table \[T4\]) from irreps $\Gamma_n^+$, with $n=1,2,4,5$: $$\begin{aligned}
V_C &=& \sum_n \langle Q_3^-({\bf q}_1 \rangle^n \Bigg[
a_n Z^+ T_n + b_n Q_4^+(0) U_n \nonumber \\ && \ + c_n Q_2^+(0) V_n
+ d_n Q_1^+(0) W_n + e_n Y^+ S_n \Biggr] \ , \nonumber\end{aligned}$$ where $T_n$, $U_n$, $V_n$, $W_n$, and $S_n$ are the modes which are Raman active by virtue of their coupling proportional to $\langle Q_3^- \rangle({\bf q}_1)^n$ to modes of the tetragonal system which are Raman active. For $n=1$ invariance with respect to ${\cal I}$, $m_d$, and $m_z$ leads to $$\begin{aligned}
{\cal I} S_1({\bf q}_1) &=& - S_1({\bf q}_1) \ , \ \ \ \
m_d S_1({\bf q}_1) = S_1({\bf q}_1) \ , \nonumber \\
m_z S_1({\bf q}_1) &=& - S_1({\bf q}_1) \ , \ \ \ \
{\cal I} T_1({\bf q}_1) = - T_1({\bf q}_1) \ , \nonumber \\
m_d T_1({\bf q}_1) &=& - T_1({\bf q}_1) \ , \ \ \ \
m_z T_1({\bf q}_1) = - T_1({\bf q}_1) \nonumber \\
{\cal I} U_1({\bf q}_1) &=& - U_1({\bf q}_1) \ , \ \ \ \
m_d U_1({\bf q}_1) = - U_1({\bf q}_1) \ , \nonumber \\
m_z U_1({\bf q}_1) &=& U_1({\bf q}_1) \ , \ \ \ \
{\cal I} V_1({\bf q}_1) = - V_1({\bf q}_1) \ , \nonumber \\
m_d V_1({\bf q}_1) &=& V_1({\bf q}_1) \ , \ \ \ \
m_z V_1({\bf q}_1) = V_1({\bf q}_1) \ , \nonumber \\
{\cal I} W_1({\bf q}_1) &=& - W_1({\bf q}_1) \ , \ \ \ \
m_d W_1({\bf q}_1) = - W_1({\bf q}_1) \ , \nonumber \\
m_z W_1({\bf q}_1) &=& W_1({\bf q}_1) \ . \nonumber\end{aligned}$$ So $$\begin{aligned}
S_1({\bf q}_1) &=& Q_2^- ({\bf q}_1) \ , \
T_1({\bf q}_1) = Q_1^-({\bf q}_1) \ , \ U_1({\bf q}_1)
= Q_3^-({\bf q}_1)\ , \nonumber \\
V_1({\bf q}_1)&=&Q_4^-({\bf q}_1)\ ,
\ \ \ \ W_1({\bf q}_1)=Q_3^-({\bf q}_1) \ . \nonumber\end{aligned}$$ Thus $$\begin{aligned}
V_C^{(n=1)} &=& \langle Q_3^-({\bf q}_1) \rangle \Biggl[
\sum_{k=1}^{n(X^-)} a^{(1)}_k Z^+ Q_1^-({\bf q}_1)_k
\nonumber \\ && \
+ \sum_{k=1}^{n(X_3^-)} b^{(1)}_k Q_4^+(0) Q_3^-({\bf q}_1)_k
\nonumber \\ && \
+ \sum_{k=1}^{n(X_4^-)} c^{(1)}_k Q_2^+(0) Q_4^-({\bf q}_1)_k
\nonumber \\ && \
+ \sum_{k=1}^{n(X_3^-)} d^{(1)}_k Q_1^+(0) Q_3^-({\bf q}_1)_k
\nonumber \\ && \
+ \sum_{k=1}^{nX_2^-)} e^{(1)}_k Y^+ Q_2^-({\bf q}_1)_k
\Biggr] \ , \nonumber\end{aligned}$$ where the sums over the number of appearances of the Raman active tetragonal modes is omitted, in analogy with Eq. (\[VC1EQ\]). The fact that $Q_4^+(0)$ and $Q_1^+(0)$ are both coupled to $X_3^-({\bf q}_1)_k$ means that each $X_{3,k}$ mode has two independent admixtures, one, $Q_4^+(0)$, giving a contribution to the Raman tensor $a(\hat i \hat j + \hat j \hat i)= a(\hat K \hat K - \hat J \hat J)$ and the other, $Q_1^+(0)$, giving a contribution to the Raman tensor $b\hat k \hat k + c(\hat i \hat i + \hat j \hat j)
= b \hat I \hat I + c(\hat J \hat J + \hat K \hat K)$. As a result, the $X_{3,k}^-({\bf q}_1)$ modes have a Raman tensor $b\hat I \hat I + (c-a)(\hat J \hat J)
+ (c+a)\hat K \hat K$, so that $X_{3,k}^-({\bf q}_1)$ has a Raman polarization we denote $(X^2,Y^2,Z^2)$.
For $n=2$ invariance with respect to ${\cal I}$, $m_d$, and $m_z$ leads to $$\begin{aligned}
{\cal I} S_2(0) &=& S_2(0) \ , \ \ m_d S_2(0) = -S_2(0) \ , \nonumber \\
m_z S_2(0) &=& -S_2(0) \ , \ \ {\cal I} T_2(0) = T_2(0) \ , \nonumber \\
m_d T_2(0) &=& T_2(0) \ , \ \ m_z T_2(0) = - T_2(0) \nonumber \\
{\cal I} U_2(0) &=& U_2(0) \ , \ \ m_d U_2(0) = U_2(0) \ , \nonumber \\
m_z U_2(0) &=& U_2(0) \ , \ \ {\cal I} V_2(0) = V_2(0) \ , \nonumber \\
m_d V_2(0) &=& -V_2(0) \ , \ \ m_z V_2(0) = V_2(0) \ , \nonumber \\
{\cal I} W_2(0) &=& W_2(0) \ , \ \ \ \ m_d W_2(0) = W_2(0) \ , \nonumber \\
m_z W_2(0) &=& W_2(0) \ . \nonumber\end{aligned}$$ So $$\begin{aligned}
S_2 &=&Y_O^+(0) \ , \ \ T_2= Z^+\ , \nonumber \\
U_2 &=& \ \alpha Q_1^+(0) + \beta Q_4^+ (0) \ , \nonumber \\
V_2 &=& \ \alpha' Q_2^+(0) +\beta' Q_3^+ (0) \ , \nonumber \\
W_2 &=& \ \alpha^{\prime \prime} Q_1^+ (0) + \beta^{\prime \prime} Q_4^+ (0)
\ . \nonumber \end{aligned}$$ The only new irrep here is $\Gamma_3^+ (0)$ with OP $Q_3^+(0)$. So we have the results of Table \[T14\]. As was the case for the absorption cross section, all the newly induced modes have their Raman scattering cross section proportional to $\langle Q_3^-({\bf q}_1)\rangle^2$ only because the low symmetry sites of I4/mmm (which allow the OP $Q_3^+(0)$ to appear) are not occupied for CMO or CTO. We emphasize that this result is not a general result for the space group I4/mmm distorting into space group Cmcm.
In Figs. \[M2\] and \[M3\] we show the new Raman modes in the Cmcm phase.
![\[M2\] (Color online) As Fig. \[M1\] except for modes of irrep $X_1^-$, left and $X_2^-$, right, both at tetragonal wave vector ${\bf q}_1$. These are the modes which become Raman active (see Table \[T14\]) due to admixtures of tetragonal Raman active zone center modes. In mode $X_1^-$ the sites on the $z$ axis do not move, so there are two modes of this symmetry corresponding to the choice of the amplitudes of the $x$ and $y$ displacements. There are 7 modes of symmetry $X_2^-({\bf q}_1)$ because there are two ways to choose the amplitude of the arrows (the displacements of the equatorial oxygens) and one way to independently choose the amplitudes of the displacements along the $z$-axis for each of the 2 a orbits and for each of the 3 e orbits of such sites. These breathing modes grossly distort the octahedra and therefore have relatively high frequency.](FIG5.eps){width="8.6"}
![\[M3\] (Color online) As Fig. \[M2\] except for modes of irrep $X_3^-$, left and $X_4^-$, right, both at tetragonal wave vector ${\bf q}_1$. The equatorial oxygens (g orbits) tilt around the \[110\] direction for irrep $X_3^-({\bf q}_1)$ and around the \[1$\overline 1$0\] direction for $X_4^-({\bf q}_1)$, in each case with a single amplitude. The other ions on the $z$-axis move along the \[1$\overline 1$0\] axis for $X_3^-({\bf q}_1)$ and along the \[110\] axis for $X_4^-({\bf q}_1)$, in each case with an independent amplitude. Thus for both irreps we have 6 modes (one for each orbit). In the notation of Refs. and $Q_3^-({\bf q}_1)$ is a $(\Phi,\Phi,0)$ rotation and $Q_4^-({\bf q}_1)$ is a $(\Phi,-\Phi,0)$ rotation. The wave vector regulates how one bilayer is structured relative to the adjacent bilayer. So these two modes have slightly different frequencies because although their intrabilayer interaction energies are the same, their interbilayer interaction energies are different.](FIG6.eps){width="8.6"}
### Temperature Dependence of Mode Energies and Absorption Intensities
Near the phase transition at temperature $T=T_>$ where the order parameter $\langle Q_3^-({\bf q}_1) \rangle$ becomes nonzero one has $$\begin{aligned}
\omega_n(T) &=& \omega_n^{(0)} (T) + \delta \omega_n (T) \ , \nonumber \\
I_n(T) &=& I_n^{(0)} (T) + \delta I_n (T) \ , \ \ \ \
\label{T3EQ} \end{aligned}$$ where the first term is the smooth temperature dependence and the second term is the anomalous term at the phase transition which is only nonzero for $T<T_>$ and will involve powers of $\langle Q_3^-({\bf q}_1) \rangle \equiv \sigma \sim |T_>-T|^\beta$, where the order parameter exponent $\beta$ is 1/2 in mean field theory, but fluctuations will cause it to be less than 1/2.
First consider the mode energies. For all the [*nondegenerate*]{} ${\bf q}=0$ modes, the perturbation of the form given in Eq. (\[VVEQ\]), will lead to an energy shift from terms with $n=1$, which, in second order perturbation theory, gives a contribution to $\delta \omega (T)$ of order $\sigma^2 \sim|T_>-T|^{2 \beta}$. For the [*degenerate*]{} $\Gamma_5^\pm(0)$ modes, $\delta \omega (T)$ has a contribution from first order perturbation theory, but in that case, since $b_1$ of Eq. (\[EQHAM\]) vanishes, the contribution to $\delta \omega (T)$ is again of order $\sigma^2 \sim |T_>-T|^{2 \beta}$. These results are shown in the left-hand panels of Fig. \[MODES\]. It should also be noted that the mode energies can be modified by quartic terms in the Hamiltonian which we have not so far considered and which are of the form\[\] $$\begin{aligned}
V_4 &=& \sum_n d_n U_n^2 Q_3^-({\bf q}_1)^2 \nonumber\end{aligned}$$ which will also lead to a contribution to $\delta \omega (T)$ of order $\sigma^2 \sim |T_>-T|^{2 \beta}$. It is plausible, but not a certainty, that the sign of the anomalous contribution would cause a stiffening of the modes as the temperature is reduced as is depicted in lower left-hand panel of Fig. \[MODES\]. If $V_4$ were to be neglected, then one would have a direct correlation between $\delta I_n(T)$ and $\delta \omega_n(T)$.\[\]
Now consider the mode intensities in the presence of admixtures written in Eq. (\[MIX\]). The standard scenario in perturbation theory is that a perturbation in first order introduces new components into the wave function without changing the unperturbed component of the wave function. In second order perturbation theory the amplitude of the unperturbed component of the wave function can be modified. Here, this tells us that if the mode $U_n$ is already dipole allowed, then the intensity will be modified at order $\sigma^2$. If the mode $U_n$ is not allowed in the absence of the perturbation then its intensity will be proportional to the square of the amplitude of the admixture. Furthermore, new modes that appear at first order in $\sigma$ will have intensity of the form of Eq. (\[T3EQ\]) with $I_n^{(0)}(T)=0$ and $\delta I (T) \sim \sigma^2 \sim |T_>-T|^{2 \beta}$. New modes that only appear at second order in $\sigma$ will have intensity of the form of Eq. (\[T3EQ\]) with $I_n^{(0)}(T)=0$ and $\delta I (T) \sim \sigma^4 \sim |T_>-T|^{4 \beta}$. From Table \[T14\] we see that for CMO or CTO there are no such modes that only appear at order $\sigma^2$. This is a special property of the RP system. Had we had a more general tetragonal system with low symmetry sites, then there would be modes which only appear at order $\sigma^2$ and which would have $\delta I (T) \sim \sigma^4 \sim |T_>-T|^{4 \beta}$. Our results are summarized in the right-hand panels of Fig. \[MODES\].
![\[MODES\] (Color online) RP modes. Left panels: mode energy versus temperature, where $T_>$ is the temperature at which $\langle Q_3^-({\bf q}_1)\rangle$ becomes nonzero. Right panels: the mode intensity versus temperature. Upper left: removal of degeneracy of the $\Gamma_5^\pm$ modes. Lower left: temperature dependence of a nondegenerate mode energy. Upper right: temperature dependence of the intensity of a nondegenerate mode which is absorption allowed in the tetragonal phase. Lower right: temperature dependence of nondegenerate mode intensity for modes which have zero intensity in the tetragonal phase. In all cases, the discontinuity in slope is drawn assuming the mean-field result, $\beta=1/2$, for the order parameter exponent. Otherwise, the lower temperature additional contribution is of order $|T_>-T|^{2 \beta}$.](FIG7.eps){width="8.6"}
Cmc2$_1$ Phase
==============
Analysis based on Cmc2$_1$ symmetry
-----------------------------------
Since Cmc2$_1$ does not have a center of inversion symmetry, parity is not a good quantum number and Raman and absorption modes are mixed. In Cmc2$_1$ the general orbit is half as large as for Cmcm. However, the unit cell then contains twice as many orbits. The sites in Cmc2$_1$ are given in Table \[T15\] and the character table for ${\bf q}=0$ is given in Table \[T16\].
\[h!\]
[||c c ||]{} $E=(x,y,z)$ & $2_z=(\overline x , \overline y, 1/2+z)$ \
$m_x=(\overline x,y,z)$ & $m_y=(x, \overline y, 1/2+z)$ \
[|| c || c | c | c | c ||c | c | c ||]{} Operator & $\Gamma_1$ & $\Gamma_2$ & $\Gamma_3$ & $\Gamma_4$ & a & e & g \
$(X,Y,Z)$ & $1$ & $1$ & $1$ & $1$ & 6 & 12 & 24\
$(\overline X , Y,Z)$ & $1$ & $1$ & $-1$ & $-1$ & 2 & 0 & 0\
$(\overline X , \overline Y ,1/2+Z)$ & $1$ & $-1$ & $1$ & $-1$ & 0 & 0 & 0\
$(X , \overline Y , 1/2+Z)$ & $1$ & $-1$ & $-1$ & $1$ & 0 & 0 & 0\
${\bf P}$ & $a\hat K$ & $b\hat J$ & None & $c\hat I$ & & &\
Using the data in Table \[T16\] we find that $$\begin{aligned}
\Gamma(a_{Cmc2_1}) &=& 2 \Gamma_1 + 2\Gamma_2 + \Gamma_3 + \Gamma_4\ ,
\nonumber \\
\Gamma(e_{Cmc2_1}) &=& 3 \sum_{n=1}^4 \Gamma_n \ , \
\Gamma(g_{Cmc2_1}) = 6 \sum_{n=1}^4 \Gamma_n \ ,
\label{CMC2EQ} \end{aligned}$$ and the dipole moments and Raman tensors of each irrep are given in Table \[T16\]. Note that once we have identified the absorption modes, the identification of the Raman active modes follows, except for those of irrep $\Gamma_3$. Using Eq. (\[CMC2EQ\]) we find the number of optically active modes in the Cmc2$_1$ phase given in Table \[T19\].
[|| c || c | c | c | c ||]{} Irrep& $\Gamma_1$ & $\Gamma_2$ & $\Gamma_3$ & $\Gamma_4$ \
${\bf P}$ & $a \hat K$ & $b \hat J$ & None & $c \hat I$\
TOT & 19 & 19 & 17 & 17\
Cmcm & 10(R), 9(P) & 7(R), 12(P) & 12(R) & 7(R), 10(P)\
Addl & 10(P) & 7(P) & 5(R) & 7(P)\
Analysis based on lowering of Cmcm symmetry
-------------------------------------------
Now we perform the analysis of mode symmetry using the mixing of modes in terms of the OP’s present in the Cmc2$_1$ phase, namely $Q_3^-({\bf q}_1)$, $Z\equiv [\Gamma_{5,1}^- + \Gamma_{5,2}^-]/\sqrt 2$, and $Q_2^+({\bf q}_1)$, where we assume the OP’s of the family of ${\bf q}_1$. (Afterwards the analogous terms associated with ${\bf q}_2$ can be obtained by applying the four-fold rotation ${\cal R}_4$ to the coupling we construct for ${\bf q}_1$.) At the lower transition we can mix using all the condensed order parameters, $Q_3^-({\bf q}_1)$, $Q_2^+ ({\bf q}_1)$, and $Z$. The first question is: can we induce $Y \equiv Q_{5,1}^-(0) - Q_{5,2}^-(0)$? The answer is no. Note that $\sigmav_d Z =Z$ and $\sigmav_d Y=-Y$. Also $\sigmav_d Q_3^-({\bf q}_1)= -\sigmav_d Q_3^-({\bf q}_1)$ and $\sigmav_d Q_2^+({\bf q}_1) =- Q_2^+({\bf q}_1)$. But for wave vector conservation we must have a total even number of powers of these two operators at ${\bf q}_1$. Hence there can be no term in the free energy which is linear in $Y$. So $Y$ is not induced by the other order parameters.
### Induced Absorption Modes
So optically active absorption modes are admixed by the perturbation $$\begin{aligned}
&& V({\bf q}_1) = \sum_{k,l,m} \langle Q_3^-({\bf q}_1) \rangle^k
\langle Q_2^+({\bf q}_1) \rangle^l
\langle Z \rangle^m
\nonumber \\ &\times& \Biggl( a_{klm} U_{klm} Z
+ b_{klm} V_{klm} Y + c_{klm} W_{klm} Q_3^-(0) \Biggr) \ ,
\nonumber \end{aligned}$$ where again the $U$’s, $V$’s, and $W$’s are the single phonon operators which will become optically active due to this mechanism and which we wish to identify. We insert the angle brackets to emphasize that these higher order terms give rise to an effective bilinear coupling of the type that mixes modes. In principle any perturbation has to be invariant under all the symmetries of the tetragonal phase. However, applying a four-fold rotation takes ${\bf q}_1$ into ${\bf q}_2$ (whose order parameters are zero) and $\langle Z \rangle$ into $\langle Y \rangle$ (which is also zero). Thus we require that $V({\bf q}_1)$ be invariant under only $m_d$, $m_z$, and ${\cal I}$, in which case $$\begin{aligned}
m_dU_{klm}&=&(-)^{k+l+1}U_{klm} \ , \ \
m_dV_{klm}=(-)^{k+l}V_{klm} \ , \nonumber \\
m_dW_{klm}&=&(-)^{k+l+1}W_{klm} \ , \ \
m_zU_{klm}=U_{klm} \ , \nonumber \\
m_zV_{klm}&=&V_{klm} \ , \ \
m_zW_{klm}=-W_{klm} \nonumber \\
{\cal I}U_{klm}&=&(-)^{k+m+1}U_{klm} \ , \ \
{\cal I}V_{klm}=(-)^{k+m+1}V_{klm} \ , \nonumber \\
{\cal I}W_{klm}&=&(-)^{k+m+1}W_{klm} \ . \nonumber\end{aligned}$$
\[h!\]
[||c || c c c c c c c ||]{} $k=$ & 0 & 0 & 1 & 1 & 1 & 0 & 1\
$l=$ & 0 & 1 & 0 & 1 & 0 & 1 & 1\
$m=$ & 1 & 0 & 0 & 0 & 1 & 1 & 1\
Case & A & B & C & D & E & F & G\
New? & $*$ & $*$ & Cmcm & $=$A & $\propto$ B & $=$C & $\propto$ C\
We use the result of Ref. that, due to the cubic coupling between the order parameters, one has $$\begin{aligned}
\langle Z \rangle = a \langle Q_2^+({\bf q}_1) \rangle
\langle Q_3^-({\bf q}_1) \rangle \ ,\end{aligned}$$ where $a$ is a constant. Note the following: a) increasing $k$, $l$, or $m$ by two does not induce an additional operator and b) solutions for $U$, $V$, and $W$, depend on $k+l$ and $k+m$. Therefore we have the results of Table \[T13\] of which we only consider cases A, B, and E. We find that $$\begin{aligned}
U_{001} &=& c Q_1^+(0) + d Q_4^+(0) \ , \ \
V_{001} = c' Q_2^+(0) + d' Q_3^+(0) \ , \nonumber \\
W_{001} &=& Z^+ \ , \ \
U_{101} = U_{010} = Q_3^-({\bf q}_1 ) \ , \nonumber \\
V_{101} &=& V_{010} = Q_4^-({\bf q}_1 ) \ , \ \
W_{101} = W_{010} = Q_1^-({\bf q}_1 ) \ .
\label{UVWEQ} \end{aligned}$$ From these three cases we get (for wave vector ${\bf q}_1$) $$\begin{aligned}
&& V({\bf q}_1) = \langle Q_2^+ ({\bf q}_1) \rangle
\langle Q_3^- ({\bf q}_1) \rangle \Biggl\{ a_1 Q_3^-(0) Z^+
+ Z [ \nonumber \\ && \ a_2 Q_1^+ (0) + a_3 Q_4^+(0)]
+ Y [ a_4 Q_2^+ (0) + a_5 Q_3^+(0)] \Biggr\} \nonumber \\ && \
+ \langle Q_2^+ ({\bf q}_1) \rangle \Biggl\{ a_6 Z Q_3^-({\bf q}_1) +
a_7 Y Q_4^- ({\bf q}_1) \nonumber \\ && \
+ a_8 Q_3^-(0) Q_1^-({\bf q}_1) \Biggr\}
+ \langle Q_2^+({\bf q}_1) \rangle
\langle Q_3^-({\bf q}_1) \rangle^2 \Biggl\{ \nonumber \\ &&
a_9 Z Q_3^-({\bf q}_1) + a _{10} Y Q_4^- ({\bf q}_1)
+ a_{11} Q_3^-(0)Q_1^-({\bf q}_1) \Biggr\} \ . \nonumber
\\ \label{EQ46} \end{aligned}$$
By applying ${\cal R}_4$ to this we get $$\begin{aligned}
&& V({\bf q}_2) = \langle Q_2^+ ({\bf q}_2) \rangle
\langle Q_3^- ({\bf q}_2) \rangle \Biggl\{ a_1 Q_3^-(0) Y^+
+ Y [ \nonumber \\ && \ a_2 Q_1^+ (0) - a_3 Q_4^+(0)]
+ Z [ a_4 Q_2^+ (0) - a_5 Q_3^+(0)] \Biggr\} \nonumber \\ && \
+ \langle Q_2^+ ({\bf q}_2) \rangle \Biggl\{ a_6 Y Q_3^-({\bf q}_2) +
a_7 Z Q_4^- ({\bf q}_2) \nonumber \\ && \
+ a_8 Q_3^-(0) Q_1^-({\bf q}_2) \Biggr\}
+ \langle Q_2^+({\bf q}_2) \rangle
\langle Q_3^-({\bf q}_2) \rangle^2 \Biggl\{ \nonumber \\ &&
a_9 Y Q_3^-({\bf q}_2) + a _{10} Z Q_4^- ({\bf q}_2)
+ a_{11} Q_3^-(0)Q_1^-({\bf q}_2) \Biggr\} \ . \nonumber
\\ \label{EQ47} \end{aligned}$$
\[h!\]
[||c | c c c c c c c | c c c c || ]{} & &\
Orbit & $Q_1^+$(0) & $Q_4^+(0)$ & $Q_2^+(0)$ & $Q_3^+(0)$ & $Z^+$ & $Q_1^-(0)$ & $Q_4^-(0)$ & $Q_3^-({\bf q}_1)$ & $Q_4^-({\bf q}_1)$ & $Q_1^-({\bf q}_1)$ & $Q_3^+({\bf q}_1)$\
a & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0\
e & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1\
g & 1 & 0 & 1 & 0 & 2 & 0 & 1 & 1 & 1 & 2 & 1\
TOT & 4 & 0 & 1 & 0 & 5 & 0 & 1 & 6 & 6 & 2 & 4\
& Z & Z & Y & Y & X & none & none & Z& Y & X & none\
[**R**]{} & 1 & 1 & YZ& YZ& XZ& none & XY & 1& YZ& XZ& XY\
### Induced Raman Modes
Finally, we carry out the same analysis to determine the new Raman modes of polarization $X_OY_O$, since these modes are not dipole active and hence are not accessible to the previous calculation. To get polarization $X_OY_O=z_t(x_t-y_t)$ we must induce the mode $Y^+$ of irrep $\Gamma_5^+$. We use the results of Table \[T13\] to restrict our attention to the coupling for the cases $k=l=m-1=0$ (together with $k=l=m+1=1$) and $k=l-1=m=0$ (together with $k=l+1=m=1$) so that $$\begin{aligned}
&& V = R_1 Y^+ [\alpha \langle Z \rangle + \beta
\langle Q_2^+({\bf q}_1) \rangle \langle Q_3^-({\bf q}_1) \rangle ]
\nonumber \\
&& + R_2 Y^+ [\gamma \langle Q_2^+({\bf q}_1) \rangle + \delta
\langle Q_3^-({\bf q}_1) \rangle \langle Z \rangle ] \ ,
\label{EQ48} \end{aligned}$$ where $R_k$ are the single phonon operators which we determine by requiring that $V$ be an invariant under ${\cal I}$, $m_d$, and $m_z$. we find that $$\begin{aligned}
R_1 &=& Q_1(0) \ {\rm or} \ Q_4^- (0) \ , \hspace{0.2 in}
R_2 = Q_3^+({\bf q}_1) \ .
\label{R1R2EQ} \end{aligned}$$ The number of modes we admix into $R_1$ is equal to the number of times either $\Gamma_1^-(0)$ or $\Gamma_4^-(0)$ appears, which from Eq. (\[TETIRREP\]) is one for each g orbit or once in all. The number of modes we admix into $R_2$ is equal to the number of times irrep $\Gamma_3^+({\bf q}_1)$ appears, which from Eq. (\[REP2EQ\]) is one for each e orbit and one for each g orbit, or four in all. So there are five additional $X_OY_O$ Raman modes induced in Cmc2$_1$.
We could give figures showing the displacements in the new optically active modes which appear in the Cmc2$_1$ phase. Actually, the new modes at wave vector ${\bf q}_1$ have already been shown in Figs. \[M1\], \[M2\], and \[M3\], but now many of these modes are both absorption active and Raman active. We leave it to the reader to draw the new modes at zero wave vector using the same technique as used above.
### *Summary of Optically Active Phonons.*
In Table \[ACTIVE\] we list the irreps of the tetragonal phase which are active or become active due to the condensation of order parameters $Q_3^-({\bf q})$, $Q_2^+({\bf q})$, and $\Gamma_5^-(0)$. However, one can show that the intensities for wave vector ${\bf q}_2$ are obtained from those for wave vector ${\bf q}_1$ by simply replacing ${\bf q}_1$ by ${\bf q}_2$.
\[h!\]
[|| c | c c c | c c ||]{} Phase & Irrep & Int & Modes &\
& $\Gamma_5^-$ & $\alpha_k$ & 7 &\
I4/mmm & $\Gamma_5^-$ & $\alpha_k$ & 7 &\
& $\Gamma_5^-$ & $\alpha_k$ & 6 &\
&&&& ${\bf q}_1$ & ${\bf q}_2$ \
& $X_1^+$ &$\sigma_{1k}$ & 5 & $Y$ & $Z$\
Cmcm & $X_2^+$ &$\sigma_{1k}$ & 2 & $Z$ & $Y$\
& $X_4^+$ &$\sigma_{1k}$ & 4 & $X$ & $X$\
& $\Gamma_1^+$ & $\sigma_{2k}$ & 4 & $Z$ & $Y$\
Cmc2$_1$ & $\Gamma_2^+$ &$\sigma_{2k}$ & 1 & $Y$ & $Z$\
& $\Gamma_5^+$ & $\sigma_{2k}$ & 5 & $X$ & $X$\
& $X_3^-$ & $\sigma_{3k}$ & 6 & $Z$ & $Y$\
Cmc2$_1$ & $X_4^-$ & $\sigma_{3k}$ & 6 & $Y$ & $Z$\
& $X_1^-$ & $\sigma_{3k}$ & 2 & $X$ & $X$\
Magnetoelastic Modes
====================
Here we investigate the mixing of dipole active phonons with magnons for the RP system containing the magnetic ion Mn$^{4+}$. The motivation for this investigation is that photon absorption by magnons normally proceeds via a magnetic dipole matrix element, whereas absorption by phonons proceeds via the very much larger electric dipole matrix element. In magnetic ferroelectrics, magnons into which phonons are mixed have been dubbed “electromagnons."\[\] Such magnons not only have a much enhanced absorption cross section, but, because their energy is often much lower than the phonon energies, they can also lead to large anomalies in the static dielectric constant.\[\] Potentially the same effect is possible here if the magnons have a nontrivial coupling to phonons.
To start we review what is known about the magnetic structure of CMO which appears below the magnetic ordering temperature of 115K.\[\] The OP’s are\[\] $G_X({\bf q}_1)$, $C_Y({\bf q}_1)$, and $F_Z$, or equivalently \[\] $G_X({\bf q}_2)$, $F_Y$, and $C_Z({\bf q}_2)$, where $G$, $F$ and $C$ are the Wollan-Koehler magnetic structural descriptors\[\] for the magnetism of a single bilayer: F denotes a ferromagnetic structure, G and C are the antiferromagnetic structures illustrated in Fig. \[WK\]. The wave vector distinguishes between the two possible stacking (of G and C) as one goes from one bilayer to the next,\[\] as shown in Fig. \[WK\]. As mentioned below Eq. (\[STRESSEQ\]) one can select the wave vector by cooling through $T_>$ in the presence of a shear stress.
\[h!\]
![\[WK\] Spin states of a bilayer of Mn ions at $z= \pm \xi$. The plus and minus signs represent the signs of any component of spin. Left: the “G" configuration. Right the “C" configuration. The uncircled symbols are in the planes at $z=\pm \xi$ and the circled symbols represent the spins in planes at $z=1/2 \pm \xi$ when ${\bf q}={\bf q}_1$. For ${\bf q}={\bf q}_2$ the circled $+$ and $-$ signs are interchanged. The tetragonal axes are shown at right.](FIG8.eps){width="8.6cm"}
The plan of our analysis is as follows. We will construct the most general trilinear coupling which involves two magnetic OP’s (in order to satisfy time-reversal invariance) and one phonon operator whose wave vector is either at the zone center or at an $X$ point. We will require invariance with respect to all the symmetries of the tetragonal lattice. (As a short-cut we fix the wave vector to be ${\bf q}_1$. The terms involving ${\bf q}_2$ can be determined later by applying a four-fold rotation to those we find for ${\bf q}_1$.) We then develop an effective bilinear coupling by replacing one of the magnetic OP’s by its value at the minimum of the free energy.
Thus we analyze in detail the cubic interaction (involving the wave vector ${\bf q}_1$) $$\begin{aligned}
V_{\rm ME}({\bf q}_1) &=& \sum_{\alpha \beta \gamma} [ A_{\alpha \beta \gamma}
C_\alpha ({\bf q}_1) M_\beta Q_\gamma^+ ({\bf q}_1) \nonumber \\ && \
+ B_{\alpha \beta \gamma}
C_\alpha ({\bf q}_1) N_\beta ({\bf q}_1) Q_\gamma^- (0)
\nonumber \\ && \ + C_{\alpha \beta \gamma}
M_\alpha N_\beta ({\bf q}_1) Q_\gamma^- ({\bf q}_1)] \nonumber \\
&\equiv& \sum_{k=1}^3 V_{\rm ME}^{(k)} ({\bf q}_1) \ ,
\label{SYMEQ} \end{aligned}$$ where ${\bf M}$ denotes ${\bf F}$, $\bf N ({\bf q})$ denotes ${\bf G}({\bf q})$, and we chose the arguments of the phonon operators so as to conserve wave vector. We also used the transformation properties of the magnetic OP’s under inversion (see Table \[SYM\]) to infer the inversion signature of the $Q$’s in Eq. (\[SYMEQ\]).
\[h!\]
[||c c| c c c |c c c |c c c ||]{} Structure & ${\bf q}$ & & &\
&& $Z$ & $Y$ & $X$ & $Z$ & $Y$ & $X$ & $Z$ & $Y$ & $X$\
F & $0$ & $+$ &$+$ &$+$ & $-$ & $-$ & $+$ & $-$ & $+$ & $-$\
G & ${\bf q}$ & $-$ &$-$ &$-$ & $+$ & $+$ & $-$ & $-$ & $+$ & $-$\
C & ${\bf q}$ & $+$ &$+$ &$+$ & $-$ & $-$ & $+$ & $-$ & $+$ & $-$\
[||c c c | c c | c c | c | c || c ||]{} & & & irrep & $P$ & Int.\
$M_Z$ & $N_Y$ & $Q^-({\bf q}_1)$ & $-$ & $+$ & $-$ & $+$ & $X_1^-$ & $X$ & $M^2\langle Q_2^+ ({\bf q}_1) \rangle^2$\
$M_Y$ & $N_X$ & $Q^-({\bf q}_1)$ & $+$ & $-$ & $-$ & $-$ & $X_3^-$ & $Z$ & $M^2 \langle Q_2^+({\bf q}_1) \rangle^2$\
$C_X$ & $N_X$ & $Q^-(0)$ & $-$ & $-$ & $+$ & $-$ & $\Gamma_3^-$ & $X$ & $C^2$\
$C_Z$ & $N_X$ & $Q^-(0)$ & $-$ & $-$ & $-$ & $-$ & $\Gamma_5^-$ & $Z$ & $C^2$\
$C_Y$ & $N_Y$ & $Q^-(0)$ & $+$ & $+$ & $-$ & $+$ & $\Gamma_3^-$ & $X$ & $C^2$\
$M_X$ & $C_Y$ & $Q^+({\bf q}_1)$ & $-$ & $+$ & $+$ & $-$ & $X_4^+$ & $X$ & $C^2M^2\langle Q_3^-({\bf q}_1) \rangle^2$\
$M_Y$ & $C_Y$ & $Q^+({\bf q}_1)$ & $+$ & $+$ & $-$ & $-$ & $X_1^+$ & $Y$ & $C^2M^2\langle Q_3^-({\bf q}_1) \rangle^2$\
$M_Z$ & $C_Z$ & $Q^+({\bf q}_1)$ & $-$ & $-$ & $-$ & $-$ & $X_1^+$ & $Y$ & $C^2M^2 \langle Q_3^-({\bf q}_1) \rangle^2$\
To get the effective coupling between magnons and phonons we only analyze special terms of Eq. (\[SYMEQ\]). We take the expectation value (that minimizes the free energy) of one of the magnetic OP’s and therefore arrive at a quadratic interaction between the other (active) magnetic OP and a phonon OP. To get a magnon-phonon coupling out of this we select the component of the active magnetic OP to be [*transverse*]{} to its equilibrium value. Since the nonzero equilibrium component of ${\bf N}({\bf q}_1)$ is its $X$ component, the transverse components of ${\bf N}({\bf q}_1)$ are its $Y$ and $Z$ components. These are the components of ${\bf N}({\bf q}_1)$ which, in the Holstein-Primakoff formulation, create spin waves. The operator $Q$ creates or destroys a phonon. These terms which mix magnons and phonons give rise to “electromagnons."\[\] We treat the other trilinear terms similarly, keeping only those in which one magnetic OP is longitudinal and the other magnetic OP is transverse. Thereby we have the cases listed in Table \[MNQ\] for wave vector ${\bf q}_1$. The results for wave vector ${\bf q}_2$ are quite similar: the polarizations $Y$ and $Z$ are interchanged and in the intensities ${\bf q}_1$ is replaced by ${\bf q}_2$, as in Table \[ACTIVE\]. It is relevant to try to estimate which of these terms are the most significant. For that purpose we keep in mind the fact that the OP’s ${\bf M}$ and ${\bf C}({\bf q}_1)$ are roughly one tenth as large as ${\bf N}({\bf q}_1)$.\[\] In addition the optical activity of the phonon depends on whether it is allowed in the tetragonal phase or depends, for its activity, on the distortions of irreps $X_3^-$ or $X_2^+$ (in this connection we ignore $\Gamma_5^-$). It seems likely, then, that the largest magnon-phonon coupling will involve coupling the transverse moment of $C({\bf q}_1)$ to either $X$ or $Z$ polarized zone center phonons. But for completeness, we list the other possibilities. Table \[MNQ\] only gives results for the case when wave vector ${\bf q}_1$ is selected. Results for wave vector ${\bf q}_2$ can be obtained by replacing $Z$ polarization by $Y$ and ${\bf q}_1$ by ${\bf q}_2$, as is done in Table \[ACTIVE\].
These couplings can be detected either by anomalies in the static dielectric constant when magnetic order appears,\[\] or by anomalous electric dipole intensity in magnons due to their coupling with optical phonons.\[\]
CONCLUSIONS
===========
In this paper the effect on the optical properties of zone center phonons of the sequential lowering of symmetry by structural distortions from the high symmetry tetragonal (I4/mmm) phase has been studied. In addition, the coupling of the magnons which appear at low temperatures in the magnetically order phase of Ca$_3$Mn$_2$O$_7$ and the phonons is explored to identify possible giant enhancements in the magnon cross section in the electromagnetic spectrum.
Our specific results include
$\bullet 1$. We have tabulated the zone center phonon modes with their polarizations which appear in the absorption spectrum and in the Raman scattering spectrum of the I4/mmm, Cmcm, and Cmc2$_1$ phases for Ca$_3$X$_2$O$_7$ systems, where X is Mn or Ti.
$\bullet 2$. For modes which become optically active by virtue of a structural distortion, we indicate the dependence of their cross sections on the newly emerging order parameters near the phase boundary where the structural symmetry is lowered. We also discuss the anomalies in the phonon frequencies as the phase boundaries are crossed.
$\bullet 3$. We discuss the symmetry of the magnon phonon interaction which can lead magnons being electric dipole active and hence having anomalously large absorption cross section and leading to analogous large magnetic contributions to the dielectric constant.
Intensity Calculation {#APPA}
=====================
Tetragonal Phase
----------------
In this appendix I discuss in detail how the intensities of the modes are calculated assuming the perturbative description relative to the tetragonal phase is justified. In the tetragonal phase we have the optically active phonons of irreps $\Gamma_5^-$ and $\Gamma_3^-$, governed by the Hamiltonian $$\begin{aligned}
{\cal H}_{\rm TET} &=& \frac{1}{2} \sum_{k=1}^{n(\Gamma_5^-)}
\kappa_{\perp,k} (Y_k^2 + Z_k^2) + \frac{1}{2}
\sum_{k=1}^{n(\Gamma_3^-)} \kappa_{\parallel,k} X_k^2 \ ,
\nonumber \end{aligned}$$ where it is conveneient to work in orthorhombic coordinates, so that modes $X$, $Y$, and $Z$ are polarized along these respective orthorhombic direction. $X_k=Q_3^-(0)_k$, $Y_k=(Q_5^-(1)_k-Q_5^-(2)_k)/\sqrt 2$, and $Z_k=(Q_5^-(1)_k+Q_5^-(2)_k)/\sqrt 2$ are the order parameters for the modes of the optically active $\Gamma_3^-$ and $\Gamma_5^-$ irreps. To be specific: we number the modes in order of incresing energy.
The coupling to photons is via the dipole moment operator which we write as $$\begin{aligned}
{\bf p} &=& \sum_{k=1}^{n(\Gamma_5^-)} p_{\perp,k} [Y_k\hat Y +Z_k \hat Z]
+ \sum_{k=1}^{n(\Gamma_3^-)} p_{\parallel,k} X_k \hat X\ ,
\nonumber \end{aligned}$$ where $\hat X$ is a unit vector in the orthorhombic $X$ direction and similarly for $\hat Y$ and $\hat Z$. So the absorption intensity in the tetragonal phase of the $k$th doubly degenerate $Y_k$-$Z_k$ mode is $Ap_{\perp,k}^2$ and that of the $k$th $X$ mode is $Ap_{\parallel,k}^2$, where here and below we set $A=1$. Since the dipole moment matrix elements are subject to a first principles calculation, it makes sense to give explicit results in terms of calculable quantities.
Cmcm Phase
----------
The Hamiltonian that describes the mixing of new optically active phonons in the Cmcm phase is $$\begin{aligned}
{\cal H}_{\rm Cmcm} &=& \frac{1}{2} \sum_{k=1}^{n(X_1^+)}
\kappa^{(X_1^+)}_k Q_1^+({\bf q}_1)_k^2
+\frac{1}{2} \sum_{k=1}^{n(X_2^+)} \kappa^{(X_2^+)}_k Q_2^+({\bf q}_1)_k^2
\nonumber \\ && \
+\frac{1}{2} \sum_{k=1}^{n(X_4^+)} \kappa^{(X_4^+)}_k Q_4^+({\bf q}_1)_k^2
\nonumber \\ && + \langle Q_3^-({\bf q}_1) \rangle
\Biggl[ \sum_{k=1}^{n[X_2^+]} \sum_{l=1}^{n[\Gamma_5^-]}
a^{(1)}_{k,l} Z_k Q_2^+ ({\bf q}_1)_l \nonumber \\ &&
+ \sum_{k=1}^{n[X_1^+]} \sum_{l=1}^{n[\Gamma_5^-]}
b^{(1)}_{k,l} Y_k Q_1^+ ({\bf q}_1)_l \nonumber \\ &&
\sum_{k=1}^{n[X_4^+]} \sum_{l=1}^{n[\Gamma_3^-]}
c^{(1)}_{k,l} Q_3^-(0)_k Q_4^+ ({\bf q}_1)_l \Biggr] \ .
\nonumber \end{aligned}$$ $Q_3^-({\bf q}_1)$ is replaced by $\langle Q_3^-({\bf q}_1)\rangle$ leading to a bilinear interaction which mixes new modes.
We will analyze the $Z$-component of the dipole moment operator. The other components are treated analagously. Due to the mixing of modes, the bare mode operator $Z_k$ is related to the true mode operators (indicated by tildes) $$\begin{aligned}
Z_k &=& {\tilde Z}_k + \sum_l \frac{a^{(1)}_{kl} {\tilde Q}_2^+({\bf q}_1)_l}
{\kappa_l^{X_2^+} - \kappa_{\perp,k}} \langle Q_3^- ({\bf q}_1) \rangle \ .
\nonumber \end{aligned}$$ Thus the $Z$-component of the dipole moment operator which creates modes ${\tilde Q}_2^+({\bf q}_1)_l$ denoted $p_{Z,Q_2^+,l}$ is $$\begin{aligned}
p_{Z,Q_2^+,l} &=& p_{\perp,k} \frac{
a^{(1)}_{kl} \langle Q_3^-({\bf q}_1)\rangle}
{ \kappa_l^{(X_2^+)} - \kappa_{\perp,k}} \langle Q_3^-({\bf q}_1) \rangle \ ,
\label{EQPP} \end{aligned}$$ and the intensity of this induced mode is $|p_{Z,Q_2^+,l}|^2.$ The quantities in Eq. (\[EQPP\]) are amenable to a first principles frozen-phonon calculation.
[**ACKNOWLEDGEMENTS.**]{} I would like to thank C. Fennie for a discussion of the properties of these systems, for communicating the results of his first principle calculations, and for pointing out many relevant references. I am also grateful to B. Campbell and H. Stokes for discussions concerning the group-subgroup relations and irreps for the star of the wave vector. I wish to thank M. Lobanov for discussions concerning the magnetic structure of the Mn compound and for alerting me to several references. I also thank G. Lawes for communicating the results of the pyroelectric measurements.
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abstract: '[The precise behavior of the quasi-hyperbolic metric near a $\mathcal C^{1,1}$-smooth part of the boundary of a domain in $\R^n$ is obtained.]{}'
address:
- |
N. Nikolov: Institute of Mathematics and Informatics\
Bulgarian Academy of Sciences\
Acad. G. Bonchev 8, 1113 Sofia, BulgariaFaculty of Information Sciences\
State University of Library Studies and Information Technologies\
Shipchenski prohod 69A, 1574 Sofia, Bulgaria
- |
Pascal J. Thomas\
Université de Toulouse\
UPS, INSA, UT1, UTM\
Institut de Mathématiques de Toulouse\
F-31062 Toulouse, France
author:
- 'Nikolai Nikolov and Pascal J. Thomas'
title: 'Boundary behavior of the quasi-hyperbolic metric'
---
Introduction and results
========================
Let $D$ be a proper subdomain of $\R^n.$ Define the [*quasi-hyperbolic metric*]{} of $D$ by $$h_D(a,b)=\inf_{\g}\int_\g\frac{||du||}{d_D(u)},\quad a,b\in D$$ where $||\cdot||$ is the Euclidean norm, $d_D=\mbox{dist}(\cdot,\partial D)$ and the infimum is taken over all rectifiable curves $\g$ in $D$ joining $a$ to $b.$ By [@GO Lemma 1], the infimum is attained, and any extremal curve is called [*quasi-hyperbolic geodesic*]{} (for short, [*geodesic*]{}). It turns out that the geodesics are $\mathcal C^{1,1}$-smooth (see [@Mar Corollary 4.8]). The quasi-hyperbolic metric arises in the theory of quasi-conformal maps.
This paper is devoted to the boundary behavior of $h_D.$ First, we point out the following general lower bound.
\[GHM\][@GHM Lemma 2.6] If $D$ is a proper subdomain of $\R^n,$ then $$h_D(a,b)\ge 2\log\frac{d_D(a)+d_D(b)+||a-b||}{2\sqrt{d_D(a)d_D(b)}},\quad a,b\in D.$$
Observe that equality occurs if $n=1$ (then $D$ is an open interval or ray).
From now, we assume that $n\ge 2.$ Throughout the paper, we will say that $\zeta$ is a *$\mathcal C^\alpha$ smooth boundary point* of $D$ if and only if it admits a neighborhood in which $\partial D$ is $\mathcal C^{\alpha}$-smooth.
Recall that a $\mathcal C^1$-smooth boundary point $\zeta$ of a domain $D$ in $\R^n$ is said to be [*Dini-smooth*]{} if the inner unit normal vector $n$ to $\partial D$ near $\zeta$ is a Dini-continuous function. This means that there exists a neighborhood $U$ of $\zeta$ such that $\ds\int_0^1\frac{\om(t)}{t}dt<+\infty,$ where $$\om(t)=\om(n,\partial D\cap U,t):=\sup\{||n_x-n_y||:||x-y||<t,\ x,y\in \partial D\cap U\}$$ is the respective modulus of continuity.
If $\ds\int_0^1\om(t)\frac{\log t}{t}dt>-\infty,$ then the point $\zeta$ is called [*log-Dini smooth.*]{}
The following relations between different notions of smoothness are clear: ${\mathcal C^{1,\varepsilon}}\Rightarrow\mbox{log-Dini}\Rightarrow\mbox{Dini}\Rightarrow{\mathcal C^1}.$
\[NA\][@NA Theorem 7] Let $\zeta$ be a Dini-smooth boundary point of a domain $D$ in $\R^n.$ Then for any constant $\ds c>1+\frac{\sqrt 2}{2}$ there exists a neighborhood $U$ of $\zeta$ such that $$h_D(a,b)\le 2\log\left(1+\frac{c||a-b||}{\sqrt{d_D(a)d_D(b)}}\right),\quad a,b\in D\cap U.$$
Since $h_D$ is an inner metric, we get an upper bound of $h_D,$ similar to the lower bound from Proposition \[GHM\].
[@NA Corollary 8] Let $D$ be a Dini-smooth bounded domain in $\R^n.$ Then there exists a constant $c>0$ such that $$h_D(a,b)\le 2\log\left(1+\frac{c||a-b||}{\sqrt{d_D(a)d_D(b)}}\right),\quad a,b\in D.$$
Set now $\ds s_D(a,b)=2\sinh^{-1}\frac{||a-b||}{2\sqrt{d_D(a)d_D(b)}}$ $$=2\log\frac{||a-b||+\sqrt{||a-b||^2+4d_D(a)d(b)}}{2\sqrt{d_D(a)d_D(b)}},\quad a,b\in D.$$
Note that $h_D=s_D$ if $D$ is a half-space in $\R^n$ (cf. [@Vuo (2.8)]).
The following sharp result holds in the $\mathcal C^1$-smooth case.
\[C1\][@NA Proposition 6(a)] If $\zeta$ is a $\mathcal C^1$-smooth boundary point of a domain $D$ in $\R^n,$ then $$\lim_{\substack{a,b\to\zeta\\a\neq b}}\frac{h_D(a,b)}{s_D(a,b)}=1.$$
Since the proof of this proposition is not long, we shall include it for completeness.
[@NA Proposition 6(b) and p.3] If $D$ is a $\mathcal C^1$-smooth bounded domain in $\R^n,$ then $$q_D(a,b)=\left\{\begin{array}{ll}\ds\frac{h_D(a,b)}{s_D(a,b)},&a,b\in D,\ a\neq b\\
1,&\mbox{otherwise}\end{array}\right.$$ is a continuous function on $\R^n\times\R^n.$
The main goal of this paper is to prove the following result related to Proposition \[C1\].
\[main\] If $\zeta$ is a $\mathcal C^{1,1}$-smooth boundary point of a domain $D$ in $\Bbb R^n$, then $$\lim_{a,b\to\zeta}(h_D(a,b)-s_D (a,b))=0.$$
Note that Theorem \[main\] and Proposition \[C1\] say the same only if $s_D$ and $1/s_D$ are bounded.
The assumption about regularity in Theorem \[main\] can be weakened in the plane.
\[plane\] If $\zeta$ is a log-Dini smooth boundary point of a domain $D$ in $\Bbb R^2,$ then $$\lim_{a,b \to \zeta}\left(h_D(a,b)-s_D (a,b)\right)=0.$$
The above results imply the following optimal version of Theorem \[NA\].
\[cor\] Let $\zeta$ be a $\mathcal C^{1,1}$-smooth boundary point of a domain $D$ in $\Bbb R^n$ or $\zeta$ be a log-Dini smooth boundary point of a domain $D$ in $\Bbb R^2.$ Then for any constant $c>1$ there exists a neighborhood $U$ of $\zeta$ such that $$h_D(a,b)\le 2\log\left(1+\frac{c||a-b||}{\sqrt{d_D(a)d_D(b)}}\right),\quad a,b\in D\cap U.$$
The rest of the paper is organized as follows: Section 2 contains the proofs of Propositions \[C1\], \[plane\] and Corollary \[cor\]. Section 3 contains the proof of Theorem \[main\]. It should be mentioned that the three proofs use different flattening maps. Section 4 contains the proof of a result analogous to Corollary \[cor\] for the Kobayashi distance.
Proofs of Propositions \[C1\], \[plane\] and Corollary \[cor\]
==============================================================
[*Proof of Proposition \[C1\]*]{}. After translation and rotation, we may assume that $\zeta=0$ and that there is a neighborhood $U$ of $0$ such that $$D':=D\cap U=\{x\in U: r(x):=x_1+f(x')>0\},$$ where points of $\R^n$ are denoted by $x=(x_1,x')$, with $x'\in \R^{n-1}$, and $f$ is a $\mathcal C^1$-smooth function in $\R^n$ with $f(0)=0$ and $\nabla f(0)=0.$
Let $c>1$ and $\theta(x)=(r(x),x').$ We may shrink $U$ such that $$\label{c}
c^{-1}||x-y||\le||\theta(x)-\theta(y)||\le c||x-y||,\quad x,y\in U.$$
Choose now a neighborhood $V\subset U$ of $0$ such that $d_{D'}=d_D$ on $D\cap V.$ The regularity of $D$ implies that it is a [*uniform domain*]{} near $\zeta$ in the sense of [@GO]. Using, for example, [@GO Corollary 2], one can find a neighborhood $W\subset V$ of $0$ such that any geodesic joining points in $\tilde D=D\cap W$ is contained in $D\cap V.$ Then $h_D=h_{D'}$ on $\tilde D^2.$
Set $\Bbb R^n_+=\{x\in\R^n:x_1>0\}.$ Using the above arguments, we may shrink $W$ such that $h_{\RRR}=h_{\theta(D')}$ on $(\theta(\tilde D))^2.$
On the other hand, implies that (cf. [@Vuo Exercise 3.17]) $$c^{-2}h_{D'}(z,w)\le h_{\theta(D')}(\theta(z),\theta(w))\le c^2h_{D'}(z,w),\quad z,w\in D'.$$
Let $z,w\in \tilde D.$ Then $$c^{-2}h_D(z,w)\le h_{\RRR}(\theta(z),\theta(w))\le c^2h_D(z,w).$$ Using again, we get that $$\begin{aligned}
h_{\RRR}(\theta(z),\theta(w))&=2\sinh^{-1}\frac{||\theta(z)-\theta(w)||}{2\sqrt{r_D(z)r_D(w)}}\\
&\le2\sinh^{-1}\frac{c^2||z-w||}{2\sqrt{d_D(z)d_D(w)}}\le c^2s_D(z,w).\end{aligned}$$ We obtain in the same way that $$h_{\RRR}(\theta(z),\theta(w))\ge c^{-2}s_D(z,w).$$ So $$c^{-4}h_D(z,w)\le s_D(z,w)\le c^4h_D(z,w)$$ which implies the desired result.
[*Proof of Proposition \[plane\]*]{}. We may find a neighborhood $U$ of $\zeta$ such that $D\cap U$ is a bounded simply connected log-Dini smooth domain. Using an argument from the previous proof, we may replace $D$ by $D\cap U.$
The Kellogg–Warschawski theorem (cf. [@Pom Theorem 3.5]) implies that there exists a conformal map $\tilde f$ from the unit disc $\Bbb D$ to $D$ which extends to a $\mathcal C^1$-diffeomorphism between $\overline{\Bbb D}$ to $\overline D$ such that $\tilde f(\zeta)=1$ and $$|\tilde f'(z)-\tilde f'(w)|\le\om^\ast(|z-w|),\quad z,w\in\Bbb D,$$ where $\ds\tom^\ast(s)=\int_0^s\frac{\tom(t)}{t}dt +s\int_s^{+\infty}\frac{\tom(t)}{t^2}dt$ ($s\ge 0$) and $\tom:\R^+\to\R^+$ is a bounded continuous function with $\ds\int_0^1\tom(t)\frac{\log t}{t}dt>-\infty.$
Then $\ds f(z)=\tilde f\left(\frac{1-z}{1+z}\right)$ maps conformally $\RR$ onto $D$ and $$|f'(z)-f'(w)|\le\om^\ast(|z-w|),\quad z,w\in G=\RR\cap\Bbb D,$$ where $\om^\ast$ is defined in the same way as $\tom^\ast.$
The equality $$f(w)-f(z)-f'(z)(w-z)=(w-z)\int_{0}^1\left(f'(z+t(w-z))-f'(z)\right)dt$$ implies that $$|f(w)-f(z)-f'(z)(w-z)|\le|w-z|\om^\ast(|w-z|)$$ (since $\om^\ast$ is an increasing function). It follows that $$\label{d}
|d_D(f(z))-|f'(z)|d_{\RR}(z)|\le d_{\RR}(z)\om^\ast(d_{\RR}(z)),\quad z\in G.$$
Since $D$ is a uniform domain, there exists a neighborhood $V$ of $\zeta$ such that any geodesic $\g$ joining points $a=f(\alpha)$ and $b=f(\beta)$ in $D\cap V$ is contained in $f(G).$ It follows by that one may find a constant $C>0$ (independent of $a$ and $b$) such that $$h_{\RR}(\alpha,\beta)\le\int_{f^{-1}\circ\g}\frac{|du|}{d_{\RR}(u)}\le\int_\g\frac{|dv|}{d_D(v)}+
C\int_\g\frac{\om^\ast(d_D(v))}{d_D(v)}|dv|.$$
The first summand is equal to $h_D(a,b).$
We claim that the second summand tends to 0 as $a,b\to\zeta.$ Indeed, denote by $t$ the natural parameter of $\g$ by arc length and by $l=l(\g)$ the Euclidean length of $\g.$ Since $D$ is a uniform domain, then [@GO Corollary 2] provides a constant $c>0$ (independent of $a$ and $b$) such that $c\cdot l\le|a-b|$ and $d_D(\g(t))\ge c\cdot \max\{t,l-t\}.$ Using that $\ds\frac{\om^\ast(s)}{s}$ is a decreasing function, we get $$\int_\g\frac{\om^\ast(d_D(v))}{d_D(v)}|dv|\le\frac{2}{c}\int_0^{cl/2}\frac{\om^\ast(t)}{t}dt.$$ It is easy to check the log-Dini condition for $\om$ is is equivalent to the fact that the last integral tends to $0$ as $l\to 0$ which implies our claim.
Hence $$\liminf_{a,b\to\zeta}(h_D(a,b)-h_{\RR}(\alpha,\beta))\ge 0.$$
The opposite inequality $$\limsup_{a,b\to\zeta}(h_D(a,b)-h_{\RR}(\alpha,\beta))\le 0$$ follows in the same way by taking the geodesic joining $\alpha$ and $\beta.$
Using , we have that $$\label{asymp}
\lim_{\substack{a,b\to\zeta\\a\neq b}}\frac{|a-b|}{2\sqrt{d_D(a)d_D(b)}}\cdot
\frac{2\sqrt{d_{\RR}(\alpha)d_{\RR}(\beta)}}{|\alpha-\beta|}=1.$$ Since $h_{\RR}=s_{\RR}$ and $\sinh^{-1}qt < \log q + \sinh^{-1}t $ for $q>1,$ $t>0,$ then $$\lim_{a,b\to\zeta}(s_D(a,b)-h_{\RR}(\alpha,\beta))=0$$ which completes the proof.
[*Proof of Corollary \[cor\]*]{}. We may assume that $c=2c'-1\in(1,3].$ By Proposition \[C1\], Theorem \[main\] and Proposition \[plane\], one may find a neighborhood $U$ of $\zeta$ such that for $a,b\in D\cap U,$ $$h_D(a,b)\le c's_D(a,b),\quad h_D(a,b)\le s_D(a,b)+\log c'.$$ Then the result follows by the inequalities $\ds \sinh^{-1}\frac{t}{2}<\log(1+t)$ ($t>0$), $(1+t)^{c'}<1+ct$ ($0<t<1$) and $c'(1+t)<1+ct$ ($t>1$).
Proof of Theorem \[main\]
=========================
Theorem \[main\] will follow from Propositions \[liminf\] and \[limsup\] below.
For convenience, we assume that $D$ is a domain in $\R^{n+1}$ ($n\ge 1$).
We first localize the problem. We choose local coordinates so that $\zeta = 0$ and $T_0\partial D = \{0\} \times \R^n$.
Denote points in $\R^{n+1}$ by $\bar x = (x_0, x) \in \R \times \R^{n}$. We also write $\R^{n+1}_+= \{\bar x\in\R^{n+1}:x_0>0\}$.
There are a ball $\mathcal U \subset \R^{n+1}$ centered at $(0,0)$ and a function $f \in \mathcal C^{1,1}(\mathcal U\cap\R^n,\R)$ such that $f(0)=0$ and $Df(0)=0$ and $$\label{localform}
D\cap\mathcal U= \left\{\bar x\in\mathcal U:x_0 > f(x) \right\}.$$
By shrinking the radius of $\mathcal U$ further we may assume that the projection which to $\bar x\in \mathcal U \cap D$ associates $\pi(\bar x)$, the closest point in $\partial D$ is well-defined, and that $\mathcal U \subset \pi^{-1} (\mathcal U \cap D)$ (see [@Fed Lemma 4.11], or the proof of Lemma \[norm\] (1) below).
\[liminf\] $\ds\liminf_{a,b \to 0} \left( h_D(a,b) - s_D (a,b) \right) \ge 0.$
We can define a map $\varphi$ on $\mathcal U$ by $$\varphi (\bar x) = \left( f(x), x \right) + x_0 n_x,$$ where $n_x$ is the inward unit normal to $\partial D$ at the point $( f(x), x)$.
\[norm\] (1) There exists a ball $\mathcal U_0 \subset \mathcal U$ centered at $0$ such that $\varphi|_{\mathcal U_0}$ is a bilipschitz homeomorphism and for any $\bar x \in \mathcal U_0 \cap \R^{n+1}_+$, $$d_D \varphi (\bar x)= \| \varphi (\bar x) - (f(x), x)\| = x_0.$$
\(2) Furthermore, if $f \in \mathcal C^{\alpha}(\mathcal U\cap\R^n, \R)$, for some $\alpha\ge 2$, then $\varphi|_{\mathcal U_0}$ is a $\mathcal C^{\alpha-1}$-diffeomorphism, and there exists a ball $\mathcal U_1 \subset \mathcal U_0$ centered at $0$ and a constant $C>0$ such that for any $\bar x \in \mathcal U_1 \cap \R^{n+1}_+$ and any vector $v\in \R^{n+1}$, $$\| D\varphi(\bar x)\cdot v\|\ge(1-Cx_0)\|v\|.$$ where $D\varphi (\bar x)$ stands for the differential of $\varphi$ taken at the point $\bar x$.
\(3) In the general case where $f \in \mathcal C^{1,1}(\mathcal U\cap\R^n, \R)$, then there exists a $C>0$ such that for any $\mathcal C^1$ curve $\gamma: [t_1,t_2] \longrightarrow \mathcal U_1 \cap \R^{n+1}_+$, $\varphi \circ \gamma$ is rectifiable and for any $F \in \mathcal C([t_1,t_2], \R_+)$, $$\int_{t_1}^{t_2} F(t) |d \varphi \circ \gamma (t)|
\ge
\int_{t_1}^{t_2} F(t) |d \gamma (t)| - C \int_{t_1}^{t_2} F(t) d_D (\gamma (t)) |d \gamma (t)|.$$
Part (1) of the lemma is classical (see [@Fed Theorem 4.8]). The main point is to prove that the domain has positive [*reach,*]{} that is to say that there exists $\delta>0$ such that if $x \in D$ and $d_D(x)<\delta$, then this distance is attained at a single point, which will be the intersection of $\partial D$ and the unique normal line to it containing $x$ (see [@Fed]). In other words, for $x \in \mathcal U$ well chosen and $x_0<\delta$, $\varphi$ is one-to-one.
We quickly recall the proof. Suppose $\| \nabla f (x) - \nabla f(x')\| \le L \|x-x'\|$ for $(0,x), (0,x') \in \mathcal U_1$, then, taking without loss of generality the projection to $\partial D$ to be $(0,0)$, for some $\theta \in (0,1)$, $$\begin{gathered}
\left\| (y_0,0) - (f(x),x)\right\|^2 = y_0^2 - 2 y_0 \nabla f (\theta x) \cdot x + f(x)^2 + \|x\|^2
\\
\ge y_0^2+ \|x\|^2 - 2 y_0 L \|x\|^2 >y_0^2\end{gathered}$$ for $y_0 < 1/2L$ and $x\neq 0$.
Notice that a lemma in [@GT Appendix], explained in detail in [@KP], shows that even though $n_x$ can only be expected to be continuous with bounded derivatives, and in general of class $\mathcal C^{\alpha-1}$ when $\varphi \in \mathcal C^{\alpha}$, the function $\bar x\mapsto d_D(\bar x)$ has the same regularity as $\varphi$.
We now prove part (2). Let $(e_0, e_1, \dots , e_n)$ be the standard basis of $\R^{n+1}$. Let $\ds \tilde e_j = \frac{\partial f}{\partial x_j} (x) e_0 + e_j$, for $1\le j\le n$. They form a basis of the tangent space to $\partial D$ at $(x,f(x))$ and $\langle n_x, \tilde e_j \rangle=0$ for $1\le j\le n$.
Then $D\varphi (\bar x) \cdot e_0 = n_x$, and $\ds D\varphi (\bar x) \cdot e_j = \tilde e_j + x_0 \frac{\partial n_x}{\partial x_j}$, for $1\le j\le n$.
Given $v= \sum_0^n v_j e_j$, $$D\varphi (\bar x) \cdot v = \left( v_0 n_x + \sum_1^n v_j \tilde e_j \right) + x_0 \sum_1^n v_j
\frac{\partial n_x}{\partial x_j} =: V_1 + V_0.$$ Clearly, $\| V_0 \| = O(x_0) \|v\|$. By the orthogonality of $n_x$ to the tangent space, $$\begin{gathered}
\| V_1 \|^2 = v_0^2 + \left\|\sum_1^n v_j \tilde e_j \right\|^2
= v_0^2 + \left\|\sum_1^n v_j e_j + \left(\sum_1^n \frac{\partial f}{\partial x_j} (x)\right)e_0\right\|^2
\\
= v_0^2 + \sum_1^n v_j^2 + \left|\sum_1^n \frac{\partial f}{\partial x_j} (x)\right|^2 \ge \|v\|^2.\end{gathered}$$
In the case where $f \in \mathcal C^{1,1}$, then $\varphi \circ \gamma$ is only a Lipschitz map. By Rademacher’s theorem (see e.g. [@Fe2 Theorem 3.1.6]), it is almost everywhere differentiable and the fundamental theorem of calculus holds. We then perform the same calculation as in case (2), where the integrands are defined a.e.
[*Proof of Proposition \[liminf\]*]{}. Using Lemma \[norm\], the proof repeats the second part of the proof of Proposition \[plane\]. Suppose that $\zeta=0$ and that the domain $D$ is given by a local representation as above. We may assume that the points $a,b\in D$ are in a small enough neighborhood of $0$ so that the geodesic $\g$ which joins them is entirely contained in the range of invertibility of $\varphi$ and Lemma \[norm\] holds; we write $a=\varphi(\bar \alpha)$, $b=\varphi(\bar \beta)$, $\g=\varphi(\tilde\g)$, where $\tilde\g$ is an arc in $\R^{n+1}_+$. Then $$h_D(a,b) =
\int_\g\frac{||du||}{d_D(u)}\ge\int_{\tilde\g}\frac{||dv||}{d_{\R^{n+1}_+}(v)}
-C \cdot l(\tilde\g)\ge h_{\R^{n+1}_+}(\bar \alpha,\bar \beta)-C'||\bar\alpha-\bar\beta||,$$ where $C'>0$ is a constant independent of $a$ and $b.$ Note that $h_{\R^{n+1}_+}=s_{\R^{n+1}_+}.$ Since the differential of $\varphi$ at $\bar x$ tends to the identity as $x\to 0,$ it follows that $$\lim_{a,b\to\zeta}(s_{\R^{n+1}_+}(\bar \alpha,\bar \beta)-s_D(a,b))=0$$ which completes the proof.
\[limsup\] $\ds\limsup_{a,b \to 0} \left( h_D(a,b) - s_D (a,b) \right) \le 0.$
The proof is similar to that of Proposition \[liminf\], using a modification of the map $\varphi$ which depends on $a$ and $b.$
We again assume that $a, b \in D$, and the geodesic connecting them, all lie in a neighborhood of $\zeta$ small enough so that any point in it has a unique closest point on $\partial D$. Let $a',b'$ be the respective closest points. We take new coordinates (and obtain a new function $f$) so that $a'=0$ (instead of $\zeta=0$ as in the proof of Proposition \[liminf\]) and $$D\cap\mathcal U=\{\bar x\in\mathcal U:x_0>f(x_1,\dots,x_n)\}.$$ We may also assume that $b'_2=\dots=b'_n=0.$ Shrinking the radius $r$ of $\mathcal U,$ we may replace $x_1$ by $\sigma_1(x_1)$ such that for $\sigma=(f(\sigma_1,0,\dots,0),\sigma_1, 0,\dots,0)$ one has $||\sigma'||=1$ (in other words, $\sigma$ is parametrized by arc length). Note that $r$ can be chosen independently of $a$ and $b.$ Let $\ell$ be the length of the curve $\sigma$ from $a'$ to $b'$, so that $\sigma (0)=a'$, $\sigma (\ell)=b'$.
Consider the map $\varphi$ from $\RR$ (near $0$) to $D$ defined by $$\varphi(x_0,x_1) = \sigma (x_1) + x_0 n_{\sigma (x_1)},$$ where $n_{\sigma (x_1)}$ is the inward unit normal to $\partial D$ at the point $\sigma (x_1).$ Then $d_D(\varphi (\bar x))=x_0$ if $x_0$ is small enough, and if $\alpha = (d_D(a), 0)$ and $\beta = (d_D(b), \ell)$, we have $\varphi (\alpha)=a$, $\varphi (\beta)=b$.
\[norm2\] There exist a neighborhood $U$ of $\zeta,$ a neighborhood $V$ of $0$ and a constant $C>0$ such that for any $a,b \in D\cap U$ and $\bar x \in\RR\cap V$ and any vector $v\in \R^2$, then $\alpha, \beta \in V$ and $$\| D\varphi(\bar x)\cdot v\|\le(1+Cx_0)\|v\|.$$
As in the proof of Lemma \[norm\] (2), in the $\mathcal C^2$-smooth case, $$D \varphi (\bar x) \cdot e_0 = n_{\sigma (x_1)}, \quad
D \varphi (\bar x) \cdot e_1 = \sigma' (x_1) + x_0 \frac{\partial n_{\sigma (x_1)}}{\partial x_1}.$$ Because $||\sigma'||=1$ and is tangent to $\partial D$, $(\sigma'(x),n_x)$ form an orthonormal system, so that $D\psi (\bar x)$ differs from a linear isometric embedding by a term bounded by $\ds \left\| \frac{\partial n_{\sigma (x_1)}}{\partial x_1} \right\| x_0 $.
Geometric considerations show that $\ds \left\| \frac{\partial n_{\sigma (x_1)}}{\partial x_1} \right\|\le\frac{1}{R} $ whenever there exist two balls $B_1, B_2$ of radius $R$, tangent to each side of $\partial D$ at $\sigma (x_1)$. The argument in the proof of Lemma \[norm\] (1) shows there exists $\delta >0$ (depending only on the neighborhood $\mathcal U_0$ mentioned in that lemma) such that there exist two such balls of radius $\delta$ at each point in $\mathcal U_0 \cap \partial D $.
As in the proof of Lemma \[norm\] (3), the $\mathcal C^{1,1}$-smooth case follows by applying Rademacher’s theorem.
The proof of Proposition \[limsup\] can be finished similarly to that of Proposition \[liminf\]. Let $\g$ be the geodesic joining $\alpha$ to $\beta$ in $\R^2_+.$ Let $U,V$ be as in Lemma \[norm2\]. Shrinking $V$ if needed so that $\varphi(V)\subset U$, we have $d_{D}(\varphi(u))=d_{\RR}(u)$ for any $u\in\g$. Since $\varphi \circ \g $ is a curve joining $a$ to $b$ in $D$, using Lemma \[norm2\], we get $$\begin{gathered}
h_D(a,b) \le
\int_0^\ell \frac{\|D\varphi (\gamma(t))\cdot \gamma'(t) \|}{d_{D}( \varphi \circ \gamma(t))} dt\\
\le h_{\RR} (\alpha,\beta)+Cl(\g)<s_{\RR} (\alpha,\beta)+C\pi||\alpha-\beta||\end{gathered}$$ (here $\pi$ is the Ludolphine number, not the projection). The differential of $\varphi$ is close to a linear isometric embedding of $\R^2$ in $\R^{n+1}$ and hence we have the asymptotic relation and $$\lim_{a,b\to\zeta}(s_{\RR}(\alpha,\beta)-s_D(a,b))=0,$$ which completes the proof.
An upper estimate for the Kobayashi distance
============================================
Let $D$ be a domain in $\Bbb C^n.$ The Kobayashi (pseudo) distance $k_D$ is obtained from the Lempert function $$\begin{gathered}
l_D(a,b)=\inf\{\tanh^{-1}|\alpha|:\exists\varphi\in\mathcal O(\Bbb D,D)
\hbox{ with }\varphi(0)=a,\varphi(\alpha)=b\},\\a,b\in D.\end{gathered}$$ The Lempert function does not always satisfy the triangle inequality, but setting $$k_D(a,b):= \inf \left\{ \sum_{j=0}^{m-1} l_D(a_j,a_{j+1}): a_j \in D, a_0=a,a_m=b, m\ge 1 \right\},$$ one does obtain a (pseudo) distance, which is the largest that is dominated by $l_D$.
Recall that $k_D$ is the integrated form of the Kobayashi (pseudo) metric $$\begin{gathered}
\kappa_D(a;X)=\inf\{|\alpha|:\exists\varphi\in\mathcal O(\Bbb D,D)\hbox{
with }\varphi(0)=a,\alpha\varphi'(0)=X\},\\a\in D,\ X\in\Bbb C^n.\end{gathered}$$
Note that Theorem \[NA\] and Proposition \[plane\] (even in the Dini-smooth case) hold for $2k_D$ instead of $h_D$ (see [@NA Theorem 7] and [@NTA Proposition 6]). Moreover, the following result corresponds to Proposition \[C1\].
[@NA Proposition 5(a)] If $\zeta$ is a $\mathcal C^1$-smooth boundary point of a domain $D$ in $\Bbb C^n,$ then $$\limsup_{\substack{a,b\to\zeta\\a\neq b}}\frac{2k_D(a,b)}{h_D(a,b)}\le 1.$$
It turns out that Corollary \[cor\] also holds for $2k_D$ instead of $h_D.$ This gives the optimal version of [@FR Proposition 2.5] in the $\mathcal C^{1,1}$-smooth case.
Let $\zeta$ be a $\mathcal C^{1,1}$-smooth boundary point of a domain $D$ in $\Bbb C^n$ or $\zeta$ be a log-Dini smooth boundary point of a domain $D$ in $\Bbb C.$ Then for any constant $c>1$ there exists a neighborhood $U$ of $\zeta$ such that $$k_D(a,b)\le\log\left(1+\frac{c||a-b||}{\sqrt{d_D(a)d_D(b)}}\right),\quad a,b\in D\cap U.$$
[*Proof.*]{} Having in mind Corollary \[cor\], it is enough to show that $$\limsup_{\substack{a,b\to\zeta\\a\neq b}}\frac{2k_D(a,b)-h_D(a,b)}{||a-b||}<+\infty.$$
Since $k_D$ is the integrated form of $\kappa_D$ and the lengths of the quasi-hyperbolic geodesics joining points in $D$ near $\zeta$ are bounded up to a multiplicative constant by the Euclidean distances between the points, the last inequality will be a consequence of the following one: $$\limsup_{\substack{a\to\zeta\\||X||=1}}\left(2\kappa_D(a;X)-\frac{1}{d_D(a)}\right)<+\infty.$$
To see this, note that there exists an $r>0$ such that any $a\in D$ near $\zeta$ is contained in a (unique) ball $\Bbb B_n(\tilde a,r)\subset D$ with $r-||a-\tilde a||=d_D(a)$ (the inner ball condition). It remains to use that for such an $a$ and $||X||=1$ one has that $$\kappa_D(a;X)\le \kappa_{\Bbb B_n(\tilde a,r)}(a;X)\le\frac{r}{r^2-||a-\tilde a||^2}
<\frac{1}{2d_D(a)}+\frac{1}{4r}.\quad\qed$$
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---
abstract: 'Social interactions and personal tastes shape our consumption behaviors of cultural products. In this study, we present a computational model of a cultural market and we aim to analyze the behavior of the consumer population as an emergent phenomena. Our results suggest that the final market shares of the cultural products dramatically depend on the consumer heterogeneity and social interaction pressure. Furthermore, the relation between the resulting market shares and the social interaction is robust with respect to a wide range of variation in the parameter values and the type of topology.'
address:
- |
Computer Engineering Department, Bogazici University\
Istanbul, Turkey\
[email protected]
- |
Computer Engineering Department, Bogazici University\
Istanbul, Turkey\
[email protected]
author:
- Amac Herdagdelen
- Haluk Bingol
title: A Cultural Market Model
---
Introduction and Relevant Studies
=================================
Social interaction is an inevitable aspect of our lives and a very strong ingredient of our decision processes. Most of our decisions depend, at least partially, on what other people think and how they behave. The extent of the society’s influence on our behaviors may range from daily decisions such as what to wear at work to political decisions such as which party to vote for in the elections. The importance of the social interaction has been reflected in the social sciences for many decades and a growing body of research continues on the intersection of various disciplines including but not limited to sociology, cognitive sciences, physics and economics. [@Farrell1985; @Gordon2005; @Granovetter1978; @Markose2006; @Schelling1973; @Schelling1978] Statistical physics has a long history of dealing with interacting particles and emergent phenomena. Some of the techniques employed by statistical physics are also applied to human populations successfully and offer us new ways to explore the dynamics in social systems. [@Durlauf1999; @Galam1997; @Holyst2000; @Michard2005; @Phan2006; @Sznajd2000]
The interaction between the macro dynamics and individual decision processes is an active area of research in computational sociology and statistical physics (i.e. sociophysics). Problems in which the agents in a community are faced with a binary decision such as to vote for or against a legislation or to buy a particular product or not are extensively studied by the statistical physicists. [@Gordon2005; @Granovetter1978; @Galam1997; @Michard2005; @Sznajd2000] The *Random Field Ising Model* (RFIM) is a commonly investigated model to analyze such situations. [@Galam1997; @Michard2005; @Phan2006; @Sethna2001; @Sethna2005; @Phan2004] There are also other problems in which the opinions of the agents are represented as multidimensional continuous vectors and are influenced by the opinions of other agents. [@Lorenz2007]
In this paper, we investigate the effect of social interactions on the consumption behaviors of people in a market. In a cultural market, there are many items competing with each other and the decision to consume an item or not is not independent of the consumption decisions of other items. The consumers have to pick one or more items from a wider pool of opportunities.
An empirical study carried out by Salganik et al. (2006) provides experimental evidence that social influence has an effect on the consumption decisions of people. [@Salganik2006] In the experiment, the subjects are faced with a web based application in which they can listen to and rate as many songs as they like among 48 songs of previously unknown bands. After they listen to and rate a song, they are offered the opportunity to download the song. The study reports the results of two different experimental conditions. The first one is called the independent condition in which the subjects only see the names of the songs without any other information and make their decisions independently from the other subjects. The second condition is called the social influence condition and in this case, the number of people who have downloaded each song so far is also given to the subjects. The number of downloads of a song is called the *success* of the song. Any significant difference in the success of the same song between the two cases can be attributed to the availability of social information since there is no other experimental difference between the two settings.
The key finding of the study is that the availability of the social information significantly affects the way people behave. In the social influence condition, variation of the success outcomes of the songs are much higher than it is in the independent condition. This suggests that in the social influence case, some songs are downloaded many more times than they are in the independent case. Another measure they report is called the *unpredictability* of a song and is found by calculating the average difference between the success values of a song over different realizations of the same condition (i.e. the experimental condition is repeated several times with different subjects). If a song tends to get the same outcomes over different realizations then its unpredictability value is low otherwise it is high. As a result of the experiment, the social influence condition leads to higher unpredictability values for the songs.
Borghesi and Bouchaud (2007) propose a generalization of the RFIM such that it allows to study multiple choices made by the agents simultaneously. [@Borghesi2006] They show that it is possible to estimate some parameters of the model from the empirical data of Salganik et al. [@Salganik2006]. Furthermore, they report that the behavior of the model changes qualitatively for low and high values of social pressure which is in alignment with the empirical findings.
Another line of research is carried out under *social percolation* models. [@Weisbuch2003] In this approach, consumers and producers are modeled as adaptive agents and the hit or failure of products are studied as a result of the information contagion between the consumers. \[cha:theory.introduction\]
Methodology
===========
The Cultural Market Model {#sec:cultural market}
-------------------------
The kind of market we are interested in is one where the items are easily reproducible so that their supply is practically unlimited and the consumers have to pick a subset of the available items to consume. An example of such a market is the cultural market where the items are cultural products such as books, movies, or music albums. Two major components of a cultural market are the set of consumers and the set of cultural items. Like Weisbuch and Solomon (2003), we use the movie market as an example in which the consumers are the customers of the movie theaters and items are the movies on the market. [@Weisbuch2003]
An important assumption about a cultural market is that one of the factors effective in the consumption decision is the social pressure exerted on the individual. Theoretically, the effect may be both negative or positive but we focus on the case where it is positive (i.e. if an item is consumed more by the others, then it will have a greater chance to be consumed by the individual). The literature of psychology and economics have a large set of findings and different reasons for such effects. [@Farrell1985]
Formally speaking, the model consists of $N$ agents and $M$ items. We index the agents by Roman labels $i$ and $j$ (both in $[1,N]$) and the items by Greek labels $\alpha$ and $\beta$ (both in $[1,M]$).
The agents are assumed to be located at the vertices of an undirected graph which incorporates the social structure of the community. If there is a link between agent $i$ and agent $j$ then these two agents are said to be neighbors of each other.
Initially, none of the items are consumed. The model advances in discrete time steps and each time step the agents form their opinions about the items. The *opinion* of an agent $i$ about an item $\alpha$ is represented by $o_{i\alpha} \in \mathbb{R}$. At each time step, each agent picks the item with the highest opinion value and consumes it. An agent can consume an item only once so the selection is carried over non-consumed items. For each agent and item pair, $o_{i\alpha}$ is calculated as a weighted average of personal preferences and social pressure as $o_{i\alpha} = \gamma s_{i\alpha}+(1-\gamma)l_{i\alpha}$.
Social component $s_{i\alpha}$ represents the social pressure exerted on the agent $i$ to consume the item $\alpha$. In our model, we use a simple function which computes, for each agent $i$, the ratio of the number of neighbors who have consumed the item $\alpha$ to the total number of neighbors. Note that $s_{i\alpha} \in [0,1]$. Such a definition of social pressure is in accordance with our previous discussions since an agent will be more inclined towards consuming an item if a majority of its neighbors have already consumed it. Note that, in reality the social pressure is a function of time ($s_{i\alpha}=s_{i\alpha}(t)$) because as the agents consume items its value changes. Our implementation of the model is a synchronized one in the sense that at the start of a time step the social pressure is calculated according to current state and updated only when the time step is completed.
Personal preferences $l_{i\alpha} \in \mathbb{R}$ are predetermined and time independent values representing the likings of agents for the items. The value $l_{i\alpha}$ corresponds to the idiosyncratic personal taste of agent $i$ for the item $\alpha$. Higher values indicate higher preferences for the item.
The value $\gamma \in [0,1]$ is the *social pressure parameter* which determines the strength of the social pressure on the decision process. The case $\gamma=0$ corresponds to the pure-individualistic community where no agent cares about what others are doing (hence basing their decisions solely on their idiosyncratic liking values), $\gamma=1$ corresponds to the pure-social environment where all decisions are based on the behaviors of others.
Initial Configuration {#sec:initial configuration}
---------------------
*Topology:* For the base case, the topology of the agents is set to a ring lattice with a coordination number of $k$ (i.e. agents are located in a ring topology and each agent is connected to its nearest $k$ neighbors). But the effect of employing other topologies is also investigated.
*Personal preferences*: Personal preference data isolated from any social effect is very hard to obtain and no publicly available dataset was known to us at the time of this study. We represent the personal preference of the agents for the items as real numbers and assume that the idiosyncratic personal liking values $l_{i\alpha}$ come from a normal distribution with mean $\mu_\alpha$ and standard deviation $\sigma_{\alpha}$. This is an assumption that can be validated (or invalidated) in the future. Without any prior knowledge, assuming the liking values for the items come from a normal distribution specific to each item seems appropriate.
To simplify the matters more, we set the mean values of the distributions equal to each other (i.e. $\mu_\alpha = \mu$ where $\mu$ is a model parameter) due to our assumption that no item has any intrinsic superiority in terms of liking values over another item a priori. Although this seems unrealistic, such an assumption can only weaken the inequalities that we observe as a result of the simulations and does not pose a problem to the conclusions made. Furthermore, it is trivial to show that the particular value of the parameter $\mu$ does not change anything in the dynamics of the model hence we can set $\mu=0$. Since only the ranking of the items are considered in the decision of the agents, an initial configuration with a non zero $\mu$ value can be safely turned into a zero-$\mu$ configuration by subtracting $\mu$ from every $l_{i\alpha}$.
The value $\sigma_{\alpha}$ is called the *intra-item liking deviation* of item $\alpha$ and denotes the standard deviation in the liking values of agents for the same item. As a further simplification for the model, we assume all items have the same intra-item deviation and let $\sigma_{\alpha} =\sigma$ for all $\alpha$ where $\sigma$ is a model parameter.
Remarks
-------
We assume that consumption rates of the agents are constant and same for all. That means at each time step, each agent consumes exactly one item. Naturally, with these assumptions, all agents will consume all items after $M$ steps. In our model, a cultural item is available on the market until the end of the simulation. In real world however, not all items stay available in the market for an indefinite time period. Considering the movie market example, not all movies are shown in the theaters long enough to allow all consumers to see them. There is a continuous entrance and exit of movies. In order to keep our model simple, we ignore this fact but decide to terminate the simulations much before the point of total consumption (i.e. after $T$ steps with $T<M$). In other words, our model is limited to a simplified version of the real world where $M$ items are put on the market at the same time and none of them leaves the market until the simulation ends. In addition, we assume that a single agent cannot consume all available items on the market but a small fraction of them.
We have seen that the liking values $l_{i\alpha}$ are chosen from a normal distribution with $\mu = 0$ and the same $\sigma$ for all $\alpha$. The fact that the expected average value of the liking values of an item over all agents is 0 does not imply that the sample mean of its liking values will be zero. Let ${\langle l_{\alpha}\rangle}= \sum_{i=1}^{N}l_{i\alpha} / N$ denote the sample mean of the liking values of item $\alpha$ over all agents. We call ${\langle l_{\alpha}\rangle}$ as the *quality* of item $\alpha$. The Central Limit Theorem states that the distribution of the quality values will follow a normal distribution with mean 0 and standard deviation $\sigma/\sqrt{N}$ for large $N$. [@Ross2002] In the limit $N\rightarrow\infty$ the deviation of the quality values approaches to 0. Although the model assumes no a priori superiority of an item over another, we should expect to see different item quality values for different realizations of the model for finite and small $N$.
Quantities of Interest
----------------------
We define the *market share*, ${\langle c_{\alpha}\rangle}$, of an item $\alpha$ as the ratio of consumers who have consumed that item so far. It is calculated as ${\langle c_{\alpha}\rangle}= \sum_{i=1}^{N}c_{i\alpha}/N$ where $c_{i\alpha}$ is 1 if agent $i$ has consumed the item $\alpha$ before the current time step and to 0 otherwise. Note that the market share of an item is defined such that the sum of the market shares of all items at a given time does not have to sum up to 1.
Item quality, ${\langle l_{\alpha}\rangle}$, is already defined as the average liking value of an item over all agents for an item $\alpha$.
*Market Inequality*, $I$, represents the difference between the market shares of the cultural items. As a measure of the inequality, we calculate the Gini index of the market shares of all items. The market inequality is defined as follows.
$$I=\frac{(\sum_{\alpha=1}^M{\sum_{\beta=1}^M|{\langle c_{\alpha}\rangle} - {\langle c_{\beta}\rangle}|}) / M^2}{(2\sum_{\alpha=1}^M {\langle c_{\alpha}\rangle})/M}$$
which can be interpreted as half the relative expected difference between the market shares of two randomly chosen items. [@Salganik2006] A perfectly equal market where all items have the same market shares will have an inequality value of 0; that is the Gini index of a perfectly equal market.
The market inequality value is a symmetrical measure with respect to the quality values of the items. It does not differentiate whether it is the low quality items that receive unfairly high market shares or the high quality items. In order to answer the question of whether there is a general trend favoring the high quality items in terms of market shares or vice versa, we employ a simple measure that is used in statistics which we call the quartile difference. First, we divide the items into four quartiles according to their quality values. The items with the top 25% quality values are placed in the upper quartile $\mathcal{U}$ and the items with the bottom 25% quality values are placed in the lower quartile $\mathcal{L}$. *Quartile difference*, Q, is simply the difference between average market shares of the two quartiles. Since the number of items in the upper and lower quartiles are one forth of $N$ ($|\mathcal{U}|=|\mathcal{L}|=N/4$), quartile difference is defined as
$$Q = \sum_{\alpha\in \mathcal{U}}\frac{{\langle c_{\alpha}\rangle}}{N/4} - \sum_{\alpha\in \mathcal{L}}\frac{{\langle c_{\alpha}\rangle}}{N/4}$$
The quartile difference is a signed value so the minimal value it can take is -1 and this occurs in the extreme case in which all items in the lower quartile have market shares of 1 and all the items in the upper quartile have market shares of 0. Similarly, the maximal value it can assume can be computed as +1.
Results and Discussions
=======================
All results reported in this section are based on values obtained by averaging over 100 independent runs of the simulation with the same parameters.
It is natural to ask if the quality of an item determines its market share at the end or not. A reasonable expectation about a cultural market is that items with high quality values should get higher market shares on the average. As a starter, we let $\gamma=0$ and $\sigma =1$ which corresponds to the pure-individualistic community where all agents base their consumption decisions solely on their personal tastes. We keep the number of agents and the number of items fixed to 100 in this set of experiments (i.e. $N=100$ and $M=100$). The sensitivity of the results on these parameters will be investigated separately. Since there is no social interaction, the underlying topology has no effect on the results because its possible effects are ruled out with the zero social pressure parameter (i.e. $\gamma=0$). We let the model run for 5, 20 and 50 steps (i.e. $T \in \{5,20,50\}$). In Fig. \[fig:quality-vs-share\], we see the scatter plot of quality versus market shares of the items for three cases.
[ \[fig:quality-vs-share-a\] ![Market shares versus quality values for no social pressure. The simulation length $T$ is (a) 5, (b) 20, and (c) 50.[]{data-label="fig:quality-vs-share"}](./bingol1a.eps "fig:") ]{} [ \[fig:quality-vs-share-b\] ![Market shares versus quality values for no social pressure. The simulation length $T$ is (a) 5, (b) 20, and (c) 50.[]{data-label="fig:quality-vs-share"}](./bingol1b.eps "fig:") ]{} [ \[fig:quality-vs-share-c\] ![Market shares versus quality values for no social pressure. The simulation length $T$ is (a) 5, (b) 20, and (c) 50.[]{data-label="fig:quality-vs-share"}](./bingol1c.eps "fig:") ]{}
Let us first consider the case with $T=5$ given in Fig. \[fig:quality-vs-share-a\]. The relation between the quality and market share has a linear form. A linear relation also holds for the other two cases with $T=20$ and $T=50$ given in Fig. \[fig:quality-vs-share-b\] and \[fig:quality-vs-share-c\] correspondingly.
Prolonging the simulation increases both total consumption and market shares. This observation is trivial given the model definition and can be verified by noting the data points are shifted up for higher $T$ values. Another expected result is that high quality items receive higher market shares and this is reflected in the positive slope values of the data points for all figures. A last observation is that the rate of market share increase differs with respect to the quality values of the items. The slopes, $m$, of the best fitting lines are significantly different for different values of $T$: $m_{T=5}=0.10$, $m_{T=20}=0.28$, and $m_{T=50}=0.40$. This means that high quality items increase their market shares at an higher rate than the low quality items.
As we already noted before, the simulation length (i.e. number of steps that the simulation will run for) is an important parameter and should be set carefully with keeping in mind the actual market to be modeled. It is obvious that values of $T$ very close to the number of items (i.e. 100 at the moment) are highly unrealistic. We set $T=20$ during the next set of experiments for simplicity of analyses. Such a setting leads to a market where each agent consumes 20 percent of all available items. Even though a choice of 20 seems arbitrary, further simulations revealed that as long as the simulation length is appreciably smaller than the number of items, the market shares and the quartile difference values do not change abruptly for differing values of $T$. [@Herdagdelen2007] Unfortunately, the plots of these simulations are omitted in this paper due to space restrictions.
An interesting extension will be introducing social pressure (i.e. $\gamma>0$). To keep things simple for the moment, we create two parameter settings: One with $\gamma=0.3$ corresponding to low social pressure and one with $\gamma=0.7$ corresponding to high social pressure. With the introduction of social pressure, the topology of the neighborhood relations becomes important. For the moment, we set the topology to a fully connected graph which corresponds to a ring lattice with a coordination number $k=N-1$. An informative way to look at the effect of social pressure is to compare the market shares obtained for low and high social pressure to the ones obtained in the community with no social pressure. Using the same personal preference values allows us to compare the market shares of the same item under different social pressure environments. In order to compare the different social pressure settings, we present two figures (Fig. \[fig:share-share-low\] and Fig. \[fig:share-share-high\]) which contain the scatter plots of the market shares of the items obtained for low social pressure versus for no social pressure and high social pressure versus no social pressure correspondingly.
It is also possible to look at the scatter plots of quality and market shares obtained in these social settings. Since we have seen that quality and market shares are linearly related for no social pressure, this comparison also helps us to see the relation between quality and the market shares obtained in social pressure settings. These plots are omitted in this paper but they are in perfect accordance with the market share comparison graphs.
[ \[fig:share-share-low\] ![The effect of introducing social pressure on market shares. (a) Low social pressure versus no social pressure. (b) High social pressure versus no social pressure. In both figures, the dashed lines are the $y=x$ lines.[]{data-label="fig:share-share"}](./bingol2a.eps "fig:") ]{} [ \[fig:share-share-high\] ![The effect of introducing social pressure on market shares. (a) Low social pressure versus no social pressure. (b) High social pressure versus no social pressure. In both figures, the dashed lines are the $y=x$ lines.[]{data-label="fig:share-share"}](./bingol2b.eps "fig:") ]{}
We obtain similar results for low social pressure and no social pressure. The data points are scattered around the $y=x$ line in Fig. \[fig:share-share-low\] indicating that the effect of setting $\gamma=0.3$ has no or limited effect on the final market shares of the items. For low social pressure, the consumption decisions remain fairly intact compared to no social pressure.
For high social pressure, the relation between the quality and market share of an item is no longer linear as Fig. \[fig:share-share-high\] suggests. The items which obtain high market shares for no social pressure obtain even higher market shares for high social pressure and the items which obtain low market shares for no social pressure obtain even lower market shares for high social pressure.
These results are compatible with the experimental data obtained in the study of Salganik et al. (2006) (see Fig 3 in Ref. ). In this experimental study, the introduction of social information causes a non linearity between the market shares obtained with the presence of social pressure or no social pressure. Our model is able to generate this phenomenon.
What is the response of our model to varying degrees of social pressure? How can we extend our findings for communities with varying degrees of heterogeneity? The model parameter $\gamma$ regulates the extent of social pressure and intra-item liking deviance $\sigma$ regulates the heterogeneity of the agents in the community. Until now, we set $\sigma =1$ but this choice is arbitrary. Smaller choices for $\sigma$ will lead to more homogeneous communities in the sense that the liking values of the agents for the same items will be closer to each other. Higher choices on the other hand, will correspond to more heterogeneous communities because the liking values of the agents for the same items will deviate more.
In order to come up with answers to the two questions we ask, we calculated the market inequality $I$ and the quartile difference $Q$ of the markets at the end of 20 steps for different pairs of $\gamma$ and $\sigma$ values. Fig. \[fig:sigma-gamma-series\] visualizes the differing values of $I$ and $Q$ correspondingly.
In Fig. \[fig:sigma-gamma-series-I\], we see the effect of increasing social pressure $\gamma$ on the market inequality for different values of $\sigma$. As we have already seen previously, higher $\gamma$ values lead to higher market inequality values but the characteristic of the effect depends on $\sigma$. We see that the marker inequality values show a sharp increase from low values (e.g. $I=0.2$) to high values (e.g. $I=0.8$) at critical values of $\gamma$. However the specific value of the critical $\gamma$ value depends on $\sigma$.
The market inequality is a measure of how varied the final market shares are. It does not tell us anything about the dependence of market shares on quality. We have already seen that higher social pressure results in increased market shares for the high quality items. Figure \[fig:sigma-gamma-series-Q\] visualizes the effect of increasing social pressure on the quartile difference $Q$ for different values of intra-item liking deviation. Similar to the market inequality value, the quartile difference also shows a sharp increase at a critical value of $\gamma$ and the specific value of $\gamma$ depends on $\sigma$. However, at higher levels of social pressure the quartile difference starts to decrease. The turning point for the quartile difference (i.e. the $\gamma$ value that it starts to decrease) is again dependent on $\sigma$. This is an interesting observation because after the turning point the inequality value continues to increase while the quartile difference starts to decrease. Considering the extreme case where $\gamma=1$ may help to understand this phenomenon. When $\gamma=1$, the agents will not consider the liking values in their decision process but only the number of neighbors who have consumed the items. Thus, the quality values of the items will be irrelevant in affecting their market shares leading to a quartile difference value of zero.
Interpreting the results for inequality and quartile difference values together we conclude that there exists a critical region of $\gamma$ and for values less then the critical region the inequality and quartile difference values are in accordance with each other. High quality items gain higher market shares as we increase the social pressure and this fact is reflected in increased inequality and quartile difference values. But once the critical turning point in social pressure is passed, the relation between the quality and market share starts to weaken, but some items still continue to arbitrarily high market shares at the cost of others. Hence the inequality value continues to increase. The positive correlation between the quality and market shares disappears suddenly as reflected by the decrease in quartile difference value.
In order to assess whether the reported findings are robust with respect to the model parameters, we repeated the simulations with different parameter setting. Due to space limitations, we are unable to give a detailed report of those simulations but the results are given in an unpublished masters thesis. [@Herdagdelen2007]
We repeated the simulations for $N\in\{100, 1000, 5000, 10000\}$ and $M\in\{100, 1000, 5000\}$ and the results were qualitatively the same with the results we report in this study. Also the effect of the number of neighbors $k$ and the topology is investigated. Even for very low values of $k$ and random topology with directed links, we did not observe qualitative differences compared to ring lattice.
Conclusions and Future Work
===========================
Empirical findings suggested that the introduction of social interaction has a profound effect on the relation between the quality and market share of an item. For low values of social pressure (i.e. $\gamma=0.3$), we observed that the linear relation between the quality and market shares remains almost intact. As we increase the social pressure parameter ($\gamma=0.7$), the linearity was disrupted and we observed nonlinear relation in favor of high quality items.
An interesting finding is that the social pressure parameter does not regulate the collective nature of the model but interacts with $\sigma$ which represents the deviation among the liking values of the agents for the same items. We found out that a given market inequality value $I$ can be reproduced by a set of ($\gamma$, $\sigma$) pairs and the interaction between the two parameters remain qualitatively intact for different number of agents $N$ and average number of neighbors $k$.
To our surprise, while the inequality in the market shares increases with increasing social pressure, the quartile difference value first increases and than decreases with a peak at a critical value of social pressure dependent on $\sigma$. As we increase social pressure, the high quality items gain more and more market shares at the cost of low quality items and this is reflected in the increased inequality and quartile difference values. But after the critical social pressure value is passed, the positive correlation between the quality and market share deteriorates. Low quality items starts to gain higher market shares and quartile difference shows a decrease. Note that we cannot observe this trend just by looking at the market inequality which continues to increase as we increase the social pressure. These findings suggest that for higher values of social pressure the dynamics of the model gets history dependent and the resulting market share of an item is determined by the initial conditions of the system rather than the quality of the item.
We carried out extended simulations to see if our results depend on specific values of the parameters or robust to different values of the parameters. We concluded that the qualitative nature of the simulations are robust with respect to different number of agents (i.e. $N \in \{100, 500, 1000, 5000\}$) and varying degrees of network connectivity (i.e. for different values of $k<N$). The topology on the other hand seems to be affecting the outcome of the cultural market. The sharp increases in the inequality and quartile difference were more resilient to varying number of agents in random topology than the ring topology.
We studied the ring topology as a base model and introduced the heterogeneity by using random topology. How the model will behave if another type of network such as scale free or small world is introduced is definitely an interesting and non-trivial question and needs to be addressed in the future studies. Also the effect of introducing directed links and asymmetrical neighboring relations can be investigated in the future.
Another important point is to study the accordance between the real world data and the model output. Unfortunately, during the writing of this paper, no relevant dataset was publicly available. With the limited information available from a previous experiment (Ref. ) we were able to show that our model can produce compatible results and it can provide an explanation for the observed differences in the market share, item quality relations and market inequality values depending on the availability of social information. Additionally, we believe that study of such computational models is meaningful given the costs of conducting a real world experiment is many times higher than building and analyzing a computational model. The results and the dynamics of the cultural market model can be of use in determining the settings of a real world experiment in the future.
Acknowledgments
===============
This work was partially supported by Bogazici University Research Fund under the grant number 07A105 and is partially based on the work performed in the framework of the FP6 project SEE-GRID-2, which is funded by the European Community (under contract number INFSO-RI-031775).
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abstract: 'We present a catalogue of the 322 damped Lyman alpha absorbers taken from the literature. All damped Lyman alpha absorbers are included, with no selection on redshift or quasar magnitude. Of these, 123 are candidates and await confirmation using high resolution spectroscopy. For all 322 objects we catalogue the radio properties of the background quasars, where known. Around 60 quasars have radio flux densities above 0.1 Jy and approximately half of these have optical magnitudes brighter than $V = 18$. This compilation should prove useful in several areas of extragalactic/cosmological research.'
author:
- |
S. J. Curran$^{1}$, J. K. Webb$^{1}$, M. T. Murphy$^{1}$, R. Bandiera$^{2}$,\
E. Corbelli$^{2}$ and V. V. Flambaum$^{1}$
title: A Catalogue of Damped Lyman Alpha Absorption Systems and Radio Flux Densities of the Background Quasars
---
=2em 2[$\chi^2$]{}
=15.5cm =26 cm =-2.0 cm =0.5cm =0.5cm bcite\#1\#2[([\#1@tempswa , \#2]{})]{} pcite\#1\#2[\#1@tempswa , \#2]{}
[$^1$ School of Physics, University of New South Wales, Sydney NSW 2052, Australia, [email protected]\
$^2$ Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5, 50125, Italy\
]{}
[**Keywords:**]{} catalogues–quasars: absorption lines–radio continuum: galaxies–\
cosmology: early universe
Introduction
============
High resolution spectroscopy of quasars reveals large numbers of absorption lines due to neutral atomic hydrogen. In some cases, the neutral hydrogen column density can be very large ($N_{\rm
HI}\gapp10^{20}$ )[^1] and gives rise to a heavily damped absorption feature (). The aim of this paper is to present a catalogue of these damped Lyman alpha absorption systems (DLAs) and, where known, the radio flux densities of the background quasar. The catalogue has been compiled from the large number of studies reported in the literature.
The nature of DLAs is still open to debate. Interpretations include galactic disks [@wtsc86], low surface brightness galaxies [@jbm99] or dwarf galaxies [@mmv97] intersecting the sight-line towards a background quasar. Despite these possible differing morphologies, DLAs provide a powerful cosmological probe. The highest redshift ($ z>
3.5$) DLAs may account for a large fraction of the baryons at high redshift, suggesting they reveal gas prior to the bulk of the star formation history of the universe ( and references therein). On the other hand, recent work [@lyp+02] seems to indicate star formation rates which continue to increase with increasing redshift up to the highest galaxy redshifts observed in the Hubble Deep Field. The discovery of a DLA with truly primordial abundances would have a major impact on our understanding of the early chemical evolution of the universe, and a crucial reality check on the ever-elusive population III. This will also be important for studies of primordial deuterium abundances (see below), since deuterium is destroyed, and never created, by star formation and evolution.
High resolution spectroscopy can be used to study high chemical abundances over a large redshift range. In particular, the difficult ionisation corrections required to derive meaningful chemical abundances in Lyman-limit absorbers (where $\log N_{\rm HI} > 17.2$ , so that they are optically thick to Lyman continuum radiation, ) can be avoided using DLAs since the observed hydrogen is probably all neutral [@twl+89; @lwt+91]. Additionally, at high neutral hydrogen column densities, species such as Zn[ii]{} and Cr[ii]{} may become detectable, which are important since depletion onto dust grains is thought to be negligible for the former, whereas the latter remains in the solid phase. This allows both the study of abundances and depletion patterns/dust reddening (see ). Some further reasons why DLAs are of interest are:
1. Studies of the higher order hydrogen Lyman series in DLAs can be used to investigate the primordial deuterium abundance [@wcip91]. The advantage of using DLAs is that the deuterium column density can be somewhat larger than typical Lyman forest absorbers. This may help to discriminate against H[i]{} interlopers mimicking the deuterium line. Two recent observational studies [@ddm01; @pb01] report such D/H measurements.
2. Radio observations of quasars with a sufficiently high radio flux density can provide information complementary to that of the DLA observations: 21cm H[i]{} measurements reveal more detailed kinematic information since line saturation is less severe and provide a direct spin temperature of the cool component of the gas. Different radio and optical morphology of the background quasar also provides the opportunity of observing along slightly different sight-lines through the same absorption complex, with the potential of learning about the relative sizes of optical/radio emission regions and the cloud size of the absorbing gas. In those rare cases where the host quasar has a sufficiently strong millimetre flux and a foreground molecular cloud occults the quasar [@wc94; @wc96], a wealth of detailed chemistry is revealed [@gpb+97; @cw99].
3. Studies of high redshift dust in DLAs gives a handle on the chemical evolution and star formation rates at various cosmological epochs [@pfh99] through the contribution of dust to quasar spectral energy distributions ().
4. Certain heavy element transitions provide cosmological probes of special interest. For example, species with ground and excited state transitions sufficiently close to each other in energy provide a unique means of measuring the cosmic microwave background temperature at high redshift [@bjl73; @mbc+86; @spl00].
5. Finally, recent detailed studies of the relative positions of heavy element atomic optical transitions and comparison with present day (laboratory) wavelengths, suggests that the fine-structure constant ($\alpha\equiv e^2/\hbar c$) may have evolved with time. Inter-comparing atomic optical transitions with H[i]{} 21cm and molecular millimetre transitions may yield an order of magnitude over the already highly sensitive optical results [@cs95; @dwbf98; @mwf+00]. However, very few such constraints are available due to the paucity of quasar absorption systems where 2 of the 3 types of transition (optical atomic, H[i]{} 21cm or molecular millimetre) exist.
It is this last point which is of interest to us: As well as providing a comprehensive list of these objects for use by the astronomical community in general (Section 2), this catalogue allows us to shortlist those DLAs most likely to exhibit radio absorption lines in order to further constrain the variation in fine-structure constant. In the final sections we present the DLAs occulting radio-loud quasars along with any radio absorption features published and outline our future plans regarding the sample.
Explanatory comments and references
===================================
In Table 1, for the column density we note that argue that when $N_{\rm HI}$ is estimated directly from the absorption line equivalent width, the value is systematically underestimated when compared to the estimate derived using Voigt profile fitting. This may be due to a biases associated with estimating the quasar continuum due the presence of Lyman forest absorption lines. The value of $N_{\rm HI}$ quoted in the table in this paper are those values reported in the original sources, and are not corrected using the relation given in Corbelli, Salpeter & Bandiera (2001).
Note also that the visual magnitude is obtained from the DLA reference, [Véron-Cetty]{} & [Véron]{} (2001) or, failing these, NED, which gives an approximate value. For the radio flux densities, $S_{0.4}$ is either the 0.33 GHz WENSS, 0.37 GHz Texas or 0.41 GHz MRC flux density (see Section 2.2), and where both flux densities are available the 365 MHz value is quoted. In the case of the Texas survey, “X” denotes that the quasar was not detected at the flux density limit of 0.25 Jy ([Douglas]{} [et al.]{} 1992). $S_{1.4}$, are the measured 1.4 GHz, continuum flux densities in Jy and for $S_{higher}$, the frequency in GHz is given in parenthesis. In the appropriate columns, “X” also designates if the object is not considered a 2.7 or 5.0 GHz radio source ([Véron-Cetty]{} & [Véron]{} 2001) and where the 22 and 37 GHz values are quoted as approximate refers to the average value from 15 years of monitoring by [Teräsranta]{} [et al.]{} (1998). In general, approximate values are used when the flux density is obtained from more than one reference and the values given do not exactly agree. Finally, “–” denotes that no information could be found.
The DLA references
------------------
The DLAs are compiled from $^{1}$, $^{2}$, $^{3}$, $^{4}$, $^5$, $^6$, $^7$, $^{8}$, $^9$, $^{10}$, $^{11}$, $^{12}$, $^{13}$, $^{14}$, $^{15}$, $^{16}$, $^{17}$, $^{18}$, $^{19}$, $^{20}$, $^{21}$, $^{22}$, $^{23}$, $^{24}$, $^{25}$, $^{26}$, $^{27}$, $^{28}$, $^{29}$, $^{30}$, $^{31}$, $^{32}$, $^{33}$, $^{34}$, $^{35}$, $^{36}$, $^{37}$, $^{38}$, $^{39}$, $^{40}$, $^{41}$, $^{42}$, $^{43}$, $^{44}$, $^{45}$, $^{46}$, $^{47}$, $^{48}$, $^{49}$, $^{50}$, $^{51}$, $^{52}$, $^{53}$, $^{54}$, $^{55}$, $^{56}$, $^{57}$, $^{58}$, $^{59}$, the radio selected QSO survey of $^{60}$ and finally the five new DLAs at $z>3$ occulting PSS quasars of $^{61}$.
Note that several of the DLA citations do not give the background quasar’s coordinates, although we did manage to get some (but not all) of these from the authors. For the remaining few we obtained the coordinates from elsewhere (NED, SIMBAD, ). However, note that when checking the coordinates for the DLAs without published magnitudes against optical images (Digitized Sky and APM Surveys)[^2], we found no optical counterparts at the limiting APM magnitudes of 21.5 (North) and 22.5 (South) for QSOs 0112–30, 0115–30, 1052+04, 1159+01 (Hazard & McMahon, unpublished) nor QSO 1338+101 (Hazard & Sargent, unpublished) and so we have no knowledge of where these coordinates originally came from. Where we have failed to find optical coordinates for any of these sources, “[*not published*]{}” is inserted into the table. We have kept these sources in the catalogue for the sake of providing a full comprehensive list and, for the sake of consistency, we do not include the best radio positions as this could prove misleading.
The radio references
--------------------
The radio parameters of the background quasar supplying the continuum emission are compiled from $^{a}$, $^{b}$, $^{c}$, $^{d}$, $^{e}$, $^{f}$, $^{g}$, $^{h}$, $^{i}$, $^{j}$, $^{k}$, $^{l}$, $^{m}$, $^{n}$, $^{o}$, $^{p}$, $^{q}$, $^{r}$, $^{s}$, $^{t}$, $^{u}$, $^{v}$, $^{w}$, $^{x}$, (and references therein)$^{y}$, $^{z}$ and current versions of the Parkes (, incorporating the 408 MHz flux densities from the MRC catalogue; )$^{P}$, Green Bank [@wb92]$^{G}$, FIRST [@bwh95]$^{F}$, Texas [@dbb+96]$^{T}$, PMN [@gwbe94; @wgbe94; @gwbe95; @wgh+96]$^{pmn}$, WENSS [@rtd+97]$^{we}$, NVSS [@ccg+98]$^{N}$, $^{K}$ and $^{V}$ catalogues, and most recently, the MAMBO survey of the highest redshift PSS quasars [@ocb+01]$^{M}$. Finally, the results of $^{W}$ give new DLAs as well as the FIRST radio luminosities of the quasars illuminating these.
Note that no radio information for the quasars in Table 1 was found in the 5 GHz catalogue of , the optically quiet quasar search of , the S5 radio source catalogue of , the 100 and 150 GHz observations of Southern flat-spectrum sources [@bcb+97], the GHz peaked sample of nor the flat-spectrum sources of .
(20,20) (17.8,21.2)
(20,20) (17.8,17.7)
(20,20) (17.8,17.2)
(20,20) (17.8,17.7)
(20,20) (17.8,15.7)
(20,20) (17.8,17.2)
(20,20)(17.8,17.2)
(20,20) (17.8,16.2)
(20,20) (17.8,17.7)
(20,20) (17.8,17.7)
(20,20) (17.8,16.2)
(20,20) (17.8,17.7)
Discussion
==========
As mentioned in the introduction, we have compiled this catalogue since the comparison between optical and radio absorption lines can provide a considerably more precise determination of $\Delta\alpha/\alpha$: To a first approximation, the ratio of two optical transition frequencies used in the many-multiplet method [@dfw98b; @wfc+98] is $\frac{\omega_1}{\omega_2} \propto
1+0.1\alpha^2$. However, the ratio of the hyperfine neutral hydrogen (21cm) to an optical resonance transition frequency is directly proportional to $\alpha^2$ about 10 times larger. Thus, a substantial improvement in the determination of any variation of $\alpha$ could be made by obtaining further statistics from optical [*and*]{} 21 cm lines in cosmological absorbers. The limit on the variation of $\alpha$ can be obtained by the comparison of the 21 cm line with any other optical or radio line (Section 3.2). However, by using redshifted 21cm together with $\alpha$-sensitive species such as iron, zinc, chromium and nickel [@dfw98b], frequently seen in DLAs, we simultaneously maximise sensitivity and take advantage of the different signs of the frequency shifts due to $\alpha$ variation to help minimise systematic effects [@mwf+01b; @wmf+01].
A new systematic effect which applies to tests for $\Delta\alpha/\alpha$ involving a & optical comparison involves the possible different spatial characteristics of the radio and optical quasar emission. Large differences can result in the radio and optical light probing slightly different lines-of-sight. However, we note that there are examples where the radio and optical emission is known to coincide spatially, and those cases are clearly of particular interest (Section 3.1). In order to minimise the spatial segregation problem, the most reliable tests will come from comparing lines with neutral atomic or molecular species, or singly ionised species where the ionisation potential is smaller than that for neutral hydrogen.
Radio-loud quasars illuminating DLAs
------------------------------------
Of the known radio-loud ($S_{\rm radio}\gapp0.1$ Jy) systems, we summarise the current state of searches for atomic and molecular hydrogen (Section 3.2) absorption features. Note that with regard to the spatial distribution of the optical and radio emission, from the NVSS catalogue [@ccg+98], [*unless otherwise stated, the 1.4 GHz emission extends to a radius of $\approx1'$ and the peak emission coincides with the given optical position (Table 1).*]{}
[**PKS 0118–272:**]{} A BL Lac object where failed to detect absorption at $z=0.5579$.
[**\[HB89\] 0149+336:**]{} A gravitational lens candidate for which we could find no reference to radio absorption features.
[**PKS 0201+113:**]{} A gravitational lens where have detected absorption at $z=3.388$.
[**\[HB89\] 0201+365:**]{} No reference to radio absorption features found.
[**\[HB89\] 0215+015:**]{} A BL Lac object where failed to detect absorption.
[**\[HB89\] 0235+164:**]{} A BL Lac object where absorption at $z=0.524$ has been detected (; ). failed to detect CO at the absorption redshift.
[**\[HB89\] 0248+430:**]{} have detected the absorption at the DLA redshift.
[**\[HB89\] 0329–255:**]{} No reference to radio absorption features found.
[**\[HB89\] 0335–122:**]{} No absorption detected [@kc02].
[**PKS 0336–017:**]{} failed to detect absorption at $z=3.0619$.
[**QSO 0347–211:**]{} No reference to radio absorption features found.
[**PKS 0405–331:**]{} As above.
[**QSO 0432–440:**]{} As above. [*No NVSS data available.*]{}
[**\[HB89\] 0438–436:**]{} failed to detect CO in the torus of this AGN ($z=2.852$). [*No NVSS data available.*]{}
[**\[HB89\] 0439–4319:**]{} Tentative absorption detected by in this low redshift source. [*No NVSS data available.*]{}
[**PKS 0454+039:**]{} No [@bw83] nor H$_2$ [@gb99] absorption has been detected. [*No optical/radio offset, but there is a second $30''$ radius radio source centered at 5 s to the West.*]{}
[**\[HB89\] 0458–020:**]{} In this blazar, have detected absorption at $z=2.03945$. No H$_2$ nor CO (molecular) absorption has been detected [@wc94a; @gb99].
[**PKS 0528–250:**]{} failed to detect absorption at $z=2.8110$, although H$_2$ absorption in this DLA [@fcb88; @sp98; @gb99] and CO emission in the $z=2.14$ DLA [@bv93] have been detected. Note that no H$_2$ or CO absorption in either DLA was detected by .
[**\[HB89\] 0537–286:**]{} No reference to radio absorption features found.
[**\[HB89\] 0552+398:**]{} Although Galactic H[i]{} [@dkvh83] and HCO$^+$ [@ll96] absorption have been observed towards this quasar, no reference to absorption at the DLA (or any cosmological) redshift could be found.
[**J074110.6+311200:**]{} In this optically variable quasar, have detected absorption at $z=0.2212$.
[**FBQS J083052.0+241059:**]{} In this blazar, have detected absorption at $z=0.5247$.
[**IRAS F08279+5255:**]{} A gravitational lens in which have detected CO 4$\rr$3 emission at $z=3.911$, the redshift of the source. [*There is a weak central radio source at optical position with two stronger diagonally opposing sources near*]{} 08h31m50s/52d$43'30''$ [*and*]{} 08h31m25s/52d$46'30''$.
[**\[HB89\] 0834–201:**]{} No reference to radio absorption features found for this blazar.
[**QSO 0913+003:**]{} No reference to radio absorption features found.
[**QSO 0933–333:**]{} As above. [*Offset from optical position at*]{} 09h35m08.6s/-33d$32'34''$.
[**\[HB89\] 0938+119:**]{} No reference to radio absorption features found. [*No offset but there is a second source to the South East near*]{} 09h41m20.5s/11d$45''00'$.
[**\[HB89\] 0952+179:**]{} have detected absorption at $z=0.2378$.
[**\[HB89\] 0957+561:**]{} A gravitational lens where no absorption has been detected [@kc02].
[**\[HB89\] 1017+109:**]{} No reference to radio absorption features found. [*Radio position offset $\approx20''$ to the West of the optical centre.*]{}
[**\[HB89\] 1021-006:**]{} No reference to radio absorption features found for this optically variable quasar.
[**RX J1028.6-0844:**]{} No absorption detected [@kc02].
[**PKS 1055–301:**]{} No reference to radio absorption features found. [*Radio position offset $\approx1'$ to the West of the optical centre.*]{}
[**2MASSi J1124428–170517 :**]{} No reference to radio absorption features found. [*Offset slightly from optical position at*]{} 11h24m41.5s/-17d$05'10''$.
[**\[HB89\] 1127–145:**]{} have detected variable absorption at $z=0.3127$ towards this blazar.
[**\[HB89\] 1157+014:**]{} have detected absorption at $z=1.94362$.
[**LBQS 1213+0922:**]{} No reference to radio absorption features found.
[**FBQS J121732.5+330538 :**]{} failed to detect CO absorption at $z=1.9984$.
[**FBQS J122824.9+312837:**]{} failed to detect absorption at $z=1.7945$.
[**PKS B1228–113:**]{} No reference to radio absorption features found.
[**QSO 1230–101:**]{} As above.
[**PKS 1251–407:**]{} As above. [*No NVSS data available.*]{}
[**3C 286:**]{} at $z=0.69215$ by but no H$_2$ absorption has yet been detected [@gb99].
[**\[HB89\] 1331+170:**]{} A blazar where have detected absorption at $z=1.7764$, but failed to detect CO absorption.
[**PKS B1354–107:**]{}. No absorption detected [@kc02] in the $z_{abs}=2.966$ DLA. [*Radio position offset $\approx15''$ to the West of the optical centre.*]{}
[**\[HB89\] 1354+258:**]{} No reference to radio absorption features found.
[**\[HB89\] 1402+044:**]{} No reference to radio absorption features found for this BL Lac.
[**PKS B1418–064:**]{} No reference to radio absorption features found.
[**\[HB89\] 1451–375:**]{} failed to detect absorption in this HST source.
[**3C 336:**]{} No reference to radio absorption features found for this optically variable quasar.
[**PMN J2130–4515:**]{} No reference to radio absorption features found. [*No NVSS data available.*]{}
[**PMN J2134–0419:**]{} No reference to radio absorption features found.
[**\[HB89\] 2136+141:**]{} No reference to radio absorption features found.
[**3C 446:**]{} A blazar not detected in absorption at the DLA [@ck00] nor CO absorption at the quasar redshift [@dcw96].
[**QSO 2311–373:**]{} No reference to radio absorption features found.
[**\[HB89\] 2314–409:**]{} As above. [*No NVSS data available.*]{}
[**MG3 J234456+3433:**]{} failed to detect absorption at $z=2.9084.$
Finally, note that absorption has been observed in the inferred (from metal lines) DLAs [**3 C196**]{}, [**LBQS 1229–0207**]{} [@wlfc95] and [**\[HB89\] 1243–072**]{} [@lb01].
Searching for new radio absorbers
---------------------------------
If we summarise the current absorption results for the DLAs (Table 2), we see that although many of the positive results have very high column densities, this does not appear to be a prerequisite for absorption (FBQS J083052.0+241059). Perhaps also of relevance is the spectral energy distributions (SEDs): Note that all of the GHz peaked sources have high column densities and have all been detected in . Of the two inverted SEDs, one DLA has a high column density whereas the other is relatively low and
[l r c c c l]{} Quasar & $\tau$ & $\log N_{\rm HI}$ & $S$ & S.I. & Notes\
PKS 0118–272 & $<0.007$ & 20.3 & 1.2 & 0.1 &\
PKS 0201+113 & 0.09,0.04 & 21.3 & 0.3 & – & GPS (2.6)\
${\rm[HB89]}$ 0215+015 & $<0.04$ & 19.9 & 0.9 &– & See caption\
${\rm[HB89]}$ 0235+164 & 0.05-0.5 & 21.6 & 1.8 & -0.2 & Inverted\
${\rm[HB89]}$ 0248+430 & 0.20 & 21.6 & 1.2 & – & GPS (2.5)\
${\rm[HB89]}$ 0335–122 & $<0.008$ & 20.8 & 0.8& 0.3 &\
PKS 0336–017 &$<0.005$ & 21.2 & 1.3 & 0.6 &\
${\rm[HB89]}$ 0439–4319 & $<0.007$ & 20.0 & 0.4 & 0.2 &\
PKS 0454+039 & $<0.01$& 20.7 & 0.4 & – & See caption\
${\rm[HB89]}$ 0458–020 & 0.3 & 21.7 & 2.5 & 0.3 &\
PKS 0528–250 & $<0.2$ & 21.2 & 1.9 & 0.5 & For $z_{abs}=2.811$ DLA\
J074110.6+311200 & 0.07 & 21.2 & 1.9 & – & GPS (2.9)\
& & & & &$z_{abs}=0.221$ DLA\
FBQS J083052.0+241059 & 0.007 & 20.3 & 0.8 & -0.2 & Inverted\
${\rm[HB89]}$ 0952+179 &0.013 & 21.3 & 1.2 & 0.3 &\
${\rm[HB89]}$ 0957+561 & $<0.004$ & 20.3 & 0.9 & 1.3 &\
RX J1028.6-0844 & $<0.03$ & 20.1 & 1.7 & 0.9 & $z_{abs}=3.42$ DLA\
${\rm[HB89]}$ 1127–145 & 0.06 & 21.7 &6.2 &– & GPS (1.4)\
${\rm[HB89]}$ 1157+014 & 0.05 & 21.8 & 1.0 & 0.8 &\
FBQS J122824.9+312837 & $<0.05$ & 19.1 & 0.3 & 0 &\
3C 286 & 0.11 & 21.3 & 19.0 & 0.6 &\
${\rm[HB89]}$ 1331+170 & 0.020 & 21.4 & 0.6 & – & See caption\
PKS B1354–107 & $<0.05$ &20.8 & 0.2 & 0 & $z_{abs}=2.996$ DLA\
${\rm[HB89]}$ 1451–375 & $<0.006$ & 20.1 & 1.8 & 0.2 &\
3C 446 &$<0.02$ & 20.9 & 7.4 & 0.5 &\
MG3 J234456+3433 &$<0.04$ & 21.2 & 0.3 & 0.2 &\
\[sum\]
both of the flat SED detections have high column densities. Finally, the two steep spectrum quasars which illuminate DLAs detected in absorption (\[HB89\] 1157+014 and 3C 286) also have high column densities.
Because of the relation between turnover frequency and source size [@ffs+90; @ob97], we may expect a higher absorption detection rate from flat and inverted SED sources, since these result from similar optical and radio lines-of-sight. However, as it stands, the statistics are too small (Table 2) and so in order to maximise our sample, it appears that the way to proceed is an unbiased search for H[i]{} in the DLAs occulting the remaining radio-loud quasars.
As mentioned in Section 1, as well as optical and comparisons, the inter-comparison of atomic and molecular lines will also give a ten-fold increase in accuracy for $\Delta\alpha/\alpha$: Due to its zero dipole moment and small moment of inertia, molecular hydrogen cannot be directly observed at radio frequencies[^3] and so it is the usual practice infer the presence of this from the millimetre rotational lines of such molecules as CO. In order to also take advantage of this, we have applied for time to search for molecular absorption lines in the DLAs occulting mm-loud quasars with the IRAM 30 metre and Swedish ESO Sub-millimetre telescopes. Recently (April 2002), we have been awarded time on the Australia Telescope Compact Array in order to obtain 90 GHz flux measurements for the whole radio-loud sample, as a means of selecting new sources in which to search for millimetre absorption. The results will be published in forthcoming papers.
Also, with regard to finding new systems in which there may be absorption (in all three frequency regimes), we see that 13 of the quasars are known to be BL Lac/optically variable/blazars[^4] and that 3 of the sources are known gravitational lenses. This may be of interest as of the four known high redshift millimetre (molecular) absorbers, two are BL Lac objects; B 0218+357 [@wc95] and PKS 1413+135 [@wc94][^5]. This may suggest several strategies for finding similar new absorbers [@sr99] which could prove useful in appending to this catalogue.
Summary
=======
We have performed an exhaustive search of the literature in order to produce a list of all known damped Lyman alpha systems and their associated radio properties. It is the 57 radio-loud systems in which we are interested as many of these have the potential to show absorption in each DLA. Of the sources searched, it is seen that several exhibit such absorption and we are involved in an ongoing project to search for this in the remaining systems. Not only will this give us radio lines for comparison with optical data in order to constrain any temporal variations in the fine structure constant, but we will have a significant sample from which we could consider why some DLAs absorb in H[i]{} whereas others do not. For example, suggest that low redshift DLAs may arise from a multitude of absorbers, and hence do not have sufficient path length for absorption, while those of higher redshift are due to more compact systems. Finally, as well as finding new H[i]{} absorbers, we hope that this catalogue will prove useful to those using damped Lyman alpha systems as part of their research.\
[**Final note:**]{} In order to retain its usefulness to the astronomical community, we have now produced an on-line version of this catalogue which will be continually updated. This is available from [*http://www.phys.unsw.edu.au/$\sim$sjc/dla*]{}.
Acknowledgments {#acknowledgments .unnumbered}
===============
We wish to thank the John Templeton Foundation for supporting this work, the two anonymous referees and Barry Madore at NED for their comments, Sandra Ricketts of the Anglo-Australian Observatory library as well as Fredrik Rantakyrö, Sara Ellison, Ken Lanzetta and especially Nissim Kanekar for giving us the preliminary results of his survey. This research has made use of the NASA/IPAC Extragalactic Database (NED)[^6] as well as the VizieR database of CDS catalogues [@obm00][^7].
[[D’Odorico]{}, [Dessauges-Zavadsky]{} & [Molaro]{}<2001>]{}
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[^1]: In the catalogue some column densities less than this are to be found. This is because in order to produce the most comprehensive list, we have also included any Lyman alpha absorbers which are designated as possibly damped in the literature (see Section 2).
[^2]: These can be found at [*http://archive.stsci.edu/dss/*]{} and [*http://www.ast.cam.ac.uk/$\sim$apmcat/*]{}, respectively.
[^3]: In the case of $z>1.8$ sources, however, the ultra-violet lines of H$_2$ are redshifted into the optical window, making molecular hydrogen readily observable at these frequencies. As well as for PKS 0528–250 (Section 3.1) molecular hydrogen has also been detected in the DLAs occulting the radio-quiet quasars \[HB89\] 0000-263 [@lmc+00a], LBQS 0013–0029 [@gb97; @psl02], \[HB89\] 0347–383 [@lddm01], LBQS 1232+0815 [@gb97; @spl00] and the inferred [@wlfc95] DLA \[HB89\] 0551–366 [@lsp02].
[^4]: BL Lacs and Optically Violent Variables are known collectively as blazars. In these radio-loud active galactic nuclei the radio jet is relativistically beamed close to the line-of-sight ().
[^5]: The former, as well as PKS 1830–211, is also a gravitational lens [@wc96].
[^6]: Operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
[^7]: [*http://vizir.u-strasbg.fr/local/cgi-bin/vizHelp?faq.htx*]{}
|
[ *”Extended Comment on the Article ”Consistent Development of a Beam-To-Beam Contact Algorithm via the Curve to Solid Beam Contact - Analysis for the Non-Frictional Case”*]{}
Christoph Meier, Alexander Popp, Wolfgang A. Wall
[**published in ”Researchgate” in October, 2018**]{}
[Alexander Konyukhov and Karl Schweizerhof]{}
[*Karlsruhe Institute of Technology. Karlsruhe. Germany.*]{}
A discussion about the existence and uniqueness of the Closest Point Projection (CPP) for surfaces as well as for the point-to-edge (curve) situations has been first formulated for surfaces in the article of Konyukhov and Schweizerhof \[1\] and in the following article \[2\] for the curve-to-curve situation. This formulation is mathematically strict, because it is based on the well known theorem about the existence and uniqueness of the solution for the operator equation in which the operator is strictly convex. As it has been proved in functional as well as in convex analysis, this statement is valid not only for functions, but also for operators formulated in differentiable Hilbert spaces. Application of this fact is widely found in different branches of physics including, of course, mechanics. Thus, in Ogden \[3\] the uniqueness of the nonlinear elasticity problem has been discussed. Namely this strict mathematical result has been employed by us to formulate the statement for the existence and uniqueness of the CPP for all contact situations, see also details in monograph \[4\]. In the case of Curve-To-Curve (CTC) contact, this criterion leads to the positivity for the second derivative of the distance function ${\mathbf{F}}(\xi^1, \xi^1) = \vert \vert {\mbox{\boldmath $\rho$}}(\xi^1) - {\mbox{\boldmath $\rho$}}(\xi^2) \vert \vert $ between two curves ${\mbox{\boldmath $\rho$}}(\xi^1)$ and ${\mbox{\boldmath $\rho$}}(\xi^2)$. For further analysis the distance function is formulated via angles $\varphi^1$ and $\varphi^2$, determining the position of the osculating planes for two spatial curves relatively to the corresponding Serret-Frenet coordinate systems attached to both curves. The positiveness of the second derivative ${\mathbf{F}}"$ is analyzed via the Sylvester criteria for the positivity of the matrix. This statement is a system of nonlinear inequalities and, therefore, should be strictly solved by using also mathematical methods for the solution of the system of inequalities. That is why in the aforementioned articles \[2\], we just geometrically interpreted this statement using the projection domain, determining the position of osculating circles for both curves and the distance between them. In this domain the solution of the CPP procedure exists and is unique.
In the article \[5\] our mathematical statement has been criticized and explained that a more general criterion has been found by them [*(Citation: “...our motivation to perform mathematically concise and rigorous investigation concerning the existence and uniqueness of the corresponding closest-point projection. On the contrary to Konyukhov et. al. who have already treated this question by means of geometrical criteria, we derive a very general analytical criterion that is valid for arbitrary contact formulations and that is based on proper and easy-to-determine control quantities.“)*]{}. However, the system of non-linear inequalities has not been solved and even more restricted criterion, which is valid only for a specific contact situation, has been developed! In order to show this, the following set of mathematically not consistent operations has been performed by authors in \[5\]:
1. the second inequality from the Sylvester criteria, namely $\det {\mathbf{F}}" > 0$ has been estimated as ”the worst estimate”. The ”correctness” of such an operation will be obvious, if one can try to solve even the algebraic inequality $(x-a)(x-b) > 0 $ by ”the worst estimate” method! Any mathematically strict method for the inequality will require, of course, consideration of the positivity for the function $\det {\mathbf{F}}"$.
2. during ”the worst estimate” both angles $\varphi^1$ and $\varphi^2$ are taken such that $\vert \cos \varphi_i \vert = 1$. This operation, however, geometrically leads to the enforcement of two osculating planes to coincide.
3. finally, during the development of the criterion and the worst estimation a ”radius of the beam $R$” (?!) representing a circular cross-section of a beam with a curve mid-line has been inserted.
By using these operations it is claimed that [*a very general analytical criterion for the uniqueness and existence*]{} has been derived. This is definitely not the case! It is rather an approximation of a solution for the criterion for straight thin beams resp. small curvature beams. This solution involves a minimal angle between tangents $\psi \equiv \alpha_{crit} $ at which a switch from Curve-To-Curve (CTC) to another algorithm Point-To-Curve (PTC) is performed. This is an adaptive switch which was claimed to provide a smooth transition between PTC and CTC algorithms providing existence and uniqueness for the CPP procedure.
In the following article \[6\], we emphasized the fact, that a very general criterion by no means can be derived by performing ad-hoc and not consistent mathematical operations. First of all, none of the well known methods for solutions of the system of nonlinear inequalities has been employed, and, second, the original problem must be formulated in differential geometry for the shortest distance between curves (which have no thickness (or radius of the beam $R$) at all!!!). In addition, already in the article \[7\] we emphasized, that the beam-to-beam contact algorithm can be based also on a Point-To-Curve (PTC) projection procedure. The latter leads to another more easier described projection domain only for one curve – this algorithm has been first derived already in \[1\]. Both statements together with other cases have been widely discussed a while ago in our book \[4\]. Further scientific discussions among researchers, working in contact mechanics (including conferences), have been rising a question – in which cases PTC or CTC algorithms are applicable and how to switch between them. Thus, Litewka in \[8\] proposed ad-hoc criteria combining PTC and CTC together based on the angle between straight lines resp. straight linear finite elements.
In our last article we ourselves still did not solve the system of inequalities in order to reach a general criterion for curves, instead of this, we have found out cases of multiple solutions for CPP for CTC which are fully independent from the introduced before critical angle $\psi \equiv \alpha_{crit} $. Such cases are general parallel curves constructed so that the distance between them is constant $ {\mathbf{F}}= const $, and, therefore, the CTC contact algorithm, requiring both ${\mathbf{F}}' \ne 0$ and ${\mathbf{F}}" \ne 0$, is not applicable. The angle $\psi$ between tangents for curves can be then arbitrary. Our presented Curve-To-Surface formulation is just an alternative formulation for beam-to-beam contact with a general elliptical cross-section, which – however – works in general for parallel curves contrary to CTC. Only mathematically strict methods are involved in this formulation, in which we further do not find any of the, so-called, ”errors” – named in the extended comment of Meier et.al. In addition we also found out that the nature of switching between PTC and CTC lays on the full 3D contact Hertz formulation – as a physical background – rather than on angle $\psi$ between tangent lines as an algorithmic background, proving then the ad-hoc criteria already proposed by Litewka in \[8\].
This and only this was a point our remark. The ABC formulation as an algorithm for contact of slightly curved beams is not criticized. In fact the ABC formulation includes as other known algorithms, see e.g. Litewka \[8\], both CTC and PTC formulations and being programmed in one package will always work, because on secure side PTC will be switched on. We refer again to our article \[7\] and book \[4\], as we have also in our programs such a scheme.
Summarizing our discussions, in [*the Extended Comment on the Article \[9\]*]{}, that we received, we again did not find any general solution of CPP including any method of solution of inequalities either. This is rather a confirmation of the specific nature of the contact criterion and the corresponding solution plus algorithmic scheme. Surprisingly, the authors started to criticize our alternative Curve-To-Solid Beam formulation, which is based on the CPP for surfaces and not based neither on CPP for PTC, nor on CPP for CTC as a closer look should have revealed. While doing so, the aforementioned authors are apparently not fully following the logic in our paper. Thus, as an example, while criticizing our parallel curves, they claim in several places that it is not possible to construct any parallel curves with an arbitrary angle $\psi$ between tangent lines. However, we can propose a simple geometrical example without any mathematics: We just have to remember that ${\mathbf{F}}(\xi^1, \xi^2)$ is a distance function between two curves – so one can easily wind a wire on a cylinder at any desired angle $\psi$ – in this case the distance between the axis of a cylinder and the mid-curve of a wire will be constant ${\mathbf{F}}=r=const$. Thus, any derivatives of ${\mathbf{F}}$ are zero (necessary condition of minimum ${\mathbf{F}}'=0$ is fulfilled) and the criterion of uniqueness ${\mathbf{F}}''=0$ is spoiled – the solution is, of course, by construction multiple – at any point the distance is constant, however the angle between the axis of a cylinder and the tangent line to a wire is definitely arbitrary. Namely, this case is fully illustrating insensibility of the introduced in \[5\] criterion based on $\alpha_{crit}$ and $\mu_{max}$.
Beyond that we do not want neither comment the counter critics nor go to further debates, but suggest to check the corresponding mathematical background in differential geometry \[10\].
We are finally again emphasizing that the problem of existence and uniqueness of CPP for CTC is not yet fully resolved and can be addressed to mathematicians experienced in differential geometry and numerical analysis.
References {#references .unnumbered}
==========
[9]{}
Konyukhov A., Schweizerhof K., On the solvability of closest point projection procedures in contact analysis: Analysis and solution strategy for surfaces of arbitrary geometry, *Computer Methods in Applied Mechanics and Engineering*, Vol. 197 (33-40), 2008, 3045–3056, https://doi.org/10.1016/j.cma.2008.02.009.
Konyukhov A., Schweizerhof K., Geometrically exact covariant approach for contact between curves, *Computer Methods in Applied Mechanics and Engineering*, Vol. 199 (37-40), 2010, 2510–2531, https://doi.org/10.1016/j.cma.2010.04.012.
Ogden R.W. *Non-Linear Elastic Deformations*. Dover Publications, Inc. Mineola New-York, 1984, 560 p.
Konyukhov A., Schweizerhof K. *Computational Contact Mechanics. Geometrically exact theory for arbitrary shaped bodies*. Springer. Heidelberg, New York, Dordrecht, London, 2013, 443 p.
Meier Ch., Popp A., Wall W., A finite element approach for the line-to-line contact interaction of thin beams with arbitrary orientation, *Computer Methods in Applied Mechanics and Engineering*, Vol. 308, 2016, 377–413, https://doi.org/10.1016/j.cma.2016.05.012.
Konyukhov A., Mrenes O., Schweizerhof K. Consistent development of a beam‐to‐beam contact algorithm via the curve‐to‐solid beam contact: Analysis for the nonfrictional case. *International Journal for Numerical Methods in Engineering*, Vol. 113 (7), 2018, 1108–1144, https://doi.org/10.1002/nme.5701
Konyukhov A., Schweizerhof K. Geometrically exact theory for contact interactions of 1D manifolds. Algorithmic implementation with various finite element models, *Computer Methods in Applied Mechanics and Engineering*, Vol. 205-208, 2012, 130–138, https://doi.org/10.1016/j.cma.2011.03.013.
Litewka P. Enhanced multiple-point beam-to-beam frictionless contact finite element, *Computational Mechanics*, 2013, Vol. 52 (6), 1365–1380, https://doi.org/10.1007/s00466-013-0881-4
Meier Ch., Popp A., Wall W., Extended Comment on the Article ”Consistent Development of a Beam-To-Beam Contact Algorithm via the Curve to Solid Beam Contact - Analysis for the Non-Frictional Case”, *Published in Researchgate, Submitted in arXiv:1810.10858 \[cs.CE\]*, https://arxiv.org/abs/1810.10858, Oct. 2018.
Kreysig E. *Differential Geometry*, Dover, Mineola, New-York, 1991, 336 p.
|
---
abstract: 'We perform a calculation of the mass distribution in the $\psi(3770) \to \gamma D \bar D$ decay, studying both the $D^+ D^- $ and $D^0 \bar D^0 $ decays. The electromagnetic interaction is such that the tree level amplitude is null for the neutral channel, which leaves the $\psi(3770) \to \gamma D^0 \bar D^0$ proceeding through a loop involving the $D^+ D^- \to D^0 \bar D^0$ scattering amplitude. We use the results for this amplitude of a theoretical model that predicts a $D \bar D$ bound state and find a $D^0 \bar D^0 $ mass distribution in the decay drastically different than phase space. The rates obtained are relatively large and the experiment is easily feasible in the present BESIII facility. The performance of this experiment could provide an answer to the issue of this much searched for state, which is the analogue of the $f_0(980)$ resonance.'
author:
- Lianrong Dai
- Genaro Toledo
- Eulogio Oset
title: 'Searching for a $D \bar D$ bound state with the $\psi(3770) \to \gamma D^0 \bar D^0$ decay'
---
Introduction
============
Molecular states made from mesons or mesons and baryons have become some of the important objects in the present plethora of hadronic states. Reviews on this topic can be found in [@ramosoller; @review] and more recently in [@olsen; @karliner; @huaxing; @fengkun]. One of the interesting predicted states is a bound state of $D \bar D$ [@dani; @pavon; @hidalgo] for which there is no clear experimental evidence so far. The state is analogous to the $K \bar K$ bound state which was claimed in [@isgur] to be a good representation of the $f_0(980)$ state. This early claim found support later with the results of the chiral unitary approach for the meson-meson interaction [@npa; @kaiser; @markushin; @juanenri], where the meson meson interaction in coupled channels was studied, and other states, as the $f_0(500)$, $K^*_0(700), a_0(980)$, were also found dynamically generated from that interaction.
There has been some search for this state and in [@daniee] it was shown that an accumulation of strength close to the $D \bar D$ threshold in the $ e^+e^-\to J/\psi D \bar D$ reaction [@pakhlov] found a natural explanation in terms of the predicted bound state of [@dani] with a mass of 3730 MeV, with large uncertainties. Hopes where raised that an update of the experiment in [@chilikin] would constrain the predictions, but it was shown in [@wangliang] that this is not the case, and there is a large ambiguity in the conclusions.
In view of this, there have been some works proposing new reactions that would give evidence for this elusive state. In [@danizou] the radiative decay of the $\psi(3770)$ resonance into $\gamma$ and the $D \bar D$ bound state was proposed and the feasibility of the reaction with present production rates of the $\psi(3770)$ was assessed. There is of course the problem of which should be the ideal channel to observe the bound state. In [@xiaothree] three different reactions were suggested to detect that state. In [@daixie] the $B^0 \rightarrow D^0 \bar{D}^0 K^0$ , $B^+ \rightarrow D^0 \bar{D}^0 K^+$ reactions were suggested to find evidence for the state, looking into the mass distribution of $D \bar D$ production close to threshold. The analysis found a good agreement with experiment for the $B^+ \rightarrow D^0 \bar{D}^0 K^+$ reaction, but it was shown there that the $B^0 \rightarrow D^0 \bar{D}^0 K^0$ reaction was better suited to search for the $D \bar D$ state, because there is no tree level contribution in this later reaction, and hence the amplitude for that mechanism is proportional to the $D \bar D$ amplitude which contains the pole of the $D \bar D$ bound state. The mass distribution then was rather different from that of phase space, and its precise measurement close to threshold should give and answer to the question.
In the present work we wish to combine the lessons drawn from the works of [@danizou] and [@daixie] and study the $D \bar D $ mass distribution close to threshold from the $\psi(3770) \to \gamma D \bar D$ decay. Anticipating the results, we will find an interesting situation in which the $D^0 \bar D^0 $ production does not proceed at tree level, while $D^+ D^- $ has contribution from tree level. As a consequence, the $D^0 \bar D^0 $ production is directly influenced by the $D \bar D$ pole below threshold and exhibits a behavior close to threshold very different from phase space. For the $D^+ D^- $ production the tree level part is very important and the behavior is quite different, and in addition it shows the infrared divergence behavior when the photon energy goes to zero, which, again, is not the case for $D^0 \bar D^0 $ production. The reaction is, thus, suited for investigation of the $D \bar D$ bound state and the present rates of $\psi(3770)$ production make the experimental investigation feasible.
Formalism
=========
The state $\psi(3770)$ decays into $D\bar D$ [@pdg] with a width $\Gamma=27.2$ MeV, 52% of which goes to $D^0 \bar D^0$ and 41% to $D^+D^-$. Its shape in $e^+e^-$ production and the decay width have been object of intense study [@bes1; @bes2; @babar; @coito; @qzhao1; @qzhao2; @qixin]. In [@qixin] the $c\bar c$ component is allowed to get hadronized into meson-meson components, and the strength of the hadronization is fitted to the $\psi(3770)$ lineshape. One of the conclusions in [@qixin] is that from the experimental data one can induce that the $\psi(3770)$ is largely a $c\bar c$ state and the weight of the meson-meson components is only of the order of 15%. The $\psi(3770)$ state bears some similarity to the $\phi(1020)$, which is assumed to be a $s\bar s$ state and decays into $K\bar K$. The decay mode $\psi(3770) \to D \bar D \gamma$ necessarily has much resemblance to the $\phi \to \gamma K \bar K, \ \gamma \pi^0 \pi^0$ decays, which have also been a subject of much study [@bramon; @kim; @colan; @achasov; @pancheri; @lucio; @marco; @palomar]. As in the $\phi \to \gamma \pi^0 \pi^0$ reaction, which does not proceed via tree level, we shall also see that $\psi(3770) \to \gamma D^0 \bar D^0 $ does not get contribution from tree level and both processes proceed via a similar loop mechanism.
Tree level for $\psi(3770) \to \gamma D^+ D^- $
-----------------------------------------------
The tree level mechanism in $\psi(3770) \to \gamma D^+ D^- $ is shown diagrammatically in Fig. \[Fig:1\].
![Mechanisms for $\psi \to \gamma D^+ D^- $: (a), (b), $D$ pole mechanisms; (c) contact term demanded by gauge invariance. In parenthesis the momenta of the particles.[]{data-label="Fig:1"}](DDfig1.pdf)
The $\psi \to D^+ D^- $ elementary vertex is given by $$\label{eq:vertex}
-it_{\psi D^+D^-} = -i g_\psi (p_{D^+}-p_{D^-})_\mu \epsilon^\mu(\psi) \,.$$
The $\psi \to D^+ D^- $ decay width is given by $$\label{eq:width}
\Gamma_\psi=\frac{1}{8\pi}\frac{1}{M_\psi^2}|{\bm q}| \overline{\sum} \sum\vert t \vert^2 \,,$$ where the sum and average of $|t|^2$ calculated from Eq. (\[eq:vertex\]) gives $$\label{eq:sum}
\overline{\sum}\sum \vert t \vert^2=\frac{4}{3} \, g_\psi^2 \,{\bm q}^2 \,,$$ and ${\bm q}$ is the $D^+$ momentum in the $\psi$ decay at rest. Adjusting to the experimental $D^+D^-$ decay width we find $$\label{eq:coupling}
g_\psi=13.7$$ Considering also the $\gamma D^+D^-$ coupling $D^+(p_{D^+}) \gamma \to D^+(p'_{D^+})$ $$\label{eq:VDDgamma}
it_{\gamma D^+D^-}=-i\,e\,(p_{D^+}+p'_{D^+})_\mu \,\epsilon^\mu(\gamma) \,,$$ with $e$ the electron charge, $e^2/4\pi=\alpha=1/137$, the $\psi \to \gamma D^+ D^- $ amplitude of the diagram of Fig. \[Fig:1\] is given by $$\begin{aligned}
\label{eq:treeamplitude}
t_a+t_b+t_c =
-2\,e\,g_\psi\,\epsilon^\mu(\psi) \epsilon^\nu(\gamma)
\left( g_{\mu\nu}+{p_2}_\mu {p_1}_\nu \frac{1}{p_1\cdot k+i\epsilon}
+{p_1}_\mu {p_2}_\nu \frac{1}{p_2\cdot k+i\epsilon}
\right) \,,\end{aligned}$$ where the term $g_{\mu\nu}$ corresponds to the diagram of Fig. \[Fig:1\](c) and is introduced to respect gauge invariance. The photon has zero coupling to $D^0 D^0$ and hence there is no tree level for $\psi \to \gamma D^0 \bar D^0$.
Loop mechanism
--------------
There is, however, a loop mechanism that allows the $\psi \to \gamma D^0 \bar D^0$ decay which is depicted in Fig. \[Fig:2\].
![Loop mechanisms for $\psi \to \gamma D \bar D $ production. In parenthesis the momenta of the particles.[]{data-label="Fig:2"}](DDfig2.pdf)
This follows exactly the same trend as in [@bramon; @kim; @colan; @achasov; @pancheri; @lucio; @marco; @palomar] for $\phi \to \gamma \pi^0 \pi^0$, where the intermediate state is $K^ + K^-$ and the final $D\bar D$ are replaced by $\pi^0 \pi^0$. The diagram of Fig. \[Fig:2\](d) is also demanded by gauge invariance of the loops. Gauge invariance plays an important role in this process and thanks to it there is an efficient computational scheme which requires only the evaluation of diagrams (a) and (b) of Fig. \[Fig:2\], which give the same contribution, and shows that the result of the loop integral is convergent [@pancheri; @pestiu; @close]. The derivation goes as follows: The full amplitude for the diagrams of Fig. \[Fig:2\] has the structure $$\label{eq:loopamp}
t_L= \epsilon_\mu(\psi)\epsilon_\nu(\gamma)\,T^{\mu\nu}$$ and $T^{\mu\nu}$ must be a tensor that can be written in terms of the two independent momenta $P$ and $k$, the momentum of the $\psi$ and $\gamma$ respectively. The most general form for $T^{\mu\nu}$ is given by $$\label{eq:Tmunu}
T^{\mu\nu}=a\,g^{\mu\nu}+ b\,P^\mu P^\nu+ c\,P^\mu k^\nu+ d\,k^\mu P^\nu +e\,k^\mu k^\nu \,.$$ Gauge invariance, substituting $\epsilon_\nu(\gamma)$ by $k_\nu$ and demanding $$\label{eq:gaugeinv}
T^{\mu\nu} k_\nu=0$$ leads to $$\label{eq:TmunuGI}
a\,k^{\mu}+ b\,P^\mu (P\cdot k)+ d\,k^\mu (P\cdot k)=0$$ which implies two independent equations $$\begin{aligned}
\label{eq:conditions}
a+d\,(P\cdot k)=0\\
b=0.\end{aligned}$$ The $b$ term does not contribute because of the Lorentz condition $\epsilon_\mu(\psi) P^\mu=0$ and the $c$ and $e$ terms do not contribute because of the Lorentz condition on the photon $\epsilon_\nu(\gamma) k^\nu=0$. Hence, only the $a$ and $d$ terms of Eq. (\[eq:Tmunu\]) contribute to the amplitude and it is enough to calculate only the $a$ or $d$ coefficient. It is easy to see that only the diagrams (a), (b) of Fig. \[Fig:2\] contribute to the $d$ coefficient and since two external momenta $P^\mu k^\nu$ are factorized out of the integral, for dimensional reasons this means two powers of $q$ less in the integral, which renders it convergent. In addition, if we work at the end, as we do, in the Coulomb gauge, $\epsilon^0(\gamma)=0$, ${\bm \epsilon}(\gamma)\cdot {\bm k}=0$, then the term $dP^i k^j \epsilon_j(\psi)\epsilon_i(\gamma)=0$ in the $\psi$ rest frame, and the whole amplitude is given by $$\label{eq:finalamp}
t=a \epsilon_\mu(\psi)\epsilon^\mu(\gamma) \,;\qquad \ a =-d\,(P \cdot k) \,$$ It is customary to perform the integration of the loop integral using Feynman parametrization, but here we must divert from this formalism because the $D\bar D \to D\bar D$ scattering matrix regularized with a cut off, $q_{max}$, transfers a structure $\Theta(q_{max}-|{\bm q}|)\Theta(q_{max}-|{\bm p}|)$ to the $T$ matrix [@danijuan] and we must implement a cut off in the loop integral. On the other hand, we can benefit from the fact that the $D$ mesons are heavy particles, they are close to on-shell in the loops and we can just keep the positive energy part of their propagators $$\begin{aligned}
D({q}) \to \frac{1}{q^2-m_D+i\epsilon} &\equiv&
\frac{1}{2\omega({\bm q})}
\left(
\frac{1}{q^0-\omega({\bm q})+i\epsilon}
-\frac{1}{q^0+\omega({\bm q})-i\epsilon}
\right)\label{eq:propagator}\\
&\to&
\frac{1}{2\omega({\bm q})}
\frac{1}{q^0-\omega({\bm q})+i\epsilon}\end{aligned}$$ with $\omega({\bm q})=\sqrt{{\bm q}^2+m_D^2}$.\
The contribution of the two diagrams of Fig. \[Fig:2\](a), \[Fig:2\](b) with $D^0\bar D^0$ in the final state is given by $$\begin{aligned}
\label{eq:amp1}
-it_L&=&2\int
\frac{d^4q}{(2\pi)^4}
(-i)g_\psi (P-q-q)_\mu \epsilon^\mu(\psi)
(-ie)(P-q+P-q-k)_\nu \epsilon^\nu(\gamma) (-i)t_{D^+D^- \to D^0\bar D^0}\nonumber\\
&&\times \frac{i}{(P-q)^2-m_D^2+i\epsilon}
\frac{i}{q^2-m_D^2+i\epsilon}
\frac{i}{(P-q-k)^2-m_D^2+i\epsilon},\end{aligned}$$
$$\begin{aligned}
\label{eq:amptot4}
t_L&=&-2e g_\psi \epsilon^\mu(\psi) \epsilon^\nu(\gamma) i \int \frac{d^4q}{(2\pi)^4} 2q_\mu (2P-2q)_\nu t_{D^+D^- \to D^0\bar D^0} \nonumber\\
&&\times \frac{1}{(P-q)^2-m_D^2+i\epsilon}\frac{1}{q^2-m_D^2+i\epsilon}\frac{1}{(P-q-k)^2-m_D^2+i\epsilon},\end{aligned}$$
where $t_{D^+D^- \to D^0\bar D^0}$ is a function of the $ D^0\bar D^0$ invariant mass. We can now take the propagators of Eq. (\[eq:propagator\]) and perform the $q^0$ integration analytically using Cauchy’s integration, and we find, keeping only the $\epsilon^i(\gamma)$ transverse components that we shall have in the Coulomb gauge ($i, j$=1, 2, 3) $$\begin{aligned}
\label{eq:amptot3}
t_L&=&-2e g_\psi t_{D^+D^- \to D^0\bar D^0} \epsilon^i (\psi) \epsilon^j (\gamma)\, 4 \int \frac{d^3q}{(2\pi)^3} \frac{1}{2\omega({\bm q})}\,\frac{1}{2\omega({\bm P}-{\bm q})} \,\frac{1}{2\omega({\bm P}-{\bm q}-{\bm k})}\nonumber\\
&\times& (q_iP_j-q_iq_j)
\frac{1}{P^0-\omega({\bm q})-\omega({\bm P}-{\bm q})+i\epsilon}
\frac{1}{P^0-\omega({\bm q})-k^0-\omega({\bm P}-{\bm q}-{\bm k})+i\epsilon}\end{aligned}$$ and in order to get the $d$ coefficient we must look at the $k^iP^j$ component of this integral. The first term in Eq. (\[eq:amptot3\]), with $q_i P_j$, provides this structure immediately since when ${\bm P}\to 0$ for the rest frame of the $\psi$, the integral only depends on $k$, hence $\int d^3q \, q_i f({\bm q},{\bm k})=k_i \int d^3q \,({\bm k} \cdot {\bm q})/ {\bm k}^2 f({\bm q},{\bm k})$. The second integral for $q_i q_j$ is a bit more involved, and since we will have ${\bm P}\to 0$ at the end, we can make an expansion for ${\bm P}$ small of all the terms. Then we have an integral at the end of the type $$\begin{aligned}
%\int d^3q \,q_i \,q_j \,q_l \,p_l f(\vec{q},\vec{k})
\int d^3q q_i q_j q_l P_l f(\bm{q},\bm{k})=P_l\{A'(\delta_{ij} k_l+\delta_{il}k_j+\delta_{jl}k_i)+B' k_i k_j k_l\}\end{aligned}$$ and only the $\delta_{jl} k_i$ will contribute to the term $k_i P_j$ that we look for. Finally we find $$\begin{aligned}
d=d_1+d_2\end{aligned}$$ with $$\begin{aligned}
d_1=-8 \, e\, g_\psi \, \frac{1}{{\bm k}^2} \, \int \frac{d^3 q}{(2\pi)^3} \, {\bm q}\cdot {\bm k} \,
\frac{1}{2 \omega_1} \, \frac{1}{2 \omega_1} \,\frac{1}{2 \omega_2}
\frac{1}{P^0-2\omega_1 +i\epsilon}
\frac{1}{P^0- k^0-\omega_1 - \omega_2 +i\epsilon} \, t_{D^+ D^-,D^0 \bar{D}^0} \,\end{aligned}$$ $$\begin{aligned}
d_2 & =& 4 \, e\, g_\psi \, \frac{1}{{\bm k}^2}\, \int \frac{d^3 q}{(2\pi)^3} \, {\bm q}\cdot {\bm k} \,
\left\{{\bm q}^2-\frac{({\bm q}\cdot {\bm k})^2}{{\bm k}^2} \right\}
\,\frac{1}{2 \omega_1} \, \frac{1}{2 \omega_1} \, \frac{1}{2 \omega_2}
\frac{1}{P^0-2\omega_1 +i\epsilon} \, \frac{1}{P^0- k^0-\omega_1 - \omega_2 +i\epsilon} \, \nonumber\\
&\times& \left\{ \frac{1}{ \omega_1^2} + \frac{1}{ \omega_2^2}-\frac{1}{\omega_1(P^0-2\omega_1 +i\epsilon)}
-\frac{1}{\omega_1(P^0- k^0-\omega_1 - \omega_2 +i\epsilon)} \right\} \, t_{D^+ D^-,D^0 \bar{D}^0} \,
\end{aligned}$$ with $\omega_1=\sqrt{\bm{q^2}+m_D^2}$ and $\omega_2=\sqrt{(\bm{q}+\bm{k})^{2}+m_D^2}$. The loop $t$ matrix for $\psi(3770) \rightarrow \gamma D^0 \bar{D}^0$ can then be put as $$\begin{aligned}
\label{eq:dpsi}
t_{\psi(3770) \rightarrow \gamma D^0 \bar{D}^0}=d \,M_{\psi}k \bm{\epsilon} (\psi) \cdot \bm{\epsilon} (\gamma) \,.\end{aligned}$$ For the case of $\psi(3770) \rightarrow \gamma D^+ D^-$ we have the same formalism for the loop substituting $t_{D^+ D^-,D^0 \bar{D}^0}$ by $t_{D^+ D^-,D^+ D^-}$, but we have to add the tree level contribution of Eq. (\[eq:treeamplitude\]).
Given the structure of the amplitude it is convenient to evaluate the phase space in terms of the energy of the photon and the $D^0$ both in the $\psi$ rest frame and we obtain at the end,
$$\begin{aligned}
\frac{d \Gamma}{d M_{\rm inv}(D^0 \bar{D}^0)}= \frac{1}{8 M^2_\psi} \,
\frac{M_{\rm inv}(D^0 \bar{D}^0)}{(2\pi)^3} \int dE_1 \overline{\sum}\sum |t|^2 \, {\Theta(1-A^2)} \, {\Theta(M_\psi-k-E_1)} \,,\end{aligned}$$
where $E_1$ is the energy of the $D^0$ and $A$ is the cosine of the angle between the photon and $D^0$ given by $$\begin{aligned}
\label{eq:Ac}
A \equiv \cos\theta{({\bm p_1},{\bm k})}=\frac{1}{2 p_1 k}\left\{(M_\psi-k-E_1)^2-m^2_D-{\bm p^2_1}-{\bm k^2}\right\}\end{aligned}$$ The sum and average over spins of $|t|^2$ is given for the case of $D^+ D^-$ production by $$\begin{aligned}
\label{eq:tt1}
\overline{\sum}\sum |t|^2 &=& \frac{1}{3}(2\,e\,g_\psi)^2
\left\{
2|1+t'^A_L+t'^B_L|^2
+{\bm p^2_2}\left(\frac{1}{p_1 \cdot k} \right)^2 \left({\bm p^2_1}-\frac{({\bm p_1}\cdot {\bm k})^2}{{\bm k}^2} \right)
+{\bm p^2_1}\left(\frac{1}{p_2 \cdot k} \right)^2 \left({\bm p^2_2}-\frac{({\bm p_2}\cdot {\bm k})^2}{{\bm k}^2} \right)\right. \nonumber\\
&+& \left. \left[{\bm p_1}\cdot {\bm p_2}-\frac{({\bm p_1}\cdot {\bm k})({\bm p_2}\cdot {\bm k})}{{\bm k^2}} \right]
\left[-2\, Re(1+t'^A_L+t'^B_L) \left(\frac{1}{p_1 \cdot k}+\frac{1}{p_2 \cdot k}\right) + 2\,{\bm p_1}\cdot {\bm p_2}\frac{1}{p_1 \cdot k}\frac{1}{p_2 \cdot k} \right] \right\}
\end{aligned}$$ where $$\begin{aligned}
\label{eq:tt2}
t'^A_L+t'^B_L \to \frac{1}{2\,e\,g_\psi} d M_{\psi} k\end{aligned}$$
For the case of $D^0 \bar{D}^0$ production only $t'^A_L+t'^B_L$ has to be kept and the $\overline{\sum}\sum |t|^2$ gives us $\frac{2}{3}|d M_{\psi} k|^2$, which is what we directly obtain from Eq. (\[eq:dpsi\]). All terms appearing in Eq. (\[eq:tt1\]) can be calculated in terms of $M_{\rm inv}(D^0 \bar{D}^0)$, $E_1$ and $A$ of Eq. (\[eq:Ac\]).
$D\bar{D} \to D\bar{D}$ transition amplitude
--------------------------------------------
We use the Bethe-Salpeter equation in coupled channels $$\begin{aligned}
T=[1-VG]^{-1}V \,,\end{aligned}$$ where the channels used are $D^+D^-$, $D^0 \bar{D}^0$, $D_s \bar{D_s}$, plus the $\eta\eta$ channel to account for the decay of the $D\bar{D}$ bound state into light meson-meson channels, as done in [@daixie]. The $V_{ij}$ coefficients between $D^+D^-$, $D^0 \bar{D}^0$, $D_s \bar{D_s}$ are taken from [@dani] and we take $V_{D^+D^-,\eta\eta}=a$, $V_{D^0 \bar{D}^0,\eta\eta}=a$ and all the other matrix elements zero [@daixie]. The value of $a$ is chosen at $a=42$ in order to obtain a width $\Gamma \simeq 36$ MeV as found in [@xiaothree]. The $G$ function is a diagonal function of the meson meson loops for which either dimensional regularization or cut off regularization can be used. If the cut off regularization is used, $q_{\rm max}=830$ MeV. As discussed above, the $q_{\rm max}$ has to be also implemented in the ${\bm q}$ integral of the loop function. It is also interesting to note that since the $\eta\eta$ channel is only introduced to produce the width, it is sufficient to take $$\begin{aligned}
G_{\eta\eta}=-i\frac{1}{8\pi}\frac{1}{M_{\rm inv}} q_{\eta} \,; \qquad q_{\eta}=\frac{\lambda^{1/2}(M^2_{\rm inv},m^2_{\eta},m^2_{\eta})}{2 M_{\rm inv}}\end{aligned}$$ The contribution of the real part of $G_{\eta\eta}$ only introduces negligible changes in the results [@wangliang].
Results
=======
First we look at the $t_{D^+D^-,D^+D^-}$ and $t_{D^+D^-,D^0 \bar{D}^0}$ amplitudes. In Fig. \[fig:GT42\] we plot $|t_{D^+D^-,D^+D^-}|^2$ and $|t_{D^+D^-,D^0 \bar{D}^0}|^2$ as a function of the $D\bar{D}$ invariant mass. We can see that the amplitudes are practically identical and have a peak around $3770$ MeV corresponding to a $D\bar{D}$ bound state. This is due to the dominance of the isospin $I=0$ contribution. The amplitudes also exhibit a fast fall around threshold corresponding to the opening of the $D\bar{D}$ decay channel (Flatté effect).
![$|t|^2$ for $D^+ D^- \to D^0\bar{D}^0$ and $D^+ D^- \to D^+ D^-$ as a function of the $D\bar{D}$ invariant mass.[]{data-label="fig:GT42"}](1-GT42.pdf){width="11.cm"}
Next we plot the coefficients $d_1$, $d_2$ in Fig. \[fig:d1d2\] given by Eqs. (\[eq:tt1\]), (\[eq:tt2\]). We see that the coefficient $d_1$ is bigger than $d_2$ by about a factor of two. At low invariant mass close to threshold they increase, reflecting the amplitude $t_{D\bar{D}, D\bar{D}}$ which is contained in the coefficients.
![The modulus squared of the coefficients $d_1$, $d_2$ as a function of the $D\bar{D}$ invariant mass.[]{data-label="fig:d1d2"}](2-ch-d1d2.pdf){width="11.5cm"}
![The differential cross section for $\psi(3770) \rightarrow \gamma D^0 \bar{D}^0$ as a function of the $D^0 \bar{D}^0$ invariant mass, $a=42$. []{data-label="fig:nudg"}](3-nu-dg.pdf){width="11.cm"}
The most important result is shown in Fig. \[fig:nudg\] where we show the results of $d\Gamma/ d M_{\rm inv}$ for the $D^0 \bar{D}^0$ distribution. We see a concentration of the strength around threshold with a peak around $3735$ MeV. In order to see that this structure is tied to the resonance below threshold we plot in Fig. \[fig:phase\] the phase space for $\psi(3770) \rightarrow \gamma D^0 \bar{D}^0$ substituting $\overline{\sum}\sum |t|^2$ of Eq. (\[eq:Ac\]) by a constant. We also show the phase space for $\psi(3770) \rightarrow \gamma D^+ D^-$ keeping the physical masses for the $D$ mesons. We can observe that the shape of the phase space distribution is drastically different from that predicted in the presence of a $D\bar{D}$ bound state. The phase space peaks around $3742$ MeV instead of $3735$ MeV for the distribution with the $D\bar{D}$ bound state. The shapes of the fall down of the two distributions are also different, the one with the $D\bar{D}$ bound state falling as a concave curve and the phase space as a convex one.
![Phase space for $\psi(3770) \rightarrow \gamma D^0 \bar{D}^0$ and $\gamma D^+ D^-$ normalized to the same area.[]{data-label="fig:phase"}](4-nphase42.pdf){width="11.cm"}
Finally we show in Fig. \[fig:chdg\] the mass distribution for $\psi(3770) \rightarrow \gamma D^+ D^-$. The shape is quite different than the one for $\psi(3770) \rightarrow \gamma D^0 \bar{D}^0$ and the reason is the contribution of the tree level, which is absent for $D^0 \bar{D}^0$ production. It is clear that $\psi(3770) \rightarrow \gamma D^0 \bar{D}^0$ is, thus, the reaction that better shows the presence of the $D\bar{D}$ bound state. Although not relevant for the present discussion, but we can see the infrared divergence associated to the limit $k \to 0$ for the photon momentum, when the $D$ propagator in the tree level amplitude becomes on shell.
The strengths of the $d\Gamma/dM_{\rm inv}$ distribution obtained fall well within present capacity at BESIII. Indeed the integrated luminosity in the $3770$ MeV region is about $5~ \rm fb^{-1}$ per year, which, together with the $e^+ e^-$ production cross section of the resonance of 7.2 nb [@refbo], leads to about $3.5 \times 10^7$ accumulated events per year. The expected final data will have an accumulated $20~ \rm fb^{-1}$ luminosity [@whitepaper] and with the planned BES upgrade $15~ \rm fb^{-1}$ per year ($1.1 \times 10^8$ events per year)[^1]. Our differential widths $d\Gamma/dM_{\rm inv}$ of the order of $10^{-6}$ are well measurable with these $\psi(3770)$ production rates [^2].
![ The differential cross section for $\psi(3770) \rightarrow \gamma D^+ D^-$ as a function of the $D^+ D^-$ invariant mass.[]{data-label="fig:chdg"}](5-ch-dg.pdf){width="11.cm"}
Conclusions
===========
We have made a study of the $\psi(3770) \to \gamma D \bar D$ decay, looking at the $D^+ \bar D^- $ and $D^0 \bar D^0 $ mass distributions in $d\Gamma/dM_{\rm inv}$ close to theshold. We saw that the production of $D^0 \bar D^0 $ is particularly suited to study the dynamics of the $D \bar D$ interaction because the tree level contribution is zero and the process goes with a loop mechanism that involves the $D^+ D^- \to D^0 \bar D^0$ scattering amplitude. We have used the results of a theory that predicts a $D \bar D$ bound state and this has as a consequence that the $D^0 \bar D^0$ mass distribution accumulates close to threshold and diverts drastically from a phase space distribution. The rates that we obtain for the mass distribution are perfectly reachable with the present BESIII facility and we encourage the performance of the experiment that could shed light on the issue of this possible $D \bar D$ bound state, and in any case would provide information on the $D \bar D \to D \bar D$ interaction.
Acknowledgments {#acknowledgments .unnumbered}
===============
LRD acknowledges the support from the National Natural Science Foundation of China (Grant Nos. 11975009, 11575076). GT acknowledges the support of PASPA-DGAPA, UNAM for a sabbatical leave. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 824093 for the STRONG-2020 project. This work is partly supported by the Spanish Ministerio de Economia y Competitividad and European FEDER funds under Contracts No. FIS2017-84038-C2-1-P B and No. FIS2017-84038-C2-2-P B.
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[^1]: Haibo Li and Changzheng Yuan, private communication.
[^2]: After completion of the work we learned that a BESIII team is presently investigating these reactions.
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abstract: 'Physically Unclonable Functions (PUFs) are lightweight cryptographic primitives for generating unique signatures from minuscule manufacturing variations. In this work, we present lightweight, area efficient and low power adaptive multi-bit SRAM topology based Current Mirror Array (CMA) analog PUF design for securing the sensor nodes, authentication and key generation. The proposed Strong PUF increases the complexity of the machine learning attacks thus making it difficult for the adversary. The design is based on scl180 library.'
author:
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bibliography:
- 'citations.bib'
title: 'Adaptive Multi-bit SRAM Topology Based Analog PUF'
---
Strong analog PUF, multi bit response, low power.
Introduction
============
Due to the whopping increase in the use of IoT devices, security has become one of the prime concerns. Physically Unclonable Functions (PUF) can be used as a low power alternative to secure sensor nodes. PUFs are mathematical models or physical structures that map the challenge words to corresponding responses which are governed by the uncertainties in the variation at the device level. Silicon PUFs have emerged as potential hardware cryptographic tools due to their ability to generate hardware-unique ‘responses’ to a given digital test word or ‘challenge’, by exploiting manufacturing process variations in circuit components in the IC. These random intrinsic variations are nearly impossible to replicate and therefore, PUFs provide extremely reliable hardware authentication and key generation. The popular applications of PUFs include hardware security and authentication such as secure RFIDs, IP protection in FPGAs and cryptographic key generation.
Related Work
------------
The silicon realization of the PUFs (SPUF) is based on the random variations in dies across a wafer, and from wafer to wafer due to the process, temperature and pressure variations during the various manufacturing steps. The first implementation of PUFs on silicon is introduced in [@Gassend:2002:SPR:586110.586132], where the delay variations of CMOS logic components are used to produce unique responses. In the delay-based PUF, the analog delay difference between two structurally identical parallel paths are compared which arises due to the manufacturing variations. The Ring Oscillator (RO) PUFs are based on digital loops, they are easy to implement and possess higher reliability. However, RO PUFs are slower and consume larger power when compared to arbiter PUF (delay based PUFs) and depends heavily on the number of RO present.
The low power current based PUF structure [@5938005] uses an automatic cut-off mechanism to stop the current flow after response evaluation to minimize the power consumption. A sub-threshold design based PUF [@5599030] exploits the sensitivity due to process variation in deep sub-micron technologies, the design consist of n stage CMOS multiplexers and delay circuits followed by an arbiter. To mitigate the power supply noise, switching noises and environmental variation a differential amplifier topology based PUF was introduced in [@Choi:2011:ICD:1960723.1960728].
Integrated circuit Identification (ICID) [@839821] is based on addressable MOSFET array which drives a load to generate random repeatable voltages based on the threshold voltage mismatches. SRAM cell and memory block based PUF exploiting the intrinsic process variations in read/write reliability of cells in static memories is implemented in [@10.1007/978-3-642-23951-9_27]. [@7870303] shows a PUF cell based on 2-T amplifier working in sub threshold region. In [@5227103], PUF design based on power grid resistance variation has been investigated. The response depends on the voltage drop at distinct locations of the IC occurred due to the introduction of a variety of stimuli. The concept based on the amplification of random transistor mismatch through two complementary current mirrors and a modified design with an addition of sense amplifier is discussed in [@7397840]. T Saha et al. [@7829547] proposed aging resistant, lightweight and low-power analog PUF which exploits the susceptibility of Threshold Voltage (Vth) of MOSFETs to process variations.
Almost every discussed PUF yield a single bit response for a unique challenge word which makes it easier for machine learning algorithms to model the PUFs with fewer challenge response pairs. We present an adaptive multi-bit SRAM topology based low power and highly robust analog PUF. The rest of the paper is organized in the following manner. Section II presents a discussion on Machine learning attacks while Section III describes the architecture of the proposed PUF. Section IV discusses the reliability, and finally, Section V concludes the paper.
Discussion on Machine Learning Based Attacks on PUF
===================================================
Most of the existing PUF circuits produce a single bit response for a challenge which can be modeled easily using machine learning algorithms. The proposed PUF adheres with the features of a Strong PUF following the property such as Many Challenges, Unpredictability and Unprotected Challenge Response Interface [@6800561]. Ruhrmair et al. proposed various modelling attacks on PUFs in [@6800562]. The primary attack model consists of two steps, the first one requires finding a function with parameters which correctly describes the PUF’s challenge response behavior (or input-output behavior), followed by selection of a machine learning (ML) algorithm to train the parameters of the chosen function to improve its prediction quality using a large set of Challenge Response Pairs (CRPs) as training set. The best existing ML algorithms for attacks on PUFs involves Logistic Regression and Evolution Strategies. Other includes Support Vector Machines (SVM) and Neural Networks. The adaptive multi-bit response of the proposed PUF fails the SVM and logistic regression and makes it extremely difficult for heuristic-based methods like Evolution Strategies thus increasing the complexity of the attack.
PUF Architecture
================
Fig. 1 depicts the architecture of the proposed PUF. As illustrated in the figure, the two 4x16 decoders are used to convert the challenge word into bit-line and word-line which selects a particular bit cell from the entire array. Once the bit cell is selected, the power gating block routes the Current Mirror biasing to that specific bit cell. Its corresponding output through the MUX is processed by the configurable ADC block which produces the final variable multi-bit response of the challenge word.
PUF Bit Cell
------------
The basic unit of the proposed SRAM topology based analog PUF called bit cell comprises of two PMOS in cascode with two NMOS. All MOSFETs in the bit cell are minimum sized, hence increasing the probability of mismatch during manufacturing and also reducing the area required for the complete 16 x 16 bit-cell array. The output of the bit cell is chosen as the drain of PM2 and NM1 attributing to the symmetry. The cascode scheme used provides maximum amplification, and hence the mismatch on the extremities in even tenth of millivolts can be amplified.
PM2 and NM1 are biased through two transmission gates which are controlled via bit-line and word-line using the power gating blocks as seen in Fig 2. The power gating block consists of two transmission gates(TG) controlled by the word line and the bit line, the TGs are tied with the current mirror biasing and the bit-cell as seen from Fig. 1. One power gating block is required per 16-bit cells (one column). This helps in switching off all unselected bit cells resulting in zero ideal state power consumption. Different switching schemes were tested and studied for various process corners. The channel length modulation results in a significant error while copying currents, especially for minimum size transistors. Thus a current mirroring scheme becomes very important. Proposed configuration is shown in Fig. 4.
Due to variations in semiconductor manufacturing processes, corners are introduced in CMOS circuits. Five process corners dealing with process variations are typical-typical (TT), slow-slow (SS), fast-fast (FF), slow-fast (SF), fast-slow (FS). TT, FF and SS are not much of concern as both the MOSFETs get equally affected in one direction but other corners like SF, FS called as skewed corners may affect the response. Considering this, various current mirror configurations were tested and wide swing cascode current mirror worked well with the bit cell for all the corners with the perfect symmetrical output. The biasing current into the bit-cell using this particular current scheme varies around the center value (at no mismatch) of 4.3 $\mu$A.
Fig. 3 shows the response of various testbed used as current mirrors where X axis represents the referred variation modeled as the additional change in the threshold voltage of PM1. We observe that the ideal response is obtained through the proposed current mirror configuration. Moreover, the former has a higher gain and is perfectly symmetric compared to the reduced headroom cascode current mirror and simple cascode current mirror configuration.
![\[fig:All\_schemes\_1\]Output Voltage Response for different Current Mirror Configurations](All_schemes_1){width="35.00000%"}
![\[fig:\] Wide Swing Cascode Current Mirror with PTAT Current Reference](currentmirror){width="40.00000%"}
Switching Configurations
------------------------
The bit-cell zero static power in the idle state is achieved by turning the single bit cell ON only when it is selected and cutting OFF the supply rail for rest of the bit cell present in the entire array. Different switching configurations were tested, and their effects on the performance were studied in detail. The most viable option being the one in Fig. 5.
![\[fig:up\_down\_switches\]Naive Switching Configuration](Up-Down-Switches){width="20.00000%"}
The configuration does not require any additional circuitry to serve the purpose of switching. However, the response was hampered considering all process corners as a substantial amount of shift was observed in the output due to small voltage drops across the switching transistors. To mitigate these effects control switches were removed from the bit cell and external control circuitry is employed (power gating blocks) which even drastically reduced the number of transistors per bit cell and works well with different process corners. The switching of PM1 and NM2 of the bit-cell will slow down the charging/discharging process from the pre-charged value to the desired value due to increased parasitic capacitance per bit line. Therefore, the two central MOSFETs, PM2 and NM1 are held with VDD and GND respectively which turns OFF the bit cell through the power gating block. Fig. 6 shows the comparison of the output voltage for different process corners in case of naive switching configuration and used switching configuration.
[0.3]{} ![\[fig1:\]Output Voltage Variation for different Process Corners in (a) Naive Switching Configuration (b) Used Switching Configuration](probable "fig:"){width="\linewidth"}
[0.3]{} ![\[fig1:\]Output Voltage Variation for different Process Corners in (a) Naive Switching Configuration (b) Used Switching Configuration](proposed "fig:"){width="\linewidth"}
ADC scheme
----------
Due to the cascode configuration, the gain from the MOSFETs at the extremities to the output is very high thus it is highly probable that the response would hit either of the power rails, i.e., VDD or GND. The impact of the random process variations on circuit behavior can be studied through Monte Carlo(MC) simulation. The MC Simulation presented in Fig. 7 shows the output voltage distribution based on both the statistical variations, i.e., process and mismatch. As expected, the probability distribution is more skewed towards the power rails thus a varying multi-bit ADC scheme can exploit the utmost variations from the circuit. The final response is obtained through an adaptive multi-bit ADC scheme shown in Fig. 8. First, we define windows or regions as in the Fig. 8 using the Lloyd-Max Algorithm based on the probability distribution from the MC Simulation. The regions obtained are as follows:
Region 1: \[0,0.1451) 8 bit ADC
Region 2: \[0.1451,0.6596) 7 bit ADC
Region 3: \[0.6596,1.3308) 6 bit ADC
Region 4: \[1.3308,1.6978) 7 bit ADC
Region 5: \[1.6978,1.8) 8 bit ADC
![\[fig:\]Monte Carlo Simulation considering Process Variation and Mismatch for Bitcell Output Voltage](monter_carlo_matlab_300x220){width="40.00000%"}
The ADC selection logic unit governs bit precision of the conversion based on the regions mentioned above, and then the response is generated using the configurable single slope ADC scheme. V1, V2, V3, and V4 in Fig. 8 are the voltage values defining the windows.
![\[fig:\]Configurable ADC scheme](ADC_Scheme_configurable_wgrid){width="40.00000%"}
### Configurable Single Slope ADC Scheme
The single slope ADC technique has a series of advantages over the Flash and SAR ADC. The most important one being lower area and low power consumption. The single slope ADC technique designed in this case utilizes an external ramp circuit, a constant current source, three separate voltage comparators, a multiplexer, a free running timer, and a latching mechanism. The adaptive bit precision technique discussed is addressed using the variable frequency clock derived from the original clock along with variable counter values which are finally latched upon comparison during ADC operation. Fig. 9 describes the topology used to eliminate the Input Common Mode Range. The scheme is capable of correctly establishing ADC operation for unknown analog voltage varying from 0 to VDD. The two comparators one with NMOS as the input pair say Comparator A and the other one with PMOS as input pair say Comparator B includes offset cancellation scheme to prevent the results being skewed towards one direction. The decision of which comparator’s output to be used is performed by the third comparator i.e Comparator C based on the clause: If the unknown voltage is higher than VDD/2, the output of Comparator A is selected else the output of Comparator B is used.
![\[fig:\]Single Slope ADC](Sudo-2.png){width="40.00000%"}
Results
=======
Reliability with Temperature Variations
---------------------------------------
The reliability of bit-cell output voltage is tested with temperature variations. Fig 10 depicts the effect on the bit-cell output voltage as the temperature varies between $0\,^{\circ}\mathrm{C}$ and $60\,^{\circ}\mathrm{C}$. For simulation purposes, the referred variation is modeled as the change in the threshold voltage of the PM1 in the bit-cell.
![\[fig:\]Bit-Cell output voltage variation across different temperatures at different variations in threshold voltages ](puf_temp){width="35.00000%"}
Comparison with other existing PUF schemes
------------------------------------------
PUFs operating in the sub-threshold region [@5599030] are shown to consume lesser power compared to those in super-threshold region but they exhibit higher delays thus reducing the speed of operation. The power gating block in the proposed PUF primarily contributes to the low power consumption without any hindrance in the operating speed. Table 1 provides the comparison with existing PUFs based on speed and power for 1 bit generation.
PUF model Power @ Speed of Operation Energy / cycle
----------------- ---------------------------- ----------------
Super-threshold 136.4$\mu$W @ 1GHz 0.136 pJ
Sub-threshold 0.047$\mu$W @ 1MHz 0.047 pJ
ICID 250$\mu$W @ 0.5MHz 500 pJ
TV-PUF 0.181$\mu$W @ 1 GHz 0.0018 pJ
Proposed PUF 306.54$\mu$W @ 6.4 GHz 0.0478 pJ
: Power and Speed Comparison with PUF’s for 1 bit generation[]{data-label="table_example"}
conclusion and future scope
===========================
This paper discusses the architecture of an adaptive multi-bit Strong Analog PUF. Proposed Analog PUF is better than delay based PUFs considering unbiased responses. The output being independent of the actual layout design, the PUF is easy to fabricate due to its less circuit complexity. Furthermore, the PUF is shown to consume low power at a faster speed of operation.
The future work on this proposed PUF may include an evaluation based on uniqueness, uniformity and reliability after chip manufacturing. NIST tests can also be performed on the chip to compare it with more state of art PUF configurations.
|
---
abstract: 'In recent years statistical physicists have developed [*discrete*]{} “particle-hopping” models of vehicular traffic, usually formulated in terms of [*cellular automata*]{}, which are similar to the microscopic models of interacting charged particles in the presence of an external electric field. Concepts and techniques of non-equilibrium statistical mechanics are being used to understand the nature of the steady states and fluctuations in these so-called “microscopic” models. In this brief review we explain, primarily to the nonexperts, these models and the physical implications of the results.'
---
\]
Introduction
============
Are you surprised to see an article on vehicular traffic in this special section of current science where physicists are supposed to report on some recent developments in the area of dynamics of nonequilibrium statistical systems? Aren’t civil engineers (or, more specifically, traffic engineers) expected to work on traffic? Solving traffic problems would become easier if one knows the fundamental laws governing traffic flow and traffic jam. For almost half a century physicists have been trying to develope a theoretical framework of traffic science extending concepts and techniques of statistical physics [@hg; @gazis; @ph; @proc1; @proc2; @helbsci]. The main aim of this brief review is to show how these attempts, particularly the recent ones, have led to deep insight in this frontier area of inter-disciplinary research.
The dynamical phases of systems driven far from equilibrium are counterparts of the stable phases of systems in equilibrium. Let us first pose some of the questions that statistical physicists have been addressing in order to discover the [*fundamental laws*]{} governing vehicular traffic. For example,\
(i) What are the various [*dynamical phases*]{} of traffic? Does traffic exhibit phase-coexistence, phase transition, criticality or self-organized criticality and, if so, under which circumstances?\
(ii) What is the nature of [*fluctuations around the steady-states*]{} of traffic?\
(iii) If the initial state is far from a stationary state of the driven system, how does it [*evolve with time*]{} to reach a truely steady-state?\
(iv) What are the effects of [*quenched disorder*]{} (i.e., time-independent disorder) on the answers to the questions posed in (i)-(iii) above?\
The microscopic models of vehicular traffic can find [ *practical applications*]{} in on-line traffic control systems as well as in the planning and design of transportation network.
There are two different conceptual frameworks for modellig vehicular traffic. In the “coarse-grained” fluid-dynamical description, the traffic is viewed as a compressible fluid formed by the vehicles but these individual vehicles do not appear explicitly in the theory. In contrast, in the “microscopic” models traffic is treated as [*a system of interacting “particles” driven far from equilibrium*]{} where attention is explicitly focussed on individual vehicles each of which is represented by a “particle”; the nature of the interactions among these particles is determined by the way the vehicles influence each others’ movement. Unlike the particles in a gas, a driver is an intelligent agent who can “think”, make individual decisions and “learn” from experience. Nevertheless, many general phenomena in traffic can be explained in general terms with these models provided the behavioural effects of the drivers are captured by only a few phenomenological parameters.
The conceptual basis of the older theoretical approaches are explained briefly in section II. Most of the “microscopic” models developed in the recent years are “particle-hopping” models which are usually formulated using the language of cellular automata (CA) [@wolfram]. The Nagel-Schreckenberg (NaSch) [@ns] model and the Biham-Middleton-Levine (BML) [@bml] model, which are the most popular CA models of traffic on idealized highways and cities, respectively, have been extended by several authors to develope more realistic models. Some of the most interesting aspects of these recent developments are discussed in the long sections III and IV. The similarities between various particle-hopping models of traffic and some other models of systems, which are also far from equilibrium, are pointed out in section V followed by the concluding section VI.
Older theories of vehicular traffic
===================================
Fluid-dynamical Theories of vehicular traffic
---------------------------------------------
In traffic engineering, the [*fundamental diagram*]{} depicts the relation between density $c$ and the [**flux**]{} $J$, which is defined as the number of vehicles crossing a detector site per unit time [@may90]. Because of the conservation of vehicles, the local density $c(x;t)$ and local flux $J(x;t)$ satisfy the equation of continuity which is the analogue of the equation of continuity in the hydrodynamic theories of fluids. In the early works[@lw] it was assumed (i) that the flux (or, equivalently, the velocity) is a function of the density and (ii) that, following any change in the local density, the local speed instantaneously relaxes to a magnitude consistent with the new density at the same location. However, for a more realistic description of traffic, in the recent fluid-dynamical treatments [@kernetal; @hongetal; @leeetal] of traffic an additional equation (the analogue of the Navier-Stokes equation for fluids), which describes the time-dependence of the velocity $V(x;t)$, has been considered. This approach, however, has its limitations; for example, viscosity of traffic is not a directly measureable quantity.
Kinetic theory of vehicular traffic
-----------------------------------
In the kinetic theory of traffic, one begins with the basic quantity $g(x,v,w;t) ~dx ~dv ~dw$ which is the number of vehicles, at time $t$, located between $x$ and $x+dx$, having [*actual*]{} velocity between $v$ and $v + dv$ and [*desired*]{} velocity between $w$ and $w + dw$. In this approach, the fundamental dynamical equation is the analogue of the Boltzmann equation in the kinetic theory of gases[@ph]. Assuming reasonable forms of “relaxation” and “interaction”, the problem of traffic is reduced to that of solving the Boltzmann-like equation, a formidable task, indeed [@helbkin; @wagkin; @nagakin]!
Car-following theories of vehicular traffic
-------------------------------------------
In the car-following theories one writes, for each individual vehicle, an equation of motion which is the analogue of the Newton’s equation for each individual particle in a system of interacting classical particles. In Newtonian mechanics, the acceleration may be regarded as the [*response*]{} of the particle to the [*stimulus*]{} it receives in the form of force which includes both the external force as well as those arising from its interaction with all the other particles in the system. Therefore, the basic philisophy of the car-following theories[@hg; @gazis] can be summarized by the equation $$[Response]_n \propto [Stimulus]_n
\label{car-follow}$$ for the $n$-th vehicle ($n = 1,2,...$). The constant of proportionality in the equation (\[car-follow\]) can be interpreted as a measure of the sensitivity coefficient of the driver; it indicates how strongly the driver responds to unit stimulus. Each driver can respond to the surrounding traffic conditions only by accelerating or decelerating the vehicle. The stimulus and the sensitivity factor are assumed to be functions of the position and speed of the vehicle under consideration and those of its leading vehicle. Different forms of the equations of motion of the vehicles in the different versions of the car-following models arise from the differences in their postulates regarding the nature of the stimulus. In general, the dynamical equations for the vehicles in the car-following theories are coupled non-linear differential equations [@bandoetal; @komatsu; @hayanaka; @mason; @sugiyama; @nagatani; @wagner] and thus, in this “microscopic” approach, the problem of traffic flow reduces to problems of nonlinear dynamics.
Cellular-automata Models of Highway-traffic
===========================================
In the car-following models space is treated as a continuum and time is represented by a continuous variable $t$ while velocities and accelerations of the vehicles are also real variables. However, most often, for numerical manipulations of the differential equations of the car-following models, one needs to discretize the continuous variables with appropriately chosen grids. In contrast, in the CA models of traffic not only time but also the position, speed, and acceleration of the vehicles are treated as [*discrete*]{} variables. In this approach, a lane is represented by a one-dimensional lattice. Each of the lattice sites represents a “cell” which can be either empty or occupied by at most one “vehicle” at a given instant of time (see fig.1).
At each [*discrete time*]{} step $t \rightarrow t+1$, the state of the system is updated following a well defined prescription.
The Nagel-Schreckenberg model of highway traffic:
-------------------------------------------------
In the NaSch model, the speed $V$ of each vehicle can take one of the $V_{max}+1$ allowed [*integer*]{} values $V=0,1,...,V_{max}$. Suppose, $X_n$ and $V_n$ denote the position and speed, respectively, of the $n$-th vehicle. Then, $d_n = X_{n+1}-X_n$, is the gap in between the $n$-th vehicle and the vehicle in front of it at time $t$. At each time step $t \rightarrow t+1$, the arrangement of the $N$ vehicles on a finite lattice of length $L$ is updated [*in parallel*]{} according to the following “rules”:
[*Step 1: Acceleration.*]{} If, $ V_n < V_{max}$, the speed of the $n$-th vehicle is increased by one, but $V_n$ remains unaltered if $V_n = V_{max}$, i.e., $V_n \rightarrow min(V_n+1,V_{max})$.
[*Step 2: Deceleration (due to other vehicles).*]{} If $d_n \le V_n$, the speed of the $n$-th vehicle is reduced to $d_n-1$, i.e., $V_n \rightarrow min(V_n,d_n-1)$.
[*Step 3: Randomization.*]{} If $V_n > 0$, the speed of the $n$-th vehicle is decreased randomly by unity with probability $p$ but $V_n$ does not change if $V_n = 0$, i.e., $V_n \rightarrow max(V_n-1,0)$ with probability $p$.
[*Step 4: Vehicle movement.*]{} Each vehicle is moved forward so that $X_n \rightarrow X_n + V_n$.
The NaSch model is a minimal model in the sense that all the four steps are necessary to reproduce the basic features of real traffic; however, additional rules need to be formulated to capture more complex situations. The step 1 reflects the general tendency of the drivers to drive as fast as possible, if allowed to do so, without crossing the maximum speed limit. The step 2 is intended to avoid collision between the vehicles. The randomization in step 3 takes into account the different behavioural patterns of the individual drivers, especially, nondeterministic acceleration as well as overreaction while slowing down; this is crucially important for the spontaneous formation of traffic jams. So long as $p \neq 0$, the NaSch model may be regarded as stochastic CA [@wolfram]. For a realistic description of highway traffic [@ns], the typical length of each cell should be about $7.5$m and each time step should correspond to approximately $1$ sec of real time when $V_{max} = 5$.
The update scheme of the NaSch model is illustrated with a simple example in fig.\[fig2\].
Space-time diagrams showing the time evolutions of the NaSch model demonstrate that no jam is present at sufficiently low densities, but spontaneous fluctuations give rise to traffic jams at higher densities (fig.3(a)). From the fig.3(b) it should be obvious that the [*intrinsic stochasticity*]{} of the dynamics[@ns], arising from non-zero $p$, is essential for triggering the jams [@ns; @nh].
The use of [*parallel*]{} dynamics is also important. In contrast to a random sequential update, it can lead to a chain of overreactions. Suppose, a vehicle slows down due the randomization step. If the density of vehicles is large enough this might force the following vehicle also to brake in the deceleration step. In addition, if $p$ is not too small, it might brake even further in Step 3. Eventually this can lead to the stopping of a vehicle, thus creating a jam. This mechanism of spontaneous jam formation is rather realistic and cannot be modelled by the random sequential update.
Relation between the NaSch model and ASEP
-----------------------------------------
In the NaSch model with $V_{max} = 1$ every vehicle moves forward with probability $q = 1-p$ in the time step $t+1$ if the site immediately in front of it were empty at the time step $t$; this, is similar to the fully asymmetric simple exclusion process ([**ASEP**]{}) [@spohn; @sz; @gs] where a randomly chosen particle can move forward with probablity $q$ if the site immediately in front is empty. But, updating is done in parallel in the NaSch model whereas that in the ASEP is done in a random sequential manner. Nevertheless, the special case of $V_{max} = 1$ for the NaSch model achieves special importance from the fact that so far it has been possible to derive exact analytical results for the NaSch model only in the special limits (a) $V_{max} = 1$ and arbitrary $p$ and (b) $p = 0$ and arbitrary $V_{max}$.
NaSch model in the deterministic limits
---------------------------------------
If $p = 0$, the system can self-organize so that at low densities every vehicle can move with $V_{max}$ and the corresponding flux is $cV_{max}$; this is, however, possible only if enough empty cells are available in front of every vehicle, i.e., for $ c \leq c_m^{det} = (V_{max} + 1)^{-1}$ and the corresponding maximum flux is $J_{max}^{det} = V_{max}/(V_{max} + 1)$. On the other hand, for $c > c_m^{det}$, the flow is limited by the density of holes. Hence, the fundamental diagram in the deterministic limit $p = 0$ of the NaSch model (for any arbitrary $V_{max}$) is given by the [*exact*]{} expression $J = min(c V_{max}, (1 - c))$.
Aren’t the properties of the NaSch model with maximum allowed speed $V_{max}$, in the deterministic limit $p = 1$, exactly identical to those of the same model with maximum allowed speed $V_{max} - 1$? The answer to the question posed above is: NO; if $p = 1$, all random initial states lead to $J = 0$ in the stationay state of the NaSch model irrespective of $V_{max}$ and $c$!
Analytical Theory for the NaSch Model
-------------------------------------
In the “site-oriented” theories one describes the state of the finite system of length $L$ by completely specifying the state of each [*site*]{}. In contrast, in the “car-oriented” theories the state of the traffic system is described by specifying the positions and speeds of all the $N$ vehicles in the system. In the naive mean-field approximation one treats the probabilities of occupation of the lattice sites as independent of each other. In this approximation, for example, the steady-state flux for the NaSch model with $V_{max} = 1$ and periodic boundary conditions, one gets [@ssni] $$J = q c (1-c)
\label{eq-mfflux}$$ It turns out[@ssni] that the naive mean-field theory underestimates the flux for all $V_{max}$. Curiously, if instead of parallel updating one uses the random sequential updating, the NaSch model with $V_{max} = 1$ reduces to the ASEP for which the equation (\[eq-mfflux\]) is known to be the [*exact*]{} expression for the corresponding flux (see, e.g., [@ns])!
What are the reasons for these differences arising from parallel updating and random sequential updating? There are “garden of Eden” (GoE) states (dynamically forbidden states) [@ss98] of the NaSch model which cannot be reached by the parallel updating whereas no state is dynamically forbidden if the updating is done in a random sequential manner.
For example, the configuration shown in fig.4 is a GoE state[[^1]]{} because it could occur at time $t$ only if the two vehicles occupied the same cell simultaneously at time $t - 1$. The naive mean-field theory mentioned above does not exclude the GoE states. The exact expression, given in the next subsection, for the flux in the steady-state of the NaSch model with $V_{max} = 1$ can be derived by merely excluding these states from consideration in the naive mean-field theory [@ss98], thereby indicating that the only source of correlation in this case is the parallel updating. But, for $V_{max} > 1$, there are other sources of correlation because of which exclusion of the GoE states merely improves the naive mean-field estimate of the flux but does not yield exact results[@ss98].
A systematic improvement of the naive mean-field theory of the NaSch model has been achieved by incorporating short-ranged correlations through cluster approximations. We define a $n$-cluster to be a collection of $n$ successive sites. In the general $n$-cluster approximation, one divides the lattice into “clusters” of length $n$ such that two neighbouring clusters have $n-1$ sites in common (see fig.5).
If $n = 1$, then the $1$-cluster approximation can be regarded as the naive mean-field approximation. You can easily verify, for example, in the special case of $V_{max} = 1$, that the state of the 2-cluster at time $t+1$ depends on the state of the 4-cluster at time $t$, which, in turn, depends on the state of a larger cluster at time $t-1$ and, so on. Therefore, one needs to make an approximation to truncate this hierarchy in a sensible manner. For example, in the 2-cluster approximation for the NaSch model with $V_{max} = 1$, the 4-cluster probabilities are approximated in terms of an appropriate product of 2-cluster probabilities. Thus, in the $n$-cluster approximation[@ssni] a cluster of $n$ neighbouring cells are treated exactly and the cluster is coupled to the rest of the system in a self-consistent way.
Carrying out the 2-cluster calculation[@ssni] for $V_{max} = 1$ one not only finds an effective particle-hole attraction (particle-particle repulsion), but also obtains the exact result $$J(c,p) = \frac{1}{2}[1 - \sqrt{1- 4qc(1-c)}]
\label{fl-2cl}$$ for the corresponding flux. But one gets only approximate results from the 2-cluster calculations for all $V_{max} > 1$ (see [@shad99] for higher order cluster calculations for $V_{max} = 2$ and comparison with computer simulation data).
Let us explain the physical origin of the generic shape of the fundamental diagrams shown in fig.6. At sufficiently low density of vehicles, practically “free flow” takes place whereas at higher densities traffic becomes “congested” and traffic jams occur. So long as $c$ is sufficiently small, the average speed $\langle V \rangle$ is practically independent of $c$ as the vehicles are too far apart to interact mutually. However, a faster monotonic decrease of $\langle V \rangle$ with increasing $c$ takes place when the forward movement of the vehicles is strongly hindred by others because of the reduction in the average separation between them. Because of this trend of variation of $\langle V \rangle$ with $c$, the flux $J = \langle c V \rangle$ exhibits a maximum [@may90] at $c_m$; for $c < c_m$, increasing $c$ leads to increasing $J$ whereas for $c > c_m$ sharp decrease of $\langle V \rangle$ with increase of $c$ leads to the overall decrease of $J$.
An interesting feature of the expression (\[fl-2cl\]) is that the flux is invariant under charge conjugation, i.e., under the operation $c \rightarrow (1-c)$ which interchanges particles and holes. Therefore, the fundamental diagram is symmetric about $c = 1/2$ when $V_{max} = 1$ (see fig.6a). Although this symmetry breaks down for all $V_{max} > 1$ (see fig.6b), the corresponding fundamental diagrams appear more realistic. Moreover, for given $p$, the magnitude of $c_m$ decreases with increasing $V_{max}$ as the higher is the $V_{max}$ the longer is the effective range of interaction of the vehicles (see fig.6b). Furthermore, for $V_{max} = 1$, flux merely decreases with increasing $p$ (see eqn(\[fl-2cl\])), but remains symmetric about $c = 1/2 =c_m$. On the other hand, for all $V_{max} > 1$, increasing $p$ not only leads to smaller flux but also lowers $c_m$.
Spatio-temporal organization of vehicles
----------------------------------------
The distance from a selected point on the lead vehicle to the same point on the following vehicle is defined as the [*distance-headway*]{} ([**DH**]{}) [@may90]. In order to get information on the spatial organization of the vehicles, one can calculate the DH distribution ${\cal P}_{dh}(\Delta X)$ by following either a site-oriented approach [@chow97a] or a car-oriented approach[@ss97] if $\Delta X_j = X_j - X_{j-1}$, i.e., if the number of empty lattice sites in front of the $j$-th vehicle is identified as the corresponding DH. At moderately high densities, ${\cal P}_{dh}(\Delta X)$ exhibits two peaks; the peak at $\Delta X = 1$ is caused by the jammed vehicles while that at a larger $\Delta X$ corresponds to the most probable DH in the free-flowing regions.
The [*time-headway*]{} is defined as the time interval between the departures (or arrivals) of two successive vehicles recorded by a detector placed at a fixed position on the highway [@may90]. The time-headway distribution contains information on the temporal organization. Suppose, ${\cal P}_m(t_1)$ is the probability that the following vehicle takes time $t_1$ to reach the detector, moving from its initial position where it was located when the leading vehicle just left the detector site. Suppose, after reaching the detector site, the following vehicle waits there for $\tau - t_1$ time steps, either because of the presence of another vehicle in front of it or because of its own random braking; the probability for this event is denoted by $Q(\tau-t_1|t_1)$. The distribution ${\cal P}_{th}(\tau)$, of the time-headway $\tau$, can be obtained from [@chow98a; @ghosh] ${\cal P}_{th}(\tau) = \sum_{t_1=1}^{\tau-1} {\cal P}_m(t_1) Q(\tau-t_1|t_1)$. The most-probable time-headway, when plotted against the density, exhibits a minimum [@ghosh]; this is consistent with the well known exact relation $J = 1/T_{av}$ between flux and the average time-headway, $T_{av}$.
Is there a phase transition from “free-flowing” to “congested” dynamical phase of the NaSch model? No satisfacory order parameter has been found so far[@eisen; @chebani], except in the deterministic limit[@vdes]. The possibility of the existence of any critical density in the NaSch model is ruled out by the observations [@eisen; @chebani; @shad99; @kn94] that, for all non-zero $p$, (a) the equal-time correlation function decays exponentially with separation, and (b) the relaxation time and lifetimes of the jams remain finite. This minimal model of highway traffic also does not exhibit any first order phase transition and two-phase co-existence[@chow98a].
Extensions of the NaSch model and practical applications
--------------------------------------------------------
In recent years some other minimal models of traffic on highways have been developed by modifying the updating rules of the NaSch model [@fukuishi; @wang98; @helbshrek]. In the cruise control limit of the NaSch model [@paczus] the randomization step is applied only to vehicles which have a velocity $V < V_{max}$ after step 2 of the update rule. Vehicles moving with their desired velocity $V_{max}$ are not subject to fluctuations. This is exactly the effect of a cruise-control which automatically keeps the velocity constant at a desired value. Interestingly, the cruise-control limit of the NaSch model exhibits self-organized criticality[@bak; @ddhar]. Besides, a continuum limit of the NaSch model has also been considered[@krauss].
The vehicles which come to a stop because of hindrance from the leading vehicle may not be able to start as soon as the leading vehicle moves out of its way; it may start with a probability $q_s < 1$. When such possibilities are incorporated in the NaSch model, the “slow-to-start” rules [@bjh; @tt; @barlovic; @sss2s; @chow99] can give rise to metastable states of very high flux and hysteresis effects as well as phase separation of the traffic into a “free-flowing” phase and a “mega-jam”.
The bottleneck created by quenched disorder of the [*highway*]{} usually slows down traffic and can give rise to jams [@cv; @chow98a] and phase segregation[@lebo; @barma]. However, a different type of quenched disorder, introduced by assigning randomly different braking probabilities $p$ to different drivers in the NaSch model, can have more dramatic effects[@ktitarev; @santen99] which are reminiscent of “Bose-Einstein-like condensation” in the fully ASEP where particle-hopping rates are quenched random variables[@kf; @evans]. In such “Bose-Einstein-like condensed” state finite fraction of the empty sites are “condensed” in front of the slowest vehicle (i.e., the driver with highest $p$).
Several attempts have been made to generalize the NaSch model to describe traffic on multi-lane highways and to simulate traffic on real networks in and around several cities[@wolf]. For [*planning and design*]{} of the transportation network [@nagel99], for example, in a metropolitan area [@duis; @dallas1; @dallas2], one needs much more than just [*micro-simulation*]{} of how vehicles move on a linear or square lattice under a specified set of vehicle-vehicle and road-vehicle interactions. For such a simulation, to begin with, one needs to specify the roads (including the number of lanes, ramps, bottlenecks, etc.) and their intersections. Then, times and places of the activities, e.g., working, shopping, etc., of individual drivers are planned. Micro-simulations are carried out for all possible different routes to execute these plans; the results give informations on the efficiency of the different routes and these informations are utilized in the designing of the transportation network [@rickertdipl; @nageldiss; @nagel99]. Some socio-economic questions as well as questions on the environmental impacts of the planned transportation infrastructure also need to be addressed during such planning and design.
Cellular-automata Models of City-traffic
========================================
The Biham-Middleton-Levin model of city traffic and its generalizations
-----------------------------------------------------------------------
In the BML model[@bml], each of the sites of a square lattice represent the crossing of a east-west street and a north-south street. All the streets parallel to the $\hat{X}$-direction of a Cartesian coordinate system are assumed to allow only [*single-lane*]{} east-bound traffic while all those parallel to the $\hat{Y}$-direction allow only single-lane north-bound traffic. In the initial state of the system, vehicles are randomly distributed among the streets. The states of east-bound vehicles are updated in parallel at every odd discrete time step whereas those of the north-bound vehicles are updated in parallel at every even discrete time step following a rule which is a simple extension of the fully ASEP: a vehicle moves forward by one lattice spacing if and only if the site in front is empty, otherwise the vehicle does not move at that time step.
Computer simulations demonstrate that a [*first order*]{} phase transition takes place in the BML model at a finite non-vanishing density $c_*$, where the average velocity of the vehicles vanishes [*discontinuously*]{} signalling complete jamming; this jamming arises from the mutual blocking of the flows of east-bound and north-bound traffic at various different crossings[@tadaki; @gupta]. Note that the dynamics of the BML model is fully [*deterministic*]{} and the randomness arises only from the [*random initial conditions*]{} [@fuishjp96].
As usual, in the naive mean-field approximation one neglects the correlations between the occupations of different sites[@mmcb95]. However, if you are not interested in detailed information on the “structure” of the dynamical phases, you can get a mean-field estimate of $c_*$ by carrying out a back-of-the-envelope calculation[@nagajp93; @nagapr93; @wfh96]. In the symmetric case $c_x = c_y$, for which $v_x = v_y = v$, $c = c_* \simeq 0.343$.
The BML model has been extended to take into account the effects of (i) asymmetric distribution of the vehicles[@nagajp93], i.e., $c_x \neq c_y$, (ii) overpasses or two-level crossings[@nagapr93] that are represented by specifically identified sites each of which can accomodate upto a maximum of two vehicles simultaneously, (iii) faulty traffic lights[@chung95] (iv) static hindrances or road blocks or vehicles crashed in traffic accident, i.e., stagnant points[@acci; @gu], (v) stagnant street where the local density $c_s$ of the vehicles is initially higher than that in the other streets[@stag] (vi) jam-avoiding drive[@jamavoid] of vehicles to a neighbouring street, parallel to the original direction, to avoid getting blocked by other vehicles in front, (vii) turning of the vehicles from east-bound (north-bound) to north-bound (east-bound) streets[@turn]. (viii) a single north-bound street cutting across east-bound streets [@nagaseno] (ix) more realistic description of junctions of perpendicular streets [@freund; @chopard], (x) green-waves [@greenwave].
Marriage of NaSch and BML models
--------------------------------
At first sight the BML model may appear very unrealistic because the vehicles seem to hop from one crossing to the next. However, it may not appear so unrealistic if each unit of discrete time interval in the BML model is interpreted as the time for which the traffic lights remain green (or red) before switching red (or green) simultaneously in a synchronized manner, and over that time scale each vehicle, which faces a green signal, gets an opportunity to move from $j$-th crossing to the $j+1$-th (or, more generally[@fukui], to the $j+r$-th where $r > 1$).
However, if one wants to develope a more detailed “fine-grained” description then one must first decorate each bond[@horiguchi] with $D-1$ ($D > 1$) sites to represent $D-1$ cells in between each pair of successive crossings thereby modelling each segment of the streets in between successive crossings in the same manner in which the entire highway is modelled in the NaSch model. Then, one can follow the prescriptions of the NaSch model for describing the positions, speeds and accelerations of the vehicles [@chopard; @simon] as well as for taking into account the interactions among the vehicles moving along the same street. Moreover, one should flip the color of the signal periodically at regular interval of $T$ ($T >> 1$) time steps where, during each unit of the discrete time interval every vehicle facing green signal should get an opportunity to move forward from one cell to the next. Such a CA model of traffic in cities has, indeed, been proposed very recently [@cs] where the rules of updating have been formulated in such a way that, (a) a vehicle approaching a crossing can keep moving, even when the signal is red, until it reaches a site immediately in front of which there is either a halting vehicle or a crossing; and (b) no grid-locking would occur in the absence of random braking.
=
A phase transition from the “free-flowing” dynamical phase to the completely “jammed” phase has been observed in this model at a vehicle density which depends on $D$ and $T$. The intrinsic stochasticity of the dynamics, which triggers the onset of jamming, is similar to that in the NaSch model, while the phenomenon of complete jamming through self-organization as well as the final jammed configurations (fig.7) are similar to those in the BML model. This model also provides a reasonable time-dependence of the average speeds of the vehicles in the “free-flowing” phase[@cs].
Relation with other systems and phenomena
=========================================
You must have noticed in the earlier sections that some of the models of traffic are non-trivial generalizations or extensions of the ASEP, the simplest of the [*driven-dissipative*]{} systems which are of current interest in non-equilibrium statistical mechanics[@sz]. Some similarities between these systems and a dynamical model of protein synthesis has been pointed out[@shubio]. Another driven-dissipative system, which is also receiving wide attention of physicists in recent years, is the granular material flowing through a pipe[@proc1; @proc2]. There are some superficial similarities between the clustering of vehicles on a highway and particle-particle (and particle-cluster) aggregation process[@ben-naim].
The NaSch model with $V_{max} = 1$ can be mapped onto stochastic growth models of one-dimensional surfaces in a two-dimensional medium. Particle (hole) movement to the right (left) correspond to local forward growth of the surface via particle deposition. In this scenario a particle evaporation would correspond to a particle (hole) movement to the left (right) which is not allowed in the NaSch model. It is worth pointing out that any quenched disorder in the rate of hopping between two adjacent sites would correspond to [*columnar*]{} quenched disorder in the growth rate for the surface[@barma].
Inspired by the recent success in theoretical studies of traffic, some studies of information traffic on the computer network (internet) have also been carried out[@csabai; @ohira; @taka].
Summary and conclusion:
=======================
Nowadays the tools of statistical mechanics are increasingly being used to study self-organization and emergent collective behaviour of [*complex systems*]{} many of which, including vehicular traffic, fall outside the traditional domain of physical systems. However, as we have shown in this article, a strong theoretical foundation of traffic science can be built on the basic principles of statistical mechanics. In this brief review we have focussed attention mainly on the progress made in the recent years using “particle-hopping” models, formulated in terms of cellular automata, and compared these with several other similar systems.
[**Acknowledgements:**]{} It is our pleasure to thank R. Barlovic, J.G. Brankov, B. Eisenblätter, K. Ghosh, N. Ito, K. Klauck, W. Knospe, D. Ktitarev, A. Majumdar, K. Nagel, V.B. Priezzhev, M. Schreckenberg, A. Pasupathy, S. Sinha, R.B. Stinchcombe and D.E. Wolf for enjoyable collaborations the results of some of which have been reviewed here. We also thank M. Barma, J. Kertesz, J. Krug, G. Schütz, D. Stauffer and J. Zittartz for useful discussions and encouragements. This work is supported by SFB341 Köln-Aachen-Jülich.
This is a modified and shortened version of a longer detailed review article to be published elsewhere.
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[^1]: The configuration shown in fig.1 is also a GoE state!
|
---
abstract: 'A fall is an abnormal activity that occurs rarely, so it is hard to collect real data for falls. It is, therefore, difficult to use supervised learning methods to automatically detect falls. Another challenge in using machine learning methods to automatically detect falls is the choice of engineered features. In this paper, we propose to use an ensemble of autoencoders to extract features from different channels of wearable sensor data trained only on normal activities. We show that the traditional approach of choosing a threshold as the maximum of the reconstruction error on the training normal data is not the right way to identify unseen falls. We propose two methods for automatic tightening of reconstruction error from only the normal activities for better identification of unseen falls. We present our results on two activity recognition datasets and show the efficacy of our proposed method against traditional autoencoder models and two standard one-class classification methods.'
address:
- 'Toronto Rehabilitation Institute, 550 University Ave, Toronto, ON, M5G 2A2, Canada'
- 'University of Toronto, Canada'
author:
- 'Shehroz S. Khan'
- Babak Taati
bibliography:
- 'references.bib'
title: 'Detecting Unseen Falls from Wearable Devices using Channel-wise Ensemble of Autoencoders'
---
fall detection,one-class classification,autoencoders,anomaly detection
Introduction
============
Falls are a major cause of both fatal and non-fatal injury and a hindrance in living independently. Each year an estimated $424, 000$ individuals die from falls globally and $37.3$ million falls require medical attention [@website:whofall1]. Experiencing a fall may lead to a fear of falling [@igual2013challenges], which in turn can result in lack of mobility, less productivity and reduced quality of life. There exist several commercial wearable devices to detect falls [@pannurat2014automatic]; most of them use accelerometers to capture motion information. They normally come with an alarm button to manually contact a caregiver if the fall is not detected by the device. However, most of the devices for detecting falls produce many false alarms [@el2013fall]. Automatic detection of falls is long sought; hence, machine learning techniques are needed to automatically detect falls based on sensor data. However, a fall is a rare event that does not happen frequently [@Stone2015Fall; @khan2014iwaal]; therefore, during the training phase, there may be very few or no fall samples. Standard supervised classification techniques may not be suitable in this type of skewed data scenario. Another issue regarding the use of machine learning methods in fall detection is the choice of features. Traditional activity recognition and fall detection methods extract a variety of domain specific features from raw sensor readings to build classification models [@Ravi:2005; @khan2016classification]. It is very difficult to ascertain the number or types of features, specially in the absence of fall specific training data to build generalizable models.
To handle the problems of lack of training data from real falls and the difficulty in engineering appropriate features, we explore the use of Autoencoders (AE) that are trained only on normal activities. AEs can learn generic features from the raw sensor readings and can be used to identify unseen falls as abnormal activities during testing based on a threshold on the reconstruction error. We present two ensembles approaches of AE that train on the raw data of the normal activities from different channels of accelerometer and gyroscope separately and the results of each AE is combined to arrive upon a final decision. Typically, while using AE, the maximum of reconstruction error on the training set is considered as the threshold to identify an activity as abnormal. However, we experimentally show that such threshold may not be appropriate for detecting falls due to noisy sensor data. We present two threshold tightening techniques to remove few outliers from the normal data. Then, either a new threshold is derived using inter-quartile range or by training a new AE on the training data with outliers removed. We show result on two activity recognition datasets that contain different normal activities along with falls from wearable sensors. The rest of the paper is organized as follows. In the next Section, we present a brief introduction to Autoencoders. Section \[sec:related\] reviews the literature on fall detection using AE and on the use of AE in general outlier detection tasks. We present the proposed channel-wise ensemble of autoencoder and two threshold tightening approaches using reconstruction error in Section \[sec:threshold\]. Experimental analysis and results are discussed in Section \[sec:results\], followed by conclusions and future work in Section \[sec:conclusions\].
Brief Introduction to Autoencoders {#sec:ae}
==================================
An AE is an unsupervised multi-layer neural network that learns compact representation of the input data [@scholz2002nonlinear]. An AE tries to learn an identity function such that its outputs are similar to its inputs. However, by putting constraint on the network, such as limiting the number of hidden neurons, it can discover compact representations of the data that can be used as features for other supervised or unsupervised learning tasks. An AE is often trained by using the backpropagation algorithm and consists of an encoder and decoder part. If there is one hidden layer, an AE takes the input $\textbf{x}\in \mathbb{R}^d$ and maps it onto $\textbf{h}\in \mathbb{R}^p$, s.t.
$$\textbf{h}=f(W\textbf{x}+b)$$
where $\textbf{W}$ is a weight matrix and $\textbf{b}$ is a bias term and $f(.)$ is a mapping function. This step is referred to as encoding or learning latent representation, after which $\textbf{h}$ is mapped back to reconstruct $\textbf{y}$ of the same shape as $\textbf{x}$, i.e.
$$\textbf{y}=g(\textbf{W$'$h}+\textbf{b$'$})$$
This step is referred to as decoding or reconstructing the input back from latent representation. An AE can be used to minimize the squared reconstruction error, $L$ i.e.,
$$\label{eq:re}
L(\textbf{x,y})=\parallel \textbf{x -y} \parallel ^2$$
AE can learn compact and useful features if $p<d$; however, it can still discover interesting structures if $p>d$. This can be achieved by imposing a sparsity constraint on the hidden units, s.t. neurons are inactive most of the time or the average activation of each hidden neuron is close to zero. To achieve sparsity, an additional sparsity parameter is added to the objective function. Multiple layers of AEs can be stacked on top of each other to learn hierarchical features from the raw data. They are called Stacked AE (SAE). During encoding of a SAE, the output of first hidden layer serves as the input to the second layer, which will learn second level hierarchical features and so on. For decoding, the output of the last hidden layer is reconstructed at the second last hidden layer, and so on until the original input is reconstructed.
Related Work {#sec:related}
============
AEs can be used both in supervised and unsupervised mode for identifying falls. In a supervised classification setting, AE is used to learn representative features from both the normal and fall activities. This step can be followed by a standard machine learning classifier trained on these compressed features [@li2014unsupervised] or by a deep network [@jokanovic2016radar]. In the unsupervised mode or One-Class Classification (OCC) [@Khan:KER:2014] setting, only data for normal activities is present during training the AE. In these situations, an AE is used to learn representative features from the raw sensor data of normal activities. This step is followed by either employing (i) a discriminative model by using one-class classifiers or (ii) a generative model with appropriate threshold based on reconstruction error, to detect falls and normal activities. The present paper follows the unsupervised AE approach with a generative model and finding an appropriate threshold to indentify unseen falls.
A lot of work has been done in evaluating the feasibility of learning generic representations through AEs for general activity recognition and fall detection tasks. Plötz et al. [@Plotz2011Feature] explore the potential of discovering universal features for context-aware application using wearable sensors. They present several feature learning approaches using PCA and AE and show their superior performance in comparison to standard features across a range of activity recognition applications. Budiman et al. [@Budiman2014Stacked] use SAEs and marginalized SAE to infer generic features in conjunction with neural networks and Extreme Learning Machines as the supervised classifiers to perform pose-based action recognition. Li et al. [@Li2014] compare SAE, Denoising AE and PCA for unsupervised feature learning in activity recognition using smartphone sensors. They show that traditional features perform worse than the generic features inferred through autoencoders. Jokanovic et al. [@jokanovic2016radar] use SAE to learn generic lower dimensional features and use softmax regression classifier to identify falls using radar signals. Other researchers [@jankowski2015deep; @wang2016recognition] have used AEs to reduce the dimensionality of domain specific features prior to applying traditional supervised classification models or deep belief networks.
AEs have also been extensively used in anomaly detection. Japkowicz et al. [@japkowicz1995novelty] present the use of AE for novelty detection. For noiseless data, they propose to use a reduced percentage of maximum of reconstruction error as a threshold to identify outliers. For noisy data, they propose to identify both the intermediate positive and negative regions and subsequently optimizing the threshold until a desired accuracy is achieved. Manevitz and Yousef [@Manevitz20071466] present an AE approach to filter documents and report better performance than traditional classifiers. They report to carry out certain type of uniform transformation before training the network to improve the performance. They discuss that choosing an appropriate threshold to identify normal documents is challenging and present several variants. The method that worked the best in their application is to tighten the threshold sufficiently to disallow the classification of the highest $25$ percentile error cases from the training set. Erfani et al. [@Erfani2016121] present a hybrid approach to combine the AEs and one-class SVM ($OSVM$) for anomaly detection in high-dimensional and large-scale applications. They first extract generic features using SAE and train an $OSVM$ with linear kernel on learned features from SAE. They also use SAE as a one-class classifier by setting the threshold to be $3$ times of standard deviation away from the mean. Their results show comparable results in comparison to AE based anomaly classifier but the training and testing time greatly reduced. Sakurda and Yairi [@sakurada2014anomaly] show the use of AE in anomaly detection task and compare it with PCA and Kernel PCA. They demonstrate that the AE can detect subtle anomalies that PCA could not and is less complex than Kernel PCA. Ensembles of AE have been used to learn diverse feature representations, mainly in the supervised settings. Ithapu et al. [@ithapu2014randomized] present an ensemble of SAE by presenting it with randomized inputs and randomized sample sets of hyper-parameters from a given hyper–parameter space. They show that their approach is more accurately related to different stages of Alzheimer’s disease and leads to efficient clinical trials with very less sample estimates. Reeve and Gavin [@Reeve2015Modular] present a modular AE approach that consists of $M$ AE modules trained separately on different data representations and the combined result is defined by taking an average of all the modules present. Their results on several benchmark datasets show improved performance in comparison to baseline of bootstrap version of the AE. Dong and Japkowicz [@Dong2016Threaded] present a supervised and unsupervised ensemble approach for stream learning that uses multi-layer neural networks and AE. They train their models from multi-threads which evolve with data streams, the ensemble of the AE is trained using only the data from positive class and is accurate when anomalous training data are rare. Their method performs better as compared to the state-of-the-art in terms of detection accuracy and training time for the datasets.
The research on using AE show that it can successfully learn generic features from raw sensor data for activity and fall recognition tasks. We observe that AE can be effectively used for anomaly detection tasks and their ensembles can perform better than a single AE. In this paper, fall detection problem is formulated as an OCC or anomaly detection, where abundant data for normal activities is available during training and none for falls. We investigate the utility of features learned through AE and their ensembles for the task of fall detection.
Autoencoder Ensemble for Detecting Unseen Falls {#sec:monolithic}
===============================================
In the absence of training data for falls, a fall can be detected by training an AE/SAE on only the normal activities to learn generic features from a wearable device. These features can be fed to standard OCC algorithms to detect a test sequence as a normal activity or not (a fall in our case). Alternatively, based on the training data, a threshold can be set on the reconstruction error of the AE/SAE to identify a test sequence as an abnormal activity (a fall in our case) if its reconstruction error is higher than a given threshold. Intuitively, this would mean that the test sequence is very different from the training data comprising of normal activities. Below, we discuss two types of AE approaches used in the paper.
Monolithic Autoencoders
-----------------------
Figure \[fig:mono\] shows the AE/SAE for training normal activities using raw sensor data from a three-channel accelerometer and gyroscope. The raw sensor readings coming from each of the channels of accelerometer ($a_x,a_y,a_z$) and gyroscope ($\omega_x, \omega_y,\omega_z$) are combined and presented as input to the AE/SAE. For a sliding window of a fixed length ($n$ samples), $\mathbf{a_x} = [a_x^1, a_x^2, ..., a_x^n]$, $\mathbf{a_y} = [a_y^1, a_y^2, ..., a_y^n]$, and so on. The feature vector for a time window is constructed by concatenating these sensor readings as $\mathbf{f} = [\mathbf{a_x}, \mathbf{a_y}, \mathbf{a_z}, \bm{\omega_x}, \bm{\omega_y}, \bm{\omega_z}]^T$. We call this feature learning approach as *monolithic* because it combines raw sensor data from different channels as one input to an AE.
![Monolithic AE for detecting unseen falls.[]{data-label="fig:mono"}](monolithic){width="7cm" height="3cm"}
[0.5]{} ![Channel-wise AE for detecting unseen falls.[]{data-label="fig:ens"}](channel "fig:"){width="8cm" height="6cm"}
[0.5]{} ![Channel-wise AE for detecting unseen falls.[]{data-label="fig:ens"}](norm "fig:"){width="8cm" height="6cm"}
Channel-wise Autoencoders
-------------------------
Li et al. [@Li2014] present the use of ensemble of SAE by extracting generic features per each of the three accelerometer channels and additional channel for the magnitude of the accelerometer vector in $3$-dimensions. They extract fixed number of features for each of these $4$ channels and concatenate them. Supervised classification methods are then used on these extracted features. This setting can work for supervised classification but not in OCC scenario. In our case, we deal with OCC scenario with only normal data available during training. Therefore, separate AEs are trained on the raw data from different channels of the sensors. Each AE can detect a test sample as an unseen fall or not based on the reconstruction error and their overall result is combined to take a final decision. We propose to use two types of channel-wise ensemble strategies for detecting unseen falls as follows:
- Six Channel Ensemble ($6\_CE$): For each of the $6$ channels of an accelerometer and a gyroscope (i.e., $\mathbf{a_x},\mathbf{a_y},\mathbf{a_z}, \bm{\omega_x}, \bm{\omega_y},\bm{\omega_z}$), $6$ separate AE/SAE are trained to learn a compact representation for each channel. A decision threshold can be employed on each of these $6$ AEs to decide whether a test sample is normal or a fall.
- Two Channel Ensemble ($2\_CE$): Alternatively, we can compute the magnitude of the $3$ accelerometer channels and that of the $3$ gyroscope channels. The magnitude vector gives direction invariant information. We train two separate AE/SAE to learn a compact representation for each of the two magnitude channels. Thresholding the reconstruction error on these two channels can be used to decide whether a test sample is normal activity or a fall.
For a given test sample, the $6\_CE$ will give $6$ different decisions and the $2\_CE$ give $2$ decisions. These decisions can be combined by majority voting to arrive at a final decision; as a convention, ties are considered as falls. For simplicity, we keep the hyper-parameters for each AE/SAE corresponding to a channel as the same. The ensemble approach can be faster than the monolithic approach because AE/SAE per channel uses less amount of data in comparison to the combined $6$ channel data to a single AE/SAE. Figure \[fig:ens\] shows the graphical representation of the $6\_CE$ and $2\_CE$ approaches.
Optimizing the Threshold on the Reconstruction Error {#sec:threshold}
=====================================================
For the fall detection problem, we assume that fall data is rare and is not present during training phase [@khan2017detecting]. Therefore, we train monlithinc and channel-wise AE and SAE on the raw sensor data to learn a compact representation of the normal activities. The next step is to identify a test sample as normal or fall based on the trained AE/SAEs. The typical approach to identify a fall as an anomaly is to set a threshold on the reconstruction error. This threshold is generally set as the maximum of the reconstruction error on the full training data. We call this threshold *MaxRE*. During testing, any sample that has a reconstruction error greater than this value can be identified as a fall. However, sensor readings are not perfect and may contain spurious data [@khan2014iwaal], which can affect this threshold. Due to the presence of a few outliers in the training data, *MaxRE* is often too large, which could result in many of the falls being missed during testing time. To handle this situation, tightening of threshold is often required (as discussed in Section \[sec:related\]). We use the approach of Erfani et al. [@Erfani2016121] that sets the threshold as $3$ standard deviations away from the mean of the training data reconstruction error. We call this threshold method as *StdRE*. The *StdRE* threshold can result in identifying more falls during testing in comparison to *MaxRE* at the cost of few false alarms because the threshold in this case is smaller in comparison to *MaxRE*. A problem with *StdRE* is that it is chosen in an adhoc manner and it may not be an appropriate choice for a given data.
We now present two new approaches to tighten the threshold on reconstruction error. These approaches derive the threshold from the training data such that it can better identify unseen falls. These methods are similar to finding an optimal operating point on an ROC curve by reducing false negatives at the cost of false alarms. However, in a OCC framework, it is difficult to adopt a traditional ROC approach because of the unavailability of the validation data for the negative class. The proposed methods overcome these difficulty by removing outliers from the training data prior to setting a threshold based on only the training data.
Reduced Reconstruction Error {#sec:rre}
----------------------------
As we discussed earlier, the raw sensor data may not be perfect and may contain spurious or incorrectly labeled readings [@khan2014iwaal]. If an AE/SAE is trained on normal activities on such data, the reconstruction error for some of the samples of the training set may be very large. In this case, choosing the maximum of reconstruction error as the threshold to identify falls may lead to accepting most of the falls as normal activities. Khan et al. [@khan2017detecting] propose to use the concept of quartiles from descriptive statistic to remove few outliers present in the normal activities class. We use a similar idea but adapt it to AE to tighten the threshold on the reconstruction error. We first train an AE on normal activites, then find the reconstruction error of each training sample. Given the reconstruction error on the training data comprising of only instances of normal activities, the lower quartile ($Q_1$), the upper quartile ($Q_3$) and the inter-quartile range ($IQR = Q_3 - Q_1$), a point $P$ is qualified as an outlier of the normal class, if $$P > Q_3 + \Omega \times IQR \quad || \quad P < Q_1 - \Omega \times IQR
\label{eqn:iqr}$$
where $\Omega$ is the rejection rate that represents the percentage of data points that are within the non-extreme limits. Based on $\Omega$, the extreme values of reconstruction error that represents spurious training data can be removed and a threshold can be chosen as the maximum of the remaining reconstruction errors. We call this method as Reduced Reconstruction Error (*RRE*). The value of $\Omega$ can be found experimentally or set to remove a small fraction of the normal activities data. We describe a cross-validation technique in Section \[sec:cv\] to find *RRE* from only the normal activities.
Inlier Reconstruction Error
---------------------------
In this method, we first train an AE/SAE on full normal data and then remove the corresponding anomalous training instances based on $\Omega$ from the training set (as discussed in the previous section). After this step, we are left with training data without the outlier instances. Then, we train a new AE on this reduced data comprising of just the inlier. The idea is that the variance of reconstruction error for such inlier data will not be too high and its maximum can serve as the new threshold. We call this method as Inlier Reconstruction Error (*IRE*).
For the channel-wise ensemble approach, each AE/SAE is trained only using the raw sensor data from a specific channel of normal activities, then various thresholds, i.e., *MaxRE*, *StdRE*, *RRE* and *IRE* are computed for each channel separately. During testing, for a given threshold method, the final decision is taken as the majority voting outcome of all the AE/SAE. The intuition behind *RRE* and *IRE* is that they should provide a better trade-off between false positive and true positive rate in comparison to *MaxRE* and *StdRE*. The threshold *MaxRE* may work better in one direction, whereas *StdRE* is an adhoc approach to minimize errors. Both *RRE* and *IRE* are derived from the data and not arbitrarily set to a fixed number. The proposed threshold tightening methods, i.e. *RRE* and *IRE*, attempt to find a threshold after removing spurious sensor data from the normal activities, this may lead to improved sensitivity in detecting falls.
Cross Validation {#sec:cv}
----------------
The parameter $\Omega$ to tighten the threshold for *RRE* and *IRE* cannot be directly optimized because there is no validation set due to the absence of fall data during training. Khan et al. [@khan2017detecting] propose to remove some outliers from the normal data and consider them as proxy for unseen falls. They show that rejected outliers from the normal activities can be used to create a validation set and tune parameters of a learning algorithm in the absence of fall data. They use hidden markov models and show that some of the proxy for falls bear resemblance to actual falls. We modify this idea with respect to the autoencoders and present a cross-validation method to optimize $\Omega$ in our setting.
Firstly, we train an AE on full normal data and compute reconstruction error of each training example. Then we reject some instances from the normal activities based on a parameter $\rho$, using their reconstruction error. The parameter $\rho$ also uses IQR technique (as discussed in Section \[sec:rre\]); however it is very different from the parameter $\Omega$. The parameter $\rho$ represents the amount of outlier data removed from normal activities to generate samples for proxy falls to create a validation set. The remaining normal activities are called non-falls. The parameter $\Omega$ represents the amount of reconstruction error removed to set a ‘threshold’ to identify unseen falls during testing. For a given value of $\rho$, several values of $\Omega$ can be tested and the best is used for further analysis. Therefore, $\rho$ is considered as a hyper-parameter and $\Omega$ as a parameter to find *RRE* and *IRE*. Then the data from both the classes (non-falls and proxy fall) is divided into $K$-folds. The non-fall data from $(K-1)$ folds is combined and an AE is trained on it. The data from $K^{th}$ fold for non-fall and proxy fall is used for testing and tuning the parameters. The process is repeated $K$ times for different values of $\Omega$, the one with the best average performance over $K$ folds is chosen for further analysis. The peformance metric is discussed in Section \[sec:perf\]. Lastly, for a given $\rho$, we retrain on the non-fall data. The maximum of the reconstruction error corresponding to the best $\Omega$ (obtained in the step discussed above for a given $\rho$) is taken as *RRE*. To compute *IRE*, we remove the outliers from the non-falls corresponding to $\Omega$; then retrain on the reduced training set and take the maximum of the reconstruction error as *IRE*. The value of hyper-parameter $\rho$ can be varied to observe an overall effect on the performance of the proposed threshold tightening methods, *RRE* and *IRE*. Intuitively, a large value of $\rho$ means less number of instances are removed from normal activities as proxy for fall, which may lead to classify a lot of test samples as normal activities and may miss to identify some falls. Whereas, a small value of $\rho$ means more instances from normal activities may be rejected as proxy for falls; thus, the normal class will be smaller that may result in identifying most of the falls but at the cost of more false alarms. In summary, we exect that with increase in $\rho$, both the true positive rate and false positive rate should reduce (fall is the positive class). By varying $\rho$, we can find an optimum range of operation with a good balance of true positives and false positives. It is to be noted that in this cross-validation method, no actual falls from the training set are used because it is only comprised of normal activities and all the parameters are tuned in the absence of actual falls.
Experimental Analysis {#sec:results}
=====================
Performance Metrics {#sec:perf}
-------------------
We consider a case for detecting falls where they are not available during training and occur only during testing. Therefore, during the testing phase, we expect a skewed distribution of falls. Hence, the standard performance metrics such as accuracy may not be appropriate because it may give an over-estimated view of the performance of the classifier. To deal such a case, we use the geometric mean ($gmean$) [@kubat1997addressing; @khan2014iwaal] as the performance metric to present the test results and optimize the parameters during cross-validation. $gmean$ is defined as the square root of the multiplication of true positive and true negative rate, i.e. $$\begin{split}
gmean &= \sqrt{TPR * TNR} \\
gmean &= \sqrt{TPR * (1-FPR)}
\end{split}$$
where $TPR$ is the true positive rate, $TNR$ is the true negative rate and $FPR$ is the false positive rate. The value of $gmean$ varies from $0$ to $1$, where a $1$ means a perfect classification among falls and normal activities and $0$ as the worst outcome. We also use the $TPR$ and $FPR$ as other performance metrics to further elaborate our results.
To evaluate the performance of the proposed approaches for fall detection, we perform leave-one-subject-out cross validation (LOOCV) [@he2009activity], where *only* normal activities from $(N-1)$ subjects are used to train the classifiers and the $N^{th}$ subject’s normal activities and fall events are used for testing. This process is repeated $N$ times and the average performance metrics are reported. This evaluation is person independent and demonstrates the generalization capabilities as the subject who is being tested is not included in training the classifiers.
Datasets
--------
We show our results on two activity recognition datasets that includes different normal activities and fall events collected via wearable devices.
### German Aerospace Center (DLR) [@dlr65511]
This dataset is collected using an Inertial Measurement Unit with a sampling frequency of $100$ Hz. The dataset contains samples from 19 people of both genders of different age groups. The data is recorded in indoor and outdoor environments under semi-natural conditions. The sensor is placed on the belt either on the right or the left side of the body or in the right pocket in different orientations. The dataset contains labelled data of the following $7$ activities: Standing, Sitting, Lying, Walking (up/downstairs, horizontal), Running/Jogging, Jumping and Falling. One of the subjects did not perform fall activity; therefore, their data is omitted from the analysis.
### Coventry Dataset (COV) [@Ojetol2015data]
This dataset is collected using two SHIMMERsensor nodes strapped to the chest and thighs of subjects with a sampling frequency of $100$ Hz. Two protocols were followed to collect data from subjects. In Protocol 1, data for six types of fall scenarios are captured (forward, backward, right, left, real fall-backward and real fall forward) and a set of ADL (standing, lying, sitting on a chair or bed, walking, crouching, near falls and lying). Protocol 2 involved ascending and descending stairs. $42$ young healthy individuals simulated various ADL and fall scenarios ($32$ in Protocol $1$ and $10$ in Protocol $2$). These data from different types of falls are joined together to make one separate class for falls. The subjects for Protocol $2$ did not record corresponding fall data; therefore, that data is not used. In our analysis, we used accelerometer and gyroscope data from the sensor node strapped to the chest.
Experimental Setup
------------------
For both the datasets, all the normal activitis are joined together to form a normal class. For COV dataset, different types of falls are joined to make a fall class. The raw sensor data is processed using a $50\%$ overlapping sliding window. The time window size is set to $1.28$ seconds for the DLR dataset and $2.56$ seconds for the COV dataset (as shown in Khan et al. [@khan2017detecting]). After pre-processing, the DLR dataset has 26576 normal activities and 84 fall segments, and the COV dataset has 12392 normal activities and 908 fall segments.
We test two types of AE for the analysis; one with a single hidden layer and other with three layered SAE. For the monolithic AE, the raw data within a time window for each of the $3$ channels of accelerometer and gyroscope is concatenated, which leads to $768 (=128\times6)$ input layer neurons for the DLR dataset and $1536 (=256\times6)$ input layer neurons for the COV dataset. The number of hidden neurons (i.e., the number of generic features learned) is set to $31$ (as suggested for the engineered features case in the work of Khan et al. [@khan2017detecting]). For the monolithic SAE, the number of hidden neurons in the first layer is chosen to be half of the number of input neurons, i.e. $384$ for DLR dataset and $768$ for COV dataset and the second layer has $31$ number of features. For the channel-wise ensemble method, each channels is fed to the AE/SAE separately. Therefore, the number of neurons in the input layer per AE is set to $128$ for DLR dataset and $256$ for COV dataset and the hidden layers has $31$ neurons. For the channel-wise SAE, the hidden neurons for first layer is half the number of input layer, i.e. $64$ for DLR dataset and $128$ for the COV dataset. The second hidden layer for both the datasets has $31$ neurons. The number of training epochs is fixed to $10$ for all the different autoencdoers. Rest of the parameters such as the sparsity parameter, activation function etc., are kept at the default values [@website:ae]. Compressed features learned through monolithic AE and SAE are further used to train $OSVM$ and One-class nearest neighbour ($OCNN$) classifiers [@Khan:KER:2014] for comparison.
### Internal Cross-Validation
For $OCNN$, the number of nearest neighbours to identify an outlier is kept as $1$. $OSVM$ has a parameter $\nu$ (or the outlier fraction), which is the expected proportion of outliers in the training data. The value of this parameter is tuned, similar to parameter optimization discussed in Section \[sec:cv\]. That is, reject a small portion of normal class data as a proxy for unseen falls for a given $\rho$ and create a validation set. Then perform a $K$ fold cross-validation for different values of $\nu$ and choose the one with the largest average $gmean$ over all the $K$-folds. The ‘KernelScale’ parameter is set to ‘auto’ and ’Standardize’ to ‘true’, other parameters are kept to default values [@website:svm].
An internal $K=3$-fold cross validation is employed to optimize the parameters $\nu$ for the $OSVM$ and $\Omega$ for the *RRE* and *IRE* thresholding methods. The parameter $\Omega$ is varied from from $[0.001, 0.01, 0.1, 0.5, 1, 1.5, 1.7239, 2, 2.5, 3]$ and $\nu$ is varied from $[0.1,0.3, 0.5, 0.7, 0.9]$. The best parameter is chosen based on the average $gmean$ over $K$ folds. To understand the effect of removing outlier data from the normal activities in building classification models for unseen falls, the hyper-parameter $\rho$ is varied from $[0.001, 0.01, 0.1, 0.5, 1, 1.5, 1.7239, 2, 2.5, 3]$.
Along with $6$ channel raw data to train different classifiers to detect unseen falls, we also use $2$ channel magnitude data from each of the datasets to train different classifiers. Therefore, in the experiment we compare the following different classifiers for two types of channels data (i.e. $6$ and $2$ channels):
- Two types of AE, i.e. single layer AE and three layered SAE.
- Four types of thresholding methods i.e. *MaxRE*, *StdRE*, *RRE* and *IRE*.
- Two types of feature learning techniques - (i) monolithic and (ii) channel-wise ensemble.
- Two one-class classifiers ($OCNN$ and $OSVM$ ) trained on features learned from AE and SAE (not for the channel-wise case).
This results in $20$ different classifiers trained per $6$ / $2$ channels input raw sensor data, we compare their performance in the next section.
Results and Discussion
----------------------
Tables \[tab:DLR6\] and \[tab:DLR2\] show the results for the DLR datasets for $6$ and $2$ channel input raw data. The results correspond to $\rho=1.5$. Tables \[tab:DLR6occ\] and \[tab:DLR2occ\] show the results when the features learned using AE and SAE are fed to OSVM and OCNN. We observe that for the $6$ channel case, the best $gmean$ is obtained for channel-wise AE with *RRE* followed by *IRE* method (see Tables \[tab:DLR6gmean\]. The traditional thresholding methods of *MaxRE* and *StdRE* does not perform well. We observe similar results for DLR dataset with $2$ channel input; however, the $gmean$ values using $6$ channel input data are higher. Both the *RRE* and *IRE* methods with channel-wise AE give good trade-off between $TPR$ (see Tables \[tab:DLR6TPR\] and \[tab:DLR2TPR\]) and $FPR$ (see Tables \[tab:DLR6FPR\] and \[tab:DLR2FPR\]).
Results on the COV dataset for the $6$ and $2$ channels input data are shown in Table \[tab:COV6\] and \[tab:COV2\]. For the $6$ channel case, channel-wise ensemble method with AE for *RRE* and *IRE* give equivalent values of $gmean$, which is higher than other methods of thresholding. Both the best methods give a good trade-off between $TPR$ and $FPR$ (see Tables \[tab:COV6TPR\] and \[tab:COV6FPR\]). For the $2$ channel case, the *IRE* threshold method for both the monolithic and channel-wise approaches for AE and SAE give equivalent performance along with monolithic SAE with *RRE*. The channel-wise approach gives more false alarms but detects more falls than the monolithic approach. For both the DLR and COV datasets, the OCNN classifier perform worse than the proposed methods because it gives large number of false alarms; whereas, OSVM classifies all the test samples as falls (see Tables \[tab:DLR6occ\], \[tab:DLR2occ\], \[tab:COV6occ\], \[tab:COV2occ\]).
By convention, we classify a test sample as a fall in case of a tie in the channel-wise approach. The probability of a tie occurring is higher in $2$ channel ensemble method than in $6$ channel ensemble; therefore, its sensitivity to detect falls is higher than the $6$ channel case with an increase in the false alarm rate. We observe this behavior for both the DLR and COV datasets (see the Channel-wise rows in Tables \[tab:DLR6FPR\], \[tab:DLR2FPR\], and \[tab:COV6FPR\], \[tab:COV2FPR\] ). From this experiment we infer that the traditional methods of thresholding, i.e., *MaxRE* and *StdRE*, are not suitable for the task of fall detection. *MaxRE* may not work properly because of the presence of noise in the sensor data that can significantly increase the reconstruction error of an AE/SAE, leading to classify most of the test samples as normal activity. The *StdRE* is an ad-hoc approach that arbitrarily chooses a threshold to identify fall and does not derive it from a given dataset. However, it can perform better than *MaxRE*, in terms of identifying more falls. Both of these methods attempt to find a discriminating threshold from the training dataset to to get a good trade-off between $TPR$ and $FPR$. Our experiments suggest that for both the datasets, the proposed threshold tightening methods *RRE* and *IRE* with channel-wise ensemble approach perform equivalently and consistently better than the traditional methods of threshold tightening.
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 0 0.106 0.825 0.757
SAE 0 0.234 0.840 0.837
Channel- AE 0 0.547
wise SAE 0 0.334 0.818 0.811
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on DLR dataset ($6$ channels) for $\rho=1.5$[]{data-label="tab:DLR6"}
\[tab:DLR6gmean\]
\
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 0 0.056 0.856 0.762
SAE 0 0.138 0.893 0.893
Channel- AE 0 0.428
wise SAE 0 0.226 0.774 0.750
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on DLR dataset ($6$ channels) for $\rho=1.5$[]{data-label="tab:DLR6"}
\[tab:DLR6TPR\]
\
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 5.9e-6 0.025 0.189 0.169
SAE 5.9e-6 0.025 0.199 0.204
Channel- AE 0 0.010
wise SAE 0 0.008 0.088 0.079
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on DLR dataset ($6$ channels) for $\rho=1.5$[]{data-label="tab:DLR6"}
\[tab:DLR6FPR\]
\
[.8]{}
**Classifier** **Autoencoder Type** $gmean$ $TPR$ $FPR$
---------------- ---------------------- --------- ------- ------- --
AE 0 1 1
SAE 0 1 1
AE 0.460 0.911 0.761
SAE 0.423 0.681 0.701
: Performance of different fall detection methods on DLR dataset ($6$ channels) for $\rho=1.5$[]{data-label="tab:DLR6"}
\[tab:DLR6occ\]
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 0 0 0.504 0.774
SAE 0 0 0.630 0.776
Channel- AE 0.013 0.487
wise SAE 0 0.446 0.678 0.655
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on DLR dataset ($2$ channels) for $\rho=1.5$[]{data-label="tab:DLR2"}
\[tab:DLR2gmean\]
\
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 0 0 0.966 0.949
SAE 0 0 0.959 0.941
Channel- AE 0.003 0.323
wise SAE 0 0.329 0.629 0.579
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on DLR dataset ($2$ channels) for $\rho=1.5$[]{data-label="tab:DLR2"}
\[tab:DLR2TPR\]
\
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 7.4e-5 0.037 0.705 0.363
SAE 3.7e-5 0.039 0.544 0.353
Channel- AE 1.0e-4 0.032
wise SAE 7.7e-5 0.033 0.099 0.094
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on DLR dataset ($2$ channels) for $\rho=1.5$[]{data-label="tab:DLR2"}
\[tab:DLR2FPR\]
\
[.8]{}
**Classifier** **Autoencoder Type** $gmean$ $TPR$ $FPR$
---------------- ---------------------- --------- ------- ------- --
AE 0 1 1
SAE 0 1 1
AE 0.459 0.815 0.719
SAE 0.318 0.317 0.449
: Performance of different fall detection methods on DLR dataset ($2$ channels) for $\rho=1.5$[]{data-label="tab:DLR2"}
\[tab:DLR2occ\]
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 0.015 0.744 0.774 0.771
SAE 0.019 0.743 0.772 0.771
Channel- AE 0.014 0.463
wise SAE 0 0.226 0.737 0.707
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on COV dataset ($6$ channels) for $\rho=1.5$[]{data-label="tab:COV6"}
\[tab:COV6gmean\]
\
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 0.004 0.589 0.744 0.740
SAE 0.007 0.588 0.738 0.738
Channel- AE 0.003 0.248
wise SAE 0 0.082 0.665 0.573
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on COV dataset ($6$ channels) for $\rho=1.5$[]{data-label="tab:COV6"}
\[tab:COV6TPR\]
\
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 1.1e-4 0.017 0.169 0.169
SAE 1.1e-4 0.017 0.166 0.167
Channel- AE 0 0.002
wise SAE 0 5.9e-5 0.128 0.078
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on COV dataset ($6$ channels) for $\rho=1.5$[]{data-label="tab:COV6"}
\[tab:COV6FPR\]
\
[.8]{}
**Classifier** **Autoencoder Type** $gmean$ $TPR$ $FPR$
---------------- ---------------------- --------- ------- ------- --
AE 0 1 1
SAE 0 1 1
AE 0.432 0.977 0.805
SAE 0.484 0.949 0.751
: Performance of different fall detection methods on COV dataset ($6$ channels) for $\rho=1.5$[]{data-label="tab:COV6"}
\[tab:COV6occ\]
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 0.041 0.743 0.668 0.785
SAE 0.019 0.724 0.784 0.784
Channel- AE 0.337 0.767 0.726
wise SAE 0.331 0.757 0.739 0.786
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on COV dataset ($2$ channels) for $\rho=1.5$[]{data-label="tab:COV2"}
\[tab:COV2gmean\]
\
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 0.012 0.587 0.729 0.698
SAE 0.007 0.557 0.677 0.674
Channel- AE 0.147 0.621 0.805
wise SAE 0.142 0.606 0.781 0.779
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on COV dataset ($2$ channels) for $\rho=1.5$[]{data-label="tab:COV2"}
\[tab:COV2TPR\]
\
[.8]{}
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
**Features Types** **Autoencoder Type**
**$\textit{MaxRE}$** **$\textit{StdRE}$** **$\textit{RRE}$** **$\textit{IRE}$**
AE 1.7e-4 0.013 0.287 0.094
SAE 1.7e-4 0.012 0.06 0.056
Channel- AE 1.1e-4 0.015 0.298
wise SAE 1.1e-4 0.016 0.255 0.182
-------------------- ---------------------- ---------------------- ---------------------- -------------------- --------------------
: Performance of different fall detection methods on COV dataset ($2$ channels) for $\rho=1.5$[]{data-label="tab:COV2"}
\[tab:COV2FPR\]
\
[.8]{}
**Classifier** **Autoencoder Type** $gmean$ $TPR$ $FPR$
---------------- ---------------------- --------- ------- ------- --
AE 0 1 1
SAE 0 1 1
AE 0.594 0.905 0.606
SAE 0.580 0.606 0.432
: Performance of different fall detection methods on COV dataset ($2$ channels) for $\rho=1.5$[]{data-label="tab:COV2"}
\[tab:COV2occ\]
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on DLR dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:DLR"}](DLRTPR "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on DLR dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:DLR"}](DLRFPR "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on DLR dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:DLR"}](DLRgmean "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on DLR-norm dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:DLR-norm"}](DLR-normTPR "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on DLR-norm dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:DLR-norm"}](DLR-normFPR "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on DLR-norm dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:DLR-norm"}](DLR-normgmean "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on COV dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:COV"}](COVTPR "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on COV dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:COV"}](COVFPR "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on COV dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:COV"}](COVgmean "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on COV-norm dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:COV-norm"}](COV-normTPR "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on COV-norm dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:COV-norm"}](COV-normFPR "fig:"){width="\textwidth"}
[0.7]{} ![Performance of top $5$ fall detection methods by varying $\rho$ on COV-norm dataset. AE - Single Layer Autoencoder, SAE - 3 Layer Stacked Autoencoder, RRE - Reduced Reconstruction Error, IRE - Inlier Reconstruction Error, CW - Channel-wise Ensemble[]{data-label="fig:COV-norm"}](COV-normgmean "fig:"){width="\textwidth"}
We vary the hyper-parameter $\rho$ to understand its impact on the performance of different thresholding techniques. Figures \[fig:DLR\] and \[fig:DLR-norm\] show the variation of $TPR$, $FPR$ and $gmean$ with increasing $\rho$. We observe that as the value of $\rho$ increases, both $TPR$ and $FPR$ reduce. The reason is that at smaller values of $\rho$, a large portion of normal data is rejected as outliers and used for parameter tuning; thus, the number of instances in the non-fall class is small. This means that the AE/SAE will learn on a smaller dataset and will reject most of the variations from this small subset of normal activities as potential falls. Consequently, many falls will also be identified correctly. The reverse behavior will happen when $\rho$ is large, thus less number of normal data is rejected as outliers and the class of normal activities will be large. This will reduce the number of false alarms but can also lead to missing to identify some falls. The experimental observation for each of the $6$ and $2$ channel datasets is consistent with this intuition discussed in Section \[sec:cv\]. Similar observation can be made for the COV dataset from Figures \[fig:COV\] and \[fig:COV-norm\]. For both datasets, we notice that at large value of $\rho$, the performance of best thresholding approaches drops slower. This experimental observation suggests that a smaller amount of data (corresponding to $\rho \ge 1.5$) may be removed from the normal activities class as outliers, which can be used as a validation set to optimize the parameters of the AE/SAE and better performance can be achieved for identifying unseen falls. We also infer that channel-wise approach outperforms monolithic in all the $6$ and $2$ channel data variants of both the datasets.
Conclusions and Future Work {#sec:conclusions}
===========================
A fall is a rare event; therefore, it is difficult to build classification models using traditional supervised algorithms in the absence of training data. An associated challenge for fall detection problem is to extract discriminative features in the absence of fall data for training generalizable classifiers. In this paper, we presented solutions to deal with these issues. Firstly, we formulated a fall detection problem as a one-class classification or outlier detection problem. Secondly, we presented the use of AE, more specifically a novel way to train separate AE for each channel of the wearable sensor, to learn generic features and create their ensemble. We proposed threshold tightening methods to identify unseen falls accurately. This work provides useful insights that an ensemble based on channels of a wearable device with optimized threshold is a useful technique to identify unseen falls. In future work, we are exploring extreme value theory and combining it with the proposed approaches to identify unseen falls.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the AGE-WELL NCE Trainee Award Program and by the Canadian Consortium on Neurodegeneration in Aging (CCNA).
References {#references .unnumbered}
==========
|
---
abstract: 'Consider a log Calabi-Yau pair $(X,D)$ consisting of a smooth del Pezzo surface $X$ of degree $\geq 3$ and a smooth anticanonical divisor $D$. We prove a correspondence between genus zero logarithmic Gromov-Witten invariants of $X$ intersecting $D$ in a single point with maximal tangency and the consistent wall structure appearing in the dual intersection complex of $(X,D)$ from the Gross-Siebert reconstruction algorithm. More precisely, the logarithm of the product of functions attached to unbounded walls in the consistent wall structure gives a generating function for these invariants.'
address: |
University of Hamburg\
Department of Mathematics\
Bundesstr. 55 20146 Hamburg\
Germany
author:
- Tim Gräfnitz
title: 'Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs'
---
Introduction {#introduction .unnumbered}
============
A smooth projective surface $X$ over the complex numbers $\mathbb{C}$ together with a reduced effective anticanonical divisor $D$ forms a *log Calabi-Yau pair* $(X,D)$, meaning that $K_X+D$ is numerically trivial. The case where $D=D_1+\ldots+D_m$ is a cycle of smooth rational curves (*maximal boundary*) has been studied in [@GHK1]. In [@GPS] it was shown that generating functions of logarithmic Gromov-Witten invariants of $X$ with maximal tangency at a single point on $D$ in this case can be read off from a certain *scattering diagram*. The statement of [@GPS] was generalized in [@Bou2] to $q$-refined scattering diagrams and generating functions of higher genus Gromov-Witten invariants.
In this work we consider the somewhat complementary case with $D$ a smooth irreducible divisor. We restrict to the case where $X$ has very ample anticanonical bundle $-K_X$, i.e., is a smooth del Pezzo surface of degree $\geq 3$. Let $Q\subset\mathbb{R}^2$ be a Fano poyltope, that is, a convex lattice polytope containing the origin and with all vertices being primitive integral vectors. From this one can construct (see §\[S:smoothing\]) a family $(\mathfrak{X}_Q\rightarrow\mathbb{A}^2,\mathfrak{D}_Q)$ such that fixing one parameter $s\neq 0$ one obtains a toric degeneration of $(X,D)$ and fixing $s=0$ gives a toric degeneration of $(X^0,D^0)$, where $X^0$ is a smooth nef toric surface admitting a $\mathbb{Q}$-Gorenstein smoothing to $X$ and $D^0=\partial X^0$ is the toric boundary.
Let $(B,\mathscr{P},\varphi)$ be the *dual intersection complex* of the toric degeneration of $(X,D)$. The affine manifold with singularities $B$ is non-compact without boundary. In [@CPS] it is described how to construct a tropical superpotential from such a triple $(B,\mathscr{P},\varphi)$, leading to a Landau-Ginzburg model[^1]. This perfectly fits into the picture, since the idea of the *Gross-Siebert program* is that toric degenerations constructed from Legendre dual polarized polyhedral affine manifolds are mirror to each other, and in fact the mirror of a Fano variety together with a choice of anticanonical divisor is believed to be a Landau-Ginzburg model. The construction involves the scattering calculations described in [@GS11], leading to a *consistent wall structure* $\mathscr{S}_\infty$ on $(B,\mathscr{P},\varphi)$. This is a collection of codimension $1$ polyhedral subsets of $B$ (*slabs* and *walls*) with attached functions describing the gluing of canonical thickenings of affine pieces necessary to obtain a toric degeneration with intersection complex $(B,\mathscr{P},\varphi)$. See [@Inv] for an overview of this construction.
The statement {#the-statement .unnumbered}
-------------
\[defi:beta1\] For an effective curve class $\underline{\beta}\in H_2^+(X,\mathbb{Z})$ let $\beta$ be the class of $1$-marked stable log maps to $X$ of genus $0$, class $\underline{\beta}$ and maximal tangency with $D$ at a single unspecified point. Let $\mathscr{M}(X,\beta)$ be the moduli space of basic stable log maps of class $\beta$ (see [@LogGW]). By the results of [@LogGW] this is a proper Deligne-Mumford stack and admits a virtual fundamental class $\llbracket\mathscr{M}(X,\beta)\rrbracket$. It has virtual dimension zero. The corresponding logarithmic Gromov-Witten invariant is defined by integration, i.e., proper pushforward to a point: $$N_\beta = \int_{\llbracket\mathscr{M}(X,\beta)\rrbracket} 1.$$
Let $\mathscr{S}_\infty$ be the consistent wall structure defined by the dual intersection complex $(B,\mathscr{P},\varphi)$ of $(X,D)$ via the Gross-Siebert algorithm [@GS11]. Figure \[fig:main\] shows $\mathscr{S}_\infty$ for $(\mathbb{P}^2,E)$ up to order $6$.
(-3,-1) rectangle (15,2); (0.000,0.500) – (0.000,6.50); (0.000,0.500) – (0.000,-5.50); (-1.50,-0.500) – (15.0,5.00); (-1.50,1.50) – (15.0,-4.00); (-1.50,1.50) – (-16.5,6.50); (-1.50,-0.500) – (-16.5,-5.50); (-6.00,-1.50) – (15.0,2.00); (-6.00,2.50) – (15.0,-1.00); (-6.00,2.50) – (-30.0,6.50); (-6.00,-1.50) – (-30.0,-5.50); (-13.5,-2.50) – (15.0,0.667); (-13.5,3.50) – (15.0,0.333); (-13.5,3.50) – (-40.5,6.50); (-13.5,-2.50) – (-40.5,-5.50); (-24.0,-3.50) – (15.0,-0.250); (-24.0,4.50) – (15.0,1.25); (-24.0,4.50) – (-48.0,6.50); (-24.0,-3.50) – (-48.0,-5.50); (-37.5,-4.50) – (15.0,-1.00); (-37.5,5.50) – (15.0,2.00); (-37.5,5.50) – (-52.5,6.50); (-37.5,-4.50) – (-52.5,-5.50); (-54.0,-5.50) – (15.0,-1.67); (-54.0,6.50) – (15.0,2.67); (0.000,0.000) – (15.0,0.000); (-3.00,-1.00) – (15.0,-1.00); (0.000,1.00) – (15.0,1.00); (-3.00,2.00) – (15.0,2.00); (-45.0,-5.00) – (15.0,-5.00); (-30.0,-4.00) – (15.0,-4.00); (-18.0,-3.00) – (15.0,-3.00); (-9.00,-2.00) – (15.0,-2.00); (-9.00,3.00) – (15.0,3.00); (-18.0,4.00) – (15.0,4.00); (-30.0,5.00) – (15.0,5.00); (-45.0,6.00) – (15.0,6.00); (1.50,0.500) – (15.0,0.500); (-4.50,-1.50) – (15.0,-1.50); (0.000,-0.500) – (15.0,-0.500); (0.000,1.50) – (15.0,1.50); (-4.50,2.50) – (15.0,2.50); (-22.5,-3.50) – (15.0,-3.50); (-12.0,-2.50) – (15.0,-2.50); (-12.0,3.50) – (15.0,3.50); (-22.5,4.50) – (15.0,4.50); (0.000,0.000) – (15.0,2.50); (-18.0,-3.00) – (15.0,-0.800); (-9.00,-2.00) – (15.0,0.000); (-3.00,-1.00) – (15.0,1.00); (0.000,1.00) – (15.0,6.00); (-3.00,2.00) – (-3.00,6.50); (-9.00,3.00) – (-19.5,6.50); (-18.0,4.00) – (-33.0,6.50); (0.000,0.000) – (15.0,-5.00); (-18.0,-3.00) – (-33.0,-5.50); (-9.00,-2.00) – (-19.5,-5.50); (-3.00,-1.00) – (-3.00,-5.50); (0.000,1.00) – (15.0,-1.50); (-3.00,2.00) – (15.0,0.000); (-9.00,3.00) – (15.0,1.00); (-18.0,4.00) – (15.0,1.80); (3.00,0.000) – (15.0,1.33); (0.000,-1.00) – (15.0,0.250); (3.00,1.00) – (15.0,3.00); (0.000,2.00) – (13.5,6.50); (3.00,0.000) – (15.0,-2.00); (0.000,-1.00) – (13.5,-5.50); (3.00,1.00) – (15.0,-0.333); (0.000,2.00) – (15.0,0.750); (3.00,0.000) – (15.0,-1.33); (3.00,1.00) – (15.0,0.000); (1.50,0.500) – (15.0,-1.00); (0.000,-0.500) – (15.0,-3.00); (0.000,1.50) – (15.0,0.250); (1.50,0.500) – (15.0,2.00); (0.000,-0.500) – (15.0,0.750); (0.000,1.50) – (15.0,4.00); (2.00,0.333) – (15.0,0.333); (2.00,0.667) – (15.0,0.667); (2.40,0.200) – (15.0,0.200); (0.000,-0.800) – (15.0,-0.800); (1.80,1.20) – (15.0,1.20); (2.40,0.800) – (15.0,0.800); (1.80,-0.200) – (15.0,-0.200); (0.000,1.80) – (15.0,1.80); (3.00,0.000) – (15.0,1.00); (3.00,1.00) – (15.0,2.33); (2.50,0.167) – (15.0,0.167); (2.50,0.833) – (15.0,0.833); (-13.0,-2.67) – (15.0,-2.67); (-5.00,-1.67) – (15.0,-1.67); (0.000,-0.667) – (15.0,-0.667); (1.00,1.33) – (15.0,1.33); (-3.00,2.33) – (15.0,2.33); (-10.0,3.33) – (15.0,3.33); (2.00,0.333) – (15.0,-0.750); (-10.0,-2.33) – (15.0,-2.33); (-3.00,-1.33) – (15.0,-1.33); (1.00,-0.333) – (15.0,-0.333); (0.000,1.67) – (15.0,1.67); (-5.00,2.67) – (15.0,2.67); (-13.0,3.67) – (15.0,3.67); (2.00,0.667) – (15.0,1.75); (0.000,0.000) – (15.0,3.33); (-3.00,-1.00) – (15.0,1.40); (0.000,1.00) – (8.25,6.50); (-3.00,2.00) – (-9.75,6.50); (0.000,0.000) – (15.0,1.67); (-9.00,-2.00) – (15.0,-0.400); (-3.00,-1.00) – (15.0,0.500); (0.000,1.00) – (15.0,3.50); (-3.00,2.00) – (10.5,6.50); (-9.00,3.00) – (-9.00,6.50); (0.000,0.000) – (8.25,-5.50); (-3.00,-1.00) – (-9.75,-5.50); (0.000,1.00) – (15.0,-2.33); (-3.00,2.00) – (15.0,-0.400); (0.000,0.000) – (15.0,-2.50); (-9.00,-2.00) – (-9.00,-5.50); (-3.00,-1.00) – (10.5,-5.50); (0.000,1.00) – (15.0,-0.667); (-3.00,2.00) – (15.0,0.500); (-9.00,3.00) – (15.0,1.40); (1.80,0.400) – (15.0,1.50); (1.80,0.600) – (15.0,-0.500); (2.25,0.250) – (15.0,0.250); (-5.25,-1.75) – (15.0,-1.75); (0.000,-0.750) – (15.0,-0.750); (1.50,1.25) – (15.0,1.25); (-2.25,2.25) – (15.0,2.25); (2.25,0.750) – (15.0,0.750); (-2.25,-1.25) – (15.0,-1.25); (1.50,-0.250) – (15.0,-0.250); (0.000,1.75) – (15.0,1.75); (-5.25,2.75) – (15.0,2.75); (1.80,0.400) – (15.0,0.400); (0.000,-0.600) – (15.0,-0.600); (0.600,1.40) – (15.0,1.40); (1.80,0.600) – (15.0,0.600); (0.600,-0.400) – (15.0,-0.400); (0.000,1.60) – (15.0,1.60); (0.000,0.000) – (15.0,1.25); (-3.00,-1.00) – (15.0,0.200); (0.000,1.00) – (15.0,2.67); (-3.00,2.00) – (15.0,5.00); (0.000,0.000) – (15.0,-1.67); (-3.00,-1.00) – (15.0,-4.00); (0.000,1.00) – (15.0,-0.250); (-3.00,2.00) – (15.0,0.800); (0.000,0.000) – (15.0,1.00); (0.000,1.00) – (15.0,2.25); (0.000,0.000) – (15.0,-1.25); (0.000,1.00) – (15.0,0.000);
The unbounded walls in $\mathscr{S}_\infty$ are all parallel in direction $m_{\textup{out}}\in \Lambda_B$. Here $\Lambda_B$ is the sheaf of integral tangent vectors on $B$ and $m_{\textup{out}}$ is the primitive vector in the unique unbounded direction of $B$ (the upward direction in Figure \[fig:main\]). Let $f_{\textup{out}}$ be the product of all functions attached to unbounded walls in $\mathscr{S}_\infty$, regarded as elements of $\mathbb{C}\llbracket x\rrbracket$ for $x:=z^{(-m_{\textup{out}},0)}\in\mathbb{C}[\Lambda_B\oplus\mathbb{Z}]$ the unique lift of the section $-m_{\text{out}}$. Then the main theorem is the following. It can be interpreted as a tropical correspondence theorem, since the wall structure $\mathscr{S}_\infty$ is combinatorial in nature and supported on the dual intersection complex $(B,\mathscr{P},\varphi)$ of $(X,D)$.
\[thm:main\] $$\textup{log }f_{\textup{out}} = \sum_{\underline{\beta}\in H_2^+(X,\mathbb{Z})} (D \cdot\underline{\beta}) \cdot N_\beta \cdot x^{D \cdot \underline{\beta}}.$$
For $(\mathbb{P}^2,E)$, this correspondence respects the torsion points on $E$: Consider the group law on $E$ with identity a flex point of $E$. The $3d$-torsion points form a subgroup $T_d$ of $S^1\times S^1$ isomorphic to $\mathbb{Z}_{3d}\times\mathbb{Z}_{3d}$. The stable log maps contributing to $N_d$ meet $E$ in such a $3d$-torsion point (Lemma \[lem:torsion\]). For $P\in\cup_{d\geq 1}T_d$, let $k(P)$ be the smallest integer such that $P\in T_{k(P)}$. Let $N_{d,k}$ be the logarithmic Gromov-Witten invariant of stable log maps contributing to $N_d$ and intersecting $E$ in a point $P$ with $k(P)=k$. In §\[S:torsion\] we will show that this is well-defined.
Let $s_{k,l}$ be the number of points in $T_d \simeq \mathbb{Z}_{3d}\times\mathbb{Z}_{3d}$ with $k(P)=k$ that are fixed by $M_l=\left(\begin{smallmatrix}1& 3l\\ 0 & 1\end{smallmatrix}\right)$, but not fixed by $M_{l'}$ for any $l'<l$. Let $r_l$ be the number of points on $S^1$ that are $3l$-torsion but not $3l'$-torsion for all $l'<l$. For an unbounded wall $\mathfrak{p}\in\mathscr{S}_\infty$ let $l(\mathfrak{p})$ be the smallest integer such that $\textup{log }f_{\mathfrak{p}}$ has non-trivial $x^{3l(\mathfrak{p})}$-coefficient. The number of walls in $\mathscr{S}_\infty$ with $l(\mathfrak{p})=l$ is $r_l$ (Lemma \[lem:rl\]).
\[thm:torsion\] Let $\mathfrak{p}$ be an unbounded wall in $\mathscr{S}_\infty$ with $l(\mathfrak{p})=l$. Then $$\textup{log }f_{\mathfrak{p}} = \sum_{d=1}^\infty 3d \left(\sum_{k: l\mid k\mid d} \frac{s_{k,l}}{r_l} N_{d,k}\right) x^{3d}.$$
Subtracting multiple cover contributions of curves of smaller degree, one obtains log BPS numbers $n_d$ and $n_{d,k}$ (see §\[S:BPS\]). Some of the $n_{d,k}$ have been calculated in [@Ta1]. The $n_d$ can be calculated from local BPS numbers (see §\[S:calc\]) which in turn can be calculated using local mirror symmetry (see [@CKYZ]).
The logarithmic Gromov-Witten invariants $N_d$ and $N_{d,k}$ and the log BPS numbers $n_d$ and $n_{d,k}$ are calculated, for $d\leq 6$, in §\[S:calcP2\].
We expect that there is a generalization of this correspondence to $q$-refined wall structures and higher genus logarithmic Gromov-Witten invariants, similar to the maximal boundary case [@Bou2]. The main argument to obtain higher genus statements in [@Bou1] and [@Bou2], the gluing and vanishing properties of $\lambda$-classes, are purely local. So one should be able to use the correspondence of [@Bou2] at each joint of the wall structure $\mathscr{S}_\infty$ and obtain a higher genus version of Theorem \[thm:main\]. However, the introduction of $q$-refinements and $\lambda$-classes as well as the higher genus calculations would go beyond the scope of this paper.
In recent work [@BFGW] on the holomorphic anomaly equation for $(\mathbb{P}^2,E)$, Bousseau, Fan, Guo and Wu studied these higher genus invariants. In [@Bou3][@Bou4] Pierrick Bousseau establishes a relation between scattering diagrams and wall-crossing phenomena of counts of coherent sheaves on $\mathbb{P}^2$, as well as a proof of [@Ta2], Conjecture 1.6. [@Bou4] uses Theorem \[thm:torsion\] above.
Very recently, Yu-Shen Lin [@Lin] worked out a symplectic analogue of the correspondence described in this paper.
Motivation {#motivation .unnumbered}
----------
The reason for an enumerative meaning of wall structures is the following. By the Strominger-Yau-Zaslow conjecture [@SYZ], mirror dual Calabi-Yau varieties admit mirror dual Lagrangian torus fibrations. To construct the mirror to a given Calabi-Yau, one first constructs the *semi-flat* mirror by dualizing the non-singular torus fibers. Then one corrects the complex structure of the semi-flat mirror such that it extends across the locus of singular fibers. It is expected that these corrections are determined by counts of holomorphic discs in the original variety with boundary on torus fibers [@SYZ][@Fuk].
Kontsevich and Soibelman [@KS] showed that in dimension two and with at most nodal singular fibers in the torus fibration, corrections of the complex structure are determined by algebraic self-consistency constraints which can be encoded by trees of gradient flow lines in the fan picture (dual intersection complex) of the degeneration, with certain automorphisms attached to the edges of the trees. From this they constructed a rigid analytic space from $B$, in dimension two.
Under the discrete Legendre transform ([@DataI], [§]{}1.4) the gradient flow lines in the fan picture become straight lines in the cone picture (intersection complex). This was used by Gross and Siebert to construct a toric degeneration from the cone picture in any dimension. In the cone picture the self-consistency calculations are describe by scattering diagrams (locally) and wall structures (globally). The fact that wall structures are used to construct a complex manifold in the cone picture and at the same time give generating functions for holomorphic curve counts of the fan picture can be seen as an explicit explanation for the connection between deformations and holomorphic curves in mirror symmetry.
In [@GHK1] Gross, Hacking and Keel use the correspondence between scattering diagrams and logarithmic Gromov-Witten invariants of a log Calabi-Yau surface $(X,D)$ with maximal boundary established in [@GPS] to define a canonical consistent scattering diagram for $(X,D)$ all at once. There is an affine singularity at the vertex of this scattering diagram. Hence, the scattering diagram only gives the mirror to $(X,D)$ away from a codimension $2$ subset. To obtain the whole mirror they use broken lines to construct theta functions – certain canonical sections of line bundles. The leads to a partial compactification giving the mirror to $(X,D)$. It can be defined explicitly as the spectrum of an explicit algebra generated by theta functions, and with multiplication rule defined by the enumerative geometry of $(X,D)$. This has led to the modern viewpoint of *intrinsic mirror symmetry* [@Intr1][@Intr2]. It circumvents the constructions of scattering diagrams and broken lines and directly defines the mirror to $(X,D)$ as the spectrum of an algebra with multiplication rule defined by certain *punctured Gromov-Witten invariants* of $(X,D)$ [@Intr1][@ACGS2].
Plan of the paper {#plan-of-the-paper .unnumbered}
-----------------
In §\[S:smoothing\] we describe how smoothing the boundary of a Fano polytope leads to a family $(\mathfrak{X}_Q\rightarrow\mathbb{A}^2,\mathfrak{D}_Q)$ as above. Fixing one parameter $s\neq 0$ gives a toric degeneration $(\mathfrak{X}\rightarrow\mathbb{A}^1,\mathfrak{D})$ of $(X,D)$. It contains logarithmic singularities lying on the central fiber, corresponding to affine singularities in the dual intersection complex $(B,\mathscr{P},\varphi)$. In §\[S:resolution\] we describe a small log resolution of these singularities, leading to a log smooth degeneration $(\tilde{\mathfrak{X}}\rightarrow\mathbb{A}^1,\tilde{\mathfrak{D}})$ of $(X,D)$. In §\[S:tropmap\] we describe tropicalizations of stable log maps to the central fiber of $(\tilde{\mathfrak{X}}\rightarrow\mathbb{A}^1,\tilde{\mathfrak{D}})$ and show that there is a finite number of them. The tropicalizations induce a refinement of $\mathscr{P}$ and hence a logarithmic modification. This is a degeneration $(\tilde{\mathfrak{X}}_d\rightarrow\mathbb{A}^1,\tilde{\mathfrak{D}}_d)$ of $(X,D)$ such that stable log maps to the central fiber are torically transverse. This enables us to use the degeneration formula of logarithmic Gromov-Witten theory in §\[S:degformula\]. It gives a description of $N_\beta$ in terms of invariants $N_V$ labeled by vertices of the tropical curves found in §\[S:tropmap\]. In §\[S:scattering\] we show that the scattering calculations of [@GS11] give a similar formula for the logarithm of functions attached to unbounded walls in the consistent wall structure $\mathscr{S}_\infty$. This ultimately leads to a proof of Theorem \[thm:main\]. In §\[S:torsion\] we explain that this correspondence respects the torsion points on $E$, leading to Theorem \[thm:torsion\]. In §\[S:calc\] we explicitly calculate some invariants for $\mathbb{P}^2$, $\mathbb{P}^1\times\mathbb{P}^1$ and the cubic surface. In Appendix \[A:artin\] we give the background to understand the logarithmic modification in §\[S:refinement\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank my supervisor Bernd Siebert for suggesting this interesting topic, as well as for his continuous support and many helpful discussions and explanations.
I thank Helge Ruddat for his support as well as for the opportunity to present this work at several seminars and at the workshop “Logarithmic Enumerative Geometry and Mirror Symmetry” [@Oberwolfach] in Oberwolfach he organized together with Michel van Garrel and Dan Abramovich. I thank Pierrick Bousseau for many comments and corrections to this paper, as well as for the suggestion and explanation of the more refined statement presented in §\[S:torsion\]. Thanks to Michel van Garrel for carefully reading a late draft of this paper and giving helpful comments, and to Yu-Shen Lin for his invitation to present this work at seminars in Boston University and Boston College. I thank Lawrence Jack Barrott, Navid Nabijou and Hülya Argüz for several discussions on topics including the subject of this paper, and Mark Gross for sending me a Maple code for local computations of scattering diagrams.
This work was supported by the DFG funded Research Training Group 1670 “Mathematics inspired by String theory and Quantum Field Theory”.
Smoothing toric degenerations {#S:smoothing}
=============================
A *smooth very ample log Calabi-Yau pair* is a log Calabi-Yau pair $(X,D)$ consisting of a smooth del Pezzo surface $X$ of degree $d\geq 3$ and a smooth very ample anticanonical divisor $D$.
\[con:family1\] Let $M\simeq\mathbb{Z}^2$ be a lattice and let $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$ be the corresponding vector space. Let $Q\subset M_{\mathbb{R}}$ be a Fano polytope, i.e., a convex lattice polytope containing the origin and with all vertices being primitive integral vectors. $Q$ can be seen as an affine manifold via its embedding into $M_{\mathbb{R}}\simeq\mathbb{R}^2$. Let $\check{\mathscr{P}}$ be the polyhedral decomposition of $Q$ by inserting edges connecting the vertices of $Q$ to the origin. Let $\check{\varphi} : Q \rightarrow \mathbb{R}$ be the strictly convex piecewise affine function on $(Q,\check{\mathscr{P}})$ defined by $\check{\varphi}(0)=0$ and $\check{\varphi}(v)=1$ for all vertices $v$ of $Q$. This means $\check{\varphi}$ is affine on the maximal cells of $\check{\mathscr{P}}$ and locally at each vertex $v$ of $\check{\mathscr{P}}$ gives a strictly convex piecewise affine function on the fan $\Sigma_v$ describing $\check{\mathscr{P}}$ locally. The triple $(Q,\check{\mathscr{P}},\check{\varphi})$ is a *polarized polyhedral affine manifold* ([@GHS], Construction 1.1). From this one obtains a toric degeneration of a toric del Pezzo surface with cyclic quotient singularities via the construction of Mumford [@Mum] (see also [@Inv], §1) as follows. Let $$Q_{\check{\varphi}} = \left\{(m,h) \in M_{\mathbb{R}} \times \mathbb{R} \ | \ h \geq \varphi(m), m \in Q\right\}$$ be the convex upper hull of $\check{\varphi}$ and let $$C(Q_{\check{\varphi}}) = \text{cl}\left(\mathbb{R}_{\geq 0} \cdot (Q_{\check{\varphi}} \times \{1\})\right) \subset M_{\mathbb{R}}\times\mathbb{R}\times\mathbb{R}$$ be the cone over $Q_{\check{\varphi}}$. The ring $\mathbb{C}[C(Q_{\check{\varphi}})\cap(M\times\mathbb{Z}\times\mathbb{Z})]$ is graded by the last component and we can define $$\mathfrak{X}^0 := \text{Proj}\left(\mathbb{C}[C(Q_{\check{\varphi}})\cap (M\times\mathbb{Z}\times\mathbb{Z})]\right).$$ By construction $\mathfrak{X}^0$ comes with an embedding into $\mathbb{P}^{N-1}\times\textup{Spec }\mathbb{C}[t]$, where $N$ is the number of lattice points of $Q$ and $t=z^{(0,1,0)}$. The sublattice of $M_{\mathbb{R}}$ generated by the vertices of $\check{\mathscr{P}}$ defines a weighted projective space $\mathbb{P}_{\check{\mathscr{P}}}\subset\mathbb{P}^{N-1}$ and we naturally have an embedding $\mathfrak{X}^0 \subset \mathbb{P}_{\check{\mathscr{P}}}\times\mathbb{A}^1$. Projection to the last coordinate gives a toric degeneration $\mathfrak{X}^0 \rightarrow \mathbb{A}^1$ of a toric del Pezzo surface with quotient singularities. The divisor $$\mathfrak{D}^0=\{z^{(0,0,0,1)}=0\} \subset \mathfrak{X}^0$$ defined by setting the coordinate corresponding to the origin in $Q$ to zero is a very ample anticanonical divisor, since it corresponds to the pullback of the line bundle $\mathcal{O}_{\mathbb{P}^{N-1}}(1)$, which is the anticanonical bundle on the general fiber of $\mathfrak{X}^0$. By construction, the polarized polyhedral affine manifold $(Q,\check{\mathscr{P}},\check{\varphi})$ is the *intersection complex* ([@DataI], [§]{}4.2) of the toric degeneration $\mathfrak{X}^0$.
One can deform $\mathfrak{X}^0$ by perturbing its defining equations. This means we add a term $t^lsf$ to each equation, where $l$ is the lowest non-trivial $t$-order in the defining equations of $\mathfrak{X}^0$, $s\in\mathbb{A}^1$ is the deformation parameter and $f$ is a general polynomial defining a section of the anticanonical bundle of the general fiber of $\mathfrak{X}^0$. We give some examples below. By [@Pri1], Theorem 1.1, this leads to a flat $2$-parameter family $$(\mathfrak{X}_Q \rightarrow \mathbb{A}^2,\mathfrak{D}_Q)$$ such that
for $s=0$ we have a toric degeneration $(\mathfrak{X}^0\rightarrow\mathbb{A}^1,\mathfrak{D}^0)$ of a toric del Pezzo surface with quotient singularities $(X^0,D^0)$;
for $s\neq 0$ we have a toric degeneration $(\mathfrak{X}\rightarrow\mathbb{A}^1,\mathfrak{D})$ of a $\mathbb{Q}$-Gorenstein smoothing $(X,D)$ of $(X^0,D^0)$.
We write the fibers of $(\mathfrak{X}_Q\rightarrow\mathbb{A}^2,\mathfrak{D}_Q)$ as $(X_t^s,D_t^s)$, where $s$ is the deformation parameter and $t$ is the parameter for the toric degeneration. We denote the $1$-parameter families defined by fixing one parameter by $(\mathfrak{X}_t\rightarrow\mathbb{A}^1,\mathfrak{D}_t)$ and $(\mathfrak{X}^s\rightarrow\mathbb{A}^1,\mathfrak{D}^s)$, respectively. When we fix a parameter different from zero, we sometimes omit the index, e.g. $X=X_t^s$ for $s,t\neq 0$. When writing $(\mathfrak{X}\rightarrow\mathbb{A}^1,\mathfrak{D})$ we will always mean the toric degeneration $(\mathfrak{X}^s\rightarrow\mathbb{A}^1,\mathfrak{D}^s)$ for $s\neq 0$. Moreover, we often supress the divisor in the notation.
Let $\check{B}$ be the affine manifold with singularities obtained from $Q$ by introducing affine singularities on the interior edges of $\check{\mathscr{P}}$ such that the boundary of $\check{B}$ is a straight line, and let $(\check{B},\check{\mathscr{P}},\check{\varphi})$ be the corresponding polarized polyhedral affine manifold. Then the intersection complex of the toric degeneration $\mathfrak{X}\rightarrow\mathbb{A}^1$ is $(\check{B},\check{\mathscr{P}},\check{\varphi})$.
The $t$-constant term in the defining equation for $X^0$ is independent of the variable $z^{(0,0,0,1)}$, so the central fiber of $\mathfrak{X}^s\rightarrow\mathbb{A}^1$ is independent of $s$. As a consequence, the maximal cells of the intersection complex of $\mathfrak{X}^s\rightarrow\mathbb{A}^1$ are the same for each $s$. So the parameter $s$ only changes the affine structure, given by the fan structures at vertices of the intersection complex. These fan structures are defined by local models for the family at zero-dimensional toric strata of the central fiber. For $s\neq 0$, locally at a zero-dimensional toric stratum of $X_0^s$ the family $\mathfrak{X}^s$ is given by $\{xy=t^l\}\subset\mathbb{A}^4$ for some $l>0$. So the fan structure at each vertex of the intersection complex of $\mathfrak{X}\rightarrow\mathbb{A}^1$ is given by the fan of $\mathbb{P}^1\times\mathbb{A}^1$. This shows that the boundary is a straight line.
\[expl:P2\] Figure \[fig:P2\] shows the intersection complex $(\check{B},\check{\mathscr{P}},\check{\varphi})$ of a toric degeneration of the log Calabi-Yau pair $(\mathbb{P}^2,E)$, where $E\subset\mathbb{P}^2$ is a smooth anticanonical divisor, i.e., an elliptic curve. This is obtained by smoothing a toric degeneration of $(\mathbb{P}^2,\partial\mathbb{P}^2)$, where $\partial\mathbb{P}^2$ is the toric boundary of $\mathbb{P}^2$. One can write down such a smoothing explicitly as follows. $$\begin{aligned}
\mathfrak{X}_Q &=& V\left(XYZ-t^3(W+sf_3)\right) \subset \mathbb{P}(1,1,1,3) \times \mathbb{A}^2 \\
\mathfrak{D}_Q &=& V(W)\subset\mathfrak{X}_Q\end{aligned}$$ Here $X,Y,Z,W$ are the coordinates of $\mathbb{P}(1,1,1,3)$, as shown in Figure \[fig:P2\], and $f_3$ is a general homogeneous degree $3$ polynomial in $X,Y,Z$.
For $t\neq 0$ we have $X_t^s=\mathbb{P}^2$, since we can eliminate $W$ by $W=t^{-3}XYZ-sf_3$. For $s\neq 0$, $D_t^s\subset\mathbb{P}^2$ is defined by a general degree $3$ polynomial, so is an elliptic curve, and $D_t^0\subset\mathbb{P}^2$ is a cycle of three lines. For $t=0$ we have $X_0^s=V(XYZ)$ in $\mathbb{P}(1,1,1,3)$. This is a union of three $\mathbb{P}(1,1,3)$ glued along toric divisors as described by the combinatorics of Figure \[fig:P2\]. $D_0^s$ is again a cycle of three lines.
\(0) at (0,0); (1) at (-1,-1); (2) at (2,-1); (3) at (-1,2); (1a) at (-0.5,-0.5); (2a) at (1,-0.5); (3a) at (-0.5,1); (1) – (2) – (3) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (1a) – (1); (2a) – (2); (3a) – (3); (a) at (0,-1); (b) at (1,-1); (c) at (-1,0); (d) at (-1,1); (e) at (1,0); (f) at (0,1);
\[expl:cubic\] Figure \[fig:cubic\] shows the intersection complex of a toric degeneration of a smooth cubic surface $X$ (del Pezzo surface of degree $3$) obtained by smoothing the Fano polytope of the toric Gorenstein del Pezzo surface $X^0 = \mathbb{P}^2/\mathbb{Z}_3$, where $\mathbb{Z}_3$ acts by $(x,y,z) \mapsto (x,\zeta y,\zeta^{-1}z)$ for $\zeta$ a nontrivial third root of unity. This can be given explicitly as follows. $$\begin{aligned}
\mathfrak{X}_Q &=& V\left(XYZ-t^3(W^3+sf_3)\right) \subset \mathbb{P}^3\times\mathbb{A}^2 \\
\mathfrak{D}_Q &=& V(W) \subset \mathfrak{X}_Q\end{aligned}$$ Again, $X,Y,Z,W$ are the projective coordinates and $f_3$ is a general homogeneous degree $3$ polynomial in $X,Y,Z$. For $t,s\neq 0$, $X_t^s$ is a smooth cubic surface, and $D_t^s$ is a hyperplane section. For $t\neq 0,s=0$, $X_t^0$ is given by $V(XYZ-tW^3)\subset\mathbb{P}^3$, thus is a $\mathbb{Z}_3$-quotient of $\mathbb{P}^2$, and $D_t^0$ is a cycle of three lines. For $t=0$ we have $X_0^s=V(XYZ)\subset\mathbb{P}^3$. This is a union of three $\mathbb{P}^2$ glued as described by the combinatorics of Figure \[fig:cubic\], and again $D_0^s$ is a cycle of three lines.
\(0) at (0,0); (1) at (1,0); (2) at (0,1); (3) at (-1,-1); (1a) at (0.5,0); (2a) at (0,0.5); (3a) at (-0.5,-0.5); (1) – (2) – (3) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (1a) – (1); (2a) – (2); (3a) – (3);
\[expl:8’a\] Figure \[fig:8’a\] shows the intersection complex of a toric degeneration of $\mathbb{P}^1\times\mathbb{P}^1$ obtained by smoothing the Fano polytope of $\mathbb{P}(1,1,2)$. This can be given explicitly as follows, with $f_2$ a general homogeneous degree $2$ polynomial in $X,Y,Z,U$ and $W$ the degree $2$ coordinate, $$\begin{aligned}
\mathfrak{X}_Q &=& V\left(XY-U^2+t^2sf_2,ZU-t^2(W+sf_2)\right)\subset\mathbb{P}(1,1,1,1,2)\times\mathbb{A}^2 \\
\mathfrak{D}_Q &=& V(W)\subset\mathfrak{X}_Q\end{aligned}$$ Indeed, $t=0$ implies $Z=0$ or $U=0$ which in turn implies $X=0$ or $Y=0$. We have $V(X)=V(Y)=\mathbb{P}(1,1,2)$ and $V(Z)=\{XY=U^2\}\subset\mathbb{P}(1,1,1,2)$ which is isomorphic to $\mathbb{P}(1,1,4)$. For $t\neq 0$ we have $X_t^0=\{XY=U^2+t^2sf_2\}\subset\mathbb{P}^3$ by elimination of $W$. For $s=0$ this is a singular quadric, so $X_t^0=\mathbb{P}(1,1,2)$. For $s\neq 0$ it is a smooth, so $X_t^s=\mathbb{P}^1\times\mathbb{P}^1$. Again, $D_t^s$ is smooth if and only if $t,s\neq 0$.
\(0) at (0,0); (1) at (0,1); (2) at (2,-1); (3) at (-2,-1); (1a) at (0,0.5); (2a) at (1,-0.5); (3a) at (-1,-0.5); (1) – (2) – (3) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (1a) – (1); (2a) – (2); (3a) – (3); (a) at (-1,0); (b) at (1,0); (c) at (-1,-1); (d) at (0,-1); (e) at (1,-1);
\[defi:toricmodel\] Let $X$ be a smooth del Pezzo surface. A *toric model* for $X$ is a toric del Pezzo surface with cyclic quotient singularities $X^0$ that admits a $\mathbb{Q}$-Gorenstein deformation to $X$.
Note that there may be different toric models $X^0$ for the same smooth del Pezzo surface $X$. In fact, the Fano polytopes $Q$ of such $X^0$ are related via *combinatorial mutations* ([@ACC+], Theorem 3, see also [@CGG+]).
$(9)$
\(0) at (0,0); (1) at (-1,-1); (2) at (2,-1); (3) at (-1,2); (1a) at (-0.5,-0.5); (2a) at (1,-0.5); (3a) at (-0.5,1); (1) – (2) – (3) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (1a) – (1); (2a) – (2); (3a) – (3); (a) at (0,-1); (b) at (1,-1); (c) at (-1,0); (d) at (-1,1); (e) at (1,0); (f) at (0,1);
$(8')$
\(0) at (0,0); (1) at (-1,-1); (2) at (1,-1); (3) at (1,1); (4) at (-1,1); (1a) at (-0.5,-0.5); (2a) at (0.5,-0.5); (3a) at (0.5,0.5); (4a) at (-0.5,0.5); (1) – (2) – (3) – (4) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (a) at (0,-1); (b) at (0,1); (c) at (-1,0); (d) at (1,0);
$(8)$
\(0) at (0,0); (1) at (-1,-1); (2) at (2,-1); (3) at (0,1); (4) at (-1,1); (1a) at (-0.5,-0.5); (2a) at (1,-0.5); (3a) at (0,0.5); (4a) at (-0.5,0.5); (1) – (2) – (3) – (4) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (a) at (0,-1); (b) at (1,-1); (c) at (-1,0); (d) at (1,0);
$(7)$
\(0) at (0,0); (1) at (-1,0); (2) at (0,-1); (3) at (2,-1); (4) at (0,1); (5) at (-1,1); (1a) at (-0.5,0); (2a) at (0,-0.5); (3a) at (1,-0.5); (4a) at (0,0.5); (5a) at (-0.5,0.5); (1) – (2) – (3) – (4) – (5) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (0) – (5a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (5a) – (5);
\
$(6)$
\(0) at (0,0); (1) at (-1,0); (2) at (0,-1); (3) at (1,-1); (4) at (1,0); (5) at (0,1); (6) at (-1,1); (1a) at (-0.5,0); (2a) at (0,-0.5); (3a) at (0.5,-0.5); (4a) at (0.5,0); (5a) at (0,0.5); (6a) at (-0.5,0.5); (1) – (2) – (3) – (4) – (5) – (6) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (0) – (5a); (0) – (6a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (5a) – (5); (6a) – (6);
$(8'a)$
\(0) at (0,0); (1) at (0,1); (2) at (2,-1); (3) at (-2,-1); (1a) at (0,0.5); (2a) at (1,-0.5); (3a) at (-1,-0.5); (1) – (2) – (3) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (1a) – (1); (2a) – (2); (3a) – (3); (a) at (-1,0); (b) at (1,0); (c) at (-1,-1); (d) at (0,-1); (e) at (1,-1);
$(7a)$
\(0) at (0,0); (1) at (1,-1); (2) at (-2,-1); (3) at (0,1); (4) at (1,0); (1a) at (0.5,-0.5); (2a) at (-1,-0.5); (3a) at (0,0.5); (4a) at (0.5,0); (1) – (2) – (3) – (4) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (a) at (0,-1); (b) at (-1,-1); (c) at (-1,0);
$(6a)$
\(0) at (0,0); (1) at (0,1); (2) at (1,0); (3) at (0,-1); (4) at (-2,-1); (1a) at (0,0.5); (2a) at (0.5,0); (3a) at (0,-0.5); (4a) at (-1,-0.5); (1) – (2) – (3) – (4) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (a) at (0,-1); (b) at (-1,-1); (c) at (-1,0);
\
$(6b)$
\(0) at (0,0); (1) at (0,1); (2) at (1,0); (3) at (-3,-2); (1a) at (0,0.5); (2a) at (0.5,0); (3a) at (-1.5,-1); (1) – (2) – (3) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (1a) – (1); (2a) – (2); (3a) – (3); (a) at (-1,0); (b) at (-1,-1); (c) at (-2,-1);
$(6c)$
\(0) at (0,0); (1) at (-1,0); (2) at (0,1); (3) at (1,0); (4) at (1,-1); (5) at (-1,-1); (1a) at (-0.5,0); (2a) at (0,0.5); (3a) at (0.5,0); (4a) at (0.5,-0.5); (5a) at (-0.5,-0.5); (1) – (2) – (3) – (4) – (5) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (0) – (5a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (5a) – (5); (a) at (0,-1);
$(5a)$
\(0) at (0,0); (1) at (0,1); (2) at (1,0); (3) at (-1,-1); (4) at (-2,-1); (1a) at (0,0.5); (2a) at (0.5,0); (3a) at (-0.5,-0.5); (4a) at (-1,-0.5); (1) – (2) – (3) – (4) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (a) at (-1,0);
$(5b)$
\(0) at (0,0); (1) at (0,1); (2) at (1,0); (3) at (0,-1); (4) at (-1,-1); (5) at (-1,0); (1a) at (0,0.5); (2a) at (0.5,0); (3a) at (0,-0.5); (4a) at (-0.5,-0.5); (5a) at (-0.5,0); (1) – (2) – (3) – (4) – (5) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (0) – (5a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4); (5a) – (5);
\
$(4a)$
\(0) at (0,0); (1) at (0,1); (2) at (-1,0); (3) at (0,-1); (4) at (1,0); (1a) at (0,0.5); (2a) at (-0.5,0); (3a) at (0,-0.5); (4a) at (0.5,0); (1) – (2) – (3) – (4) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4);
$(4b)$
\(0) at (0,0); (1) at (0,1); (2) at (1,0); (3) at (-1,-1); (4) at (-1,0); (1a) at (0,0.5); (2a) at (0.5,0); (3a) at (-0.5,-0.5); (4a) at (-0.5,0); (1) – (2) – (3) – (4) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (0) – (4a); (1a) – (1); (2a) – (2); (3a) – (3); (4a) – (4);
$(4c)$
\(0) at (0,0); (1) at (0,1); (2) at (1,0); (3) at (-2,-1); (1a) at (0,0.5); (2a) at (0.5,0); (3a) at (-1,-0.5); (1) – (2) – (3) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (1a) – (1); (2a) – (2); (3a) – (3); (a) at (-1,0);
$(3a)$
\(0) at (0,0); (1) at (0,1); (2) at (1,0); (3) at (-1,-1); (1a) at (0,0.5); (2a) at (0.5,0); (3a) at (-0.5,-0.5); (1) – (2) – (3) – (1); (0) – (1a); (0) – (2a); (0) – (3a); (1a) – (1); (2a) – (2); (3a) – (3);
To every smooth del Pezzo surface $X$ with very ample anticanonical class there exists a toric model $X^0$ with at most Gorenstein singularities.
For any $\mathbb{Q}$-Gorenstein deformation $\mathfrak{X}\rightarrow\mathbb{A}^1$, the relative canonical class $K_{\mathfrak{X}/\mathbb{A}^1}$ is Cartier. Hence, the degree of $X$ equals the degree of any of its toric models $X^0$. We need to show that the degrees of the given toric models are the ones shown in Figure \[fig:list\]. By definition, the degree of $X^0$ is the self-intersection of its (anti)canonical class. The Fano polytope $Q$ of $X^0$ is exactly the Newton polytope of its anticanonical class. By the duality between subdivisions of Newton polytopes and tropical curves, the self intersection can easily be computed by intersecting two tropical curves dual to the Fano polytope $Q$. For (3a), i.e., $X^0=\mathbb{P}^2/\mathbb{Z}_3$ as in Example \[expl:cubic\], the intersection of tropical curves is the following:
(0,0) – (-1,-1); (0,0) – (-1,2); (0,0) – (2,-1); (3,1) – (1,-1); (3,1) – (2,3); (3,1) – (5,0);
The determinant of primitive tangent vectors at the intersection point is $|\textup{det}\left(\begin{smallmatrix}\textup{ }\ 1&2\\-1&1\end{smallmatrix}\right)|=3$. Indeed, this is the degree of $\mathbb{P}^2/\mathbb{Z}_3$. Similarly one computes the degrees of the other cases in Figure \[fig:list\].
There are two smooth del Pezzo surfaces of degree $8$, the blow up of $\mathbb{P}^2$ at a point and $\mathbb{P}^1\times\mathbb{P}^1$. The del Pezzo surface $X^0$ in case (8’a) is a toric model for $\mathbb{P}^1\times\mathbb{P}^1$ by Example \[expl:8’a\]. The other cases are determined by the degree.
Let $Q$ be a Fano polytope and let $\mathfrak{X}_Q\rightarrow\mathbb{A}^2$ be the family from Construction \[con:family1\]. Let $(\check{B},\check{\mathscr{P}},\check{\varphi})$ be the intersection complex of $\mathfrak{X}\rightarrow\mathbb{A}^1$, i.e., one of the polarized polyhedral affine manifolds in Figure \[fig:list\]. Performing the discrete Legendre transform ([@DataI], §1.4) we obtain another polarized polyhedral affine manifold that is the dual intersection complex ([@DataI], §4.1) of $\mathfrak{X}\rightarrow\mathbb{A}^1$.
\[defi:sigma0\] Let $\sigma_0$ be the unique bounded maximal cell of the dual intersection complex of $\mathfrak{X}\rightarrow\mathbb{A}^1$.
\[con:family\] Refine the dual intersection complex of $\mathfrak{X}\rightarrow\mathbb{A}^1$ by introducing rays starting at the origin and pointing to the integral points of the bounded maximal cell. This yields another polarized polyhedral affine manifold $(B,\mathscr{P},\varphi)$, as shown in Figure \[fig:listb\]. A refinement of the dual intersection complex gives a logarithmic modification of $\mathfrak{X}_Q\rightarrow\mathbb{A}^2$ (see §\[A:artin\]). Since the deformation parameter $s$ is not part of the logarithmic data, the logarithmic modification does not change the general fiber $X=X_{t\neq0}^{s\neq0}$. It can be constructed as follows.
Blow up $\mathfrak{X}_Q\subset\mathbb{P}_{\check{\mathscr{P}}}\times\mathbb{A}^2$ at $X_{\sigma_0}\times\{(0,0)\}$, where $X_{\sigma_0}$ is the point corresponding to $\sigma_0$. This corresponds to a refinement of $\sigma_0$ by edges from the origin to the corners of $\sigma_0$.
Introducing the ray starting at the origin and pointing in the direction of an integral vector on the interior of bounded edge $\omega$ of $\mathscr{P}$ corresponds to a blow up at $X_\omega\times\mathbb{A}^1\times\{s=0\}$, where $X_\omega=\mathbb{P}^1$ is the component corresponding to $\omega$, i.e., the line through the points corresponding to the bounded maximal cell and the unbounded maximal cell containing $\omega$, respectively.
In cases where $X^0$ is not smooth we refine the asymptotic fan of $\mathscr{P}$. This corresponds to a toric blow up of the toric model $X^0$. This blow up is nef but not necessarily ample. By [@KM], Proposition A.2, the deformation of such a nef toric model still is $(X,D)$ and has Picard group isomorphic to $\text{Pic}(X)$. Note that $\text{Pic}(X)$ is isomorphic to the group of effective curve classes $H_2^+(X,\mathbb{Z})$.
By abuse of notation, from now on $\mathfrak{X}_Q\rightarrow\mathbb{A}^2$ will denote the logarithmic modification obtained by the refinement of the dual intersection complex from Construction \[con:family\]. Note that $X^0$ is smooth, toric and nef, but not necessarily ample. We call it a *smooth toric model* of $X$. If $X^0$ is ample it coincides with the toric model of $X$ constructed from Q (Definition \[defi:toricmodel\]).
$(9)$
\(0) at (0,0); (1) at (1,0); (2) at (0,1); (3) at (-1,-1); (1a) at (2,0); (2a) at (0,2); (3a) at (-2,-2); (12) at (0.5,0.5); (12a) at (0.5,2); (12b) at (2,0.5); (23) at (-0.5,0); (23a) at (-0.5,2); (23b) at (-2,-1.5); (31) at (0,-0.5); (31a) at (2,-0.5); (31b) at (-1.5,-2); (1) – (2) – (3) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (12) – (12a) – (12b) – (12); (23) – (23a) – (23b) – (23); (31) – (31a) – (31b) – (31); (12a) – (12) – (12b); (23a) – (23) – (23b); (31a) – (31) – (31b); (12) node\[fill,cross,inner sep=1pt,rotate=45\]; (23) node\[fill,cross,inner sep=1pt,rotate=63.43\]; (31) node\[fill,cross,inner sep=1pt,rotate=26.57\];
$(8')$
\(0) at (0,0); (1) at (1,0); (2) at (0,1); (3) at (-1,0); (4) at (0,-1); (1a) at (2,0); (2a) at (0,2); (3a) at (-2,0); (4a) at (0,-2); (12) at (0.5,0.5); (12a) at (0.5,2); (12b) at (2,0.5); (23) at (-0.5,0.5); (23a) at (-0.5,2); (23b) at (-2,0.5); (34) at (-0.5,-0.5); (34a) at (-2,-0.5); (34b) at (-0.5,-2); (41) at (0.5,-0.5); (41a) at (2,-0.5); (41b) at (0.5,-2); (1) – (2) – (3) – (4) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (12) – (12a) – (12b) – (12); (23) – (23a) – (23b) – (23); (34) – (34a) – (34b) – (34); (41) – (41a) – (41b) – (41); (12a) – (12) – (12b); (23a) – (23) – (23b); (34a) – (34) – (34b); (41a) – (41) – (41b); (12) node\[fill,cross,inner sep=1pt,rotate=45\]; (23) node\[fill,cross,inner sep=1pt,rotate=45\]; (34) node\[fill,cross,inner sep=1pt,rotate=45\]; (41) node\[fill,cross,inner sep=1pt,rotate=45\];
$(8)$
\(0) at (0,0); (1) at (-1,-1); (2) at (0,-1); (3) at (1,0); (4) at (0,1); (1a) at (-2,-2); (2a) at (0,-2); (3a) at (2,0); (4a) at (0,2); (12) at (-0.5,-1); (12a) at (-0.5,-2); (12b) at (-1.5,-2); (23) at (0.5,-0.5); (23a) at (0.5,-2); (23b) at (2,-0.5); (34) at (0.5,0.5); (34a) at (2,0.5); (34b) at (0.5,2); (41) at (-0.5,-0); (41a) at (-2,-1.5); (41b) at (-0.5,2); (1) – (2) – (3) – (4) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (12) – (12a) – (12b) – (12); (23) – (23a) – (23b) – (23); (34) – (34a) – (34b) – (34); (41) – (41a) – (41b) – (41); (12a) – (12) – (12b); (23a) – (23) – (23b); (34a) – (34) – (34b); (41a) – (41) – (41b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=45\]; (34) node\[fill,cross,inner sep=1pt,rotate=45\]; (41) node\[fill,cross,inner sep=1pt,rotate=63.43\];
$(7)$
\(0) at (0,0); (1) at (-1,-1); (2) at (0,-1); (3) at (1,0); (4) at (1,1); (5) at (0,1); (1a) at (-2,-2); (2a) at (0,-2); (3a) at (2,0); (4a) at (2,2); (5a) at (0,2); (12) at (-0.5,-1); (12a) at (-0.5,-2); (12b) at (-1.5,-2); (23) at (0.5,-0.5); (23a) at (0.5,-2); (23b) at (2,-0.5); (34) at (1,0.5); (34a) at (2,0.5); (34b) at (2,1.5); (45) at (0.5,1); (45a) at (1.5,2); (45b) at (0.5,2); (51) at (-0.5,0); (51a) at (-2,-1.5); (51b) at (-0.5,2); (1) – (2) – (3) – (4) – (5) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (5a); (12) – (12a) – (12b) – (12); (23) – (23a) – (23b) – (23); (34) – (34a) – (34b) – (34); (45) – (45a) – (45b) – (45); (51) – (51a) – (51b) – (51); (12a) – (12) – (12b); (23a) – (23) – (23b); (34a) – (34) – (34b); (45a) – (45) – (45b); (51a) – (51) – (51b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=45\]; (34) node\[fill,cross,inner sep=1pt,rotate=0\]; (45) node\[fill,cross,inner sep=1pt,rotate=0\]; (51) node\[fill,cross,inner sep=1pt,rotate=63.43\];
\
$(6)$
\(0) at (0,0); (1) at (1,1); (2) at (0,1); (3) at (-1,0); (4) at (-1,-1); (5) at (0,-1); (6) at (1,0); (1a) at (2,2); (2a) at (0,2); (3a) at (-2,0); (4a) at (-2,-2); (5a) at (0,-2); (6a) at (2,0); (12) at (0.5,1); (12a) at (0.5,2); (12b) at (1.5,2); (23) at (-0.5,0.5); (23a) at (-0.5,2); (23b) at (-2,0.5); (34) at (-1,-0.5); (34a) at (-2,-0.5); (34b) at (-2,-1.5); (45) at (-0.5,-1); (45a) at (-1.5,-2); (45b) at (-0.5,-2); (56) at (0.5,-0.5); (56a) at (2,-0.5); (56b) at (0.5,-2); (61) at (1,0.5); (61a) at (2,0.5); (61b) at (2,1.5); (1) – (2) – (3) – (4) – (5) – (6) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (5a); (0,0) – (6a); (12) – (12a) – (12b) – (12); (23) – (23a) – (23b) – (23); (34) – (34a) – (34b) – (34); (45) – (45a) – (45b) – (45); (56) – (56a) – (56b) – (56); (61) – (61a) – (61b) – (61); (12a) – (12) – (12b); (23a) – (23) – (23b); (34a) – (34) – (34b); (45a) – (45) – (45b); (56a) – (56) – (56b); (61a) – (61) – (61b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=45\]; (34) node\[fill,cross,inner sep=1pt,rotate=0\]; (45) node\[fill,cross,inner sep=1pt,rotate=0\]; (56) node\[fill,cross,inner sep=1pt,rotate=45\]; (61) node\[fill,cross,inner sep=1pt,rotate=0\];
$(8'a)$
\(0) at (0,0); (1) at (-1,-1); (2) at (1,-1); (3) at (0,1); (1a) at (-2,-2); (2a) at (2,-2); (3a) at (0,2); (12) at (-0.5,-1); (12a) at (-1.5,-2); (12b) at (-0.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (23) at (0.5,0); (23a) at (0.5,2); (23b) at (2,-1.5); (31) at (-0.5,0); (31a) at (-0.5,2); (31b) at (-2,-1.5); (1) – (2) – (3) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (0,-2); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (31) – (31a) – (31b) – (31); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (31a) – (31) – (31b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=26.57\]; (31) node\[fill,cross,inner sep=1pt,rotate=63.43\]; (b) at (0,-1);
$(7a)$
\(0) at (0,0); (1) at (1,-1); (2) at (-1,-1); (3) at (-1,0); (4) at (0,1); (1a) at (2,-2); (2a) at (-2,-2); (3a) at (-2,0); (4a) at (0,2); (12) at (0.5,-1); (12a) at (1.5,-2); (12b) at (0.5,-2); (12’) at (-0.5,-1); (12’a) at (-0.5,-2); (12’b) at (-1.5,-2); (23) at (-1,-0.5); (23a) at (-2,-1.5); (23b) at (-2,-0.5); (34) at (-0.5,0.5); (34a) at (-2,0.5); (34b) at (-0.5,2); (41) at (0.5,0); (41a) at (2,-1.5); (41b) at (0.5,2); (1) – (2) – (3) – (4) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (0,-2); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (34) – (34a) – (34b) – (34); (41) – (41a) – (41b) – (41); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (34a) – (34) – (34b); (41a) – (41) – (41b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=0\]; (34) node\[fill,cross,inner sep=1pt,rotate=45\]; (41) node\[fill,cross,inner sep=1pt,rotate=26.57\]; (b) at (0,-1);
$(6a)$
\(0) at (0,0); (1) at (1,-1); (2) at (-1,-1); (3) at (-1,1); (4) at (0,1); (1a) at (2,-2); (2a) at (-2,-2); (3a) at (-2,2); (4a) at (0,2); (12a) at (1.5,-2); (12b) at (0.5,-2); (12’) at (-0.5,-1); (12’a) at (-0.5,-2); (12’b) at (-1.5,-2); (23) at (-1,0.5); (23a) at (-2,1.5); (23b) at (-2,0.5); (23’) at (-1,-0.5); (23’a) at (-2,-0.5); (23’b) at (-2,-1.5); (34) at (-0.5,1); (34a) at (-0.5,2); (34b) at (-1.5,2); (41) at (0.5,0); (41a) at (0.5,2); (41b) at (2,-1.5); (1) – (2) – (3) – (4) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (0,-2); (0,0) – (-2,0); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (23’) – (23’a) – (23’b) – (23’); (34) – (34a) – (34b) – (34); (41) – (41a) – (41b) – (41); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (23’a) – (23’) – (23’b); (34a) – (34) – (34b); (41a) – (41) – (41b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=0\]; (23’) node\[fill,cross,inner sep=1pt,rotate=0\]; (34) node\[fill,cross,inner sep=1pt,rotate=0\]; (41) node\[fill,cross,inner sep=1pt,rotate=63.43\]; (b) at (-1,0); (c) at (0,-1);
\
$(6b)$
\(0) at (0,0); (1) at (-1,-1); (2) at (1,-1); (3) at (-1,2); (1a) at (-2,-2); (2a) at (2,-2); (3a) at (-2,4); (12) at (-0.5,-1); (12a) at (-1.5,-2); (12b) at (-0.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (23) at (0,0.5); (23a) at (2,-1.5); (23b) at (-1.75,4); (31) at (-1,-0.5); (31a) at (-2,-1.5); (31b) at (-2,-0.5); (31’) at (-1,0.5); (31’a) at (-2,0.5); (31’b) at (-2,1.5); (31”) at (-1,1.5); (31”a) at (-2,2.5); (31”b) at (-2,3.5); (1) – (2) – (3) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (0,-2); (0,0) – (-2,0); (0,0) – (-2,2); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (31) – (31a) – (31b) – (31); (31’) – (31’a) – (31’b) – (31’); (31”) – (31”a) – (31”b) – (31”); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (31a) – (31) – (31b); (31’a) – (31’) – (31’b); (31”a) – (31”) – (31”b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=33.69\]; (31) node\[fill,cross,inner sep=1pt,rotate=0\]; (31’) node\[fill,cross,inner sep=1pt,rotate=0\]; (31”) node\[fill,cross,inner sep=1pt,rotate=0\]; (b) at (-1,0); (c) at (-1,1); (d) at (0,-1);
$(6c)$
\(0) at (0,0); (1) at (-1,-1); (2) at (1,-1); (3) at (1,0); (4) at (0,1); (5) at (-1,0); (1a) at (-2,-2); (2a) at (2,-2); (3a) at (2,0); (4a) at (0,2); (5a) at (-2,0); (12) at (-0.5,-1); (12a) at (-0.5,-2); (12b) at (-1.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (23) at (1,-0.5); (23a) at (2,-1.5); (23b) at (2,-0.5); (34) at (0.5,0.5); (34a) at (2,0.5); (34b) at (0.5,2); (45) at (-0.5,0.5); (45a) at (-0.5,2); (45b) at (-2,0.5); (51) at (-1,-0.5); (51a) at (-2,-1.5); (51b) at (-2,-0.5); (1) – (2) – (3) – (4) – (5) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (5a); (0,0) – (0,-2); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (34) – (34a) – (34b) – (34); (45) – (45a) – (45b) – (45); (51) – (51a) – (51b) – (51); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (34a) – (34) – (34b); (45a) – (45) – (45b); (51a) – (51) – (51b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=0\]; (34) node\[fill,cross,inner sep=1pt,rotate=45\]; (45) node\[fill,cross,inner sep=1pt,rotate=0\]; (51) node\[fill,cross,inner sep=1pt,rotate=0\]; (b) at (0,-1);
$(5a)$
\(0) at (0,0); (1) at (1,-1); (2) at (-1,-1); (3) at (-1,2); (4) at (0,1); (1a) at (2,-2); (2a) at (-2,-2); (3a) at (-2,4); (4a) at (0,4); (12) at (-0.5,-1); (12a) at (-1.5,-2); (12b) at (-0.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (23) at (-1,1.5); (23a) at (-2,3.5); (23b) at (-2,2.5); (23’) at (-1,0.5); (23’a) at (-2,1.5); (23’b) at (-2,0.5); (23”) at (-1,-0.5); (23”a) at (-2,-0.5); (23”b) at (-2,-1.5); (34) at (-0.5,1.5); (34a) at (-0.5,4); (34b) at (-1.75,4); (41) at (0.5,0); (41a) at (0.5,4); (41b) at (2,-1.5); (1) – (2) – (3) – (4) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (0,-2); (0,0) – (-2,0); (0,0) – (-2,2); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (23’) – (23’a) – (23’b) – (23’); (23”) – (23”a) – (23”b) – (23”); (34) – (34a) – (34b) – (34); (41) – (41a) – (41b) – (41); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (23’a) – (23’) – (23’b); (23”a) – (23”) – (23”b); (34a) – (34) – (34b); (41a) – (41) – (41b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=0\]; (23’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23”) node\[fill,cross,inner sep=1pt,rotate=0\]; (34) node\[fill,cross,inner sep=1pt,rotate=0\]; (41) node\[fill,cross,inner sep=1pt,rotate=26.57\]; (b) at (-1,1); (c) at (-1,0); (d) at (0,-1);
$(5b)$
\(0) at (0,0); (1) at (-1,-1); (2) at (1,-1); (3) at (1,0); (4) at (0,1); (5) at (-1,1); (1a) at (-2,-2); (2a) at (2,-2); (3a) at (2,0); (4a) at (0,2); (5a) at (-2,2); (12) at (-0.5,-1); (12a) at (-0.5,-2); (12b) at (-1.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (23) at (1,-0.5); (23a) at (2,-1.5); (23b) at (2,-0.5); (34) at (0.5,0.5); (34a) at (2,0.5); (34b) at (0.5,2); (45) at (-0.5,1); (45a) at (-0.5,2); (45b) at (-1.5,2); (51) at (-1,0.5); (51a) at (-2,1.5); (51b) at (-2,0.5); (51’) at (-1,-0.5); (51’a) at (-2,-1.5); (51’b) at (-2,-0.5); (1) – (2) – (3) – (4) – (5) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (5a); (0,0) – (0,-2); (0,0) – (-2,0); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (34) – (34a) – (34b) – (34); (45) – (45a) – (45b) – (45); (51) – (51a) – (51b) – (51); (51’) – (51’a) – (51’b) – (51’); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (34a) – (34) – (34b); (45a) – (45) – (45b); (51a) – (51) – (51b); (51’a) – (51’) – (51’b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=0\]; (34) node\[fill,cross,inner sep=1pt,rotate=45\]; (45) node\[fill,cross,inner sep=1pt,rotate=0\]; (51) node\[fill,cross,inner sep=1pt,rotate=0\]; (51’) node\[fill,cross,inner sep=1pt,rotate=0\]; (b) at (0,-1); (c) at (-1,0);
\
$(4a)$
\(0) at (0,0); (1) at (1,-1); (2) at (-1,-1); (3) at (-1,1); (4) at (1,1); (1a) at (2,-2); (2a) at (-2,-2); (3a) at (-2,2); (4a) at (2,2); (12) at (0.5,-1); (12a) at (1.5,-2); (12b) at (0.5,-2); (12’) at (-0.5,-1); (12’a) at (-0.5,-2); (12’b) at (-1.5,-2); (23) at (-1,0.5); (23a) at (-2,1.5); (23b) at (-2,0.5); (23’) at (-1,-0.5); (23’a) at (-2,-0.5); (23’b) at (-2,-1.5); (34) at (-0.5,1); (34a) at (-0.5,2); (34b) at (-1.5,2); (34’) at (0.5,1); (34’a) at (0.5,2); (34’b) at (1.5,2); (41) at (1,0.5); (41a) at (2,0.5); (41b) at (2,1.5); (41’) at (1,-0.5); (41’a) at (2,-0.5); (41’b) at (2,-1.5); (1) – (2) – (3) – (4) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (0,-2); (0,0) – (-2,0); (0,0) – (0,2); (0,0) – (2,0); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (23’) – (23’a) – (23’b) – (23’); (34) – (34a) – (34b) – (34); (34’) – (34’a) – (34’b) – (34’); (41) – (41a) – (41b) – (41); (41’) – (41’a) – (41’b) – (41’); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (23’a) – (23’) – (23’b); (34a) – (34) – (34b); (34’a) – (34’) – (34’b); (41a) – (41) – (41b); (41’a) – (41’) – (41’b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=0\]; (23’) node\[fill,cross,inner sep=1pt,rotate=0\]; (34) node\[fill,cross,inner sep=1pt,rotate=0\]; (34’) node\[fill,cross,inner sep=1pt,rotate=0\]; (41) node\[fill,cross,inner sep=1pt,rotate=0\]; (41’) node\[fill,cross,inner sep=1pt,rotate=0\]; (b) at (-1,0); (c) at (0,-1); (d) at (1,0); (e) at (0,1);
$(4b)$
\(0) at (0,0); (1) at (1,-1); (2) at (-1,-1); (3) at (-1,2); (4) at (1,0); (1a) at (2,-2); (2a) at (-2,-2); (3a) at (-2,4); (4a) at (2,0); (12) at (-0.5,-1); (12a) at (-1.5,-2); (12b) at (-0.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (23) at (-1,1.5); (23a) at (-2,3.5); (23b) at (-2,2.5); (23’) at (-1,0.5); (23’a) at (-2,1.5); (23’b) at (-2,0.5); (23”) at (-1,-0.5); (23”a) at (-2,-0.5); (23”b) at (-2,-1.5); (34) at (-0.5,1.5); (34a) at (-0.5,4); (34b) at (-1.75,4); (34’) at (0.5,0.5); (34’a) at (0.5,4); (34’b) at (2,0.5); (41) at (1,-0.5); (41a) at (2,-0.5); (41b) at (2,-1.5); (1) – (2) – (3) – (4) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (4a); (0,0) – (0,-2); (0,0) – (-2,0); (0,0) – (0,4); (0,0) – (-2,2); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (23’) – (23’a) – (23’b) – (23’); (23”) – (23”a) – (23”b) – (23”); (34) – (34a) – (34b) – (34); (34’) – (34’a) – (34’b) – (34’); (41) – (41a) – (41b) – (41); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (23’a) – (23’) – (23’b); (23”a) – (23”) – (23”b); (34a) – (34) – (34b); (34’a) – (34’) – (34’b); (41a) – (41) – (41b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=0\]; (23’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23”) node\[fill,cross,inner sep=1pt,rotate=0\]; (34) node\[fill,cross,inner sep=1pt,rotate=45\]; (34’) node\[fill,cross,inner sep=1pt,rotate=45\]; (41) node\[fill,cross,inner sep=1pt,rotate=0\]; (b) at (-1,1); (c) at (-1,0); (d) at (0,1); (e) at (0,-1);
$(4c)$
\(0) at (0,0); (1) at (-1,-1); (2) at (1,-1); (3) at (-1,3); (1a) at (-2,-2); (2a) at (2,-2); (3a) at (-2,6); (12) at (-0.5,-1); (12a) at (-1.5,-2); (12b) at (-0.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (23) at (0.5,0); (23a) at (2,-1.5); (23b) at (0.5,6); (23’) at (-0.5,2); (23’a) at (-1.833,6); (23’b) at (-0.5,6); (31) at (-1,-0.5); (31a) at (-2,-1.5); (31b) at (-2,-0.5); (31’) at (-1,0.5); (31’a) at (-2,0.5); (31’b) at (-2,1.5); (31”) at (-1,1.5); (31”a) at (-2,2.5); (31”b) at (-2,3.5); (31”’) at (-1,2.5); (31”’a) at (-2,4.5); (31”’b) at (-2,5.5); (1) – (2) – (3) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (0,-2); (0,0) – (-2,0); (0,0) – (0,6); (0,0) – (-2,2); (0,0) – (-2,4); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (23) – (23a) – (23b) – (23); (23’) – (23’a) – (23’b) – (23’); (31) – (31a) – (31b) – (31); (31’) – (31’a) – (31’b) – (31’); (31”) – (31”a) – (31”b) – (31”); (31”’) – (31”’a) – (31”’b) – (31”’); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (23a) – (23) – (23b); (23’a) – (23’) – (23’b); (31a) – (31) – (31b); (31’a) – (31’) – (31’b); (31”a) – (31”) – (31”b); (31”’a) – (31”’) – (31”’b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=45\]; (23’) node\[fill,cross,inner sep=1pt,rotate=45\]; (31) node\[fill,cross,inner sep=1pt,rotate=0\]; (31’) node\[fill,cross,inner sep=1pt,rotate=0\]; (31”) node\[fill,cross,inner sep=1pt,rotate=0\]; (31”’) node\[fill,cross,inner sep=1pt,rotate=0\]; (b) at (-1,0); (c) at (-1,1); (d) at (-1,2); (e) at (0,-1); (f) at (0,1);
$(3a)$
\(0) at (0,0); (1) at (-1,-1); (2) at (2,-1); (3) at (-1,2); (1a) at (-2,-2); (2a) at (4,-2); (3a) at (-2,4); (12) at (-0.5,-1); (12a) at (-1.5,-2); (12b) at (-0.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (12”) at (1.5,-1); (12”a) at (2.5,-2); (12”b) at (3.5,-2); (23) at (-0.5,1.5); (23a) at (-0.5,4); (23b) at (-1.75,4); (23’) at (0.5,0.5); (23’a) at (4,0.5); (23’b) at (0.5,4); (23”) at (1.5,-0.5); (23”a) at (4,-0.5); (23”b) at (4,-1.75); (31) at (-1,-0.5); (31a) at (-2,-1.5); (31b) at (-2,-0.5); (31’) at (-1,0.5); (31’a) at (-2,0.5); (31’b) at (-2,1.5); (31”) at (-1,1.5); (31”a) at (-2,2.5); (31”b) at (-2,3.5); (1) – (2) – (3) – (1); (0,0) – (1a); (0,0) – (2a); (0,0) – (3a); (0,0) – (0,-2); (0,0) – (-2,0); (0,0) – (0,4); (0,0) – (4,0); (0,0) – (2,-2); (0,0) – (-2,2); (12) – (12a) – (12b) – (12); (12’) – (12’a) – (12’b) – (12’); (12”) – (12”a) – (12”b) – (12”); (23) – (23a) – (23b) – (23); (23’) – (23’a) – (23’b) – (23’); (23”) – (23”a) – (23”b) – (23”); (31) – (31a) – (31b) – (31); (31’) – (31’a) – (31’b) – (31’); (31”) – (31”a) – (31”b) – (31”); (12a) – (12) – (12b); (12’a) – (12’) – (12’b); (12”a) – (12”) – (12”b); (23a) – (23) – (23b); (23’a) – (23’) – (23’b); (23”a) – (23”) – (23”b); (31a) – (31) – (31b); (31’a) – (31’) – (31’b); (31”a) – (31”) – (31”b); (12) node\[fill,cross,inner sep=1pt,rotate=0\]; (12’) node\[fill,cross,inner sep=1pt,rotate=0\]; (12”) node\[fill,cross,inner sep=1pt,rotate=0\]; (23) node\[fill,cross,inner sep=1pt,rotate=45\]; (23’) node\[fill,cross,inner sep=1pt,rotate=45\]; (23”) node\[fill,cross,inner sep=1pt,rotate=45\]; (31) node\[fill,cross,inner sep=1pt,rotate=0\]; (31’) node\[fill,cross,inner sep=1pt,rotate=0\]; (31”) node\[fill,cross,inner sep=1pt,rotate=0\]; (b) at (-1,0); (c) at (-1,1); (d) at (0,-1); (e) at (1,-1); (f) at (1,0); (g) at (0,1);
\[expl:8’a2\] Consider the smoothing of $\mathbb{P}(1,1,2)$ to $\mathbb{P}^1\times\mathbb{P}^1$ (case (8’a)) from Example \[expl:8’a\]. The logarithmic modification from Construction \[con:family\] is a $2$-parameter family $\mathfrak{X}_Q\rightarrow\mathbb{A}^2$ such that $\mathfrak{X}\rightarrow\mathbb{A}^1$ is a toric degeneration of $\mathbb{P}^1\times\mathbb{P}^1$ and $\mathfrak{X}^0\rightarrow\mathbb{A}^1$ is a toric degeneration of the Hirzebruch surface $\mathbb{F}_2$, the $\mathbb{P}^1$-bundle over $\mathbb{P}^1$ given by $\mathbb{F}_2=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(2))$. This is the smooth surface obtained by blowing up the singular point on $\mathbb{P}(1,1,2)$, corresponding the the subdivision of the asymptotic fan given in Figure \[fig:8’a2\].
The Picard group $\text{Pic}(\mathbb{F}_2)\simeq H_2^+(\mathbb{F}_2,\mathbb{Z})$ is generated by the class of a fiber $F$ and the class of a section, e.g., the exceptional divisor $E$ of the blow up. The intersection numbers are $F^2=0$, $E^2=-2$ and $E\cdot F=1$. The anticanonical bundle is $-K_{\mathbb{F}_2}=2F+S+E=4F+2E$, where $S=2F+E$ is the class of a section different from the exceptional divisor. The classes of curves corresponding to the rays in the fan of $\mathbb{F}_2$ are given in Figure \[fig:8’a2\].
The Picard group of $\mathbb{P}^1\times\mathbb{P}^1$ is generated by the class of a bidegree $(1,0)$ curve $L_1$ and a bidegree $(0,1)$ curve $L_2$, with intersection numbers $L_1^2=0$, $L_2^2=0$ and $L_1\cdot L_2=1$. There is an isomorphism $$\text{Pic}(\mathbb{F}_2) {\xrightarrow{\raisebox{-0.7ex}[0ex][0ex]{$\sim$}}}\text{Pic}(\mathbb{P}^1\times\mathbb{P}^1), \ F \mapsto L_2, \ E \mapsto L_1-L_2.$$ Note that there is another isomorphism by the symmetry $L_1\leftrightarrow L_2$ and we made a choice here. We will use this isomorphism in §\[S:calc8’a\] to calculate the logarithmic Gromov-Witten invariants of $\mathbb{P}^1\times\mathbb{P}^1$ in an alternative way.
\(0) at (0,0); (1) at (-1,-1); (2) at (1,-1); (3) at (0,1); (1a) at (-2,-2); (2a) at (2,-2); (3a) at (0,2); (12) at (-0.5,-1); (12a) at (-1.5,-2); (12b) at (-0.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (23) at (0.5,0); (23a) at (0.5,2); (23b) at (2,-1.5); (31) at (-0.5,0); (31a) at (-0.5,2); (31b) at (-2,-1.5); (1) – (2) – (3) – (1); (0,0) – (1a) node\[below\][$F\mapsto L_2$]{}; (0,0) – (2a) node\[below\][$F\mapsto L_2$]{}; (0,0) – (3a) node\[above\][$S=2F+E\mapsto L_1+L_2$]{}; (0,0) – (0,-2) node\[below\][$E\mapsto L_1-L_2$]{}; (b) at (0,-1);
Affine charts {#S:affinecharts}
-------------
Figure \[fig:listb\] shows the dual intersection complexes $(B,\mathscr{P},\varphi)$ in the chart of $\sigma_0$ (Definition \[defi:sigma0\]). The shaded regions are cut out and the dashed lines are mutually identified, so in fact all unbounded edges are parallel.
\[defi:mout\] Let $m_{\text{out}}\in\Lambda_B$ denote the primitive integral tangent vector pointing in the unique unbounded direction on $B$.
Figure \[fig:dualb\] shows the dual intersection complex $(B,\mathscr{P},\varphi)$ of $(\mathbb{P}^2,E)$ in the chart of an unbounded maximal cell. Intuitively, this picture can be obtained by mutually gluing the dashed lines in Figure \[fig:listb\], (9). The two horizontal dashed lines are identified. The monodromy transformation by passing across the upper horizontal dashed line is given by $\Lambda_B \rightarrow \Lambda_B, m \mapsto \left(\begin{smallmatrix}1&9\\ 0&1\end{smallmatrix}\right) \cdot m$.
\(1) at (0,0); (2) at (0,1); (3) at (-3,-1); (4) at (-3,2); (0a) at (-2,2); (0b) at (1,1); (0c) at (1,0); (3) – (1) – (2) – (4); (1) – (5,0); (2) – (5,1); (3) – (5,-1); (4) – (5,2); (-1.5,-0.25) – (0,0.5) – (-1.5,1.25); (-3,-0.8) – (-1.5,-0.5) – (-0.75,0.25); (-3,1.8) – (-1.5,1.5) – (-0.75,0.75); (-1.5,-0.25) – (0,0.5) – (-1.5,1.25) – (-1.5,-0.25) – cycle; (-3,1.8) – (-1.5,1.5) – (-0.75,0.75) – (-0.75,0.25) – (-1.5,-0.5) – (-3,-0.8); (5) at (0,0.5); (6) at (-1.5,-0.5); (7) at (-1.5,1.5); (0,0) – (-1,0); (0,1) – (-1,1); (1,2) – (4,1.667); (1,-1) – (1,-0.667); (6,0) – (6,1);
We can extend the description of the affine structure across the horizontal dashed line by giving a chart of a discrete covering space $\bar{B}$ of $B$ (Figure \[fig:dualc\]). Passing from one fundamental domain to an adjacent one amounts to applying the monodromy transformation by crossing the horizontal dashed line in $B$.
\(1) at (0,0); (2) at (0,1); (3) at (-3,-1); (4) at (-3,2); (3) – (1) – (2) – (4); (1) – (4.5,0); (2) – (4.5,1); (3) – (4.5,-1); (4) – (4.5,2); (3) – (-9,-2) – (-9.9,-2.1); (4) – (-9,3) – (-9.9,3.1); (-9,-2) – (4.5,-2); (-9,3) – (4.5,3); (-9.9,-2.1) – (4.5,-2.1) – (4.5,-1) – (-9.9,-1) – (-9.9,-2.1); (-9.9,3.1) – (4.5,3.1) – (4.5,2) – (-9.9,2) – (-9.9,3.1); (-1.5,-0.25) – (0,0.5) – (-1.5,1.25); (-6,-1.4) – (-1.5,-0.5) – (-0.75,0.25); (-6,2.4) – (-1.5,1.5) – (-0.75,0.75); (-3,-0.75) – (-6,-1.5) – (-9.9,-1.9875); (-3,1.75) – (-6,2.5) – (-9.9,2.9875); (-1.5,-0.25) – (0,0.5) – (-1.5,1.25) – (-1.5,-0.25); (-6,2.4) – (-1.5,1.5) – (-0.75,0.75) – (-0.75,0.25) – (-1.5,-0.5) – (-6,-1.4); (-9.9,-1.9875) – (-6,-1.5) – (-3,-0.75) – (-3,1.75) – (-6,2.5) – (-9.9,2.9875); (5) at (0,0.5); (6) at (-1.5,-0.5); (7) at (-1.5,1.5); (8) at (-6,-1.5); (9) at (-6,2.5); (0,0) – (-1,0); (0,1) – (-1,1); (0,-1) – (-4,-1); (0,2) – (-4,2); (4,1.667) – (-2,2.333); (1,-0.667) – (1,-1.333);
This gives a trivialization $\Lambda_{\bar{B}} \simeq M = \mathbb{Z}^2$ on $\bar{B}\setminus(\textup{Int}(\bar{\sigma_0})\cup\bar{\Delta})$, where $\bar{\sigma_0}$ and $\bar{\Delta}$ are the preimages of the bounded maximal cell $\sigma_0$ and the discriminant locus $\Delta$, respectively. We will see in Lemma \[lem:global\] that the consistent wall structure $\mathscr{S}_\infty$ defined by $(B,\mathscr{P},\varphi)$ has support disjoint from $\sigma_0$. Hence, the whole scattering procedure can be described in this affine chart of the covering space $\bar{B}$. This allows for a simple implementation of the scattering algorithm (see §\[S:calc\]).
Resolution of log singularities {#S:resolution}
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Let $Q$ be a Fano polytope and consider the family $\mathfrak{X}_Q\rightarrow\mathbb{A}^2$ from Construction \[con:family\]. Equip $\mathbb{A}^2$ with the divisorial log structure by $V(t)\subset\mathbb{A}^2$ and $\mathfrak{X}_Q$ with the divisorial log structure by $\mathfrak{X}_0\cup\mathfrak{D}_Q\subset\mathfrak{X}_Q$, that is, the sheaf of monoids $$\mathcal{M}_{(\mathfrak{X}_Q,\mathfrak{X}_0\cup\mathfrak{D}_Q)} := (j_\star\mathcal{O}_{\mathfrak{X}_Q\setminus (\mathfrak{X}_0\cup\mathfrak{D}_Q)}^\times)\cap\mathcal{O}_{\mathfrak{X}_Q}, \quad j : \mathfrak{X}_Q\setminus (\mathfrak{X}_0\cup\mathfrak{D}_Q) \hookrightarrow \mathfrak{X}_Q.$$ For an introduction to log structures see e.g. [@Kat1] or [@Gr10], [§]{}3.
If we consider the fibers $X_r^s$ or the families $\mathfrak{X}_t\rightarrow\mathbb{A}^1$ or $\mathfrak{X}^s\rightarrow\mathbb{A}^1$ as log schemes, we always mean equipped with the log structure by restriction of the above log structure. Now $\mathfrak{X}^0\rightarrow\mathbb{A}^1$ is log smooth, and $\mathfrak{X}\rightarrow\mathbb{A}^1$ is log smooth away from finitely many points on the central fiber, corresponding to the affine singularities of the dual intersection complex $(B,\mathscr{P},\varphi)$. At these points $\mathfrak{X}^s$ is locally given by $\textup{Spec }\mathbb{C}[x,y,w,t]/(xy-t^l(w+s))$ with log structure given by $V(t)\cup V(w)$. This is isomorphic to $\text{Spec }\mathbb{C}[x,y,\tilde{w},t]/(xy-t^l\tilde{w})$ with $\tilde{w}=w+s$. The log structure is given by $V(t)\cup V(\tilde{w})$ for $s=0$ and by $V(t)$ for $s\neq 0$. Arguments as in [@Gr10], Example 3.20, show that for $s\neq 0$ this is not fine at the point given by $x=y=w=t=0$. Following [@DataII], Lemma 2.12, we describe a small log resolution $\tilde{\mathfrak{X}}^s \rightarrow \mathfrak{X}^s$ such that $\tilde{\mathfrak{X}}^s$ is fine and log smooth over $\mathbb{A}^1$.
The local picture
-----------------
$\text{Spec }\mathbb{C}[x,y,\tilde{w},t]/(xy-t^l\tilde{w})$ is the affine toric variety defined by the cone $\sigma$ generated by $(0,0,1)$, $(0,1,0)$, $(1,0,1)$ and $(l,1,0)$. In fact, $$\begin{aligned}
\textup{Spec }\mathbb{C}[\sigma^\vee \cap \mathbb{Z}^3] &=& \textup{Spec }\mathbb{C}[z^{(1,0,0)},z^{(-1,l,1)},z^{(0,0,1)},z^{(0,1,0)}] \\
&=& \textup{Spec }\faktor{\mathbb{C}[x,y,\tilde{w},t]}{(xy-\tilde{w}t^l)}.\end{aligned}$$ We obtain a toric blow up by subdividing the fan consisting of the single cone $\sigma$. There are two ways of doing this and they are related by a *flop*. We choose the subdivision $\Sigma$ as in [@DataII], Lemma 2.12, with maximal cones $\sigma_1$ generated by $(0,0,1)$, $(1,0,1)$ and $(0,1,0)$, and $\sigma_2$ generated by $(l,1,0)$, $(1,0,1)$ and $(0,1,0)$.
(0,0) node\[below\][$(0,1,0)$]{} – (3,0) node\[below\][$(l,1,0)$]{} – (1,1) node\[above\][$(1,0,1)$]{} – (0,1) node\[above\][$(0,0,1)$]{} – (0,0); (0,0) – (1,1);
These cones define affine toric varieties $$\begin{aligned}
X_{\sigma_1} &=& \textup{Spec }\mathbb{C}[z^{(1,0,0)},z^{(0,1,0)},z^{(-1,0,1)}] = \textup{Spec }\mathbb{C}[x,t,u] = \mathbb{A}^3, \\
X_{\sigma_2} &=& \textup{Spec }\mathbb{C}[z^{(-1,l,1)},z^{(0,0,1)},z^{(0,1,0)},z^{(1,0,-1)}] = \textup{Spec }\mathbb{C}[y,\tilde{w},t,v]/(yv-t^l), \\
X_{\sigma_{12}} &=& \textup{Spec }\mathbb{C}[z^{(1,0,0)},z^{(0,1,0)},z^{\pm(-1,0,1)}] = \textup{Spec }\mathbb{C}[x,t,u^{\pm1}] = \mathbb{A}^2 \times \mathbb{G}_m.\end{aligned}$$ The toric variety $X_\Sigma$ defined by $\Sigma$ is obtained by gluing $X_{\sigma_1}$ and $X_{\sigma_2}$ along $X_{\sigma_{12}}$. This is the fibered coproduct (with $u=U/V$ and $v=V/U$) $$X_\Sigma = X_{\sigma_1} \amalg_{X_{\sigma_{12}}} X_{\sigma_2} = \textup{Proj }\faktor{\mathbb{C}[x,y,\tilde{w},t][U,V]}{(\tilde{w}V-xU,yV-t^lU)}.$$ Note that we take $\textup{Proj}$ of the polynomial ring with variables $U,V$ over the ring $\mathbb{C}[x,y,\tilde{w},t]$, so only $U,V$ are projective coordinates. The grading is given by degree in $U$ and $V$. The exceptional set of the resolution $X_\Sigma\rightarrow X_\sigma$ is a line contained in the irreducible component of the central fiber $X_{\Sigma,0}$ given by $y=0$.
(-2,-0.6) – (0,-0.6) – (2.4,0.6); (0,-0.6) – (0,0.6); (0,0) – (0.4,0) – (0.8,0); (0.8,0) – (1,0.1) node\[right\][$x$]{}; (0,0) – (-0.2,0) node\[below\][$y$]{}; (0,0) – (0,0.2) node\[right\][$w$]{}; (0.8,0) – (0.6,0) node\[above\][$u$]{}; (0,0) – (0.2,0) node\[below\][$v$]{};
Equip $X_\Sigma$ with the divisorial log structure by its central fiber $X_{\Sigma,0}$ and pull this log structure back to $X_{\Sigma,0}$. Then $X_{\sigma_1}$ and $X_{\sigma_2}$ are log smooth with respect to the restriction of this log structure, since they are simple normal crossings. They form an affine cover of $X_{\Sigma,0}$, so $X_{\Sigma,0}$ is log smooth.
Similarly, if we make the opposite choice of subdivision, the exceptional line is contained in the irreducible component of the central fiber given by $x=0$.
The global picture
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Let $Q$ be a Fano polytope and consider the $2$-parameter family $\mathfrak{X}_Q\rightarrow\mathbb{A}^2$ from Construction \[con:family\].
\[con:deg2\] For each log singularity on $\mathfrak{X}_Q$ we have two choices of small resolution, fixed by choosing which irreducible component of $\tilde{X}_0^{s\neq 0}$ contains the exceptional line. We make a symmetric choice such that we have one exceptional line on each irreducible component of $\tilde{X}_0$ (see Figure \[fig:duald\]). The only reason for doing so is to avoid distinction of cases. We obtain a log smooth family $(\mathfrak{X}_Q,\mathfrak{D}_Q)\rightarrow \mathbb{A}^2$. Since we only change the fibers $X_0^s$, we still have that $(\tilde{\mathfrak{X}}^0,\mathfrak{D}^0)\rightarrow\mathbb{A}^1$ is a degeneration of $X^0$ and $\mathfrak{X}\rightarrow\mathbb{A}^1$ is a degeneration of $X$, but these are no toric degenerations.
The small resolution does not change the local toric models at generic points of toric strata. As a consequence, the dual intersection complex $\tilde{B}$ of $\tilde{\mathfrak{X}}\rightarrow\mathbb{A}^1$ is homeomorphic to the dual intersection complex $B$ of $\mathfrak{X}\rightarrow{A}^1$. But there is one difference here. The irreducible components of $\tilde{X}_0$ are non-toric, so there is no natural fan structure at the vertices. Further, $\tilde{X}_0$ has no log singularities. Hence, there is no focus-focus singularity on bounded edges in the dual intersection complex. However, the gluing is still in such a way that the unbounded edges are parallel, leading to affine singularities at the vertices of $\tilde{B}$, coming from the gluing. This gives a triple $(\tilde{B},\mathscr{P},\varphi)$ as in Figure \[fig:duald\]. Affine manifolds with singularities at vertices have been considered by Gross-Hacking-Keel [@GHK1] to construct mirrors to log Calabi-Yau surfaces.
\(1) at (0,0); (2) at (0,1); (3) at (-3,-1); (4) at (-3,2); (-3,-1) – (0,0) – (0,1) – (-3,2); (3) – (1) – (2) – (4); (1) – (5,0); (2) – (5,1); (3) – (5,-1); (4) – (5,2); (0,1) – (-0.6,1.2) node\[above\][$m_{v,+}$]{}; (0,1) – (0,0.5) node\[right\][$m_{v,-}$]{}; (0,1) – (0.5,1) node\[above\][$m_{\textup{out}}$]{}; (0,1) – (-0.3,1.1); (0,0) – (0,0.316); (-3,-1) – (-2.7,-0.9); (5) at (0,0); (6) at (0,1); (7) at (-3,-1); (8) at (-3,2);
\[defi:Lexc\] For a vertex $v$ of $\mathscr{P}$, let $L^{\textup{exc}}_v$ be the unique exceptional line contained in the irreducible component $\tilde{X}_v$ of $\tilde{X}_0$ corresponding to $v$.
For later convenience we indicate the choices of small resolutions by red stubs attached to the vertices of $\mathscr{P}$. The stub at a vertex $v$ points in the direction corresponding to the toric divisor of $X_v$ intersecting $L^{\textup{exc}}_v$. Denote the primitive vector in the direction of the red stub adjacent to $v$ by $m_{v,+}\in\Lambda_{\tilde{B},v}$. Denote the primitive vector in the direction of the other edge of $\sigma_0$ adjacent to $v$ by $m_{v,-}\in\Lambda_{\tilde{B},v}$. Further, $m_{\textup{out}}$ is the unique unbounded direction (Definition \[defi:mout\]).
Logarithmic Gromov-Witten invariants
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Logarithmic Gromov-Witten invariants have been defined in [@Che1][@AC] and [@LogGW] as counts of stable log maps. A stable log map is a stable map defined in the category of log schemes with additional logarithmic data at the marked points, allowing for specification of contact orders. This leads to a generalization of Gromov-Witten theory in log smooth situations. For example, Gromov-Witten invariants relative to a (log) smooth divisor can be defined in this context, avoiding the target expansion of relative Gromov-Witten theory [@Li][@Li2]. This is the case of interest to us.
Let $\tilde{\mathfrak{X}}_Q\rightarrow\mathbb{A}^2$ be the log smooth family from Construction \[con:deg2\]. Note that $\tilde{X}_{t\neq 0}=X$. For the definition of stable log maps and their classes, see [@LogGW].
\[defi:beta\] For an effective curve class $\underline{\beta}\in H_2^+(X,\mathbb{Z})\simeq H_2^+(X^0,\mathbb{Z})$ define a class $\beta$ of stable log maps to $\tilde{\mathfrak{X}}_Q \rightarrow \mathbb{A}^2$ as follows:
genus $g=0$;
$k=1$ marked point $p$;
fibers have curve class $\underline{\beta}$;
contact data $u_p = (D \cdot \underline{\beta}) m_{\textup{out}}$, that is, full tangency with $D$ at the marked point. Here $m_{\textup{out}}\in\Lambda_{\tilde{B}}$ is the primitive integral tangent vector pointing in the unbounded direction on $\tilde{B}$ (Definition \[defi:mout\]).
One comment is in order about the space in which $u_p$ lives. By definition ([@LogGW], Discussion 1.8(ii)), $u_p$ is an element of $P_p^\vee:=\textup{Hom}(f^\star\overline{\mathcal{M}}_{\tilde{\mathfrak{X}}_Q}|_p,\mathbb{N})$ and $m_{\textup{out}}$ is an element of $\Lambda_{\tilde{B}}$, the sheaf of integral tangent vectors on the dual intersection complex $\tilde{B}$. Let $v$ be the vertex of $\mathscr{P}$ corresponding to the irreducible component of $\tilde{X}_0^s$ to which the marked point is mapped. Then $m_{\textup{out}}$ defines an element of $\Lambda_{\tilde{B},v}$ that we also write as $m_{\textup{out}}$. Both, $P_p^\vee$ and $\Lambda_{\tilde{B},v}$, are submonoids of $\Lambda_{\Sigma(\tilde{X}_0^s),v}$ and their intersection is given by $\mathbb{N}\cdot m_{\textup{out}}\subseteq\Lambda_{\tilde{B},v}$. Thus $\mathbb{N}\cdot m_{\textup{out}}$ can be viewed as a submonoid of $P_p^\vee$, so the definition above makes sense.
Define $w_{\textup{out}} = \textup{min}\{D\cdot\underline{\beta} \ | \ \underline{\beta}\in H_2^+(X,\mathbb{Z}) \}$.
For $(X,D) = (\mathbb{P}^2,E)$ we have $w_{\textup{out}}=3$, since $E$ has degree $3$. For $X=\mathbb{P}^1\times\mathbb{P}^1$ we have $w_{\text{out}}=2$ and for the smooth cubic we have $w_{\text{out}}=1$.
\[defi:Nd\] Let $\mathscr{M}(\tilde{\mathfrak{X}}_Q,\beta)$ be the moduli space of basic stable log maps to $\tilde{\mathfrak{X}}_Q\rightarrow\mathbb{A}^2$ of class $\beta$. By the results of [@LogGW] this is a proper Deligne-Mumford stack and admits a virtual fundamental class $\llbracket\mathscr{M}(\tilde{\mathfrak{X}}_Q,\beta)\rrbracket$. The virtual dimension of $\mathscr{M}(\tilde{\mathfrak{X}}_Q,\beta)$ is zero, so we can define the logarithmic Gromov-Witten invariant $$N_\beta = \int_{\llbracket\mathscr{M}(\tilde{\mathfrak{X}}_Q,\beta)\rrbracket} 1.$$ Moreover, for $d>0$ define $$N_d = \sum_{\substack{\underline{\beta} \in H_2^+(X,\mathbb{Z}) \\ D\cdot\underline{\beta}=dw_{\textup{out}}}} N_\beta.$$
\[rem:constant\] Logarithmic Gromov-Witten invariants are constant in log smooth families ([@MR], Appendix A). This means the following. Let $\gamma : \{\text{pt}\} \rightarrow \mathbb{A}^2$ be a point and let $\gamma^!$ be the corresponding refined Gysin homomorphism ([@Ful], [§]{}6.6). Then $\gamma^!\beta$ defines a class of stable log maps to the fiber $\tilde{X}_t^s$ that, by abuse of notation, we also write as $\beta$. We get a moduli space and a virtual fundamental class $\llbracket\mathscr{M}(\tilde{X}_t,\beta)\rrbracket$. Then, for all $(t,s)\in\mathbb{A}^2$, $$N_\beta = \int_{\llbracket\mathscr{M}(\tilde{X}_t^s,\beta)\rrbracket}1.$$ This shows that $N_\beta$ equals the logarithmic Gromov-Witten invariant $N_\beta$ defined in the introduction. Moreover, as shown in [@AMW], $N_\beta$ equals the relative Gromov-Witten invariant of the smooth pair $(X,D)$ as defined in [@Li].
Log BPS numbers {#S:BPS}
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The logarithmic Gromov-Witten invariants $N_\beta$ are not integers but rather rational numbers. The fractional part comes from multiple cover contributions of curves with class $\beta'$ such that $\underline{\beta}=k\cdot\underline{\beta}'$.
The $k$-fold cover of an irreducible curve of class $\beta'$ contributes the following factor to $N_{k\cdot\beta'}$: $$M_{\beta'}[k] = \frac{1}{k^2}\binom{k(D\cdot\underline{\beta}'-1)-1}{k-1}$$
We use the same formula for reducible curves, though it may be unclear how to interpret this as a multiple cover contribution.
Define numbers $n_\beta$ by subtracting multiple cover contributions: $$N_\beta = \sum_{\beta' : \underline{\beta}=k\cdot\underline{\beta}'} M_{\beta'}[k] \cdot n_{\beta'}.$$ They are called *Gopakumar-Vafa invariants* or *log BPS numbers* as they are related to BPS state counts in string theory [@GV].
\[rem:local\] The logarithmic Gromov-Witten invariants $N_\beta$ are related to local Gromov-Witten invariants $N_\beta^{\textup{loc}}$ of the total space of the canonical bundle $K_X$ of $X$ by the formula $N_\beta = (-1)^{D\cdot\underline{\beta}-1}(D\cdot\underline{\beta})N_\beta^{\textup{loc}}$. This was conjectured by Takahashi ([@Ta2], Remark 1.11) and proven by Gathmann ([@Ga], Example 2.2) and more generally by van Garrel, Graber and Ruddat [@vGGR]. The log BPS numbers $n_d$ were shown to be integers in [@vGWZ], using integrality of local BPS numbers.
Tropical curves and refinement {#S:tropmap}
==============================
In this section we analyze how tropicalizations of stable log maps contributing to $N_d$ look like. We prove that for each $d$ there are only finitely many such tropical curves (Corollary \[cor:finite\]). Choosing a subdivision of the dual intersection complex $(\tilde{B},\mathscr{P},\varphi)$ such that tropicalizations contributing to $N_d$ are contained in the $1$-skeleton of the polyhedral decomposition leads to a logarithmic modification $\tilde{\mathfrak{X}}_d$ of $\tilde{\mathfrak{X}}$ (Construction \[con:deg3\]) with the property that stable log maps to the central fiber $Y$ of $\tilde{\mathfrak{X}}_d$ contributing to $N_d$ are torically transverse.
Tropicalization of stable log maps
----------------------------------
\[defi:Cones\] Define $\textbf{Cones}$ to be the category whose objects are pairs $(\sigma_{\mathbb{R}},M)$ where $M\cong\mathbb{Z}^n$ is a lattice and $\sigma_{\mathbb{R}}\subseteq M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$ is a top-dimensional strictly convex rational polyhedral cone. A morphism of cones $\varphi : \sigma_1\rightarrow \sigma_2$ is a homomorphism $\varphi : M_1 \rightarrow M_2$ which takes $\sigma_{1\mathbb{R}}$ into $\sigma_{2\mathbb{R}}$. It is a *face morphism* if it identifies $\sigma_{1\mathbb{R}}$ with a face of $\sigma_{2\mathbb{R}}$ and $M_1$ with a saturated sublattice of $M_2$. A *generalized cone complex* is a topological space with a presentation as the colimit of an arbitrary finite diagram in the category $\textbf{Cones}$ with all morphisms being face morphisms.
\[defi:trop\] Let $X$ be a fine saturated log scheme with log structure defined in the Zariski topology. For the generic point $\eta$ of a stratum of $X$, its characteristic monoid $\overline{\mathcal{M}}_{X,\eta}$ defines a dual monoid $(\overline{\mathcal{M}}_{X,\eta})^\vee := \textup{Hom}(\overline{\mathcal{M}}_{X,\eta},\mathbb{N})$ lying in the group $(\overline{\mathcal{M}}_{X,\eta})^\star := \textup{Hom}(\overline{\mathcal{M}}_{X,\eta},\mathbb{Z})$, hence a dual cone $$\sigma_\eta := \left((\overline{\mathcal{M}}_{X,\eta})_{\mathbb{R}}^\vee,(\overline{\mathcal{M}}_{X,\eta})^\star\right).$$ If $\eta$ is specialization of $\eta'$, there is a well-defined generization map $\overline{\mathcal{M}}_{X,\eta}\rightarrow \overline{\mathcal{M}}_{X,\eta'}$. Dualizing, we obtain a face morphism $\sigma_{\eta'}\rightarrow\sigma_\eta$. This gives a diagram of cones indexed by strata of $X$ with face morphisms, hence gives a generalized cone complex $\Sigma(X)$, the *tropicalization* of $X$. This construction is functorial.
Let $Q$ be a Fano polytope and let $\tilde{\mathfrak{X}}\rightarrow\mathbb{A}^1$ be the log smooth degeneration (Construction \[con:deg2\]) of the corresponding smooth very ample log Calabi-Yau pair $(X,D)$. Let $\underline{\beta}\in H_2^+(X,\mathbb{Z})$ be an effective curve class and consider a basic stable log map $f:C/\textup{pt}_Q\rightarrow\tilde{X}_0/\textup{pt}_{\mathbb{N}}$ of class $\beta$ (Definition \[defi:beta\]). Here $Q$ is the basic monoid[^2] of $f$ ([@LogGW], [§]{}1.5). We will see in Corollary \[cor:finite\] that $Q=\mathbb{N}$, so $$\label{eq:stablelog}
\begin{xy}
\xymatrix{
C \ar[r]^f \ar[d]^\gamma & \tilde{X}_0 \ar[d]^{\tilde{\pi}_0} \\
\textup{pt}_{\mathbb{N}} \ar[r]^g & \textup{pt}_{\mathbb{N}}
}
\end{xy}$$ Tropicalization of gives a diagram of generalized cone complexes. Note that $\Sigma(\textup{pt}_{\mathbb{N}})=\mathbb{R}_{\geq 0}$. The fiber $\Sigma(\tilde{\pi}_0)^{-1}(1)$ is homeomorphic to the dual intersection complex $\tilde{B}$ of $\tilde{X}_0$. Similarly, for all $b\neq 0$, the fiber $\Sigma(\gamma)^{-1}(b)$ is homeomorphic to the dual intersection graph $\Gamma_C$ of $C$. Hence, tropicalization of the above diagram and restriction to the fiber over $1\in\Sigma(\textup{pt}_{\mathbb{N}})=\mathbb{R}_{\geq 0}$ gives a continuous map $$\label{eq:Delta}
\tilde{h} : \Gamma_C \rightarrow \tilde{B}.$$
There is additional data on $\Gamma_C$ making $\tilde{h} : \Gamma_C\rightarrow \tilde{B}$ into a tropical curve in the sense of [@ACGS1]. Note that such a tropical curve only fulfills a modified version of the balancing condition ([@LogGW], Proposition 1.15). In §\[S:balancing\] we will see what this means in our case.
To make the connection with scattering diagrams in §\[S:scattering\] it is useful to consider tropical curves on $B$ (not $\tilde{B}$) that are balanced in the usual sense but may have some bounded legs.
There are many slightly different definitions of parametrized tropical curves, depending on the context in which they are used. The following definition is a synthesis of the definition in [@ACGS1] and [@Gr10], Definition 1.32. In [@ACGS1] only tropical curves with no bounded legs are considered, while in [@Gr10] tropical curves are required to be balanced.
\[defi:tropical\] Let $B$ be a $2$-dimensional integral affine manifold with singularities. A *(parametrized) tropical curve on $B$* is a map $h : \Gamma \rightarrow B$ where $\Gamma$ is a graph, possibly with some non-compact edges (*legs*), together with
a non-negative integer $g_V$ (*genus*) for each vertex $V$;
a non-negative integer $\ell_E$ (*length*) for each compact edge $E$;
a *weight vector* $u_{(V,E)}\in\Lambda_{B,h(V)}$ for every vertex $V$ and edge or leg $E$ adjacent to $V$. Here $\Lambda_{B,h(V)}$ is the stalk of the sheaf $\Lambda_B$ of integral affine tangent vectors on $B$ at the point $h(V)\in B$. The index of $u_{(V,E)}$ in the lattice $\Lambda_{B,h(V)}$ is called the *weight* $w_E$ of $E$;
such that
if $E$ is a compact edge with vertices $V_1$, $V_2$, then $h$ maps $E$ affine linearly[^3] to the line segment connecting $h(V_1)$ and $h(V_2)$, and $h(V_2)-h(V_1)=\ell_Eu_{(V_1,E)}$. In particular, $u_{(V_1,E)} = -u_{(V_2,E)}$. Moreover, if $E$ is a loop, then $u_{(V,E)}=0$;
if $E$ is a leg with vertex $V$, then $h$ maps $E$ affine linearly either to the ray $h(V)+\mathbb{R}_{\geq 0}u_{(V,E)}$ or to the line segment $[h(V),\delta)$ for $\delta$ an affine singularity of $B$ such that $\delta-h(V)\in\mathbb{R}_{>0} u_{(V,E)}$, i.e., $u_{(V,E)}$ points from $h(V)$ to $\delta$.
We write the set of compact edges of $\Gamma$ as $E(\Gamma)$, the set of legs as $L(\Gamma)$, the set of legs mapped to a ray (*unbounded legs*) as $L_\infty(\Gamma)$ and the set of legs mapped to an open line segment (*bounded legs*) as $L_\Delta(\Gamma)$ (since such edges end at the singular locus $\Delta$ of $B$).
The *genus* of a parametrized tropical curve $h : \Gamma \rightarrow B$ is defined by $$g_h := g_\Gamma + \sum_{V\in V(\Gamma)}g(V),$$ where $g_\Gamma$ is the genus (first Betti number) of the graph $\Gamma$.
Note that if $E$ is a leg of $\Gamma$, then $h(E)$ must be parallel to the edge of $\mathscr{P}$ containing $\delta$, since there is only one tangent direction at $\delta$, i.e., $\Lambda_{B,\delta}\simeq\mathbb{Z}$.
We will only consider tropical curves of genus $0$. In particular, our tropical curves will have no loops.
Let $(B,\mathscr{P})$ be a $2$-dimensional polyhedral affine manifold. A tropical curve $h : \Gamma\rightarrow B$ is *on $(B,\mathscr{P})$* if
the edges of $\Gamma$ do not extend across several maximal cells of $\mathscr{P}$. In other words, we have a well-defined map $E(\Gamma) \cup L(\Gamma) \rightarrow \mathscr{P}$ associating to an edge or leg $E$ the minimal cell of $\mathscr{P}$ containing it.
there are no bivalent vertices in $\Gamma$ mapped to a maximal cell of $\mathscr{P}$.
\[con:tropical\] Let $\tilde{h} : \Gamma_C \rightarrow \tilde{B}$ be the continuous map from . We describe additional data making $\tilde{h}$ a tropical curve on $(\tilde{B},\mathscr{P})$.
For each vertex $V$, the genus is $g_V=0$.
Let $E\in E(\Gamma_C)$ be a compact edge with vertices $V_1,V_2$, corresponding to a node $q\in C$. Then $\overline{\mathcal{M}}_{C,q}$ is isomorphic to the submonoid $S_{e_q}$ of $\mathbb{N}^2$ generated by $(e_q,0)$, $(0,e_q)$ and $(1,1)$ for some $e_q\in\mathbb{N}_{>0}$ ([@Kat2], 1.8). Moreover, there is an equation $\tilde{h}(V_2)-\tilde{h}(V_1)=\pm e_qu_q$ for some $u_q\in\Lambda_{\tilde{B}}$ (see [@LogGW], Discussions 1.8, 1.13). Then the length of $E$ is $\ell_E=e_q$ and the weight vectors are $u_{(V_i,E)}=\pm u_q$, with sign chosen such that $u_{(V_i,E)}$ points away from $\tilde{h}(V_i)$.
Let $E\in L_\infty(\Gamma_C)$ be an unbounded leg with vertex $V$, corresponding to a marked point $p\in C$. Then $\overline{\mathcal{M}}_{C,p}$ is isomorphic to $\mathbb{N}\oplus\mathbb{N}$ and $f^\star\overline{\mathcal{M}}_{X}|_p\rightarrow\overline{\mathcal{M}}_{C,p}$ is determined by an element of $P_p^\vee=\textup{Hom}(f^\star\overline{\mathcal{M}}_{X}|_p,\mathbb{N})$, inducing an element $u_p\in\Lambda_{\tilde{B},\tilde{h}(V)}$. The weight vector is $u_{(V,E)}=u_p$.
The properties for $\tilde{h} : \Gamma_C \rightarrow \tilde{B}$ to be on $(\tilde{B},\mathscr{P})$ can be achieved by (1) inserting vertices at points mapping to vertices or edges of $\mathscr{P}$ and (2) removing bivalent vertices mapping to a maximal cell of $\mathscr{P}$, by replacing a chain of edges connected via bivalent vertices with a single edge. The latter is possible, since vertices of $\Gamma_C$ not mapping to vertices of $\mathscr{P}$ are balanced by Proposition \[prop:balancing\], (I), below.
Let $B$ be a $2$-dimensional integral affine manifold with singularities and $m\in\Lambda_B$ an integral tangent vector. A tropical curve $h:\Gamma\rightarrow B$ is called *of degree $d$ relative to $m$* if there is exactly one unbounded leg $E_{\textup{out}}\in L_\infty(\Gamma)$, and its weight vector is $u_{(V_{\textup{out}},E_{\textup{out}})}=d \cdot m$. Here $V_{\textup{out}}$ is the unique vertex of $E_{\textup{out}}$.
\[prop:tropical\] Given a stable log map $f:C/\textup{pt}_{\mathbb{N}}\rightarrow \tilde{X}_0/\textup{pt}_{\mathbb{N}}$ contributing to $N_d$, the continuous map $\tilde{h}:\Gamma_C\rightarrow \tilde{B}$ from together with the additional data defined in Construction \[con:tropical\] is a tropical curve on the dual intersection complex $(\tilde{B},\mathscr{P})$ without bounded legs of genus $0$ and degree $dw_{\textup{out}}$ relative to $m_{\textup{out}}\in\Lambda_{\tilde{B}}$ (Definition \[defi:mout\]). By abuse of notation, we call it the *tropicalization* of $f$.
The properties (i) and (ii) of Definition \[defi:tropical\] follow by the structure of $f : C \rightarrow \tilde{X}_0$ on the level of ghost sheaves (see [@LogGW], Discussions 1.8, 1.13). Hence, $\tilde{h}$ is a tropical curve. Moreover, these discussions show that $\tilde{h}$ has no bounded legs. $\tilde{h}$ is of degree $dw_{\textup{out}}$ relative to $m_{\textup{out}}$ by Definition \[defi:beta\], (4).
There is one issue here, since we lost some information by smoothing $(X^0,D^0)$. An effective curve class $\beta\in H_2^+(X^0,\mathbb{Z})$ is determined by its intersection numbers $d_1,\ldots,d_k$ with the components of $D^0=D_1,\ldots,D_k$. After smoothing $D^0$ we only see the sum $d=d_1,\ldots,d_k$. In particular, if $X^0$ is a smooth toric del Pezzo surface with Picard number $>1$, i.e., different from $(\mathbb{P}^2,E)$, then we only see the total degree, not the multi-degree. One could solve this problem using non-trivial gluing data capturing information of the divisor $D^0$. We will give a more geometric solution in §\[S:degdiv\] by looking at the limit of curves under $s\rightarrow 0$, where $s\in\mathbb{A}^2$ is the deformation parameter of the family $\mathfrak{X}_Q\rightarrow\mathbb{A}^2$ from Construction \[con:family\].
Types of vertices {#S:balancing}
-----------------
Let $f:C/\textup{pt}_{\mathbb{N}}\rightarrow\tilde{X}_0/\textup{pt}_{\mathbb{N}}$ be a stable log map contributing to $N_d$ and let $\tilde{h}:\Gamma_C\rightarrow \tilde{B}$ be the corresponding tropical curve.
\[prop:balancing\] Let $C_V$ be an irreducible component of $C$, corresponding to a vertex $V$ of $\Gamma_C$. The following three cases can occur:
If $C_V$ is mapped to a $0$- or $1$-dimensional toric stratum of $\tilde{X}_0$, i.e., if $V$ is not mapped to a vertex of $\mathscr{P}$, then the ordinary balancing condition holds: $$\sum_{E\ni V}u_{(V,E)} = 0.$$ The sum is over all edges or legs $E\in E(\Gamma_C)\cup L(\Gamma_C)$ containing $V$.
If $C_V$ is mapped onto an exceptional divisor $L^{\textup{exc}}_v$ on some component $\tilde{X}_v$ of $\tilde{X}_0$ (Definition \[defi:Lexc\]), then $C_V$ is a $k$-fold multiple cover of $L^{\textup{exc}}_v\simeq\mathbb{P}^1$ for some $k>0$. It is fully ramified at the point $p=L^{\textup{exc}}_v\cap\partial\tilde{X}_v$, where $\partial\tilde{X}_v$ is the proper transform of the toric boundary $\partial X_v$ under the resolution from §\[S:resolution\]. The vertex $V$ is mapped to the vertex $v$ of $\mathscr{P}$. It is $1$-valent with adjacent edge $E$ mapped onto the edge of $\mathscr{P}$ containing the red stub adjacent to $v$. The balancing condition reads (with $m_{v,+}$ as in Figure \[fig:duald\]) $$u_{(V,E)} = km_{v,+}.$$
Otherwise, $V$ is mapped to a vertex $v$ of $\mathscr{P}$ and has exactly one adjacent edge or leg $E_{V,\textup{out}}$ that is not mapped onto a compact edge of $\mathscr{P}$. All other edges (possibly none) are compact with other vertex of type (II) above. In this case, for some $k\geq 0$, the following balancing condition holds: $$\sum_{E\ni V} u_{(V,E)} + km_{v,+} = 0.$$
If $C_V$ does not intersect an exceptional line, the log structure on $\tilde{X}_0$ along the image of $C_V$ is the toric one. Hence for any section $m\in\Gamma(C_V,f^\star\overline{\mathcal{M}}_X|_{C_V})$ the associated $\mathcal{O}_{C_V}$-torsor $\kappa^{-1}(m)$ is trivial. Here $\kappa : f^\star\mathcal{M} \rightarrow f\star\overline{\mathcal{M}}_X$. Then by [@LogGW], Proposition 1.15, the ordinary balancing condition holds. This proves (I).
If $C_V$ is mapped onto an exceptional line $L^{\textup{exc}}_v\simeq\mathbb{P}^1$ on some component $\tilde{X}_v$, it is a $k$-fold multiple cover for some $k>0$. Suppose it is not fully ramified at the point where $L^{\textup{exc}}_v$ meets $\partial \tilde{X}_v$, i.e., $V$ has valency $>1$. Let $E_1,E_2$ be two distinct edges adjacent to $V$. We have $V \neq V_{\textup{out}}$, since $C_V$ does not meet the toric divisor of $\tilde{X}_v$ belonging to $\tilde{D}_0$. By Proposition \[prop:tropical\], $\Gamma_C$ has only one leg, and this leg is attached to $V_{\textup{out}}$. Thus $E_1$ and $E_2$ are bounded. Let $V_1,V_2$ be the vertex of $E_1,E_2$ different from $V$, respectively. There is a chain of vertices and edges (probably the trivial one) connecting $V_1$ to $V_{\textup{out}}$ and similarly for $V_2$. These two chains form a cycle of the graph $\Gamma_C$, so $g(\Gamma_C)>0$ in contradiction with rationality of $C$. Hence, there is a unique bounded edge $E$ adjacent to $V$. Let $V'$ be its other vertex. Then $h(V')-h(V)$ points in the direction of $m_{v,+}$, since the only special point (node) of $C_V$ is mapped to $L^{\textup{exc}}_v\cap\partial\tilde{X}_v$. It follows by Definition \[defi:tropical\] that $u_{(V,E)}$ points in the direction of $m_{v,+}$. Its affine length, the weight $w_E$, is the multiplicity of the node which is the ramification order $k$. This proves (II).
Let $C_V$ be a component of $C$ that intersects an exceptional divisor $L^{\textup{exc}}_v$ on some component $\tilde{X}_v$ but is not mapped onto it. Let $m\in\Gamma(C_V,f^\star\overline{\mathcal{M}}_X|_{C_V})$ be the generator of the submonoid $\mathbb{N}$ of $\Gamma(C_V,f^\star\overline{\mathcal{M}}_X|_{C_V})$ corresponding to the exponent of the degeneration parameter $t$. The $\mathcal{O}_{C_V}$-torsor associated to $m$ is $f^\star\nu^\star\mathcal{O}_{X_v}(-kD_{v,+})$, where $\nu : \tilde{X}_v \rightarrow X_v$ is the resolution and $D_{v,+}$ is the toric divisor of $X_v$ whose proper transform $\tilde{X}_v$ intersects $L_v^{\textup{exc}}$. This is because the condition that $C_v$ intersects $L_v^{\textup{exc}}$ is equivalent to a point condition on $D_{v,+}$. The corresponding contact data is $km_{v,+}$. This shows $\sum_{E\ni V} u_{(V,E)} + km_{v,+} = 0$, where $k$ is the sum of the affine lengths of the additional $u_p$, proving (III).
\[defi:types\] Denote the set of vertices of the given types in Proposition \[prop:balancing\] by $V_{I}(\Gamma_C)$, $V_{II}(\Gamma_C)$ and $V_{III}(\Gamma_C)$, respectively.
\[defi:tildeH\] Let $\tilde{\mathfrak{H}}_d$ be the set of tropical curves $\tilde{h}:\tilde{\Gamma}\rightarrow \tilde{B}$ on $(\tilde{B},\mathscr{P})$ of genus $0$ and degree $dw_{\textup{out}}$ relative to $m_{\textup{out}}\in\Lambda_{\tilde{B}}$, without bounded legs and with vertices of one of the types (I)-(III) above.
(-7,-1) node\[fill,circle,inner sep=1.5pt\]– (-6,0) node\[fill,circle,inner sep=1.5pt\] – (-5.8,0.67) node\[fill,circle,inner sep=1.5pt\] – (-6,1) node\[fill,circle,inner sep=1.5pt\] – (-6.5,0) node\[fill,circle,inner sep=1.5pt\]; (-5.8,0.67) – (-5.3,0.833) node\[below\][$9$]{} – (-4.8,1); (-4.5,0.5) – (-3.5,0.5); (1) at (0,0); (2) at (0,1); (3) at (-3,-1); (4) at (-3,2); (-3,-1) – (0,0) – (0,1) – (-3,2); (3) – (1) – (2) – (4); (1) – (5,0); (2) – (5,1); (3) – (5,-1); (4) – (5,2); (0,0.05) node\[fill,circle,inner sep=1.5pt,label=left:(II)\] – (0,1) node\[fill,circle,inner sep=1.5pt,label=above:(III)\] – (2,0.67) node\[fill,circle,inner sep=1.5pt,label=below:(I)\] – (0,-0) node\[fill,circle,inner sep=1.5pt,label=below:(III)\] – (-3,-1) node\[fill,circle,inner sep=1.5pt,label=below:(II)\]; (2,0.67) – (3.5,0.67) node\[below\][$9$]{} – (5,0.67); (0,1) – (-0.3,1.1); (0,0) – (0,0.316); (-3,-1) – (-2.7,-0.9); (5) at (0,0); (6) at (0,1); (7) at (-3,-1); (8) at (-3,2);
\[lem:finite\] Let $\tilde{h}:\tilde{\Gamma}\rightarrow \tilde{B}$ be a tropical curve in $\tilde{\mathfrak{H}}_d$ for some $d>0$. Then $\tilde{h}(\tilde{\Gamma})$ is disjoint from the interior of $\sigma_0$ (Definition \[defi:sigma0\]).
Let $\tilde{h}:\tilde{\Gamma}\rightarrow \tilde{B}$ be a tropical curve in $\tilde{\mathfrak{H}}_d$. Give $\tilde{\Gamma}$ the structure of a rooted tree by defining the root vertex to be the vertex $V_{\textup{out}}$ of the unique unbounded leg $E_{\textup{out}}$. Let $V$ be a vertex of $\tilde{\Gamma}$ and let $E_{V,\textup{out}}$ be the edge connecting $V$ with its parent, or $E_{V,\textup{out}}=E_{\textup{out}}$ if $V$ is the root vertex $V_{\textup{out}}$. By Proposition \[prop:balancing\], if $V$ is of type (II) or (III) mapped to a vertex $v$ of $\mathscr{P}$, then $E_{V,\textup{out}}$ is mapped to the subset $\mathbb{R}_{\leq 0}m_{v,+}+\mathbb{R}_{\leq 0}m_{v,-}$ of $\tilde{B}$, and if $V$ is of type (I), then $E_{V,\textup{out}}$ is mapped to the subset $\cup_{V'} \mathbb{R}_{\leq 0}m_{\tilde{h}(V'),+}+\mathbb{R}_{\leq 0}m_{\tilde{h}(V'),-}$ of $\tilde{B}$, where the union is over all vertices of type (II) and (III) in the subgraph of $\tilde{\Gamma}$ with root $V$. In particular, $\tilde{h}(\tilde{\Gamma})$ is contained in $$\bigcup_{V'\in V_{II}(\tilde{\Gamma})\cup V_{III}(\tilde{\Gamma})} \mathbb{R}_{\leq 0}m_{\tilde{h}(V'),+}+\mathbb{R}_{\leq 0}m_{\tilde{h}(V'),-}$$ This is disjoint from the interior of $\sigma_0$.
Balanced tropical curves
------------------------
We describe a procedure to obtain tropical curves in $\tilde{\mathfrak{H}}_d$ from tropical curves to $B$ (not $\tilde{B}!$) that are balanced in the usual sense. This makes the connection to scattering diagrams in §\[S:scattering\] more transparent. Moreover, the degeneration formula gets more symmetric when expressed in invariants labeled by balanced tropical curves (see Theorem \[thm:degmax\]).
\[defi:balancedtrop\] Let $\mathfrak{H}_d$ be the set of tropical curves $h:\Gamma\rightarrow B$ on $(B,\mathscr{P})$, possibly with bounded legs, of genus $0$ and degree $dw_{\textup{out}}$ relative to $m_{\textup{out}}$, satisfying the ordinary balancing condition at each vertex $V$ of $\Gamma$: $$\sum_{E\ni V}u_{(V,E)} = 0,$$
\[lem:tilde\] There is a surjective map $\mathfrak{H}_d \rightarrow \tilde{\mathfrak{H}}_d$.
The map is constructed as follows. Let $h:\Gamma\rightarrow B$ be a tropical curve in $\mathfrak{H}_d$. Let $E\in L_\Delta(\Gamma)$ be a bounded leg with vertex $V$. Then $E$ is mapped to the line segment $[h(V),\delta)$ for $\delta$ an affine singularity on an edge $\omega$ of $\mathscr{P}$. Since $\Lambda_{B,\delta}$ is one-dimensional, $h(E)$ is parallel to $\omega$. Since $h$ is on $(B,\mathscr{P})$ and by the balancing condition, $h(V)$ must be a vertex $v$ of $\mathscr{P}$. Let $m_{v,\delta}$ be the primitive integral tangent vector pointing from $v$ to $\delta$ and let $m_{v,+},m_{v,-}$ be as in Figure \[fig:duald\]. Two cases can occur.
If $m_{v,\delta}=m_{v,+}$, i.e., if $E$ is mapped in the direction of the red stub attached to $v$, then remove $E$ from $\Gamma$.
Otherwise, $m_{v,\delta}=m_{v,-}$. Then add a vertex $V'$ to $E$ to obtain a compact edge $\tilde{E}$. Define $u_{(V',E)}=-u_{(V,E)}$ and $\tilde{h}(\tilde{E})=\omega$, such that $\tilde{h}(V')=v'$ is a vertex of $\mathscr{P}$. This determines the length $\ell_{\tilde{E}}$ by Definition \[defi:tropical\], (i).
Let $\tilde{h} : \tilde{\Gamma} \rightarrow \tilde{B}$ be a tropical curve in $\tilde{\mathfrak{H}}_d$. We can construct a preimage of $\tilde{h}$ as follows. (1) For each vertex $V\in V_{III}(\tilde{\Gamma})$, add a bounded leg $E$ with vertex $V$ and weight vector $u_{(V,E)}=-\sum_{E'\ni V} u_{(V,E')}$. The image of $E$ is specified by Definition \[defi:tropical\], (ii). (2) For each vertex $V\in V_{II}(\tilde{\Gamma})$, let $E$ be the unique adjacent edge. It is a bounded edge and we remove the vertex $V$ from $E$ to obtain a bounded leg. This shows that the map $\mathfrak{H}_d\rightarrow\tilde{\mathfrak{H}}_d$ is surjective. Note that in step (1) we could also add several bounded legs with weights a partition of $\sum_{E'\ni V} u_{(V,E')}$, so the number of preimages of $\tilde{h}$ is the number of such partitions.
Let $(\bar{B},\bar{\mathscr{P}})$ be the covering space of $(B,\mathscr{P})$ described in §\[S:affinecharts\]. Let $\bar{\mathfrak{H}}_d$ be the set of balanced tropical curves on $(\bar{B},\bar{\mathscr{P}})$ of genus $0$ and degree $dw_{\textup{out}}$ relative to $m_{\textup{out}}$ satisfying the ordinary balancing condition and such that the image of $E_{\textup{out}}$ lies in a particular fundamental domain.
\[lem:bij\] The map $\bar{\mathfrak{H}}_d \rightarrow \mathfrak{H}_d$ defined by sending $\bar{h} : \Gamma \rightarrow \bar{B}$ to $h : \Gamma \rightarrow B$, where $h$ is the composition of $\bar{h}$ with the covering map $\bar{B} \rightarrow B$ is bijective.
The inverse map is given as follows. Let $h : B \rightarrow \Gamma$ be a tropical curve in $\mathfrak{H}_d$ and choose an unbounded maximal cell of $\mathscr{P}$. Choose a fundamental domain of $\bar{B} \rightarrow B$ and let $\bar{h}(E_{\textup{out}})$ be the preimage of $h(E_{\textup{out}})$ in that fundamental domain. Whenever the image $h(V)$ of an edge $V$ lies on the horizontal dashed line in Figure \[fig:dualb\] with respect to the chart on the unbounded maximal cell chosen, we change the fundamental domain and apply the monodromy transformation.
(-6.25,-0.5) – (-6,0) node\[fill,circle,inner sep=1.5pt\] – (-5.8,0.67) node\[fill,circle,inner sep=1.5pt\] – (-6,1) node\[fill,circle,inner sep=1.5pt\] – (-6.25,0.5); (-6,1) – (-6.25,1.5) node\[right\][$2$]{}; (-5.8,0.67) – (-5.3,0.833) node\[below\][$9$]{} – (-4.8,1); (-4.5,0.5) – (-3.5,0.5); (1) at (0,0); (2) at (0,1); (3) at (-3,-1); (4) at (-3,2); (3) – (1) – (2) – (4); (1) – (5,0); (2) – (5,1); (3) – (5,-1); (4) – (5,2); (-1.5,-0.25) – (0,0.5) – (-1.5,1.25); (-3,-0.8) – (-1.5,-0.5) – (-0.75,0.25); (-3,1.8) – (-1.5,1.5) – (-0.75,0.75); (-1.5,-0.25) – (0,0.5) – (-1.5,1.25) – (-1.5,-0.25) – cycle; (-3,1.8) – (-1.5,1.5) – (-0.75,0.75) – (-0.75,0.25) – (-1.5,-0.5) – (-3,-0.8); (5) at (0,0.5); (6) at (-1.5,-0.5); (7) at (-1.5,1.5); (0,0) – (-1,0); (0,1) – (-1,1); (0,0) circle (1.5pt); (0,1) circle (1.5pt); (2,0.66) circle (1.5pt); (-1.5,1.5) – (-0.75,1.25) node\[above\][$2$]{} – (0,1) – (2,0.66) – (0,0) – (-1.5,-0.5); (2,0.66) – (3.5,0.66) node\[above\][$9$]{} – (5,0.66); (0,1) – (0,0.5);
\[lem:finite2\] The set $\bar{\mathfrak{H}}_d$ is finite.
Let $\bar{h}:\Gamma\rightarrow \bar{B}$ be a tropical curve in $\bar{\mathfrak{H}}_d$. We show that the set of weight vectors of bounded legs $\{u_{(V,E)} \ | \ V\in E\in L_\Delta(\Gamma)\}$ determines the image of $\bar{h}$. By Lemma \[lem:finite\] and the constructions of the maps in Lemmas \[lem:bij\] and \[lem:tilde\] the image of $\bar{h}$ is disjoint from $\bar{\sigma}_0$, so we can work in the chart from §\[S:affinecharts\]. Give $\Gamma$ the structure of a rooted tree by defining the root vertex to be the vertex $V_{\textup{out}}$ of the unique unbounded leg $E_{\textup{out}}$.
Let $V$ be a vertex of $\Gamma$. We show by induction on the height (maximal length of chains connecting $V$ with a leaf of $\Gamma$) of $V$ that $\{u_{(V,E)} \ | \ V\in E\in L_\Delta(\Gamma)\}$ determines $\bar{h}(V)$. For the base case, let $E\in L_\Delta(\Gamma)$ be a bounded leg. Then $\bar{h}$ maps $E$ onto a line segment $[v,\delta)$ for $v$ a vertex of $\bar{\mathscr{P}}$ an $\delta$ an affine singularity of $\bar{B}$. As the tangent vectors of these line segments are all different, the weight vector $u_{(V,E)}$ determines the image of $E$. Hence, the images of all leaves are determined by $\{u_{(V,E)} \ | \ V\in E\in L_\Delta(\Gamma)\}$. For the induction step, let $V$ be a vertex of height $k>0$ and assume the image of all vertices of height $<k$ is determined. Then the images of all childs $V_1,\ldots,V_n$ of $V$ are determined. By the balancing condition we know the weight vectors $u_{(V_i,E_i)}$ of all edges $E_i$ connecting $V$ with its childs. This gives, for each child of $V$, a ray in $\bar{B}$ with base $\bar{h}(V_i)$ and direction $m_i$, the direction of $-u_{(V_i,E_i)}$. Since $\bar{h}$ is a tropical curve, all these rays must intersect in a point. This is the image of $V$.
This shows that $\{u_{(V,E)} \ | \ V\in E\in L_\Delta(\Gamma)\}$ determines the images of all vertices and bounded legs of $\bar{h}$. The image of $E_{\textup{out}}$ is determined, since we know $\bar{h}(V_{\textup{out}})$ and the direction $m_{\textup{out}}$ of the image of $E_{\textup{out}}$. Hence, $\{u_{(V,E)} \ | \ V\in E\in L_\Delta(\Gamma)\}$ determines the image of $\bar{h}$. By the balancing condition, $$\sum_{E\in L_\Delta(\Gamma)} u_{(V,E)} = u_{(V_{\textup{out}},E_{\textup{out}})} = dw_{\textup{out}}m_{\textup{out}}.$$ Hence, for finite $d$, there is a finite number of possible sets of weight vectors of bounded legs $\{u_{(V,E)} \ | \ V\in E\in L_\Delta(\Gamma)\}$. This shows that $\mathfrak{H}_d$ is finite.
\[cor:finite\] The sets $\mathfrak{H}_d$ and $\tilde{\mathfrak{H}}_d$ are finite. In particular, tropical curves in $\tilde{\mathfrak{H}}_d$ are rigid and the basic monoid of stable log maps contributing to $N_d$ equals $\mathbb{N}$.
The limit $s\rightarrow 0$ {#S:degdiv}
--------------------------
Let $\underline{\beta}\in H_2^+(X,\mathbb{Z})$ be an effective curve class and let $f : \mathfrak{C} \rightarrow \tilde{\mathfrak{X}}_Q$ be a stable log map of class $\beta$ (Definition \[defi:beta\]). Figure \[fig:bigpicture\] shows the fibers of $f$ and its image on the log singular family $\mathfrak{X}_Q$.
The fibers $f_t^0 : C_t^0 \rightarrow X_t^0$ map entirely onto the divisor $D_t^0$.
Suppose there is an irreducible component of $C_t^0$ not mapped onto $D_t^0$. Then, since the marked point is mapped to $D_t^0$, there exists an irreducible component of $C_t^0$ that is not mapped onto $D_t^0$ and is not contracted to a point. But then the tropicalization of $f_t^0$ must have at least two legs, since the balancing condition implies balancing of the legs. This means $C_t^0$ must have at least two marked points, in contradiction with the definition of $\beta$ (Definition \[defi:beta\]).
(2,0.5) – (2.5,0.5) node\[above\][$s\rightarrow 0$]{} – (3,0.5); (0.5,3) – (0.5,2.5) node\[right\][$t\rightarrow 0$]{} – (0.5,2); (2,4.75) – (2.5,4.75) node\[above\][$s\rightarrow 0$]{} – (3,4.75); (5.5,3) – (5.5,2.5) node\[left\][$t\rightarrow 0$]{} – (5.5,2); (7,2.5) – (8,2.5); (0) at (0,0); (1) at (-1,-1); (2) at (2,-1); (3) at (-1,2); (1a) at (-0.5,-0.5); (2a) at (1,-0.5); (3a) at (-0.5,1); (1) – (2) – (3) – (1); (0) – (1); (0) – (2); (0) – (3); (-0.67,0.25) ellipse (9.5pt and 24pt); (4) at (1,-1); (5) at (1.2,-1.4); (5) node\[below\][$\mathbb{N}^2$]{} to \[bend right=10\] (4); (6) at (-1,-1.4); (6) node\[below\][$\mathbb{N}^3$]{} to \[bend right=0\] (1); (7) at (0,-1.46); (7) node\[below\][$\mathbb{N}$]{} to \[bend right=30\] (1a); (0) at (5,0); (1) at (4,-1); (2) at (7,-1); (3) at (4,2); (1a) at (4.5,-0.5); (2a) at (6,-0.5); (3a) at (4.5,1); (1) – (2) – (3) – (1); (0) – (1); (0) – (2); (0) – (3); (1) – (3); (4) at (6,-1); (5) at (6.2,-1.4); (5) node\[below\][$\mathbb{N}^2$]{} to \[bend right=10\] (4); (6) at (4,-1.4); (6) node\[below\][$\mathbb{N}^2$]{} to \[bend right=0\] (1); (7) at (5,-1.4); (7) node\[below\][$\mathbb{N}^2$]{} to \[bend right=30\] (1a); (-1,6) – (-1,3.5) – (1.5,3.5) – (1.5,6) – (-1,6); plot (,[sqrt((2\*)\^3+1)/2+4.75]{}); plot (,[-sqrt((2\*)\^3+1)/2+4.75]{}); (-0.75,5.25) – (1,5.25); (4) at (0,4.25); (5) at (0.5,4.5); (5) node\[right\][$\mathbb{N}$]{} to \[bend right=10\] (4); (4,6) – (4,3.5) – (6.5,3.5) – (6.5,6) – (4,6); (4.5,3.75) – (4.5,5.75); (4.25,4) – (6.25,4); (4.25,5.75) – (6.25,3.75); (4) at (4.5,5.5); (5) at (5,5.5); (5) node\[right\][$\mathbb{N}$]{} to \[bend right=10\] (4); (6) at (5.2,4.8); (7) at (5.7,5); (7) node\[right\][$\mathbb{N}$]{} to \[bend right=10\] (6);
(2,0.5) – (2.5,0.5) node\[above\][$s\rightarrow 0$]{} – (3,0.5); (0.5,3) – (0.5,2.5) node\[right\][$t\rightarrow 0$]{} – (0.5,2); (2,4.75) – (2.5,4.75) node\[above\][$s\rightarrow 0$]{} – (3,4.75); (5.5,3) – (5.5,2.5) node\[left\][$t\rightarrow 0$]{} – (5.5,2); (0) at (0,0); (1) at (-1,-1); (2) at (2,-1); (3) at (-1,2); (1a) at (-0.5,-0.5); (2a) at (1,-0.5); (3a) at (-0.5,1); (1) – (2) – (3) – (1); (0) – (1); (0) – (2); (0) – (3); (1a) – (0,-0.6); (2a) – (0.69,-0.11); (3a) – (-0.62,0.52); (-0.65,0.05) ellipse (10pt and 17pt); (1a) – (0,-0.6); (4) at (1,-1); (5) at (1.2,-1.4); (5) node\[below\][$\mathbb{N}^2$]{} to \[bend right=10\] (4); (6) at (-1,-1.4); (6) node\[below\][$\mathbb{N}^3$]{} to \[bend right=0\] (1); (7) at (0,-1.4); (7) node\[below\][$\mathbb{N}^2$]{} to \[bend right=30\] (1a); (0) at (5,0); (1) at (4,-0.6); (2) at (6.2,-1); (3) at (4.4,1.8); (1a) at (4.5,-1); (2a) at (6.65,-0.65); (3a) at (4,1.5); (1a) – (2); (2a) – (3); (3a) – (1); (0) – (1); (0) – (2); (0) – (3); (1) to \[bend left=20\] (1a); (2) to \[bend left=20\] (2a); (3) to \[bend left=20\] (3a); (4) at (5.7,-1); (5) at (6.4,-1.4); (5) node\[below\][$\mathbb{N}^2$]{} to \[bend right=10\] (4); (6) at (4,-1.4); (6) node\[below\][$\mathbb{N}^3$]{} to \[bend right=0\] (1); (7) at (5.6,-1.4); (7) node\[below\][$\mathbb{N}^2$]{} to \[bend right=10\] (1a); (8) at (4.25,-0.8); (9) at (4.8,-1.46); (9) node\[below\][$\mathbb{N}$]{} to \[bend left=20\] (8); (-1,6) – (-1,3.5) – (1.5,3.5) – (1.5,6) – (-1,6); plot (,[sqrt((2\*)\^3+1)/2+4.75]{}); plot (,[-sqrt((2\*)\^3+1)/2+4.75]{}); (-0.75,5.25) – (1,5.25); (4) at (0,4.25); (5) at (0.5,4.5); (5) node\[right\][$\mathbb{N}$]{} to \[bend right=10\] (4); (4,6) – (4,3.5) – (6.5,3.5) – (6.5,6) – (4,6); (4.35,4.2) node\[fill,circle,inner sep=0.8pt\] – (4.35,5.75); (4.5,3.85) – (6.05,3.85) node\[fill,circle,inner sep=0.8pt\]; (4.6,5.65) node\[fill,circle,inner sep=0.8pt\] – (6.25,4); (1) at (4.35,4.2); (2) at (6.05,3.85); (3) at (4.6,5.65); (1a) at (4.75,3.75); (2a) at (6.25,4.25); (3a) at (4.25,5.5); (1) to \[bend left=20\] (1a); (2) to \[bend left=20\] (2a); (3) to \[bend left=20\] (3a); (4) at (4.65,5.625); (5) at (5,5.7); (5) node\[right\][$\mathbb{N}^2$]{} to \[bend left=10\] (4); (6) at (5.15,5.1); (7) at (5.8,5.3); (7) node\[right\][$\mathbb{N}$]{} to \[bend right=20\] (6); (8) at (6.125,4.1); (9) at (6.2,4.5); (9) node\[above=-3pt\][$\mathbb{N}$]{} to \[bend right=10\] (8); (10) at (6.1,4); (11) at (5.7,4.2); (11) node\[left=-3pt\][$\{0\}$]{} to \[bend left=10\] (10);
\[defi:G\] Let $\Gamma_{D_t^0}$ be the dual intersection graph of $D_t^0$. This is a cycle with $r$ vertices. Let $\mathfrak{G}_d$ be the set of graph morphisms $g : \Gamma \rightarrow \Gamma_{X_t^0}$ where $\Gamma$ is a tree (genus $0$ graph) with vertices $V$ decorated $d_V\in\mathbb{N}_{>0}$ such that
if two vertices $V$ and $V'$ are connected by an edge, they are mapped to different vertices of $\Gamma_{X_t^0}$
$\sum_V d_V = d$
For $g : \Gamma \rightarrow \Gamma_{X_t^0}$ in $\mathfrak{G}_d$, the graph $\Gamma$ consists of a single path.
Hence, there is an identification $$\mathfrak{G}_d \simeq \left\{[i;d_1,\ldots,d_k] \ \big| \ i,k,d_i \in \mathbb{N}_{>0}, \sum_{i=1}^k d_i = d \right\}$$ as follows. Label $\Gamma_{D_t^0}$ cyclically by $1,\ldots,r$ and label the vertices of $\Gamma$ by $V_1,\ldots,V_k$ such that for all $j=1,\ldots,k-1$ the vertex $V_j$ is connected to $V_{j+1}$ by an edge and such that $l_{g(V_j)}=l_{g(V_{j+1})}-1 \text{ mod } r$, where $l_v$ is the label of a vertex $v$ of $\Gamma_{X_t^0}$. Then define $i:=l_{g(V_1)}$ and $d_j:=d_{V_j}$.
\[defi:HG\] Note that projection to the unique unbounded direction defines a map $B \rightarrow \Gamma_{X_t^0}$, where vertices of $\Gamma_{X_t^0}$ correspond to unbounded edges of $B$. Define a surjective map $$\mathfrak{H}_d \rightarrow \mathfrak{G}_d$$ by composing $h : \Gamma \rightarrow B$ with this projection and collapsing vertices connected by an edge an mapped to the same vertex of $\Gamma_{X_t^0}$. The label at a vertices of the image is given by the sum of degrees of classes attached to its preimages.
Let $f : \mathfrak{C} \rightarrow \mathfrak{X}_Q$ be a stable log map with tropicalization $\tilde{h}$ mapping to $h\in\mathfrak{H}_d$ under the map from Lemma \[lem:tilde\]. The limit of $f$ with respect to the family $\mathfrak{X}_t^0\rightarrow\mathbb{A}^1$ has dual graph that, after collapsing vertices that are connected by an edge and mapped to the same vertex, is given by the image of $h$ under the map from Definition \[defi:HG\].
Let $f : \mathfrak{C} \rightarrow \mathfrak{X}_Q$ be a stable log map with tropicalization $\tilde{h}$ mapping to $h : \Gamma \rightarrow B$. Consider the fiber $f_0^0 : C_0^0 \rightarrow X_0^0$. If a vertex $V$ of $\Gamma$ is mapped to a vertex $v$ of $\mathscr{P}$ or the unbounded edge adjacent to $v$, then the corresponding irreducible component $C_V$ of $C_0^0$ is mapped to the irreducible component $X_v$ of $X_0^0$ corresponding to $v$. But then, for $t\neq 0$, the corresponding irreducible component $C_V$ of $C_t^0$ is mapped to the irreducible component of $D_t^0$ corresponding to $v$. This is precisely the image of $V$ under the map from Definition \[defi:HG\].
\(1) at (0,0); (2) at (0,1); (3) at (-3,-1); (4) at (-3,2); (3) – (1) – (2) – (4); (1) – (5,0); (2) – (5,1); (3) – (5,-1); (4) – (5,2); (-1.5,-0.25) – (0,0.5) – (-1.5,1.25); (-3,-0.8) – (-1.5,-0.5) – (-0.75,0.25); (-3,1.8) – (-1.5,1.5) – (-0.75,0.75); (-1.5,-0.25) – (0,0.5) – (-1.5,1.25) – (-1.5,-0.25) – cycle; (-3,1.8) – (-1.5,1.5) – (-0.75,0.75) – (-0.75,0.25) – (-1.5,-0.5) – (-3,-0.8); (5) at (0,0.5); (6) at (-1.5,-0.5); (7) at (-1.5,1.5); (0,0) – (-1,0); (0,1) – (-1,1); (0,0) circle (1.5pt); (0,1) circle (1.5pt); (2,0.66) circle (1.5pt); (-1.5,1.5) – (-0.75,1.25) node\[above\][$2$]{} – (0,1) – (2,0.66) – (0,0) – (-1.5,-0.5); (2,0.66) – (3.5,0.66) node\[below\][$9$]{} – (5,0.66); (0,1) – (0,0.5); (6,0.5) – (7,0.5); (8,-0.9) – (8,2); (8,0) node\[right\][$1$]{} – (8,1) node\[right\][$2$]{}; (a) at (8,-1); (b) at (8,0); (c) at (8,1); (d) at (8,2);
\[defi:GH\] As above fix a cyclic ordering of the irreducible components of $D_t^0=D_1+\ldots+D_r$. Define a surjective map $$\mathfrak{G}_d \rightarrow H_2^+(X,\mathbb{Z})$$ by sending $[i;d_0,\ldots,d_k]$ to the curve class $\underline{\beta}$ defined by $$D_{i+j} \cdot \underline{\beta} = \sum_{\substack{\lambda\in\mathbb{N}_{>0} \\ \lambda r + j \leq k}} d_{\lambda r + j}.$$ This is well-defined by the balancing condition and since $H_2^+(X,\mathbb{Z})\simeq H_2^+(X^0,\mathbb{Z})$, where $X^0$ is a toric variety.
Let $\mathfrak{H}_\beta$ be the set of tropical curves mapping to $\underline{\beta}$ under the decomposition of the maps from Definitions \[defi:HG\] and \[defi:GH\]. Let $\tilde{\mathfrak{H}}_\beta$ be the preimage of $\mathfrak{H}_\beta$ under the map from Lemma \[lem:tilde\].
The tropicalization of a stable log map of class $\beta$ is in $\tilde{\mathfrak{H}}_\beta$.
Refinement and logarithmic modification {#S:refinement}
---------------------------------------
To apply the degeneration formula in §\[S:degformula\] we need a degeneration of $(X,D)$ such that all stable log maps to the central fiber are torically transverse. We achieve this as follows.
\[con:deg3\] Let $\mathscr{P}_d$ be a refinement of $\mathscr{P}$ such that each tropical curve in $\mathfrak{H}_{\leq d}=\cup_{d'\leq d}\mathfrak{H}_{d'}$ (or equivalently in $\tilde{\mathfrak{H}}_{\leq d}$) is contained in the $1$-skeleton of $\mathscr{P}_d$. This defines a refinement of the generalized cone complex $\Sigma(\tilde{X}_0)$ by taking cones over cells of $\mathscr{P}_d$. In turn, it induces a logarithmic modification $\tilde{\mathfrak{X}}_d\rightarrow\mathbb{A}^1$ of $\tilde{\mathfrak{X}}\rightarrow\mathbb{A}^1$ (see §\[A:artin\]) without changing the generic fiber. By making a base change $\mathbb{A}^1 \rightarrow \mathbb{A}^1, t \mapsto t^e$ we can scale $\mathscr{P}_d$ and thus assume it has integral vertices (c.f. [@NS], Proposition 6.3).
\[rem:transverse\] The dual intersection complex of the central fiber $Y$ of $\tilde{\mathfrak{X}}_d$ is given by $(\tilde{B},\mathscr{P}_d,\varphi)$. Hence, all stable log maps to $Y\rightarrow\textup{pt}_{\mathbb{N}}$ contributing to $N_d$ are torically transverse, since tropicalizations of such maps are contained in the $1$-skeleton of $\mathscr{P}_d$, with vertices mapping to vertices of $\mathscr{P}_d$.
It was shown in [@AW] that Gromov-Witten invariants are invariant under logarithmic modifications. Hence, $$N_\beta = \int_{\llbracket\mathscr{M}(Y,\beta)\rrbracket}1.$$ In the next section we will apply the degeneration formula of logarithmic Gromov-Witten theory to get a formula for $N_d$ in terms of logarithmic Gromov-Witten invariants of irreducible components of $Y$.
The degeneration formula {#S:degformula}
========================
Consider a semi-stable degeneration $\pi : \mathfrak{X} \rightarrow T = \textup{Spec }R$, for $R$ a discrete valuation ring. This is a projective surjection such that the generic fiber is smooth and the fiber $X=\pi^{-1}(0)$ over the closed point $0\in T$ is simple normal crossings with two smooth connected (hence irreducible) components $X_1,X_2$ meeting in a smooth connected divisor $D$. In the logarithmic language this means that $\pi$ is log smooth when $T$ and $\mathfrak{X}$ carry the divisorial log structures given by $0\in T$ and $X\subseteq\mathfrak{X}$, respectively. The degeneration formula relates invariants (relative or logarithmic Gromov-Witten invariants) on the generic fiber of $\pi : \mathfrak{X} \rightarrow T$ to invariants on the components $X_1,X_2$ of the special fiber.
The degeneration formula was proven for stable relative maps, in symplectic geometry [@LiRu][@IoPa] and in algebraic geometry [@Li2][@AF], as well as for stable log maps using expanded degenerations [@Che2]. A pure log-geometric version avoiding the target expansions of relative Gromov-Witten theory was worked out by Kim, Lho and Ruddat in [@KLR], using the logarithmic Gromov-Witten theory of [@LogGW].
While the formula in [@KLR] is presented for the setup above, it easily generalizes to the case where $X_1,X_2$ are quasi-projective and log smooth rather than projective and smooth. In particular, $X_1,X_2$ might be reducible. Applying the formula repeatedly, we get a degeneration formula for several log smooth components $X_1,\ldots,X_n$ intersecting in smooth connected divisors such that $X_i\cap X_j\cap X_k=\emptyset$ for all $i\neq j\neq k\in\{1,\ldots,n\}$ and with dual intersection graph a tree.
Fix an integer $d>0$ and let $\tilde{\mathfrak{X}}_d\rightarrow\mathbb{A}^1$ be the refined log smooth degeneration of $(X,D)$ from Construction \[con:deg3\]. Write the central fiber as $Y$ and let $\beta$ be a class of stable log maps to $Y$ contributing to $N_d$ (Definition \[defi:Nd\]). Let $Y^\circ$ be the complement of zero-dimensional toric strata in $Y$ and write $$\mathscr{M}_\beta := \mathscr{M}(Y,\beta).$$
Since $\mathscr{M}(Y^\circ,\beta)$ is canonically isomorphic to the moduli space of torically transverse stable log maps to $Y$ of class $\beta$ and all such maps are torically transverse (Remark \[rem:transverse\]), we have $$\mathscr{M}_\beta \cong \mathscr{M}(Y^\circ,\beta).$$ For a vertex $v\in\mathscr{P}_d^{[0]}$, let $Y_v^\circ$ be the complement of the $0$-dimensional toric strata of the irreducible component $Y_v$ of $Y$ corresponding to $v$. Then $Y^\circ$ is a union of finitely many log smooth schemes $Y_v^\circ$ over $\textup{pt}_{\mathbb{N}}$, with $Y_v^\circ \cap Y_{v'}^\circ = \emptyset$ if there is no edge connecting $v$ and $v'$, and $D_E^\circ:=Y_v^\circ \cap Y_{v'}^\circ$ log smooth and a divisor of both $Y_v^\circ$ and $Y_{v'}^\circ$ if there is an edge $E$ connecting $v$ and $v'$. The intersection of any triple of components is empty. Hence, we can apply the degeneration formula.
Toric invariants {#S:toricinv}
----------------
We introduce logarithmic Gromov-Witten invariants of toric varieties with point conditions on the toric boundary, following [@GPS].
Let $M\simeq\mathbb{Z}^2$ be a lattice and let $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$ be the associated vector space. Let $(m_1,\ldots,m_n)$ be an $n$-tuple of distinct nonzero primitive vectors in $M$ and let $\textbf{w}=(\textbf{w}_1,\ldots,\textbf{w}_n)$ be an $n$-tuple of weight vectors $\textbf{w}_i=(w_{i1},\ldots,w_{il_i})$ with $l_i>0$, $w_{ij}\in \mathbb{N}$ such that $$\sum_{i=1}^n|\textbf{w}_i|m_i = w_{\textup{out}} m_{\textup{out}}$$ for $0\neq m\in M$ primitive and $w_{\textup{out}}> 0$. Here $|\textbf{w}_i| := \sum_{j=1}^{l_i} w_{ij}$. Let $\Sigma$ be the complete rational fan in $M_{\mathbb{R}}$ whose rays are generated by $-m_1,\ldots,-m_n,m_{\textup{out}}$ and let $X_\Sigma$ be the corresponding toric surface over $\mathbb{C}$. By refining $\Sigma$ if necessary, we can assume that $X_\Sigma$ is nonsingular. Let $D_1,\ldots,D_n,D_{\textup{out}}\subseteq X_\Sigma$ be the toric divisors corresponding to the given rays. Let $X_\Sigma^\circ$ be the complement of the $0$-dimensional torus orbits in $X_\Sigma$, and let $D_i^\circ=D_i\cap X_\Sigma^\circ$, $D_{\textup{out}}^\circ = D_{\textup{out}}\cap X_\Sigma^\circ$. Then define a class $\beta_{\textbf{w}}$ of stable log maps to $X_\Sigma$ as follows.
genus $g=0$;
$k=l_1+\ldots+l_n+1$ marked points $p_{ij}, i=1,\ldots,n, j=1,\ldots,l_i$ and $p$;
$\underline{\beta}_{\textbf{w}}\in H_2(X_\Sigma,\mathbb{Z})$ defined by intersection numbers with toric divisors, $$D_i \cdot \underline{\beta}_{\textbf{w}} = |\textbf{w}_i|, \quad D_{\textup{out}}\cdot\underline{\beta}_{\textbf{w}} = w_{\textup{out}};$$
contact data $u_{p_{ij}}=w_{ij}m_i$ and $u_p=w_{\textup{out}}m_{\textup{out}}$.
By restriction we get a class of stable log maps to $X_\Sigma^\circ$ that we also denote by $\beta_{\textbf{w}}$. The moduli space $\mathscr{M}(X_\Sigma^\circ,\beta_{\textbf{w}})$ in general is not proper, since $X_\Sigma^\circ$ is not proper. However, the evaluation map $$\label{eq:evcirc}
\textup{ev}^\circ : \mathscr{M}(X_\Sigma^\circ,\beta_{\textbf{w}}) \rightarrow \prod_{i=1}^n(D_i^\circ)^{l_i}$$ is proper ([@GPS], Proposition 4.2) and we obtain a proper moduli space via base change to a point. To be precise, let $\gamma : \textup{Spec }\mathbb{C} \rightarrow \prod_{i=1}^n(D_i^\circ)^{l_i}$ be a point. Then $$\mathscr{M}_\gamma := \textup{Spec }\mathbb{C} \times_{\prod_{i=1}^n(D_i^\circ)^{l_i}} \mathscr{M}(X_\Sigma^\circ,\beta_{\textbf{w}})$$ is a proper Deligne-Mumford stack admitting a virtual fundamental class, and we can define the logarithmic Gromov-Witten invariant $$\label{eq:Ntor}
N_{\textbf{m}}(\textbf{w}) := \int_{\mathscr{M}_\gamma}\gamma^!\llbracket\mathscr{M}(X_\Sigma^\circ,\beta_{\textbf{w}})\rrbracket.$$ Since the codimension of the inclusion $\gamma$ equals the virtual dimension of $\mathscr{M}(X_\Sigma^\circ,\beta_{\textbf{w}})$, this definition makes sense.
Note that we may add further primitive vectors $m_i$ to $\textbf{m}$, with weight vectors $\textbf{w}_i=0$. This leads to a subdivision of $\Sigma$, hence to a toric blow up of $X_\Sigma$, but the logarithmic Gromov-Witten invariants do not change.
The decomposition formula {#S:decomposition}
-------------------------
By the decomposition formula for stable log maps ([@ACGS1], Theorem 1.2), the moduli space $\mathscr{M}_\beta$ decomposes into moduli spaces indexed by certain decorated tropical curves. Here decorated means that there are classes of stable log maps $\beta_V$ attached to the vertices. In this section we show that a tropical curve in $\tilde{\mathfrak{H}}_d$ automatically carries such decorations.
\[prop:decorations\] Let $f : C/\textup{pt}_{\mathbb{N}} \rightarrow Y/\textup{pt}_{\mathbb{N}}$ be a stable log map in $\mathscr{M}_\beta$ with tropicalization $\tilde{h} : \tilde{\Gamma} \rightarrow \tilde{B}$. For each vertex $V\in\tilde{\mathscr{P}}_d^{[0]}$, the class $[C_V]\in H_2^+(Y_{\tilde{h}(V)},\mathbb{Z})$ is uniquely determined by the intersection numbers of $C_V$ with components of $\partial Y_{\tilde{h}(V)}$, i.e., by $\tilde{h}$.
If $V$ is of type (I) as in Definition \[defi:types\], then $Y_{\tilde{h}(V)}$ is a toric variety, so the statement is true. If $V$ is of type (II), then $C_V$ is a multiple cover of some exceptional line $L^{\textup{exc}}_v$ (Definition \[defi:Lexc\]). Its intersection with $\partial Y_v$ determines the degree $d$ of the multiple cover, hence the curve class $[C_V] = d[L^{\textup{exc}}_v]\in H_2^+(Y_v,\mathbb{Z})$. Let V be a vertex of type (III). It is mapped to a vertex $v$ of $\mathscr{P}$. Let $X_v$ be the corresponding component of $X_0$. This is a toric variety. By Proposition \[prop:balancing\], (III), we know the intersection of the image of $C_V$ under the resolution $\nu : \tilde{\mathfrak{X}} \rightarrow \mathfrak{X}$ from §\[S:resolution\] with the toric divisors of $X_v$, hence the curve class $[\nu(C_V)]\in H_2^+(X_v,\mathbb{Z})$. But this determines $[C_V] = [\nu(C_V)] - k[L^{\textup{exc}}_v] \in H_2^+(Y_v,\mathbb{Z})$, where $k$ is as in Proposition \[prop:balancing\], (III).
For a tropical curve $\tilde{h} : \tilde{\Gamma} \rightarrow \tilde{B}$ in $\tilde{\mathfrak{H}}_d$, let $\mathscr{M}_{\tilde{h}}$ be the moduli space of stable log maps with tropicalization $\tilde{h}$. This is proper by [@ACGS1], Proposition 4.4.2. Then the decomposition formula of logarithmic Gromov-Witten theory ([@ACGS1], Theorem 1.2) gives:
\[prop:dec\] $$\llbracket\mathscr{M}_\beta\rrbracket = \sum_{\tilde{h}\in\tilde{\mathfrak{H}}_\beta}\frac{l_{\tilde{\Gamma}}}{|\textup{Aut}({\tilde{\Gamma}})|} F_\star\llbracket\mathscr{M}_{\tilde{h}}\rrbracket,$$ where $l_{\tilde{\Gamma}} := \textup{lcm}\{w_E \ | \ E\in E(\tilde{\Gamma})\}$ and $F:\mathscr{M}_{\tilde{h}}\rightarrow\mathscr{M}_\beta$ is the forgetful map.
We apply [@ACGS1], Theorem 1.2, to $\llbracket\mathscr{M}_\beta\rrbracket$. By Proposition \[prop:decorations\] and rigidity of tropical curves in $\tilde{\mathfrak{H}}_\beta$, the summation over decorated rigid tropical curves obtained from [@ACGS1], Theorem 1.2, is just summation over $\tilde{\mathfrak{H}}_\beta$. Moreover, the multiplicity $m_{\tilde{\tau}}$ in that formula is the index of the image of the lattice $\Sigma(\textup{pt}_Q)=\mathbb{N}$ inside the lattice $\Sigma(\textup{pt}_\mathbb{N})=\mathbb{N}$. Since the affine length of the image of an edge $E\in E(\tilde{\Gamma})$ is $\ell_Ew_E$ by Definition \[defi:tropical\], (i), this is precisely $l_{\tilde{\Gamma}} := \textup{lcm}\{w_E \ | \ E\in E(\tilde{\Gamma})\}$.
Contributions of the vertices {#S:contributions}
-----------------------------
Let $\tilde{h} : \tilde{\Gamma} \rightarrow \tilde{B}$ be a tropical curve in $\tilde{\mathfrak{H}}_d$. Define $$\mathscr{M}_V^\circ := \mathscr{M}(X_{\tilde{h}(V)}^\circ,\beta_V),$$ where $X_{\tilde{h}(V)}^\circ$ is the complement of the $0$-dimensional toric strata in $X_{\tilde{h}(V)}^\circ$.
For $V\in V_{II}(\tilde{\Gamma})$ (Definition \[defi:types\]) with adjacent edge $E$ the moduli space $\mathscr{M}_V^\circ$ is proper of virtual dimension zero, since it is isomorphic to the moduli space of $w_E$-fold multiple covers of $\mathbb{P}^1$ totally ramified at a point.
For $V\in V(\tilde{\Gamma})\setminus V_{II}(\tilde{\Gamma})$ we obtain a proper moduli space as follows. Again, $\tilde{\Gamma}$ is a rooted tree with root vertex $V_{\textup{out}}$. There is a natural orientation of the edges of $\tilde{\Gamma}$ by choosing edges to point from a vertex to its parent. For each vertex $V\in V(\tilde{\Gamma})\setminus V_{II}(\tilde{\Gamma})$ there is an evaluation map $$\textup{ev}_{V,-}^\circ : \mathscr{M}_V^\circ \rightarrow \prod_{E\rightarrow V}D_E^\circ,$$ where the product is over all edges of $\tilde{\Gamma}$ adjacent to $V$ and pointing towards $V$.
The evaluation map $\textup{ev}_{V,-}^\circ$ is proper.
For $V\in V_I(\tilde{\Gamma})$ this is [@GPS], Proposition 4.2. For $V\in V_{III}(\tilde{\Gamma})$ it is a combination of the proofs of [@GPS], Propositions 4.2 and 5.1.
Since properness of morphisms is stable under base change, we obtain a proper moduli space by base change to a point $$\gamma_V : \textup{Spec }\mathbb{C} \rightarrow \prod_{E\rightarrow V}D_E^\circ,$$ that is, $$\mathscr{M}_{\gamma_V} := \textup{Spec }\mathbb{C} \times_{\prod_{E\rightarrow V}D_E^\circ} \mathscr{M}_V^\circ$$ is a proper Deligne-Mumford stack.
\[defi:NV\] For a vertex $V$ of $\tilde{\Gamma}$ define $$N_V := \begin{cases} \int_{\llbracket\mathscr{M}_V^\circ\rrbracket}1, & V \in V_{II}(\tilde{\Gamma}); \\ \int_{\mathscr{M}_{\gamma_V}}\gamma_V^!\llbracket\mathscr{M}_V^\circ\rrbracket, & V \in V_I(\tilde{\Gamma})\cup V_{III}(\tilde{\Gamma}). \end{cases}$$ This is a finite number, since for $V \in V_{II}(\tilde{\Gamma})$, the moduli space $\mathscr{M}_V^\circ$ has virtual dimension $0$ and for $V\in V_I(\tilde{\Gamma}) \cup V_{III}(\tilde{\Gamma})$ the codimension of $\gamma$ equals the virtual dimension of $\mathscr{M}_V^\circ$. The properness of $\textup{ev}_{V,-}^\circ$ implies that $N_V$ is independent of $\gamma$.
\[prop:N\] We give $N_V$ for the different types of vertices (Definition \[defi:types\]).
For $V\in V_I(\tilde{\Gamma})$ let $e_1,\ldots,e_n$ be the edges of $\mathscr{P}_d$ adjacent to $\tilde{h}(V)$ and let $m_1,\ldots,m_n$ be the corresponding primitive vectors. Let $\textbf{w}_{i}=(w_{i1},\ldots,w_{il_i})$ be the weights of edges of $\tilde{\Gamma}$ mapping to $e_i$ and write $\textbf{w}=(\textbf{w}_1,\ldots,\textbf{w}_n)$. Then $N_V$ is the toric invariant from $$N_V = N_\textbf{m}(\textbf{w}),$$
If $V\in V_{II}(\tilde{\Gamma})$, then $$N_V = \frac{(-1)^{w_E-1}}{w_E^2},$$ where $E_V$ is the unique edge adjacent to $V$.
If $V\in V_{III}(\tilde{\Gamma})$, then $$N_V = \sum_{\textbf{w}_{V,+}} \frac{N_{\textbf{m}}(\textbf{w})}{|\textup{Aut}(\textbf{w}_{V,+})|} \prod_{i=1}^l\frac{(-1)^{w_{V,i}-1}}{w_{V,i}}.$$ The sum is over all weight vectors $\textbf{w}_{V,+}=(w_{V,1},\ldots,w_{V,l_V})$ such that $|\textbf{w}_{V,+}| := \sum_{i=1}^{l_V} w_{V,i} = k$, with $k$ as in Proposition \[prop:balancing\], (III). Further, $N_{\textbf{m}}(\textbf{w})$ is as in with $\textbf{m}=(m_{v,-},m_{v,+})$ and $\textbf{w}=((w_E)_{E\in E_{V,-}(\tilde{\Gamma})},\textbf{w}_{V,+})$, where $E_{V,-}$ is the set of edges adjacent to $V$ and mapped to direction $m_{v,-}$.
\(I) is by the definition of $N_{\textbf{m}}(\textbf{w})$ in . (II) is the genus $0$ part of [@BP], Theorem 5.1. See also [@GPS], Propositions 5.2 and 6.1. For (III) we apply [@GPS], Proposition 5.3. We only blow up one point on the divisor $D_{v,+}$, so we have $\textbf{P}=(P_+)$ with $P_+ = k$ in the notation of [@GPS], Proposition 5.3, with $k$ as in Proposition \[prop:balancing\], (III). Note that our $i$ is called $j$ in [@GPS] and the $i$ of [@GPS] is equal to $1$ here. Further, $R_{\textbf{P}_+|\textbf{w}_+} = \prod_{i=1}^{l_i}\frac{(-1)^{w_i-1}}{w_i^2}$ by [@GPS], Proposition 5.2 and the discussion thereafter. Then [@GPS], Proposition 5.3, gives (III).
Gluing
------
Define $\bigtimes_{V\in V(\tilde{\Gamma})}\mathscr{M}_V^\circ$ to be the moduli space of stable log maps in $\prod_V\mathscr{M}_V^\circ$ matching over the divisors $D_E^\circ$, $E\in E(\tilde{\Gamma})$, i.e., the fiber product $$\begin{xy}
\xymatrix{
\displaystyle\bigtimes_V\mathscr{M}_V^\circ \ar[rr]\ar[d] && \displaystyle\prod_V\mathscr{M}_V^\circ \ar[d]^{\textup{ev}^\circ} \\
\displaystyle\prod_{E\in E(\tilde{\Gamma})} D_E^\circ \ar[rr]^\delta && \displaystyle\prod_V\prod_{\substack{E\in E(\tilde{\Gamma}) \\ V \in E}} D_E^\circ
}
\end{xy}$$ Here $\textup{ev}^\circ$ is the product of evaluation maps to common divisors (labeled by compact edges) and $\delta$ is the diagonal map.
There is an étale morphism $\textup{cut} : \mathscr{M}_{\tilde{h}} \rightarrow \bigtimes_V\mathscr{M}_V^\circ$ by cutting a curve along its gluing nodes. Its degree is $$\textup{deg}(\textup{cut}) = \frac{\prod_{E\in E(\tilde{\Gamma})}w_E}{l_{\tilde{\Gamma}}},$$ where $l_{\tilde{\Gamma}} = \textup{lcm}\{w_E\}$.
This is [@KLR], Lemma 9.2, (4).
\[prop:gluing\] We have $$\llbracket\mathscr{M}_{\tilde{h}}\rrbracket = \textup{cut}^\star\delta^!\prod_{V\in V(\tilde{\Gamma})}\llbracket\mathscr{M}_V^\circ\rrbracket.$$
This is a combination of [@KLR], equations (9.11), (9.13) and (9.14). See also [@Bou1], Proposition 20.
\[prop:pieces\] We have $$\int_{\delta^!\prod_{V}\llbracket\mathscr{M}_V^\circ\rrbracket} 1 = \prod_{V\in V(\tilde{\Gamma})}N_V.$$
This is similar to the proof of [@Bou1], Proposition 22. By definition of $\delta$ we have $$\int_{\delta^!\prod_{V}\llbracket\mathscr{M}_V^\circ\rrbracket} 1 = \int_{\prod_{V}\llbracket\mathscr{M}_V^\circ\rrbracket} (\textup{ev}^\circ)^\star[\delta],$$ where $$[\delta] = \prod_{E\in E(\tilde{\Gamma})}(\textup{pt}_E \times 1 + 1 \times \textup{pt}_E)$$ is the class of the diagonal $\prod_E D_E^\circ$. As before we give $\tilde{\Gamma}$ the structure of a rooted tree by choosing the root vertex to be the vertex $V_{\textup{out}}$ of the unique unbounded leg $E_{\textup{out}}$. For a bounded edge $E$ let $V_{E,+}$ and $V_{E,-}$ be the vertices of $E$ such that $V_{E,+}$ is the parent of $V_{E,-}$.
We will show by dimensional arguments that the only term of $$(\textup{ev}^\circ)^\star[\delta] = \prod_{E\in E(\tilde{\Gamma})} \left((\textup{ev}_{V_{E,-}})^\star[\textup{pt}_E] + (\textup{ev}_{V_{E,+}})^\star[\textup{pt}_E]\right)$$ giving a nonzero contribution after integration over $\prod_{V}\llbracket\mathscr{M}_V^\circ\rrbracket$ is $\prod(\textup{ev}_{V_{E,+}})^\star[\textup{pt}_E]$. In other words:
For each compact edge $E$, a term of $(\textup{ev}^\circ)^\star[\delta]$ giving a nonzero contribution after integration over $\prod_{V}\llbracket\mathscr{M}_V^\circ\rrbracket$ does not contain a factor $(\textup{ev}_{V_{E,-}})^\star[\textup{pt}_E]$.
Let $E$ be a compact edge with $V_{E,-}$ a vertex of type (II) as in Definition \[defi:types\]. By Proposition \[prop:N\], (II), the virtual dimension of $\mathscr{M}_{V_{E,-}}$ is zero. Hence, $(\textup{ev}_{V_{E,-}})^\star[\textup{pt}_E]=0$, since its insertion over $\mathscr{M}_{V_{E,-}}$ defines an enumerative problem of virtual dimension $-1$.
Now consider a compact edge $E$ with $V_{E,-}$ of type (III). Let $E_i, i\in I$ be the edges adjacent to $V_{E,-}$ and different from $E$ (possibly $I=\emptyset$). By Proposition \[prop:balancing\], (III), the edges $E_i$ connect $V_{E,-}$ with a vertex $V_i$ of type (II). The terms in $(\textup{ev}^\circ)^\star[\delta]$ containing a factor $(\textup{ev}_{V_{E_i,-}})^\star[\textup{pt}_{E_i}]$ give zero after integration over $\llbracket\mathscr{M}_{V_i}\rrbracket$ by the dimensional argument above. Hence, to give a nonzero contribution, a term of $(\textup{ev}^\circ)^\star[\delta]$ must contain the factor $\prod_{i\in I}(\textup{ev}_{V_{E_i,+}})^\star[\textup{pt}_{E_i}]$. By Proposition \[prop:N\], (III), the virtual dimension of $\mathscr{M}_V$ is $|I|$, so the insertion of $\prod_{i\in I}(\textup{ev}_{V_{E_i,+}})^\star[\textup{pt}_{E_i}]$ in $\mathscr{M}_V$ defines an enumerative problem of virtual dimension $0$. Any further insertion would reduce the virtual dimension to $-1$, so a term of $(\textup{ev}^\circ)^\star[\delta]$ giving a nonzero contribution does not contain the factor $(\textup{ev}_{V_{E,-}})^\star[\textup{pt}_E]$.
We will show the claim for compact edges $E$ with $V_{E,-}$ a vertex of type (I) by induction on the height of $V_{E,-}$, that is, the maximal length of chains connecting $V_{E,-}$ with a leaf of $\tilde{\Gamma}$. By Proposition \[prop:balancing\], (I), a vertex of type (I) fulfills the ordinary balancing condition. In particular, it must have more than one adjacent edge, hence cannot be a leaf. This shows the set of leaves of $\tilde{\Gamma}$ is contained in $V_{II}(\tilde{\Gamma})\cup V_{III}(\tilde{\Gamma})$. Thus we have already shown that the claim is true for compact edges $E$ with $V_{E,-}$ of height $0$. This is the base case. For the induction step assume that the claim is true for compact edges $E$ with $V_{E,-}$ of height $\leq k$ for some $k\in\mathbb{N}$ and consider a compact edge $E$ with $V_{E,-}$ of height $k+1$. Assume that $V_{E,+}$ is of type (I), since otherwise the claim is true by the above arguments. Let $E_i,i\in I$ be the edges connecting $V_{E,-}$ with its childs. By Proposition \[prop:N\], (I), the virtual dimension of $\mathscr{M}_{V_{E,-}}$ is $|I|$. By the induction hypothesis, a term of $(\textup{ev}^\circ)^\star[\delta]$ giving a nonzero contribution must contain the factor $\prod_{i\in I}(\textup{ev}_{V_{E_i,+}})^\star[\textup{pt}_{E_i}]$. Inserting this over $\mathscr{M}_{V_{E,-}}$ gives an enumerative problem of virtual dimension $0$. Again, for dimensional reasons, a term of $(\textup{ev}^\circ)^\star[\delta]$ giving a nonzero contribution cannot contain the factor $(\textup{ev}_{V_{E,-}})^\star[\textup{pt}_E]$, hence it must contain the factor $(\textup{ev}_{V_{E,+}})^\star[\textup{pt}_E]$. This proves the claim. Now $$\int_{\prod_{V\in V(\tilde{\Gamma})}\llbracket\mathscr{M}_V^\circ\rrbracket} \prod_{E\in E(\tilde{\Gamma})}(\textup{ev}_{V_{E,+}})^\star[\textup{pt}_E] = \prod_{V\in V(\tilde{\Gamma})} \int_{\llbracket\mathscr{M}_V\rrbracket}(\textup{ev}_{E\rightarrow V})^\star [\textup{pt}] = \prod_{V\in V(\tilde{\Gamma})}N_V,$$ completing the proof.
The degeneration formula {#the-degeneration-formula}
------------------------
Combining the decomposition formula and the gluing formula, we obtain the *degeneration formula*, expressing $N_\beta$ in terms of logarithmic Gromov-Witten invariants $N_V$ labeled by vertices of tropical curves.
\[prop:deg\] $$N_\beta = \sum_{\tilde{h}\in\tilde{\mathfrak{H}}_\beta} \frac{1}{|\textup{Aut}(\tilde{\Gamma})|} \cdot \prod_{E\in E(\tilde{\Gamma})}w_E \cdot \prod_{V\in V(\tilde{\Gamma})} N_V.$$
Combining Propositions \[prop:dec\], \[prop:gluing\] and \[prop:pieces\] we get $$\label{eq:deg}
\llbracket\mathscr{M}_\beta\rrbracket = \sum_{\tilde{h}\in\tilde{\mathfrak{H}}_\beta} \frac{l_{\tilde{\Gamma}}}{|\textup{Aut}(\tilde{\Gamma})|}F_\star\textup{cut}^\star\delta^!\prod_{V\in V(\tilde{\Gamma})}\llbracket\mathscr{M}_V^\circ\rrbracket.$$ Since the virtual dimension of $\mathscr{M}_d$ is zero, integration (i.e., proper pushforward to a point) gives a numerical version of the degeneration formula, $$N_\beta = \sum_{\tilde{h}\in\tilde{\mathfrak{H}}_\beta} \frac{1}{|\textup{Aut}(\tilde{\Gamma})|} \cdot \prod_{E\in E(\tilde{\Gamma})}w_E \cdot \int_{\delta^!\prod_{V}\llbracket\mathscr{M}_V^\circ\rrbracket} 1,$$ Here we used that $\textup{cut}_\star\textup{cut}^\star$ is multiplication by $\textup{deg}(\textup{cut})=\frac{\prod w_E}{l_{\tilde{\Gamma}}}$. By Proposition \[prop:pieces\] we get the above formula.
As mentioned earlier, summation over balanced tropical curves in $\mathfrak{H}_\beta$ will give a more symmetric version of the above formula:
\[defi:Ntor\] Let $h : \Gamma \rightarrow B$ be a tropical curve in $\mathfrak{H}_\beta$ and let $V$ be a vertex of $\Gamma$. Then the image of $V$ under the map from Lemma \[lem:tilde\] is a vertex of $\tilde{\Gamma}$ of type (I) or (III). Let $\textbf{m}$ and $\textbf{w}$ be as in the respective case of Proposition \[prop:N\] and define $$N_V^{\textup{tor}} := N_{\textbf{m}}(\textbf{w}).$$ Note that $N_V^{\textup{tor}}=N_V$ for vertices of type (I).
\[defi:Nh\] For a tropical curve $h : \Gamma \rightarrow B$ in $\mathfrak{H}_d$ for some $d$ define $$N_h := \left(\frac{1}{|\textup{Aut}(\Gamma)|} \cdot \prod_{E\in E(\Gamma)}w_E\cdot \prod_{E\in L_\Delta(\Gamma)}\frac{(-1)^{w_E-1}}{w_E}\cdot\prod_{V\in V(\Gamma)} N_V^{\textup{tor}}\right),$$ where $L_\Delta(\Gamma)$ is the set of bounded legs (see Definition \[defi:tropical\]).
\[thm:degmax\] $$N_\beta = \sum_{h\in\mathfrak{H}_\beta} N_h.$$
Using Propositions \[prop:deg\] and \[prop:N\], we have $$\begin{aligned}
N_d &=& \sum_{\tilde{h}\in\tilde{\mathfrak{H}}_\beta}\left( \frac{1}{|\textup{Aut}(\tilde{\Gamma})|} \cdot \prod_{E\in E(\tilde{\Gamma})} w_E \cdot \prod_{V\in V_I(\tilde{\Gamma})}N_V^{\textup{tor}} \cdot \prod_{V\in V_{II}(\tilde{\Gamma})}\frac{(-1)^{w_{E_V}-1}}{w_{E_V}^2}\right. \\
&& \textup{ } \hspace{3cm} \left.\cdot \prod_{V\in V_{III}(\tilde{\Gamma})}\left(\sum_{\textbf{w}_{V,+}} N_V^{\textup{tor}} \frac{1}{|\textup{Aut}(\textbf{w}_{V,+})|}\prod_{i=1}^{l_V}\frac{(-1)^{w_{V,i}-1}}{w_{V,i}}\right)\right).\end{aligned}$$ Canceling the $w_{E_V}$ for vertices of type (II) in the first product against the ones in the denominator of the third product and factoring out the second sum we get
$$\begin{aligned}
N_d &=& \sum_{\tilde{h}\in\tilde{\mathfrak{H}}_\beta}\sum_{(\textbf{w}_{V,+})_{V\in V_{III}(\tilde{\Gamma})}} \left(\frac{1}{|\textup{Aut}(\tilde{\Gamma})||\textup{Aut}(\textbf{w}_{V,+})|} \cdot \prod_{E\in E(\tilde{\Gamma})\setminus\cup_{V\in V_{II}(\tilde{\Gamma})}\{E_V\}}w_E \right. \\
&& \textup{ } \hspace{0cm} \left. \cdot \prod_{V\in V_I(\tilde{\Gamma})\cup V_{III}(\tilde{\Gamma})} N_V^{\textup{tor}} \cdot \prod_{V\in V_{II}(\tilde{\Gamma})}\frac{(-1)^{w_{E_V}-1}}{w_{E_V}}\cdot \prod_{V\in V_{III}(\tilde{\Gamma})}\prod_{i=1}^{l_V}\frac{(-1)^{w_{V,i}-1}}{w_{V,i}}\right).\end{aligned}$$
Now by the construction of the map $\mathfrak{H}_d \rightarrow \tilde{\mathfrak{H}}_d$ in the proof of Lemma \[lem:tilde\], the two summations can be replaced by a summation over $\mathfrak{H}_d$. Note that for $\tilde{h}\in\tilde{\mathfrak{H}}_d$ we have $$\sum_{h\mapsto\tilde{h}}\frac{1}{|\textup{Aut}(\Gamma)|} = \frac{1}{|\textup{Aut}(\tilde{\Gamma})|}\sum_{(\textbf{w}_{V,+})_{V\in V_{III}(\tilde{\Gamma})}} \frac{1}{|\textup{Aut}(\textbf{w}_{V,+})|},$$ where the sum is over all $h\in\mathfrak{H}_d$ giving $\tilde{h}$ via the map from Lemma \[lem:tilde\]. Moreover, note that $V(\Gamma)=V_I(\tilde{\Gamma})\cup V_{III}(\tilde{\Gamma})$ and $E(\Gamma)=E(\tilde{\Gamma})\setminus\cup_{V\in V_{II}(\tilde{\Gamma})}\{E_V\}$, where, for a vertex $V$ of type (II), $E_V$ is the unique edge containing the vertex $V$. Then $$\begin{aligned}
N_d &=& \sum_{h\in\mathfrak{H}_\beta}\left(\frac{1}{|\textup{Aut}(\Gamma)|} \cdot \prod_{E\in E(\Gamma)} w_E \cdot \prod_{V\in V(\Gamma)} N_V^{\textup{tor}} \right. \\
&& \textup{ } \hspace{2cm} \left. \cdot \prod_{V\in V_{II}(\tilde{\Gamma})}\frac{(-1)^{w_{E_V}-1}}{w_{E_V}}\cdot \prod_{V\in V_{III}(\tilde{\Gamma})}\prod_{i=1}^{l_V}\frac{(-1)^{w_{V,i}-1}}{w_{V,i}} \right).\end{aligned}$$ Using that $$\prod_{E\in L_\Delta(\Gamma)} \frac{(-1)^{w_E-1}}{w_E} = \prod_{V\in V_{II}(\tilde{\Gamma})}\frac{(-1)^{w_{E_V}-1}}{w_{E_V}}\cdot \prod_{V\in V_{III}(\tilde{\Gamma})}\prod_{i=1}^{l_V}\frac{(-1)^{w_{V,i}-1}}{w_{V,i}}$$ completes the proof.
$$N_d = \sum_{h\in\mathfrak{H}_d} N_h.$$
Invariants with prescribed limit under $s\rightarrow 0$
-------------------------------------------------------
For an element $[i;d_1,\ldots,d_k] \in \mathfrak{G}_d$ (Definition \[defi:G\]) define $$N_{[i;d_1,\ldots,d_k]} = \sum_{h\mapsto[i;d_1,\ldots,d_k]} N_h,$$ where the sum is over all tropical curves $h \in \mathfrak{H}_d$ mapping to $[i;d_1,\ldots,d_k]$ under the map from Definition \[defi:HG\].
Barrott and Nabijou [@BN] define invariants indexed by $\mathfrak{G}_d$ by only looking at the family $\mathfrak{X}_t \rightarrow \mathbb{A}^1$ for $t\neq 0$. We conjecture that these coincide with the invariants defined above.
Scattering calculations {#S:scattering}
=======================
In this section we recall the notions of wall structures and scattering diagrams from [@GS11], restricting to the $2$-dimensional case with trivial gluing data. We then explain how the dual intersection complex $(B,\mathscr{P},\varphi)$ of the toric degeneration $(\mathfrak{X},\mathfrak{D})\rightarrow\mathbb{A}^1$ of $(X,D)$ defines a consistent wall structure $\mathscr{S}_\infty$. For toric varieties the correspondence between scattering diagrams (wall structures with one vertex) and logarithmic Gromov-Witten invariants was shown in [@GPS]. We use the relation between scattering diagrams and tropical curves from [@GPS], Theorem 2.8, (see Lemma \[lem:2.8\]) to extend this correspondence to our non-toric case.
Scattering diagrams {#S:scatteringdiag}
-------------------
Let $M\simeq\mathbb{Z}^2$ be a lattice and write $M_{\mathbb{R}}=M\otimes_{\mathbb{Z}}\mathbb{R}$. Let $\Sigma$ be a fan in $M_{\mathbb{R}}$ and let $\varphi$ be an integral strictly convex piecewise affine function on $\Sigma$ with $\varphi(0)=0$. Note that the rays of $\Sigma$ form the corner locus of $\varphi$. Let $P_\varphi$ be the monoid of integral points in the upper convex hull of $\varphi$, $$P_\varphi = \{p=(\overline{p},h)\in M\oplus\mathbb{Z} \ | \ h \geq \varphi(\overline{p})\}.$$ Write $t:=z^{(0,1)}$ and let $R_\varphi$ be the $\mathbb{C}\llbracket t\rrbracket$-algebra obtained by completion of $\mathbb{C}[P_\varphi]$ with respect to $(t)$, $$R_\varphi = \varprojlim\mathbb{C}[P]/(t^k).$$
A *ray* for $\varphi$ is a half-line $\mathfrak{d} = \mathbb{R}_{\geq 0}\cdot m_{\mathfrak{d}} \subseteq M_{\mathbb{R}}$, with $m_{\mathfrak{d}}\in M$ primitive, together with an element $f_{\mathfrak{d}}\in R_\varphi$ such that
each exponent $p=(\overline{p},h)$ in $f_{\mathfrak{d}}$ satisfies $\overline{p}\in\mathfrak{d}$ or $-\overline{p}\in\mathfrak{d}$. In the first case the ray is called *incoming*, in the latter it is called *outgoing*;
if $m_{\mathfrak{d}}$, is not a ray generator of $\Sigma$, then $f_{\mathfrak{d}} \equiv 1 \textup{ mod }(z^{m_{\mathfrak{d}}}t)$;
A *scattering diagram* for $\varphi$ is a set $\mathfrak{D}$ of rays for $\varphi$ such that for every power $k>0$ there are only a finite number of rays $(\mathfrak{d},f_{\mathfrak{d}})\in\mathfrak{D}$ with $f_{\mathfrak{d}} \not\equiv 1 \textup{ mod }(t^k)$.
Note that [@GPS] deals with the case $\varphi\equiv 0$, where the $t$-order of an element $z^{(\overline{p},h)}$ is simply given by $h$. In our case, the $t$-order is $\varphi(-\overline{p})+h\geq 0$: $$z^{(\overline{p},h)} = \left(z^{(-\overline{p},\varphi(-\overline{p}))}\right)^{-1} t^{\varphi(-\overline{p})+h}.$$
Let $\mathfrak{D}$ be a scattering diagram for $\varphi$. Let $\gamma : [0,1] \rightarrow M_{\mathbb{R}}$ be a closed immersion not meeting the origin and with endpoints not contained in any ray of $\mathfrak{D}$. Then, for each power $k>0$, we can find numbers $0<r_1\leq r_2\leq\ldots\leq r_s<1$ and rays $\mathfrak{d}_i=(\mathbb{R}_{\geq 0}m_i,f_i)\in\mathfrak{D}$ with $f_i \not\equiv 1 \textup{ mod }(t^k)$ such that (1) $\gamma(r_i)\in\mathfrak{d}_i$, (2) $\mathfrak{d}_i\neq\mathfrak{d}_j$ if $r_i=r_j$ and $i\neq j$, and (3) $s$ is taken as large as possible.
For each ray $\rho\in\Sigma^{[1]}$ write $f_{\rho}=1+z^{(m_\rho,\varphi(m_\rho))}$ for $m_\rho\in M$ the primitive generator of $\rho$, and define $$\tilde{R}_\varphi^k = \left(R_\varphi/(t^{k+1})\right)_{\prod_{\rho}f_\rho}.$$ For each $i$, define a $\mathbb{C}\llbracket t\rrbracket$-algebra automorphism of $\tilde{R}_\varphi^k$ by $$\theta_{\mathfrak{d}_i}(z^p)^k = f_i^{-\braket{n_i,\overline{p}}}z^p,$$ where $n_i\in N=\text{Hom}(M,\mathbb{Z})$ is the unique primitive vector satisfying $\braket{n_i,m_i} = 0$ and $\braket{n_i,\gamma'(r_i)} > 0$. Define $$\theta_{\gamma,\mathfrak{D}}^k = \theta^k_{\mathfrak{d}_s} \circ \ldots \circ \theta^k_{\mathfrak{d}_1}.$$ If $r_i=r_j$, then $\theta_{\mathfrak{d}_i}$ and $\theta_{\mathfrak{d}_j}$ commute. Hence, $\theta_{\gamma,\mathfrak{D}}^k$ is well-defined. Moreover, define $$\theta_{\gamma,\mathfrak{D}} = \lim_{k\rightarrow\infty}\theta_{\gamma,\mathfrak{D}}^k.$$
A scattering diagram $\mathfrak{D}$ is *consistent to order $k$* if, for all $\gamma$ such that $\theta_{\gamma,\mathfrak{D}}^k$ is defined, $$\theta_{\gamma,\mathfrak{D}}^k \equiv 1 \textup{ mod }(t^{k+1}).$$ It is *consistent to any order*, or simply *consistent*, if $\theta_{\gamma,\mathfrak{D}} = 1$.
For each scattering diagram $\mathfrak{D}$ there exists a consistent scattering diagram $\mathfrak{D}_\infty$ containing $\mathfrak{D}$ such that $\mathfrak{D}_\infty\setminus\mathfrak{D}$ consists only of outgoing rays.
Two scattering diagrams $\mathfrak{D}$, $\mathfrak{D}'$ are *equivalent* if $\theta_{\gamma,\mathfrak{D}}=\theta_{\gamma,\mathfrak{D}'}$ for any closed immersion $\gamma$ for which both sides are defined.
A scattering diagram $\mathfrak{D}$ is called *minimal* if
any two rays $\mathfrak{d},\mathfrak{d}'$ in $\mathfrak{D}$ have distinct support , i.e., $m_{\mathfrak{d}}\neq m_{\mathfrak{d}'}$;
it contains no *trivial* ray, i.e., with $f_{\mathfrak{d}}=1$.
Every scattering diagram $\mathfrak{D}$ is equivalent to a unique minimal scattering diagram. In fact, if $\mathfrak{d},\mathfrak{d}'$ have the same support, then we can replace these two rays with a single ray with the same support and attached function $f_{\mathfrak{d}}\cdot f_{\mathfrak{d}'}$. Moreover, we can remove any trivial ray without affecting $\theta_{\gamma,\mathfrak{D}}$.
Scattering diagrams and toric invariants
----------------------------------------
In [@GPS], Theorem 2.4, a bijective correspondence between certain scattering diagrams and tropical curves is established, leading to an enumerative correspondence ([@GPS], Theorem 2.8). Combining this with the tropical correspondence theorem for torically transverse stable log maps with point conditions on the toric boundary ([@GPS], Theorems 3.4, 4.4), we get the following.
\[lem:2.8\] Let $\textbf{m}=(m_1,\ldots,m_n)$ be an $n$-tuple of distinct primitive vectors of $M$. Let $\Sigma$ be a fan in $M_{\mathbb{R}}$ and let $\varphi$ be an integral strictly convex piecewise affine function on $\Sigma$ such that $\varphi(0)=0$. Let $\mathfrak{D}$ be a scattering diagram for $\varphi$ consisting of a number of lines[^4], one in each direction $m_i$ and with attached function $f_i\in\mathbb{C}[z^{(-m_i,0)}]\subseteq P_\varphi$. Write the logarithm of $f_i$ as $$\textup{log }f_i = \sum_{w=1}^\infty a_{iw}z^{(-wm_i,0)}, \quad a_{iw}\in\mathbb{C}.$$ Let $\mathfrak{D}_\infty$ be the associated minimal consistent scattering diagram and let $\mathfrak{d}\in \mathfrak{D}_\infty\setminus\mathfrak{D}$ be a ray in direction $m_{\mathfrak{d}}$ with attached function $f_{\mathfrak{d}}$. Then $$\textup{log }f_{\mathfrak{d}} = \sum_{w=1}^n\sum_{\textbf{w}}w\frac{N_{\textbf{m}}(\textup{\textbf{w}})}{|\textup{Aut}(\textup{\textbf{w}})|} \left(\prod_{\substack{1\leq i\leq n\\ 1\leq j\leq l_i}} a_{iw_{ij}}\right) z^{(-wm_{\mathfrak{d}},0)},$$ where the sum is over all $n$-tuples of weight vectors $\textbf{w}=(\textbf{w}_1,\ldots,\textbf{w}_n)$ satisfying $$\sum_{i=1}^n |\textbf{w}_i|m_i = wm_{\mathfrak{d}}.$$ Here $N_{\textbf{m}}(\textbf{w})$ is the toric logarithmic Gromov-Witten invariant defined in §\[S:toricinv\].
Moreover, let $m\in\mathbb{Q}_{>0}m_1+\ldots+\mathbb{Q}_{>0}m_n$ be a primitive vector. If there is no ray $\mathfrak{d}\in\mathfrak{D}_\infty$ in direction $m$, then $N_{\textbf{m}}(\textbf{w})=0$ for all $\textbf{w}$ satisfying $\sum_{i=1}^n |\textbf{w}_i|m_i = wm_{\mathfrak{d}}$.
Wall structures
---------------
Let $(B,\mathscr{P},\varphi)$ be a $2$-dimensional polarized polyhedral affine manifold. Note that for each $x\in B\setminus\Delta$ (where $\Delta$ is the singular locus of $B$), $\varphi$ defines an integral strictly convex piecewise affine function $$\varphi_x : \Lambda_{B,x} \simeq M \rightarrow \mathbb{R}$$ on $\Sigma_x$, the fan describing $(B,\mathscr{P})$ locally at $x$. If $\tau\in\mathscr{P}$ is the smallest cell containing $x$, then this is given by $$\Sigma_x = \{K_{\tau_x}\sigma \ | \ \tau\subseteq\sigma\in\mathscr{P}\},$$ where $K_\tau\sigma$ is the cone generated by $\sigma$ relative to $\tau$, i.e., $$K_\tau\sigma = \mathbb{R}_{\geq 0}(\sigma-\tau) = \{m \in M_{\mathbb{R}} \ | \ \exists m_0\in\tau, m_1\in\sigma,\lambda\in\mathbb{R}_{\geq 0} : m=\lambda(m_1-m_0)\}.$$ As in §\[S:scatteringdiag\] this defines a monoid by the integral points in the upper convex hull of $\varphi_x$, $$\label{eq:Px}
P_x := P_{\varphi_x} = \{p=(\overline{p},h)\in\Lambda_{B,x}\oplus\mathbb{Z} \ | \ h\geq\varphi_x(\overline{p})\}.$$ Note that $\textup{Spec }\mathbb{C}[P_x]$ gives a local toric model for the toric degeneration defined by $(B,\mathscr{P},\varphi)$ at a point on the interior of the toric stratum corresponding to $\tau_x$.
\[defi:structure\] Let $(B,\mathscr{P},\varphi)$ be a $2$-dimensional polarized polyhedral affine manifold such that $(B,\mathscr{P})$ is simple. For $x,x'\in B$ integral points, let $m_{xx'}\in\Lambda_B$ denote the primitive vector pointing from $x$ to $x'$. For a $1$-cell $\rho$ and $x\in\rho\setminus\Delta$ let $v[x]$ be the unique[^5] vertex of the irreducible component of $\rho\setminus\Delta$ containing $x$.
A *slab* $\mathfrak{b}$ on $(B,\mathscr{P},\varphi)$ is a $1$-dimensional rational polyhedral subset of a $1$-cell $\rho_{\mathfrak{b}}\in\mathscr{P}^{[1]}$ together with elements $f_{\mathfrak{b},x}\in\mathbb{C}[P_x]$, one for each $x\in\mathfrak{b}\setminus\Delta$, satisfying the following conditions.
$f_{\mathfrak{b},x} \equiv 1 \textup{ mod } (t)$ if $\rho_{\mathfrak{b}}$ does not contain an affine singularity;
$f_{\mathfrak{b},x} \equiv 1+z^{(m_{v[x]\delta},\varphi(m_{v[x]\delta}))} \textup{ mod } (t)$ if $\rho_{\mathfrak{b}}$ contains an affine singularity $\delta$;
$f_{\mathfrak{b},x} = z^{(m_{v[x]v[x']},\varphi(m_{v[x]v[x']}))}f_{\mathfrak{b},x'}$ for all $x,x'\in\mathfrak{b}\setminus\Delta$.
Note that conditions (1) and (2) are compatible with (3).
A *wall* $\mathfrak{p}$ on $(B,\mathscr{P},\varphi)$ is a $1$-dimensional rational polyhedral subset of a maximal cell $\sigma_{\mathfrak{p}}\in\mathscr{P}^{[2]}$ with $\mathfrak{p}\cap\textup{Int}(\sigma_{\mathfrak{p}})$ together (i) a *base point* ($\mathfrak{p})\in\mathfrak{p}\setminus\partial\mathfrak{p}$, (ii) an *exponent* $p_{\mathfrak{p}} \in \Gamma(\sigma_{\mathfrak{p}},\Lambda \oplus \underline{\mathbb{Z}})$ such that $p_{\mathfrak{p},x}=(\overline{p}_{\mathfrak{p},x},h_{\mathfrak{p},x})\in P_x$ for all $\mathfrak{p}\setminus\Delta$ with $h_{\mathfrak{p},x}>\varphi(\overline{p}_{\mathfrak{p},x})$ for $x\neq\textup{Base}(\mathfrak{p})$, and (iii) $c_{\mathfrak{p}}\in\mathbb{C}$, such that $$\mathfrak{p} = (\textup{Base}(\mathfrak{p}) - \mathbb{R}_{\geq 0}\overline{p}_{\mathfrak{p}}) \cap \sigma_{\mathfrak{p}}.$$ For each $x\in\mathfrak{p}\setminus\Delta$ this defines a function $$f_{\mathfrak{p},x} = 1+c_{\mathfrak{p}}z^{p_{\mathfrak{p}}} \in \mathbb{C}[P_x].$$
A *wall structure* $\mathscr{S}$ on $(B,\mathscr{P},\varphi)$ is a locally finite set of slabs and walls with a polyhedral decomposition $\mathscr{P}_{\mathscr{S}}$ of its support $|\mathscr{S}|=\cup_{\mathfrak{b}\in\mathscr{S}}\mathfrak{b}$ such that
The map mapping a slab $\mathfrak{b}\in\mathscr{S}$ to its underlying $1$-cell of $\mathscr{P}$ is injective;
Each closure of a connected component of $B\setminus|\mathscr{S}|$ (*chamber*) is convex and its interior is disjoint from any wall;
Any wall in $\mathscr{S}$ is a union of elements of $\mathscr{P}_{\mathscr{S}}$;
Any maximal cell of $\mathscr{P}$ contains only finitely many slabs or walls in $\mathscr{S}$.
A *joint* $\mathfrak{j}$ of a wall structure $\mathscr{S}$ on $(B,\mathscr{P},\varphi)$ is a vertex of $\mathscr{P}_{\mathscr{S}}$. At each joint $\mathfrak{j}$, the wall structure defines a scattering diagram $\mathfrak{D}_{\mathfrak{j}}$ for $\varphi_{\mathfrak{j}}$.
\[defi:S0\] A polarized polyhedral affine manifold $(B,\mathscr{P},\varphi)$ induces an *initial wall structure* $\mathscr{S}_0$ consisting only of slabs as follows. For each $1$-cell $\rho$ containing an affine singularity $\delta$ there is a slab $\mathfrak{b}$ with underlying polyhedral subset $\rho$ and with $$f_{\mathfrak{b},v} = 1+z^{(m_{v\delta},\varphi(m_{v\delta}))},$$ where $m_{v\delta}\in \Lambda_{B,v}$ is the primitive vector pointing from $v$ to $\delta$.
A wall structure $\mathscr{S}$ is *consistent (to order $k$) at a joint $\mathfrak{j}$* if the associated scattering diagram $\mathfrak{D}_{\mathfrak{j}}$ is consistent (to order $k$). It is *consistent (to order $k$)* if it is consistent (to order $k$) at any joint.
Two wall structures $\mathscr{S}$, $\mathscr{S}'$ are *compatible to order $k$* if the following conditions hold.
If $\mathfrak{p}\in\mathscr{S}$ is a wall with $c_{\mathfrak{p}}\neq 0$ and $f_{\mathfrak{p},x} \not\equiv 1 \textup{ mod }(t^{k+1})$ for some $x\in\mathfrak{p}\setminus\Delta$, then $\mathfrak{p}\in\mathscr{S}'$, and vice versa.
If $x\in \textup{Int}(\mathfrak{b})\cap\textup{Int}(\mathfrak{b}')$ for slabs $\mathfrak{b}\in\mathscr{S},\mathfrak{b}'\in\mathscr{S}'$, then $f_{\mathfrak{b},x} \equiv f_{\mathfrak{b}',x} \textup{ mod }(t^{k+1})$.
\[prop:Sk\] If $(B,\mathscr{P},\varphi)$ is a polarized polyhedral affine manifold with $(B,\mathscr{P})$ simple, then there exists as sequence of wall structures $(\mathscr{S}_k)_{k\in\mathbb{N}}$ such that
$\mathscr{S}_0$ is the initial wall structure defined by $(B,\mathscr{P},\varphi)$;
$\mathscr{S}_k$ is consistent to order $k$;
$\mathscr{S}_k$ and $\mathscr{S}_{k+1}$ are compatible to order $k$.
By [@GS11], Remark 1.29, if $(B,\mathscr{P})$ is simple, then the central fiber of the corresponding toric degeneration is locally rigid for any choice of open gluing data. In this case, the existence of a sequence of wall structures as claimed is the main part ([§]{}3, [§]{}4) of [@GS11].
Proof of the main theorem
-------------------------
Let $Q$ be a Fano polytope and let $(B,\mathscr{P},\varphi)$ be the dual intersection complex of the corresponding toric degeneration $\mathfrak{X}\rightarrow\mathbb{A}^1$ of $(X,D)$. Let $\mathscr{S}_\infty$ be the consistent wall structure defined by $(B,\mathscr{P},\varphi)$, i.e., the limit of the $\mathscr{S}_k$ in Proposition \[prop:Sk\]. Let $\sigma_0$ be as in Definition \[defi:sigma0\].
\[lem:global\] The support $|\mathscr{S}_\infty|$ is disjoint from the interior of $\sigma_0$.
$(B,\mathscr{P},\varphi)$ defines an initial wall structure $\mathscr{S}_0$ as in Definition \[defi:S0\]. The joints of $\mathscr{S}_0$ are the vertices of $\mathscr{P}$. Let $v$ be such a vertex. By [@GS11] Proposition 3.9, the walls in $\mathscr{S}_\infty$ with base point $v$ lie in the cone $v + \mathbb{R}_{\leq 0}m_{v\delta} + \mathbb{R}m_{v\delta'} \subseteq B$. Here $\delta,\delta'$ are the affine singularities on edges adjacent to $v$ and $m_{\delta v}$ is the primitive integral tangent vector on $B$ pointing from $v$ to $\delta$.
By inductively using [@GS11], Proposition 3.9, all walls in $\mathscr{S}_\infty$ lie in the union of these cones, i.e., $$|\mathscr{S}_\infty| \subseteq \bigcup_{v\in\mathscr{P}^{[0]}} v + \mathbb{R}_{\leq 0}m_{v\delta} + \mathbb{R}_{\leq 0}m_{v\delta'}.$$ In particular, there are no walls in $\mathscr{S}_\infty$ supported on the interior of $\sigma_0$.
The unbounded walls in $\mathscr{S}_\infty$ are all parallel in the direction $m_{\textup{out}}\in \Lambda_B$. Let $f_{\textup{out}}$ be the product of all functions $f_{\mathfrak{p}}$ attached to unbounded walls $\mathfrak{p}$ in $\mathscr{S}_\infty$. Then $f_{\text{out}}$ can be regarded as an element of $\mathbb{C}\llbracket x\rrbracket$ for $x:=z^{(-m_{\textup{out}},0)}\in\mathbb{C}[\Lambda_B\oplus\mathbb{Z}]$ the unique lift of the section $-m_{\text{out}}$.
\[thm:scattering\] $$\textup{log }f_{\textup{out}} = \sum_{d=1}^\infty (D\cdot\underline{\beta}) \cdot N_d \cdot x^{D\cdot\underline{\beta}}.$$
In fact, we will prove a more general statement, giving an enumerative meaning for the function attached to any wall in $\mathscr{S}_\infty$. For this we need the following.
A *tropical disk* $h : \Gamma \rightarrow B$ is a tropical curve with the choice of univalent vertex $V_\infty$, adjacent to a unique edge $E_\infty$, such that $h$ is balanced for all vertices $V \neq V_\infty$.
Let $\mathfrak{p}\in\mathscr{S}_\infty$ be a wall and choose $x\in\textup{Int}(\mathfrak{p})$. Define $\mathfrak{H}_{\mathfrak{p},w}$ to be the set of all tropical disks $h : \Gamma \rightarrow B$ with $h(V_\infty)=x$ and $u_{(V_\infty,E_\infty)}=-w\cdot m_{\mathfrak{p}}$.
Note that the sets $\mathfrak{H}_{\mathfrak{p},w}$ are in bijection for different choices of $x$. Moreover, for an unbounded wall $\mathfrak{p}$ the set $\mathfrak{H}_{\mathfrak{p},w}$ is empty for $w_{\text{out}}\nmid w$ and there is an injective map $\mathfrak{H}_{\mathfrak{p},dw_{\text{out}}} \hookrightarrow \mathfrak{H}_d$ by removing $V_\infty$ and extending $E_\infty$ to infinity (giving $E_{\text{out}}$).
\[prop:scattering\] For a wall $\mathfrak{p}$ of $\mathscr{S}_\infty$ we have, with $N_V^{\textup{tor}}$ as in Definition \[defi:Ntor\], $$\textup{log }f_{\mathfrak{p}} = \sum_{w=1}^\infty\sum_{h\in\mathfrak{H}_{\mathfrak{p},w}} \left(\frac{1}{|\textup{Aut}(\Gamma)|}\prod_{E\in E(\Gamma)}w_E\cdot\prod_{V\neq V_{\textup{out}}} N_V^{\textup{tor}} \cdot \prod_{E\in L_\Delta(\Gamma)}\frac{(-1)^{w_E-1}}{w_E}\right)z^{(wm_{\mathfrak{p}},0)}.$$
We want to prove this by induction, so we need a well-ordered set. For a wall $\mathfrak{p}$ of $\mathscr{S}_\infty$, define a set $$\textup{Parents}(\mathfrak{p}) = \{\mathfrak{p}'\in\mathscr{S}_\infty \ | \ \mathfrak{p}\cap\mathfrak{p}'=\textup{Base}(\mathfrak{p}) \neq \textup{Base}(\mathfrak{p}')\}.$$ Here $\textup{Base}(\mathfrak{p})$ is the base point of $\mathfrak{p}$ (see Definition \[defi:structure\], (2)). Note that $\textup{Base}(\mathfrak{p}')$ is only defined for walls, so we define the condition $\textup{Base}(\mathfrak{p})\neq\textup{Base}(\mathfrak{p}')$ to be always satisfied when $\mathfrak{p}'$ is a slab. Then define inductively $$\textup{Ancestors}(\mathfrak{p}) = \{\mathfrak{p}\} \cup \bigcup_{\mathfrak{p}'\in\textup{Parents}(\mathfrak{p})} \textup{Ancestors}(\mathfrak{p}').$$ For each $k$, the set of walls in $\mathscr{S}_k$ is finite and totally ordered by $$\mathfrak{p}_1 \leq \mathfrak{p}_2 :\Leftrightarrow \mathfrak{p}_1 \in \textup{Ancestors}(\mathfrak{p}_2).$$ Hence, it is well-ordered and we can use induction. The set of smallest elements with respect to this ordering is $$\{\mathfrak{p}\in\mathscr{S}_\infty \textup{ wall}\ | \ \textup{Base}(\mathfrak{p})\in\mathscr{P}_{\mathscr{S}_0}^{[0]}=\mathscr{P}^{[0]}\}.$$
For such $\mathfrak{p}$, the set $\textup{Ancestors}(\mathfrak{p})$ consists of $\mathfrak{p}$ and two slabs $\mathfrak{b}_1,\mathfrak{b}_2$. This defines a scattering diagram at the joint $\textup{Base}(\mathfrak{p})$ that is equivalent to the scattering diagram obtained from two lines in the directions $m_1,m_2$ of the slabs $\mathfrak{b}_1,\mathfrak{b}_2$ with attached functions $f_1=1+z^{(-m_1,0)}$ and $f_2=1+z^{(-m_2,0)}$, respectively. Note that $$\textup{log }f_i = \sum_{w=1}^\infty \frac{(-1)^{w-1}}{w}z^{(wm_i,0)}.$$ Then Lemma \[lem:2.8\] gives $$\textup{log }f_{\mathfrak{p}} = \sum_{w=1}^n\sum_{\textbf{w}}w\frac{N_{\textbf{m}}(\textup{\textbf{w}})}{|\textup{Aut}(\textup{\textbf{w}})|} \left(\prod_{\substack{1\leq i\leq n\\ 1\leq j\leq l_i}} \frac{(-1)^{w_{ij}-1}}{w_{ij}}\right) z^{(-wm_{\mathfrak{p}},0)},$$ where the sum is over all $n$-tuples of weight vectors $\textbf{w}=(\textbf{w}_1,\ldots,\textbf{w}_n)$ satisfying $$\sum_{i=1}^n |\textbf{w}_i|m_i = wm_{\mathfrak{p}}.$$ Here $N_{\textbf{m}}(\textbf{w})$ is the toric logarithmic Gromov-Witten invariant defined in §\[S:toricinv\]. Tropical disks $h$ in $\mathfrak{H}_{\mathfrak{p},w}$ have a $1$-valent vertex $V_\infty$ mapping to the interior of $\mathfrak{p}$ and another vertex $V$ with one compact edge of weight $w$ and several bounded legs in directions $m_1$ or $m_2$ with weights $\textbf{w}_1=(w_{11},\ldots,w_{1l_1})$ and $\textbf{w}_2=(w_{21},\ldots,w_{2l_2})$, respectively, such that $\sum_{i=1}^n |\textbf{w}_i|m_i = wm_{\mathfrak{p}}$. Hence, the set $\mathfrak{H}_{\mathfrak{p},w}$ is in bijection with the set of pairs $\textbf{w}$ as above, with $w_{ij}$ being the weights of the bounded legs of the corresponding tropical disk $h$. Moreover, $|\textup{Aut}(\Gamma)|=|\textup{Aut}(\textbf{w})|$ and $N_V^{\textup{tor}}=N_{\textbf{m}}(\textbf{w})$ by definition. Hence, the summation over $\textbf{w}$ can be replaced by a summation over $\mathfrak{H}_{\mathfrak{p},w}$, and we obtain $$\textup{log }f_{\mathfrak{p}} = \sum_{w=1}^\infty\sum_{h\in\mathfrak{H}_{\mathfrak{p},w}} w\frac{N_V^{\textup{tor}}}{|\textup{Aut}(\Gamma)|}\left(\prod_{E\in L_\Delta(\Gamma)}\frac{(-1)^{w_E-1}}{w_E}\right)z^{(wm_{\mathfrak{p}},0)}.$$ This is precisely the claimed formula for such $\mathfrak{p}$, completing the base case.
For the induction step, let $\mathfrak{p}$ be a wall of $\mathscr{S}_\infty$ and assume the claimed formula holds for all walls $\mathfrak{p}'\in\textup{Ancestors}(\mathfrak{p})\setminus\{\mathfrak{p}\}$. By Lemma \[lem:2.8\] and using the induction hypothesis, $$\begin{aligned}
\textup{log }f_{\mathfrak{p}} &=& \sum_{w=1}^n\sum_{\textbf{w}}\left(w\frac{N_{\textbf{m}}(\textup{\textbf{w}})}{|\textup{Aut}(\textup{\textbf{w}})|} \prod_{\substack{\mathfrak{p}'\in\textup{Parents}(\mathfrak{p})\\ 1\leq j\leq l_{\mathfrak{p}'}}} \sum_{h\in\mathfrak{H}_{\mathfrak{p}',w_{\mathfrak{p}'j}}} \left(\frac{1}{|\textup{Aut}(\Gamma)|} \vphantom{\prod_{E\in L_\Delta(\Gamma)}\frac{(-1)^{w_E-1}}{w_E}} \right.\right. \\
&& \textup{ }\hspace{2cm} \left.\left. \cdot \prod_{E\in E(\Gamma)}w_E\cdot\prod_{V\neq V_{\textup{out}}} N_V^{\textup{tor}} \cdot \prod_{E\in L_\Delta(\Gamma)}\frac{(-1)^{w_E-1}}{w_E}\right) \vphantom{\prod_{\substack{\mathfrak{p}'\in\textup{Parents}(\mathfrak{p})\\ 1\leq j\leq l_{\mathfrak{p}'}}}} \right) z^{(-wm_{\mathfrak{p}},0)},\end{aligned}$$ where the second sum is over all tuples $\textbf{w}=(\textbf{w}_{\mathfrak{p}'})_{\mathfrak{p}'\in\textup{Parents}(\mathfrak{p})}$ of weight vectors $\textbf{w}_{\mathfrak{p}'}=(w_{\mathfrak{p}'1},\ldots,w_{\mathfrak{p}'l_{\mathfrak{p}'}})$ with $$\sum_{\mathfrak{p}'\in\textup{Parents}(\mathfrak{p})}|\textbf{w}_{\mathfrak{p}'}|m_{\mathfrak{p}'} = wm_{\mathfrak{p}}.$$ Factoring out, we can replace the product over $\textup{Parents}(\mathfrak{p})$ and $1\leq j \leq l_{\mathfrak{p}'}$ and the sum over $h\in\mathfrak{H}_{\mathfrak{p}',w_{\mathfrak{p}'j}}$ by a sum over tuples $(h\in\mathfrak{H}_{\mathfrak{p}',w_{\mathfrak{p}'j}})_{\substack{\mathfrak{p}'\in\textup{Parents}(\mathfrak{p}) \\ 1\leq j \leq l_{\mathfrak{p}'}}}$.
Further, we can replace the summations over $\textbf{w}$ and such tuples by a summation over $\mathfrak{H}_{\mathfrak{p},w}$, since this is precisely the data that determines a tropical disk in $\mathfrak{H}_{\mathfrak{p},w}$. We get the claimed formula by using that, for $h\in\mathfrak{H}_{\mathfrak{p},w}$, we have $$\frac{1}{|\textup{Aut}(\Gamma)|} = \sum_{\mathfrak{\textbf{w}}} \frac{1}{|\textup{Aut}(\textbf{w})|} \sum_{\substack{\mathfrak{p}'\in\textup{Parents}(\mathfrak{p})\\ 1\leq j\leq l_{\mathfrak{p}'}}}\sum_{h'\in\mathfrak{H}_{\mathfrak{p}',w_{\mathfrak{p}'j}}} \frac{1}{|\textup{Aut}(\Gamma')|}.$$ Now Theorem \[thm:scattering\] follows by summation over all outgoing walls $\mathfrak{p}$ in $\mathscr{S}_\infty$.
Torsion points {#S:torsion}
==============
In this section we consider $(\mathbb{P}^2,E)$. Choose a flex point $O$ on the elliptic curve $E$ and consider the group law on $E$ with $O$ the identity. An *$m$-torsion point* on $E$ is a point $P$ such that $m \cdot P = O$. As a topological group, the elliptic curve is a torus $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. The $m$-torsion points form a group $\mathbb{Z}_m \times \mathbb{Z}_m$.
\[lem:torsion\] If $C$ is a rational degree $d$ curve intersecting $E$ in a single point $P$, then $P$ is a $3d$-torsion point.
Let $C$ be a rational degree $d$ curve intersecting $E$ in a single point $P$ and let $L$ be the line tangent to $O$. Then the cycle $C-dL$ has degree $0$, so it is linearly equivalent to zero, since $\textup{Pic }\mathbb{P}^2 \cong \mathbb{Z}$ by the degree map. Moreover, it intersects $E$ in the cycle $3d(P-O)$ which in turn is linearly equivalent to zero.
Let $T_d\simeq\mathbb{Z}_{3d}\times\mathbb{Z}_{3d}$ be the set of $3d$-torsion points on $E$ and let $\beta_d$ be the class of degree $d$ stable log maps (Definition \[defi:beta\]). Then we have a decomposition $$\mathscr{M}(\mathbb{P}^2,\beta_d) = \coprod_{P\in T_d} \mathscr{M}(\mathbb{P}^2,\beta_d)_P,$$ where $\mathscr{M}(\mathbb{P}^2,\beta_d)_P$ is the subspace of $\mathscr{M}(\mathbb{P}^2,\beta_d)$ of maps intersecting $E$ in $P$. Let $\llbracket\mathscr{M}(\mathbb{P}^2,\beta_d)_P\rrbracket$ be the restriction of $\llbracket\mathscr{M}(\mathbb{P}^2,\beta_d)\rrbracket$ to $\mathscr{M}(\mathbb{P}^2,\beta_d)_P$ and define $$N_{d,P} := \int_{\llbracket\mathscr{M}(\mathbb{P}^2,\beta_d)_P\rrbracket}1.$$
For $P\in\cup_{d\geq 1}T_d$ denote by $k(P)$ the smallest integer $k\geq 1$ such that $P$ is $3k$-torsion.
$N_{d,P}$ only depends on $P$ through $k(P)$.
This was shown in [@Bou4], Lemma 1.2, using ideas from [@CvGKT2]. The freedom of choice of $O$ and the fact that the monodromy of the family of smooth cubics in $\mathbb{P}^2$ maps surjectively to $SL(2,\mathbb{Z})$ acting on $T_k \simeq \mathbb{Z}_{3k} \times \mathbb{Z}_{3k}$ implies that two points $P,P'$ with $k(P)=k(P')$ are related to each other via a monodromy transformation. Then the deformation invariance of logarithmic Gromov-Witten invariants shows that $N_{d,P}=N_{d,P'}$ for all $d$.
Write $N_{d,k}$ for $N_{d,P}$ with $P$ such that $k(P)=k$.
Under the toric degeneration $\mathfrak{X} \rightarrow \mathbb{A}^1$ different $3d$-torsion points may map to the same point on the central fiber $X_0$, and even to a $0$-dimensional stratum. However, the limits of the $3$-torsion points all lie on the $1$-dimensional strata. The intersection points with $3d$-torsion correspond to the unbounded walls $\mathfrak{p}$ in $\mathscr{S}_\infty$ with non-zero $t^{3d}$-coefficient of $\textup{log }f_{\mathfrak{p}}$. Their number is exactly $3d$ and they are distributed as the $3l$-torsion points of a circle (see Figure \[fig:main\] and Figure \[fig:sage2\]).
This can be explained as follows. The $2$-dimensional torus is a $S^1$-fibration over $S^1$. In the SYZ limit, the $S^1$-fiber shrinks to a point and we only see the $S^1$-basis, which is the tropicalization. In this limit, the $\mathbb{Z}_{3k}$-fibers of $T_k\simeq\mathbb{Z}_{3k}\times\mathbb{Z}_{3k}$ are identified and we only see the $\mathbb{Z}_m$ in the basis. The toric degeneration of divisors $\mathfrak{D}\rightarrow\mathbb{A}^1$ is an elliptic fibration. It contains a singular fiber that is a cycle of three $\mathbb{P}^1$, i.e., an $I_3$ fiber in Kodaira’s classification of singular elliptic fibers. Then the monodromy acting on the first cohomology class of the general fiber is given by $M_1=\left(\begin{smallmatrix}1&3\\ 0&1\end{smallmatrix}\right)$ up to conjugation. Now the action of $M_1$ on $\mathbb{Z}_3\times\mathbb{Z}_3$ is trivial. This means that each $3$-torsion point really defines a section of the family, which will have a limit on the special fiber. Such limit is necessarily on the smooth part of the special fiber, i.e., on a $1$-dimensional toric stratum (see [@SS], Theorem 6.3). The action of $M_1$ on $\mathbb{Z}_6\times\mathbb{Z}_6$ has some fixed points, corresponding to $6$-torsion points on $E$ that define sections with limit on the smooth part of the special fiber, i.e., on $1$-dimensional strata. The other $6$-torsion points are permuted by the action of $M_1$, so they only define multisections with limit on the singular part of the special fiber, i.e., on $0$-dimensional strata.
Now consider the refined degeneration $\tilde{\mathfrak{X}}_d\rightarrow\mathbb{A}^1$. By construction of the refinement, the limits of all $3d$-torsion points lie on $1$-dimensional strata of the central fiber $X:=Y$. Indeed, the central fiber of the elliptic fibration $\tilde{\mathfrak{D}}\rightarrow\mathbb{A}^1$ is a cycle of $3d$ lines, i.e., an $I_{3d}$-fiber. Then the monodromy acting on $\mathbb{Z}_{3k}\times\mathbb{Z}_{3k}$ is given by $\left(\begin{smallmatrix}1&3d\\ 0&1\end{smallmatrix}\right)$ which is the identity for all $k\mid d$.
Let $s_{k,l}$ be the number of points P on $E$ with $k(P)=k$, fixed by the action of $M_l:=\left(\begin{smallmatrix}1&3l\\ 0&1\end{smallmatrix}\right)$, but not fixed by the action of $M_{l'}$ for all $l'<l$. Note that $s_k:=\sum_{l\mid k}s_{k,l}$ is the number of points $P$ on $E$ with $k(P)=k$ and that $\sum_{k\mid d}s_k=(3d)^2$ is the number of $3d$-torsion points.
-------------- -------------- -------------- -------------- --------------- ---------------
$s_{1,1}=9$
$s_{2,1}=9$ $s_{2,2}=18$
$s_{3,1}=18$ $s_{3,3}=54$
$s_{4,1}=18$ $s_{4,2}=18$ $s_{4,4}=72$
$s_{5,1}=36$ $s_{5,5}=180$
$s_{6,1}=18$ $s_{6,2}=36$ $s_{6,3}=54$ $s_{6,6}=108$
-------------- -------------- -------------- -------------- --------------- ---------------
: The number $s_{k,l}$ of points $P$ on $E$ with $k(P)=k$, fixed by $M_l$, but not fixed by $M_{l'}$ for $l'<l$.
\[tab:s\]
For a wall $\mathfrak{p}\in\mathscr{S}_\infty$ let $l(\mathfrak{p})$ be the smallest number such that $\textup{log }f_{\mathfrak{p}}$ has non-trivial $t^{3l(\mathfrak{p})}$-coefficient. Let $r_l$ be the number of walls with $l(\mathfrak{p})=l$.
\[lem:rl\] The number $r_l$ can be defined recursively by $$r_1 = 3, \quad r_l = 3l - \sum_{l'\mid l}r_{l'}.$$
For a wall $\mathfrak{p}\in\mathscr{S}_\infty$, the condition $l(\mathfrak{p})=l$ means that the corresponding toric stratum of $X:=Y$ contains the limits of the points on $E$ that are fixed by the action of $M_l$ but not fixed by $M_{l'}$ for $l'<l$. Since in the SYZ limit we only see the basis of the fibration, $r_l$ equals the number of points on a circle $S^1$ that are $3l$-torsion but not $3l'$-torsion. This number can be defined as above.
$r_1=3$ $r_2=3$ $r_3=6$ $r_4=6$ $r_5=12$ $r_6=6$
--------- --------- --------- --------- ---------- ---------
: The number $r_l$ of walls $\mathfrak{p}$ with $l(\mathfrak{p})=l$.
\[tab:r\]
Note that $s_{k,l}/r_l$ is the number of points $P$ with $k(P)=k$ and with limit on the stratum corresponding to a particular wall $\mathfrak{p}$ with $l(\mathfrak{p})=l$. A direct consequence of Proposition \[prop:scattering\] is the following.
\[cor:torsion\] Let $\mathfrak{p}$ be an unbounded wall of order $l$ in $\mathscr{S}_\infty$. Then $$\textup{log }f_{\mathfrak{p}} = \sum_{d=1}^\infty 3d \left(\sum_{k: l\mid k\mid d} \frac{s_{k,l}}{r_l} N_{d,k}\right) x^{3d}.$$
Similarly to §\[S:BPS\] we can define integers $n_{d,k}$ recursively by $$n_{d,d} = N_{d,d}, \quad n_{d,k} = N_{d,k} - \sum_{d':k\mid d'\mid d} M_{d'}[d/d'] \cdot n_{d',k}.$$
Some of the numbers $n_{d,k}$ have been calculated by Takahashi [@Ta1]. Their relation to local BPS numbers is studied in [@CvGKT][@CvGKT2][@CvGKT3].
\[rem:combine\] Unfortunately we are not able to apply the methods of this section to the refined situation of §\[S:degdiv\] in order to calculate the contributions to $N_{[i;d_1,\ldots,d_m]}$ with prescribed torsion: Let $N_{[i;d_1,\ldots,d_m],k}$ be the logarithmic Gromov-Witten invariant of stable log maps contributing to $N_{[i;d_1,\ldots,d_m]}$ and meeting $E$ in a fixed point of order $3k$, and let $n_{[i;d_1,\ldots,d_m],k}$ be the corresponding log BPS number. Define $l([i;d_1,\ldots,d_m]):=l(\mathfrak{p})$ for $\mathfrak{p}$ the unbounded wall for $[i;d_1,\ldots,d_m]$. Then $$\begin{aligned}
\sum_{k=1}^d \frac{s_{k,l}}{r_l} n_{[i;d_1,\ldots,d_m],k} &=& n_{[i;d_1,\ldots,d_m]} \quad \text{for } l=l([i;d_1,\ldots,d_m])\\
\sum_{l([i;d_1,\ldots,d_m])=l} n_{[i;d_1,\ldots,d_m],k} &=& n_{d,k} \qquad\quad\ \ \text{for all } l|k\end{aligned}$$ This gives a system of linear equations for the indeterminates $n_{[i;d_1,\ldots,d_m],k}$. In general the number of equations will be smaller than the number of indeterminates, so there will be no unique solution. However, for $d\leq 3$ we indeed have enough equations to determine the numbers $n_{[i;d_1,\ldots,d_m],k}$ as we will show in §\[S:calcP2c\].
Explicit calculations {#S:calc}
=====================
In this section we will calculate some logarithmic Gromov-Witten invariants and log BPS numbers explicitly. To this end, I wrote a sage code for calculating scattering diagrams and wall structures. It can be found on my webpage[^6].
$(\mathbb{P}^2,E)$ {#S:calcP2}
------------------
We want to calculate the numbers $N_{d,k}$ and $n_{d,k}$ for $d\leq 6$ as well as the numbers $n_{[i;d_1,\ldots,d_k]}$ for $d\leq 4$. Loading the code into a sage shell and typing `tex(scattering(diag(6,"P2"),6,"P2"),diagP2(6),(1,0))` one can produce a TikZ code that under some small changes gives Figure \[fig:sage\]. It shows the part of the wall structure $\bar{\mathscr{S}}_6$ on the discrete covering space $\bar{B}$ (see §\[S:affinecharts\]) that is relevant for computing the functions on the central maximal cell. The full $\bar{\mathscr{S}}_6$ would be symmetric, carrying much more walls on the outer area.
(0.000,0.500) – (0.000,6.50); (0.000,0.500) – (0.000,-5.50); (-1.50,-0.500) – (15.0,5.00); (-1.50,1.50) – (15.0,-4.00); (-1.50,1.50) – (-16.5,6.50); (-1.50,-0.500) – (-16.5,-5.50); (-6.00,-1.50) – (15.0,2.00); (-6.00,2.50) – (15.0,-1.00); (-6.00,2.50) – (-30.0,6.50); (-6.00,-1.50) – (-30.0,-5.50); (-13.5,-2.50) – (15.0,0.667); (-13.5,3.50) – (15.0,0.333); (-13.5,3.50) – (-40.5,6.50); (-13.5,-2.50) – (-40.5,-5.50); (-24.0,-3.50) – (15.0,-0.250); (-24.0,4.50) – (15.0,1.25); (-24.0,4.50) – (-48.0,6.50); (-24.0,-3.50) – (-48.0,-5.50); (-37.5,-4.50) – (15.0,-1.00); (-37.5,5.50) – (15.0,2.00); (-37.5,5.50) – (-52.5,6.50); (-37.5,-4.50) – (-52.5,-5.50); (-54.0,-5.50) – (15.0,-1.67); (-54.0,6.50) – (15.0,2.67); (0.000,0.000) – (15.0,0.000); (-3.00,-1.00) – (15.0,-1.00); (0.000,1.00) – (15.0,1.00); (-3.00,2.00) – (15.0,2.00); (-45.0,-5.00) – (15.0,-5.00); (-30.0,-4.00) – (15.0,-4.00); (-18.0,-3.00) – (15.0,-3.00); (-9.00,-2.00) – (15.0,-2.00); (-9.00,3.00) – (15.0,3.00); (-18.0,4.00) – (15.0,4.00); (-30.0,5.00) – (15.0,5.00); (-45.0,6.00) – (15.0,6.00); (1.50,0.500) – (15.0,0.500); (-4.50,-1.50) – (15.0,-1.50); (0.000,-0.500) – (15.0,-0.500); (0.000,1.50) – (15.0,1.50); (-4.50,2.50) – (15.0,2.50); (-22.5,-3.50) – (15.0,-3.50); (-12.0,-2.50) – (15.0,-2.50); (-12.0,3.50) – (15.0,3.50); (-22.5,4.50) – (15.0,4.50); (0.000,0.000) – (15.0,2.50); (-18.0,-3.00) – (15.0,-0.800); (-9.00,-2.00) – (15.0,0.000); (-3.00,-1.00) – (15.0,1.00); (0.000,1.00) – (15.0,6.00); (-3.00,2.00) – (-3.00,6.50); (-9.00,3.00) – (-19.5,6.50); (-18.0,4.00) – (-33.0,6.50); (0.000,0.000) – (15.0,-5.00); (-18.0,-3.00) – (-33.0,-5.50); (-9.00,-2.00) – (-19.5,-5.50); (-3.00,-1.00) – (-3.00,-5.50); (0.000,1.00) – (15.0,-1.50); (-3.00,2.00) – (15.0,0.000); (-9.00,3.00) – (15.0,1.00); (-18.0,4.00) – (15.0,1.80); (3.00,0.000) – (15.0,1.33); (0.000,-1.00) – (15.0,0.250); (3.00,1.00) – (15.0,3.00); (0.000,2.00) – (13.5,6.50); (3.00,0.000) – (15.0,-2.00); (0.000,-1.00) – (13.5,-5.50); (3.00,1.00) – (15.0,-0.333); (0.000,2.00) – (15.0,0.750); (3.00,0.000) – (15.0,-1.33); (3.00,1.00) – (15.0,0.000); (1.50,0.500) – (15.0,-1.00); (0.000,-0.500) – (15.0,-3.00); (0.000,1.50) – (15.0,0.250); (1.50,0.500) – (15.0,2.00); (0.000,-0.500) – (15.0,0.750); (0.000,1.50) – (15.0,4.00); (2.00,0.333) – (15.0,0.333); (2.00,0.667) – (15.0,0.667); (2.40,0.200) – (15.0,0.200); (0.000,-0.800) – (15.0,-0.800); (1.80,1.20) – (15.0,1.20); (2.40,0.800) – (15.0,0.800); (1.80,-0.200) – (15.0,-0.200); (0.000,1.80) – (15.0,1.80); (3.00,0.000) – (15.0,1.00); (3.00,1.00) – (15.0,2.33); (2.50,0.167) – (15.0,0.167); (2.50,0.833) – (15.0,0.833); (-13.0,-2.67) – (15.0,-2.67); (-5.00,-1.67) – (15.0,-1.67); (0.000,-0.667) – (15.0,-0.667); (1.00,1.33) – (15.0,1.33); (-3.00,2.33) – (15.0,2.33); (-10.0,3.33) – (15.0,3.33); (2.00,0.333) – (15.0,-0.750); (-10.0,-2.33) – (15.0,-2.33); (-3.00,-1.33) – (15.0,-1.33); (1.00,-0.333) – (15.0,-0.333); (0.000,1.67) – (15.0,1.67); (-5.00,2.67) – (15.0,2.67); (-13.0,3.67) – (15.0,3.67); (2.00,0.667) – (15.0,1.75); (0.000,0.000) – (15.0,3.33); (-3.00,-1.00) – (15.0,1.40); (0.000,1.00) – (8.25,6.50); (-3.00,2.00) – (-9.75,6.50); (0.000,0.000) – (15.0,1.67); (-9.00,-2.00) – (15.0,-0.400); (-3.00,-1.00) – (15.0,0.500); (0.000,1.00) – (15.0,3.50); (-3.00,2.00) – (10.5,6.50); (-9.00,3.00) – (-9.00,6.50); (0.000,0.000) – (8.25,-5.50); (-3.00,-1.00) – (-9.75,-5.50); (0.000,1.00) – (15.0,-2.33); (-3.00,2.00) – (15.0,-0.400); (0.000,0.000) – (15.0,-2.50); (-9.00,-2.00) – (-9.00,-5.50); (-3.00,-1.00) – (10.5,-5.50); (0.000,1.00) – (15.0,-0.667); (-3.00,2.00) – (15.0,0.500); (-9.00,3.00) – (15.0,1.40); (1.80,0.400) – (15.0,1.50); (1.80,0.600) – (15.0,-0.500); (2.25,0.250) – (15.0,0.250); (-5.25,-1.75) – (15.0,-1.75); (0.000,-0.750) – (15.0,-0.750); (1.50,1.25) – (15.0,1.25); (-2.25,2.25) – (15.0,2.25); (2.25,0.750) – (15.0,0.750); (-2.25,-1.25) – (15.0,-1.25); (1.50,-0.250) – (15.0,-0.250); (0.000,1.75) – (15.0,1.75); (-5.25,2.75) – (15.0,2.75); (1.80,0.400) – (15.0,0.400); (0.000,-0.600) – (15.0,-0.600); (0.600,1.40) – (15.0,1.40); (1.80,0.600) – (15.0,0.600); (0.600,-0.400) – (15.0,-0.400); (0.000,1.60) – (15.0,1.60); (0.000,0.000) – (15.0,1.25); (-3.00,-1.00) – (15.0,0.200); (0.000,1.00) – (15.0,2.67); (-3.00,2.00) – (15.0,5.00); (0.000,0.000) – (15.0,-1.67); (-3.00,-1.00) – (15.0,-4.00); (0.000,1.00) – (15.0,-0.250); (-3.00,2.00) – (15.0,0.800); (0.000,0.000) – (15.0,1.00); (0.000,1.00) – (15.0,2.25); (0.000,0.000) – (15.0,-1.25); (0.000,1.00) – (15.0,0.000);
We have $$\begin{aligned}
\text{log }f_{\text{out}} &=& 27x^3 + \frac{405}{2}x^6 + 2196x^9 + \frac{110997}{4}x^{12} \\
&& \text{ } \qquad\ \ + \frac{2051892}{5}x^{15} + 5527710x^{18} + \mathcal{O}(x^{20})\end{aligned}$$ This yields the following logarithmic Gromov-Witten invariants:
$N_1=9$ $N_2=\frac{135}{4}$ $n_3=244$ $n_4=\frac{36999}{16}$ $n_5=\frac{635634}{25}$ $n_6=307095$
--------- --------------------- ----------- ------------------------ ------------------------- --------------
: The invariants $N_d$ of $(\mathbb{P}^2,E)$ for $d \leq 6$.
\[tab:N\]
Subtracting multiple cover contributions we get the following log BPS numbers:
$n_1=9$ $n_2=27$ $n_3=234$ $n_4=2232$ $n_5=25380$ $n_6=305829$
--------- ---------- ----------- ------------ ------------- --------------
: The log BPS numbers $n_d$ of $(\mathbb{P}^2,E)$ for $d \leq 6$.
\[tab:n\]
They are related to the local BPS numbers $n_d^{\text{loc}}$, shown in [@CKYZ], Table 1, by Remark \[rem:local\] and [@CKYZ], (2.1).
### Torsion points {#S:calcP2a}
Write $f_l$ for the function $f_{\mathfrak{p}}$ attached to a wall $\mathfrak{p}$ with $l(\mathfrak{p})=l$ (see Figure \[fig:sage2\]). The sage code gives the following: $$\begin{aligned}
\textup{log }f_1 &= 9x^{3}+\frac{63}{2}x^{6}+246x^{9}+\frac{9279}{4}x^{12}+\frac{175464}{5}x^{15}+307041x^{18} + \mathcal{O}(x^{21}) \\
\textup{log }f_2 &= 36x^{6} + 2322x^{12} + 307164x^{18} + \mathcal{O}(x^{21}) \\
\textup{log }f_3 &= 243x^9 + \frac{614061}{2}x^{18} + \mathcal{O}(x^{21}) \\
\textup{log }f_4 &= 2304x^{12} + \mathcal{O}(x^{21}) \\
\textup{log }f_5 &= 25425x^{15} + \mathcal{O}(x^{21}) \\
\textup{log }f_6 &= 307152x^{18} + \mathcal{O}(x^{21})\end{aligned}$$ From this we calculate the numbers $n_{d,k}$ for $d\leq 6$ below to be the following:
--------------- --------------- --------------- -------------- --------------- ---------------
$n_{1,1}=1$
$n_{2,1}=0$ $n_{2,2}=1$
$n_{3,1}=2$ $n_{3,3}=3$
$n_{4,1}=14$ $n_{4,2}=14$ $n_{4,4}=16$
$n_{5,1}=108$ $n_{5,5}=113$
$n_{6,1}=927$ $n_{6,2}=938$ $n_{6,3}=936$ $n_{6,6}=948$
--------------- --------------- --------------- -------------- --------------- ---------------
: The numbers $n_{d,k}$ calculated below.
\[tab:results\]
The $n_{d,d}$ coincide with the $m_d$ in [@Ta2], Theorem 1.4. The numbers $n_{d,k}$ for $d\leq 3$ are calculated in [@Ta1]. The sum $\sum_{k\mid d}s_kn_{d,k}$ is the log BPS number $n_d$ of $(\mathbb{P}^2,E)$. From $n_5$ and $n_{5,5}$ one also obtains $n_{5,1}$. To the best of my knowledge the numbers $n_{4,1}$, $n_{4,2}$, $n_{6,1}$, $n_{6,2}$ and $n_{6,3}$ are new.
(0,0) rectangle (16,1); (0.000,0.500) – (0.000,6.50); (0.000,0.500) – (0.000,-5.50); (-1.50,-0.500) – (15.0,5.00); (-1.50,1.50) – (15.0,-4.00); (-1.50,1.50) – (-16.5,6.50); (-1.50,-0.500) – (-16.5,-5.50); (-6.00,-1.50) – (15.0,2.00); (-6.00,2.50) – (15.0,-1.00); (-6.00,2.50) – (-30.0,6.50); (-6.00,-1.50) – (-30.0,-5.50); (-13.5,-2.50) – (15.0,0.667); (-13.5,3.50) – (15.0,0.333); (-13.5,3.50) – (-40.5,6.50); (-13.5,-2.50) – (-40.5,-5.50); (-24.0,-3.50) – (15.0,-0.250); (-24.0,4.50) – (15.0,1.25); (-24.0,4.50) – (-48.0,6.50); (-24.0,-3.50) – (-48.0,-5.50); (-37.5,-4.50) – (15.0,-1.00); (-37.5,5.50) – (15.0,2.00); (-37.5,5.50) – (-52.5,6.50); (-37.5,-4.50) – (-52.5,-5.50); (-54.0,-5.50) – (15.0,-1.67); (-54.0,6.50) – (15.0,2.67); (0.000,0.000) – (15.0,0.000) node\[above left = 0 and -0.15\][$f_1$]{}; (-3.00,-1.00) – (15.0,-1.00); (0.000,1.00) – (15.0,1.00) node\[above right = 0 and -0.15\][$f_1$]{}; (-3.00,2.00) – (15.0,2.00); (-45.0,-5.00) – (15.0,-5.00); (-30.0,-4.00) – (15.0,-4.00); (-18.0,-3.00) – (15.0,-3.00); (-9.00,-2.00) – (15.0,-2.00); (-9.00,3.00) – (15.0,3.00); (-18.0,4.00) – (15.0,4.00); (-30.0,5.00) – (15.0,5.00); (-45.0,6.00) – (15.0,6.00); (1.50,0.500) – (15.0,0.500) node\[above\][$f_2$]{}; (-4.50,-1.50) – (15.0,-1.50); (0.000,-0.500) – (15.0,-0.500); (0.000,1.50) – (15.0,1.50); (-4.50,2.50) – (15.0,2.50); (-22.5,-3.50) – (15.0,-3.50); (-12.0,-2.50) – (15.0,-2.50); (-12.0,3.50) – (15.0,3.50); (-22.5,4.50) – (15.0,4.50); (0.000,0.000) – (15.0,2.50); (-18.0,-3.00) – (15.0,-0.800); (-9.00,-2.00) – (15.0,0.000); (-3.00,-1.00) – (15.0,1.00); (0.000,1.00) – (15.0,6.00); (-3.00,2.00) – (-3.00,6.50); (-9.00,3.00) – (-19.5,6.50); (-18.0,4.00) – (-33.0,6.50); (0.000,0.000) – (15.0,-5.00); (-18.0,-3.00) – (-33.0,-5.50); (-9.00,-2.00) – (-19.5,-5.50); (-3.00,-1.00) – (-3.00,-5.50); (0.000,1.00) – (15.0,-1.50); (-3.00,2.00) – (15.0,0.000); (-9.00,3.00) – (15.0,1.00); (-18.0,4.00) – (15.0,1.80); (3.00,0.000) – (15.0,1.33); (0.000,-1.00) – (15.0,0.250); (3.00,1.00) – (15.0,3.00); (0.000,2.00) – (13.5,6.50); (3.00,0.000) – (15.0,-2.00); (0.000,-1.00) – (13.5,-5.50); (3.00,1.00) – (15.0,-0.333); (0.000,2.00) – (15.0,0.750); (3.00,0.000) – (15.0,-1.33); (3.00,1.00) – (15.0,0.000); (1.50,0.500) – (15.0,-1.00); (0.000,-0.500) – (15.0,-3.00); (0.000,1.50) – (15.0,0.250); (1.50,0.500) – (15.0,2.00); (0.000,-0.500) – (15.0,0.750); (0.000,1.50) – (15.0,4.00); (2.00,0.333) – (15.0,0.333) node\[above\][$f_3$]{}; (2.00,0.667) – (15.0,0.667) node\[above\][$f_3$]{}; (2.40,0.200) – (15.0,0.200) node\[above\][$f_5$]{}; (0.000,-0.800) – (15.0,-0.800); (1.80,1.20) – (15.0,1.20); (2.40,0.800) – (15.0,0.800) node\[above\][$f_5$]{}; (1.80,-0.200) – (15.0,-0.200); (0.000,1.80) – (15.0,1.80); (3.00,0.000) – (15.0,1.00); (3.00,1.00) – (15.0,2.33); (2.50,0.167) – (15.0,0.167) node\[above\][$f_6$]{}; (2.50,0.833) – (15.0,0.833) node\[above\][$f_6$]{}; (-13.0,-2.67) – (15.0,-2.67); (-5.00,-1.67) – (15.0,-1.67); (0.000,-0.667) – (15.0,-0.667); (1.00,1.33) – (15.0,1.33); (-3.00,2.33) – (15.0,2.33); (-10.0,3.33) – (15.0,3.33); (2.00,0.333) – (15.0,-0.750); (-10.0,-2.33) – (15.0,-2.33); (-3.00,-1.33) – (15.0,-1.33); (1.00,-0.333) – (15.0,-0.333); (0.000,1.67) – (15.0,1.67); (-5.00,2.67) – (15.0,2.67); (-13.0,3.67) – (15.0,3.67); (2.00,0.667) – (15.0,1.75); (0.000,0.000) – (15.0,3.33); (-3.00,-1.00) – (15.0,1.40); (0.000,1.00) – (8.25,6.50); (-3.00,2.00) – (-9.75,6.50); (0.000,0.000) – (15.0,1.67); (-9.00,-2.00) – (15.0,-0.400); (-3.00,-1.00) – (15.0,0.500); (0.000,1.00) – (15.0,3.50); (-3.00,2.00) – (10.5,6.50); (-9.00,3.00) – (-9.00,6.50); (0.000,0.000) – (8.25,-5.50); (-3.00,-1.00) – (-9.75,-5.50); (0.000,1.00) – (15.0,-2.33); (-3.00,2.00) – (15.0,-0.400); (0.000,0.000) – (15.0,-2.50); (-9.00,-2.00) – (-9.00,-5.50); (-3.00,-1.00) – (10.5,-5.50); (0.000,1.00) – (15.0,-0.667); (-3.00,2.00) – (15.0,0.500); (-9.00,3.00) – (15.0,1.40); (1.80,0.400) – (15.0,1.50); (1.80,0.600) – (15.0,-0.500); (2.25,0.250) – (15.0,0.250) node\[above\][$f_4$]{}; (-5.25,-1.75) – (15.0,-1.75); (0.000,-0.750) – (15.0,-0.750); (1.50,1.25) – (15.0,1.25); (-2.25,2.25) – (15.0,2.25); (2.25,0.750) – (15.0,0.750) node\[above\][$f_4$]{}; (-2.25,-1.25) – (15.0,-1.25); (1.50,-0.250) – (15.0,-0.250); (0.000,1.75) – (15.0,1.75); (-5.25,2.75) – (15.0,2.75); (1.80,0.400) – (15.0,0.400) node\[above\][$f_5$]{}; (0.000,-0.600) – (15.0,-0.600); (0.600,1.40) – (15.0,1.40); (1.80,0.600) – (15.0,0.600) node\[above\][$f_5$]{}; (0.600,-0.400) – (15.0,-0.400); (0.000,1.60) – (15.0,1.60); (0.000,0.000) – (15.0,1.25); (-3.00,-1.00) – (15.0,0.200); (0.000,1.00) – (15.0,2.67); (-3.00,2.00) – (15.0,5.00); (0.000,0.000) – (15.0,-1.67); (-3.00,-1.00) – (15.0,-4.00); (0.000,1.00) – (15.0,-0.250); (-3.00,2.00) – (15.0,0.800); (0.000,0.000) – (15.0,1.00); (0.000,1.00) – (15.0,2.25); (0.000,0.000) – (15.0,-1.25); (0.000,1.00) – (15.0,0.000);
\
From Corollary \[cor:torsion\] we get the following:
For $l=6$ we get $$307041 = 18 \cdot \frac{108}{6} \cdot N_{6,6},$$ hence $N_{6,6}=948$. There are no multiple cover contributions, as there are no curves of degree $<6$ meeting a point with $l=6$. This shows $n_{6,6}=948$:
For $l=5$ we get $$25425 = 15 \cdot \frac{180}{12} \cdot N_{5,5},$$ hence $N_{5,5}=113$. Again, there are no multiple cover contributions, so $n_{5,5}=113$.
For $l=4$ we get $$2304 = 12 \cdot \frac{72}{6} \cdot N_{4,4},$$ hence $N_{4,4}=n_{4,4}=16$.
For $l=3$ we get $$243 = 9 \cdot \frac{54}{6} \cdot N_{3,3},$$ hence $N_{3,3}=n_{3,3}=3$. Moreover, $$\frac{614061}{2} = 18 \cdot \left(\frac{54}{6} \cdot N_{6,3} + \frac{54}{6} \cdot 948\right),$$ hence $N_{6,3}=\frac{3789}{4}$. Subtracting $M_3[2]\cdot n_{3,3}=\frac{15}{4}\cdot 3$ we get $n_{6,3}=936$.
For $l=2$ we get $$36 = 6 \cdot \frac{18}{3} \cdot N_{2,2},$$ hence $N_{2,2}=n_{2,2}=1$, and $$2322 = 12\cdot\left(\frac{18}{3}\cdot N_{4,2}+\frac{18}{3}\cdot 16\right),$$ hence $N_{4,2}=\frac{65}{4}$. Subtracting $M_1[4]\cdot n_{1,1}+M_2[2]n_{2,1}=\frac{35}{16}\cdot 1+\frac{9}{4}\cdot 0$ we get $n_{4,1}=14$. Moreover, $$307164 = 18\cdot\left(\frac{18}{3}\cdot N_{6,2}+\frac{36}{3}\cdot 948\right),$$ hence $N_{6,2}=\frac{8533}{9}$. Subtracting $M_2[3]\cdot n_{2,2}=\frac{91}{9}\cdot 1$ we get $n_{6,2}=938$.
For $l=1$ we get $$9 = 3 \cdot \frac{9}{3} \cdot N_{1,1},$$ hence $N_{1,1}=n_{1,1}=1$, and $$\frac{63}{2} = 6\cdot\left(\frac{9}{3}\cdot N_{2,1} + \frac{9}{3}\cdot 1\right),$$ hence $N_{2,1}=\frac{3}{4}$. Subtracting $M_1[2]\cdot n_{1,1}=\frac{3}{4}\cdot 1$ we get $n_{2,1}=0$. Moreover, $$246 = 9\cdot\left(\frac{9}{3}\cdot N_{3,1}+\frac{18}{3}\cdot 3\right),$$ hence $N_{3,1}=\frac{28}{9}$. Subtracting $M_1[3]\cdot n_{1,1}=\frac{10}{9}\cdot 1$ we get $n_{3,1}=2$. Finally, $$307041 = 18 \cdot \left(\frac{9}{3}\cdot N_{6,1}+\frac{9}{3}\cdot\frac{8533}{9}+\frac{18}{3}\cdot\frac{3789}{4}+\frac{18}{3}\cdot 948\right),$$ hence $N_{6,1}=\frac{2842}{3}$. Subtracting $M_1[6]\cdot n_{1,1}+M_2[3]\cdot n_{2,1}+M_3[2]\cdot n_{3,1}=\frac{77}{6}\cdot 1+\frac{91}{9}\cdot 0+\frac{15}{4}\cdot 2$ we get $n_{6,1}=927$.
### Degenerating the divisor {#S:calcP2b}
Since the irreducible components of the divisor $D_t^0=D_1+D_2+D_3$ smoothing to $D_t^s=E$ are isomorphic to each other, we can fix any divisor $D_i$ and write $[d_1,\ldots,d_k]:=[i;d_1,\ldots,d_k]$. The tropical curves contributing to the invariants $N_{[d_1,\ldots,d_k]}$ for $d\leq 4$ are shown in Figure \[fig:colors\]. The only $[d_1,\ldots,d_k]$ with more than one associated tropical curve is $[2,2]$ which has two tropical curves.
The attached functions are as follows: $$\begin{aligned}
f_{[1]} &=&1+9x^3 + \mathcal{O}(x^6) \\
f_{[2]} &=&(1+8x^3)(1+72x^6) + \mathcal{O}(t^9) \\
f_{[1,1]} &=& 1+36x^6 + \mathcal{O}(x^9) \\
f_{[3]} &=& (1+9x^3)(1+72x^6)(1-78x^9) + \mathcal{O}(x^{12}) \\
f_{[2,1]} &=& 1+243x^9 + \mathcal{O}(x^{12}) \\
f_{[1,1,1]} &=& 1+81x^9 + \mathcal{O}(x^{12}) \\
f_{[4]} &=& (1+9x^3)(1+72x^6)(1-78x^9)(1+5256x^{12}) + \mathcal{O}(x^{15}) \\
f_{[3,1]} &=& 1+1872x^{12} + \mathcal{O}(x^{15}) \\
f_{[2,2]} &=& (1+36x^6)(1+1296x^{12})(1+1530x^{12}) + \mathcal{O}(x^{15}) \\
f_{[2,1,1]} &=& 1+432x^{12} + \mathcal{O}(x^{15}) \\
f_{[1,2,1]} &=& 1+1296x^{12} + \mathcal{O}(x^{15})\end{aligned}$$
From this one easily calculates the log Gromov-Witten invariants $N_{[d_1,\ldots,d_k]}$. The invariant $N_{[d]}$ has a $d$-fold cover contribution from $n_{[d]}$ and $N_{[2,2]}$ has a $2$-fold cover contribution from $n_{[1,1]}$. This gives the following log BPS numbers:
-------------- ----------------- ----------------- ------------------ ------------------- --------------------
$n_{[1]}=3$
$n_{[2]}=3$ $n_{[1,1]}=6$
$n_{[3]}=15$ $n_{[2,1]}=27$ $n_{[1,1,1]}=9$
$n_{[4]}=72$ $n_{[3,1]}=156$ $n_{[2,2]}=168$ $n_{[2,1,1]}=36$ $n_{[1,2,1]}=108$ $n_{[1,1,1,1]}=12$
-------------- ----------------- ----------------- ------------------ ------------------- --------------------
: The log BPS numbers of $(\mathbb{P}^2,E)$ with fixed limit under degeneration of $E$.
\[tab:colors\]
\
(-7.2,-1.7) – (-3,-1) – (0,0) – (0,1) – (-3,2) – (-7.2,2.7); (-3,-1) – (10,-1); (-3,2) – (10,2); (0,0) – (10,0); (0,1) – (10,1); (-1.5,-0.5) – (0,0) – (0,0.5); (0,0) – (2.25,0.25) – (-1.5,1.5); (2.25,0.25) – (10,0.25); (-1.5,-0.5) – (0,0) – (0,0.5); (0,0) – (10,0); (-6,2.5) – (3,1) – (-1.5,-0.5); (3,1) – (10,1); (-1.5,-0.5) – (1.5,0.5) – (-1.5,1.5); (1.5,0.5) – (10,0.5); (0,0) – (3,0.5) – (0,1); (3,0.5) – (10,0.5); (-1.5,-0.5) – (0,0) – (0,0.5); (0,0) – (4.5,0.75) – (-6,2.5); (4.5,0.75) – (10,0.75); (-6,-1.5) – (6,0.5) – (-6,2.5); (6,0.5) – (10,0.5); (-1.5,-0.5) – (0,0) – (0,0.5); (0,0) – (2,0.33) – (-1.5,1.5); (2,0.33) – (10,0.33); (0,0.5) – (0,1) – (-1.5,1.5); (-6,2.5) – (3,1) – (-1.5,-0.5); (0,1) – (10,1); (0) at (0,0.5); (0) at (-1.5,-0.5); (0) at (-1.5,1.5); (0) at (-6,-1.5); (0) at (-6,2.5);
### Combining the methods {#S:calcP2c}
As indicated in Remark \[rem:combine\] in general it is not possible to combine the above methods and calculate invariants with prescribed limit under degeneration of the divisor and meeting a points on $E$ with prescribed torsion. However, for $d\leq 3$ this is indeed possible as we will show now.
For $d=1$ we have $\mathcal{G}_1=\{[1]\}$, so $n_{[1],1}=n_{1,1}=1$. For $d=2$ there are two types of tropical curves. One of them contributes to $N_{[1,1]}$ and corresponds to stable log maps meeting $E$ in a point of order $6$, so $n_{[1,1],1}=0$ and $n_{[1,1],2}=r_2 \cdot n_{2,2}=3$. The other one contributes to $N_{[2]}$, so $n_{[2],1}=r_1 \cdot n_{1,1}=0$.
For $d=3$ we have $l([3])=1$, $l([2,1])=3$ and $l([1,1,1])=1$. So $n_{[2,1],1}=0$ and the equations in Remark \[rem:combine\] are the following: $$\begin{aligned}
3n_{[3],1} + 6n_{[3],3} &= 15 & n_{[3],1} + n_{[1,1,1],1} &= 2 \\
9n_{[2,1],3} &= 27 & n_{[3],3} + n_{[1,1,1],3} &= 3 \\
3n_{[1,1,1],1} + 6n_{[1,1,1],3} &= 9 & n_{[2,1],3} &= 3\end{aligned}$$ This system of linear equations has the unique solution:
$n_{[3],1}=1$ $n_{[2,1],1}=0$ $n_{[1,1,1],1}=2$
--------------- ----------------- -------------------
$n_{[3],3}=1$ $n_{[2,1],3}=3$ $n_{[1,1,1],3}=1$
: The log BPS numbers of $(\mathbb{P}^2,E)$ for $d=3$ with fixed limit under degeneration of $E$ and prescribed torsion.
\[tab:P1xP1\]
\
If we try the same method for $d>3$, we have more indeterminates than independent equations, so there is no unique solution.
$\mathbb{P}^1\times\mathbb{P}^1$
--------------------------------
Similarly, executing the code for $\mathbb{P}^1\times\mathbb{P}^1$ gives the following functions: $$\begin{aligned}
\text{log }f_{\text{out}} = 16x^2 + 72x^4 + 352x^6 + 3108x^8 + \frac{120016}{5}x^{10} + 198384x^{12} + \mathcal{O}(x^{14})\end{aligned}$$ This gives:
$N_1=8$ $N_2=18$ $N_3=\frac{1312}{3}$ $N_4=\frac{777}{2}$ $N_5=\frac{60008}{25}$ $N_6=16532$
--------- ---------- ---------------------- --------------------- ------------------------ -------------
: The invariants $N_d$ of $\mathbb{P}^1\times\mathbb{P}^1$ for $d \leq 6$.
\[tab:N\]
$n_1=8$ $n_2=16$ $n_3=72$ $n_4=368$ $n_5=2400$ $n_6=16320$
--------- ---------- ---------- ----------- ------------ -------------
: The log BPS numbers $n_d$ of $\mathbb{P}^1\times\mathbb{P}^1$ for $d \leq 6$.
\[tab:n\]
Moreover, the functions and invariants with prescribed limits under $s\rightarrow 0$ are: $$\begin{aligned}
f_{[1]} &=& 1+4x^2 + \mathcal{O}(x^4) \\
f_{[2]} &=& (1+4x^2)(1+10x^4) + \mathcal{O}(x^6) \\
f_{[1,1]} &=& 1+16x^4 + \mathcal{O}(x^6) \\
f_{[3]} &=& (1+4x^2)(1+10x^4)(1-20x^6) + \mathcal{O}(x^8) \\
f_{[2,1]} &=& 1+36x^6 + \mathcal{O}(x^8) \\
f_{[1,1,1]} &=& 1+36x^6 + \mathcal{O}(x^8) \\
f_{[4]} &=& (1+4x^2)(1+10x^4)(1-20x^6)(1+115x^8) + \mathcal{O}(x^{10}) \\
f_{[3,1]} &=& 1+64x^8 + \mathcal{O}(x^{10}) \\
f_{[2,2]} &=& (1+16x^4)(1+64x^8)(1+264x^8) + \mathcal{O}(x^{10}) \\
f_{[2,1,1]} &=& 1+64x^8 + \mathcal{O}(x^{10}) \\
f_{[1,2,1]} &=& 1+256x^8 + \mathcal{O}(x^{10}) \\
f_{[1,1,1,1]} &=& 1+64x^8 + \mathcal{O}(x^{10})\end{aligned}$$
------------- --------------- ----------------- ----------------- ------------------ -------------------
$n_{[1]}=2$
$n_{[2]}=0$ $n_{[1,1]}=4$
$n_{[3]}=0$ $n_{[2,1]}=6$ $n_{[1,1,1]}=6$
$n_{[4]}=0$ $n_{[3,1]}=8$ $n_{[2,2]}=15$ $n_{[2,1,1]}=8$ $n_{[1,2,1]}=32$ $n_{[1,1,1,1]}=8$
------------- --------------- ----------------- ----------------- ------------------ -------------------
: The log BPS numbers of $\mathbb{P}^1\times\mathbb{P}^1$ with fixed limit under degeneration of the divisor.
\[tab:P1xP1\]
From this we compute the log BPS numbers with given curve class (bidegree) as follows. The factors are the number of unbounded walls in the wall structure contributing to $n_{[i;d_1,\ldots,d_m]}$ for different choices of $i$. $$\begin{aligned}
n_{(1,0)} &= 2 \cdot n_{[1]} &= 2 \\
n_{(2,0)} &= 2 \cdot n_{[2]} &= 0 \\
n_{(1,1)} &= 4 \cdot n_{[1,1]} &= 16 \\
n_{(3,0)} &= 2 \cdot n_{[3]} &= 0 \\
n_{(2,1)} &= 4 \cdot n_{[2,1]} + 2 \cdot n_{[1,1,1]} &= 36 \\
n_{(4,0)} &= 2 \cdot n_{[4]} &= 0 \\
n_{(3,1)} &= 4 \cdot n_{[3,1]} + 4 \cdot n_{[2,1,1]} &= 768 \\
n_{(2,2)} &= 4 \cdot n_{[2,2]} + 4 \cdot n_{[1,2,1]} + 4 \cdot n_{[1,1,1,1]} &= 1152\end{aligned}$$
### Deforming $\mathbb{F}_2$ {#S:calc8'a}
Consider the toric degeneration of $\mathbb{P}^1\times\mathbb{P}^1$ by deformation of the Hirzebruch surface $\mathbb{F}_2$ (case (8’a) in Figure \[fig:listb\]) from Example \[expl:8’a2\].
For $d=1$ it is clear by the symmetry $L_1\leftrightarrow L_2$ that $n_{(1,0)}=n_{L_1}=n_{L_2}=4$. For $d=2$ we have $n_{(2,0)}=n_{2L_1}=n_{2L_2}$ and $n_{(1,1)}=n_{L_1+L_2}$. Figure \[fig:calc8’a\] shows the tropical curves corresponding to stable log maps of class $\underline{\beta}=L_1+L_2$. The first one has multiplicity $4$, the second one $-4$, the third one $8$ and the last one $4$. By symmetry there are two tropical curves similar to the red ones at the lower vertex, again with multiplicities $-4$ and $8$. This gives $n_{(1,1)}=n_{L_1+L_2}=16$ and in turn $n_{(2,0)}=0$. One can proceed similarly for higher degrees.
(-8,3.5) – (-9,4) node\[fill,circle,inner sep=1pt\] – (-10,4.25); (-9,4) – (-8.25,4) node\[above\][$4$]{} – (-7.5,4); (-10,2.5) – (-9.5,2.25) node\[above\][$2$]{} – (-9,2) node\[fill,circle,inner sep=1pt\] – (-8.9,1.5); (-9,2) – (-9.1,1.5) node\[fill,circle,inner sep=1pt\] – (-9.2,1); (-9,2) – (-8.25,2) node\[above\][$4$]{} – (-7.5,2); (-10,0.4) – (-9,0) node\[fill,circle,inner sep=1pt\] – (-8.9,-0.5); (-9,0) – (-9.1,-0.5) node\[fill,circle,inner sep=1pt\] – (-9.2,-1); (-9,0) – (-8.25,0) node\[above\][$4$]{} – (-7.5,0); (-10,0.6) – (-9,0); (-10,-2.5) – (-9,-2) node\[fill,circle,inner sep=1pt\] – (-10,-1.5); (-9,-2) – (-8.25,-2) node\[above\][$4$]{} – (-7.5,-2); (-7,1) – (-6,1); (-5.6,3.6) – (-2,3); (-2,3) – (0,2) – (0,0) – (-2,-1); (-2,-1) – (-5.6,-1.6); (-2,3) – (6,3) node\[right\][$L_1+L_2$]{}; (0,2) – (6,2) node\[right\][$L_2$]{}; (0,1) – (6,1) node\[right\][$L_1-L_2$]{}; (0,0) – (6,0) node\[right\][$L_2$]{}; (-2,-1) – (6,-1); (0) at (-5,3.5); (0) at (-1,2.5); (0) at (0,1.5); (0) at (0,0.5); (0) at (-1,-0.5); (0) at (-5,-1.5); (-1,2.5) – (-2,3) – (2,3) node\[above\][$4$]{} – (6,3); (-2,3) – (-5,3.5); (-1,2.5) – (0,2) node\[left\][$2$]{} – (0,1) node\[left\][$1$]{} – (0,0.5); (0,2) – (3,2) node\[above\][$4$]{} – (6,2); (-1,2.5) – (2,1) – (-1,-0.5); (2,1) – (4,1) node\[above\][$4$]{} – (6,1);
Cubic surface {#S:calccubic}
-------------
The dual intersection complex of the cubic surface (case (3a) in Figure \[fig:listb\]) is quite similar to the one of $(\mathbb{P}^2,E)$. The only differences are that for each vertex the determinant of primitive generators of adjacent bounded edges is $1$ instead of $3$ and that there are three affine singularities on each bounded edge. As a consequence, by the change of lattice trick ([@GHK2], Proposition C.13), the wall structure of $(X,D)$ is in bijection with the wall structure of $(\mathbb{P}^2,E)$, and the functions attached to walls in direction $m_{\text{out}}$, in particular the unbounded walls, coincide. This immediately implies $$N_d(X,D) = 3 \cdot N_d(\mathbb{P}^2,E).$$ Subtracting multiple covers we get (note that $w_{\text{out}}=1$ and the multiple cover contributions of of degree $1$ curves are $M_1[k]=\frac{1}{k^2}\binom{-1}{k-1}=\frac{(-1)^{k-1}}{k^2}$):
$n_1=27$ $n_2=108$ $n_3=729$ $n_4=6912$ $n_5=76275$ $n_6=920727$
---------- ----------- ----------- ------------ ------------- --------------
: The log BPS numbers $n_d$ of the cubic surface for $d \leq 6$.
\[tab:n\]
### Curve classes
A smooth cubic surface $X$ can be given by blowing up six general points on $\mathbb{P}^2$. Let $e_1,\ldots,e_1$ be the classes of the exceptional divisors and let $\ell$ be the pullback of the class of a line in $\mathbb{P}^2$. Then $\ell,e_1,\ldots,e_6$ generate $\text{Pic}(X)\simeq H_2^+(X,\mathbb{Z})\simeq\mathbb{Z}^7$.
The dual intersection complex of a smooth nef toric surface $X^0$ deforming to $X$ is shown in Figure \[fig:fan\]. Its asymptotic fan is the fan of $X^0$. Denote the curves corresponding to the rays of this fan by $C_1,\ldots,C_9$, labelled as in Figure \[fig:fan\].
\(0) at (0,0); (1) at (-1,-1); (2) at (2,-1); (3) at (-1,2); (1a) at (-2,-2); (2a) at (4,-2); (3a) at (-2,4); (12) at (-0.5,-1); (12a) at (-1.5,-2); (12b) at (-0.5,-2); (12’) at (0.5,-1); (12’a) at (0.5,-2); (12’b) at (1.5,-2); (12”) at (1.5,-1); (12”a) at (2.5,-2); (12”b) at (3.5,-2); (23) at (-0.5,1.5); (23a) at (-0.5,4); (23b) at (-1.75,4); (23’) at (0.5,0.5); (23’a) at (4,0.5); (23’b) at (0.5,4); (23”) at (1.5,-0.5); (23”a) at (4,-0.5); (23”b) at (4,-1.75); (31) at (-1,-0.5); (31a) at (-2,-1.5); (31b) at (-2,-0.5); (31’) at (-1,0.5); (31’a) at (-2,0.5); (31’b) at (-2,1.5); (31”) at (-1,1.5); (31”a) at (-2,2.5); (31”b) at (-2,3.5); (1) – (2) – (3) – (1); (0,0) – (1a) node\[below\][$C_9$]{}; (0,0) – (0,-2) node\[below\][$C_1$]{}; (0,0) – (2,-2) node\[below\][$C_2$]{}; (0,0) – (2a) node\[below\][$C_3$]{}; (0,0) – (4,0) node\[right\][$C_4$]{}; (0,0) – (0,4) node\[above\][$C_5$]{}; (0,0) – (3a) node\[left\][$C_6$]{}; (0,0) – (-2,2) node\[left\][$C_7$]{}; (0,0) – (-2,0) node\[left\][$C_8$]{}; (b) at (-1,0); (c) at (-1,1); (d) at (0,-1); (e) at (1,-1); (f) at (1,0); (g) at (0,1);
Then (see [@KM], §4) an isomorphism $\text{Pic}(X^0) \simeq \text{Pic}(X)$ is given as follows: $$\begin{aligned}
[C_1] &\mapsto e_2-e_5 & [C_2] &\mapsto \ell-e_2-e_3-e_6 & [C_3] &\mapsto e_6 \\
[C_4] &\mapsto e_3-e_6 & [C_5] &\mapsto \ell-e_1-e_3-e_4 & [C_6] &\mapsto e_4 \\
[C_7] &\mapsto e_1-e_4 & [C_8] &\mapsto \ell-e_1-e_2-e_5 & [C_9] &\mapsto e_5 \\\end{aligned}$$\
Now we know the curve classes of the cubic surface $X$ corresponding to the unbounded edges in the dual intersection complex $(B,\mathscr{P},\varphi)$. In turn, we are able to associate to each tropical curve in $(B,\mathscr{P},\varphi)$ the curve class of the corresponding stable log maps, by composition of the maps from Definitions \[defi:HG\] and \[defi:GH\].
As shown in [@Hos], the Weyl group $W_{E_6}$ of type $E_6$ acts on $\text{Pic}(X)$ as symmetries of configurations of the $27$ lines and this action preserves the local Gromov-Witten invariants $N_\beta^{\text{loc}}$ of $X$. Hence, by the log-local correspondence [@vGGR], it preserves the logarithmic Gromov-Witten invariants $N_\beta$ of $X$ considered here. The curve classes $\underline{\beta}$ of the cubic $X$ giving a nonzero contribution $N_\beta^{\text{loc}}$, up to action of $W_{E_6}$, are given in [@KM], Table 1, along with the corresponding local BPS number $n_\beta^{\text{loc}}$.
For $d=1$ and $d=2$ there is, up to the action of $W_{E_6}$, only one curve class giving a nonzero contribution, so this is trivial. For $d=1$ this is $\underline{\beta}=e_6$ and the length of its orbit is $27$, so $n_\beta=1$. For $d=2$ it is $\underline{\beta}=\ell-e_1$, with orbit length $27$, so $n_\beta=4$. For $d=3$ there are two equivalence classes giving a nonzero contribution, with representatives $\ell$ and $3\ell-\sum_{i=1}^6 e_i$, respectively.
(-5,-4) – (-3,-3); (-3,-3) – (0,0) – (0,3) – (-3,6); (-3,6) – (-5,7); (-3,6) – (10,6) node\[right\][$e_5$]{}; (-2,5) – (10,5) node\[right\][$e_2-e_5$]{}; (-1,4) – (10,4) node\[right\][$\ell-e_2-e_3-e_6$]{}; (0,3) – (10,3) node\[right\][$e_6$]{}; (0,2) – (10,2) node\[right\][$e_3-e_6$]{}; (0,1) – (10,1) node\[right\][$\ell-e_1-e_3-e_4$]{}; (0,0) – (10,0) node\[right\][$e_4$]{}; (-1,-1) – (10,-1) node\[right\][$e_1-e_4$]{}; (-2,-2) – (10,-2) node\[right\][$\ell-e_1-e_2-e_5$]{}; (-3,-3) – (10,-3); (0) at (-4,6.5); (0) at (-0.5,3.5); (0) at (-1.5,4.5); (0) at (-2.5,5.5); (0) at (0,0.5); (0) at (0,1.5); (0) at (0,2.5); (0) at (-0.5,-0.5); (0) at (-1.5,-1.5); (0) at (-2.5,-2.5); (0) at (-4,-3.5); (-2.5,5.5) – (-2,5) node\[left\][$1$]{} – (-1,4) node\[left\][$2$]{} – (0,3) node\[left\][$3$]{} – (0,2) node\[left\][$2$]{} – (0,1) node\[left\][$1$]{} – (0,0.5); (0,3) – (5,3) node\[above\][$3$]{} – (10,3); (-4,6.5) – (3,3) – (-2.5,-2.5); (3,3) – (7,3) node\[above\][$3$]{} – (10,3);
The red tropical curve in Figure \[fig:calccubic\] corresponds to the class $$1 \cdot (e_2-e_5) + 2 \cdot (\ell-e_2-e_3-e_6) + 3 \cdot e_6 + 2 \cdot (e_3-e_6) + 1 \cdot (\ell-e_1-e_3-e_4) = 3\ell-\sum_{i=1}^6e_i$$ and similarly for the blue tropical curve. Changing the affine singularities in which the bounded legs end may change the curve class. It turns out that for the red tropical curve any change leads to the class $\underline{\beta}=\ell$ or to a class giving a nonzero contribution. Its multiplicity is $18$, so together with the choice of outgoing edge this gives a contribution of $54$ to $n_{3\ell-\sum_{i=1}^6e_i}$. For the blue tropical curve there are two changes leading again to $\underline{\beta}=3\ell-\sum_{i=1}^6e_i$ and six changes leading to $\underline{\beta}=\ell$. The multiplicity of any of these tropical curves is $3$. Together with the choice of outgoing edge this gives a contribution of $3 \cdot 3 \cdot 3 = 27$ to $n_{3\ell-\sum_{i=1}^6e_i}$. The orbit length is $1$, so $n_{3\ell-\sum_{i=1}^6e_i}=81$. The orbit length of $\ell$ is $72$, so $n_\ell = (729-81)/72 = 9$. So, in agreement with [@KM], Table 1, we have:
$n_{e_i}=1$ $n_{\ell-e_i}=4$ $n_\ell=9$ $n_{3\ell-\sum_{i=1}^6e_i}=81$
------------- ------------------ ------------ --------------------------------
: The log BPS numbers $n_\beta$ of the cubic surface for $d \leq 3$.
\[tab:n\]
\
Figure \[fig:calccubic2\] shows tropical curves corresponding to stable log maps of class $\underline{\beta}=\ell$. In particular, any change of bounded legs of the green tropical curves still leads to class $\underline{\beta}=\ell$. For instance, the red tropical curve has class $$1 \cdot (e_2-e_5) + 2 \cdot (\ell-e_2-e_3-e_6) + 3 \cdot e_6 + 1 \cdot (e_3-e_6) = 2\ell-e_2-e_3-e_5$$ Under the action of $W_{E_6}$ this is equivalent to $\underline{\beta}=2\ell-e_1-e_2-e_3$ and in turn (via the map $s_6$ from [@KM], §3) to $$2 \cdot (2\ell-e_1-e_2-e_3)-(\ell-e_2-e_3)-(\ell-e_1-e_3)-(\ell-e_1-e_2) = \ell.$$ Similarly, one computes the classes of the other tropical curves.
(-5,-4) – (-3,-3); (-3,-3) – (0,0) – (0,3) – (-3,6); (-3,6) – (-5,7); (-3,6) – (10,6) node\[right\][$e_5$]{}; (-2,5) – (10,5) node\[right\][$e_2-e_5$]{}; (-1,4) – (10,4) node\[right\][$\ell-e_2-e_3-e_6$]{}; (0,3) – (10,3) node\[right\][$e_6$]{}; (0,2) – (10,2) node\[right\][$e_3-e_6$]{}; (0,1) – (10,1) node\[right\][$\ell-e_1-e_3-e_4$]{}; (0,0) – (10,0) node\[right\][$e_4$]{}; (-1,-1) – (10,-1) node\[right\][$e_1-e_4$]{}; (-2,-2) – (10,-2) node\[right\][$\ell-e_1-e_2-e_5$]{}; (-3,-3) – (10,-3); (0) at (-4,6.5); (0) at (-0.5,3.5); (0) at (-1.5,4.5); (0) at (-2.5,5.5); (0) at (0,0.5); (0) at (0,1.5); (0) at (0,2.5); (0) at (-0.5,-0.5); (0) at (-1.5,-1.5); (0) at (-2.5,-2.5); (0) at (-4,-3.5); (-2.5,5.5) – (-2,5) node\[left\][$1$]{} – (-1,4) node\[left\][$2$]{} – (0,3) node\[left\][$3$]{} – (0,2) node\[left\][$1$]{} – (0,1.5); (0,3) – (5,3) node\[above\][$3$]{} – (10,3); (-4,6.5) – (3,3) – (-1.5,-1.5); (3,3) – (7,3) node\[above\][$3$]{} – (10,3); (-0.5,3.5) node\[right\][$2$]{} – (0,3) – (0,2.5); (0,3) – (2,2) – (-0.5,-0.5); (2,2) – (6,2) node\[below\][$3$]{} – (10,2);
Artin fans and logarithmic modifications {#A:artin}
========================================
An *Artin fan* is a logarithmic algebraic stack that is logarithmically étale over a point. Artin fans were introduced in [@AW] to prove the invariance of logarithmic Gromov-Witten invariants under *logarithmic modifications*, that is, proper birational logarithmically étale morphisms. We will briefly summarize this subject.
To any fine saturated log smooth scheme $X$ one can define an associated Artin fan $\mathcal{A}_X$. It has an étale cover by finitely many *Artin cones* – stacks of the form $[V/T]$, where $V$ is a toric variety and $T$ its dense torus. In this way, $\mathcal{A}_X$ encodes the combinatorial structure of $X$. A *subdivision* of the Artin fan $\mathcal{A}_X$ induces a logarithmic modification of $X$ via pull-back. Moreover, all logarithmic modifications of $X$ arise this way. This ultimately leads to a proof of the birational invariance in logarithmic Gromov-Witten theory [@AW].
Olsson [@Ol] showed that a logarithmic structure on a given underlying scheme $\underline{X}$ is equivalent to a morphism $\underline{X}\rightarrow\underline{\textbf{Log}}$, where $\underline{\textbf{Log}}$ is a zero-dimensional algebraic stack – the moduli stack of logarithmic structures. It carries a universal logarithmic structure whose associated logarithmic algebraic stack we denote by $\textbf{Log}$ – providing a universal family of logarithmic structures $\textbf{Log}\rightarrow\underline{\textbf{Log}}$. As shown in [@AW], if $X$ is a fine saturated log smooth scheme, then the morphism $X\rightarrow\textbf{Log}$ factors through an initial morphism $X\rightarrow\mathcal{A}_X$, where $\mathcal{A}_X$ is an Artin fan and $\mathcal{A}_X\rightarrow\textbf{Log}$ is étale and representable. While this serves as a definition of the associated Artin fan $\mathcal{A}_X$, there is a more explicit description of $\mathcal{A}_X$ in terms of the *tropicalization* of $X$, given below.
Let $S$ be a fine saturated log scheme.
\[defi:Log\] The *moduli stack of log structures over $S$* is the category $\underline{\textbf{Log}_S}$ fibered over the category of $\underline{S}$-schemes defined as follows. The objects over a scheme morphism $\underline{X}\rightarrow\underline{S}$ are the log morphisms $X\rightarrow S$ over $\underline{X}\rightarrow\underline{S}$. The morphisms from $X\rightarrow S$ to $X'\rightarrow S$ are the log morphisms $h : X \rightarrow X'$ over $S$ for which $h^\star\mathcal{M}_{X'}\rightarrow\mathcal{M}_X$ is an isomorphism.
$\underline{\textup{\textbf{Log}}_S}$ is an algebraic stack locally of finite presentation over $\underline{S}$.
An *Artin fan* is a logarithmic algebraic stack that is logarithmically étale over a point. An *Artin cone* is a logarithmic algebraic stack isomorphic to $[V/T]$, where $V$ is a toric variety and $T$ its dense torus.
\[rem:artin\] If a logarithmic algebraic stack has a strict representable étale cover by Artin cones, then it is an Artin fan. In fact, in [@AW] Artin fans were defined this way. Later the definition was generalized to the one above.
\[lem:artin\] An algebraic stack that is representable and étale over $\textup{\textbf{Log}}$ has a strict étale cover by Artin cones.
\[prop:artin\] Let $X$ be a logarithmic algebraic stack that is locally connected in the smooth topology. Then there is an initial factorization of $X\rightarrow \textbf{Log}$ through a strict étale morphism $\mathcal{A}_X \rightarrow \textbf{Log}$ which is representable by algebraic spaces.
Let $X$ be a fine saturated log smooth scheme. The *Artin fan of $X$* is the stack $\mathcal{A}_X$ from Proposition \[prop:artin\]. Indeed, this is an Artin fan by Lemma \[lem:artin\] and Remark \[rem:artin\].
We now give a more explicit description of the Artin fan $\mathcal{A}_X$ of a fine saturated log smooth scheme $X$. By Lemma \[lem:artin\] and Proposition \[prop:artin\], $\mathcal{A}_X$ has a strict étale cover by Artin cones. In fact, $\mathcal{A}_X$ is a colimit of Artin cones $\mathcal{A}_\sigma$ corresponding to the cones $\sigma$ in the tropicalization $\Sigma(X)$ of $X$.
For a cone $\sigma\subseteq N_{\mathbb{R}}$, let $P=\sigma^\vee\cap M$ be the corresponding monoid. The *Artin cone defined by $\sigma$* is the logarithmic algebraic stack $$\mathcal{A}_\sigma = \left[\faktor{\textup{Spec }\mathbb{C}[P]}{\textup{Spec }\mathbb{C}[P^{\textup{gp}}]}\right]$$ with the toric log structure coming from the global chart $P \rightarrow \mathbb{C}[P]$.
Let $\Sigma$ be a generalized cone complex (Definition \[defi:Cones\]) that is a colimit of a diagram of cones $s : I \rightarrow \textup{\textbf{Cones}}$. Then define $\mathcal{A}_\Sigma$ to be the colimit as sheaves over $\textup{Log}$ of the corresponding diagram of sheaves given by $I \ni i \mapsto \mathcal{A}_{s(i)}$.
Let $X$ be a fine saturated log smooth scheme with tropicalization (Definition \[defi:trop\]) a generalized cone complex $\Sigma(X)$. Then $$\mathcal{A}_X \cong \mathcal{A}_{\Sigma(X)}.$$
A *subdivision* of an Artin fan $\mathcal{X}$ is a morphism of Artin fans $\mathcal{Y}\rightarrow\mathcal{X}$ whose base change via any map $\mathcal{A}_\sigma\rightarrow\mathcal{X}$ is isomorphic to $\mathcal{A}_\Sigma$ for some subdivision $\Sigma$ of $\sigma$.
A *logarithmic modification* of fine saturated log smooth schemes is a proper surjective logarithmically étale morphism.
Let $X$ be a fine saturated log smooth scheme with tropicalization $\Sigma(X)$. Then a subdivision $\tilde{\Sigma}(X)$ of $\Sigma(X)$ gives a subdivision $\mathcal{A}_{\tilde{\Sigma}(X)} \rightarrow \mathcal{A}_X$ of the Artin fan of $X$. The pull back $\tilde{X} := \mathcal{A}_{\tilde{\Sigma}(X)} \times_{\mathcal{A}_X} X \rightarrow X$ is a logarithmic modification. Moreover, all logarithmic modifications of $X$ arise this way:
If $Y\rightarrow X$ is a logarithmic modification of fine saturated log smooth schemes, then $Y\rightarrow\mathcal{A}_Y \times_{\mathcal{A}_X} X$ is an isomorphism.
\[thm:AW\] Let $h : Y \rightarrow X$ be a logarithmic modification of log smooth schemes. This induces a projection $\pi : \bar{\mathscr{M}}(Y) \rightarrow \bar{\mathscr{M}}(X)$ with $$\pi_\star\llbracket\bar{\mathscr{M}}(Y)\rrbracket = \llbracket\bar{\mathscr{M}}(X)\rrbracket,$$ where $\bar{\mathscr{M}}(X)$ is the stack of stable log maps to $X$.
Logarithmic Gromov-Witten invariants are invariant under logarithmic modifications.
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[^1]: In fact, there is additional information captured in so called *gluing data* ([@DataI], Definition 2.25). In this paper we always make the trivial choice, setting $s_e=1$ for any inclusion $e : \omega \rightarrow \tau$ of cells $\omega,\tau\in\mathscr{P}$, and will not mention gluing data.
[^2]: The basic monoid $Q$ has the property that $\Sigma(\textup{pt}_Q)=\textup{Hom}(Q,\mathbb{R}_{\geq 0})$ is the moduli space of deformations of $\Sigma(C)$ as a tropical curve preserving its combinatorial type ([@LogGW], Remark 1.21).
[^3]: The affine structure on $\Gamma$ is given by the lengths $\ell_E$ of its edges.
[^4]: This means that the rays in $\mathfrak{D}$ come in pairs, one incoming and one outgoing, with the same attached function
[^5]: This is unique by the construction of discriminant locus $\Delta$ for tropical affine manifolds, see [@GS11], p.10.
[^6]: <https://www.math.uni-hamburg.de/en/personen/graefnitz/>
|
---
abstract: 'We show that the dynamical behavior of a coupled map lattice where the individual maps are Bernoulli shift maps can be solved analytically for integer couplings. We calculate the invariant density of the system and show that it displays a nontrivial spatial behavior. We also introduce and calculate a generalized spatiotemporal correlation function.'
address: |
Condensed Matter Physics 114-36 and Neural Systems Program 139-74\
California Institute of Technology, Pasadena CA 91125
author:
- 'R. O. Grigoriev and H. G. Schuster\*'
title: A Solvable Model for Spatiotemporal Chaos
---
psfig
Introduction
============
The study of temporal chaos in low-dimensional systems, some of which can be described by low-dimensional maps [@frish; @schuster], was extremely beneficial for the understanding of turbulence. In 1984 coupled map lattices were introduced into the physical literature as a tool for studying spatiotemporal chaos in spatially extended, i.e. high-dimensional systems [@waller]. They consist of spatially coupled low-dimensional maps and represent dynamical systems that are discrete in space and time, but continuous in the state variable. They serve as models for coupled Josephson junctions, excitable media, population dynamics, neural dynamics and turbulence [@kaneko]. Although Bunimovich and Sinai mathematically proved a number of statements regarding the appearance of coherent structures from spatiotemporal chaos [@sinai], most results in the field have been obtained by numerical simulations [@kaneko; @politi]. While the study of temporal chaos has greatly profited from the existence of simple maps like the Bernoulli shift map and the cat map [@schuster; @arnold], which can be solved explicitly (for integer expansion rates), thereby making the mechanisms of mixing and temporal chaos understandable, no investigation of this type has been provided up to now for the problem of spatiotemporal chaos. Here we present a solvable model for spatiotemporal chaos which can be solved in the sense of the Bernoulli map.
We start from the model equations [@waller] $$\label{eq_cml_gen}
z_{i}^{t+1}=(1-\varepsilon )f(z_{i}^{t})+\frac{\varepsilon }{2}\left[
f(z_{i+1}^{t})+f(z_{i-1}^{t})\right],$$ where the index $i$ runs over the $N$ sites of a discrete lattice, $\varepsilon$ measures the strength of the spatial coupling between neighboring sites and $f(z_i^t)$ is a local map which determines the evolution of the continuous variable $z_i^t$ at discrete time-steps $t=0,1,2,\cdots$. The extension of eq. (\[eq\_cml\_gen\]) to more than nearest neighbor coupling and arbitrary dimensions is straightforward. We first write it in the new variables $x_i^t=f(z_i^t)$ as $$\label{eq_cml}
x_{i}^{t+1}=f\left[(1-\varepsilon)x_{i}^{t}+\frac{\varepsilon}{2}
\left(x_{i+1}^{t}+x_{i-1}^{t}\right) \right].$$
In order to obtain a solvable model we choose $f(x)=(ax)\,{\rm mod}\;1$, i.e. the Bernoulli shift map with integer stretching factor $a$. To make our approach for the spatially extended system more transparent we will calculate in Section \[sec\_single\] the invariant density and a temporal correlation function for the single Bernoulli shift map. In section \[sec\_lattice\] we will show that the coupled map lattice is solvable for special values of the coefficients $a$ and $\varepsilon$ in eq. (\[eq\_cml\]) and we will also calculate the invariant density and a spatial and a spatiotemporal correlation functions. Finally in section \[sec\_discuss\] we discuss our results and indicate directions of further research.
Properties of a single Bernoulli shift map {#sec_single}
==========================================
First we recall that the single map $x^{t+1}=(ax^{t})\,{\rm mod}\;1$ can be solved as $x^{t}=(a^{t}x^{0})\,{\rm mod}\;1$ because $$\begin{aligned}
x^{2}=&&\left\{ a\left[ (ax^{0})\,{\rm mod}\;1\right] \right\}
{\rm mod}\;1\cr =&&\left\{ a\left[ ax^{0}-k^{0}\right] \right\}
\,{\rm mod}\;1=(a^{2}x^{0})\,{\rm mod}\;1
\end{aligned}$$ where $k^0$ is an integer which represents the action of the modulo function. For the last equality sign in to hold we needed the fact that $a$ is an integer such that $a k^0$ becomes again an integer, which can be dropped within the last modulo function.
Since the modulo function confines the variable $x^{t}$ to a circle we could view the Bernoulli shift map as a linear map $x^{t+1}=ax^{t}$ where the variables live on a unit circle, i.e. on a 1-torus. We shall see below that we can view our coupled map system as a linear map acting on variables confined to an $N$-torus, where $N$ is the number of lattice sites.
The invariant density $\rho(x)$ of the simple Bernoulli shift map measures the distribution of $x$ values on the attractor generated by the map and is well known to be a constant [@schuster]. We can obtain this result by noting that $\rho (x)$ is defined on an unit circle, i.e. it is periodic in $x$ and therefore can be represented as a Fourier series $$\label{eq_four_sing}
\rho (x)=\sum_k\hat{\rho}(k)e^{2\pi ikx},$$ where $k$ takes only integer values $k=0,\pm 1,\pm 2,\cdots$. The invariant density $\rho(x)$ evolves from an initial distribution $\rho^0(x)$ according to the Frobenius-Perron equation [@schuster] $$\label{eq_dens_sing}
\rho^t\left( x\right)=\int_{0}^{1}dx^{\prime }\delta
\left[ x-\left( a^{t}x'\right)\,
\mathop{\rm mod}\;1\right] \rho^0\left( x^{\prime }\right)$$ and is defined as the long-time limit $\rho(x)=\lim_{t\rightarrow \infty}
\rho^t(x)$. In order to solve eq. (\[eq\_dens\_sing\]) we use (\[eq\_four\_sing\]) and the fact that the Bernoulli shift map becomes a linear map on a torus, i.e. $\exp(2\pi i\left[(a^tx)\,{\rm
mod}\;1\right])=\exp(2\pi ia^{t}x)$, and obtain $$\hat{\rho}^t(k)=\hat{\rho}^0(a^t k).$$
If we make the reasonable assumption that the initial distribution $\rho^0(x)$ is non-singular, then $\lim_{k\rightarrow\pm\infty}
\hat{\rho}^0(k)=0$. This means that in eq. (\[eq\_four\_sing\]) all Fourier coefficients of $\hat{\rho}^t(k)$ tend to zero in the infinite-time limit, except the one, which belongs to $k=0$. Since $$\hat{\rho}^0(0)=\int_0^1 dx\rho^0(x)=1$$ this yields $\hat{\rho}(k)=\delta_{k,0}$ and $\rho(x)=1$.
In a similar fashion we can now define and calculate the time correlation function on the 1-torus. The usual time correlation function is defined via $$\left\langle xx^t\right\rangle=\int_0^1 dx\rho(x)xf^t(x),$$ where the time evolution of $x$ is given by the map $f(x)$, but here we introduce the time correlation function $G(t)$ $$\label{eq_sing_cor}
G(t)=\int_0^1 dx^0\rho(x^0)e^{2\pi i(x^0-x^t)},$$ where $$\label{eq_sing_sol}
x^{t}=(a^{t}x^0)\,{\rm mod}\;1,$$ which respects the fact that the variable $x$ is an angular variable on a torus [@jose].
For $\rho(x)=1$ we obtain from eqs. (\[eq\_sing\_cor\],\[eq\_sing\_sol\]) $$G(t)=\int_{0}^{1}dx\rho \left( x\right)
e^{2\pi i\left( 1-a^{t}\right)x}=\delta_{1,a^{t}}.$$
In the following section we will demonstrate what changes have to be made in order to compute in a similar fashion as above the solution of the dynamical equations, the invariant density and the time correlation function for our coupled map lattice.
A lattice of coupled Bernoulli shift maps. {#sec_lattice}
==========================================
For the Bernoulli shift map $f(x)=(ax)\,{\rm mod}\;1$ the time evolution for the variables $x_{i}^{t}$ of the coupled map lattice becomes according to eq. (\[eq\_cml\]) $$\label{eq_map_nnc}
x_{i}^{t+1}=\left(a\left[(1-\varepsilon)x_{i}^{t}+\frac{\varepsilon}{2}
(x_{i+1}^{t}+x_{i-1}^{t})\right]\right)\,{\rm mod}\;1.$$
If the parameters $a=m+2n$ and $\varepsilon=2n/(m+2n)$ are such that both $(1-\varepsilon)a$ and $a\varepsilon/2$ take integer values $m$ and $n$, then the equation of motion for the coupled map system can be written in the compact form $$\label{eq_map_mat}
x_{i}^{t+1}=\left(\sum_{j}A_{ij}x_{j}^{t}\right){\rm mod}\;1,$$ where the coupling matrix $A$ has integer elements $$\label{eq_mat_coup}
A_{ij}=m\delta_{i,j}+n(\delta_{i,i+1}+\delta_{i,i-1}).$$
We will now free ourselves from the specific form (\[eq\_mat\_coup\]) for $A_{ij}$, which was physically motivated by the nearest neighbor lattice model (\[eq\_map\_nnc\]) and show that eq. (\[eq\_map\_mat\]) can be solved for [*any*]{} matrix $A$, which has [*integer valued elements*]{} $A_{ij}$. In order to see this we write (\[eq\_map\_mat\]) in vector notation as $$\label{eq_map_vec}
{\bf x}^{t+1}=\left( A{\bf x}^{t}\right){\rm mod}\;1,$$ where ${\bf x}^{t}=(x_{1}^{t},\cdots,x_{N}^{t})$ and the modulo is taken for each component of the vector $A{\bf x}^t$. Then we obtain by iterating from the initial condition: $${\bf x}^{t+1}=\left(A{\bf x}^t\right){\rm mod}\;1
= A{\bf x}^t-{\bf k}^t,$$ where ${\bf k}^t$ is a vector with integer components which represents the action of the modulo function. This yields $$\begin{aligned}
\label{eq_mod_iter}
{\bf x}^{t+2} = &&\left( A{\bf x}^{t+1}\right) {\rm mod}\;1
=\left( A\left[ A{\bf x}^t-{\bf k}^t\right] \right){\rm mod}\;1
\cr = &&\left( AA{\bf x}^t-A{\bf k}^t\right){\rm mod}\;1
=\left( A^2{\bf x}^t\right){\rm mod}\;1,
\end{aligned}$$ where the last equality sign only holds because all elements of the matrix $A$ are integers, such that $A{\bf k}^t$ is a vector with integer components which can be dropped under the last modulo function. Since (\[eq\_mod\_iter\]) holds for any $t$, we obtain the closed-form solution as a function of the initial value $$\label{eq_solution}
{\bf x}^{t}=\left( A^t{\bf x}^0\right){\rm mod}\;1.$$
Eq. (\[eq\_mod\_iter\]) shows that we can solve not only our coupled map lattice problem (\[eq\_map\_nnc\]), but all linearly coupled systems, where the coupling occurs via a matrix $A$ with integer elements $A_{ij}$ and the nonlinearity is provided by the modulo function. The solution can be obtained by first solving the linear problem, i.e. by obtaining $A^t{\bf x}^0$ and then taking the modulo, which is the same as having the linear map acting on an $N$-torus in analogy to the famous Arnold’s cat map in two dimensions [@arnold].
Next we investigate the invariant density and the spatiotemporal correlation function of the coupled map lattice. The first quantity gives us information about the measurable time averaged spatial structures in the system and the second one tells us about the measurable spatiotemporal structures.
The Invariant Density {#sec_inv_dens}
---------------------
The invariant density $\rho({\bf x})$ yields the distribution of points on the attractor generated by the map ${\bf x}^{t+1}=\left(A{\bf x}^t\right)
{\rm mod}\;1$. By starting from an initial distribution $\rho^0({\bf x})$ it could be obtained as the infinite-time limit of $\rho^t({\bf x})$ in the Frobenius-Perron equation $$\label{eq_dens_ev}
\rho^t({\bf x}) = \int d{\bf x}' \delta \left[{\bf x}-
\left( A^t{\bf x}'\right){\rm mod}\;1\right] \rho^0({\bf x}')$$
Since all quantities involved in eq. (\[eq\_dens\_ev\]) are periodic on an $N$-torus, the Fourier decomposition of $\rho^t({\bf x})$ contains only wavevectors ${\bf k}$ with integer components, i.e. $$\label{eq_dens_four}
\rho^t({\bf x})=\sum_{\bf k}\hat{\rho}^t({\bf k})
e^{2\pi i\,{\bf k}\cdot {\bf x}}.$$ By using the equality $\exp(2\pi i\,{\bf k}\cdot[(A^t{\bf x})\,{\rm
mod}\;1]) = \exp(2\pi i[(A^t)^T{\bf k}]\cdot{\bf x})$ eq. (\[eq\_dens\_four\]) yields $$\label{eq_fou_comp}
\hat{\rho}^t({\bf k})=
\hat{\rho}^0\left(\left(A^t\right)^T{\bf k}\right).$$ If the initial distribution $\rho^0({\bf x})$ is non-singular, all Fourier coefficients vanish for large values of the wavevector, i.e. $$\lim_{|{\bf k}|\rightarrow \infty }\hat{\rho}^0({\bf k})=0.$$
For a completely expanding map, where all eigenvalues of the matrix $A$ have an absolute value larger than one, $\lim_{t\rightarrow\infty}\left(
A^{t}\right)^{T}{\bf k}=\infty$ for each ${\bf k}\ne 0$ and the only non-vanishing Fourier component becomes $$\hat{\rho}^0(0) =\int d{\bf x}\rho^0({\bf x})=1,$$ which yields a constant invariant density $$\rho({\bf x})=1.$$ This result is completely analogous to the single map case. However we may obtain different results for the invariant density if there are contracting directions in the phase space.
Indexing the stable and unstable eigenvalues $\lambda$ and right (left) eigenvectors ${\bf e}$ ($\tilde{\bf e}$) of the matrix $A^T$ with indices “s” and “u” respectively, we have $$\label{eq_eig_exp}
\left(A^t\right)^T{\bf k}=
\sum_s\lambda_s^t(\tilde{\bf e}^s\cdot{\bf k}){\bf e}^s+
\sum_u\lambda_u^t(\tilde{\bf e}^u\cdot{\bf k}){\bf e}^u.$$ According to the above, we will only obtain results which differ from the trivial expanding case, if there exists at least one ${\bf k}\ne 0$, such that its components along the unstable directions are all zero, i.e. it is contained in the stable manifold $W^s$ of the fixed point ${\bf k}=0$ of the “conjugate” map $$\label{eq_conj_map}
{\bf k}^{t+1}=A^T{\bf k}^t.$$
In fact the above argument does not take into account one particular feature of the system. Specifically, the map (\[eq\_conj\_map\]) might not be hyperbolic, i.e. it might possess the [*central*]{} manifold, defined by the eigenvectors corresponding to $|\lambda|=1$. If this is the case (and it is for the coupled map lattice (\[eq\_map\_nnc\]) as we shall see below), we assume that the invariant density, that we calculate corresponds to the [*physical*]{} (or Kolmogorov) [*measure*]{} [@eckmann]. The latter is calculated by introducing a small amount of noise into the system and then taking the zero-noise limit.
Assuming that the physical measure is unique, we can obtain the invariant density starting from $\rho^0({\bf x})=1$, which is equivalent to $\hat{\rho}^0({\bf k})=\delta_{{\bf k},0}$. Then only the stable manifold contributes to the invariant density and the central manifold can be treated as the unstable one. This essentially means, that the averaging over the central directions is accomplished by the infinitely small noise present in the system.
On the other hand, eqs. (\[eq\_fou\_comp\],\[eq\_eig\_exp\]) tell us that it is not enough to have contracting eigenvalues in order to get a non-constant invariant density.
Let us first consider the case with a single stable direction ${\bf e}^s$. Since all components $k_i$ of a vector ${\bf k}$ are integer, it is contained within the stable manifold $W^s$ only if $\nu{\bf k}={\bf e}^s$. This in turn means that the components $\{e^s_1,\cdots,e^s_N\}$ should be mutually rational, i.e. $$e_1^s:e_2^s:\cdots:e_N^s=k_1:k_2:\cdots:k_N.$$
An example where we have one contracting and one expanding direction is the cat map $$A=\left(\matrix{1 & 1 \cr 1 & 2}\right).$$ Although the eigenvalue corresponding to the contracting direction is $\lambda_s=(3-\sqrt{5})/2<1$, this map still has a constant invariant density because the components of the eigenvector ${\bf e}^s=(2,1-
\sqrt{5})$ belonging to $\lambda_s$ have a non-rational ratio (see fig. \[fig\_cat\]) leading to $$\lim_{t\rightarrow \infty }\hat{\rho}^t({\bf k})=\delta_{{\bf k},0}.$$
Generally, in order to get a non-constant invariant density our model must possess a stable manifold, which in turn should contain at least one vector with mutually rational components. Every vector ${\bf k}^*$ with integer components, which is pulled in the long time limit into the origin, according to eq. (\[eq\_eig\_exp\]), $$\label{eq_pull_in}
\lim_{t\rightarrow \infty }\hat{\rho}^t({\bf k}^*)=
\lim_{t\rightarrow \infty }\hat{\rho}^0
\left(\left( A^{t}\right)^T {\bf k}^*\right)=\hat{\rho}^0(0)=1$$ can be represented as a linear combination of a (usually small) number of basis vectors ${\bf f}^j$, $j=1,\cdots,M$ with integer coefficients $n_j$, i.e. ${\bf k}^*=\sum_j n_j{\bf f}^j$. The invariant density will only contain non-vanishing Fourier components with such wavevectors ${\bf
k}^*$, all with weight 1: $$\begin{aligned}
\label{eq_inv_dens}
\rho({\bf x})&=&\sum_{{\bf k}^*}e^{2\pi i\,{\bf k}^*\cdot{\bf x}}
=\prod_{j=1}^M\sum_{n_j}e^{2\pi i n_j\,{\bf f}^j\cdot{\bf x}}\cr
&=&\prod_{j=1}^M\sum_{n_j}\delta({\bf f}^j\cdot{\bf x}-n_j).
\end{aligned}$$
The Coupled Map Lattice {#sec_coup_map}
-----------------------
Up to now our conclusions have been completely general for any coupling matrix $A$ with integer elements. Let us now consider the condition (\[eq\_pull\_in\]) in more detail for our 1-dimensional nearest neighbor model (\[eq\_map\_nnc\]). The corresponding matrix (\[eq\_mat\_coup\]) can be diagonalized by Fourier transformation in the space variables $i$, leading for periodic boundary conditions to eigenvalues $$\label{eq_evalue}
\lambda_q=m+2n\cos(q)$$ and the corresponding eigenvectors $$\begin{aligned}
\label{eq_evector}
{\bf e}_c^q&=&N^{-1}(\cos(q),\cos(2q),\cdots,\cos(Nq)),\cr
{\bf e}_s^q&=&N^{-1}(\sin(q),\sin(2q),\cdots,\sin(Nq)),
\end{aligned}$$ where $q=2\pi p/N$ and $p=0,\cdots,N/2$.
Of these only a few have mutually rational components. For instance, both $\cos(q):1$ and $\sin(2q):\sin(q)$ are rational only if $\cos(q)$ is rational, which immediately restricts the allowed wavevectors $q=2\pi p/N$ to a set of 5 values: $q^*=0,\pi/3,\pi/2,2\pi/3,\pi$. Each $q^*$ generates several basis vectors, provided $|\lambda_{q^*}|<1$: $$\begin{aligned}
\label{eq_basis}
{\bf f}^0=&&(1,\cdots,1);\cr
{\bf f}^{\pi/3}_1=&&(1,-1,-2,-1,1,2,\cdots,2);\cr
{\bf f}^{\pi/3}_2=&&(-1,-2,-1,1,2,1,\cdots,1);\cr
{\bf f}^{\pi/2}_1=&&(0,-1,0,1,\cdots,1);\cr
{\bf f}^{\pi/2}_2=&&(1,0,-1,0,\cdots,0);\cr
{\bf f}^{2\pi/3}_1=&&(-1,-1,2,\cdots,2);\cr
{\bf f}^{2\pi/3}_2=&&(-1,2,-1,\cdots,-1);\cr
{\bf f}^\pi=&&(-1,1,\cdots,1).
\end{aligned}$$
Rationality of $\cos(q^*)$ in not an unexpected result, e.g. choosing $\cos(q^*)=-m/2n$ results in the eigenvalue $\lambda_{q^*}=0$, according to (\[eq\_evalue\]), which requires $$({\bf f}\cdot{\bf x}^t)\,{\rm mod}\;1=0,\quad \forall t>0,$$ where we defined ${\bf f}=\kappa_{q^*}{\bf e}^{q^*}$ with $\kappa_{q^*}=N$ if $q^*=0,\pi/2,\pi$ and $2N$ otherwise. This in turn, requires $\rho({\bf x})\sim \delta(({\bf f}\cdot{\bf x})\,{\rm mod}\;1)$, which is seen to be the case by rewriting (\[eq\_inv\_dens\]) as $$\rho({\bf x})=
\prod_{j=1}^M\delta(({\bf f}^j\cdot{\bf x})\,{\rm mod}\;1).$$
It is useful to define the projection of the invariant density $\rho({\bf
x})$ on a chosen direction ${\bf g}$: $$\label{eq_project}
\rho_{\bf g}(s)=\int\delta(s-{\bf g}\cdot{\bf x})\rho({\bf x})\,d{\bf x}.$$ For example, if $g_i=\delta_{ij}$, eq. (\[eq\_project\]) gives the distribution of the $j$-th map variable $\rho(x_j)=1$.
If ${\bf g}$ coincides with one of the basis directions, i.e. ${\bf
g}=\nu{\bf f}^l$ for some $l$, the projection $$\begin{aligned}
\label{eq_delta}
\rho_{\bf g}(s)&=&
\int_{I^N}\delta(s-\nu{\bf f}^l\cdot{\bf x})
\prod_{j=1}^M\sum_p\delta(p-{\bf f}^j\cdot{\bf x})\,d{\bf x}\cr
&=&\sum_p D_p\,\delta(s-\nu p),
\end{aligned}$$ ($I^N$ denotes the unit $N$-dimensional cube) becomes singular: we get a series of $\delta$-functions with an envelope $$D_p=\int_{I^N}\delta(p-{\bf f}^l\cdot{\bf x})
\prod_{j\ne l}\sum_n\delta(n-{\bf f}^j\cdot{\bf x})\,d{\bf x}.$$ Otherwise, the projection (\[eq\_project\]) is a continuous, non-singular function of parameter $s$. In other words, only the projection on the directions defined by the basis vectors ${\bf f}^j$ is singular.
In particular, the eigenvector ${\bf e}^q$ defines a basis direction ${\bf
f}^j$ if and only if the projection (\[eq\_project\]) on this eigenvector (we define $\rho^q(s)=\rho_{\bf g}(s)$ for ${\bf g}={\bf
e}^q$), $$\label{eq_eig_proj}
\rho^q(s)=
\int_{I^N}\delta(s-{\bf e}^q\cdot{\bf x})
\prod_{j=1}^M\sum_{p_j}\delta(p_j-{\bf f}^j\cdot{\bf x})\,d{\bf x},$$ is singular. This implies that ${\bf e}^q=\nu_q{\bf f}^j$ for some $j$.
One can trivially verify that the projection (\[eq\_eig\_proj\]) has the average $$s_q=\int s\,\rho^q(s)\,ds=\frac{1}{2}\delta_{q,0}$$ and the dispersion given by $$\sigma_q^2=\int (s-s_q)^2\rho^q(s)\,ds
=\frac{1}{24N}(1+\delta_{q,0}+\delta_{q,\pi})$$ for all $q\ne q^*$ and almost always for $q=q^*$. A few degenerate cases like $\rho({\bf x})=\delta(x_1-x_2)$ or $\rho({\bf x})=\delta(x_1+x_2-1)$ for $N=2$, or $\rho({\bf x})=\delta(x_1-x_3)\,\delta(x_2-x_4)$ for $N=4$ give different dispersions.
As expected, numerically calculating the projection $\rho^q(s)$ on the stable and unstable directions (\[eq\_evector\]), we only get a singular distribution for $q=q^*=0,\pi/3,\pi/2,2\pi/3,\pi$ (fig. \[fig\_narrow\](b)), provided that the respective eigenvector is stable ($|\lambda_q|<1$). Otherwise a smooth Gaussian-like distribution is obtained (fig. \[fig\_narrow\](a)).
Indeed one can easily see that in the large-length limit both the continuous distribution and the envelope of the singular distribution (\[eq\_eig\_proj\]) become Gaussian: $$\label{eq_gauss}
\rho^q(s)\approx\cases{
\frac{1}{\sigma_q}\phi\left(\frac{s-s_q}{\sigma_q}\right)
& if $\forall j,\, {\bf e}^q\times{\bf f}^j\ne 0$, \cr
% & if $\forall j,\, {\bf e}^q\not\,\parallel{\bf f}^j$, \cr
\frac{\nu_q}{\sigma_q}\phi\left(\frac{s-s_q}{\sigma_q}\right)
\delta(s-\nu_q p) & if $\exists j: {\bf e}^q=\nu_q{\bf f}^j$,
}$$ where $\phi(t)=(2\pi)^{-1/2}\exp(-t^2/2)$ is the normalized Gaussian and $\nu_q=\kappa_q^{-1}$.
The Spatiotemporal Correlations {#sec_spa_temp}
-------------------------------
The standard spatial correlation function is trivially calculated to yield $$\label{eq_corr_fun}
C(r)=\langle x_i x_{i+r}\rangle-\langle x_i\rangle\langle x_{i+r}\rangle
=\frac{1}{12}\,\delta_{r,0}$$ for the completely expanding case with $\rho({\bf x})=1$ (here $\langle\cdot\rangle$ denotes the average taken with $\rho({\bf x})$).
If there are contracting directions, we rewrite (\[eq\_corr\_fun\]) as $$\begin{aligned}
C(r)&=&\sum_q(\sigma_{s,q}^2+\sigma_{c,q}^2)e^{iqr}\cr
&=&\sum_q(\sigma_{s,q}^2+\sigma_{c,q}^2-\frac{1}{12N})e^{iqr}+
\frac{1}{12}\,\delta_{r,0},
\end{aligned}$$ where $\sigma_{s,q}=\sigma_{c,q}=\sigma_q$ for all $q$ except $\sigma_{s,0}=\sigma_{s,\pi}=0$. Since $\sigma_q^2=
(1+\delta_{q,0}+\delta_{q,\pi})/24N$ for all $q\ne q^*$, $$\begin{aligned}
C(r)&=&\frac{1}{12}\,\delta_{r,0}+
(\sigma_0^2-\frac{1}{12N})+(\sigma_\pi^2-\frac{1}{12N})(-1)^r\cr
&+&2\sum_{q=\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3}}
(\sigma_{s,q}^2+\sigma_{c,q}^2-\frac{1}{12N})\cos(qr).
\end{aligned}$$
This reduces to a $\delta$-correlation (which coincides with the result (\[eq\_corr\_fun\]) obtained for $\rho({\bf x})=1$) in all but a few special cases, when $\sigma_q^2 \ne (1+\delta_{q,0}+\delta_{q,\pi})/24N$. For instance, choosing $m=0$ and $n=\pm 1$ yields for $N=4$ $$\rho({\bf x})=\delta(x_1-x_3)\delta(x_2-x_4)$$ and therefore $\sigma_{s,\pi/2}=\sigma_{c,\pi/2}=0$ and $\sigma^2_{c,0}=\sigma^2_{c,\pi}=1/24$, resulting in $$\label{eq_corr_4}
C(r)=\frac{1}{24}+\frac{(-1)^r}{24}=\cases{
\frac{1}{12} & if $r=0,2$,\cr
0 & if $r=1,3$.}$$
Since the invariant density, although being nontrivial, does not tell us much about the spatiotemporal structures in the system, next we introduce a spatiotemporal correlation function $G_i(r,t)$, which is a straightforward generalization of the time correlation function (\[eq\_sing\_cor\]): $$\label{eq_time_cor}
G_i(r,t) =\int d{\bf x}^0\rho({\bf x}^0)
e^{2\pi i(x_i^0-x_{i+r}^t)}.$$
By expanding $\rho({\bf x})$ into Fourier series we obtain in analogy to (\[eq\_fou\_comp\]): $$\label{eq_time_four}
G_i(r,t)=\sum_{{\bf k}}\hat{\rho}({\bf k})
\prod_{j=1}^N\delta(k_j-A_{i+r,j}^t+\delta_{i,j}).$$ Since only the non-vanishing Fourier components $\hat{\rho}({\bf k}^*)=1$ (where ${\bf k}^*=\sum_l n_l{\bf f}^l$) of the invariant density (\[eq\_inv\_dens\]) contribute, (\[eq\_time\_four\]) reduces to $$G_i(r,t)=\sum_{n_1,\cdots,n_M}\prod_{j=1}^N\delta\left(
\sum_{l=1}^M n_l f^l_j-A_{i+r,j}^t+\delta_{i,j}\right).$$ In a translationally invariant system $G_i(r,t)$ does not depend on $i$, so we drop the index and fix $i$ (set $i=1$ to be specific).
It can be easily verified that the correlation (\[eq\_time\_cor\]) is short-ranged in both space and time. First we note that it vanishes if the vector ${\bf k}_r^t$ with components $k_j=A_{1+r,j}^t-\delta_{1,j}$ does not lie on the stable manifold $W^s$. According to (\[eq\_eig\_exp\]) $$A_{1+r,j}^t=\lambda_1^t\left(e^1_{1+r} e^1_j+\left(
\frac{\lambda_2}{\lambda_1}\right)^t e^2_{1+r} e^2_j+\cdots\right),$$ where $\lambda_1$ is the largest and $\lambda_2$ — the next largest eigenvalue and ${\bf e}^1$ and ${\bf e}^2$ are the respective eigenvectors. For increasing $t$ the vector ${\bf k}_r^t$ asymptotically approaches the direction defined by ${\bf e}^1$ and therefore cannot lie on the stable manifold for $t\ge\tau$, where $
\tau$ is some finite (and typically small) integer.
On the other hand for $t=0$ we have $$G(r,0)=\sum_{n_1,\cdots,n_M}\prod_{j=1}^N\delta\left(
\sum_{l=1}^M n_l f^l_j-\delta_{1+r,j}+\delta_{1,j}\right).$$ Since all basis vectors (\[eq\_basis\]) are periodic in space with periods 1, 2, 3, 4 or 6, any linear combination of these will also be periodic with period of at most 12. Since the vector with components $k_j=
\delta_{1+r,j}-\delta_{1,j}$ is [*not*]{} periodic for $r\ne 0$, the maximal size of the system with non-trivial correlation is limited by $N=12$. Again, choosing $m=0$ and $n=\pm 1$ for $N=4$ as an example, we have ${\bf f}^1=(0,-1,0,1)$ and ${\bf f}^2=(1,0,-1,0)$ as basis vectors and consequently $$G(r,0)=\cases{1 & if $r=0,2$,\cr 0 & if $r=1,3$,}$$ i.e. we retrieve the result (\[eq\_corr\_4\]).
Discussion {#sec_discuss}
==========
To summarize, we have shown that the solution for the dynamical behavior of a lattice of Bernoulli maps that are coupled by a matrix $A$ with integer coefficients can be given in the closed form as ${\bf
x}^t=(A^t{\bf x}^0){\rm mod}\;1$: the dynamical behavior of the coupled map system can be described by the repeated action of a linear map $A^t{\bf x}^0$ on variables that are confined to an $N$-torus. This picture explains that the essentials of the dynamical behavior are dictated by the eigenvalues and eigenvectors of $A$.
The invariant density $\rho({\bf x})$ of this system displays Fourier coefficients that are different from zero, i.e. is non-constant, whenever the stable manifold of the zero wavevector contains a non-empty basis of directions ${\bf f}^j$ with mutually rational components, generating the infinite asymptotically contracting set of wavevectors. For nearest neighbor couplings in a 1-dimensional lattice (given by eq. \[eq\_mat\_coup\]) the maximal number of basis vectors is eight (actually even less, since, e.g. $|\lambda_0-\lambda_\pi|=4|n|>2$).
We have calculated the standard spatial correlation function $C(r)$ for the model with nearest neighbor couplings and shown that it is given by $C(r)=\delta_{r,0}/12$ almost always. A few special cases exist however for sufficiently small lattices, where the spatial correlations are different. Nevertheless, $C(r)$ always vanishes at sufficiently large distances.
The invariant density of this system and the spatial correlation function display little structure as compared to the Lyapunov spectrum, which is, for the nearest neighbor coupling, given by $\Lambda_q=\log|m+2n\cos(q)|$. This result shows that the time averaged spatial behavior is [*not*]{} simply a straightforward reflection of the Lyapunov spectrum (see related work listed in [@daido]).
We have also calculated the measurable spatiotemporal correlation function $G(r,t)$ for the translationally invariant model and shown that it too is short-ranged in both space and time.
It is instructive to compare our results with the general results obtained by Bunimovich and Sinai [@sinai], who proved that, for sufficiently small coupling (in our case determined by parameter $\varepsilon$), certain expanding coupled map systems with finite-range coupling possess an absolutely continuous invariant measure $\mu({\bf x}):\ d\mu({\bf x})=
\rho({\bf x})\,d{\bf x}$, and also that the time and space correlation functions decay exponentially (not slower than exponentially, to be exact).
Our results indicate, that for larger coupling, the invariant measure still exists, but might not be absolutely continuous due to the fact, that large coupling often causes the appearance of contracting directions, even if the isolated local maps $f(x)$ are expanding. The space and time correlations in our model are seen to decay even faster then exponentially, but the few special cases, giving non-trivial correlations imply that there might be some general relationship between the continuity of the invariant measure and the appearance of coherent structures in the system.
Let us finally point out several directions of further research.
One open problem is the extension of our results to higher dimensions and to couplings which have a longer range. In the 1-dimensional case the eigenvectors remain also valid for longer-ranged couplings, only the eigenvalues change. This means that a model with long, but finite, range will have no more structure in the invariant density than the short-ranged model. This is of course a peculiarity of the Bernoulli shift map, but should again be taken as a warning for making conclusions from the spatial range of the coupling onto the observable spatial patterns.
Although our solution for the dynamics and the correlation functions hold for general dimensions it would be interesting to see what the restrictions on the wavevectors that generate the basis of the invariant density look like in two and three dimensions.
Finally, one could investigate the dynamical behavior of a system, whose time dependence is given a priori by the equation (\[eq\_solution\]) also for matrices $A$ with [*non-integer*]{} elements. By doing so one will loose the property of the original map, that the relation relation (\[eq\_map\_vec\]) holds step by step, but the trajectories generated by equation (\[eq\_solution\]) are well defined.
H.G.S. thanks C. Koch for the kind hospitality extended to him at Caltech and the Volkswagen Foundation for financial support. The authors thank M. C. Cross for the careful reading of the manuscript. This research has also been partially supported by the NSF through grant No. DMR-9013984.
Permanent Address: Institute für Theoretische Physik, Universität Kiel, Olshausenstrasse 40, D 24098 Kiel, Germany
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---
abstract: 'We introduce a systematic construction of a gapless symmetry protected topological phase in one dimension by “decorating” the domain walls of Luttinger liquids. The resulting strongly interacting phases provide a concrete example of a gapless symmetry protected topological (gSPT) phase with robust symmetry-protected edge modes. Using boundary conformal field theory arguments, we show that while the bulks of such gSPT phases are identical to conventional Luttinger liquids, their boundary critical behavior is controlled by a different, strongly-coupled renormalization group fixed point. Our results are checked against extensive density matrix renormalization group calculations.'
author:
- 'Daniel E. Parker'
- Thomas Scaffidi
- Romain Vasseur
bibliography:
- 'gapless\_SPT.bib'
title: Topological Luttinger Liquids from Decorated Domain Walls
---
Introduction
============
Topological materials offer a rich and diverse set of condensed matter systems, extending beyond the Landau paradigm of symmetry-breaking order. The discovery of topological insulators and superconductors in the past decade [@PhysRevLett.95.226801; @Bernevig1757; @Konig766; @PhysRevLett.98.106803; @PhysRevB.75.121306; @PhysRevB.79.195322; @Hsieh:2008aa; @Hasan2010; @Rasche2013; @story2012] led to the general concept of Symmetry-Protected Topological phases (SPTs) which, as the name suggests, are gapped quantum phases of matter with topological properties protected by symmetry [@PhysRevB.80.155131; @PhysRevB.84.235128; @PhysRevB.83.075102; @PhysRevB.83.075103; @Chen2011b; @Pollmann2012; @YuanMing2012; @Levin2012; @Chen1604; @Chen2011]. They feature short-range entanglement and spectral gaps, and have gapless edge states protected by the presence of certain symmetries. (For a review, see [@doi:10.1146/annurev-conmatphys-031214-014740].) Examples include the experimentally accessible Haldane phase in quantum spin chains [@PhysRevLett.61.1029; @Kennedy1992; @Buyers1986]. Non-perturbative techniques have led to an essentially exhaustive classification of gapped bosonic [@Chen1604; @Chen2011; @YuanMing2012; @PhysRevB.91.134404] and, to some extent, fermionic SPT phases [@PhysRevB.81.134509; @PhysRevB.83.075103; @PhysRevB.89.201113; @Wang629; @PhysRevB.90.115141; @Gu2015]. A key feature in the definition of SPTs is the spectral gap to bulk excitations, whose presence is generally believed to be necessary to ensure protection of the edge states.
Attention has recently turned to gapless topological materials, including Weyl and Dirac semi-metals with topologically-protected Fermi arc surface states [@PhysRevB.83.205101; @PhysRevX.5.031013; @Lu622; @Xuaaa9297; @doi:10.1146/annurev-conmatphys-031113-133841]. In previous work [@scaffidi2017gapless], the authors showed that the assumption of a spectral gap for strongly interacting SPT systems is unnecessary (see also [@2012arXiv1206.1332G; @PhysRevLett.118.087201]). We provided a general construction of systems —in any dimension— that are gapless in the bulk with symmetry-protected topological edge modes, which we dub gapless SPTs (gSPTs). Through exactly solvable examples, we demonstrated that many of the tools regularly applied to gapped SPTs — including unitary twists [@Chen2011], entanglement spectra [@PhysRevLett.101.010504], strange correlators [@Cenke2014], and trial wavefunctions [@PhysRevB.93.115105] — carry over to the gapless case. In general, our construction works by “twisting” [@Chen2011] a gapless topologically trivial system, or by “decorating” its domain walls [@AshvinDecorated]. The resultant non-trivial gSPT is as stable as the underlying gapless system. Therefore twisting can turn critical points or lines into points or lines of gSPTs, but also gapless phases into gSPT phases.
In this work we systematically study an exactly solvable gSPT phase created by twisting a Luttinger liquid [@0022-3719-14-19-010; @giamarchi2003quantum] (LL). The resulting phase, which we call “LL$^{\star}$” is a topologically non-trivial phase with gapless edge modes. Just as for a regular SPT, the edge states are robust to all perturbations that preserve certain protecting symmetries. Additionally, we show that while the LL and LL$^{\star}$ phases are identical in the bulk, the boundary critical behavior of the LL$^{\star}$ state realizes a different strongly-coupled renormalization group fixed point. The spectrum of boundary critical exponents is exactly described within boundary CFT [@cardy_conformal_1984; @cardy_boundary_1989], and the topological edge states lead to an exotic “superposition” of conformally invariant boundary conditions.
The goal of this work is two-fold. First, we wish to put the notion of topological Luttinger liquids in a more systematic context. Similar kinds of topological Luttinger liquids have been studied previously in several works [@Tsui2015330; @PhysRevB.91.235309; @ruhman2017topological] with a variety of setups — relying essentially on the spin-charge separation property of Luttinger liquids. By examining a topological Luttinger liquid as a gSPT, we can construct such gapless topological states in a more systematic way, and we can understand their topological nature using the powerful tools and techniques of SPTs. Second, by studying this one-dimensional example — where exact results are available through CFT and explicit numerical confirmation is possible — we may completely understand this example of a gSPT. We expect that many of the results and physical pictures developed here have close analogues in higher-dimensional gSPTs.
This paper is organized as follows. In Section II we introduce our model and briefly describe its topological and spectral properties. Section III establishes the existence of a gSPT phase and analyzes its phase diagram. The surface critical behavior of this problem is analyzed in terms of BCFT in Section IV, along with extensive numerical validation. We then conclude with possible generalizations to higher dimensions and implications for gapless topological systems.
Model and topological edge modes {#Sec:Model}
================================
Consider a spin-$1/2$ chain with two alternating species, which we call $\sigma$ and $\tau$, as shown in Figure \[fig:phase\_diagram\] (c). We will follow the systematic procedure for constructing a gSPT from our previous work [@scaffidi2017gapless]: start with a gapless phase and then twist it by a local unitary operator to arrive at a topologically non-trivial phase. This is directly analogous to how the ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ gapped SPT [@Chen2011; @AshvinDecorated; @Bahri:2015aa; @PhysRevB.93.155131], the simplest one-dimensional bosonic SPT, is constructed in the decorated domain wall picture [@AshvinDecorated].
We start from a topologically trivial phase which is the XXZ model for the $\sigma$ spins and a paramagnet for the $\tau$ spins: $$\begin{aligned}
H^{(Z)}_\text{gTrivial} \ &=\ \sum_i \sigma_i^x \sigma_{i+2}^x + \sigma_i^y \sigma_{i+2}^y + \Delta \sigma_i^z \sigma_{i+2}^z - \tau^x_i + H_\text{int}\\
H_\text{int} \ &=\ \sum_i \alpha \sigma_i^z \tau_{i+1}^x \sigma_{i+2}^z -
g_\tau \tau^z_i \tau^z_{i+2} - u_\tau \tau^x_{i} \tau^x_{i+2}.\\
\end{aligned}
\label{eq:H_gTrivial}$$ This has a $U(1) \rtimes {\mathbb{Z}}_2^{(\sigma)} \times {\mathbb{Z}}_2^{(\tau)}$ global symmetry, generated by $U_\theta = \prod_{i=1,3,\dots} e^{i \theta \sigma^z_i/2}$, $\mathcal{C}_\sigma = \prod_{i=1,3,\dots} \sigma_i^x$ and $\mathcal{C}_\tau = \prod_{i=2,4,\dots} \tau_i^x$ respectively.
The choice of anisotropy in the $Z$-direction is arbitrary; taking the anisotropy in the $Y$-direction gives the same gTrivial state, but we will show below that it will produce a slightly different gSPT. Because this choice amounts to an onsite unitary $\sigma_i^y \leftrightarrow \sigma_i^z$, we will, for the most part, treat both models simultaneously. For brevity, we refer to the Hamiltonian in Eq. as the $(Z)$-model, and the one obtained from Eq. by doing $\sigma_i^y \leftrightarrow \sigma_i^z$ as the $(Y)$-model: $$\begin{aligned}
H^{(Y)}_\text{gTrivial} \ &=\ \sum_i \sigma_i^x \sigma_{i+2}^x + \sigma_i^z \sigma_{i+2}^z + \Delta \sigma_i^y \sigma_{i+2}^y - \tau^x_i + H_\text{int}\\
H_\text{int} \ &=\ \sum_i \alpha \sigma_i^y \tau_{i+1}^x \sigma_{i+2}^y -
g_\tau \tau^z_i \tau^z_{i+2} - u_\tau \tau^x_{i} \tau^x_{i+2}.\\
\end{aligned}
\label{eq:H_gTrivialY}$$ However, the slight differences between the two models will help to elucidate several subtle issues, so we will contrast them at several points below.
The symmetry protecting the topological properties is given by ${\mathbb{Z}}_2^{(\sigma)} \times {\mathbb{Z}}_2^{(\tau)}$, while the $U(1)$ symmetry enforces a gapless bulk in the Luttinger liquid phase. Interactions in $H_\text{int}$ are chosen to be the simplest ones compatible with the symmetry. The perturbation $g_\tau$ takes the $\tau$’s from a simple paramagnet to an Ising model, and $u_\tau$ takes the Ising model away from integrability. The interaction $\alpha$ couples the $\sigma$ and $\tau$ sectors together by the simplest term compatible with all symmetries.
To twist the model into a non-trivial gapless SPT, we use the local unitary operator $U$ (also used to construct the gapped ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ SPT) $$U = \prod_{{\rm DW}(\sigma)} (-1)^{\frac{1-\tau_i^z}{2}},
\label{eq:twist_operator}$$ where the product runs over all the domain walls of the $\sigma$ spins in the $z$ basis. One can think of $U$ as attaching a charge of ${\mathbb{Z}}_2^{(\tau)}$ to the domain walls of ${\mathbb{Z}}_2^{(\sigma)}$ [@AshvinDecorated; @PhysRevB.91.195117]. From a more concrete perspective, $U$ gives a factor of $(-1)$ for each instance of two consecutive down spins (in the $z$ basis) in a classical spin configuration, as shown in Figure \[fig:phase\_diagram\] (c).
The non-trivial gapless SPT is defined by $$H_{\text{gSPT}} = U H_\text{gTrivial} U.
\label{eq:H_gSPT}$$ For the $(Z)$-model, the $U(1)$ symmetry is invariant under conjugation by $U$, but for the $Y$-model, the $U(1)$ symmetry is also twisted in the process, and is therefore slightly less natural as a physical symmetry after twisting. With periodic boundary conditions, $H_\text{gSPT}$ clearly has the same spectrum as $H_\text{gTrivial}$. However, for open boundary conditions, we will see that $H_\text{gSPT}$ has gapless edge modes. This manifests as a two-fold exponential degeneracy of all the low-energy states, as shown in Figure \[fig:phase\_diagram\] (b). Remarkably, we will see that these topological edge modes (and the associated two-fold degeneracy of the spectrum) are robust to symmetry-preserving perturbations, even if they are strongly coupled to the bulk gapless degrees of freedom.
To demonstrate the presence of edge modes, let us first restrict ourselves to a limit were they can be found exactly: a semi-infinite chain $i\ge 1$, starting with $\sigma_1$, without interactions, i.e. $\alpha = g_\tau = u_\tau = 0$. The action of $U$ on the Pauli matrices is $\sigma_i^x \to \tau_{i-1}^z \sigma_i^x \tau_{i+1}^z, \sigma_i^y \to \tau_{i-1}^z \sigma_i^y \tau_{i+1}^z, \sigma_i^z \to \sigma_i^z$, and similarly for the $\tau$’s. Hence, the first few terms in $H_\text{gSPT}$ for the $(Z)$-model are given by $$H_\text{gSPT}^{\text{open}} =
\Delta\sigma_1^z \sigma_3^z
+ \tau_2^z \sigma_3^x \sigma_5^x \tau_6^z
+ \tau_2^z \sigma_3^y \sigma_5^y \tau_6^z
- \sigma_1^z \tau_2^x \sigma_3^z
+\cdots$$ (Terms such as $\sigma_1^x \sigma_3^x\tau_4^z$ and $\sigma_1^y \sigma_3^y\tau_4^z$ are not compatible with the symmetries and cannot be included.) In this simple case, the edge mode $\Psi$ is simply the first spin; $\Psi = \sigma_1^z$ commutes with the Hamiltonian and thus indexes a double degeneracy of every state. Even in this simple, exactly solvable limit, the edge mode does [*not*]{} completely decouple from the gapless degrees of freedom. In particular, because of the $\Delta \sigma_1^z \sigma_3^z $ term in the Hamiltonian, enforcing $\sigma_1^z = \pm 1$ induces an effective magnetic field $\pm \Delta$ acting on $\sigma_3^z$. The effects of this boundary magnetic field on the gapless degrees of freedom will be discussed in Section \[sec:BCFT\]. We emphasize that this is dramatically different from edge modes at the boundary of [*gapped*]{} SPTs: for gapped SPTs, one can construct local operators that can flip the edge mode from $\sigma_1^z = \pm 1$ to $\sigma_1^z = \mp 1$, while in the gapless case above this is not possible because of the coupling to the gapless bulk modes. (In particular, this is why there is only a two-fold degeneracy of the low-energy states for an open system of finite length $L$, instead of the four-fold degeneracy expected for gapped 1d SPT with two spin-$\frac{1}{2}$ edge modes.)
The edge modes survive in the presence of generic interactions compatible with the symmetry. In particular, we add $\alpha, g_\tau$, and $u_\tau$, as well as additional ${\mathbb{Z}}_2^{(\sigma)} \times {\mathbb{Z}}_2^{(\tau)}$-preserving boundary perturbations such as $\sigma_1^x$. (We allow terms breaking the $U(1)$ symmetry at the boundary since this symmetry only protects the gaplessness of the bulk, and plays no role in the topological properties of the system.) Due to the spectral gap in the $\tau$ sector, local channels to flip the edge mode are exponentially suppressed. Thus the only effect of interactions is to dress $\Psi$, which remains largely localized to the first site. In a system with finite length $L$, open boundary conditions and say, $u_\tau=\alpha=0$ but $g_\tau \ll 1$, the amplitude for flipping the edge modes at both ends scales as ${\sim} g_\tau^L = {\rm e}^{L \log g_\tau }$ and is thus exponentially small in $L$. As for gapped SPTs, we thus expect the edge modes to remain protected by the gap of the $\tau$ spins away from the exactly solvable limit described above. This picture is confirmed numerically below.
We may thus consider $H_\text{gSPT}$ as a gapless symmetry-protected topological state. We emphasize that our construction was completely systematic, following the general construction from previous work [@scaffidi2017gapless] and employing the same unitary transformation used for the standard ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ SPT. We should therefore expect it to inherit many of the properties of normal SPTs. As a practical matter, this implies $H_\text{gSPT}$ should be robust to perturbations and therefore constitute a phase. We confirm this in the next section.
Phase diagram
=============
In this section we study the phase diagram of $H_\text{gSPT}$, and show that it has a genuine gapless SPT *phase* protected by the symmetry $U(1) \rtimes {\mathbb{Z}}_2^{(\sigma)} \times {\mathbb{Z}}_2^{(\tau)}$. Since, for periodic boundary conditions, $H_\text{gSPT}$ and $H_\text{gTrivial}$ are related by a local unitary transformation, they share the same bulk properties. In particular, their phase boundaries are the same; the LL$^{\star}$ state is as stable as the original (untwisted) LL phase [@scaffidi2017gapless]. We therefore analyze Eq. to understand the phase diagram of $H_\text{gSPT}$ (which is the same for both the $(Y)$- and $(Z)$-models).
A numerical phase diagram is shown in Figure \[fig:phase\_diagram\] (d), computed with DMRG [@white92; @ITensor] and finite-size scaling. To avoid fine-tuning, we include all perturbations from $H_\text{int}$. It is clear that LL$^\star$ is a genuine phase and not just a critical point. By working in various limits and approximations, we may identify the various proximate phases and many of the phase transitions. First, the line $\alpha = 0$ is the same as the well-known XXZ model, which is a ferromagnet for $\Delta < -1$ and an antiferromagnet for $\Delta > 1$. The middle region $-1 < \Delta < 1$ is a Luttinger liquid, whose low-energy properties are given by the compact free boson conformal field theory, a fact we employ in the following section.
When $\alpha \to \infty$, the Hamiltonian is dominated by the interaction $\alpha \sigma^z \tau^x \sigma^z$. This has $2^{L/2}$ degenerate ground states: any classical configuration of $\sigma^z$ is permitted, and the $\sigma$ spins fix the $\tau$ spins by the requirement $\braket{\sigma^z_i \tau^x_{i+1} \sigma^z_{i+2}} = -1$. Adding the other terms at first order in perturbation theory gives an effective Hamiltonian for this low-energy sector of $H_\text{eff} = \sum_i \left( \Delta +1 \right) \sigma^z_i \sigma_{i+2}^z$. This implies the ground state is ferromagnetic for $\Delta < -1$ and undergoes a direct transition to the anti-ferromagnetic phase at $\Delta > -1$. Indeed, we see a transition near $\Delta = -1$ for $\alpha \gg 0$ in the phase diagram.
For $\Delta \to \infty$, the situation is trivial and we always have an anti-ferromagnet for $\sigma$ and a paramagnet for the $\tau$’s. In the opposite limit $\Delta \to - \infty$, we may assume the $\sigma$’s are perfectly ferromagnetic, and hence $\braket{\sigma^z_i \sigma^z_{i+2}} = 1$. This gives an effective Hamiltonian $H_\text{eff} = -\sum_i (1-\alpha) \tau^x_{i} + g_\tau \tau^z_i \tau^z_{i+2} + u_\tau \tau^x_i \tau^x_{i+2}$ for the $\tau$ spins. When $u_\tau = 0$, this gives Ising transitions for the $\tau$’s at $\alpha = 1 \pm g_\tau$. Adding in $u_\tau$ breaks the integrability of the Ising model, moving the transitions to $\alpha = 1 \pm \left( g_\tau - O(u_\tau) \right)$.
Lastly, we can understand the phase transitions out of the LL$^{\star}$ phase in an effective theory. Integrating out the gapped $\tau$ spins gives $\Delta_\text{eff}(\alpha) = \Delta + \alpha \braket{\tau^x} \approx\Delta + \alpha$. This gives phase transitions at $\Delta_\text{eff}(\alpha) = \pm 1$, i.e. $\Delta(\alpha) = \pm 1 - \alpha$. Figure \[fig:phase\_diagram\] (d) shows this argument accurately predicts the phase boundaries for LL$^{\star}$ up to $\alpha \approx 1$.
We have established that LL$^\star$ is a genuine phase within our choice of parameters and does not rely on fine-tuning. The stability of LL$^\star$ is a separate question, which we defer to Section \[sec:BCFT\_LLstar\]. For now, note that both the LL and LL$^\star$ phases can be gapped out by dimerizing the $\sigma$ spins, so translation invariance by a single unit cell should also be included in the symmetry group protecting the gaplessness of the system.
Boundary criticality {#sec:BCFT}
====================
Since $H_\text{gSPT}$ is a gapless model in 1+1 dimensions, its universal properties are described by conformal field theory [@DiFran]. In this section we will show that boundary CFT (BCFT) [@cardy_conformal_1984; @cardy_boundary_1989] predicts the exact low-energy spectrum (below the paramagnetic gap) in the LL$^\star$ phase, including degeneracies. We will see that $H_\text{gSPT}$ realizes a superposition of boundary conditions that would otherwise require symmetry-breaking fields in non-topological (gTrivial) systems.
BCFT for LL
-----------
To start, let us review some facts about the standard (non-twisted) XXZ model, which will apply to $H_\text{gTrivial}$. Through bosonization, and integrating out the $\tau$ spins, the low-energy physics of this model can be captured by a single gapless bosonic field (Luttinger liquid) $${\cal L}_{\rm LL} = \frac{g}{4 \pi} \left( \partial_\mu \varphi \right)^2,
\label{eq:XXZ_bosonized}$$ where $\varphi$ is a free scalar field compactified onto a circle of radius $2\pi$ (i.e. $\varphi \equiv \varphi + 2\pi$), and $g = \pi^{-1} \arccos(-\Delta_{\rm eff})$ is the Luttinger parameter [@giamarchi2003quantum] and $\Delta_{\rm eff}$ is the effective anisotropy of the $\sigma$ spins after integrating out the gapped $\tau$ spins. (For our choice of parameters, $\Delta_\text{eff} \approx \Delta +\alpha \braket{\tau^x} $.) For simplicity, we rescaled the space coordinate $x$ to set the Fermi velocity $v_F = 2\pi \frac{\sqrt{1-\Delta_{\rm eff}^2}}{\arccos \Delta_{\rm eff}}$ to unity in the Lagrangian. The correspondence between spins and the bosonization fields is [@giamarchi2003quantum] $$\begin{aligned}
\sigma_k^z &\simeq \frac{1}{ \pi} \partial_x \varphi+ A (-1)^k \sin \varphi, \notag\\
\sigma_k^x \pm i \sigma_k^y &\simeq e^{\pm i \theta} \left( B \cos \varphi + C (-1)^k \right),
\label{eq:spin_field_correspondence}\end{aligned}$$ where $A,B,C$ are non-universal constants, and $\theta$ is the dual variable with commutation relation $[\varphi(x),\theta(y)] = i \pi {\operatorname{sgn}}(x-y)$. Free boundary conditions on the $\sigma$ spins correspond to the Dirichlet boundary conditions $\varphi(x=0) = 0 = \varphi(x=L)$ [@oshikawa1997boundary]. The $U(1) \rtimes {\mathbb{Z}}_2$ symmetry of the $\sigma$ spins acts as $\theta \to \theta + \alpha$, and $(\varphi,\theta) \to (-\varphi,-\theta)$, respectively. The gaplessness of the XXZ chain is also protected by translation invariance, which acts on the field theory [@PhysRevB.46.10866; @giamarchi2003quantum] as a discrete symmetry $(\varphi,\theta) \to (\varphi+\pi,\theta+\pi)$ (or by a mirror symmetry $(\varphi,\theta) \to (\pi-\varphi,\theta+ \varphi)$). With these protecting symmetries, the leading perturbation is $\cos 2 \varphi$ with scaling dimension $2/g$, which is irrelevant for $\Delta_{\rm eff}<1$ and opens a Néel gap for $\Delta_{\rm eff}>1$. The same stability analysis applies to the LL$^\star$ (gSPT) state.
The partition function of this gTrivial model can be computed using standard tools of boundary conformal field theory and is summarized in the Appendix. Ignoring non-universal contributions, it reads $$Z_\text{gTrivial}(g) = \operatorname{Tr}\ {\rm e}^{-\beta H_{\rm gTrivial}} = \frac{1}{\eta(q)} \sum_{e \in {\mathbb{Z}}} q^{g e^2},
\label{eq:Z_gTrivial}$$ where $q = e^{- \beta\pi v_F/L}$ and $\eta(q) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$ is the Dedekind $\eta$-function. This partition function can be interpreted as a generating function $q^{-1/24} \sum_i q^{\Delta_i}$ for the boundary scaling dimensions $\Delta_{e,n} = g e^2 + n$ ($e\in {\mathbb{Z}}, n>1$) of the theory, whose multiplicity is $p(n)$, the number of integer partitions of $n$ [@DiFran]. In CFT language, these are the scaling dimensions of the electric operators for the massless free boson with Dirichlet boundary conditions (and their descendants). The value of $e$ for a state corresponds to total spin quantum number $\braket{\sigma^z} = \sum_i \braket{\sigma^z_i}$. These critical exponents can also be extracted numerically from the finite size scaling of the energy gaps of $H_{\rm gTrivial}$ $$E_{e,n}(L)-E_0(L) = \frac{\pi v_F}{L}\Big[ g(\Delta_\text{eff}) e^2 + n \Big],\text{ for } e \in {\mathbb{Z}},\; n > 1,
\label{eq:gTrivial_spectrum}$$ with $E_0(L)$ is the groundstate energy and $E_{k,n}(L)$ the energy of an excited state. To compare with numerics, the standard technique is to determine the operator dimensions via extrapolation from finite size data. In practice, one extracts the operator dimension $x$ by fitting $\frac{\pi v_F}{L} \left[ E(L) - E_0(L) \right] = x + a_1 L^{-1} + a_2 L^{-2} + \cdots$. We perform this in Figures \[fig:phase\_diagram\] (a) and \[fig:spectrum\_comparison\] (a). Figure \[fig:spectrum\_comparison\] (a) shows matches the numerical data essentially exactly, with minor deviations for ${\left| \Delta_{\rm eff} \right|} \approx 1$ where additional operators become marginal.
Numerics were performed with DMRG using the ITensor Library [@ITensor]. For each $\Delta_{\rm eff}$, the lowest 50 eigenvalues were computed for $L = 20,24,28,32$ with bond dimension up to $\chi = 400$ or until the error, estimated by $\braket{\psi|H^2|\psi} - \braket{\psi|H|\psi}^2$, was less than $10^{-6}$ for each state. The Hamiltonian parameters used were $\alpha = 0.1$, $g_\tau = 0.3, u_\tau = 0.1$. To improve convergence under finite size scaling, the gap to the $\tau$ sector was increased by taking $-\tau_i^x \to -10 \tau_i^x$. The data shown in Figure \[fig:phase\_diagram\] (a) and (b) is a slice of Figure \[fig:spectrum\_comparison\] at $\Delta = 0.3$. The degeneracies, as well as the operator dimensions, match CFT predictions. Away from ${\left| \Delta \right|} \approx 1$, errors are ${\sim} 1\%$, and are due to a combination of finite-size fitting error, failure to match corresponding eigenvalue correct at different finite sizes, and DMRG errors. The last is increasingly prevalent beyond eigenvalue 30 where errors in previous states begin to accumulate significantly.
BCFT for LL$^{\star}$ {#sec:BCFT_LLstar}
---------------------
We now consider the case of the non-trivial gSPT Luttinger Liquid (LL$^\star$). Because $H_\text{gTrivial}$ and $H_\text{gSPT}$ have identical spectra with periodic boundary conditions, the difference between the two Hamiltonians can only manifest itself at the boundaries. Hence the low-energy part of the spectrum of $H_\text{gSPT}$ should again be described by the effective field theory — but now with different boundary conditions. In this section we restrict ourselves to the $(Y)$-model and treat the $(Z)$-model subsequently.
To determine the correct boundary conditions, let us consider the physical effect of the topological edge modes. From the exactly solvable limit studied in Sec. \[Sec:Model\], we expect LL$^\star$ to have localized topological edge modes that strongly overlap with $\sigma^z$ at the edges. To leading order, an edge mode in the states $\ket{\uparrow}$ or $\ket{\downarrow}$ will induce an effective boundary field $\pm h_b$ for the gapless $\sigma$ spins in the $z$ (easy-plane) direction. Using the bosonization formulas (with the cyclic relabeling $X \to Z$, $Y \to X$, $Z \to Y$), we thus find the effective action for the LL$^\star$ phase on an interval $[0,L]$ $$\begin{aligned}
S_\text{gSPT} &= \frac{g}{4 \pi} \int_0^L dx \int d\tau \left( \partial_\mu \varphi \right)^2 \mp \lambda_L \int d\tau \cos \theta(x=0) \notag\\
& \mp \lambda_R \int d\tau \cos \theta(x=L),\end{aligned}$$ where the $\mp$ signs correspond to the edge modes on the left and right of the chain being in the state $\ket{\uparrow}$ or $\ket{\downarrow}$ (in the $z$-basis), respectively. With our bosonization convention, we have $\cos \theta \to -\cos \theta$ under the ${\mathbb{Z}}_2^{(\sigma)}$ symmetry, so that the $\cos \theta$ terms would not be allowed by themselves in a ${\mathbb{Z}}_2^{(\sigma)} \times {\mathbb{Z}}_2^{(\tau)}$-symmetric action. The coefficients $\lambda_R$ and $\lambda_L$ are non-universal functions of the boundary field $h_b$ induced by the edge modes. They are simply proportional to $h_b$ for small fields. From our microscopic model, we expect $h_b$ to be the same order as the single-particle bandwidth, so the $\lambda$ couplings are not perturbatively small. In any case, the $\cos \theta$ boundary perturbations have scaling dimension $h=g <1$ and are therefore always relevant [@0305-4470-31-12-003]. At low energy, $\lambda_R$ and $\lambda_L$ flow to strong coupling so the cosines will pin down the $\theta$ field at the boundary. This induce a flow from Dirichlet ($\left. \varphi \right|_\partial = 0$) to Neumann boundary conditions ($\left. \theta \right|_\partial = 0$ or $\left. \theta\right|_\partial = \pi$ depending on the $\pm$ signs).
The total partition function in this case is given by the superposition of these conformally invariant boundary conditions corresponding to the four configurations of the two edge modes $$Z_\text{gSPT} = Z_{0,0} + Z_{0,\pi} + Z_{\pi,0} + Z_{\pi,\pi},
\label{eq:Z_gSPT_I}$$ where $Z_{a,b}$ denotes the boundary partition function of a free boson with $\theta(x=0)=a$ and $\theta(x=L)=b$. Using standard boundary CFT tools (see Appendix), we find that $$\begin{aligned}
Z_\text{gSPT}(g) &= \frac{2}{\eta(q)} \sum_{m \in {\mathbb{Z}}} \left( q^{m^2/g} + q^{\left( m-\frac{1}{2} \right)^2/g} \right) \notag \\
&= 2 Z_\text{gTrivial}\left(\frac{1}{4g} \right).
\label{eq:Z_gSPT_II}\end{aligned}$$ where, as above, $g = \pi^{-1} \arccos(-\Delta_\text{eff})$. This immediately implies the spectrum for $H_\text{gSPT}$ is $$E_{m,n}-E_0 = \frac{\pi v_F}{L}\Big[ \frac{m^2}{4g} + n \Big],\text{ for } m \in {\mathbb{Z}},\; n > 1,
\label{eq:gSPT_spectrum}$$ now with multiplicity $2p(n)$. This is the same as with $g \to 1/4g$ and the double degeneracy. In the special case $\Delta_{\rm eff} = 0$ (i.e. $g = \tfrac{1}{2}$), the spectrum of $H_\text{gSPT}$ is exactly a doubly degenerate version of $H_\text{gTrivial}$. (This non-interacting limit agrees with previous results [@2012arXiv1206.1332G; @Grover280; @scaffidi2017gapless].) For generic $\Delta_{\rm eff}$, however, the spectrum of boundary critical exponents will change. Employing the same procedure as for the trivial case, Figure \[fig:spectrum\_comparison\] (b) compares to numerical results. One can see that the correspondence is excellent, away from ${\left| \Delta_{\rm eff} \right|} \approx 1$. Note that realizing these boundary conditions in a topologically trivial Luttinger liquid would require introducing symmetry-breaking magnetic fields at the boundary.
Note that the groundstate of $H_\text{gSPT}$ (as well as all low-lying excited states) is doubly degenerate, in contrast with ordinary gapped SPTs with spin-$\frac{1}{2}$ edge modes that exhibit a 4-fold degeneracy, such as the Haldane chain. This is because edge modes in the configurations $\Ket{\uparrow_L \downarrow_R}$ and $\Ket{\downarrow_L \uparrow_R}$ in the $z$ basis (corresponding to $Z_{0,\pi}$ and $Z_{\pi,0}$) induce a kink of $\pi$ in $\theta(x)$. In boundary CFT language, this corresponds to the insertion of a boundary condition changing (BCC) operator. This modifies the critical exponents by a factor of $-\tfrac{1}{2}$, visible in Eq. . The states with $\Ket{\uparrow_L \downarrow_R}$, $\Ket{\downarrow_L \uparrow_R}$ are hence power-law (${\sim} 1/L$) split from states with $\Ket{\uparrow_L \uparrow_R}$, $\Ket{\downarrow_L \downarrow_R}$ in any finite size system. Of course, finite-size eigenstates should preserve the symmetry $\mathcal{C}_\sigma = \prod_i \sigma^x_i$ and will therefore come in cat-state superpositions of the edge modes $\Ket{\uparrow_L \downarrow_R} \pm \Ket{\downarrow_L \uparrow_R}$ and $\Ket{\uparrow_L \uparrow_R} \pm \Ket{\downarrow_L \downarrow_R}$. At finite sizes, these cat states are not exactly degenerate but are exponential split ${\sim} {\rm e}^{-L/\xi}$ since the edge modes are exponentially localized, as the degeneracy between the states is broken at $L$th order in perturbation theory. In practice, this exponential splitting is completely negligible compared to the $1/L$ critical scaling in Eq. .
The presence of cat states of edge modes has interesting consequences for the edge magnetization in the groundstate. As discussed above, in the doubly degenerate (up to exponentially small correction) ground state of $H_\text{gSPT}$, the edge modes the ends of the chain are in linear combinations of $\ket{\uparrow_L \downarrow_R} \pm \ket{\downarrow_L \uparrow_R}$ (in the $z$-basis). As in usual symmetry breaking, any magnetic field larger than the exponential splitting between these cat states and much smaller than the CFT finite size gap ${\sim} 1/L$ is enough to pick out one of the two states breaking the symmetry at the boundary. This creates a spontaneous edge magnetization for the $\sigma$ spins in the $z$ direction, which decays algebraically into the bulk (see Fig. \[fig:magnetization\] (a) and (b)). The ${\mathbb{Z}}_2^{(\sigma)}$ symmetry is therefore spontaneously broken at the edge. In the language of surface criticality, this would correspond to an extraordinary transition, in which the edge breaks the symmetry before the bulk does [@cardy1996scaling]. Note that such an extraordinary transition would be forbidden in a regular 1D system since the edge is zero-dimensional. The boundary fixed point $Z_{\rm gSPT}$ is therefore extremely unusual and strongly relies on SPT physics.
BCFT for the $(Z)$-model {#sec:whyY}
------------------------
![(Top) Edge magnetization for gSPT and gTrivial systems at $\Delta = \alpha = 0, g_\tau = 0.3,u_\tau = 0.1$ on 16 sites, via exact diagonalization. (Middle) Power-law decay of edge magnetization into the (antiferromagnetic) bulk via DMRG on 80 sites under an applied magnetic field $h_z = 10^{-5}$. The power law depends on $\Delta$; $g_\tau = 0.3, u_\tau = 0.1, \alpha = 0$. (Bottom) Edge magnetization for $H_\text{gSPT}^{(Z)}$ with the same parameters. The magnetization is now non-universal and changes between algebraic antiferromagnetism for $\Delta = 0.8$ and exponentially decaying ferromagnetism for $\Delta =-0.8$. []{data-label="fig:magnetization"}](magnetization_comp)
Now that the boundary CFT picture of the edge modes is well-established for the $(Y)$-model, let us comment briefly on the $(Z$)-model. The same techniques apply, but the boundary critical exponents happen to be non-universal in this case. The ground states of $H_\text{gSPT}^{(\text{$Z$})}$ also have pinned edge spins along the $z$ axis, which are antiferromagnetic (i.e. $\ket{\uparrow_L \downarrow_R}$ and $\ket{\downarrow_L \uparrow_R}$) for $\Delta_{\rm eff} > 0$ and ferromagnetic (i.e. $\ket{\uparrow_L \uparrow_R}$ and $\ket{\downarrow_L \downarrow_R}$) for $\Delta_{\rm eff} < 0$. This is shown in Figure \[fig:magnetization\].
The edge field is now in the direction of anisotropy, so by the bosonization dictionary , it is a perturbation $\partial_x \varphi$ at the boundary. This is an exactly marginal perturbation, which can be absorbed by a singular gauge transformation leading to a line of RG fixed points with a non-universal critical exponents [@0305-4470-31-12-003]. Phenomenologically, this gives the same result: we still have a “superposition” of boundary conditions, giving rise to double degneracy and edge magnetization. Now, however, the edge magnetization can have either algebraic or exponential decay, as shown in Figure \[fig:magnetization\] (c), which is a feature of boundary fields in the $z$ direction for the XXZ spin chain. Both the $(Y)$- and $(Z)$-models are gapless SPTs with gapless edge modes. The primary difference is that the $(Y)$-model is more appealing from the field theory perspective, while the $(Z)$-model has more natural protecting symmetries (since in that case the $U(1)$ symmetry commutes with the SPT twisting unitary $U$).
Conclusions
===========
We have demonstrated the existence of a gapless symmetry-protected topological phase, LL$^{\star}$. It has the crucial phenomenological feature of an SPT — topological edge states — so long as the protecting symmetries are preserved. Using standard field theory and numerical techniques, we have shown LL$^{\star}$ is a genuine phase, and thus robust to small perturbations. Using the language of boundary CFT, we have exactly solved the low-energy part of the model, which realizes an exotic “superposition” of conformal boundary conditions. These boundary conditions give rise to unusual edge effects in physical observables.
From a field theory point of view, gapped SPTs can be understood as non-linear sigma models with a bulk topological theta term with $\theta=2\pi$ [@PhysRevB.91.134404]. Gapless SPTs can then be obtained by tuning a subset of the protecting symmetries to criticality. It would be interesting to study how the kind of exotic surface criticality reported here arises in this language.
There are multiple ways to interpret gSPTs. On the one hand, they provide a host of novel examples of surface criticality, where the anomalous edge properties produce effects similar to the so-called extraordinary transitions. The universality classes realized by these should be commonplace in the quantum phase transitions separating gapped SPTs and broken-symmetry phases. On the other hand, gSPTs can be thought of as new type of gapless topological matter. For example, twisting $U(1)$ gauge theories in 3D can lead to gapless topological phases in higher dimensions [@PhysRevX.6.011034]. One could also partially gauge the symmetries [@Levin2012], which might produce gapless fractionalized matter. We therefore expect gSPTs to be a useful concept which paves the way to a complete understanding of gapless topological matter.
We thank Ehud Altman and Ashvin Vishwanath for insightful discussions and Johannes Motruk for invaluable advice on DMRG. We acknowledge support from the Emergent Phenomena in Quantum Systems initiative of the Gordon and Betty Moore Foundation (T.S.) and NSF DMR-1507141 (D.P.).
Boundary Conformal Field Theory Partition Functions
===================================================
This Appendix will compute the partition functions of the free boson BCFT and derive the equation in Section \[sec:BCFT\]. The computation is standard and relatively straightforward, but is detailed here for completeness. We will follow closely the references [@oshikawa1997boundary; @blumenhagen2009introduction].
We would like to compute the partition function for the compactified free boson with various (conformally invariant) boundary conditions. Initially, we will work with Dirichlet boundary conditions, fixing $\varphi$ to $\varphi_a$ and $\varphi_b$ on the two ends. In order to follow more closely the CFT literature, we choose to work with a boson with compactification radius $r = \sqrt{g/2}$. Since time does not enter the Hamiltonian we may rescale $x \to x/v_F$ to eliminate the Fermi velocity and instead work on an interval of length $L/v_F$. With this convention, the Hamiltonian density is $\mathcal{H} = \frac{1}{2\pi}\left[ \left( \partial_x \theta \right)^2 + (\partial_x \varphi)^2 \right]$ and via the standard relation $\partial_x \theta = - \pi \Pi_\varphi = \partial_t \varphi$ [@giamarchi2003quantum], this equivalent to the Lagrangian density $\mathcal{L} = \frac{1}{2\pi} \left( \partial_\mu \varphi \right)^2$.
The Dirichlet-Dirichlet partition function may now be computed on a cylinder with circumference $\beta$ and length $L/v_F$. It is convenient to use conformal invariance to reverse time and space: this is called the “direct channel” in the boundary CFT language. Define $\varphi = \varphi(x,t) \equiv \varphi(x+\beta,t)$ for $0 \le x \le \beta$ and $0 \le t \le L/v_F$. Since imaginary time is propagating along the original spatial direction, the partition function in this picture is not a trace, but is given by the amplitude $$Z^{DD}(\Delta \varphi) = \Braket{D(\varphi_a)|e^{- \frac{L}{v_F} H_\beta}|D(\varphi_b)},
\label{eq:Z_DD}$$ where $\ket{D(\varphi_{a,b})}$ are the conformal boundary states associated with Dirichlet boundary conditions, $\Delta \varphi = \varphi_a - \varphi_B$, and $H_\beta$ is the Hamiltonian for the free boson on the circle with circumference $\beta$.
To proceed, expand the chiral components $\varphi(x,t) = \varphi_L(x^+) + \varphi_L(x^-)$ as $$\begin{aligned}
\varphi_L(x^+) \ &=\ \frac{\hat{a_0}}{2} + \frac{\pi}{\beta}\left( r w + \frac{\hat{a^\dagger}}{2} \right)x^+\\
\ &\quad +\ \frac{1}{2} \sum_{n=1}^\infty \left[ \frac{a_n}{\sqrt{n}} e^{-i n x^{+} \frac{2\pi}{\beta}} \right] + \frac{a_n^\dagger}{\sqrt{n}} e^{inx^+ \frac{2\pi}{\beta}},\\
\varphi_R(x^-) \ &=\ \frac{\hat{a_0}}{2} + \frac{\pi}{\beta}\left( -r w + \frac{\hat{a^\dagger}}{2} \right)x^-\\
\ &\quad +\ \frac{1}{2} \sum_{n=1}^\infty \left[ \frac{b_n}{\sqrt{n}} e^{-i n x^{-} \frac{2\pi}{\beta}} \right] + \frac{b_n^\dagger}{\sqrt{n}} e^{inx^- \frac{2\pi}{\beta}}.\\\end{aligned}$$ where $w$ is an integer winding number and $x^\pm =\pm x + t$. Since the constant mode $\hat{x}$ also obeys $\hat{x} \equiv \hat{x} +2 \pi \beta$, the conjugate momentum $p$ is quantized to an integer multiple of $1/r$. The new operators obey commutation relations $[a_n, a_m^\dagger] = [b_n, b_m^\dagger] = \delta_{nm}$ and $[\hat{x}, \hat{p}] = i$. We may then rewrite the Hamiltonian, in terms of the modes, either in the CFT picture as the generator of translations in the $t$ direction, or by putting the modes back into the Hamiltonian: $$ H_\beta =\ \frac{2\pi}{\beta}\left[ (rw)^2 + \left( \frac{\hat{p}}{2} \right)^2 + \sum_{n=1}^\infty n a_n^\dagger a_n + n b_n^\dagger b_n - \frac{1}{12} \right].
\label{eq:mode_Hamiltonian}$$ The Dirichlet boundary states also take a simple form in terms of these modes [@oshikawa1997boundary] $$\ket{D(\varphi_0)} = \frac{1}{\sqrt{2r}}\sum_{k \in {\mathbb{Z}}} e^{-ik \varphi_0/r} \exp\left[ -\sum_{n = 1}^{\infty} a_n^\dagger b_n^\dagger \right] \ket{(0,k)},
\label{eq:Dirichlet_boundary_state}$$ where $\ket{(w,k)}$ is the vacuum state with winding number $w$ and $\hat{p} = k/r$. For Neumann boundary conditions, by contrast, which can also be interpreted as Dirichlet boundary conditions for the dual field $\theta$, the boundary state is $$\ket{N(\varphi_0)} =\sqrt{r} \sum_{k \in {\mathbb{Z}}} e^{-i2rk \varphi_0} \exp\left[ +\sum_{n = 1}^{\infty} a_n^\dagger b_n^\dagger \right] \ket{(w,0)}.
\label{eq:Neumann_boundary_state}$$ These are related by the duality transformation $\varphi \leftrightarrow \theta$, $r \leftrightarrow 1/(2r)$ (or $g \leftrightarrow 1/g$ in the Luttinger parameter convention).
We may now directly evaluate $\eqref{eq:Z_DD}$, and following [@blumenhagen2009introduction], we introduce the basis $$\ket{{\boldsymbol{m}}} = \ket{(m_1, m_2,\dots)} = \prod_{n=1}^\infty \frac{\left( a_n^\dagger \right)^{m_n}}{\sqrt{m_n!}} \ket{\Omega},
\label{eq:clever_basis}$$ where $\ket{\Omega}$ is the vacuum, and the factors were chosen so that $\braket{{\boldsymbol{n}}|{\boldsymbol{m}}} = \delta_{{\boldsymbol{n}}{\boldsymbol{m}}}$. Similarly, define $\ket{\widetilde{{\boldsymbol{m}}}}$ with $a^\dagger_n \to b^\dagger_n$. It is also convenient to introduce the anti-unitary charge conjugation operator, $\mathcal{C}$ , whose action on the modes $\mathcal{C} a_n \mathcal{C}^{-1} = - a_n$, $\mathcal{C} b_n \mathcal{C}^{-1} = - b_n$ and performs complex conjugation $\mathcal{C} \, z\, \mathcal{C}^{-1} = z^*$ for $z \in {\mathbb{C}}$. Using these states, we find $$\exp\left[ -\sum_{n=1}^\infty a_n^\dagger b_n^\dagger \right] \ket{\Omega} = \sum_{\{ {\boldsymbol{m}} \}} \ket{{\boldsymbol{m}}} \otimes \ket{ \mathcal{C}\, \widetilde{{\boldsymbol{m}}}\,},
\label{eq:clever_basis_simplification}$$ where $\{ {\boldsymbol{m}} \}$ runs over all configurations of $m_n$. Notice that the state naturally separated into two chiral sectors.
Combining Equations , , and yields $$\begin{aligned}
Z^{DD}( \Delta \varphi) \ &=\ \frac{1}{2r} \sum_{k \in {\mathbb{Z}}, \{ {\boldsymbol{m}}\}} e^{-ik \Delta \varphi/r} A \times B \times C,\\
A \ &=\ \braket{k|\exp\left( -\frac{2\pi L}{v_F \beta}\left[ \left( rw \right)^2 + \left( \frac{\hat{p}}{2} \right)^2 - \frac{1}{12} \right] \right)|k}, \notag\\
B \ &=\ \braket{{\boldsymbol{m}}|\exp\left( -\frac{2\pi L}{v_F \beta}\left[ \sum_{n=1}^\infty n a_n^\dagger a_n \right] \right)|{\boldsymbol{m}}},\notag\\
C \ &=\ \braket{\mathcal{C} \,\widetilde{{\boldsymbol{m}}}\,|\exp\left( -\frac{2\pi L}{v_F \beta}\left[ \sum_{n=1}^\infty n b_n^\dagger b_n \right] \right)|\mathcal{C}\,\widetilde{{\boldsymbol{m}}}\,},\notag\end{aligned}$$ where we have used the orthogonality property of the $\ket{k}$’s $\ket{{\boldsymbol{m}}}$’s and $\ket{\,\widetilde{{\boldsymbol{m}}}\,}$’s. We can now easily compute each of these terms. Since the winding number is always zero, $A = e^{-\frac{2\pi L}{v_F \beta}\left[ \left( \frac{k}{2r} \right)^2 - \frac{1}{12} \right]}$. Since $a_n^\dagger a_n$ is the number operator, we have $\exp\left( \sum_{n=1}^\infty n a_n^\dagger a_n \right) \ket{{\boldsymbol{m}}} = \exp\left( \sum_{n=1}^\infty n m_n \right) \ket{{\boldsymbol{m}}}$, which implies $B = e^{-\frac{2\pi L}{v_F \beta}\left[ \sum_{n=1}^\infty n m_n \right]}$. Of course, the same logic applies to the other chiral sector, so $C = B$. The partition function is therefore $$Z^{DD}(\Delta \varphi) = \frac{1}{2r} \sum_{k, \{ {\boldsymbol{m}} \}} e^{i k \Delta \varphi/r} \tilde{q}^{\frac{k^2}{8r^2}} \tilde{q}^{-1/24} \prod_{n=1}^\infty \tilde{q}^{n m_n},$$ using the modular parameter $\tilde{q} = e^{2\pi i \tilde{\tau}} = e^{-\frac{4 \pi L}{v_F \beta}}$. The $k$ and ${\boldsymbol{m}}$ parts separate out. Focusing on the ${\boldsymbol{m}}$ part, $$\sum_{{\boldsymbol{m}}} \tilde{q}^{-\frac{1}{24}} \prod_{n=1}^\infty \tilde{q}^{n m_n} = \tilde{q}^{-\frac{1}{24}} \prod_{n=1}^\infty \sum_{m_n = 0}^\infty \tilde{q}^{nm_n} = \frac{1}{\eta(\tilde{q})}.$$ where $\eta(q) = q^{\frac{1}{24}} \prod_{n=1}^\infty (1- q^n)$ is the Dedekind $\eta$-function. The $k$-part is naturally expressed in terms of the Jacobi $\theta$-function [@elias2003complex] $$\Theta(z | \tau) = \sum_{k \in {\mathbb{Z}}} q^{\frac{1}{2} k^2} e^{2\pi i k z}
= \sum_{k \in {\mathbb{Z}}} \left( e^{2\pi i \tau }\right)^{\frac{1}{2} k^2} e^{2\pi i k z}.
\label{eq:Jacobi_theta_function}$$ Hence the partition function is $$Z^{DD}(\Delta \varphi) = \frac{1}{2r} \frac{1}{\eta(\tilde{q})} \Theta\left(\frac{\Delta \varphi}{2\pi r} \; \Bigg| \; -\frac{1}{2r^2} \frac{L}{v_F \beta} \right).
\label{eq:Z_DD_II}$$
Finally, we use the modular properties of the $\eta$ and $\Theta$ functions to rewrite the partition function as a sum of Boltzmann weights to read off the operator content of the theory. For $\operatorname{Im} \tau > 0$ [@elias2003complex] $$\begin{aligned}
\Theta(z | -1/\tau) \ &=\ \sqrt{\tau/i} \; e^{i \pi \tau z^2} \Theta\left( z \tau | \tau \right),\\
\eta(-1/\tau) \ &=\ \sqrt{\tau/i} \eta(\tau).\end{aligned}$$ Applying these and simplifying yields $$Z^{DD}(\Delta \varphi) = \frac{1}{\eta(q)} \sum_{k \in {\mathbb{Z}}} q^{g \left( k+z \right)^2},
\label{eq:Z_DD_III}$$ with conjugate modular parameter $q = e^{-\beta \frac{\pi v_F}{L}}$, $z = \Delta \varphi/(2\pi r)$, and we have substituted back in the Luttinger parameter $g = 2r^2$. This is precisely Equation .
In the CFT language, the state-operator correspondence tells us each term in the sum represents one operator, whose scaling dimension is $g (k+z)^2$. Notice that if we increase $z \to z + 1$, the partition function is unchanged, which comports with the fact that $\varphi$ is compactified. The role of $\eta$ is to generate descendant fields. It is the generating function for partitions numbers $\prod_{n=1}^\infty \frac{1}{1-q^n} = \sum_{n=1}^\infty p(n) q^n$ [@elias2003complex].
Expanding the partition function as $$Z^{DD}(\Delta \varphi) = \sum_{k \in {\mathbb{Z}}, n\ge 1} p(n) q^{-\frac{1}{24} + g(k+z)^2 + n},
\label{eq:Z_DD_IV}$$ and interpreting these as Boltzmann weights, we find the boundary spectrum of the theory is $$E_{k,n} = \frac{\pi v_F}{L} \left[ -\frac{1}{24} + g(k+z)^2 + n \right],
\label{eq:spectrum}$$ with multiplicity $p(n)$.
Now that we have done the Dirichlet case, the case with Neumann boundary conditions at both ends is simple. We simply substitute $\Delta \varphi \to \Delta \theta$ and $g \to 1/g$ in Equation to find $$Z^{NN}(\Delta \theta) = \frac{1}{\eta(q)} \sum_{k \in {\mathbb{Z}}} q^{\left( k+ w \right)^2/g},
\label{eq:Z_NN_I}$$ where $w = \Delta \theta r/ \pi$. When we pin a spin at the edge to point up (down), that is equivalent to pinning $\theta = 0$ ($\theta = \pi/2r$). The superposition of up and down at both ends is therefore $$Z_\text{gSPT} = Z^{NN}(0) + Z^{NN}(0) + Z^{NN}(\pi/2r) + Z^{NN}(-\pi/2r),$$ so that $$Z_\text{gSPT} = \frac{2}{\eta(q)} \sum_{m \in {\mathbb{Z}}} \left( q^{m^2/g} + q^{\left( m+\frac{1}{2} \right)^2/g} \right).$$ This is precisely Eq. .
|
---
abstract: 'We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight $0$ and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin’s conjecture, we classify the even $2$-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by Doud and Moore [@DM06], of conductor $1951$. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of Huxley [@Hux85] from 1985.'
address:
- 'Howard House, University of Bristol, Queens Ave., Bristol BS8 1SN, United Kingdom'
- 'Department of Mathematics, Box 480, Uppsala University, 751 06 Uppsala, Sweden'
author:
- 'Andrew R. Booker, Min Lee and Andreas Strömbergsson'
title: 'Twist-minimal trace formulas and the Selberg eigenvalue conjecture'
---
[^1]
Introduction
============
In [@BS07], the first and third authors derived a fully explicit version of the Selberg trace formula for cuspidal Maass newforms of squarefree conductor, and applied it to obtain partial results toward the Selberg eigenvalue conjecture and the classification of $2$-dimensional Artin representations of small conductor. In this paper we remove the restriction to squarefree conductor, with the following applications:
\[thm:SEC\] The Selberg eigenvalue conjecture is true for $\Gamma_1(N)$ for $N\leq 880$, and for $\Gamma(N)$ for $N\leq 226$.
\[thm:Artin\] Assuming Artin’s conjecture, Table \[tab:Artin\] is the complete list, up to twist, of even, nondihedral, irreducible, $2$-dimensional Artin representations of conductor $\le2862$.
[|rrrrrrrrrrrrrrrr|]{}\
[$163$](http://www.lmfdb.org/ArtinRepresentation/2.163.8t12.1c1) & [$277$](http://www.lmfdb.org/ArtinRepresentation/2.277.8t12.1c1) & [$349$](http://www.lmfdb.org/ArtinRepresentation/2.349.8t12.1c1) & [$397$](http://www.lmfdb.org/ArtinRepresentation/2.397.8t12.1c1) & [$547$](http://www.lmfdb.org/ArtinRepresentation/2.547.8t12.1c1) & [$549$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_61.8t12.1c1) & [$607$](http://www.lmfdb.org/ArtinRepresentation/2.607.8t12.1c1) & [$679$](http://www.lmfdb.org/ArtinRepresentation/2.7_97.8t12.1c1) & [$703$](http://www.lmfdb.org/ArtinRepresentation/2.19_37.8t12.1c1) & [$709$](http://www.lmfdb.org/ArtinRepresentation/2.709.8t12.1c1) & [$711$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_79.8t12.1c1) & [$763$](http://www.lmfdb.org/ArtinRepresentation/2.7_109.8t12.1c1) & [$853$](http://www.lmfdb.org/ArtinRepresentation/2.853.8t12.1c1) & [$937$](http://www.lmfdb.org/ArtinRepresentation/2.937.8t12.1c1) & [$949$](http://www.lmfdb.org/ArtinRepresentation/2.13_73.8t12.1c1) & [$995$]{}\
[$1009$](http://www.lmfdb.org/ArtinRepresentation/2.1009.8t12.1c1) & [$1073$]{} & [$1143$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_127.8t12.1c1) & [$1147$](http://www.lmfdb.org/ArtinRepresentation/2.31_37.8t12.1c1) & [$1197$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_7_19.8t12.1c1) & [$1267$](http://www.lmfdb.org/ArtinRepresentation/2.7_181.8t12.1c1) & [$1267$]{} & [$1333$](http://www.lmfdb.org/ArtinRepresentation/2.31_43.8t12.1c1) & [$1343$]{} & [$1368$]{} & [$1399$](http://www.lmfdb.org/ArtinRepresentation/2.1399.8t12.1c1) & [$1413$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_157.8t12.1c1) & [$1699$](http://www.lmfdb.org/ArtinRepresentation/2.1699.8t12.1c1) & [$1773$]{} & [$1777$](http://www.lmfdb.org/ArtinRepresentation/2.1777.8t12.1c1) & [$1789$](http://www.lmfdb.org/ArtinRepresentation/2.1789.8t12.1c1)\
[$1879$](http://www.lmfdb.org/ArtinRepresentation/2.1879.8t12.1c1) & [$1899$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_211.8t12.1c1) & [$1899$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_211.8t12.2c1) & [$1935$]{} & [$1951$](http://www.lmfdb.org/ArtinRepresentation/2.1951.8t12.1c1) & [$1953$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_7_31.8t12.1c1) & [$1957$](http://www.lmfdb.org/ArtinRepresentation/2.19_103.8t12.1c1) & [$1984$]{} & [$2051$]{} & [$2077$](http://www.lmfdb.org/ArtinRepresentation/2.31_67.8t12.1c1) & [$2097$]{} & [$2131$](http://www.lmfdb.org/ArtinRepresentation/2.2131.8t12.1c1) & [$2135$]{} & [$2169$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_241.8t12.1c1) & [$2169$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_241.8t12.2c1) & [$2223$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_13_19.8t12.1c1)\
[$2311$](http://www.lmfdb.org/ArtinRepresentation/2.2311.8t12.1c1) & [$2353$](http://www.lmfdb.org/ArtinRepresentation/2.13_181.8t12.1c1) & [$2439$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_271.8t12.1c1) & [$2456$]{} & [$2587$](http://www.lmfdb.org/ArtinRepresentation/2.13_199.8t12.1c1) & [$2639$]{} & [$2689$](http://www.lmfdb.org/ArtinRepresentation/2.2689.8t12.1c1) & [$2709$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_7_43.8t12.1c1) & [$2743$](http://www.lmfdb.org/ArtinRepresentation/2.13_211.8t12.1c1) & [$2763$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_307.8t12.1c1) & [$2797$](http://www.lmfdb.org/ArtinRepresentation/2.2797.8t12.1c1) & [$2803$](http://www.lmfdb.org/ArtinRepresentation/2.2803.8t12.1c1) & [$2817$](http://www.lmfdb.org/ArtinRepresentation/2.3e2_313.8t12.1c1) & & &\
\
$785$ & $1345$ & $1940$ & $2159$ & $2279$ & $2313$ & $2364$ & $2424$ & $2440$ & $2713$ & $2777$ & $2777$ & $2777$ & $2857$ & &\
\
$1951$ & $1951$ & $2141$ & $2141$ & $2804$ & $2804$ & & & & & & & & & &\
- As we pointed out in [@BS07], in the case of squarefree level the Selberg trace formula becomes substantially cleaner if one sieves down to *newforms*, and that also helps in numerical applications by thinning out the spectrum. For nonsquarefree level this is no longer the case, as the newform sieve results in more complicated formulas in many cases.
Our main innovation in this paper is to introduce a further sieve down to *twist-minimal* forms, i.e. those newforms whose conductor cannot be reduced by twisting. Although there are many technical complications to overcome in the intermediate stages, in the end we find that the twist-minimal trace formula is again significantly cleaner and helps to improve the numerics.
A natural question to explore in further investigations is whether one can skip the intermediate stages and derive the twist-minimal trace formula directly. In particular, some of the most intricate parts of the present paper have to do with explicitly describing the Eisenstein series in full generality, but it often turns out that the corresponding terms of the trace formula are annihilated by the twist-minimial sieve. A direct proof might shed light on why this is so and avoid messy calculations.
- Theorem \[thm:SEC\] for $\Gamma(N)$ improves a 30-year-old result of Huxley [@Hux85], who proved the Selberg eigenvalue conjecture for groups of level $N\le18$. Treating nonsquarefree conductors is essential for this application, since a form of level $N$ can have conductor as large as $N^2$. Moreover, the reduction to twist-minimal spaces yields a substantial improvement in our numerical results by essentially halving the spectrum in the critical case of forms of prime level $N$ and conductor $N^2$ (see below). This partially explains why our result for $\Gamma(N)$ is within a factor of $4$ of that for $\Gamma_1(N)$, despite the conductors being much larger.
- By the Langlands–Tunnell theorem, the Artin conjecture is true for tetrahedral and octahedral representations, so the conclusion of Theorem \[thm:Artin\] holds unconditionally for those types. In the icosahedral case, by [@Boo03] it is enough to assume the Artin conjecture for all representations in a given Galois conjugacy class; i.e., if there is a twist-minimal, even icosahedral representation of conductor $\le2862$ that does not appear in Table \[tab:Artin\], then Artin’s conjecture is false for at least one of its Galois conjugates.
The entries of Table \[tab:Artin\] were computed by Jones and Roberts [@JR17] by a thorough search of number fields with prescribed ramification behavior, and we verified the completeness of the list via the trace formula. In principle the number field search by Jones and Roberts [@JR14] is exhaustive, so it should be possible to prove Theorem \[thm:Artin\] unconditionally with a further computation, but that has not yet been carried out to our knowledge.
- Theorem \[thm:SEC\] for $\Gamma_1(N)$ improves on the result from [@BS07] by extending to nonsquarefree $N$ and increasing the upper bound from $854$ to $880$. To accomplish the latter, we computed a longer list of class numbers of the quadratic fields ${\mathbb{Q}}(\sqrt{t^2\pm4})$ using the algorithm from [@BBJ]. Nothing (other than limited patience of the user) prevents computing an even longer list and increasing the bounds in Theorem \[thm:SEC\] a bit more. However, as explained in [@BS07 §6], our method suffers from an exponential barrier to increasing the conductor, so that by itself is likely to yield only marginal improvements. Some ideas for surmounting this barrier are described in [@BS07 §6]; in any case, as Theorem \[thm:Artin\] shows, the first even icosahedral representation occurs at conductor $N=1951$,[^2] so the bounds in Theorem \[thm:SEC\] cannot be improved unconditionally beyond $1950$.
- All of our computations with real and complex numbers were carried out using the interval arithmetic package [`Arb`](http://arblib.org/) [@Joh17]. Thus, modulo bugs in the software and computer hardware, our results are rigorous. The reader is invited to inspect our source code at [@code].
We conclude the introduction with a brief outline of the paper. In §\[sec:prelim\]–\[ss:twist-minimal\_TF\] we define the space of twist-minimal Maass forms and state our version of the Selberg trace formula for it. In §\[sec:fullTF\] we state and prove the full trace formula in general terms, and then specialize it to $\Gamma_0(N)$ with nebentypus character. In §\[sec:sieve\] we apply the sieving process to pass from the full space to newforms, and then to twist-minimal forms. In §\[sec:galois\], we describe some details of the application of the trace formula to $\Gamma(N)$ and to Artin representations. Finally, in §\[sec:numerics\] we make a few remarks on numerical aspects of the proofs of Theorems \[thm:SEC\] and \[thm:Artin\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
We are grateful to Abhishek Saha for teaching us about twist-minimal representations, in particular Lemma \[lem:twistconductor\]. We also thank John Jones and David Roberts for their efforts to find Artin representations for all of the twist-equivalence classes in Table \[tab:Artin\] and sharing their data in the [`LMFDB`](http://www.lmfdb.org/ArtinRepresentation/).
Preliminaries on twist-minimal spaces of Maass forms {#sec:prelim}
----------------------------------------------------
Let ${\mathbb{H}}=\{z=x+iy\in{\mathbb{C}}:y>0\}$ denote the hyperbolic plane. Given a real number $\lambda>0$ and an even Dirichlet character $\chi\pmod*{N}$, let ${\mathcal{A}}_\lambda(\chi)$ denote the vector space of Maass cusp forms of eigenvalue $\lambda$, level $N$ and nebentypus character $\chi$, i.e. the set of smooth functions $f:{\mathbb{H}}\to{\mathbb{C}}$ satisfying
1. $f(\frac{az+b}{cz+d})=\chi(d)f(z)$ for all $\begin{psmallmatrix}a&b\\c&d\end{psmallmatrix}\in\Gamma_0(N)$;
2. $\int_{\Gamma_0(N)\backslash{\mathbb{H}}}|f|^2\frac{dx\,dy}{y^2}<\infty$;
3. $-y^2\left(\frac{\partial^2}{\partial x^2}
+\frac{\partial^2}{\partial y^2}\right)f=\lambda f$.
We omit the level $N$ from the notation ${\mathcal{A}}_\lambda(\chi)$ since it is determined implicitly as the modulus of $\chi$ (which might differ from its conductor, i.e. $\chi$ need not be primitive). Any $f\in{\mathcal{A}}_\lambda(\chi)$ has a Fourier expansion of the form $$f(x+iy)=\sqrt{y}\sum_{n\in{\mathbb{Z}}\setminus\{0\}}
a_f(n)K_{\sqrt{\frac14-\lambda}}(2\pi|n|y)e^{2\pi inx},$$ for certain coefficients $a_f(n)\in{\mathbb{C}}$, where $K_s(y)=\frac12\int_{\mathbb{R}}e^{st-y\cosh{t}}\,dt$ is the $K$-Bessel function.
For any $n\in{\mathbb{Z}}\setminus\{0\}$ coprime to $N$, let $T_n:{\mathcal{A}}_\lambda(\chi)\to{\mathcal{A}}_\lambda(\chi)$ denote the Hecke operator defined by $$(T_nf)(z)=\frac1{\sqrt{|n|}}
\sum_{\substack{a,d\in{\mathbb{Z}}\\d>0,ad=n}}
\chi(a)\sum_{b\pmod*{d}}
\begin{cases}
f\!\left(\frac{az+b}{d}\right)&\text{if }n>0,\\
f\!\left(\frac{a\bar{z}+b}{d}\right)&\text{if }n<0.
\end{cases}$$ We say that $f\in{\mathcal{A}}_\lambda(\chi)$ is a *normalized Hecke eigenform* if $a_f(1)=1$ and $f$ is a simultaneous eigenfunction of $T_n$ for every $n$ coprime to $N$. In this case, one has $T_nf=a_f(n)f$.
Let $\operatorname{cond}(\chi)$ denote the conductor of $\chi$. For any $M\in{\mathbb{Z}}_{>0}$ with $\operatorname{cond}(\chi)\mid M$, let $\chi|_M$ denote the unique character mod $M$ such that $\chi|_M(n)=\chi(n)$ for all $n$ coprime to $MN$. Then for any $M$ with $\operatorname{cond}(\chi)\mid M\mid N$ and any $d\mid\frac{N}{M}$, we have a linear map $\ell_{M,d}:{\mathcal{A}}_\lambda(\chi|_M)\to{\mathcal{A}}_\lambda(\chi)$ defined by $(\ell_{M,d}f)(z)=f(dz)$. Let $${\mathcal{A}}^{\rm old}_\lambda(\chi)=\sum_{\substack{M,d\in{\mathbb{Z}}_{>0}\\
\operatorname{cond}(\chi)\mid M\mid N,\,M<N\\d\mid\frac{N}{M}}}
\ell_{M,d}{\mathcal{A}}_\lambda(\chi|_M)$$ denote the span of the images of all lower level forms under these maps, and let ${\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)\subseteq{\mathcal{A}}_\lambda(\chi)$ denote the orthogonal complement of ${\mathcal{A}}^{\rm old}_\lambda(\chi)$ with respect to the Petersson inner product $$\langle f,g\rangle=\int_{\Gamma_0(N)\backslash{\mathbb{H}}}
f\bar{g}\frac{dx\,dy}{y^2}.$$ We call this the space of *newforms* of eigenvalue $\lambda$, conductor $N$ and character $\chi$.
By strong multiplicity one, we have ${\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)\cap{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi')=\{0\}$ unless $\chi$ and $\chi'$ have the same modulus and satisfy $\chi(n)=\chi'(n)$ for all $n$. In particular, any nonzero $f\in{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)$ uniquely determines its conductor, which we denote by $\operatorname{cond}(f)$. Moreover, the Hecke operators $T_n$ map ${\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)$ to itself, and ${\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)$ has a unique basis consisting of normalized Hecke eigenforms.
Suppose that $f\in{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)$ is a normalized Hecke eigenform. Then for any Dirichlet character $\psi\pmod*{q}$, there is a unique $M\in{\mathbb{Z}}_{>0}$ and a unique $g\in{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi\psi^2|_M)$ such that $a_g(n)=a_f(n)\psi(n)$ for all $n\in{\mathbb{Z}}\setminus\{0\}$ coprime to $q$. We write $f\otimes\psi$ to denote the corresponding $g$. We say that $f$ is *twist minimal* if $\operatorname{cond}(f\otimes\psi)\ge\operatorname{cond}(f)$ for every Dirichlet character $\psi$.
Let ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)\subseteq{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)$ denote the subspace spanned by twist-minimal normalized Hecke eigenforms. Clearly any normalized Hecke eigenform can be expressed as $f\otimes\psi$ for some twist-minimal form $f$. In turn, for any normalized Hecke eigenform $f\in{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)$, we have $$\operatorname{cond}(f\otimes\psi)=\operatorname{lcm}(\operatorname{cond}(f),\operatorname{cond}(\psi)\operatorname{cond}(\chi\psi)),$$ as implied by the following lemma.
\[lem:twistconductor\] Let $F$ be a nonarchimedean local field, $\pi$ an irreducible, admissible, generic representation of $\operatorname{GL}_2(F)$ with central character $\chi$, and $\psi$ a character of $F^\times$. Let $a(\pi)$ and $a(\psi)$ denote the respective conductor exponents of $\pi$ and $\psi$ (written additively). Then we have $$\label{eq:twistconductor}
a(\pi\otimes\psi)\le\max\{a(\pi),a(\psi)+a(\chi\psi)\},$$ with equality if $\pi$ is twist minimal or $a(\pi)\ne a(\psi)+a(\chi\psi)$.
There are three cases to consider:
1. $\pi\cong\chi_1\boxplus\chi_2$ is a principal series representation, in which case $$a(\pi\otimes\psi)=a(\chi_1\psi)+a(\chi_2\psi),\quad
a(\pi)=a(\chi_1)+a(\chi_2),\quad\text{and}\quad\chi=\chi_1\chi_2.$$
2. $\pi\cong\operatorname{St}\otimes\chi_1$ is a twist of the Steinberg representation, in which case $$a(\pi\otimes\psi)=\max\{1,2a(\chi_1\psi)\},\quad
a(\pi)=\max\{1,2a(\chi_1)\},\quad\text{and}\quad\chi=\chi_1^2.$$
3. $\pi$ is supercuspidal.
In the first two cases, we verify by a straightforward case-by-case analysis using the observation that, for any characters $\xi_1$ and $\xi_2$, $a(\xi_1\xi_2)\le\max\{a(\xi_1),a(\xi_2)\}$, with equality if $a(\xi_1)\ne a(\xi_2)$. This also shows that equality holds in when $a(\pi)\ne a(\psi)+a(\chi\psi)$.
In the third case, Tunnell [@Tun78 Prop. 3.4] showed that $a(\pi\otimes\psi)\le\max\{a(\pi),2a(\psi)\}$, with equality if $a(\psi)\ne\frac12a(\pi)$, and that $a(\chi)\le\frac12a(\pi)$. It follows that $\max\{a(\pi),2a(\psi)\}=\max\{a(\pi),a(\psi)+a(\chi\psi)\}$, so Tunnell’s theorem implies . If, in addition, we have $a(\pi)\ne a(\psi)+a(\chi\psi)$, then $\min\{a(\psi),a(\chi\psi)\}<\frac12a(\pi)$. Making use of the isomorphism $\pi\cong\pi^\vee\otimes\chi$, we may replace $(\pi,\psi)$ by $(\pi^\vee,\chi\psi)$ if necessary, so as to assume that $a(\psi)<\frac12a(\pi)$. Thus, Tunnell’s theorem implies that equality holds in in this case.
Finally, if $\pi$ is twist minimal then $a(\pi\otimes\psi)\ge a(\pi)$, so equality holds in even if $a(\pi)=a(\psi)+a(\chi\psi)$.
Thus, $f\mapsto f\otimes\psi$ extends to an injective linear map from ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)$ to ${\mathcal{A}^{\mathrm{new}}}_\lambda(\chi\psi^2|_M)$, where $M=\operatorname{lcm}(N,\operatorname{cond}(\psi)\operatorname{cond}(\chi\psi))$.
In light of this, it is enough to consider the trace formula for twist-minimal spaces of forms. In fact, since some twist-minimal spaces are trivial, and for others a given form can have more than one representation as the twist of a twist-minimal form (i.e. it is possible to have $\operatorname{cond}(f\otimes\psi)=\operatorname{cond}(f)$ and $f\otimes\psi\ne f$), a further reduction of the nebentypus character is possible, as follows.
\[def:minimal\] Let $\chi=\prod_{p\mid N}\chi_p$ be a Dirichlet character modulo $N$, and put $e_p=\operatorname{ord}_p{N}$, $s_p=\operatorname{ord}_p\operatorname{cond}(\chi)$. We say that $\chi$ is *minimal* if the following statement holds for every prime $p\mid N$: $$\begin{aligned}
&p>2\text{ and }\bigl[s_p\in\{0,e_p\}\text{ or }
\chi_p\text{ has order }2^{\operatorname{ord}_2(p-1)}\bigr]\\
\text{or }&p=2\text{ and }s_p\in\{\lfloor{e_p/2}\rfloor,e_p\}\cup\begin{cases}
\{0\}&\text{if }e_p\le 3,\\
\{0,2\}&\text{if }e_p>3\text{ and }2\nmid e_p,\\
\emptyset&\text{if }e_p>3\text{ and }2\mid e_p.
\end{cases}\end{aligned}$$
Note that for an odd prime $p$, there are $2^{\operatorname{ord}_2(p-1)-1}$ choices of $\chi_p\pmod*{p^{e_p}}$ of order $2^{\operatorname{ord}_2(p-1)}$, and for any such $\chi_p$ we have $s_p=1$ and $\chi_p(-1)=-1$; in particular, when $p\equiv3\pmod*{4}$, the Legendre symbol $\left(\frac{\cdot}{p}\right)$ is the unique such character. (For $p\equiv1\pmod*{4}$, the spaces resulting from different choices of $\chi_p$ of order $2^{\operatorname{ord}_2(p-1)}$ are twist equivalent, but there is no canonical choice. Similarly, for $p=2$ the characters of conductor $2^{\lfloor{e_2/2}\rfloor}$ and fixed parity yield twist-equivalent spaces.)
\[lem:twistminimal\_isom\] Let $\chi\pmod*{N}$ be a Dirichlet character, and suppose that ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)\ne\{0\}$. Then there exists $\psi\pmod*{N}$ such that $\chi\psi^2$ is minimal and ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)\xrightarrow{\otimes\psi}
{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi\psi^2)$ is an isomorphism.
Since the map $f\mapsto f\otimes\psi$ is injective, by Lemma \[lem:twistconductor\] it suffices to show that there is a $\psi\pmod*{N}$ such that $\chi\psi^2$ is minimal and $\operatorname{cond}(\psi)\operatorname{cond}(\chi\psi)\mid N$. Writing $\psi=\prod_{p\mid N}\psi_p$, this is equivalent to finding $\psi_p\pmod*{p^{\operatorname{ord}_p{N}}}$ such that $\chi_p\psi_p^2$ is minimal and $$\label{eq:ordpsum}
\operatorname{ord}_p\operatorname{cond}(\psi_p)+\operatorname{ord}_p\operatorname{cond}(\chi_p\psi_p)\le\operatorname{ord}_p{N}$$ for each prime $p\mid N$.
Fix $p\mid N$ and set $e=\operatorname{ord}_p{N}$, $s=\operatorname{ord}_p\operatorname{cond}(\chi_p)$. If $e\in\{s,1\}$ then $\chi_p$ is minimal, so we can take $\psi_p$ equal to the trivial character mod $p^e$. Hence we may assume that $e>\max\{s,1\}$. In this case, since ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)$ is nonzero, it follows from [@AL78 Theorem $4.3'$] that $s\le\frac12e$.
Suppose that $p$ is odd, and let $g$ be a primitive root mod $p^e$. Then $\chi_p(g)=e(a/\varphi(p^e))$ for a unique $a\in{\mathbb{Z}}\cap[1,\varphi(p^e)]$. Set $$b=\begin{cases}
-a/2&\text{if }2\mid{a},\\
[\varphi(p^e)2^{-\operatorname{ord}_2(p-1)}-a]/2&\text{if }2\nmid a,
\end{cases}$$ and let $\psi_p$ be the character defined by $\psi_p(g)=e(b/\varphi(p^e))$. Then $\chi_p\psi_p^2$ is minimal, and since $p^{\operatorname{ord}_p{a}}\mid b$ and $e>1$, it follows that $\operatorname{ord}_p\operatorname{cond}(\psi_p)\le\frac12e$, which implies the desired inequality .
Suppose now that $p=2$. Since $s\le\frac12e$ and there is no character of conductor $2$, $\chi_2$ is already minimal if $e\le 3$. Also, by the analogue of [@AL78 Theorem 4.4(iii)] for Maass forms, if $e\ge4$ is even then we must have $s=e/2$, so $\chi_2$ is again minimal. Hence, we may assume that $e$ is an odd number exceeding $3$.
If $s\in\{0,2,\lfloor{e/2}\rfloor\}$ then $\chi_2$ is minimal, so we may assume that $3\le s\le(e-3)/2$. Recalling that $({\mathbb{Z}}/2^e{\mathbb{Z}})^\times$ is generated by $-1$ and $5$, we have $\chi_p(5)=e(a/2^{s-2})$ for a unique odd number $a\in[1,2^{s-2})$. Let $\psi_2$ be the character defined by $\psi_2(-1)=1$ and $\psi_2(5)=e(-a/2^{s-1})$. Then $\chi_2(5)\psi_2(5)^2=1$, so $\chi_2\psi_2^2$ has conductor at most $4$. Moreover, $\operatorname{cond}(\psi_2)=\operatorname{cond}(\chi_2\psi_2)=2^{s+1}$, so that $$\operatorname{ord}_2(\psi_2)+\operatorname{ord}_2(\chi_2\psi_2)=2s+2<e,$$ as desired.
Thus, we may restrict our attention to the spaces ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)$ for minimal characters $\chi$.
The twist-minimal trace formula {#ss:twist-minimal_TF}
-------------------------------
Let $g:{\mathbb{R}}\to{\mathbb{C}}$ be even, continuous and absolutely integrable, with Fourier transform $h(r)=\int_{\mathbb{R}}g(u)e^{iru}\,du$. We say that the pair $(g,h)$ is of *trace class* if there exists $\delta>0$ such that $h$ is analytic on the strip $\Omega=\{r\in{\mathbb{C}}:|\Im(r)|<\frac12+\delta\}$ and satisfies $h(r)\ll(1+|r|)^{-2-\delta}$ for all $r\in\Omega$. Let $\chi\pmod*{N}$ be a Dirichlet character, and write $\chi=\prod_{p\mid N}\chi_p$, where each $\chi_p$ is a character modulo $p^{\operatorname{ord}_p{N}}$. The trace formula is an expression for $\sum_{\lambda>0}\operatorname{tr}T_n|_{{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)}
h\Bigl(\sqrt{\lambda-\tfrac14}\Bigr)$ in terms of $g$ and $\chi$. Its terms are linear functionals of $g$ with coefficients that are multiplicative functions of $\chi$, i.e. functions $F$ satisfying $F(\chi)=\prod_{p\mid N}F(\chi_p)$.
In what follows we fix a prime $p$ and a character $\chi\pmod*{p^e}$ of conductor $p^s$, and define the local factor at $p$ for various terms appearing in the trace formula. As a notational convenience, for any proposition $P$ we write $\delta_P$ to denote the characteristic function of $P$, i.e. $\delta_P=1$ if $P$ is true and $\delta_P=0$ if $P$ is false.
### Identity term {#identity-term .unnumbered}
Define $$\label{eq:Mchi}
M(\chi)=\begin{cases}
p^{e-1}(p+1) & \text{if }s=e, \\
\frac{p-\chi(-1)}{\gcd(2,p-1,e)}\varphi(p^{e-1})
& \text{if }s<e\le2,\\
\frac{p^2-1}{\gcd(2,p-1,e)}\varphi(p^{e-2})
& \text{if }e>\max\{s,2\}.
\end{cases}$$
### Constant eigenfunction {#constant-eigenfunction .unnumbered}
Define $$\mu(\chi)=\begin{cases}
-1&\text{if }s<e=1,\\
0&\text{otherwise.}
\end{cases}$$
### Parabolic terms {#parabolic-terms .unnumbered}
For $m\ge1$ and $n\in\{\pm1\}$, define $$\begin{aligned}
\label{Phimndef}
\Phi_{m,n}(\chi)=\begin{cases}
\overline{\chi(m)}+\chi(n)\chi(m)&\text{if }s=e,\\
-1&\text{if }s<e=1\text{ and }m=p^k\text{ for some }k>0,\\
0&\text{otherwise}
\end{cases}\end{aligned}$$ and $$\Psi_n(\chi)=\begin{cases}
1+\chi(n) & \text{if }s=e\text{ or }n=e=1,\\
\frac{p-\chi(-1)}{\gcd(2,p-1,e)} & \text{if } n=1, s<e=2, \\
\frac{p^{\lfloor\frac{e-3}2\rfloor}+p^{\lceil\frac{e-3}2\rceil}}
{\gcd(2,p-1,e)}(p-1)
& \text{if }n=1, e>\max\{s,2\},\\
\frac32 & \text{if }n=-1, p=2, s<e=1, \\
\frac12 & \text{if }n=-1, p=2, s<e\in\{2, 3\},\\
1 & \text{if }n=-1, p>2, s<e=1,\\
0 & \text{otherwise.}
\end{cases}$$
### Elliptic and hyperbolic terms {#elliptic-and-hyperbolic-terms .unnumbered}
Following the notation in [@BL17 §1.1], let ${\mathcal{D}}$ denote the set of discriminants, that is $${\mathcal{D}}=\{D\in{\mathbb{Z}}:D\equiv 0\mbox{ or }1\pmod*{4}\}.$$ Any nonzero $D\in{\mathcal{D}}$ may be expressed uniquely in the form $d\ell^2$, where $d$ is a fundamental discriminant and $\ell>0$. We define $\psi_D(n)=\left(\frac{d}{n/\gcd(n,\ell)}\right)$, where $\left(\frac{\;\;}{\;\;}\right)$ denotes the Kronecker symbol. Note that $\psi_D$ is periodic modulo $D$, and if $D$ is fundamental then $\psi_D$ is the usual quadratic character mod $D$. Set $$L(z,\psi_D)=\sum_{n=1}^{\infty}\frac{\psi_D(n)}{n^z}
\quad\mbox{for }\Re(z)>1.$$ Then it is not hard to see that $$L(z,\psi_D)=L(z,\psi_d)\prod_{p\mid\ell}
\Biggl[1+\bigl(1-\psi_d(p)\bigr)\sum_{j=1}^{\operatorname{ord}_p\ell}p^{-jz}\Biggr],$$ so that $L(z,\psi_D)$ has analytic continuation to ${\mathbb{C}}$, apart from a simple pole at $z=1$ when $D$ is a square. In particular, if $D$ is not a square then we have $$L(1,\psi_D)=L(1,\psi_d)\cdot\frac1{\ell}\prod_{p\mid\ell}
\left[1+\bigl(p-\psi_d(p)\bigr)\frac{p^{\operatorname{ord}_p\ell}-1}{p-1}\right].$$
Let $t\in{\mathbb{Z}}$ and $n\in\{\pm1\}$ with $D=t^2-4n$ not a square. Then $D\in{\mathcal{D}}$, so we may write $D=d\ell^2$ as above. Define $r=\operatorname{ord}_p(D/2)+1$, $$\alpha=\frac{\psi_d(p)-1}
{1+(p-\psi_d(p))\frac{p^{\operatorname{ord}_p\ell}-1}{p-1}}
\quad\text{and}\quad
\omega=\begin{cases}
1&\text{if }2\nmid t\text{ or }p=2\text{ and }r=2s,\\
0&\text{otherwise.}
\end{cases}$$ For $s=e$ we set $$\label{e:Htn_s=e}
H_{t,n}(\chi)=\begin{cases}
\chi\bigl(\frac{t+p^s\omega}{2}\bigr)
\bigl(2+\alpha\frac{p^e+p^{e-1}-2}{p-1}\bigr)
& \text{if }r\geq 2e,\\
\chi\bigl(\frac{t+\sqrt{d}\ell}{2}\bigr) +
\chi\bigl(\frac{t-\sqrt{d}\ell}{2}\bigr)
& \text{if }r<2e\text{ and }\psi_d(p)=1, \\
0 & \text{otherwise,}
\end{cases}$$ where $\sqrt{d}$ denotes a square root of $d$ modulo $4p^e$. For $s<e$ we set $$\begin{gathered}
\label{e:Htn_s<e}
H_{t,n}(\chi)=\delta_{e=1\text{ or } \left(\frac{n}{p}\right)=1}
\cdot\delta_{r\geq e-1}\cdot\frac{\alpha}{\gcd(2,e)}
\chi\!\left(\frac{t+p^s\omega}{2}\right) p^{e-3}\\
\cdot\bigg[p\bigl(p-\delta_{e=2}\chi(-1)
-\delta_{r=e-1}(p+\delta_{2\mid e})\bigr)-\delta_{e>2}\bigg]\end{gathered}$$ when $p>2$, and $$\label{e:boldS2_min_e>s}
H_{t,n}(\chi)=\alpha\chi\!\left(\frac{t+2^{r/2}\omega}{2}\right)2^{e-3}
\begin{cases}
3 & \text{if }e\ge3, r>e, \\
1+2(-1)^e & \text{if }e\ge 3, r=e, \\
1-2(-1)^d & \text{if }e\ge3, r=e-1, 2\nmid e, \\
2 & \text{if }e=2, r\geq e, \\
-1 & \text{if }e=2, r<e, \\
4 & \text{if } e=1,\\
0 & \text{otherwise}
\end{cases}$$ when $p=2$.
With the notation in place, we can now state the trace formula for $T_n$ acting on ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)$.
\[THMmintf\] Let $\chi\pmod*{N}$ be a minimal character of conductor $q$, $n\in\{\pm1\}$, and $(g,h)$ a pair of test functions of trace class. Then $$\begin{aligned}
\sum_{\lambda>0}&\operatorname{tr}T_n|_{{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)}
h\Bigl(\sqrt{\lambda-\tfrac14}\Bigr)\\
&=-\delta_{n=1}\biggl[
\frac{M(\chi)}{12}\int_{\mathbb{R}}\frac{g'(u)}{\sinh(u/2)}\,du
+\mu(\chi)\int_{\mathbb{R}}g(u)\cosh(u/2)\,du\biggr]\\
&+\sum_{\substack{t\in {\mathbb{Z}}\\D=t^2-4n\\\sqrt{D}\notin{\mathbb{Q}}}}
H_{t,n}(\chi)L(1,\psi_{D})
\begin{cases}
g\!\left(2\log\frac{|t|+\sqrt{D}}2\right)
&\text{if }D>0,\\
\frac{\sqrt{|D|}}{\pi}\int_{\mathbb{R}}\frac{g(u)\cosh(u/2)}{4\sinh^2(u/2)+|D|}\,du
&\text{if }D<0
\end{cases}
\\
&+\Phi_{1,n}(\chi)\bigg\{
\biggl(\gamma+\log\frac{2\pi}{N}+\frac12\log\frac{2}{(2,N)}\biggr)g(0)
-\int_0^\infty\log\Bigl(\sinh\frac{u}{2}\Bigr)g'(u)\,du\\
&+\delta_{n=1}\biggl[
\biggl(\gamma+\log\frac{2}{N}+\frac12\log(2,N)
+\frac12\sum_{p\mid{N}}\log{p}\biggr)g(0)
-\int_0^\infty\log\Bigl(\tanh\frac{u}{4}\Bigr)g'(u)\,du
\biggr]\bigg\}\\
&+2\sum_{m=2}^\infty \frac{\Lambda(m)}{m}\Phi_{m,n}(\chi)g(2\log m)
-\Lambda(N/q)\Psi_n(\chi)g(0)-\delta_{N=1}\frac14\int_{\mathbb{R}}g(u)\,du.\end{aligned}$$
Statement and proof of the full Selberg trace formula {#sec:fullTF}
=====================================================
Selberg trace formula in general form from [@BS07] {#ss:STF_general}
--------------------------------------------------
The group of all isometries (orientation preserving or not) of ${\mathbb{H}}$ can be identified with $\operatorname{G}= \operatorname{PGL}_2({\mathbb{R}})$, where the action is defined by $$T(z)
=
\begin{cases}
\frac{az+b}{cz+d} & \text{if } ad-bc>0, \\
\frac{a\bar{z}+b}{c\bar{z}+d} & \text{if } ad-bc<0,
\end{cases}
\quad
\text{for } T={\begin{pmatrix}}a & b\\ c& d{\end{pmatrix}}\in \operatorname{G}.$$ The group of orientation preserving isometries, $\operatorname{G}^+ = \operatorname{PSL}_2({\mathbb{R}})$, is a subgroup of index $2$ in $\operatorname{G}$. We write $\operatorname{G}^-=\operatorname{G}\setminus \operatorname{G}^+$ for the other coset in $\operatorname{G}$. Let $\Gamma$ be a discrete subgroup of $\operatorname{G}$ such that the surface $\Gamma{\backslash}{\mathbb{H}}$ is noncompact but of finite area, and let $\chi$ be a (unitary) character on $\Gamma$. We set $\Gamma^+:= \Gamma\cap \operatorname{G}^+$ and assume $\Gamma^+\neq \Gamma$. We let $L^2(\Gamma{\backslash}{\mathbb{H}}, \chi)$ be the Hilbert space of functions $f: {\mathbb{H}}\to {\mathbb{C}}$ satisfying the automorphy relation $$f(\gamma z) = \chi(\gamma) f(z), \quad\forall \gamma \in \Gamma,$$ and $$\int_{\Gamma{\backslash}{\mathbb{H}}} \left|f\right|^2 \; d\mu < \infty.$$ We let $\{\phi_j\}_{j\geq 1}$ be any orthonormal basis of the discrete spectrum of the Laplace operator $\Delta = -y^2 \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)$ on $L^2(\Gamma{\backslash}{\mathbb{H}}, \chi)$, i.e.$\phi_j \in C^\infty({\mathbb{H}})\cap L^2(\Gamma{\backslash}{\mathbb{H}}, \chi)$ and $\Delta \phi_j = \lambda_j \phi_j$, say with increasing eigenvalues $0\leq \lambda_1\leq\lambda_2\leq\cdots$. We also let $$r_j = \sqrt{\lambda_j -\tfrac{1}{4}} \in {\mathbb{R}}^+\cup i\left[-\tfrac{1}{2}, 0\right].$$ The trace formula from [@BS07 Theorem 2 and (2.37)] reads as follows. The even analytic function $h$ and its Fourier transform $g$ are given as in Section \[ss:twist-minimal\_TF\]. The trace formula for $(\Gamma, \chi)$ is $$\sum_{j\geq 1} h(r_j)
=
\operatorname{I}(\Gamma, \chi)+ \operatorname{NEll}(\Gamma, \chi) + \operatorname{Ell}(\Gamma, \chi) + \operatorname{C}(\Gamma, \chi) + \operatorname{Eis}(\Gamma, \chi),$$ where $$\label{e:I}
\operatorname{I}(\Gamma, \chi) = \frac{\operatorname{Area}(\Gamma{\backslash}{\mathbb{H}})}{4\pi} \int_{\mathbb{R}}h(r) r\tanh(\pi r) \; dr,$$ $$\label{e:NEl}
\operatorname{NEll}(\Gamma, \chi)
=
\sum_{\{T\}\subset M, \text{ non-ell}}
\frac{\log N(T_0)}{[Z_{\Gamma}(T): [T_0]]}
\frac{\chi(T)}{N(T)^{\frac{1}{2}}- \operatorname{sgn}(\det T) N(T)^{-\frac{1}{2}}}
g(\log N(T)),$$ $$\label{e:El}
\operatorname{Ell}(\Gamma, \chi)
=
\sum_{\{T\}\subset M, \text{ elliptic}} \frac{\chi(T)}{2|Z_\Gamma(T)| \sin(\theta(T))}
\int_{\mathbb{R}}\frac{e^{-2\theta(T) r}}{1+e^{-2\pi r}} h(r) \; dr,$$ $$\begin{gathered}
\label{e:Cu}
\operatorname{C}(\Gamma, \chi)
=
\bigg(\sum_{j\in C_{\Gamma, \chi}, k(j) = j} \chi(T_{j, 0})\bigg)
\bigg\{ \frac{1}{8} h(0)
+ \frac{1}{4\pi} \int_{\mathbb{R}}h(r) \bigg(\frac{\Gamma'}{\Gamma}\!\left(\frac{1}{2}+ir\right)
-\frac{\Gamma'}{\Gamma}(1+ir) \bigg) \; dr
\bigg\}
\\
+
\frac{g(0)}{4}
\bigg( \sum_{1\leq j\leq \kappa, k(j) = j} \sum_{v\in\{0, 1\}} \chi(T_{j, v}) \log {\mathbf{c}}_{j, v}
- 2\sum_{1\leq j\leq \kappa, j\notin C_{\Gamma, \chi}} \log |1-\chi(T_j)|
\bigg)
\\
+
\frac{|C_{\Gamma, \chi}|}{2}
\bigg\{
\frac{1}{4} h(0)
-
g(0)\log 2
-
\frac{1}{2\pi}\int_{\mathbb{R}}h(r) \frac{\Gamma'}{\Gamma}(1+ir) \; dr
\bigg\}\end{gathered}$$ and $$\label{e:Eis}
\operatorname{Eis}(\Gamma, \chi)
=
\frac{1}{4\pi} \int_{\mathbb{R}}h(r)
\frac{(\varphi^{\Gamma})'}{\varphi^{\Gamma}}\!\left(\frac{1}{2}+ir\right) \; dr
- \frac{1}{4} h(0) \operatorname{tr}\Phi^{\Gamma}\!\left(\frac{1}{2}\right).$$ Note that all the sums and integrals are absolutely convergent (see [@BS07 Proposition 2.2]). In the remainder of this section we explain the notation appearing in , , , and . The set $M$ in $\operatorname{NEll}(\Gamma, \chi)$ and $\operatorname{Ell}(\Gamma, \chi)$ is given as $$M = \{T\in \Gamma\;:\; T\text{ has no cusp of $\Gamma^+$ as a fixpoint}\},$$ i.e. $M$ is the set of all $T\in \Gamma$ which do not fix any cusp of $\Gamma^+$ and “$\{T\}\subset M$" denotes that we add over a set of representatives for the $\Gamma$-conjugacy classes in $M$. We write $Z_\Gamma(T)$ for the centralizer of $T$ in $\Gamma$. The sum in $\operatorname{NEll}(\Gamma, \chi)$ in is over all non-elliptic conjugacy classes in $M$. Thus $T$ is hyperbolic, reflection or a glide reflection, and let $\alpha\in (-\infty, -1] \cup (1, \infty)$ be the unique number such that $T$ is conjugate within $\operatorname{G}$ to ${\left(\begin{smallmatrix}}\alpha & 0 \\ 0 & 1{\end{smallmatrix}\right)}$. Then we write $$N(T) = |\alpha|.$$ We denote by $T_0$ some hyperbolic element or a glide reflection in $Z_\Gamma(T)$. Then the infinite cyclic group $$[T_0] = \{T_0^n\;:\; n\in {\mathbb{Z}}\}$$ has finite index $[Z_\Gamma(T): [T_0]]$ in $Z_\Gamma(T)$ and the ratio $$\frac{\log N(T_0)}{[Z_\Gamma(T):[T_0]]}$$ depends only on $T$ and not on our choice of $T_0$. The sum in $\operatorname{Ell}(\Gamma, \chi)$ in is over all elliptic conjugate classes in $M$, and we write $\theta(T)$ for the unique number $\theta\in (0, \frac{\pi}{2}]$ such that $T$ is conjugate within $\operatorname{G}$ to ${\left(\begin{smallmatrix}}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta{\end{smallmatrix}\right)}$. Now we explain the notations appearing in $\operatorname{C}(\Gamma, \chi)$ in . Let $$\eta_1, \ldots, \eta_\kappa \in \partial{\mathbb{H}}= {\mathbb{R}}\cup\{\infty\}$$ be a set of representatives of the cusps of $\Gamma^+{\backslash}{\mathbb{H}}$, one from each $\Gamma^+$-equivalence class. For each $j\in \{1, \ldots, \kappa\}$ we choose ${\mathsf{N}}_j\in \operatorname{G}^+$ such that ${\mathsf{N}}_j(\eta_j) = \infty$ and the stabalizer $\Gamma_{\eta_j}^+$ is $[T_j]$ where $$T_j = {\mathsf{N}}_j^{-1} {\begin{pmatrix}}1 & -1 \\ 0 & 1{\end{pmatrix}}{\mathsf{N}}_j.$$ We write $C_{\Gamma, \chi}$ for the set of open cusp representatives, viz. $$C_{\Gamma, \chi} = \{j\in \{1, \ldots, \kappa\}\;:\; \chi(T_j) = 1\}.$$ We fix, once and for all, an element $V\in \Gamma-\Gamma^+$. For each $j\in\{1, \ldots, \kappa\}$ there exists $k(j)\in\{1, \ldots, \kappa\}$ and $U_j\in \Gamma^+$ such that $$V \eta_j = U_j \eta_{k(j)}.$$ Note that $k(j)$ is uniquely determined by the condition that $\eta_{k(j)}$ is the representative of the cusp $V\eta_j$ and then $U_j$ is determined up to the right shifts with $T_{k(j)}$. By [@BS07 p.116 and (2.5)], $U_j$ satisfies $$\label{e:Uj}
U_j = V{\mathsf{N}}_j^{-1} {\begin{pmatrix}}-1 & x_j \\ 0 & 1{\end{pmatrix}}{\mathsf{N}}_{k(j)} \in \Gamma^+, \text{ for some }x_j\in {\mathbb{R}}.$$ We note in particular that $k(k(j)) = j$ and that $k(j)\in C_{\Gamma, \chi}$ if and only if $j\in C_{\Gamma, \chi}$. For each $j\in\{1, \ldots, \kappa\}$ with $k(j) = j$, and each $v\in {\mathbb{Z}}$, we set $$\label{e:Tjv}
T_{j, v} = {\mathsf{N}}_j^{-1} {\begin{pmatrix}}-1 & x_j+v\\ 0 & 1{\end{pmatrix}}{\mathsf{N}}_j = U_j^{-1} V T_j^v\in \Gamma.$$ The last identity follows from . Then $T_{j, v}$ is a reflection fixing the point $\eta_j$. Also the other fixpoint of $T_{j, v}$ in $\partial{\mathbb{H}}$ must be a cusp; we write it as $V_2\eta_k$ for some $V_2\in \Gamma^+$ and $k\in \{1, \ldots, \kappa\}$, and then define the number ${\mathbf{c}}_{j, v}>0$ by the relation (cf. [@BS07 (2.26)]) $$\label{e:mbc}
{\mathsf{N}}_j V_2 {\mathsf{N}}_k^{-1} = {\begin{pmatrix}}* & * \\ {\mathbf{c}}_{j, v} & * {\end{pmatrix}},
\text{ where } \det({\mathsf{N}}_j V_2 {\mathsf{N}}_k^{-1}) = 1.$$
It now remains to explain the notations $\varphi^\Gamma$ and $\Phi^\Gamma$ appearing in $\operatorname{Eis}(\Gamma, \chi)$ in . For $j\in C_{\Gamma, \chi}$, let $E_j(z, s, \chi)$ be the Eisenstein series for $\langle\Gamma^+, \chi\rangle$ associated to the cusp $\eta_j$: $$\label{e:Eis_j}
E_j(z, s, \chi)
=
\sum_{W\in [T_j]{\backslash}\Gamma^+}
\chi(W^{-1}) (\Im({\mathsf{N}}_j W z))^s,
\text{ for } z\in {\mathbb{H}}\text{ and } \Re(s)>1,$$ continued meromorphically to all $s\in {\mathbb{C}}$. There is a natural $\Gamma$-analog of $E_j(z, s, \chi)$, which for every $j\in C_{\Gamma, \chi}$ is given by (cf. [@BS07 (2.8)–(2.10)]): $$\label{e:Eis_j_Gamma}
E_j^{\Gamma}(z, s, \chi)
=
E_j(z, s, \chi)
+
\chi(V^{-1}) E_j (Vz, s, \chi).$$ We fix, once and for all, a subset $R_{\Gamma, \chi} \subset C_{\Gamma, \chi}$ such that $$\label{e:R_Gammachi}
\forall j\in C_{\Gamma, \chi}:
\begin{cases}
\text{ exactly one of } j, k(j)\in R_{\Gamma, \chi} & \text{if } k(j)\neq j, \\
j\in R_{\Gamma, \chi} \iff \chi(V^{-1}U_j)=1 & \text{if } k(j)=j.
\end{cases}$$ For each $j\in C_{\Gamma, \chi}$ with $k(j) = j$, it holds that $\chi(V^{-1} U_j) = \pm1$. Hence if $j\notin R_{\Gamma, \chi}$ then $E_j^{\Gamma}(z, s, \chi) \equiv 0$. In [@BS07 (2.32)], meromorphic functions $\varphi_{j, \ell}^{\Gamma}(s)$ for $j, \ell\in R_{\Gamma, \chi}$ are constructed so that for each $j\in R_{\Gamma, \chi}$, $$\label{e:Eis_avr_FE}
E_j^{\Gamma}(z, 1-s, \chi)
=
\sum_{\ell\in R_{\Gamma, \chi}} \varphi_{j, \ell}^{\Gamma}(1-s) E_\ell^{\Gamma}(z, s, \chi).$$ The relations together determine the functions $\varphi_{j, \ell}^{\Gamma}(s)$ uniquely (cf. [@BS07 p.125(bottom)]). We also note that all $\varphi_{j, \ell}^{\Gamma}(s)$ are holomorphic along the line $\Re(s) = \frac{1}{2}$. The $|R_{\Gamma, \chi}|\times |R_{\Gamma, \chi}|$ matrix $$\Phi^{\Gamma}(s) = (\varphi_{j, \ell}^{\Gamma}(s))_{j, \ell\in R_{\Gamma, \chi}}$$ is called the scattering matrix for $\Gamma$ and we write $$\varphi^{\Gamma}(s) = \det\Phi^\Gamma(s).$$ We point out that this determinant $\varphi^{\Gamma}(s)$, as well as the trace of $\Phi^{\Gamma}(s)$ which also appears in are independent of which ordering of $R_{\Gamma, \chi}$ we chose when defining the matrix $\Phi^{\Gamma}(s)$.
Trace formula for $\Gamma_0(N)$, $\chi$
---------------------------------------
Throughout this section we will use the convention that all matrix representatives for elements in $\operatorname{G}= \operatorname{PGL}_2({\mathbb{R}})$ are taken to have determinant $1$ or $-1$. Let $N$ be an arbitrary positive integer and set $\Gamma^+= \Gamma_0(N)\subset \operatorname{G}^+$. Fix $$V = {\begin{pmatrix}}-1 &0 \\ 0 & 1{\end{pmatrix}}\in \operatorname{G}$$ and note that $V^2 =1_2$, the identity matrix, and $V\Gamma^+V^{-1} = \Gamma^+$. Hence $\Gamma = \Gamma_0^\pm (N) = \langle\Gamma^+, V\rangle\subset \operatorname{G}$ is a supergroup of $\Gamma^+$ of index $2$. We will use the standard notation $\omega(N)$ for the number of primes dividing $N$. Let $\chi$ be a Dirichlet character modulo $N$. For any divisor $\alpha\mid N$ satisfying $\gcd(\alpha, N/\alpha)=1$, we define $$\chi_\alpha(x) = \chi(y)$$ for $y\equiv_{\alpha} x$ and $y\equiv_{N/\alpha}1$. By the Chinese remainder theorem, such $y$ is uniquely determined modulo $N$ and $\chi_\alpha$ is a Dirichlet character modulo $\alpha$. It follows that $\chi = \chi_\alpha \chi_{N/\alpha}$ for all such divisors $\alpha$. For any prime $p$, we set $$\chi_p = \chi_{\gcd(N, p^{\infty})}.$$ It follows that $\chi = \prod_{p\mid N} \chi_p$ (identity of functions on ${\mathbb{Z}}$); also $\chi_p \equiv 1$ for all $p\nmid N$. Let us agree to all $\chi$ pure if $\chi_p(-1) = 1$ for every odd prime $p$. Note that if $\chi$ is even then $\chi$ is pure if and only if $\chi_p(-1)=1$ for all primes $p$. From now on we fix $\chi$ to be an even Dirichlet character modulo $N$, and we set $q=\operatorname{cond}(\chi)$. We introduce two indicator functions $I_\chi$ and $I_{q, 4}$ through $$\begin{aligned}
& I_{\chi} = I(\chi\text{ is pure});\\
& I_{q, 4} = I(p\equiv_41 \text{ for every prime }p\mid q).\end{aligned}$$ We view $\chi$ as a character on $\Gamma^+$ via $$\chi\!\left({\left(\begin{smallmatrix}}a & b\\ c & d{\end{smallmatrix}\right)}\right) = \chi(d) \text{ for } {\left(\begin{smallmatrix}}a & b\\ c& d{\end{smallmatrix}\right)}\in\Gamma^+.$$ This character can be extended in two ways to a character of $\Gamma$: either by $\chi(V)=1$ or $\chi(V)=-1$. These extensions are explicitly given by $$\label{e:chi-epsilon}
\chi^{({\varepsilon})}(T) = \det(T)^{{\varepsilon}} \chi(d), \text{ for } T={\left(\begin{smallmatrix}}a & b\\ c& d{\end{smallmatrix}\right)}\in \Gamma \text{ where } {\varepsilon}\in\{0, 1\}.$$ Let $\operatorname{Tr}(\operatorname{G}, \chi^{({\varepsilon})})$ be the trace formula for $(\Gamma_0^\pm (N), \chi^{({\varepsilon})})$. For $n\in \{\pm1\}$, we have $$\label{e:Tn_epsilon}
\sum_{\lambda\geq0} \operatorname{tr}T_n|_{{\mathcal{A}}_\lambda(\chi)} h\left(\sqrt{\lambda-\tfrac{1}{4}}\right)
=
\operatorname{Tr}(\operatorname{G}, \chi^{(0)})+n \operatorname{Tr}(\operatorname{G}, \chi^{(1)}).$$
For each prime $p\mid N$, define $$\label{e:Psi1}
\Psi_1(p^{e_p}, p^{s_p})
=
\begin{cases}
2p^{e_p-s_p} & \text{if } e_p < 2s_p, \\
p^{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}}+ p^{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}} & \text{if } e_p \geq 2s_p,
\end{cases}$$ $$\label{e:Psi2}
\Psi_2(p^{e_p}, p^{s_p})
=
\begin{cases}
1 & \text{if } e_p=0, \\
\begin{cases}
2e_p & \text{if } e_p>0 \text{ and } s_p=0, \\
2(e_p-2s_p+1) & \text{if } e_p\geq 2s_p \text{ and } s_p>0, \\
\end{cases}
& \text{if } p \text{ is odd}, \\
\begin{cases}
e_p+1 & \text{if } e_p\in\{1, 2\} \text{ and } s_p=0, \\
2(e_p-1) & \text{if } e_p\in\{4, 3\} \text{ and } s_p=0, \\
4(e_p-3) & \text{if } e_p\geq 5 \text{ and } s_p=0, \\
4(e_p-2s_p-1) & \text{if } e_p\geq 2s_p+1 \text{ and } s_p\geq 3, \\
\end{cases}
& \text{if } p=2, \\
0 & \text{otherwise,}
\end{cases}$$ $$\label{e:tPsi2}
{\tilde{\Psi}}_2(p^{e_p}, \chi_p)
=
\begin{cases}
1 & \text{if } e_p=0, \\
\min\{e_2+1, 4\} & \text{if } p=2 \text{ and } s_2=0, \\
2 & \text{if } p\equiv_4-1 \text{ and } s_p=0, \\
\chi_p(\sqrt{-1}) \Psi_2(p^{e_p}, p^{s_p}) & \text{if } p\equiv_41 \text{ and } \chi_p \text{ even}, \\
0 & \text{otherwise}
\end{cases}$$ (where $\sqrt{-1}$ is a square root of $-1$ in ${\mathbb{Z}}/p{\mathbb{Z}}$), and $$\label{e:Psi3}
\Psi_3(p^{e_p}, p^{s_p})
=
\begin{cases}
0 & \text{if } e_p=0, \\
p^{e_p-s_p}(4s_p-1) & \text{if } e_p < 2s_p, \\
\left(p^{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}}+p^{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}}\right)
\left(e_p-\frac{1}{p-1}\right)
+
\frac{2}{p-1} + \delta_{s_p>0} \big(p^{s_p-1} +2\frac{p^{s_p-1}-1}{p-1}\big)
& \text{if } e_p\geq 2s_p.
\end{cases}$$ For $j\in \{1, 2\}$ we set $\Psi_j(N, q) = \prod_{p\mid N} \Psi_j(p^{e_p}, p^{s_p})$. Similarly, ${\tilde{\Psi}}_2(N,\chi)=\prod_{p\mid N}{\tilde{\Psi}}_2(p^{e_p}, \chi_p)$. Let $$\label{e:Omega1}
\Omega_1(N, q)
=
\begin{cases}
1 & \text{if } 0\leq e_2 \leq 1 \text{ or } s_2=e_2, \\
\frac{3}{2} & \text{if } e_2=2\text{ and } s_2 < e_2, \\
2 & \text{if } e_2\geq 3 \text{ and } s_2 < e_2,
\end{cases}$$ $$\label{e:Omega2}
\Omega_2(N, q)
=
\begin{cases}
\frac{1}{2}-\frac{3}{4}e_2
& \text{if } 0\leq e_2 \leq 2, \\
-2 & \text{if } e_2\geq 3, s_2=0, \\
-2s_2 & \text{if } e_2> s_2 \geq 3, \\
-e_2 & \text{if } e_2=s_2\geq 3.
\end{cases}$$ Finally we define $$\label{e:Phi}
\Phi_\pm(p^{e_p}, m)
=
\Psi_1(p^{e_p}, p^{w_\pm(m)})
\qquad\text{with }\:
w_\pm(m) = \max\{s_p, e_p-\operatorname{ord}_p(\pm m^2-1)\}.$$
\[thm:STF\_N\] For $n\in\{\pm1\}$ we have $$\label{e:STF_N}
\sum_{\lambda\geq0} \operatorname{tr}T_n|_{{\mathcal{A}}_\lambda(\chi)} h\left(\sqrt{\lambda-\tfrac{1}{4}}\right)
=
\operatorname{I}(\Gamma, \chi; n)
+
(\operatorname{NEll}+\operatorname{Ell})(\Gamma, \chi; n)
+
(\operatorname{C}+\operatorname{Eis})(\Gamma, \chi; n),$$ where $$\operatorname{I}(\Gamma, \chi; n)
=
\frac{1+n}{2} \frac{N\prod_{p\mid N} (1+p^{-1})}{12}
\int_{\mathbb{R}}rh(r) \tanh(\pi r) \; dr,$$ $$(\operatorname{NEll}+\operatorname{Ell})(\Gamma, \chi; n)
=
\sum_{\substack{t\in {\mathbb{Z}}, t^2-4n=d\ell^2, \\ \sqrt{t^2-4n}\notin {\mathbb{Q}}}}
A_{t, n}[h] {\bold{h}}^+(d)
\prod_{p} {\bold{S}}_p(p^{e_p}, \chi_p; t, n),$$ where $A_{t, n}[h]$ is defined in and ${\bold{S}}_p(p^{e_p}, \chi_p; t, n)$ is given in and Lemma \[lem:boldSp\], $$\begin{gathered}
\label{e:Cu+Eis_N_1}
(\operatorname{C}+\operatorname{Eis})(\Gamma, \chi; 1)
\\
=
\Psi_1(N, q) \bigg\{ \frac{1}{4} h(0)
-\frac{1}{2\pi} \int_{\mathbb{R}}h(r) \bigg(\frac{\Gamma'}{\Gamma}\!\left(\frac{1}{2}+ir\right) + \frac{\Gamma'}{\Gamma}(1+ir)\bigg) \; dr
\bigg\}
- \frac{1}{4} I_\chi \Psi_2(N, q) h(0)
\\
+
\bigg\{ \Psi_1(N, q) \log\!\left(\frac{\pi}{2}\right)
-
\sum_{p\mid N} \Psi_3(p^{e_p}, p^{s_p}) \prod_{p'\mid N, p'\neq p} \Psi_1({p'}^{e_{p'}}, {p'}^{s_{p'}}) \log p\bigg\}
g(0)
\\
+
2\sum_{m\geq1, \gcd(m,q)=1}\frac{\Lambda(m)}{m} \prod_{p\mid N,p\nmid m} \Big[(\Re\chi_p(m))\Phi_+(p^{e_p}, m)\Big]\,
g(2\log m)\end{gathered}$$ and $$\begin{gathered}
\label{e:Cu+Eis_N_-1}
(\operatorname{C}+ \operatorname{Eis}) (\Gamma, \chi; -1)
=
\frac{1}{4}I_\chi \big\{ 2^{\omega(N)} \Omega_1(N, q) - I_{q, 4} {\tilde{\Psi}}_2(N, \chi)\big\} h(0)
\\
-
I_\chi 2^{\omega(N)}\Omega_1(N, q)
\frac{1}{2\pi} \int_{\mathbb{R}}h(r) \frac{\Gamma'}{\Gamma}(1+ir) \; dr\\
+
I_\chi 2^{\omega(N)}
\bigg\{
\Omega_1(N, q) \bigg(\log \pi - \sum_{p\mid N, p>2} \max\{\tfrac{1}{2}, s_p\} \log p\bigg)
+
\Omega_2(N, q) \log 2
\bigg\} g(0)
\\
+
2\sum_{m\geq1, \gcd(m,q)=1} \frac{\Lambda(m)}{m} \prod_{p\mid N,p\nmid m}
\Big[(\{\Re/i\Im\}\chi_p(m)) \Phi_-(p^{e_p}, m)\Big]\,
g(2\log m),\end{gathered}$$ where $\{\Re/i\Im\}\chi_p(m) = \frac{1}{2}\big(\overline{\chi_p(m)}+\chi_p(-m)\big)$.
Identity $I(\Gamma, \chi)$
--------------------------
\[lem:I\_N\] For $n\in \{\pm1\}$, we have $$I(\Gamma, \chi; n)
=
\frac{1+n}{2} \frac{N \prod_{p\mid N} (1+p^{-1})}{12}
\int_{\mathbb{R}}rh(r) \tanh(\pi r) \; dr.$$
It is well known that $\operatorname{Area}(\Gamma^+{\backslash}{\mathbb{H}}) = \frac{\pi}{3} N\prod_{p\mid N} (1+p^{-1})$ (cf. [@Miy89 Theorem 4.2.5(2)]), and the area of $\Gamma{\backslash}{\mathbb{H}}$ is half as large. By , we have $$\operatorname{I}(\Gamma, \chi^{({\varepsilon})}) = \frac{N \prod_{p\mid N}(1+p^{-1})}{24} \int_{\mathbb{R}}h(r) r\tanh(\pi r) \; dr
\qquad\text{for }\:{\varepsilon}=\pm1.$$
Non-cuspidal contributions $\operatorname{NEll}+\operatorname{Ell}(\Gamma, \chi)$
---------------------------------------------------------------------------------
The aim of this section is to prove the following proposition.
\[prop:NEll+Ell\_N\] For $n\in \{\pm1\}$, we have $$\label{e:NEll+Ell_N}
\operatorname{NEll}(\Gamma, \chi; n)+\operatorname{Ell}(\Gamma, \chi; n)
=
\sum_{\substack{ t\in {\mathbb{Z}}, t^2-4n=d\ell^2, \\\sqrt{t^2-4n}\notin {\mathbb{Q}}}}
A_{t, n}[h]\, {\bold{h}}^+(d) \prod_p {\bold{S}}_p(p^{e_p}, \chi_p; t, n),$$ where $d$ is a fundamental discriminant and $\ell\in {\mathbb{Z}}_{>0}$.
The method we use to enumerate the conjugacy classes appearing in these sums is well known, cf., e.g., [@Eic56], [@Vig80] and [@Miy89]. Here we will follow the setup in [@Miy89] fairly closely (as was done in [@BS07] in the case of $N$ squarefree). We start by introducing some notations. Set $$R = \left\{
{\begin{pmatrix}}a & b\\ c& d{\end{pmatrix}}\in \operatorname{M}_2({\mathbb{Z}})\;:\; c\equiv_N0
\right\}$$ and $$R^1=\{T\in R\;:\; \det(T)=1\}.$$ Let $$P = \left\{\langle n, t\rangle\in \{\pm1\} \times {\mathbb{Z}}\;:\;
\sqrt{t^2-4n}\notin {\mathbb{Q}},\;
\exists T_{n, t}\in R, \; \det(T_{n, t})=n \text{ and } \operatorname{tr}(T_{n, t}) = t\right\}.$$ For each $\langle n, t\rangle\in P$, we fix one such element $T_{n, t}\in R$. Given $\langle n, t\rangle\in P$, we let $d$ and $\ell$ be the unique integers such that $t^2-4n=d\ell^2$, $\ell\in {\mathbb{Z}}_{\geq 1}$ and $d$ is a fundamental discriminant (that is $d\equiv_41$, is squarefree and $d\neq 1$ or else $d\equiv_40$, $d/4$ is squarefree and $d/4\not\equiv_41$). Then the subalgebra ${\mathbb{Q}}[T_{n, t}]\subset \operatorname{M}_2({\mathbb{Q}})$ is isomorphic to the quadratic field ${\mathbb{Q}}(\sqrt{d})$. For any positive integer $f$, we write ${\mathfrak{r}}[f]$ for the order ${\mathbb{Z}}+f\frac{d+\sqrt{d}}{2}{\mathbb{Z}}$ in ${\mathbb{Q}}(\sqrt{d})$. Using ${\mathbb{Q}}[T_{n, t}]\cong {\mathbb{Q}}(\sqrt{d})$ (either of the two possible isomorphisms), we let ${\mathfrak{r}}[f]$ denote also the corresponding order in ${\mathbb{Q}}[T_{n, t}]$. For any order ${\mathfrak{r}}$ in ${\mathbb{Q}}[T_{n, t}]$ we set $$C(T_{n, t}, {\mathfrak{r}})
=
\{ \delta T_{n, t} \delta^{-1}\;:\;
\delta\in \operatorname{GL}_2({\mathbb{Z}}), \; {\mathbb{Q}}[T_{n, t}]\cap \delta^{-1} R\delta={\mathfrak{r}}\}.$$ Note that $C(T_{n, t}, {\mathfrak{r}})$ is closed under conjugation by elements in $R^1$. We denote by $$C(T_{n, t}, {\mathfrak{r}})//R^1 = R^1 {\backslash}C(T_{n, t}, {\mathfrak{r}}) / R^1$$ a set of representatives for the inequivalent $R^1$-conjugacy classes in $C(T_{n, t}, {\mathfrak{r}})$. Also for $U={\left(\begin{smallmatrix}}a& b\\ c& d{\end{smallmatrix}\right)}\in C(T_{n, t}, {\mathfrak{r}})$, we write $$\chi^{({\varepsilon})}(U) = \det(U)^{{\varepsilon}} \chi(d).$$ For every $f\mid \ell$, it holds that $C(T_{n, t}, {\mathfrak{r}}[f])\subset R$ and using this one checks that $\chi^{({\varepsilon})}(U)$ for $U\in C(T_{n, t}, {\mathfrak{r}}[f])$ only depends on the $R^1$-conjugacy class of $U$. Let ${\bold{h}}({\mathfrak{r}}[f])$ be the narrow class number for ${\mathfrak{r}}[f]$. We also define $$\label{e:Atnh}
A_{t, n}[h] =
\begin{cases}
\frac{\log \epsilon_1}{\sqrt{t^2-4n}} g\left(\log \frac{(|t|+\sqrt{t^2-4n})^2}{4}\right),
& \text{ if } t^2-4n>0, \\
\frac{2}{|{\mathfrak{r}}[1]^1| \sqrt{4n-t^2}} \int_{\mathbb{R}}\frac{e^{-2r\cdot \arccos(|t|/2)}}{1+e^{-2\pi r}} h(r) \; dr,
& \text{ if } t^2-4n < 0,
\end{cases}$$ where in the first case $\epsilon_1>1$ is the proper fundamental unit in ${\mathbb{Q}}(\sqrt{d})$.
Following the discussion in [@BS07 pp. 136-137], we see that $\operatorname{NEll}(\Gamma, \chi^{({\varepsilon})}) + \operatorname{Ell}(\Gamma, \chi^{({\varepsilon})})$ can be collected as $$\begin{gathered}
\label{e:NEl+El_N_1}
\operatorname{NEll}(\Gamma, \chi^{({\varepsilon})}) + \operatorname{Ell}(\Gamma, \chi^{({\varepsilon})})
\\
=\frac12
\sum_{\langle n, t\rangle\in P}
\bigg\{\sum_{f\mid \ell}
\left[{\mathfrak{r}}[1]^1:{\mathfrak{r}}[f]^1\right]
\bigg( \sum_{U\in C(T_{n, t}, {\mathfrak{r}}[f])// R^1} \chi^{({\varepsilon})}(U)\bigg)
\bigg\}
A_{t, n}[h]
\begin{cases}
1 & \text{if } t^2-4n>0, \\
\frac{1}{2} & \text{if } t^2-4n<0.
\end{cases}\end{gathered}$$
The expression would of course look slightly nicer if we did not include the factor “$2$" in the definition of $A_{t, n}[h]$ in the case $t^2-4n<0$. However our definition makes the final expression slightly simpler.
The following lemma generalizes [@Str00 Lemma 3.4].
\[lem:ntP\] Let $n, t$ be any integers satisfying $\sqrt{t^2-4n}\notin {\mathbb{Q}}$, and let $d, \ell$ be the unique integers such that $t^2-4n=d\ell^2$, $\ell>0$ and $d$ is a fundamental discriminant. Then there exists $T\in R$ satisfying $\det(T)=n$ and $\operatorname{tr}(T)=t$ if and only if, for each prime $p\mid N$, we have $$\label{e:ntP}
\operatorname{ord}_p(\ell) \geq {\left\lceil}\frac{e_p}{2}{\right\rceil}\text{ or }
\left(\frac{d}{p}\right)=1
\text{ or }
\left[ 2\nmid e_p, \; \operatorname{ord}_p(\ell) \geq {\left\lfloor}\frac{e_p}{2}{\right\rfloor}\text{ and } p\mid d\right].$$ Here $e_p = \operatorname{ord}_p(N)$.
Such an element $T\in R$ exists if and only if there is some $a\in {\mathbb{Z}}$ satisfying $a(t-a)-n\equiv_N0$, and this holds if and only if the same congruence equation is solvable modulo $p^{e_p}$ for each prime $p\mid N$. The lemma now follows by completing the square in the expression $a(t-a)-n$, and splitting into the classes $2\mid e_p$ and $2\nmid e_p$, as well as $p=2$ and $p>2$.
The $R^1$-conjugacy classes in each $C(T_{n, t}, {\mathfrak{r}})$ can be enumerated using a local-to-global principle. Let $T=T_{n, t}$ and ${\mathfrak{r}}= {\mathfrak{r}}[f]$ for some $f\mid \ell$. For each prime $p$, we set $$C_p(T, {\mathfrak{r}})
=
\left\{x T x^{-1}\;:\; x\in \operatorname{GL}_2({\mathbb{Q}}_p), \; {\mathbb{Q}}_p[T]\cap x^{-1}R_p x={\mathfrak{r}}_p\right\},$$ where $$R_p = \left\{{\begin{pmatrix}}a& b\\ c& d{\end{pmatrix}}\in \operatorname{M}_2({\mathbb{Z}}_p)\;:\;
\gamma\equiv_N0\right\}$$ is the closure of $R$ in $\operatorname{M}_2({\mathbb{Q}}_p)$ and ${\mathfrak{r}}_p$ is the closure of ${\mathfrak{r}}$ in ${\mathbb{Q}}_p(\sqrt{d})\cong{\mathbb{Q}}_p[T]$. One checks that $C_p(T, {\mathfrak{r}})$ is closed under $R_p^\times$-conjugation. We also set $$C_\infty(T, {\mathfrak{r}}) = \left\{xT x^{-1}\;:\; x\in \operatorname{GL}_2({\mathbb{R}})\right\},$$ which is closed under $R_\infty^\times$-conjugation, where we have set $R_\infty^\times = \operatorname{GL}_2^+({\mathbb{R}})$. Let us now mildly alter our previous definition of $C(T, {\mathfrak{r}})//R^1$ (a set of representatives), so as to instead let $C(T, {\mathfrak{r}})//R^1$ denote the set of $R^1$-conjugacy classes in $C(T, {\mathfrak{r}})$. Similarly for each place $v$ of ${\mathbb{Q}}$, we let $C_v(T, {\mathfrak{r}})//R_v^\times$ be the set of $R_v^\times$-conjugacy classes in $C_v(T, {\mathfrak{r}})$. Clearly we have a natural map $$\label{e:theta_map}
\theta: C(T, {\mathfrak{r}}) // R^1 \to \prod_{v} C_v(T, {\mathfrak{r}})//R_v^\times.$$ By [@Miy89 Lemma 6.5.2] (trivially generalized as noted in [@BS07]), the map $\theta$ is surjective, and in fact $\theta$ is exactly ${\bold{h}}({\mathfrak{r}})$-to-$1$, where ${\bold{h}}({\mathfrak{r}})$ is the (narrow) class number for ${\mathfrak{r}}$. It remains to understand each factor in the right hand side of . For $v=\infty$, we have, as in [@Miy89 (6.6.1)], $$\left|C_\infty(T, {\mathfrak{r}}[f])//R_\infty^\times\right|
=
\begin{cases}
1 & \text{if } d>0, \\
2 & \text{if } d<0.
\end{cases}$$ Also for primes $p\nmid N$, we have $R_p \subset \operatorname{M}_2({\mathbb{Z}}_p)\subset \operatorname{M}_2({\mathbb{Q}}_p)$ and thus $$\left|C_p(T, {\mathfrak{r}})// R_p^\times\right| = 1$$ (cf., e.g., [@Miy89 Theorem 6.6.7]). Finally, if $p\mid N$, let $e = \operatorname{ord}_p(N)$ and $\rho = \operatorname{ord}_p(\ell/f)$. For $\alpha\in {\mathbb{Z}}_{\geq 1}$ let $$\Omega(p^{\alpha}; n, t)
=
\left\{ \xi\in {\mathbb{Z}}_p\;:\; \xi^2-t\xi+n\equiv_{p^{\alpha}} 0\right\}.$$ Then by [@Miy89 Theorem 6.6.6] [^3], a complete set of representatives for $C_p(T, {\mathfrak{r}})//R_p^\times$ (where $T=T_{n, t}$, ${\mathfrak{r}}= {\mathfrak{r}}[f]$ and $f\mid \ell$) is given by $$\begin{gathered}
\left\{{\begin{pmatrix}}\xi & p^{\rho}\\-\frac{\xi^2-t\xi+n}{p^{\rho}} & t-\xi{\end{pmatrix}}\;:\;
\xi\in \Omega(p^{e+2\rho}; n, t)/p^{e+\rho}\right\}
\\
\bigcup
\begin{cases}
\left\{{\begin{pmatrix}}t-\xi & -\frac{t^2-t\xi+n}{p^{e+\rho}}\\ p^{e+\rho} & \xi{\end{pmatrix}}\;:\;
\xi\in \Omega(p^{e+2\rho+1}; n, t)/p^{e+\rho}\right\}
& \text{if } p^{2\rho+1} \mid \ell^2 d, \\
\emptyset & \text{otherwise.}
\end{cases}\end{gathered}$$ The sets $\Omega(p^{e+2\rho}; n, t)$ and $\Omega(p^{e+2\rho+1}; n, t)$ are both closed under addition with any element from $p^{e+\rho}{\mathbb{Z}}_p$, and $\Omega(p^{e+2\rho}; n, t)/p^{e+\rho}$ and $\Omega(p^{e+2\rho+1}; n, t)/p^{e+\rho}$ denote complete sets of representatives for $\Omega(p^{e+2\rho}; n, t)\mod p^{e+\rho} {\mathbb{Z}}_p$ and $\Omega(p^{e+2\rho+1}; n, t) \mod p^{e+\rho}{\mathbb{Z}}_p$, respectively. Using the above facts together with $\chi = \prod_{p\mid N} \chi_p$ and the fact that $\xi^2-t\xi+n = \xi(\xi-t)+n$ is invariant under $\xi\mapsto t-\xi$, it follows that, for any $f\mid \ell$, $$\begin{gathered}
\label{e:sum_chi(U)}
\sum_{U\in C(T_{n, t}, {\mathfrak{r}}[f])// R^1} \chi^{({\varepsilon})}(U)
=
\prod_{p\mid N}
\bigg(
\sum_{\xi\in \Omega(e^{e+2\rho}; n, t)/p^{e+\rho}} \chi_p(\xi)
+
\delta_{p^{2\rho+1}\mid \ell^2d} \sum_{\xi\in \Omega(p^{e+2\rho+1}; n, t) / p^{e+\rho}} \chi_p(\xi)
\bigg)
\\
\times
{\bold{h}}({\mathfrak{r}}[f]) n^{{\varepsilon}}
\begin{cases}
1 & \text{if } d>0, \\
2 & \text{if } d<0.
\end{cases}\end{gathered}$$ For $p\mid N$ with $e=\operatorname{ord}_p(N)$ and $f=\operatorname{ord}_p(\ell)$, set $$\begin{gathered}
\label{e:boldSp_0}
{\bold{S}}_p(p^{e}, \chi; t, n)
=
\sum_{\xi\in \Omega(p^{e+2f}; n, t)/p^{e+f}} \chi(\xi)
+
\delta_{p\mid d} \sum_{\xi\in \Omega(p^{e+2f+1}; n, t)/p^{e+f}} \chi(\xi)
\\
+
\left(p-\left(\frac{d}{p}\right)\right) \sum_{\beta\in \{0, 1\}}
\sum_{k=1}^f p^{k-1}
\sum_{\xi\in \Omega(p^{e+2f-2k+\beta}; n, t)/p^{e+f-k}} \chi(\xi).\end{gathered}$$ Furthermore for $p\nmid N$, set $$\label{e:boldSp_e=0_0}
{\bold{S}}_p(1, 1; t, n) = 1+\left(p-\left(\frac{d}{p}\right)\right)
\frac{p^{\operatorname{ord}_p(\ell)}-1}{p-1}.$$ Since $${\bold{h}}({\mathfrak{r}}[f])\left[{\mathfrak{r}}[1]^1:{\mathfrak{r}}[f]^1\right]
=
{\bold{h}}^+(d) f\prod_{p\mid f} p^{-1}\left(p-\left(\frac{d}{p}\right)\right),$$ for each $\langle n, t\rangle\in P$, we have $$\label{e:sum_f|ell_prod}
\sum_{f\mid \ell}
\left[{\mathfrak{r}}[1]^1:{\mathfrak{r}}[f]^1\right]
\left( \sum_{U\in C(T_{n, t}, {\mathfrak{r}}[f])// R^1} \chi^{({\varepsilon})}(U)\right)
=
\bigg(\prod_p {\bold{S}}_p(p^{e_p}, \chi_p; t, n)\bigg)
n^{{\varepsilon}} {\bold{h}}^+(d)
\begin{cases}
1 & \text{if } d>0, \\
2 & \text{if } d<0.
\end{cases}$$ Applying to , then by , $$\label{e:NEl+El_N_2}
\operatorname{NEll}(\Gamma, \chi; n) + \operatorname{Ell}(\Gamma, \chi; n)
=
\sum_{\substack{t\in{\mathbb{Z}}\\(\langle n, t\rangle\in P)}}
A_{t, n}[h] {\bold{h}}^+(d) \prod_p {\bold{S}}_p(p^{e_p}, \chi_p; t, n).$$ In the remainder of this section, we prove Lemma \[lem:boldSp\] and get an explicit formula for ${\bold{S}}_p(p^e, \chi; t, n)$. Let us first record the general solution to the congruence equation $\xi^2-t\xi+n\equiv0$ modulo a prime power in the following lemma.
\[lem:Omega\_p\] Let $p$ be a prime and $\alpha$ a positive integer. Then $$\begin{gathered}
\label{e:Omega_p}
\Omega(p^{\alpha}; n, t)
\\
=
\begin{cases}
\left\{ \frac{t+p^{{\left\lceil}\frac{\alpha}{2}{\right\rceil}} \omega}{2} + p^{{\left\lceil}\frac{\alpha}{2}{\right\rceil}}u\;:\; u\in {\mathbb{Z}}_p\right\}
& \text{if } \operatorname{ord}_p(t^2-4n) \geq \alpha+\max\{\operatorname{ord}_p(d)-1, 0\}, \\
\left\{ \frac{t}{2}\pm \frac{\sqrt{d}\ell}{2} + p^{\alpha-\operatorname{ord}_p(\ell)}u\;:\;
u\in {\mathbb{Z}}_p\right\}
& \text{if } \operatorname{ord}_p(t^2-4n) < \alpha \text{ and } \left(\frac{d}{p}\right)=1, \\
\emptyset & \text{otherwise,}
\end{cases}\end{gathered}$$ where $$\omega =
\begin{cases}
1 & \text{if } p=2, \alpha=\operatorname{ord}_p(t^2-4n)-1 \text{ and } 8\nmid d, \\
& \text{or } p=2, \alpha=\operatorname{ord}_p(t^2-4n) \text{ and } 2\nmid d, \\
& \text{or } p\text{ and } t \text{ odd}, \\
0 & \text{otherwise.}
\end{cases}$$ and $t^2-4n = d\ell^2$ with $d$ a fundamental discriminant and $\ell\in {\mathbb{Z}}$.
Suppose first that $p$ is odd. For $\alpha\geq 1$, since $\xi\in \Omega(p^{\alpha}; n, t)$, we have $$\xi^2-t\xi+n \equiv \left(\xi-\frac{t}{2}\right)^2 -\frac{t^2}{4}+n
\equiv 0\pmod*{p^{\alpha}},$$ so that $$\label{e:eq_p_comp}
\left(\xi-\frac{t}{2}\right)^2 \equiv \frac{d\ell^2}{4} \pmod*{p^{\alpha}}.$$ When $\operatorname{ord}_p(d\ell^2) \geq \alpha$ this yields $\left(\xi-\frac{t}{2}\right)^2\equiv 0\pmod*{p^{\alpha}}$. For some $u\in {\mathbb{Z}}_{p}$, we have $$\xi = \frac{t}{2}+p^{{\left\lceil}\frac{\alpha}{2}{\right\rceil}} u.$$ Next consider the case when $\operatorname{ord}_p(d\ell^2) < \alpha$. We must have $p\nmid d$ and $\operatorname{ord}_p(d\ell^2)=2\operatorname{ord}_p(\ell) < \alpha$ for $\xi$ to satisfy . In this case has a solution only if $\left(\frac{d}{p}\right)=1$, and then $\operatorname{ord}_p\left(\xi-\frac{t}{2}\right)=\operatorname{ord}_p(\ell) = \frac{\operatorname{ord}_p(d\ell^2)}{2}$. Therefore, for some $u\in {\mathbb{Z}}_p$, $$\xi = \frac{t}{2}\pm \frac{\sqrt{d}\ell}{2} + p^{\alpha-\operatorname{ord}_p(\ell)} u.$$ Suppose now that $p=2$. Since $n\in \{\pm1\}$ and $$\xi^2-t\xi+n = \xi(\xi-t) +n\equiv 0\pmod*{2^{\alpha}},$$ both $\xi$ and $\xi-t$ must be odd, so $t$ is even. Note that when $\alpha=1$ and $t$ is even, $$\Omega(2; n, t)
=
\left\{1+2u\;:\; u\in {\mathbb{Z}}_2\right\}.$$ From now on, we assume that $\alpha\geq 2$ and $t$ is even. Then $d\ell^2 = t^2-4n = 4\left(\frac{t^2}{4}-n\right)$, so $4\mid d$ or $2\mid \ell$. Therefore $$\label{e:eq_2_comp}
\left(\xi-\frac{t}{2}\right)^2 \equiv \frac{t^2-4n}{4} \equiv
\frac{d\ell^2}{4} \pmod*{2^{\alpha}}.$$ If $\operatorname{ord}_2(d\ell^2) -2 \geq \alpha$, we have $\left(\xi-\frac{t}{2}\right)^2\equiv 0\pmod*{2^{\alpha}}$, so $$\Omega(2^{\alpha}; n, t)
=
\left\{
\frac{t}{2}+2^{{\left\lceil}\frac{\alpha}{2}{\right\rceil}} u\;:\; u\in {\mathbb{Z}}_2
\right\}.$$ Next consider the case when $\operatorname{ord}_2(d\ell^2)-2 < \alpha$. When $4\mid d$, we have $(\operatorname{ord}_2(d)-2) + 2\operatorname{ord}_2(\ell) < \alpha$. Dividing both sides of by $2^{2\operatorname{ord}_2(\ell)}$, we get $$\left(\frac{\xi-\frac{t}{2}}{2^{\operatorname{ord}_2(\ell)}}\right)^2
\equiv
\frac{d}{4}\left(\frac{\ell}{2^{\operatorname{ord}_2(\ell)}}\right)^2
\pmod*{2^{\alpha-2\operatorname{ord}_2(\ell)}}.$$ Note that $d/4\equiv_4 2$ or $3$, i.e. $d/4$ is not a square modulo $4$, which implies that $\Omega(2^{\alpha}; n, t)=\emptyset$ if $\alpha-2\operatorname{ord}_2(\ell)\geq 2$. If $\alpha-2\operatorname{ord}_2(\ell)=1$ then $$\frac{\xi-\frac{t}{2}}{2^{\operatorname{ord}_2(\ell)}}
\equiv
\frac{d}{4} \pmod*{2}.$$ Thus, when $\alpha=2\operatorname{ord}_2(\ell)+1$ and $4\mid d$, we get $$\Omega(2^{\alpha}; n, t)
=
\left\{\frac{t}{2}+2^{\operatorname{ord}_2(\ell)}\left(\frac{d}{4} + 2u\right)\;:\;
u\in {\mathbb{Z}}_2\right\}.$$ When $d\equiv 1\pmod*{4}$, we have $2\operatorname{ord}_2(\ell) -2< \alpha$. Dividing both sides of by $2^{2\operatorname{ord}_2(\ell) -2}$, we get $$\left(\frac{\xi-\frac{t}{2}}{2^{\operatorname{ord}_2(\ell)-1}}\right)^2
\equiv
d\left(\frac{\ell}{2^{\operatorname{ord}_2(\ell)}}\right)^2
\pmod*{2^{\alpha-2\operatorname{ord}_2(\ell)+2}}.$$ When $\alpha=2\operatorname{ord}_2(\ell)-1$ or $\alpha=2\operatorname{ord}_2(\ell)$, we have $$\Omega(2^{\alpha}; n, t)
=
\left\{\frac{t}{2}+2^{\operatorname{ord}_2(\ell)-1} (1+2u)\;:\; u\in {\mathbb{Z}}_2\right\}.$$ For $\alpha>2\operatorname{ord}_2(\ell)$, following a similar procedure to the case when $p$ is odd, we get $$\Omega(2^{\alpha}; n, t)
=
\left\{ \frac{t}{2}\pm \frac{\sqrt{d}\ell}{2}+2^{\alpha-\operatorname{ord}_2(\ell)} u\;:\; u\in {\mathbb{Z}}_2 \right\}.$$
Next, by Lemma \[lem:ntP\], if $\langle n, t\rangle\in \{\pm1\}\times {\mathbb{Z}}$ satisfies $\sqrt{t^2-4n}\notin {\mathbb{Q}}$ and $\langle n, t\rangle\notin P$, then there is a prime $p\mid N$ such that either
- $\left(\frac{d}{p}\right)=-1$ and $e_p>2\operatorname{ord}_p(\ell)$; or
- $p\mid d$ and $e_p > 2\operatorname{ord}_p(\ell) + 1$.
By inspection in , we see that the summation condition “$\langle n, t\rangle\in P$" in may be replaced by summation over all $\langle n, t\rangle\in\{\pm1\}\times {\mathbb{Z}}$ satisfying $\sqrt{t^2-4n}\notin {\mathbb{Q}}$, as in .
\[lem:avg\_char\] Let $p$ be a prime, $s\in {\mathbb{Z}}_{\geq 0}$ and $\psi$ a primitive Dirichlet character of conductor $p^s$. For $a, b\in {\mathbb{Z}}_{\geq 0}$, $a+b\geq s$, we have $$\label{e:avg_char}
\sum_{u\pmod*{p^{b}}} \psi(x+p^{a} u)
=
\begin{cases}
p^b \psi(x) & \text{if } s\leq a, \\
0 & \text{otherwise.}
\end{cases}$$
Since $\psi$ is primitive, $\tau(\overline{\psi})\neq 0$ and we have $$\psi(x+p^a u)
=
\frac{1}{\tau(\overline{\psi})}
\sum_{\alpha\pmod*{p^s}} \overline{\psi(\alpha)} e^{2\pi i (x+p^a u) \frac{\alpha}{p^s}}.$$ Then $$\sum_{u\pmod*{p^b}} \psi(x+p^a u)
=
\frac{1}{\tau(\overline{\psi})}
\sum_{\alpha\pmod*{p^s}} \overline{\psi(\alpha)} e^{2\pi ix\frac{\alpha}{p^s}}
\begin{cases}
\sum_{u\pmod*{p^b}} e^{2\pi i\alpha\frac{u}{p^{s-a}}} & \text{if } s> a, \\
p^b & \text{if } s\leq a.
\end{cases}$$ Since $a+b\geq s$, if $s>a$, $$\sum_{u\pmod*{p^b}} e^{2\pi i \alpha \frac{u}{p^{s-a}}}
=
\sum_{u\pmod*{p^b}} e^{2\pi i \alpha p^{a+b-s} \frac{u}{p^b}}
=
\begin{cases}
p^b & \text{if } p^{s-a}\mid \alpha, \\
0 & \text{otherwise.}
\end{cases}$$ Since $\psi(\alpha)=0$ when $s>0$ and $p\mid \alpha$, we get .
\[lem:boldSp\] For $t\in{\mathbb{Z}}$ and $n\in \{\pm1\}$, let $t^2-4n = d\ell^2$ where $d$ is a fundamental discriminant and $\ell\in{\mathbb{Z}}_{>0}$. For $e=\operatorname{ord}_p(N)\geq 1$ and $s=\operatorname{ord}_p(\operatorname{cond}(\chi))$, let $h = \max\{2s-1, e\}$, $g=\operatorname{ord}_p(t^2-4n)$ and $f=\frac{g-\operatorname{ord}_p(d)}{2}=\operatorname{ord}_p(\ell)$. When $p$ is odd, we have $$\begin{gathered}
\label{e:boldSp_odd}
{\bold{S}}_p(p^e, \chi; t, n)
\\
=
\delta_{g\geq h>0} \chi\!\left(\frac{t+p^s \delta_{2\nmid t}}{2}\right)
\bigg\{
p^{e-1} \big(p^{f-{\left\lfloor}\frac{h-1}{2}{\right\rfloor}}+p^{f-{\left\lfloor}\frac{h}{2}{\right\rfloor}}\big)
+
\left(1-\left(\frac{d}{p}\right)\right) \frac{p^{e-1}}{p-1}
\big(p^{f-{\left\lfloor}\frac{h-1}{2}{\right\rfloor}} + p^{f-{\left\lfloor}\frac{h}{2}{\right\rfloor}} - p-1\big)
\bigg\}
\\
+
\delta_{\substack{g\leq h-1, \\ \left(\frac{d}{p}\right)=1}}
\left(\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right)+ \chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)\right)
p^{f+\min\{e-s, f\}}.\end{gathered}$$ When $p=2$ we have ${\bold{S}}_2(2^e, \chi; t, n)=0$ if $t$ is odd, while if $t$ is even then $$\begin{gathered}
\label{e:boldSp_even}
{\bold{S}}_2(2^e, \chi; t, n)
\\
=
\chi\!\left(\frac{t}{2}\right)
\begin{cases}
\delta_{2\nmid d} 2^{e-1}
\big(2^{\frac{g}{2}-{\left\lfloor}\frac{h-1}{2}{\right\rfloor}} +2^{\frac{g}{2}-{\left\lfloor}\frac{h}{2}{\right\rfloor}}\big)
\\
+
\left(1-\left(\frac{d}{2}\right)\right) 2^{e-1}
\big(2^{{\left\lfloor}\frac{g}{2}{\right\rfloor}-{\left\lfloor}\frac{h-1}{2}{\right\rfloor}} + 2^{{\left\lfloor}\frac{g}{2}{\right\rfloor}-{\left\lfloor}\frac{h}{2}{\right\rfloor}}-3\big)
& \text{if } g\geq h+1 \text{ and } 2\nmid g, \\
& \text{or } g\geq \max\{h, 2s+2\} \text{ and } 2\mid g, \\
-\delta_{2\nmid d} 2^{e+1} - \left(1-\left(\frac{d}{2}\right)\right) 2^{e-1}
& \text{if } g=2s\geq e+1, \\
-3\cdot 2^{e-1} & \text{if } g=2s=e \text{ and } 2\nmid d, \\
0 & \text{otherwise}
\end{cases}
\\
+
\delta_{\substack{g\leq h-1, \\ \left(\frac{d}{2}\right)=1}}
\left(\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)\right)
2^{f+\min\{e-s, f\}}.\end{gathered}$$
By Lemma \[lem:Omega\_p\], setting $\alpha=e+2(f-k)+\beta$ for $\beta\in\{0, 1\}$ and $0\leq k \leq f$ in , we get $$\begin{gathered}
\Omega(p^{e+2(f-k)+\beta}; n, t)/p^{e+f-k}
\\
=
\begin{cases}
\left\{\frac{t+p^{{\left\lceil}\frac{e+\beta}{2}{\right\rceil}+f-k} \omega}{2} +
p^{{\left\lceil}\frac{e+\beta}{2}{\right\rceil}+f-k} u\;:\; u\pmod*{p^{{\left\lceil}\frac{e+\beta}{2}{\right\rceil}-\beta}}\right\}
& \text{if } 2k+\operatorname{ord}_p(d) \geq e+\beta+\delta, \\
\left\{\frac{t\pm \sqrt{d}\ell}{2}+p^{e+\beta+f-2k} u\;:\; u\pmod*{p^{k-\beta}}\right\}
& \text{if } 2k \leq e+\beta-1 \text{ and } \left(\frac{d}{p}\right)=1, \\
\emptyset & \text{otherwise,}
\end{cases}\end{gathered}$$ where $\delta=\max\{\operatorname{ord}_p(d)-1, 0\}$ and $$\omega = \begin{cases}
1
& \text{if } p=2, 2\nmid e+\beta, k=\frac{e+\beta+1}{2}-\frac{\operatorname{ord}_2(d)}{2}\leq f
\text{ and } 8\nmid d, \\
& \text{or } p=2, 2\mid e+\beta, k=\frac{e+\beta}{2}\leq f
\text{ and } 2\nmid d, \\
& \text{or } p\neq 2 \text{ and } t\text{ odd, }\\
0 & \text{otherwise.}
\end{cases}$$ When $p$ is odd, applying Lemma \[lem:avg\_char\], $$\begin{gathered}
\sum_{\xi\in \Omega(p^{e+2(f-k)+\beta}; n, t)/p^{e+f-k}} \chi(\xi)
\\
=
\begin{cases}
p^{{\left\lfloor}\frac{e+\beta}{2}{\right\rfloor}-\beta}
\chi\!\left(\frac{t+\delta_{2\nmid t} p^s}{2}\right)
& \text{if } {\left\lceil}\frac{e+\beta-\operatorname{ord}_p(d)}{2}{\right\rceil}\leq k\leq f+{\left\lceil}\frac{e+\beta}{2}{\right\rceil}-{\left\lceil}\frac{h+\beta}{2}{\right\rceil}, \\
p^{k-\beta}
\left(\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)\right)
& \text{if } k\leq\min\{{\left\lfloor}\frac{e+\beta+f-s}{2}{\right\rfloor}, {\left\lfloor}\frac{e-1+\beta}{2}{\right\rfloor}, f\}
\text{ and } \left(\frac{d}{p}\right)=1, \\
0 & \text{otherwise.}
\end{cases}\end{gathered}$$ Recalling , we have $$\begin{gathered}
{\bold{S}}_p(p^e, \chi; t, n)
\\
=
\delta_{\left(\frac{d}{p}\right)=1}
\left(\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)\right)
\bigg\{
1+\sum_{\beta\in \{0, 1\}} p^{-1-\beta}(p-1)
\sum_{k-1}^{{\left\lfloor}\frac{\min\{e+\beta+f-s, e-1+\beta, 2f\}}{2}{\right\rfloor}}
p^{2k}
\bigg\}
\\
+
\chi\!\left(\frac{t+\delta_{2\nmid t} p^s}{2}\right)
\left(p-\left(\frac{d}{p}\right)\right)
\sum_{\beta\in \{0, 1\}} p^{{\left\lfloor}\frac{e+\beta}{2}{\right\rfloor}-\beta-1}
\sum_{k={\left\lceil}\frac{e+\beta-\operatorname{ord}_p(d)}{2}{\right\rceil}}^{f+{\left\lceil}\frac{e+\beta}{2}{\right\rceil}-{\left\lceil}\frac{h+\beta}{2}{\right\rceil}}
p^k.\end{gathered}$$ Since $$\begin{gathered}
\label{e:sum_d/p=1}
1+\sum_{\beta\in \{0, 1\}} p^{-1-\beta}(p-1)
\sum_{k-1}^{{\left\lfloor}\frac{\min\{e+\beta+f-s, e-1+\beta, 2f\}}{2}{\right\rfloor}}
p^{2k}
\\
=
p^{\min\{e+f-s, e-1, 2f\}}
=
\begin{cases}
p^{e-1} & \text{if } g\geq h, \\
p^{\min\{e+f-s, 2f\}} & \text{if } g\leq h-1
\end{cases}\end{gathered}$$ and $$\begin{gathered}
\sum_{\beta\in \{0, 1\}} p^{{\left\lfloor}\frac{e+\beta}{2}{\right\rfloor}-\beta-1}
\sum_{k={\left\lceil}\frac{e+\beta-\operatorname{ord}_p(d)}{2}{\right\rceil}}^{f+{\left\lceil}\frac{e+\beta}{2}{\right\rceil}-{\left\lceil}\frac{h+\beta}{2}{\right\rceil}}
p^k
\\
=
p^{e-1} \frac{p^{\max\{0, {\left\lfloor}\frac{g}{2}{\right\rfloor}-{\left\lceil}\frac{h}{2}{\right\rceil}+1\}}-1}{p-1}
+
p^{e-1-\operatorname{ord}_p(d)} \frac{p^{\max\{0, {\left\lceil}\frac{g}{2}{\right\rceil}-{\left\lfloor}\frac{h}{2}{\right\rfloor}\}}-1}{p-1}
\\
=
\begin{cases}
p^{e-1} \frac{p^{{\left\lfloor}\frac{g}{2}{\right\rfloor}-{\left\lceil}\frac{h}{2}{\right\rceil}+1}-1}{p-1}
+
p^{e-1-\operatorname{ord}_p(d)} \frac{p^{{\left\lceil}\frac{g}{2}{\right\rceil}-{\left\lfloor}\frac{h}{2}{\right\rfloor}}-1}{p-1}
& \text{if } g\geq h, \\
0 & \text{if } g\leq h-1,
\end{cases}\end{gathered}$$ we get . Now assume $p=2$. If $t$ is odd then $d\equiv5$ mod $8$; thus $\bigl(\frac d2\bigr)=-1$, and $\Omega(p^\alpha;n,t)=\emptyset$ for all $\alpha\geq1$ by Lemma \[lem:Omega\_p\]; therefore all sums in are empty, and so ${\bold{S}}_2(2^e, \chi; t, n)=0$. From now on we assume $t$ is even. Applying Lemma \[lem:avg\_char\], we get $$\begin{gathered}
\sum_{\xi\in \Omega(2^{e+2(f-k)+\beta}; n, t) /2^{e+f-k}} \chi(\xi)
\\
=
\begin{cases}
2^{{\left\lfloor}\frac{e+\beta}{2}{\right\rfloor}-\beta} \chi\!\left(\frac{t}{2}\right)
& \text{if } {\left\lceil}\frac{e+\beta-\delta_{4\mid d}}{2}{\right\rceil}\leq k \leq f+{\left\lceil}\frac{e+\beta}{2}{\right\rceil}-{\left\lceil}\frac{h+\beta}{2}{\right\rceil}, \\
& \text{and } \left[2\nmid g \text{ or } 2\mid g \text{ and } \frac{g}{2}-1\right], \\
2^{{\left\lfloor}\frac{e+\beta}{2}{\right\rfloor}-\beta} \chi\!\left(\frac{t}{2}+2^{s-1}\right)
& \text{if } g=2s, k={\left\lceil}\frac{e+\beta}{2}{\right\rceil}\leq f\text{ and } 2\nmid d, \\
& \text{or } g=2s, 2\nmid e+\beta-1, k=\frac{e+\beta-1}{2}\leq f\text{ and } 4\mid d, \\
2^{k-\beta} \left(\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)\right)
& \text{if } 2k \leq \min\{e+\beta+f-s, e+\beta-1, 2f\}
\text{ and } \left(\frac{d}{2}\right)=1, \\
0 & \text{otherwise.}
\end{cases}\end{gathered}$$ Note that if $2\nmid d$ then $d\equiv_41$. So for $s=2$ and $\left(\frac{d}{2}\right)=1$, $\sqrt{d}\pmod*{4}$ is well defined. Furthermore, for $2\nmid d$, since $4\mid d\ell^2$, so $f=\operatorname{ord}_p(\ell) \geq 1$. Also note that since $(x+2^{s-1})^2\equiv x^2\pmod*{2^{s}}$ for $s\geq 2$, we have $\chi(x+2^{s-1}) = \pm \chi(x)$. Because $\chi$ is primitive, we get $$\chi(x+2^{s-1}) = -\chi(x).$$ By , $$\begin{gathered}
{\bold{S}}_2(2^e, \chi; t, n)
\\
=
\delta_{\left(\frac{d}{2}\right)=1}
\left(\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)\right)
\bigg\{ 1+ \sum_{\beta\in \{0, 1\}} \sum_{k=1}^{{\left\lfloor}\frac{\min\{e+\beta+f-s, e+\beta-1, 2f\}}{2}{\right\rfloor}}
2^{2k-1-\beta}\bigg\}\\
+
\delta_{\substack{2\nmid g \\ \text{ or } 2\mid g \text{ and } s\leq \frac{g}{2}-1}}
\left(2-\left(\frac{d}{2}\right)\right)
\chi\!\left(\frac{t}{2}\right)
\sum_{\beta\in\{0, 1\}} 2^{{\left\lfloor}\frac{e+\beta}{2}{\right\rfloor}-\beta}
\sum_{k={\left\lceil}\frac{e+\beta-\delta_{4\mid d}}{2}{\right\rceil}}^{f+{\left\lceil}\frac{e+\beta}{2}{\right\rceil}-{\left\lceil}\frac{h+\beta}{2}{\right\rceil}} 2^{k-1}
\\
-
\delta_{g=2s} \chi\!\left(\frac{t}{2}\right)
\left(2-\left(\frac{d}{2}\right)\right)
\sum_{\beta\in \{0, 1\}} 2^{{\left\lfloor}\frac{e+\beta}{2}{\right\rfloor}-\beta}
\begin{cases}
2^{{\left\lceil}\frac{e+\beta}{2}{\right\rceil}-1} & \text{if } {\left\lceil}\frac{e+\beta}{2}{\right\rceil}\leq f \text{ and } 2\nmid d, \\
2^{\frac{e+\beta-1}{2}-1} & \text{if } \frac{e+\beta-1}{2}\leq f \text{ and } 4\mid d, \\
0 & \text{otherwise.}
\end{cases}\end{gathered}$$ Applying and $$\begin{gathered}
\sum_{\beta\in \{0, 1\}} 2^{{\left\lfloor}\frac{e+\beta}{2}{\right\rfloor}-\beta-1}
\sum_{k={\left\lceil}\frac{e+\beta-\delta_{4\mid d}}{2}{\right\rceil}}^{f+{\left\lceil}\frac{e+\beta}{2}{\right\rceil}-{\left\lceil}\frac{h+\beta}{2}{\right\rceil}} 2^k
\\
=
\begin{cases}
2^{e-1-\delta_{4\mid d} }
\big(2^{{\left\lfloor}\frac{g}{2}{\right\rfloor}-{\left\lfloor}\frac{h-1}{2}{\right\rfloor}} + 2^{{\left\lfloor}\frac{g}{2}{\right\rfloor}-{\left\lfloor}\frac{h}{2}{\right\rfloor}} -2-\delta_{4\mid d}\big)
& \text{if } g\geq h+\delta_{4\mid d}, \\
0 & \text{otherwise,}
\end{cases}\end{gathered}$$ we get .
Cusps {#ss:cusps}
-----
One easily verifies that every cusp of $\Gamma^+ = \Gamma_0(N)$ is equivalent to a cusp of the form $$\label{e:cusp_N}
\tfrac{a}{c}, \; a, c\in {\mathbb{Z}}, \; c\geq 1, \; c\mid N \text{ and } \gcd(c, a)=1.$$ Any two such cusps $\frac{a_1}{c_1}$ and $\frac{a_2}{c_2}$ are $\Gamma^+$-equivalent if and only if $$c_1 = c_2 \text{ and } a_1\equiv_{\gcd(c_1, N/c_1)} a_2$$ and we then write $\frac{a_1}{c_1}\sim_{\Gamma^+} \frac{a_2}{c_2}$. We fix, once and for all, a set $C_\Gamma$ containing exactly one representative $\frac{a}{c}$ satisfying for each cusp class, i.e.$$C_\Gamma
=
\left\{\tfrac{a}{c}\;:\; a, c\in {\mathbb{Z}}, \; c\geq 1, \; c\mid N, \; \gcd(a, c)=1\right\}
/\sim_{\Gamma^+}.$$ We make our choice of $C_\Gamma$ in such a way that $$\label{e:CGamma_gcd}
\begin{cases}
\text{for every $\frac ac\in C_\Gamma$ with $\gcd(c,N/c)>2$, also $\:\frac{-a}c\in C_\Gamma$;}
\\[5pt]
\text{for every $c\mid N$ with $\gcd(c,N/c)\leq2$, $\:\frac 1c\in C_\Gamma$.}
\end{cases}$$ This is possible since $\frac{a}{c}\not\sim_{\Gamma^+} \frac{-a}{c}$ whenever $\gcd(c,N/c)>2$. Note also that it follows from our choice of $C_\Gamma$ that for every $c\mid N$ with $\gcd(c,N/c)\leq2$, $\frac1c$ is the unique cusp in $C_\Gamma$ with denominator $c$.
For each $\frac{a}{c}\in C_{\Gamma}$, we fix a matrix $$W_{a/c} = {\begin{pmatrix}}a & b\\ c& d{\end{pmatrix}}\in \operatorname{SL}_2({\mathbb{Z}}).$$ We make these choices so that $$W_{-a/c} = {\begin{pmatrix}}-a & b\\ c& -d{\end{pmatrix}}\text{ for } \gcd(c, N/c)>2
\text{ and }
W_{1/c} = {\begin{pmatrix}}1 & 0 \\ c & 1{\end{pmatrix}}\text{ for } \gcd(c, N/c)\leq 2$$ holds for any $\frac{a}{c}\in C_{\Gamma}$ (cf. ). Note that $W_{a/c}(\infty) = \frac{a}{c}$, so $W_{a/c}^{-1}(\frac{a}{c}) = \infty$. We also set $$\label{e:Na/c}
{\mathsf{N}}_{a/c}
=
{\begin{pmatrix}}\sqrt{\frac{\gcd(c^2, N)}{N}} & 0 \\ 0 & \sqrt{\frac{N}{\gcd(c^2, N)}}{\end{pmatrix}}W_{a/c}^{-1}$$ and $$\label{e:Ta/c}
T_{a/c} = {\mathsf{N}}_{a/c}^{-1} {\begin{pmatrix}}1 & -1\\ 0 & 1{\end{pmatrix}}{\mathsf{N}}_{a/c}\in \Gamma^+.$$ We then verify that the fixator subgroup of $\frac{a}{c}$ in $\Gamma^+$ is the cyclic subgroup generated by $T_{a/c}$. That is, for the cusp $\eta_j = \frac{a}{c}$, ${\mathsf{N}}_{a/c}$ and $T_{a/c}$ correspond to ${\mathsf{N}}_j$ and $T_j$ in §\[ss:STF\_general\].
\[lem:chi\_Ta/c\] For any $\frac{a}{c}\in C_{\Gamma}$, $$T_{a/c} =
{\begin{pmatrix}}1+ac\frac{N}{\gcd(c^2, N)} & -a^2 \frac{N}{\gcd(c^2, N)}\\
N\frac{c^2}{\gcd(c^2, N)} & 1-ac\frac{N}{\gcd(c^2, N)}{\end{pmatrix}}$$ and $$\label{e:chi_Ta/c}
\chi(T_{a/c})
=
\chi\!\left(1-ac\frac{N}{\gcd(c^2, N)}\right).$$
This lemma follows from the definition of $T_{a/c}$ in .
For later use, we prove the following lemma and then compute the set of open cusps. From now on, for each prime $p\mid N$, set $e_p = \operatorname{ord}_p(N)$ and $s_p = \operatorname{ord}_p(q)$, where $q=\operatorname{cond}(\chi)$.
\[lem:varphi\_gcd\] For every $x\in{\mathbb{R}}$, $$\begin{gathered}
\label{e:varphi_gcd}
\sum_{c\mid N, q\mid \frac{N}{\gcd(c, N/c)}}
\varphi(\gcd(c, N/c)) (\gcd(c, N/c))^x
\\
=
\prod_{p\mid N}
\bigg[2+
(p-1)p^{x}\frac{p^{\min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, e_p-s_p\}(x+1)}+p^{\min\{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}, e_p-s_p\}(x+1)}-2}{p^{x+1}-1}\bigg].\end{gathered}$$
By the multiplicativity of the Euler $\varphi$-function, the left hand side of becomes a product over primes dividing $N$: $$\prod_{p\mid N} \left(\sum_{\substack{j\geq0, \\ e_p-s_p\geq \min\{j, e_p-j\}}}
\varphi(p^{\min\{j, e_p-j\}}) p^{x\min\{j, e_p-j\}}\right).$$ For each prime $p\mid N$, the inner sum is $$\begin{gathered}
\label{e:varphi_gcd_p}
\sum_{\substack{j\geq0, \\ e_p-s_p\geq \min\{j, e_p-j\}}}
\varphi(p^{\min\{j, e_p-j\}}) p^{x\min\{j, e_p-j\}}
\\
=
\sum_{j=0}^{\min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, e_p-s_p\}} \varphi(p^j) p^{jx}
+
\sum_{j=0}^{\min\{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}, e_p-s_p\}} \varphi(p^j) p^{jx}
\\
=
2+ (p-1)p^{x} \frac{p^{\min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, e_p-s_p\}(x+1)}-1}{p^{x+1}-1}
+
(p-1)p^{x} \frac{p^{\min\{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}, e_p-s_p\}(x+1)}-1}{p^{x+1}-1}
.\end{gathered}$$
\[lem:open\_cusp\] The set of open cusps is $$\label{e:open_cusp_N}
C_{\Gamma, \chi}
=
\left\{\tfrac{a}{c}\in C_{\Gamma}\;:\; q\mid \frac{N}{\gcd(c, N/c)}\right\}.$$ Its cardinality is $$|C_{\Gamma, \chi}| = \sum_{c\mid N, q\mid \frac{N}{\gcd(c, N/c)}}
\varphi(\gcd(c, N/c))
=
\Psi_1(N, q).$$ Here $q=\operatorname{cond}(\chi)$ and $\Psi_1(N, q)$ is given in .
Using and $\operatorname{cond}(\chi)=q$ we have $$\chi(T_{a/c}) = \chi\!\left(1-a\tfrac{N}{\gcd(c, N/c)}\right)=1
\iff
q\mid \tfrac{N}{\gcd(c, N/c)}.$$ By Lemma \[lem:varphi\_gcd\], taking $x=0$, we get $$|C_{\Gamma, \chi}|
=
\sum_{c\mid N, q\mid\frac{N}{\gcd(c, N/c)}}\varphi(\gcd(c, N/c))
=
\prod_{p\mid N}\bigg[p^{\min\{{\left\lfloor}\frac{e_p}{s}{\right\rfloor}, e_p-s_p\}}+p^{\min\{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}, e_p-s_p\}}\bigg]
=
\Psi_1(N, q).$$
Adapting the notation from §\[ss:STF\_general\] to our present explicit setting, for each $\frac{a}{c}\in C_{\Gamma}$ we write $k(a/c)$ for the representative in $C_\Gamma$ for the cusp $V\frac{a}{c} = \frac{-a}{c}$. Then by , $$\label{e:ka/c}
k(a/c) = \begin{cases}
\frac{-a}{c} & \text{if } \gcd(c, N/c) >2, \\
\frac{1}{c} & \text{if } \gcd(c, N/c)\leq 2.
\end{cases}$$ Furthermore, as in we have fixed a choice of an element $U_{a/c}\in \Gamma^+$ satisfying $$\label{e:Ua/c}
U_{a/c} = V {\mathsf{N}}_{a/c}^{-1} {\begin{pmatrix}}-1 & u \\ 0 & 1{\end{pmatrix}}{\mathsf{N}}_{k(a/c)}$$ for some $u\in {\mathbb{R}}$ (which may depend on $\frac ac$).
\[lem:Ua/c\] For $\frac{a}{c}\in C_{\Gamma}$, when $\gcd(c, N/c)>2$ we have $$U_{a/c} =
V{\mathsf{N}}_{a/c}^{-1} {\begin{pmatrix}}-1 & u \\ 0 & 1{\end{pmatrix}}{\mathsf{N}}_{-a/c}
=
{\begin{pmatrix}}-1 + au\frac{N}{\gcd(c, N/c)} & a^2 u \frac{N/c}{\gcd(c, N/c)}\\
-Nu \frac{c}{\gcd(c, N/c)} & -1-au \frac{N}{\gcd(c, N/c)}{\end{pmatrix}},$$ for some $u\in {\mathbb{Z}}$. When $\gcd(c, N/c)\leq 2$, $$\label{e:U1/c}
U_{1/c} =
V {\mathsf{N}}_{1/c}^{-1}
{\begin{pmatrix}}-1 & u_1\frac{\gcd(c, N/c)}{N/c} \\ 0 & 1{\end{pmatrix}}{\mathsf{N}}_{1/c}$$ for some $u_1\in {\mathbb{Z}}$ satisfying $cu_1\equiv_{N/c} -2$. Furthermore, for every $\frac{a}{c}\in C_{\Gamma, \chi}$, $$\label{e:chi_U1/c_1}
\chi(U_{a/c})= \chi_{\alpha(N, c)}(-1),$$ where $$\label{e:alpha}
\alpha(N, c)
=
\begin{cases}
N & \text{if } \gcd(c, N/c)>2, \\
\frac{N}{c} & \text{if } \gcd(c, N/c)=1, \\
\frac{N}{2c} & \text{if } \gcd(c, N/c) = 2 \text{ and } \frac{N}{2c} \text{ is odd, }\\
\frac{2N}{c} & \text{if } \gcd(c, N/c) = 2 \text{ and } \frac{N}{2c} \text{ is even.}
\end{cases}$$
Given $\frac{a}{c}\in C_{\Gamma}$, recalling , $$U_{a/c} = V {\mathsf{N}}_{a/c}^{-1} {\begin{pmatrix}}-1 & u \\ 0 & 1{\end{pmatrix}}{\mathsf{N}}_{k(a/c)} \in \Gamma^+,$$ for some $u\in {\mathbb{R}}$, which is uniquely determined modulo one. If $\gcd(c, N/c)>2$, then $k(a/c) = -a/c$ and we get $$U_{a/c} = {\begin{pmatrix}}-1 + acu \frac{N}{\gcd(c^2, N)} & a^2u \frac{N}{\gcd(c^2, N)}\\
-Nu\frac{c^2}{\gcd(c^2, N)} & -1-acu \frac{N}{\gcd(c^2, N)}{\end{pmatrix}}.$$ One easily checks that $U_{a/c}\in \Gamma^+=\Gamma_0(N)$ if and only if $u\in {\mathbb{Z}}$. If $\frac{a}{c}\in C_{\Gamma, \chi}$, then $q\mid \frac{N}{\gcd(c, N/c)}$. Because $$-1-acu\frac{N}{\gcd(c^2, N)} = -1-au\frac{N}{\gcd(c, N/c)} \equiv_q -1$$ and $\chi$ is even, we get $\chi(U_{a/c}) = \chi(-1) = 1$. If $\gcd(c, N/c) \leq 2$, then $$U_{1/c} = {\begin{pmatrix}}1+ cu\frac{N}{\gcd(c^2, N)} & -u \frac{N}{\gcd(c^2, N)}\\
-2c-c^2u \frac{N}{\gcd(c^2, N)} & 1+ cu\frac{N}{\gcd(c^2, N)}{\end{pmatrix}}.$$ Let $u_1 = u\frac{N}{\gcd(c^2, N)}$. Then $U_{1/c}\in \Gamma_0(N)$ if and only if $u_1\in {\mathbb{Z}}$ and satisfying $cu_1\equiv_{N/c}-2$. Then $1+cu_1 \equiv_{N/c}-1$. Moreover, $$\chi(U_{1/c}) = \chi(1+cu_1).$$ If $\gcd(c, N/c)=1$, then $\chi = \chi_c \chi_{N/c}$, so $$\chi(U_{1/c}) = \chi_{N/c}(1+cu_1) = \chi_{N/c}(-1).$$ Assume that $\gcd(c,N/c)=2$ and $\frac{1}{c}\in C_{\Gamma, \chi}$, so that $q\mid \frac{N}{2}$. If $\frac{N}{2c}$ is odd then $\gcd(2c, \frac{N}{2c})=1$ and $\chi = \chi_{2c} \chi_{\frac{N}{2c}}$. Since $q = \operatorname{cond}(\chi_{2c}) \operatorname{cond}(\chi_{\frac{N}{2c}})$ divides $\frac{N}{2}$ and $\operatorname{cond}(\chi_{\frac{N}{2c}})\mid \frac{N}{2c}$, we have $\operatorname{cond}(\chi_{2c})\mid c$. Thus $$\chi(U_{1/c}) = \chi_{2c}(1+cu_1) \chi_{\frac{N}{2c}}(1+cu_1)
=
\chi_{\frac{N}{2c}} (-1).$$ If $\frac{N}{2c}$ is even, then $\frac{c}{2}$ is odd, so $\gcd\!\left(\frac{2N}{c}, \frac{c}{2}\right)=1$ and $\chi = \chi_{\frac{2N}{c}} \chi_{\frac{c}{2}}$. Since $q=\operatorname{cond}(\chi_{\frac{2N}{c}}) \operatorname{cond}(\chi_{\frac{c}{2}})$ divides $\frac{N}{2}$, $\operatorname{cond}(\chi_{\frac{2N}{c}})$ must divide $\frac{N}{c}$, so we get $$\chi(U_{1/c}) =\chi_{\frac{c}{2}}(1+cu_1) \chi_{\frac{2N}{c}} (1+cu_1) = \chi_{\frac{2N}{c}}(-1).$$
Note that in our present setting, says that for each $c\mid N$ with $\gcd(c, N/c)\leq 2$, and each $v\in {\mathbb{Z}}$, we set $$\label{e:T1/cv}
T_{1/c, v} = U_{1/c}^{-1} V T_{1/c}^v.$$ Also, for any such $c$ and $v$, we have a number ${\mathbf{c}}_{1/c,v}>0$ defined by . The following two lemmas evaluate $\chi^{({\varepsilon})}(T_{1/c, v})$ and ${\mathbf{c}}_{1/c,v}$. Recall that $\chi^{({\varepsilon})}(V) = (-1)^{{\varepsilon}}$.
\[lem:chi\_T1/cv\] For $c\mid N$ and $\gcd(c, N/c)\leq 2$, we have $$\label{e:chi_T1/cv}
\chi^{({\varepsilon})}(T_{1/c, v}) =
\begin{cases}
(-1)^{{\varepsilon}} \chi_{\alpha(N, c)}(-1) & \text{if } \frac{1}{c}\in C_{\Gamma, \chi}, \\
(-1)^{{\varepsilon}+v+w_c} & \text{if } \frac{1}{c}\notin C_{\Gamma, \chi},
\end{cases}$$ where $w_c\in\{0, 1\}$ is a constant determined by $(-1)^{w_c} = \chi(U_{1/c})$. Here $\alpha(N, c)$ as in .
It follows from that $\chi^{({\varepsilon})}(T_{1/c, v}) = (-1)^{{\varepsilon}} \overline{\chi(U_{1/c})} \chi(T_{1/c})^v$. If $\frac{1}{c}\in C_{\Gamma, \chi}$ then $\chi(T_{1/c})=1$, and using we have $$\chi^{({\varepsilon})}(T_{1/c, v}) = (-1)^{{\varepsilon}} \chi_{\alpha(N, c)}(-1),$$ for all $v\in {\mathbb{Z}}$. Now assume $\frac{1}{c}\notin C_{\Gamma, \chi}$. Then $\gcd(c, N/c)=2$ and $q\nmid \frac{N}{2}$, and by Lemma \[lem:chi\_Ta/c\] we have $$\chi(T_{1/c}) = \chi\!\left(1-\tfrac{N}{2}\right)=-1,$$ since $\left(1-\frac{N}{2}\right)^2 \equiv_N1$ and $q\nmid \frac{N}{2}$. By Lemma \[lem:Ua/c\], $\chi(U_{1/c}) = \chi(1+cu_1)$ for some $u_1\in {\mathbb{Z}}$ satisfying $cu_1\equiv_{N/c}-2$. Then $$(1+cu_1)^2 = 1+ 2cu_1\left(1+\tfrac{c}{2} u_1\right) \equiv_N1,$$ so $\chi(U_{1/c})^2 = 1$ and $\chi(U_{1/c})\in \{\pm1\}$. Choosing $w_c\in\{0, 1\}$ so that $\chi(U_{1/c}) = (-1)^{w_c}$, we get $$\chi^{({\varepsilon})}(T_{1/c, v}) = (-1)^{{\varepsilon}+v+w_c}.$$
\[lem:mbc\] Assume that $c\mid N$ and $\gcd(c, N/c)\leq 2$. If $4\mid N$ then ${\mathbf{c}}_{1/c, v}=\sqrt N$ for all $v$. On the other hand, if $4\nmid N$, then there is some $s\in\{0,1\}$ (which depends on $c$ and $U_{1/c}$) such that for all $v$, $$\label{e:mbc_1/c_v}
{\mathbf{c}}_{1/c, v}
=
\begin{cases}
\sqrt{N} & \text{if } v\equiv_2 s,\\
\sqrt{2N} & \text{if } v\not\equiv_2 s\text{ and }2\mid N\\
2\sqrt{N} & \text{if } v\not\equiv_2 s\text{ and }2\nmid N.
\end{cases}$$
For given $c$ and $v$, following the definition of ${\mathbf{c}}_{1/c,v}$ in , we recall that $T_{1/c,v}$ is a reflection fixing the point $\frac1c$, and that also the other fixpoint of $T_{1/c,v}$ in $\partial{\mathbb{H}}$ must be a $\Gamma^+$-cusp, which we call $\eta$. We choose $V_2\in\Gamma^+$ and $\frac{a'}{c'}\in C_\Gamma$ so that $\eta=V_2(\frac{a'}{c'})$. Then $\frac{a'}{c'}$ is a fixpoint of $V_2^{-1}T_{\frac 1c,v}V_2\in\Gamma^+ V$, and this forces $k(\frac{a'}{c'})=\frac{a'}{c'}$, i.e. $\gcd(c',N/c')\leq2$ and $a'=1$.
Set $M=\gcd(c,N/c)\leq2$. From and we get $$\begin{aligned}
\label{lem:mbcPF1}
{\mathsf{N}}_{1/c} T_{1/c, v} {\mathsf{N}}_{1/c}^{-1}= {\begin{pmatrix}}-1 & u_1\frac{M}{N/c}+v \\ 0 & 1{\end{pmatrix}},\end{aligned}$$ where $u_1$ is an integer satisfying $cu_1\equiv_{N/c} -2$; and the two fixpoints in $\partial{\mathbb{H}}$ of the reflection in are $\infty$ and $\frac12(u_1\frac M{N/c}+v)$. Hence $$\eta
=
{\mathsf{N}}_{1/c}^{-1} \biggl(\frac12\Bigl(u_1\frac{M}{N/c} +v\Bigr)\biggr)
=
{\begin{pmatrix}}1 & 0\\ c& 1{\end{pmatrix}}\frac{u_2}{2} = \frac{u_2}{cu_2+2},
\qquad\text{with }\:u_2:=u_1+v\frac{N/c}{M}.$$ But we know from above that the $C_\Gamma$-representative for the cusp $\eta$ is $\frac1{c'}$, and one easily verifies that the $C_\Gamma$-representative of an arbitrary $\Gamma^+$-cusp $\frac{\alpha}{\gamma}$ with $\alpha, \gamma\in {\mathbb{Z}}$ has denominator $\gcd\bigl(N, \frac{\gamma}{\gcd(\alpha, \gamma)}\bigr)$. Hence, letting $u_3:=\gcd(u_2,cu_2+2)=\gcd(u_2,2)\in\{1,2\}$, we have $$c'=\gcd\Bigl(N,\frac{cu_2+2}{u_3}\Bigr)=\frac1{u_3}\gcd\bigl(Nu_3,cu_2+2\bigr)=\frac N{cu_3}\gcd\Bigl(cu_3,\frac{cu_2+2}{N/c}\Bigr).$$ In the last equality we used the fact that $\frac Nc\mid cu_2+2$, since $cu_1\equiv_{N/c} -2$. To make the last expression more explicit, note that $\gcd(cu_3,\frac{cu_2+2}{N/c})\in\{1,2\}$, since $\gcd(cu_3,cu_2+2)=\gcd(cu_3,2)\in\{1,2\}$. Therefore, $$\label{e:c'}
c'=\begin{cases}
N/c&\text{if }\: u_2\equiv\frac{cu_2+2}{N/c}\pmod* 2,
\\
N/(2c)&\text{if }\: 2\mid u_2\text{ and }2\nmid\frac{cu_2+2}{N/c},
\\
2N/c&\text{if }\: 2\nmid u_2\text{ and }2\mid\frac{cu_2+2}{N/c}.
\end{cases}$$
Let us write $W_{1/c} V_2 W_{1/c'}={\left(\begin{smallmatrix}}\alpha & \beta\\ \gamma & \delta {\end{smallmatrix}\right)}\in\operatorname{SL}_2({\mathbb{Z}})$ with $\gamma\geq0$. Since $\frac{\alpha}{\gamma}=W_{1/c} V_2 W_{1/c'} (\infty) = \frac{u_2}{2}$, we get $\gamma = \frac{2}{u_3}$. Using this in the definition of ${\mathbf{c}}_{1/c,v}$, , we get $$\label{e:mbc_1/c_v_0A}
{\mathbf{c}}_{1/c,v}=\frac2{u_3}\sqrt{\frac{N/c}{\gcd(c,N/c)}}\sqrt{\frac{N/c'}{\gcd(c',N/c')}}.$$ Let us set $f=\operatorname{ord}_2(c)$ and $g=\operatorname{ord}_2(N/c)$; then $\gcd(c,N/c)\leq2$ implies $\min(f,g)\leq1$. Using , the formula can be re-expressed as ${\mathbf{c}}_{1/c,v}=\sqrt N\cdot 2^{\rho/2}$, where $$\label{e:mbc_1/c_v_0B}
\rho=\begin{cases}
-2\min(f,g)& \text{if } 2\mid u_2 \text{ and } 2\mid\frac{cu_2+2}{N/c},
\\
1-\min(f,g)-\min(f+1,g-1)& \text{if } 2\mid u_2 \text{ and } 2\nmid\frac{cu_2+2}{N/c},
\\
1-\min(f,g)-\min(f-1,g+1)& \text{if } 2\nmid u_2 \text{ and } 2\mid \frac{cu_2+2}{N/c},
\\
2-2\min(f,g) & \text{if } 2\nmid u_2 \text{ and } 2\nmid \frac{cu_2+2}{N/c}.
\end{cases}$$
Let us first assume $\min(f,g)=1$, i.e.both $c$ and $\frac Nc$ are even. In this case, if either $4\mid c$ or $2\mid u_2$, then $cu_2+2\equiv_42$, which implies $2\nmid \frac{cu_2+2}{N/c}$ and $4\nmid\frac Nc$. Hence, the first case in cannot occur, and if $f\geq2$ ($\Rightarrow g=1$) then also the third case in is excluded, while if $g\geq2$ ($\Rightarrow f=1$) then the second case in is excluded. By inspection, it then follows that $\rho=0$. Next assume $\min(f,g)=0$. Note that if $f\geq1$ ($\Rightarrow g=0$) then $2\mid\frac{cu_2+2}{N/c}$, and on the other hand if if $g\geq1$ ($\Rightarrow f=0$) then the condition $cu_1\equiv_{N/c}-2$ forces $u_1$ to be even, and then also $u_2=u_1+v\frac{N/c}M$ is even. If $f\geq2$ or $g\geq2$ then these observations imply $\rho=0$, by . We have thus proved the lemma in the case $4\mid N$, and it only remains to consider the three cases with $f,g\leq1$, not both $1$. If $f=1$ ($\Rightarrow g=0$) then implies $\rho=\delta_{2\nmid u_2}$, and so, recalling $u_2=u_1+v\frac{N/c}M$, it follows that holds with $s\equiv_2 u_1$. If $g=1$ ($\Rightarrow f=0$) then implies $\rho=\delta_{2\nmid \frac{cu_2+2}{N/c}}
=\delta_{4\mid u_2}$, and so holds with $s\equiv_2 1+\frac{u_1}2$. Finally if $f=g=0$, i.e. $2\nmid c$ and $2\nmid\frac Nc$, then $u_2\equiv_2\frac{cu_2+2}{N/c}$, and thus implies $\rho=2\delta_{2\nmid u_2}$; therefore again holds with $s\equiv_2u_1$.
Cuspidal contributions $\operatorname{C}(\Gamma, \chi)$
-------------------------------------------------------
Recall that we write $\operatorname{C}(N,\chi;n)$ for the cuspidal contribution in the trace formula in Theorem \[thm:STF\_N\]; thus $\operatorname{C}(N,\chi;n)=\operatorname{C}(\Gamma, \chi^{(0)})+n\operatorname{C}(\Gamma, \chi^{(1)})$ where $C(\Gamma,\chi^{({\varepsilon})})$ is the cuspidal contribution (cf. ) in the trace formula for $(\Gamma_0^\pm (N), \chi^{({\varepsilon})})$. Our aim in this section is to prove the following proposition, giving an explicit formula for $\operatorname{C}(N,\chi;n)$. Let $$\label{e:veN}
{\varepsilon}_N
=
\begin{cases}
1 & \text{if } N \text{ is odd}, \\
\frac{1}{2} & \text{if } 2\parallel N, \\
0 & \text{otherwise,}
\end{cases}$$ and for each prime $p\mid N$, define $$\label{e:Psi4}
\Psi_4(p^{e_p}, p^{s_p})
=
p^{e_p-s_p}
\max\{2s_p-e_p-1, 0\}.$$
\[prop:Cu\] We have $$\begin{gathered}
\label{e:Cu_N_1}
\operatorname{C}(N, \chi; 1)
=
\Psi_1(N, q)
\bigg\{ \frac{1}{4} h(0) - \frac{1}{2\pi} \int_{\mathbb{R}}h(r) \frac{\Gamma'}{\Gamma}(1+ir)\; dr
\bigg\}
\\
-
\bigg\{\Psi_1(N, q) \sum_{p\mid N} \frac{\Psi_4(p^{e_p}, p^{s_p})}{\Psi_1(p^{e_p}, p^{s_p})} \log p
+ \Psi_1(N, q) \log 2\bigg\}
g(0)\end{gathered}$$ and $$\begin{gathered}
\label{e:Cu_N_-1}
\operatorname{C}(N, \chi; -1)
=
I_\chi 2^{\omega(N)} \Omega_1(N, q)
\bigg\{
\frac{1}{4} h(0)
+ \frac{1}{2\pi}
\int_{\mathbb{R}}h(r) \bigg(\frac{\Gamma'}{\Gamma}\!\left(\frac{1}{2}+ir\right) - \frac{\Gamma'}{\Gamma}(1+ir)\bigg)
\; dr
\bigg\}
\\
+
\bigg\{I_\chi 2^{\omega(N)-1} \big(\Omega_1(N, q) \log N + {\varepsilon}_N \log 2\big)
\bigg\}
g(0).\end{gathered}$$ Here $\Psi_1$, $\Psi_4$, $\Omega_1$ and ${\varepsilon}_N$ are given in , , and respectively.
The proposition will be proved by evaluating the various sums appearing in in our explicit setting. By , we can write $$\begin{aligned}
&
\sum_{j\in C_{\Gamma, \chi}, k(j)=j} \chi(T_{j, 0})
=
\sum_{c\mid N, \gcd(c, N/c)\leq 2, q\mid \frac{N}{\gcd(c, N/c)}}
\chi^{({\varepsilon})}(T_{1/c, 0});
\\
&
\sum_{1\leq j \leq\kappa, k(j)=j} \sum_{v\in \{0, 1\}}
\chi(T_{j, v}) \log{\mathbf{c}}_{j, v}
=
\sum_{c\mid N, \gcd(c, N/c)\leq 2}
\sum_{v\in\{0, 1\}}
\chi^{({\varepsilon})}(T_{1/c, v}) \log {\mathbf{c}}_{1/c, v}; \\
&
\sum_{\substack{1\leq j\leq\kappa, \\ j\notin C_{\Gamma, \chi}}} \log\left|1-\chi(T_j)\right|
=
\sum_{\substack{\frac{a}{c}\in C_{\Gamma}, \\ q\nmid \frac{N}{\gcd(c, N/c)}}}
\log \left|1-\chi(T_{a/c})\right|.\end{aligned}$$ We get formulas for each of these sums in the following lemmas.
\[lem:sum\_chiT1/c0\] $$\sum_{c\mid N, \gcd(c, N/c)\leq 2, q\mid \frac{N}{\gcd(c, N/c)}}
\chi^{({\varepsilon})}(T_{1/c, 0})
=
(-1)^{{\varepsilon}} I_\chi 2^{\omega(N)} \Omega_1(N, q),$$ where $\Omega_1(N, q)$ is given in .
By Lemma \[lem:chi\_T1/cv\] and the definition of $\alpha(N,c)$ in , we have $$\begin{gathered}
\sum_{c\mid N, \gcd(c, N/c)\leq 2, q\mid \frac{N}{\gcd(c, N/c)}}
\chi^{({\varepsilon})}(T_{1/c, 0})
=
(-1)^{{\varepsilon}}
\sum_{c\mid N, \gcd(c, N/c)\leq 2, q\mid \frac{N}{\gcd(c, N/c)}}
\chi_{\alpha(N, c)} (-1)
\\
=
(-1)^{{\varepsilon}}
\biggl\{
\sum_{c\mid N, \gcd(c, N/c)=1} \chi_{N/c}(-1)
+
\delta_{4\mid N, q\mid \frac{N}{2}}
\sum_{c\mid N, \gcd(c, N/c)=2}
\big\{
\delta_{2\nmid\frac{N}{2c}}\,
\chi_{\frac{N}{2c}}(-1)
+
\delta_{2\mid\frac{N}{2c}}\,
\chi_{\frac{2N}{c}}(-1)
\big\}\biggr\}.\end{gathered}$$ For $\gcd(c, N/c)=1$, we have $$\label{e:avr_chi_gcd1}
\sum_{c\mid N, \gcd(c, N/c)=1}
\chi_{N/c}(-1)
=
\prod_{p\mid N} (1+\chi_p(-1))
=
I_\chi 2^{\omega(N)}.$$ When $4\mid N$, $q\mid\frac{N}{2}$ and $\gcd(c,N/c)=2$, then either $\frac{N}{2c}$ is odd so $2^{e_2-1}\parallel c$ or $\frac{N}{2c}$ is even, $8\mid N$ and $2\parallel c$. For $\frac{N}{2c}$ odd, we have $$\sum_{c\mid N, \gcd(c, N/c)=2, 2\nmid \frac{N}{2c}}
\chi_{\frac{N}{2c}}(-1)
=
\prod_{p\mid N, p>2} (1+\chi_p(-1))
=
I_\chi 2^{\omega(N)-1}.$$ For $8\mid N$, $q\mid \frac{N}{2}$ and $\frac{N}{2c}$ even, we have $$\sum_{c\mid N, \gcd(c, N/c)=2, 2\mid \frac{N}{2c}}
\chi_{\frac{2N}{c}}(-1)
=
\chi_2(-1) \prod_{p\mid N, p>2} (1+\chi_p(-1))
=
I_\chi 2^{\omega(N)-1}.$$ Combining these, we obtain the formula stated in the lemma.
\[lem:sum\_T1/cv\_mbc\] $$\sum_{c\mid N, \gcd(c, N/c)\leq 2}
\sum_{v\in\{0, 1\}}
\chi^{({\varepsilon})}(T_{1/c, v}) \log {\mathbf{c}}_{1/c, v}
=
(-1)^{{\varepsilon}} I_\chi 2^{\omega(N)}
\big(
\Omega_1(N, q)\log N + {\varepsilon}_N \log 2
\big).$$ Here $\Omega_1(N, q)$ and ${\varepsilon}_N$ are given in and respectively.
Recalling and , when $4\mid N$, we get $$\begin{gathered}
\sum_{c\mid N, \gcd(c, N/c)\leq 2}
\sum_{v\in\{0, 1\}}
\chi^{({\varepsilon})}(T_{1/c, v}) \log {\mathbf{c}}_{1/c, v}
\\
=
(-1)^{{\varepsilon}}
\log N
\left\{
\sum_{c\mid N, \gcd(c, N/c)=1}
\chi_{\alpha(N, c)}(-1)
+
\frac{1}{2}
\sum_{c\mid N, \gcd(c, N/c)=2}
\begin{cases}
2\chi_{\alpha(N, c)}(-1)
& \text{if } q\mid \frac{N}{2}, \\
\sum_{v\in \{0, 1\}} (-1)^{v+w_c}
& \text{if } q\nmid \frac{N}{2}
\end{cases}
\right\}
\\
=
(-1)^{{\varepsilon}}
\log N
\sum_{c\mid N, \gcd(c, N/c)\leq 2, q\mid \frac{N}{\gcd(c, N/c)}}
\chi_{\alpha(N, c)}(-1)
=(-1)^{{\varepsilon}} I_\chi 2^{\omega(N)} \Omega_1(N, q) \log N,\end{gathered}$$ where the last equality holds by Lemma \[lem:sum\_chiT1/c0\] (and its proof).
For $4\nmid N$, by and , we have $$\begin{gathered}
\sum_{c\mid N, \gcd(c, N/c)\leq 2}
\sum_{v\in\{0, 1\}}
\chi^{({\varepsilon})}(T_{1/c, v}) \log {\mathbf{c}}_{1/c, v}
\\
=
(-1)^{{\varepsilon}}
\sum_{c\mid N, \gcd(c, N/c)=1}
\chi_{\alpha(N, c)}(-1)
\bigg( \log N + \frac{1}{\gcd(2, N)} \log 2\bigg)
\\
=(-1)^{{\varepsilon}}I_\chi 2^{\omega(N)} \Omega_1(N, q)\bigg( \log N + \frac{1}{\gcd(2, N)} \log 2\bigg),\end{gathered}$$ where we again used Lemma \[lem:sum\_chiT1/c0\]. The desired formula follows if we also note that $4\nmid N$ implies $\Omega_1(N, q)=1$.
\[lem:sum\_notinCGamachi\] $$\sum_{\substack{\frac{a}{c}\in C_{\Gamma}, \\ q\nmid \frac{N}{\gcd(c, N/c)}}}
\log \left|1-\chi(T_{a/c})\right|
=
\Psi_1(N, q)
\sum_{p\mid N} \frac{\Psi_4(p^{e_p}, p^{s_p})}{\Psi_1(p^{e_p}, p^{s_p})}
\log p.$$ Here $\Psi_1$ and $\Psi_4$ are given in and .
Recalling , and writing $M=\gcd(c,N/c)$, we have $$\begin{aligned}
\label{lem:sum_notinCGamachipf2}
\sum_{\substack{\frac{a}{c}\in C_{\Gamma}, \\ q\nmid \frac{N}{M}}} \log \left|1-\chi(T_{a/c})\right|
=
\sum_{c\mid N, q\nmid \frac{N}{M}}
\sum_{a\in ({\mathbb{Z}}/M{\mathbb{Z}})^\times}
\log \left|1-\chi\!\left(1-a\tfrac{N}{M}\right)\right|.\end{aligned}$$ Now let $c$ be fixed subject to the conditions $c\mid N$ and $q\nmid\frac NM$. Note that $N\mid (N/M)^2$, and therefore the map $a\mapsto 1-aN/M$ is a homomorphism from the additive group ${\mathbb{Z}}/M{\mathbb{Z}}$ to the multiplicative group $({\mathbb{Z}}/N{\mathbb{Z}})^\times$. In particular there is some $A\in{\mathbb{Z}}$ (uniquely determined modulo $M$) such that $\chi(1-aN/M)=e(aA/M)$ for all $a\in{\mathbb{Z}}$, and we have $M\nmid A$ since $q\nmid\frac NM$. It follows that, with $B=M/\gcd(A,M)>1$, $$\begin{aligned}
\sum_{a\in ({\mathbb{Z}}/M{\mathbb{Z}})^\times}
\log \left|1-\chi\!\left(1-a\tfrac{N}{M}\right)\right|
&=\frac{\varphi(M)}{\varphi(B)}\,\log\biggl(\,\prod_{y\in({\mathbb{Z}}/B{\mathbb{Z}})^\times}\left|1-e\left(\tfrac yB\right)\right|\biggr)
\hspace{100pt}&
\\
&=\frac{\varphi(M)}{\varphi(B)}\begin{cases}
\log p&\text{ if $B=p^r$ for some prime $p$ and $r\geq1$,}
\\
0&\text{ otherwise.}
\end{cases}\end{aligned}$$ (cf., e.g., [@Nag51 §46(7)]). Note that for $a\in{\mathbb{Z}}$ we have $\chi(1-aN/M)=e(aA/M)=1$ if and only if $B\mid a$; hence in fact $$\begin{aligned}
\label{lem:sum_notinCGamachipf1}
B=\frac{q}{\gcd(N/M,q)}.\end{aligned}$$
For any prime $p\mid N$, let $f_p=\operatorname{ord}_p(c).$ It follows from that $B=p^r$ holds for some prime $p$ and some $r\geq1$ if and only if $\min\{f_p,e_p-f_p\}>e_p-s_p$ while $\min\{f_{p'},e_{p'}-f_{p'}\}\leq e_{p'}-s_{p'}$ for every prime $p'\neq p$. Furthermore, when this holds, we have $$\frac{\varphi(M)}{\varphi(B)}
=
p^{e_p-s_p}
\prod_{p'\mid N, p'\neq p} \varphi\!\left({p'}^{\min\{e_{p'}-f_{p'}, f_{p'}\}}\right).$$ Hence the expression in equals $$\begin{aligned}
\sum_{p\mid N}
\bigg[
\prod_{p'\mid N, p'\neq p}
\bigg( \sum_{\substack{ 0\leq f_{p'}\leq e_{p'}, \\
\min\{f_{p'},e_{p'}-f_{p'}\}\leq e_{p'}-s_{p'} }}
\varphi\!\left({p'}^{\min\{e_{p'}-f_{p'}, f_{p'}\}}\right)
\bigg)
\bigg]
\,
p^{e_p-s_p}\log p
\sum_{\substack{0\leq f_p\leq e_p\\
\min\{f_p,e_p-f_p\}>e_p-s_p}}1.\end{aligned}$$ Here the sum over $f_p$ equals $\max\{2s_p-e_p-1,0\}$, and each sum over $f_{p'}$ can be evaluated using (with $x=0$). This leads to the statement of the lemma.
The proposition follows from $$\operatorname{C}(N,\chi;n)=\operatorname{C}(\Gamma, \chi^{(0)})+n\operatorname{C}(\Gamma, \chi^{(1)})$$ by evaluating the various sums appearing in using the above three lemmas and also the formula $|C_{\Gamma, \chi}|=\Psi_1(N, q)$ (cf. Lemma \[lem:open\_cusp\]).
Eisenstein series
-----------------
Huxley [@Hux84] gave explicit expressions (involving Dirichlet $L$-functions) for the scattering matrix and its determinant for the congruence subgroups $\Gamma^0(N)$, $\Gamma^1(N)$ and $\Gamma(N)$ of the modular group, for arbitrary level $N$. Note that $\Gamma^0(N)$ is conjugate to the group $\Gamma_0(N)$ which we consider here. The case of squarefree $N$ had previously been considered in [@Hej83 Ch. 11]. In our setting, the Eisenstein series for $\langle\Gamma^+, \chi\rangle$ associated to an open cusp $\frac{a}{c}\in C_{\Gamma, \chi}$ is given by (cf. ) $$\label{e:Eis_a/c}
E_{a/c}(z, s, \chi) = \sum_{U\in [T_{a/c}]{\backslash}\Gamma^+} \overline{\chi(U)} (\Im({\mathsf{N}}_{a/c} Uz))^s,
\text{ for } z\in {\mathbb{H}}\text{ and } \Re(s)>1.$$ Substituting $U=W_{a/c} {\left(\begin{smallmatrix}}\alpha & \beta\\ \gamma & \delta{\end{smallmatrix}\right)}$, using and , and writing $z=x+iy$ and $W_{a/c} = {\left(\begin{smallmatrix}}a & b\\ c& d{\end{smallmatrix}\right)}\in \operatorname{SL}_2({\mathbb{Z}})$, one sees that can be rewritten as $$\label{e:Eis_a/c_2}
E_{a/c}(z, s, \chi)
=
\frac{\gcd(c^2, N)^{s}}{2N^s}
\sum_{\langle\gamma, \delta\rangle}
\chi(a\alpha+b\gamma) \frac{y^{s}}{|\gamma z+\delta|^{2s}},$$ where now the sum runs over all $\langle\gamma, \delta\rangle\in {\mathbb{Z}}^2$ satisfying $\gcd(\gamma, \delta)=1$, $\gcd(\gamma, N)=c$ and $\delta\frac{\gamma}{c}\equiv_{\gcd(c, N/c)} -a$. Also in , $\alpha=\alpha(\gamma, \delta)\in {\mathbb{Z}}$ denotes any integer satisfying both $\alpha\delta\equiv_\gamma 1$ and $\alpha\equiv_{N/c} -d\frac{\gamma}{c}$. Such $\alpha$ exists for every relevant pair $\langle\gamma, \delta\rangle$. We wish to make the factor $\chi(a\alpha+b\gamma)$ somewhat more explicit. For a prime $p$, from now on we are using the following notations: $$e_p = \operatorname{ord}_p(N), \quad s_p = \operatorname{ord}_p(q)\quad \text{ and }\quad f_p = \operatorname{ord}_p(c).$$ Since $\frac{a}{c}\in C_{\Gamma, \chi}$, by Lemma \[lem:open\_cusp\], we have $s_p\leq \max\{e_p-f_p, f_p\}$. Also $\gcd(\gamma, N)=c$ implies that $\operatorname{ord}_p(\gamma)\geq f_p$ with equality unless $f_p=e_p$. Now the conditions on $\alpha=\alpha(\gamma, \delta)$ imply $$a\alpha+b\gamma
\equiv
\begin{cases}
a\delta^{-1} & \mod{p^{f_p}}, \\
a(-d\frac{\gamma}{c}) + b\gamma \equiv -\frac{\gamma}{c} & \mod{p^{e_p-f_p}}.
\end{cases}$$ Hence $$\label{e:chip_ab}
\chi_p(a\alpha+b\gamma)
=
\begin{cases}
\chi_p(a) \overline{\chi_p(\delta)} & \text{if } e_p \leq 2f_p, \\
\chi_p(-\gamma/c) & \text{if } e_p> 2f_p.
\end{cases}$$ Note that both relations are valid in the special case $e_p=2f_p$. As in [@Hux84] we now introduce a family of sums similar to but simpler than . It will turn out that these sums can be expressed as linear combinations of the Eisenstein series $E_{a/c}(z, s, \chi)$ (cf. Lemma \[lem:E\_chi1chi2\] below), and a key step in computing the scattering matrix will be to invert these linear relations. Throughout this and the next section, we let $\chi_1$ and $\chi_2$ denote primitive Dirichlet characters satisfying $\chi_1(-1) = \chi_2(-1)$. For $j\in \{1, 2\}$, we write $q_j = \operatorname{cond}(\chi_j)$ and $\chi_{j, p} = (\chi_j)_p$ for any prime $p$. As in [@Hux84 p.143], for $\Re(s)>1$ and $z=x+iy$, we define $$B_{\chi_1}^{\chi_2}(z, s) = \sum_{(c, d)\in {\mathbb{Z}}^2-\{(0, 0)\}} \chi_1(c) \chi_2(d) \frac{y^s}{|cz+d|^{2s}}$$ and $$E_{\chi_1}^{\chi_2}(z, s) = \sum_{(c, d)\in {\mathbb{Z}}^2, \gcd(c, d)=1} \chi_1(c) \chi_2(d) \frac{y^{s}}{|cz+d|^{2s}}
=
L(2s, \chi_1\chi_2)^{-1} B_{\chi_1}^{\chi_2}(z, s).$$ As in [@Hux84], we have $$\label{e:B_chi1chi2_modular}
B_{\chi_1}^{\chi_2}\!\left({\left(\begin{smallmatrix}}A& B\\ C& D{\end{smallmatrix}\right)}z, s\right)
=
\chi_1(D) \chi_2(A) B_{\chi_1}^{\chi_2}(z, s),$$ for all ${\left(\begin{smallmatrix}}A& B\\ C& D{\end{smallmatrix}\right)}\in \Gamma_0(1)$ with $q_1\mid C$ and $q_2\mid B$. Similarly, $$\label{e:Bchi1chi2_V_fe}
B_{\chi_1}^{\chi_2}(Vz, s) = \chi_1(-1) B_{\chi_1}^{\chi_2}(z, s).$$ The same transformation formulas also hold for $E_{\chi_1}^{\chi_2}(z, s)$. We let $$\label{e:F}
F = \left\{(m, \chi_1, \chi_2)\;:\; m\in {\mathbb{Z}}_{\geq 1},\; mq_1 \mid N, \; q_2\mid m, \; \operatorname{cond}(\chi\chi_2\overline{\chi_1}) = 1\right\}.$$ Note that $\operatorname{cond}(\chi\chi_2\overline{\chi_1})=1$ if and only if $\chi\chi_2(x) = \chi_1(x)$ for all $x\in ({\mathbb{Z}}/N{\mathbb{Z}})^\times$; however this does not imply $\chi\chi_2=\chi_1$ since the product $\chi\chi_2$ is not necessarily primitive. It follows from that for every $(m, \chi_1, \chi_2)\in F$, the function $E_{\chi_1}^{\chi_2}(mz, s)$ is $\langle\Gamma^+, \chi\rangle$-invariant, that is $E_{\chi_1}^{\chi_2}(mTz, s) = \chi(T) E_{\chi_1}^{\chi_2}(mz, s)$ for all $T\in \Gamma^+$. In fact, each such function $E_{\chi_1}^{\chi_2}(mz, s)$ may be expressed as a linear combination of the Eisenstein series for $\langle\Gamma^+, \chi\rangle$ as follows.
\[lem:E\_chi1chi2\] For any $(m, \chi_1, \chi_2)\in F$, we have $$E_{\chi_1}^{\chi_2}(mz, s)
=
\frac{2N^s}{m^s}
\sum_{\frac{a}{c}\in C_{\Gamma, \chi}} \frac{\gcd(m, c)^{2s}}{\gcd(c^2, N)^s}
\,\,\chi_1\Bigl(-\frac{c}{\gcd(m, c)}\Bigr)\, \chi_2\Bigl(a\frac{m}{\gcd(m, c)}\Bigr)
\,E_{a/c}(z, s, \chi).$$
For $(m, \chi_1, \chi_2)\in F$, by [@Hux84 p.146(top)], $E_{\chi_1}^{\chi_2}(mz, s)$ can be expressed as $$\label{e:Echi1chi2_Hux}
\sum_{m=gh}\,\,
\sum_{\substack{e\pmod*{N/h}, \\ \gcd(e, g)=1}}
\sum_{\substack{f\pmod*{N}, \\ \gcd(f, he, N)=1}}
\chi_2(g) \chi_1(e) \chi_2(f) \Bigl(\frac{h}{g}\Bigr)^{\!s}
\sum_{\gamma\in he+N{\mathbb{Z}}}
\sum_{\substack{\delta\in f+N{\mathbb{Z}}, \\ \gcd(\gamma, \delta)=1}}
\frac{y^s}{|\gamma z+\delta|^{2s}}.$$ Given any $g, h, e, f$ as in this sum, let $a$ and $c$ be the uniquely determined integers such that $$c = \gcd(he, N), \quad
a\equiv_{\gcd(c, N/c)} -f\frac{he}{c}
\:\:\text{ and }\:\: \tfrac{a}{c}\in C_{\Gamma}.$$ One easily checks that this transformation gives a bijection between the set of tuples $$(g, h, e, f, \gamma, \delta)$$ appearing in the sum, and the set $$\left\{(a, c, \gamma, \delta)\;:\; \tfrac{a}{c}\in C_{\Gamma},
\gamma, \delta\in {\mathbb{Z}}, \gcd(\gamma, \delta)=1, \gcd(\gamma, N)=c,
\delta\equiv_{\gcd(c, N/c)} -a\left(\frac{\gamma}{c}\right)^{-1}
\right\}.$$ The inverse map is given by $$h=\gcd(m, c), \quad
g=\frac{m}{h}, \quad
e\equiv_{N/h} \frac{\gamma}{h}
\text{ and }
f\equiv_N\delta.$$ Now let $(a, c, \gamma, \delta)$ be any tuple in the above set, with corresponding $(g, h, e, f, \gamma, \delta)$. We then have, for each prime $p\mid N$, $$\label{e:chi2chi1chi2_p}
\chi_{2, p}(g) \chi_{1, p}(e) \chi_{2, p}(f)
=
\chi_{2, p}(g) \chi_{2, p}(a) \chi_{1, p}\left(-\frac{c}{h}\right)
\begin{cases}
\chi_p(a) \overline{\chi_p(\delta)} & \text{if } \operatorname{ord}_p(N/c) \leq \operatorname{ord}_p(c), \\
\chi_p(-\gamma/c) & \text{if } \operatorname{ord}_p(N/c) > \operatorname{ord}_p(c).
\end{cases}$$ To prove this, first note that if $\operatorname{ord}_p(q_2) > \operatorname{ord}_p(c)$ then $g = \frac{m}{\gcd(c, m)}\equiv_p 0$, and thus $\chi_{2, p}(g)=0$ so holds. Hence from now on we may assume $\operatorname{ord}_p(q_2) \leq \operatorname{ord}_p(c)$.
Let us first assume that $\operatorname{ord}_p(c) > \operatorname{ord}_p(m)$. Then $p\mid e$ but $p\nmid a$, $p\nmid \delta$ (since $\gcd(a, c)=1$ and $\gcd(\gamma, \delta)=1$). If $p\mid q_1$ then both sides of are $0$. On the other hand, if $p\nmid q_1$, viz. $\chi_{1, p}\equiv 1$ then $\operatorname{cond}(\chi_p\chi_{2, p})=1$ and so $\chi_{2, p}(f) = \overline{\chi_p(f)} = \overline{\chi_p(\delta)}$ and $\chi_{2, p}(a) = \overline{\chi_p(a)}$; and if $\operatorname{ord}_p(N/c) > \operatorname{ord}_p(c)$ then also $\delta \equiv_{\gcd(c, p^\infty)} -a\left(\frac{\gamma}{c}\right)^{-1}$; hence holds. It remains to consider the case $\operatorname{ord}_p(c) \leq \operatorname{ord}_p(m)$. Then since $mq_1\mid N$ we have $\operatorname{ord}_p(q_1) \leq \operatorname{ord}_p(N/c)$ and hence $\chi_{1, p}(e) = \chi_{1, p}(-\gamma/c) \chi_{1, p}(-c/\gcd(m, c))$. Also, using $$f\equiv \delta\equiv -a\left(\frac{\gamma}{c}\right)^{-1} \pmod*{\gcd(c, N/c)}$$ we get $$\begin{cases}
\chi_{1, p}(-\gamma/c) = \chi_{1, p}(a) \overline{\chi_{1, p}(\delta)} & \text{if } \operatorname{ord}_p(N/c) \leq \operatorname{ord}_p(c); \\
\chi_{2, p}(f) = \chi_{2, p}(a) \overline{\chi_{2, p}(-\gamma/c)} & \text{if } \operatorname{ord}_p(N/c) > \operatorname{ord}_p(c).
\end{cases}$$ The second identity holds since we are assuming $\operatorname{ord}_p(q_2) \leq \operatorname{ord}_p(c)$ from start. From this follows and so we have proved that holds in all cases. Next let us note that $$\label{e:chi2chi1=0}
\chi_2(g)\chi_1(e)=0 \text{ whenever } \tfrac{a}{c}\notin C_{\Gamma, \chi}.$$ To prove this, assume that $\frac{a}{c}\notin C_{\Gamma, \chi}$; then by Lemma \[lem:open\_cusp\], there is a prime $p\mid N$ such that $$\operatorname{ord}_p(q) > \operatorname{ord}_p\!\left(\tfrac{N}{\gcd(c, N/c)}\right)=\max\{\operatorname{ord}_p(N/c), \operatorname{ord}_p(c)\}.$$ As noted above, if $\operatorname{ord}_p(q_2) > \operatorname{ord}_p(c)$ then $\chi_{2, p}(g) = 0$ and holds. On the other hand, if $\operatorname{ord}_p(q_2) \leq \operatorname{ord}_p(c)$ then using $\operatorname{cond}(\chi\chi_2\overline{\chi_1})=1$ we get $\operatorname{ord}_p(q_1) = \operatorname{ord}_p(q)$ and since $mq_1\mid N$ we have $$\operatorname{ord}_p(m) < \operatorname{ord}_p(\gcd(c, N/c))\leq \operatorname{ord}_p(c),$$ hence $p\mid e$ so $\chi_{1, p}(e)=0$ and holds. This lemma is now a direct consequence of , the bijection between $\{(g, h, e, f, \gamma, \delta)\}$ and $\{(a, c, \gamma, \delta)\}$ and , , and .
Next, recalling the definition of $F$ in , for any $m\mid N$ we now set $$\label{e:Fm}
F_m = \{ (\chi_1, \chi_2) \;:\; (m, \chi_1, \chi_2)\in F\}.$$ The following lemma gives an explicit enumeration of the set $F_m$. For any $c\in {\mathbb{Z}}_{\geq 1}$, we write $$X_c = \left\{\psi \text{ primitive Dirichlet character }:\; \operatorname{cond}(\psi)\mid c\right\}.$$ This set is in one-to-one correspondence with the dual of the group $({\mathbb{Z}}/c{\mathbb{Z}})^\times$. In particular $|X_c| = \varphi(c)$. Recall that $q=\operatorname{cond}(\chi)$. Let us write $$G_{N, q} = \left\{m\mid N\;:\; q\mid\tfrac{N}{\gcd(m, N/m)}\right\}.$$
\[lem:map\_Fm\_X\] Given $m\mid N$, the set $F_m$ is nonempty if and only if $m\in G_{N, q}$. When this holds, the map $$\label{e:map_Fm_X}
(\chi_1, \chi_2) \mapsto \prod_{p\mid N}
\begin{cases}
\chi_{1, p} & \text{if } \operatorname{ord}_p(q) \leq \operatorname{ord}_p(m), \\
\chi_{2, p} & \text{if } \operatorname{ord}_p(q) > \operatorname{ord}_p(m)
\end{cases}$$ is a bijection from $F_m$ onto $X_{\gcd(m, N/m)}$. In particular $|F_m| = \varphi(\gcd(m, N/m))$.
By the definition of $F$, two primitive Dirichlet characters $\chi_1$ and $\chi_2$ satisfy $(\chi_1, \chi_2)\in F_m$ if and only if, for every prime $p\mid N$, $$\label{e:Fm_p}
s_2 \leq \operatorname{ord}_p(m), \quad s_1 \leq \operatorname{ord}_p(N/m) \quad \text{ and } \quad
\operatorname{cond}(\chi_p \overline{\chi_{1, p}} \chi_{2, p})=1,$$ Here $s_j = \operatorname{ord}_p(q_j)$ for $j\in \{1, 2\}$. Let $s = \operatorname{ord}_p(q)$. Note that $\operatorname{cond}(\chi_p \overline{\chi_{1, p}} \chi_{2, p})=1$ implies that $s\leq \max\{s_1, s_2\}$. Hence by , $$s \leq \max\{\operatorname{ord}_p(m), \operatorname{ord}_p(N/m)\} = \operatorname{ord}_p\!\left(\tfrac{N}{\gcd(m, N/m)}\right).$$ Since this must hold for every $p\mid N$, it follows that $F_m$ can be nonempty only if $m\in G_{N,q}$. From now on we assume $m\in G_{N, q}$. We will prove that the map in is a bijection from $F_m$ onto $X_{\gcd(m, N/m)}$. In particular this will imply that $F_m$ is non-empty, with $|F_m| = \varphi(\gcd(m, N/m))$.
First consider any prime $p\mid N$ satisfying $s\leq\operatorname{ord}_p(m)$ (with $s=\operatorname{ord}_p(q)$). For such $p$, the first and third relations in imply $s_1\leq\operatorname{ord}_p(m)$, which together with the second relation in implies $s_1 \leq \operatorname{ord}_p(\gcd(m, N/m))$. Conversely if $\chi_{1,p}$ is *any* primitive character with conductor $p^{s_1}$ subject to $s_1\leq\operatorname{ord}_p(\gcd(m,N/m))$, and if $\chi_{2,p}$ is the unique primitive character satisfying $\operatorname{cond}(\chi_p\overline\chi_{1,p}\chi_{2,p})=1$, then all the conditions in are fulfilled.
Next consider any $p\mid N$ for which $s>\operatorname{ord}_p(m)$. Recall that we are assuming $m\in G_{N,q}$; hence $s\leq\max\{\operatorname{ord}_p(m),\operatorname{ord}_p(N/m)\}$ and so $s\leq\operatorname{ord}_p(N/m)$. Then the two last relations in together imply $s_2\leq\operatorname{ord}_p(N/{m})$, and in combination with the first relation this gives $s_2\leq\operatorname{ord}_p (\gcd({m},N/{m}))$. Conversely if $\chi_{2,p}$ is *any* primitive character with conductor $p^{s_2}$ subject to $s_2\leq\operatorname{ord}_p(\gcd(m,N/m))$, and if $\chi_{1,p}$ is the unique primitive character satisfying $\operatorname{cond}(\chi_p\overline\chi_{1,p}\chi_{2,p})=1$, then all the conditions in are fulfilled.
The above observations imply that the map in is indeed a bijection as stated.
We now turn to the full group $\Gamma = \Gamma_0^\pm (N)$. Let $\frac{a}{c}\in C_{\Gamma, \chi}$ be given. By Lemma \[lem:Ua/c\] and [@BS07 (2.8)], we have $$\label{e:Eis_a/c_V}
E_{a/c}(Vz, s, \chi)
=
\chi_{\alpha(N, c)}(-1) E_{k(a/c)}(z, s, \chi),$$ where $\alpha(N, c)$ is as in and $k(a/c)$ as in . Hence gives $$\label{e:Eis_a/c_Gamma}
E_{a/c}^{\Gamma}(z, s, \chi^{({\varepsilon})})
=
\begin{cases}
E_{a/c}(z, s, \chi) + (-1)^{{\varepsilon}} E_{-a/c}(z, s, \chi) & \text{if } \gcd(c, N/c)>2, \\
\big(\chi_{\alpha(N, c)}(-1)(-1)^{{\varepsilon}} + 1\big) E_{1/c}(z, s, \chi)
& \text{if } \gcd(c, N/c)\leq 2.
\end{cases}$$ It is also natural to introduce, for any primitive Dirichlet characters $\chi_1$ and $\chi_2$ satisfying $\chi_1(-1) = \chi_2(-1)$ as before, $${{^{\varepsilon}\! B}}_{\chi_1}^{\chi_2}(z, s)
=
B_{\chi_1}^{\chi_2}(z, s) + \chi^{({\varepsilon})}(V^{-1}) B_{\chi_1}^{\chi_2}(Vz, s)
=
\begin{cases}
2B_{\chi_1}^{\chi_2}(z, s) & \text{if } \chi_1(-1) = (-1)^{{\varepsilon}}, \\
0 & \text{otherwise}
\end{cases}$$ (cf. ). This function is $\langle\Gamma, \chi\rangle$-invariant, since one easily checks that it is $\langle\Gamma^+,\chi\rangle$-invariant and also transforms correctly under $V$. We also set $$\label{e:eE_chi1chi2}
{{^{\varepsilon}\! E}}_{\chi_1}^{\chi_2}(z, s)
=
L(2s, \chi_1\chi_2)^{-1}\, {{^{\varepsilon}\! B}}_{\chi_1}^{\chi_2}(z, s)
=
\begin{cases}
2E_{\chi_1}^{\chi_2}(z, s) & \text{if } \chi_1(-1) = (-1)^{{\varepsilon}}, \\
0 & \text{otherwise.}
\end{cases}$$ Recall from that we have fixed a subset $R_{\Gamma, \chi}\subset C_{\Gamma, \chi}$, which in our present notation (cf. also and Lemma \[lem:Ua/c\]) has the property that for every $\frac{a}{c}\in C_{\Gamma, \chi}$ with $\gcd(c, N/c)\leq 2$, $\frac{a}{c}$ lies in $R_{\Gamma, \chi}$ if and only if $\chi_{\alpha(N, c)}(-1) = (-1)^{{\varepsilon}}$. Set $$\begin{gathered}
F^{{\varepsilon}}
= \{(m, \chi_1, \chi_2)\in F\;:\; \chi_1(-1)=(-1)^{{\varepsilon}}\}
\\
= \{(m, \chi_1, \chi_2)\;:\; m\in {\mathbb{Z}}_{\geq 1}, \; q_2\mid m,\; mq_1\mid N, \; \operatorname{cond}(\chi\chi_2\overline{\chi_1})=1,\; \chi_1(-1) = (-1)^{{\varepsilon}}\}.\end{gathered}$$ We now have the following analogue of Lemma \[lem:E\_chi1chi2\].
\[lem:eE\_chi1chi2\] For any $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$, $${{^{\varepsilon}\! E}}_{\chi_1}^{\chi_2}(mz, s)
=
\frac{2N^s}{m^s}
\sum_{\frac{a}{c}\in R_{\Gamma, \chi}} \frac{\gcd(m, c)^{2s}}{\gcd(c^2, N)^{s}}
\,\chi_1\Bigl(-\frac{c}{\gcd(m, c)}\Bigr)
\,\chi_2\Bigl(\frac{am}{\gcd(m, c)}\Bigr)
I_c
\,E_{a/c}^\Gamma(z, s, \chi^{({\varepsilon})}),$$ where $I_c = 1+\delta_{\gcd(c, N/c) >2}$.
Fix $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$. Then ${{^{\varepsilon}\! E}}_{\chi_1}^{\chi_2}(mz, s) = 2E_{\chi_1}^{\chi_2}(mz, s)\not\equiv0$ and this is expressed in Lemma \[lem:E\_chi1chi2\] as a linear combination of $E_{a/c}(z, s, \chi)$ over all $\frac{a}{c}\in C_{\Gamma, \chi}$. For $\frac{a}{c}\in C_{\Gamma, \chi}$ with $\gcd(c, N/c)>2$, we see immediately that the combined contribution from $\frac{a}{c}\in C_{\Gamma, \chi}$ and $\frac{-a}{c}\in C_{\Gamma, \chi}$ to the sum in Lemma \[lem:E\_chi1chi2\] (doubled) is: $$\frac{4N^s}{m^s}
\frac{\gcd(m, c)^{2s}}{\gcd(c^2, N)^s}
\chi_1\!\left(-\frac{c}{\gcd(m, c)}\right) \chi_2\!\left(a\frac{m}{\gcd(m, c)}\right)
E_{a/c}^\Gamma(z, s, \chi^{({\varepsilon})}),$$ since $\chi_2(-1) = \chi_1(-1) = (-1)^{{\varepsilon}}$. Next take $\frac{a}{c}\in C_{\Gamma, \chi}$ with $\gcd(c, N/c)\leq 2$ (thus $a=1$). If $\chi_{\alpha(N, c)}(-1) = -(-1)^{{\varepsilon}}$, then as we will prove below $$\label{e:chi1chi2=0}
\chi_1\!\left(-\frac{c}{\gcd(m, c)}\right) \chi_2\!\left(\frac{m}{\gcd(m, c)}\right)=0,$$ so that the contribution from $E_{1/c}(z, s, \chi)$ in Lemma \[lem:E\_chi1chi2\] vanishes. If $\chi_{\alpha(N, c)}(-1) = (-1)^{{\varepsilon}}$, then the contribution to the doubled sum is $$\frac{4N^s}{m^s} \frac{\gcd(c, m)^{2s}}{\gcd(c^2, N)^{s}}
\chi_1\!\left(-\frac{c}{\gcd(m, c)}\right) \chi_2\!\left(\frac{m}{\gcd(m, c)}\right)
\frac{1}{2} E_{1/c}^\Gamma(z, s, \chi^{({\varepsilon})}),$$ since $E_{1/c}^{\Gamma}(z, s, \chi^{({\varepsilon})}) = 2E_{1/c}(z, s, \chi)$. Hence, adding up the contributions, we obtain . It remains to prove the claim for every $\frac{1}{c}\in C_{\Gamma, \chi}$ satisfying $\gcd(c,N/c)\leq 2$ and $\chi_{\alpha(N, c)}(-1) = -(-1)^{{\varepsilon}}$. To do so, let us fix $s>1$. Using and $\chi_1(-1)=(-1)^{{\varepsilon}}$, we see that $E_{\chi_1}^{\chi_2}(mVz, s) \equiv (-1)^{{\varepsilon}} E_{\chi_1}^{\chi_2}(mz, s)$ and applying Lemma \[lem:E\_chi1chi2\] and , this leads to a linear relation between the functions $z\mapsto E_{a/c}(z, s, \chi)$ for $\frac{a}{c}$ running through $C_{\Gamma, \chi}$. However these functions are linearly independent, since $E_{a/c}(z, s,\chi)$ grows like $(\Im({\mathsf{N}}_{a/c}(z)))^s$ when $z$ approaches the cusp $\frac{a}{c}$ but is bounded by $O((\Im({\mathsf{N}}_{a'/c'}(z)))^{1-s})$ as $z$ approaches any other cusp $\frac{a'}{c'}\in C_{\Gamma}$; cf. [@Hej83 p.280 (Prop. 3.7)]. It follows that all coefficients in the linear relation which we have obtained mush vanish. In particular for $\frac{1}{c}\in C_{\Gamma, \chi}$ with $\gcd(c, N/c)\leq 2$ this implies $$(\chi_{\alpha(N, c)}(-1)-(-1)^{{\varepsilon}}) \chi_1\!\left(-\frac{c}{\gcd(m,
c)}\right) \chi_2\!\left(\frac{m}{\gcd(m, c)}\right)
=0,$$ and follows.
The claim for every $\frac{1}{c}\in C_{\Gamma, \chi}$ satisfying $\gcd(c, N/c)\leq 2$ and $\chi_{\alpha(N, c)}(-1) = -(-1)^{{\varepsilon}}$ may alternatively be proved directly from the definition of $\alpha(N, c)$, using no facts about Eisenstein series. However we have not found any way of doing this without going through a somewhat lengthy case-by-case analysis.
Mimicking now [@Hux84 p.147], we let ${\mathbf{B}}(s)$ be the vector of the functions $${\mathbf{B}}(s) = ({{^{\varepsilon}\! B}}_{\chi_1}^{\chi_2}(mz, s))_{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}}.$$ So far our discussion has been for $\Re(s)>1$. However the functions $B_{\chi_1}^{\chi_2}(z, s)$ can be meromorphically continued to all $s\in {\mathbb{C}}$; hence so can $E_{\chi_1}^{\chi_2}(z, s)$, ${{^{\varepsilon}\! B}}_{\chi_1}^{\chi_2}(z, s)$ and ${{^{\varepsilon}\! E}}_{\chi_1}^{\chi_2}(z, s)$; and we have the functional equation (cf. [@Hux84 p.145]) $${\mathbf{B}}(s) = {\mathbf{D}}(s) P\, {\mathbf{B}}(1-s),$$ where $P$ is a permutation which in each row $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$ has an entry one in column $(q_1m/q_2, \overline{\chi_2}, \overline{\chi_1}))\in F^{{\varepsilon}}$ and zeros elsewhere, and ${\mathbf{D}}(s)$ is a diagonal matrix with entries $$\frac{q_1^{1-s}}{q_2^{s}} \pi^{2s-1} \frac{\tau(\chi_2)}{\tau(\overline{\chi_1})}
\frac{\Gamma(1-s)}{\Gamma(s)}.$$ Using the fact that the map $(m, \chi_1, \chi_2)\mapsto (q_1m/q_2, \overline{\chi_2}, \overline{\chi_1})$, which is the permutation given by $P$, is an involution of the set $F^{{\varepsilon}}$, we obtain $$\label{e:detP}
\det(P) = (-1)^{\frac{|F^{{\varepsilon}}|-|F_0^{{\varepsilon}}|}{2}},$$ where $$\label{e:Fve0}
F_0^{{\varepsilon}} = \{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}\;:\; \chi_1=\overline{\chi_2}\}$$ is the set of fixpoints of the involution. Making use of the same permutation we also see that $$\label{e:detD}
\det({\mathbf{D}}(s))
=
\left(\frac{\pi^{2s-1} \Gamma(1-s)}{\Gamma(s)}\right)^{|F^{{\varepsilon}}|}
\prod_{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}} q_1^{1-2s}.$$ Let ${\mathbf{E}}^{\Gamma}(s)$ be the vector of the functions $E_{a/c}^{\Gamma}(z, s, \chi^{({\varepsilon})})$ for all $\frac{a}{c}\in R_{\Gamma, \chi}$. By and Lemma \[lem:eE\_chi1chi2\], we have $$\label{e:vecB_vecEGamma}
{\mathbf{B}}(s) = L(s) M(s) {\mathbf{E}}^{\Gamma}(s),$$ where $L(s)$ is the $|F^{{\varepsilon}}|\times |F^{{\varepsilon}}|$ diagonal matrix with entries $L(2s, \chi_1\chi_2)$ and $M(s)$ is an $|F^{{\varepsilon}}|\times |R_{\Gamma, \chi}|$ matrix, whose entry in row $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$, column $\frac{a}{c}\in R_{\Gamma, \chi}$ is $$\label{e:Ms}
\frac{2N^s}{m^s} \frac{\gcd(m, c)^{2s}}{\gcd(c^2, N)^{s}}
\,\chi_1\!\left(-\frac{c}{\gcd(m, c)}\right) \chi_2\!\left(a\frac{m}{\gcd(m, c)}\right) I_c.$$ At first holds for $\Re(s)>1$; however ${\mathbf{E}}^{\Gamma}(s)$ can be meromorphically continued to all $s\in {\mathbb{C}}$ (cf. and [@Hej83 Ch. 6.11 and Ch. 8.3]) and remains valid for $s\in {\mathbb{C}}$. Let $Q(s)$ be the $|F^{{\varepsilon}}|\times |F^{{\varepsilon}}|$ matrix whose entry in row $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$ and column $(m', \chi_1', \chi_2')\in F^{{\varepsilon}}$ is $$\label{e:Qs}
\begin{cases}
\mu\!\left(\frac{m}{m'}\right) \chi_2\!\left(\frac{m}{m'}\right) \left(\frac{m}{m'}\right)^{-s}
& \text{if } m'\mid m, \chi_1' = \chi_1 \text{ and }\chi_2' = \chi_2, \\
0 & \text{otherwise.}
\end{cases}$$ Then the product matrix $Q(s) M(s)$ has the following entry in row $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$ and column $\frac{a}{c}\in R_{\Gamma, \chi}$: $$\begin{gathered}
\label{e:QsMs}
I_c \sum_{m'\mid m} \mu\!\left(\frac{m}{m'}\right) \chi_2\!\left(\frac{m}{m'}\right) \left(\frac{m}{m'}\right)^{-s}
\frac{2N^s}{{m'}^s} \frac{\gcd(c, m')^{2s}}{\gcd(c^2, N)^s}
\,\chi_1\!\left(-\frac{c}{\gcd(c, m')}\right) \chi_2\!\left(a\frac{m'}{\gcd(c, m')}\right)
\\
=
\frac{2N^s}{m^s \gcd(c^2, N)^{s}} I_c
\sum_{d\mid m} \mu(d) \gcd(m/d, c)^{2s}
\chi_1 \!\left(-\frac{c}{\gcd(m/d, c)}\right) \chi_2\!\left(a\frac{m/d}{\gcd(m/d, c)}\right)
\\
=
\frac{2N^s}{m^s} \frac{\gcd(c, m)^{2s}}{\gcd(c^2, N)^{s}}
\, I_c\, \chi_1\!\left(-\frac{c}{\gcd(c, m)}\right) \chi_2\!\left(a\frac{m}{\gcd(c, m)}\right)
\\
\times
\prod_{p\mid m}
\left(1-\begin{cases}
1 & \text{if } \operatorname{ord}_p(c) < \operatorname{ord}_p(m), \\
\chi_1(p) \chi_2(p) p^{-2s} & \text{if } \operatorname{ord}_p(c) \geq \operatorname{ord}_p(m)
\end{cases}
\right)
\\
=
\begin{cases}
0 & \text{if } m\nmid c, \\
\frac{2N^s m^s}{\gcd(c^2, N)^s}
I_c \chi_1\!\left(-\frac{c}{m}\right) \chi_2(a) \prod_{p\mid m} (1-\chi_1(p)\chi_2(p) p^{-2s})
& \text{if } m\mid c.
\end{cases}\end{gathered}$$ The following lemma is immediate from Lemma \[lem:open\_cusp\] and the definition of $R_{\Gamma, \chi}$ in .
\[lem:|RGammachi|\] For every $c\in G_{N, q}$, we have $$\left|\left\{a\;:\; \tfrac{a}{c}\in R_{\Gamma, \chi}\right\}\right|
=
\frac{1}{2}
\begin{cases}
\varphi\!\left(\gcd(c, N/c)\right)
& \text{if } \gcd(c, N/c)>2, \\
(-1)^{{\varepsilon}} \chi_{\alpha(N, c)}(-1)+1 & \text{if } \gcd(c, N/c)\leq 2.
\end{cases}$$ Here $\chi^{({\varepsilon})}(V) = (-1)^{{\varepsilon}}$ and $\alpha(N, c)$ is given in .
For any $m\mid N$, we set $$F_m^{{\varepsilon}} = \{(\chi_1, \chi_2)\;:\; (m, \chi_1, \chi_2)\in F^{{\varepsilon}}\}
=
\{(\chi_1, \chi_2)\in F_m\;:\; \chi_1(-1) = (-1)^{{\varepsilon}}\}.$$
\[lem:|Fm\_ve|\] Assume $m\in G_{N, q}$. Then for any $(\chi_1, \chi_2)\in F_m$, $(\chi_1, \chi_2)\in F_m^{{\varepsilon}}$ holds if and only if $$\label{e:varpi_chi_-1}
\varpi(-1) \prod_{p\mid N, \operatorname{ord}_p(q) > \operatorname{ord}_p(m)} \chi_p(-1) = (-1)^{{\varepsilon}},$$ where $\varpi\in X_{\gcd(m, N/m)}$ is the image of $(\chi_1, \chi_2)$ under the map . It follows that $$\label{e:|Fm_ve|}
|F_m^{{\varepsilon}}|
=
\frac{1}{2}
\begin{cases}
\varphi(\gcd(m, N/m)) & \text{if } \gcd(m, N/m) >2, \\
(-1)^{{\varepsilon}} \chi_{\alpha(N, m)}(-1)+1 & \text{if } \gcd(m, N/m)\leq 2,
\end{cases}$$
Using and $\chi_{1, p}(-1) = \chi_{2, p}(-1) \chi_p(-1)$, which holds for every prime $p$ since $q(\chi\chi_2\overline{\chi_1}) = 1$, we see that for the left hand side of equals to $\chi_1(-1)$. This proves the first statement of the lemma. It follows that $|F_m^{{\varepsilon}}| = \frac{1}{2} \varphi(\gcd(m, N/m))$ if $\gcd(m, N/m)>2$ since then exactly half of the characters of the group $({\mathbb{Z}}/\gcd(m, N/m){\mathbb{Z}})^\times$ are even and half are odd. It remains to prove the statement in the case $\gcd(m, N/m)\leq 2$. In this case $X_{\gcd(m, N/m)}$ consists of the trivial character only, and we see that our task is to prove that $$\prod_{p\mid N, \operatorname{ord}_p(q) > \operatorname{ord}_p(m)} \chi_p(-1) = \chi_{\alpha(N, m)}(-1).$$ But $\chi_p(-1)=1$ whenever $p\nmid q$. Hence we are done if we can show that for all primes $p$ $$\begin{aligned}
\label{lem:|Fm_ve|PF1}
\operatorname{ord}_p(\alpha(N, m)) = \begin{cases}
\operatorname{ord}_p(N) & \text{if } \operatorname{ord}_p(q) > \operatorname{ord}_p(m), \\
0 & \text{if } 1\leq \operatorname{ord}_p(q) \leq \operatorname{ord}_p(m).
\end{cases}\end{aligned}$$ When $p>2$ this claim is immediate from the definition of $\alpha(N, m)$ in , since it follows from $\gcd(m, N/m) \leq 2$ that $\operatorname{ord}_p(m)\in \{0, \operatorname{ord}_p(N)\}$ for every such prime $p$. Similarly the claim is immediate also for $p=2$ when $\gcd(m, N/m)=1$. Finally assume $p=2$ and $\gcd(m, N/m)=2$. Then $\operatorname{ord}_p(N) \geq 2$, $\operatorname{ord}_p(m)\in\{1, \operatorname{ord}_p(N)-1\}$ and implies that $\operatorname{ord}_p(\alpha)=0$ if $\operatorname{ord}_p(m) = \operatorname{ord}_p(N)-1$, otherwise $\operatorname{ord}_p(\alpha)=\operatorname{ord}_p(N)$. Also $\operatorname{ord}_p(q) < \operatorname{ord}_p(N)$ since $m\in G_{N,q}$; and recall that $\operatorname{ord}_p(q)$ cannot equal to $1$ since there is no primitive character modulo $2$. From these observations we see that holds also when $p=2$ and $\gcd(m, N/m)=2$.
It follows from Lemma \[lem:|RGammachi|\] and Lemma \[lem:|Fm\_ve|\] that $$\label{e:|Fve|_sumG}
|F^{{\varepsilon}}| = \sum_{m\in G_{N, q}} |F_m^{{\varepsilon}}| = |R_{\Gamma, \chi}|.$$ In particular $M(s)$ is a square matrix.
\[lem:detM\] We have $\det(Q(s)) = 1$ and $$\label{e:detM}
\det(M(s)) = c(N, \chi) \prod_{m\in G_{N, q}} \left(\frac{N}{\gcd(m, N/m)}\right)^{|F_v^{{\varepsilon}}|s}
\prod_{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}} \prod_{p\mid m}
(1-\chi_1\chi_2(p) p^{-2s}),$$ for some nonzero constant $c(N, \chi)\in {\mathbb{C}}$. Here $Q(s)$ and $M(s)$ are given as in and
Let us fix an ordering $F^{{\varepsilon}}$ such that $(m, \chi_1, \chi_2)$ comes before $(m', \chi_1', \chi_2')$ whenever $m< m'$. Giving both the rows and columns of $Q(s)$ this ordering, then $Q(s)$ is lower triangular by definition, and has all diagonal entries equal to one. Therefore $\det(Q(s))=1$. It also follows that $\det(M(s)) = \det(Q(s) M(s))$. Now $Q(s) M(s)$ is a square matrix whose rows (respectively columns) are naturally indexed by $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$ (respectively $\frac{a}{c}\in R_{\Gamma, \chi})$. Let us view $Q(s)M(s)$ as a block matrix, with the blocks indexed by $m$ (row) and $c$ (column). Note that Lemma \[lem:|RGammachi|\] and Lemma \[lem:|Fm\_ve|\] then show that each diagonal block is a square matrix; furthermore, implies that $Q(s)M(s)$ is upper block triangular. Hence, again using , we see that the determinant is given by the expression in the right hand side of , for some constant $C = c(N, \chi)\in {\mathbb{C}}$. It also follows that in order to prove $C\neq 0$, it suffices to check that for any $c\in G_{N, q}$ for which $$\label{e:Rc}
R_c = \left\{a\;:\; \tfrac{a}{c}\in R_{\Gamma, \chi}\right\}$$ is nonempty, the $|R_c|\times |R_c|$-matrix $(\chi_2(a))$, with rows indexed by $(\chi_1, \chi_2)\in F_c$ and columns indexed by $a\in R_c$ (recall $|F_c| = |R_c|$) has nonvanishing determinant. This is trivial if $\gcd(c, N/c) \leq 2$ since then $R_c=\{1\}$ (if $R_c\neq \emptyset$). Now assume that $\gcd(c, N/c) >2$, and set $t=|F_c|=|R_c|=\frac{1}{2}\varphi(\gcd(c, N/c))$. Using Lemma \[lem:map\_Fm\_X\] and Lemma \[lem:|Fm\_ve|\] we find that by multiplying the columns with appropriate constants of absolute value one, the determinant can be transformed into $\det(\psi_i(g_j))$ where $g_1,
\ldots, g_t$ are the elements and $\psi_1, \ldots, \psi_t$ are the characters of the Abelian group $({\mathbb{Z}}/\gcd(c, N/c){\mathbb{Z}})^{\times}/\{\pm1\}$. Such determinant is nonvanishing, since multiplying the matrix with its conjugate transpose gives $t$ times the $t\times t$ identity matrix (cf. [@Hux84 p.149]).
Now by and we have $${\mathbf{E}}^{\Gamma}(s) = M(s)^{-1} L(s)^{-1} D(s) P L(1-s) M(1-s) {\mathbf{E}}^{\Gamma}(1-s)$$ and hence (by the same type of uniqueness argument as in [@BS07 below (2.32)]) the “$\Gamma$-scattering matrix" introduced in [@BS07 (2.33)] is $$\Phi^{\Gamma}(s) = M(s)^{-1} L(s)^{-1} D(s) P L(1-s) M(1-s).$$ Hence by , and Lemma \[lem:detM\], $$\begin{gathered}
\label{e:detPhiGamma}
\det(\Phi^{\Gamma}(s))
=
(-1)^{\frac{|F^{{\varepsilon}}|-|F_0^{{\varepsilon}}|}{2}}
\left(\prod_{m\in G_{N, q}} \left(\frac{N}{\gcd(m, N/m)}\right)^{|F_m^{{\varepsilon}}|}\right)^{1-2s}
\left(\frac{\pi^{2s-1} \Gamma(1-s)}{\Gamma(s)}\right)^{|F^{{\varepsilon}}|}
\\
\times
\prod_{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}} \left(q_1^{1-2s} \frac{L(2-2s, \chi_1\chi_2\omega_m)}{L(2s, \chi_1\chi_2\omega_m)}\right),\end{gathered}$$ where $\omega_m$ denotes the trivial Dirichlet character modulo $m$. We note that for each $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$, $\chi_1\chi_2\omega_m$ is a Dirichlet character modulo $mq_1$.
By an entirely similar discussion one computes the determinant of the scattering matrix $\Phi(s)$ for $\langle\Gamma^+, \chi\rangle$. We will not need this formula, but we state it here for possible future reference: $$\begin{gathered}
\label{e:detPhi}
\det(\Phi(s)) = (-1)^{\frac{|F|-|F_0|}{2}} \left(\prod_{m\in G_{N, q}}\left(\frac{N}{\gcd(m, N/m)}\right)^{\varphi(\gcd(m, N/m))}\right)^{1-2s}
\left(\frac{\pi^{2s-1} \Gamma(1-s)}{\Gamma(s)}\right)^{|F|}
\\
\times
\prod_{(m, \chi_1, \chi_2)\in F} \left(q_1^{1-2s} \frac{L(2-2s, \chi_1\chi_2\omega_m)}{L(2s, \chi_1\chi_2\omega_m)}\right).\end{gathered}$$
From we obtain: $$\begin{gathered}
\label{e:varphiGamma_ratio}
\frac{(\varphi^{\Gamma})'}{\varphi^\Gamma}(s)
=
-2\sum_{m\in G_{N, q}} |F_m^{{\varepsilon}}| \log\!\left(\frac{N}{\gcd(m, N/m)}\right)
+
|F^{{\varepsilon}}| \left(2\log \pi - \frac{\Gamma'}{\Gamma}(1-s) - \frac{\Gamma'}{\Gamma}(s)\right)
\\
-2\sum_{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}} \left(\log q_1 + \frac{L'}{L}(2-2s, \chi_1\chi_2\omega_m)
+
\frac{L'}{L}(2s, \chi_1\chi_2\omega_m)\right).\end{gathered}$$ We can now evaluate the integral appearing in :
\[lem:Eis\_integral\] $$\begin{gathered}
\label{e:Eis_integral}
\frac{1}{4\pi} \int_{{\mathbb{R}}} h(r)
\frac{(\varphi^{\Gamma})'}{\varphi^{\Gamma}}\!\left(\frac{1}{2}+ir\right)\; dr
\\
=
\left(|F^{{\varepsilon}}| \log \pi - \sum_{m\in G_{N, q} } |F_m^{{\varepsilon}}| \log\!\left(\frac{N}{\gcd(m, N/m)}\right)
-
\sum_{(m, \chi_1,\chi_2)\in F^{{\varepsilon}}} \log q_1\right)
g(0)
\\
-
\frac{|F^{{\varepsilon}}|}{2\pi} \int_{\mathbb{R}}h(r) \frac{\Gamma'}{\Gamma}\!\left(\frac{1}{2}+ir\right) \; dr
-
\frac{|F_0^{{\varepsilon}}|}{2} h(0)
+
2\sum_{r=1}^\infty \frac{\Lambda(r) \{\chi\}_{{\varepsilon}}(r)}{r} g(2\log r),\end{gathered}$$ where $$\label{e:chi_ve}
\{\chi\}_{{\varepsilon}}(r) = \sum_{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}} \chi_1\chi_2\omega_m(r).$$
This follows from using $$\begin{gathered}
\frac{1}{4\pi} \int_{\mathbb{R}}h(r) \left(\frac{L'}{L} (1-2ir, \chi_1\chi_2\omega_m)
+ \frac{L'}{L} (1+2ir, \chi_1\chi_2\omega_m) \right) \; dr
\\
=
-\sum_{r=1}^\infty \frac{\Lambda(r) \chi_1\chi_2\omega_m(r)}{r} g(2\log r)
+
\delta_{\chi_1=\overline{\chi_2}} \frac{1}{4} h(0)\end{gathered}$$ which follows by imitating [@Hux85 p.509] and .
We stress that our $\{\chi\}_{{\varepsilon}}(r)$ (cf. ) is not exactly a generalization of the number which was denoted by that symbol in [@BS07 (2.41)]. Rather, in the special case when $N$ is squarefree, “$\{\chi\}_{{\varepsilon}}(r)$" in [@BS07] equals $2^{1-\omega(N/\gcd(r,N))}\{\chi\}_{{\varepsilon}}(r)$ in our present notation.
In order to evaluate , we also need to compute $\operatorname{tr}\Phi^{\Gamma}(\frac{1}{2})$.
\[lem:TrPhiGamma\_1/2\] We have $$\label{e:TrPhiGamma_1/2}
\operatorname{tr}\Phi^{\Gamma}\!\left(\tfrac{1}{2}\right) = -|F_0^{{\varepsilon}}|.$$
We have $$\begin{gathered}
\operatorname{tr}\Phi^{\Gamma}\!\left(\frac{1}{2}\right)
=
\lim_{s\to \frac{1}{2}} \operatorname{tr}(M(s)^{-1} L(s)^{-1} D(s) P L(1-s) M(1-s))
\\
=
\lim_{s\to \frac{1}{2}} \operatorname{tr}(L(s)^{-1} D(s) P L(1-s) M(1-s) M(s)^{-1}),\end{gathered}$$ and using the definition of $L(s)$, $D(s)$ and $P$, we find that $L(s)^{-1} D(s) P L(1-s)$ is the $|F^{{\varepsilon}}|\times |F^{{\varepsilon}}|$ matrix which in row $(m, \chi_1, \chi_2)\in F^{{\varepsilon}}$ has all entries zero except in column $(q_1 m/q_2, \overline{\chi_2}, \overline{\chi_1})$, there the entry is $$\label{e:trPhiGamma_ratio_L}
\frac{q_1^{1-s}}{q_2} \pi^{2s-1}
\frac{\tau(\chi_2)}{\tau(\overline{\chi_1})}
\frac{\Gamma(1-s)}{\Gamma(s)}
\frac{L(2-2s, \overline{\chi_1\chi_2})}{L(2s, \chi_1\chi_2)}.$$ If $\chi_1\neq \overline{\chi_2}$ then $\chi_1\chi_2$ is nontrivial Dirichlet character and thus $L(s, \chi_1\chi_2)$ is holomorphic at $s=1$ with $L(1, \chi_1\chi_2)\neq 0$ (cf., e.g., [@Coh80 §X.11]), and similarly $L(1, \overline{\chi_1\chi_2})\neq 0$. However if $\chi_1=\overline{\chi_2}$, then $$L(2s, \chi_1\chi_2) = L(2s, \overline{\chi_1\chi_2})
= \zeta(2s) \prod_{p\mid q_1}(1-p^{-2s})
= (\prod_{p\mid q_1} (1-p^{-1}))(2s-1)^{-1}+O(1)$$ as $s\to \frac{1}{2}$. Hence when $s\to \frac{1}{2}$, then number in tends to $$\begin{cases}
-1 & \text{if } \chi_1=\overline{\chi_2}, \\
\sqrt{\frac{q_1}{q_2}} \frac{\tau(\chi_2)}{\tau(\overline{\chi_1})}
\frac{L(1, \overline{\chi_1\chi_2})}{L(1, \chi_1\chi_2)}
& \text{if } \chi_1\neq \overline{\chi_2}.
\end{cases}$$ Also note that $M(1-s) M(s)^{-1}$ tends to the $|F^{{\varepsilon}}|\times |F^{{\varepsilon}}|$ identity matrix as $s\to \frac{1}{2}$. Indeed, $M(s)$ is by definition a holomorphic function of $s$ in the entire complex plane, and Lemma \[lem:detM\] shows that $M(s)^{-1}$ is also holomorphic when $\Re(s)\neq 0$. Hence we obtain the stated formula.
Continuous spectrum $\operatorname{Eis}(\Gamma, \chi)$
------------------------------------------------------
Recall that we write $\operatorname{Eis}(N,\chi;n)$ for the contribution from the continuous spectrum in the trace formula in Theorem \[thm:STF\_N\]; thus $\operatorname{Eis}(N,\chi;n)=\operatorname{Eis}(\Gamma, \chi^{(0)})+n\operatorname{Eis}(\Gamma, \chi^{(1)})$ where $\operatorname{Eis}(\Gamma,\chi^{({\varepsilon})})$ is the corresponding contribution (cf. ) in the trace formula for $(\Gamma_0^\pm (N), \chi^{({\varepsilon})})$. Our aim in this section is to prove the following proposition, giving an explicit formula for $\operatorname{Eis}(N,\chi;n)$. Let $$\begin{gathered}
\label{e:Psi5}
\Psi_5(p^{e_p},p^{s_p})
=
\frac{\Psi_1(p^{e_p}, p^{s_p})-2}{p-1}
\\
+\max\left\{{\left\lceil}\tfrac{e_p}{2}{\right\rceil}, s_p\right\} p^{\min\left\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, e_p-s_p\right\}}
+
\max\left\{{\left\lceil}\tfrac{e_p+1}{2}{\right\rceil}, s_p\right\} p^{\min\left\{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}, e_p-s_p\right\}}.\end{gathered}$$ Also let $$B_p(x) =
\begin{cases}
xp^x - 2\frac{p^x-1}{p-1} & \text{if } x\in {\mathbb{Z}}_{\geq 0}, \\
0 & \text{if } x< 0.
\end{cases}$$ and $$\label{e:Psi6}
\Psi_6(p^{e_p}, p^{s_p})
=
\begin{cases}
B_p(e_p-s_p) + s_p p^{e_p-s_p} & \text{if } e_p < 2s_p, \\
B_p\!\left({\left\lfloor}\tfrac{e_p}{2}{\right\rfloor}\right) + B_p\!\left({\left\lfloor}\tfrac{e_p-1}{2}{\right\rfloor}\right)
- B_p(s_p-1) + s_pp^{s_p-1}
& \text{if } e_p \geq 2s_p.
\end{cases}$$ Finally, let $$\label{e:Psi7}
\Psi_7(p^{e_p}, p^{s_p})
=
\begin{cases}
\begin{cases}
-e_p+1 & \text{if } s_p=0, \\
-e_p+2s_p & \text{if } s_p\geq 1, \\
\end{cases}
& \text{if } p>2, \\
\begin{cases}
-e_p+3 & \text{if } s_p=0\text{ and } e_p\geq 3, \\
-e_p+2s_p+1 & \text{if } e_p > s_p\geq 2, \\
e_p & \text{if } e_p=s_p\geq 2, \\
0 & \text{otherwise,}
\end{cases}
& \text{if } p=2.
\end{cases}$$
\[prop:Eis\_N\] $$\begin{gathered}
\label{e:Eis_N_1}
\operatorname{Eis}(\Gamma, \chi; 1)
=
\bigg\{
\Psi_1(N, q) \log\pi
-
\Psi_1(N, q) \sum_{p\mid N} \frac{\Psi_5(p^{e_p}, p^{s_p})+\Psi_6(p^{e_p}, p^{s_p})}{\Psi_1(p^{e_p}, p^{s_p})} \log p
\bigg\} g(0)
\\
-
\Psi_1(N, q) \frac{1}{2\pi} \int_{\mathbb{R}}h(r) \frac{\Gamma'}{\Gamma}\!\left(\frac{1}{2}+ir\right) \; dr
-
\frac{1}{4} I_\chi \Psi_2(N, q) h(0)
\\
+
2\sum_{m=1}^\infty \frac{\Lambda(m)}{m}
\prod_{p\mid N} \Big[(\Re\chi_p(m)) \Phi_+(p^{e_p}; m) \Big]\,
g(2\log m)\end{gathered}$$ and $$\begin{gathered}
\label{e:Eis_N_-1}
\operatorname{Eis}(\Gamma, \chi; -1)
\\
=
-I_\chi
\bigg\{
2^{\omega(N)} \big(\Omega_1(N, q) (\log N - \log (2\pi)) + \log 2\big)
+
\Omega_1(N, q) 2^{\omega(N)-1} \sum_{p\mid N} \Psi_7(p^{e_p}, p^{s_p}) \log p
\bigg\} g(0)
\\
-
I_\chi 2^{\omega(N)} \Omega_1(N, q)
\frac{1}{2\pi} \int_{\mathbb{R}}h(r) \frac{\Gamma'}{\Gamma}\!\left(\frac{1}{2}+ir\right) \; dr
\\
-
\frac{1}{4} I_\chi
\bigg\{
I_{q, 4}\, \chi(\sqrt{-1}) \,
\min\{e_2+1, 4\}
\prod_{p\mid N, p\equiv_41} \Psi_2(p^{e_p}, p^{s_p})
\prod_{p\mid N, p\equiv_4-1} 2
\bigg\} h(0)
\\
+
2\sum_{m=1}^\infty \frac{\Lambda(m)}{m}
\prod_{p\mid N}
\Big[(\{\Re/i\Im\}\chi_p(m))\Phi_-(p^{e_p}; m) \Big]\,
g(2\log m).\end{gathered}$$ Here $\Psi_1$, $\Psi_5$, $\Psi_6$, $\Psi_7$, $\Psi_2$ and $\Phi_\pm$ are given in , , , , and respectively. Also $\sqrt{-1}$ denotes a square root of $-1$ in ${\mathbb{Z}}/q{\mathbb{Z}}$ (as exists when $I_{q,4}=1$; and $\chi(\sqrt{-1})$ is independent of the choice of the square root when $I_\chi=1$).
The proposition will be proved by making the right hand sides in the formulas in Lemma \[lem:Eis\_integral\] and Lemma \[lem:TrPhiGamma\_1/2\] more explicit. We will carry this out in a sequence of lemmas.
\[lem:|Fm\_ve|log\] Recall that ${\varepsilon}\in \{0, 1\}$ is given by $\chi^{({\varepsilon})}(V) = (-1)^{{\varepsilon}}$. We have $$\begin{gathered}
\label{e:|Fm_ve|log}
\sum_{m\in G_{N,q}} |F_m^{{\varepsilon}}| \log\!\left(\frac{N}{\gcd(m, N/m)}\right)
=
\frac{1}{2}\Psi_1(N, q) \sum_{p\mid N}
\frac{\Psi_5(p^{e_p}, p^{s_p})}{\Psi_1(p^{e_p}, p^{s_p})}\log p
\\
+
\frac{1}{2} (-1)^{{\varepsilon}}
I_\chi 2^{\omega(N)}
\bigg\{
\Omega_1(N, q)\log N - (\Omega_1(N, q) -1)\log 2
\bigg\}.\end{gathered}$$ Here $\Psi_1$, $\Psi_5$ and $\Omega_1$ are given in , and respectively. Furthermore, $$\label{e:|Fve|}
|F^{{\varepsilon}}|
=
\frac{1}{2}\Psi_1(N, q)
+
\frac{1}{2} (-1)^{{\varepsilon}}
I_\chi 2^{\omega(N)} \Omega_1(N, q).$$
By , we have $$\begin{gathered}
\sum_{m\in G_{N,q}} |F_m^{{\varepsilon}}| \log\left(\frac{N}{\gcd(m, N/m)}\right)
=
\frac{1}{2}\sum_{m\mid N, q\mid \frac{N}{\gcd(m, N/m)}}
\varphi(\gcd(m, N/m)) \log\left(\frac{N}{\gcd(m, N/m)}\right)
\\
+
\frac{1}{2} (-1)^{{\varepsilon}}
\sum_{\substack{m\mid N, \gcd(m, N/m)\leq 2, \\
q\mid \frac{N}{\gcd(m, N/m)}}}
\chi_{\alpha(N, m)}(-1) \log\!\left(\frac{N}{\gcd(m, N/m)}\right).\end{gathered}$$ For the first term, recalling Lemma \[lem:varphi\_gcd\], let $$f(x) = \sum_{m\mid N, q\mid \frac{N}{\gcd(m, N/m)}} \varphi(\gcd(m, N/m)) (\gcd(m,N/m))^{x}.$$ Then we have $$\label{e:f0_Psi1}
f(0) = \sum_{m\mid N, q\mid \frac{N}{\gcd(m, N/m)}} \varphi(\gcd(m, N/m))
=
\Psi_1(N, q)$$ and $$f'(0) = \sum_{m\mid N, q\mid \frac{N}{\gcd(m, N/m)}}
\varphi(\gcd(m, N/m)) \log(\gcd(m,N/m)).$$ By , $$\begin{gathered}
f'(0)
=
\sum_{p\mid N}
\bigg(\prod_{p'\mid N, p'\neq p}\Psi_1({p'}^{e_{p'}}, {p'}^{s_{p'}})\bigg)
\log p
\bigg[\frac{-\Psi_1(p^{e_p}, p^{s_p})+2}{p-1}
\\
+
\min\{{\left\lfloor}\tfrac{e_p}{2}{\right\rfloor}, e_p-s_p\} p^{\min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, e_p-s_p\}}
+
\min\{{\left\lfloor}\tfrac{e_p-1}{2}{\right\rfloor}, e_p-s_p\} p^{\min\{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}, e_p-s_p\}}
\bigg].\end{gathered}$$ Note that by definition in , $\Psi_1(p^{e_p}, p^{s_p})\neq 0$. We have $$\begin{gathered}
\sum_{m\mid N, q\mid \frac{N}{\gcd(m, N/m)}}
\varphi(\gcd(m, N/m)) \log\left(\frac{N}{\gcd(m, N/m)}\right)
=
\log N f(0)-f'(0)
\\
=
\Psi_1(N, q)
\sum_{p\mid N}
\frac{\log p}{\Psi_1(p^{e_p}, p^{s_p})}\Psi_5(p^{e_p}, p^{s_p}).\end{gathered}$$ For the second term, we get $$\begin{gathered}
\sum_{\substack{m\mid N, \gcd(m, N/m) \leq 2, \\ q\mid \frac{N}{\gcd(m, N/m)}}}
\chi_{\alpha(N, m)} (-1) \log\!\left(\frac{N}{\gcd(m, N/m)}\right)
\\
=
\log N
\sum_{\substack{m\mid N, \gcd(m, N/m) \leq 2, \\ q\mid \frac{N}{\gcd(m, N/m)}}}
\chi_{\alpha(N, m)}(-1)
-
\log 2
\sum_{\substack{m\mid N, \gcd(m, N/m) = 2, \\ q\mid \frac{N}{\gcd(m, N/m)}}}
\chi_{\alpha(N, m)}(-1)
.\end{gathered}$$ As in Lemma \[lem:sum\_chiT1/c0\] and its proof, we have $$\label{e:sum_chialpha_2}
\sum_{\substack{m\mid N, \gcd(m, N/m) \leq 2, \\ q\mid \frac{N}{\gcd(m, N/m)}}}
\chi_{\alpha(N, m)}(-1)
=
I_\chi 2^{\omega(N)} \Omega_1(N, q).$$ and $$\sum_{\substack{m\mid N, \gcd(m, N/m) = 2, \\ q\mid \frac{N}{\gcd(m, N/m)}}}
\chi_{\alpha(N, m)}(-1)
=
I_\chi 2^{\omega(N)} (\Omega_1(N, q)-1).$$ Similarly, recalling and , $$|F^{{\varepsilon}}| = \sum_{m\in G_{N, q}} |F_m^{{\varepsilon}}|
=
\frac{1}{2}\sum_{m\mid N, q\mid \frac{N}{\gcd(m, N/m)}}
\varphi(\gcd(m, N/m))
+
\frac{1}{2} (-1)^{{\varepsilon}} \sum_{\substack{m\mid N, \gcd(m, N/m)\leq 2, \\ q\mid \frac{N}{\gcd(m, N/m)}}}
\chi_{\alpha(N, m)}(-1),$$ so by and , we get .
For $p\mid N$, let $$\label{e:Sp}
S_p = \left\{0\leq f \leq e_p\;:\; \min\{f, e_p-f\} \leq e_p-s_p\right\}.$$ Then note that $$S_p
=
\left\{0\leq f\leq e_p\;:\; f\leq e_p-s_p \text{ or } f\geq s_p\right\}.$$
\[lem:Fp\] Fix a prime $p\mid N$. For $f\in S_p$, if $f\geq s_p$, $$F_{p^{f}} =
\left\{(\chi_1, \chi_2)\;:\; \chi_1\in X_{p^{\min\{f, e_p-f\}}},
\chi_2\text{ primitive, determined by }
\operatorname{cond}(\chi_p\chi_2\overline{\chi_1})=1\right\}$$ and if $f<s_p$, $$F_{p^{f}}
=
\left\{(\chi_1, \chi_2) \;:\; \chi_2\in X_{p^{\min\{f, e_p-f\}}}, \chi_1\text{ primitive, determined by }
\operatorname{cond}(\chi_p\chi_2\overline{\chi_1})=1\right\}.$$ Here $F_{p^f}$ is given in .
Similar to the proof of Lemma \[lem:map\_Fm\_X\].
\[lem:|Fve0|\] $$|F_0^{{\varepsilon}}|
=
\frac{1}{2} I_\chi
\bigg\{
\Psi_2(N, q)
+
(-1)^{{\varepsilon}} I_{q, 4}\, \chi(\sqrt{-1}) \min\{e_2+1, 4\}
\prod_{p\mid N, p\equiv_4 1} \Psi_2(p^{e_p}, p^{s_p})
\prod_{p\mid N, p\equiv_4{-1}} 2
\bigg\}$$ Here $\Psi_2$ is given in , and $\sqrt{-1}$ denotes a square root of $-1$ in ${\mathbb{Z}}/q{\mathbb{Z}}$ (as exists when $I_{q,4}=1$; and $\chi(\sqrt{-1})$ is independent of the choice of the square root when $I_\chi=1$).
Recalling , we can write $$F_0^{{\varepsilon}} = \left\{(m, \chi_1) \;:\;
m\in {\mathbb{Z}}_{\geq 1}, \; m\mid N, \; q_1\mid \gcd(m, N/m), \; \operatorname{cond}(\chi\overline{\chi_1}^2)=1, \; \chi_1(-1)=(-1)^{{\varepsilon}}\right\}.$$ Note that $F_0^{{\varepsilon}}=\emptyset$ unless $\chi$ is square in the sense that there exists a primitive character $\chi_1$ such that $\chi \overline{\chi_1}^2(n) = 1$ for any $n$ with $\gcd(n, q)=1$. There exists such a primitive character if and only if for every prime $p\mid N$, $\chi_p(-1)=1$, i.e. $\chi$ is pure. From now on, we assume that $\chi$ is pure. Let $$F_0(N)
= \left\{(m, \chi_1)\;:\; m\in {\mathbb{Z}}_{\geq 1}, \; m\mid N,\; q_1 \mid \gcd(m, N/m), \; \operatorname{cond}(\chi\overline{\chi_1}^2)=1\right\}$$ and $$S_0^{\pm} = \sum_{(m, \chi_1)\in F_0(N)}\chi_1(\pm1).$$ Then $$|F_0^{{\varepsilon}}| = \frac{1}{2} \big(S_0^+ + (-1)^{{\varepsilon}} S_0^-\big).$$ Note that $S_0^\pm$ is multiplicative. One can write $$S_0^\pm
=
\prod_{p\mid N}
\bigg(\sum_{f\in S_p} \sum_{(p^f, \chi_1)\in F_0(p^{e_p})} \chi_1(\pm 1)\bigg).$$ So for $p\mid N$, let $$S_{0, p}^\pm =
\sum_{f\in S_p} \sum_{(p^f, \chi_1)\in F_0(p^{e_p})} \chi_1(\pm 1).
$$ For every prime $p\mid N$, fix a primitive Dirichlet character $\psi_p$ of minimal conductor subject to $\operatorname{cond}(\chi_p\overline{\psi_p}^2)=1$. We further assume that $$\operatorname{ord}_p(\operatorname{cond}(\psi_p))
=
\begin{cases}
0 & \text{if } p=2 \text{ and } s_p=0, \\
s_p+1 & \text{if } p=2 \text{ and } s_p\geq 3, \\
s_p & \text{if } p>2. \\
\end{cases}$$ Note that $s_2$ cannot be $1$ or $2$. When $p$ is odd, let $\xi_p$ be the quadratic character modulo $p$. When $p=2$ let $\xi_4$ be the quadratic character modulo $4$ and $\xi_8$ be the even quadratic character modulo $8$. Then $\xi_4\xi_8$ is another quadratic character, and $\xi_8$, $\xi_4\xi_8$ are the only primitive characters modulo $8$. When $\chi_1$ is the primitive character such that $\operatorname{cond}(\chi_p\overline{\chi_1}^2)=1$, then $$\chi_1\in\{\psi_p \xi_p^{u_p}\;:\; u_p\in \{0, 1\}\},$$ when $p$ is odd, and $$\chi_1\in\{\psi_2 \xi_4^{u_2} \xi_8^{v_2}\;:\; u_2, v_2\in \{0, 1\}\},$$ when $p=2$. Here $\psi_p$ is as fixed above, so $q(\chi\overline{\chi_1}^2)=1$. Note that $$\operatorname{ord}_p(q_1) = \begin{cases}
\max\{s_p, u_p\} & \text{if } p\text{ odd }, \\
s_p+1 & \text{if } p=2\text{ and } s_p\geq 3, \\
\max\{2u_2, 3v_2\} & \text{if } p=2 \text{ and } s_p=0.
\end{cases}$$ When $p$ is odd, we have $$\begin{gathered}
S_{0, p}^\pm
=
\sum_{u_p\in\{0, 1\}}
\sum_{\max\{s_p, u_p\} \leq \min\{f, e_p-f\}}
\psi_p\xi_p^{u_p}(\pm 1)
\\
=
\psi_p(\pm 1)
\begin{cases}
(e_p-1)(1+\xi_p(\pm 1)) + 2 & \text{if } s_p=0, \\
(e_p-2s_p+1)(1+\xi_p(\pm 1)) & \text{if } s_p\geq 1\text{ and }2s_p \leq e_p, \\
0 & \text{if } 2s_p > e_p.
\end{cases}\end{gathered}$$ Note that $1+\xi_p(-1)=0$ if $s_p>0$ and $p\equiv_4-1$. So $$S_{0, p}^+ = \Psi_2(p^{e_p}, p^{s_p})$$ and $$S_{0, p}^- = \psi_p(-1) \begin{cases}
2 & \text{if } s_p=0 \text{ and } p\equiv_{4}-1, \\
\Psi_2(p^{e_p}, p^{s_p}) & \text{if } p\equiv_41, \\
0 & \text{if } s_p>0 \text{ and } p\equiv_{4}-1.
\end{cases}$$ When $p=2$, we have $$S_{0, p}^\pm
=
\sum_{u_2, v_2\in\{0, 1\}}
\begin{cases}
\sum_{\max\{2u_2, 3v_2\} \leq \min\{f, e_p-f\}} \xi_4^{u_2} \xi_8^{v_2}(\pm 1)
& \text{if } s_p=0, \\
\sum_{s_p+1 \leq \min\{f, e_p-f\}} \psi_p \xi_4^{u_2} \xi_8^{v_2}(\pm 1)
& \text{if } s_p \geq 3.
\end{cases}$$ So we get $$S^\pm_{0, p}
=
\psi_p(\pm 1)
\begin{cases}
e_p+1 + \max\{e_p-5, 0\} (1+\xi_4(\pm 1)) + \max\{e_p-3, 0\}\xi_4(\pm 1)
& \text{if } s_p=0, \\
2\max\{e_p-2s_p-1, 0\} \psi_p(\pm 1) (1+\psi_4(\pm 1))
& \text{if } s_p\geq 3.
\end{cases}$$ Hence $$S_{0, p}^+ = \Psi_2(p^{e_p}, p^{s_p})$$ and $$S_{0, p}^- =
\begin{cases}
4 & \text{if } e_p\geq 3 \text{ and } s_p=0, \\
e_p+1 & \text{if } e_p\in\{1, 2\} \text{ and } s_p=0, \\
0 & \text{otherwise.}
\end{cases}$$
\[lem:sum\_logq1\] $$\begin{gathered}
\sum_{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}}
\log q_1
=
\frac{1}{2}
\Psi_1(N, q) \sum_{p\mid N} \frac{\Psi_6(p^{e_p}, p^{s_p})}{\Psi_1(p^{e_p}, p^{s_p})} \log p
\\
+
(-1)^{{\varepsilon}} I_\chi 2^{\omega(N)-2} \Omega_1(N, q)
\sum_{p\mid N} \Psi_7(p^{e_p}, p^{s_p}) \log p.\end{gathered}$$ Here $\Psi_1$, $\Omega_1$, $\Psi_6$ and $\Psi_7$ are given in , , and .
Let $$S^{\pm} = \sum_{(m, \chi_1, \chi_2)\in F} \chi_1(\pm 1) \log q_1.$$ Then $$\sum_{(m, \chi_1, \chi_2)\in F^{{\varepsilon}}} \log q_1
=
\frac{1}{2}\big(S^++ (-1)^{{\varepsilon}} S^-).$$ For $n\in \{\pm1\}$, let $$f_n (x)
=
\sum_{(m, \chi_1, \chi_2)\in F}
\chi_1(n) q_1^x.$$ Then $$f_n'(0) = \begin{cases}
S^+ & \text{if } n=1, \\
S^- & \text{if } n=-1.
\end{cases}$$ Recalling , $$f_n(x)
=
\prod_{p\mid N}
\bigg(
\sum_{f\in S_p}
\sum_{(\chi_1, \chi_2)\in F_{p^{f}}} \chi_1(n) q_1^x
\bigg).$$ For each prime $p\mid N$, $$f_{n, p}(x) = \sum_{f\in S_p}
\sum_{(\chi_1, \chi_2)\in F_{p^{f}}} \chi_1(n) q_1^x.$$ Then $$f_n'(0)
=
\sum_{p\mid N} f_{n, p}'(0) \prod_{p'\mid N, p'\neq p} f_{n, p'}(0).$$ For each prime $p\mid N$ and $f\in S_p$, by Lemma \[lem:Fp\], we have $$\sum_{(\chi_1, \chi_2)\in F_{p^{f}}} \chi_1(n) q_1^x
=
\begin{cases}
\sum_{\ell=0}^{\min\{f, e_p-f\}}
\big(\sum_{\substack{\psi\text{ primitive}\\ \operatorname{cond}(\psi) = p^\ell}}
\psi(n) \big)
p^{\ell x}
& \text{if } f\geq s_p, \\
\sum_{\ell=0}^{\min\{f, e_p-f\}}
\big(\sum_{\substack{\psi\text{ primitive}\\ \operatorname{cond}(\psi) = p^\ell}}
\chi_p \psi(n) \big)
p^{s_p x}
& \text{if } f< s_p.
\end{cases}$$ Note that $$\sum_{\substack{\psi\text{ primitive}\\ \operatorname{cond}(\psi) = p^\ell}}
\psi(n)
=
\begin{cases}
1 & \text{if } \ell=0, \\
p^{\ell-2}(p-1)^2 & \text{if } n=1 \text{ and } \ell \geq 2, \\
p-2 & \text{if } n=1 \text{ and } \ell=1, \\
-1 & \text{if } p>2, n=-1 \text{ and } \ell=1, \\
& \text{ or } p=2, n=-1 \text{ and } \ell=2, \\
0 & \text{otherwise.}
\end{cases}$$ So when $f\geq s_p$, $$\begin{gathered}
\sum_{(\chi_1, \chi_2)\in F_{p^{f}}} \chi_1(n) q_1^{x}
\\
=
\begin{cases}
1 & \text{if } \min\{f, e_p-f\}=0, \\
& \text{or } p=2, n=-1 \text{ and } \min\{f, e_p-f\}=1, \\
\frac{(p-1)^2 p^{\min\{f, e_p-f\}(x+1)+x-1} - (p^x-1)^2}{p^{x+1}-1}
& \text{if } n=1 \text{ and } \min\{f, e_p-f\}\geq 1, \\
1-p^{x(1+\delta_{p=2})} & \text{if } n=-1 \text{ and } \min\{f, e_p-f\} \geq 1+\delta_{p=2}.
\end{cases}\end{gathered}$$ When $f< s_p$, we have $$\begin{gathered}
\sum_{(\chi_1, \chi_2)\in F_{p^{f}}} \chi_1(n) q_1^{x}
\\
=
\chi_p(n)p^{s_px}
\begin{cases}
1 & \text{if } \min\{f, e_p-f\}=0, \\
& \text{or } p=2, n=-1 \text{ and } \min\{f, e_p-f\}=1, \\
(p-1) p^{\min\{f,e_p-f\}-1}
& \text{if } n=1\text{ and } \min\{f, e_p-f\} \geq 1, \\
0 & \text{if } n=-1 \text{ and } \min\{f, e_p-f\}\geq 1+\delta_{p=2}.
\end{cases}\end{gathered}$$ For $n=1$ and $s_p> {\left\lfloor}\frac{e_p}{2}{\right\rfloor}$, we have $$\begin{gathered}
f_{1, p}(x)
=
\sum_{f\in S_p}
\sum_{(\chi_1, \chi_2)\in F_{p^{f}}} \chi_1(n) q_1^x
\\
=
p^{s_p x + e_p-s_p}+1
+
\frac{1}{p^{x+1}-1}
\bigg(
(p-1)^2 p^{2x} \frac{p^{(e_p-s_p)(x+1)}-1}{p^{x+1}-1}
-
(p^x-1)^2(e_p-s_p+1)
\bigg).\end{gathered}$$ For $s_p \leq {\left\lfloor}\frac{e_p}{2}{\right\rfloor}$, we have $$\begin{gathered}
f_{1, p}(x)
=
p^{s_p(x+1)-1}+1
+
\frac{(p-1)^2 p^{x-1}}{(p^{x+1}-1)^2}
\big(
p^{({\left\lfloor}\frac{e_p}{2}{\right\rfloor}+1)(x+1)} + p^{{\left\lceil}\frac{e_p}{2}{\right\rceil}(x+1)}
- p^{s_p(x+1)} - p^{x+1}
\big)
\\
-(e-s)\frac{(p^x-1)^2}{p^{x+1}-1},\end{gathered}$$ So we have $$f_{1, p}(0) =
\begin{cases}
2p^{e_p-s_p} & \text{if } e_p< 2s_p, \\
p^{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}} + p^{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}} & \text{if } e_p \geq 2s_p
\end{cases}
=
\Psi_1(p^{e_p}, p^{s_p})$$ and $$f_{1, p}'(0)
=
\log p
\begin{cases}
B_p(e_p-s_p) + s_p p^{e_p-s_p} & \text{if } e_p < 2s_p, \\
B_p\!\left({\left\lfloor}\tfrac{e_p}{2}{\right\rfloor}\right) + B_p\!\left(\tfrac{e_p-1}{2}\right)
- B_p(s_p-1) + s_pp^{s_p-1}
& \text{if } e_p \geq 2s_p.
\end{cases}$$ Hence $$S^+
=
\Psi_1(N, q) \sum_{p\mid N} \frac{\Psi_6(p^{e_p}, p^{s_p})}{\Psi_1(p^{e_p}, p^{s_p})} \log p.$$ For $n=-1$ and $p>2$, we have $$f_{-1, p}(x)
=
\begin{cases}
2+(e_p-1)(1-p^x) & \text{if } s_p=0, \\
1+\chi_p(-1) p^{s_p x} + (e_p-s_p) (1-p^x) & \text{if } s_p > 0.
\end{cases}$$ So $$f_{-1, p}(0) =
1+\chi_p(-1)$$ and $$f_{-1, p}'(0) = \log p
\begin{cases}
-e_p+1 & \text{if } s_p=0, \\
s_p (1+\chi_p(-1))-e_p & \text{if } s_p>0.
\end{cases}$$ For $n=-1$ and $p=2$, we have $$f_{-1, p}(x)
=
\begin{cases}
1+\chi_p(-1) p^{s_p x} & \text{if } e_p=s_p, \\
2(1+\chi_p(-1) p^{s_px}) + (e_p-s_p-1)(1-p^{2x}) & \text{if } e_p>s_p \geq 2, \\
2 & \text{if } s_p=0 \text{ and } e_p=1, \\
3 & \text{if } s_p=0 \text{ and } e_p=2, \\
4+(e_p-3)(1-p^{2x}) & \text{if } s_p=0 \text{ and } e_p\geq 3.
\end{cases}$$ So we get $$f_{-1, p}(0)
=
(1+\chi_p(-1))
\begin{cases}
1 & \text{if } e_p=s_p \geq 2 \text{ or } e_p=1, \\
2 & \text{if } e_p>s_p \text{ and }e_p\geq 3, \\
\frac{3}{2} & \text{if } s_p=0 \text{ and } e_p=2, \\
\end{cases}
$$ and $$f_{-1, p}'(0)
=
2\log p
\begin{cases}
\frac{1}{2} s_p \chi_p(-1) & \text{if } e_p=s_p\geq 2, \\
s_p \chi_p(-1) -e_p+s_p+1 & \text{if } e_p>s_p\geq 2, \\
-e_p+3 & \text{if } s_p=0 \text{ and } e_p\geq 3, \\
0 & \text{otherwise.}
\end{cases}$$ Since $1+\chi_p(-1)=0$ unless $\chi_p$ is even, we can conclude that $S^-=0$ unless $\chi$ is pure. Hence we get $$S^-
=
\sum_{p\mid N} f_{-1, p}'(0) \prod_{p'\mid N, p'\neq p} f_{-1, p'}(0)
=
I_\chi 2^{\omega(N)-1} \Omega_1(N, q)
\sum_{p\mid N} \Psi_7(p^{e_p}, p^{s_p}) \log p.$$
\[lem:chi\_ve\] Let $r\in{\mathbb{Z}}$. If $\gcd(r,q)>1$ then $\{\chi\}_{{\varepsilon}}(r)=0$, while if $\gcd(r,q)=1$ then $$\{\chi\}_{{\varepsilon}}(r)
=
\frac{1}{2}
\bigg\{
\prod_{p\mid N,p\nmid r} (\Re\chi_p(r)) \Phi_+(p^{e_p}, r)
+
(-1)^{{\varepsilon}} \prod_{p\mid N,p\nmid r} (\{\Re/i\Im\}\chi_p(r)) \Phi_-(p^{e_p}, r)
\bigg\},$$ where $\{\Re/i\Im\}\chi_p(r) = \frac{1}{2}(\overline{\chi_p(r)}+\chi_p(-r))$, and $\Phi_+$ and $\Phi_-$ are given in .
Let $$f_\chi^{\pm}(r)
=
\sum_{(m, \chi_1, \chi_2)\in F} \chi_1(\pm 1) \chi_1\chi_2\omega_m(r).$$ Then, recalling , $$\{\chi\}_{{\varepsilon}}(r) = \frac{1}{2}\big(f_\chi^+(r) + (-1)^{{\varepsilon}} f_{\chi}^-(r)\big).$$ Note that $f_\chi^{\pm}(r)$ is multiplicative. For a prime $p\mid N$, let $$f_{\chi, p}^{\pm}(r)
=
\sum_{f\in S_p} \omega_{p^f}(r)
\sum_{(\chi_1, \chi_2)\in F_p} \chi_1(\pm 1) \chi_1\chi_2(r).$$ Then $f_{\chi}^\pm = \prod_{p\mid N} f_{\chi, p}^{\pm}$. When $p\mid r$, since $\omega_{p^f}(r) = 0$ unless $f=0$, so $f_{\chi, p}^{\pm}(r)=0$ unless $s_p=0$. From now on, assume that $p\nmid r$.
Recalling and Lemma \[lem:Fp\], $$f_{\chi, p}^\pm(r)
=
\sum_{f\in S_p}
\begin{cases}
\overline{\chi_p(r)} \sum_{\psi \in X_{p^{\min\{f, e_p-f\}}}} \psi (\pm r^2)
& \text{if } f\geq s_p, \\
\chi_p(\pm r) \sum_{\psi \in X_{p^{\min\{f, e_p-f\}}}} \psi (\pm r^2)
& \text{if } f< s_p.
\end{cases}$$ Set $\operatorname{ord}_p(0)=\infty$. Then $$\sum_{\psi \in X_{p^{\min\{f, e_p-f\}}}} \psi (\pm r^2)
=
\begin{cases}
\varphi(p^{\min\{f, e_p-f\}}) & \text{if } \min\{f, e_p-f\} \leq \operatorname{ord}_p(\pm r^2-1), \\
0 & \text{otherwise.}
\end{cases}$$ Note that $\sum_{f=0}^k \varphi(p^f) = p^k$. Let $u_\pm = \operatorname{ord}_p(\pm r^2-1)$. When $e_p < 2s_p$, we have $$f_{\chi, p}^{\pm}(r)
=
(\chi_p(\pm r) +\overline{\chi_p(r)} )
\sum_{f=0}^{\min\{e_p-s_p, u_\pm \}} \varphi(p^f)
=
(\chi_p(\pm r) +\overline{\chi_p(r)} ) p^{e_p-w_{\pm}(r)}.$$ When $e_p \geq 2s_p$, we have $$f_{\chi, p}^\pm(r)
=
\chi_p(\pm r) \sum_{f=0}^{\min\{s_p-1, u_\pm\}} \varphi(p^f)
+
\overline{\chi_p(r)}
\bigg( \sum_{f=s_p}^{\min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, u_\pm\}} \varphi(p^{f})
+
\sum_{f=0}^{\min\{u_\pm, {\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}\}}\varphi(p^{f})
\bigg).$$ For the second term, we have two cases: $s_p> \min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, u_\pm\}$ or $s_p \leq \min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, u_\pm\}$.
When $s_p> \min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, u_\pm\}$, since $e_p\geq 2s_p$, we get $u_\pm \leq s_p-1$. So we have $$f_{\chi, p}^{\pm} (r) = (\chi_p(\pm r) + \overline{\chi_p(r)}) \sum_{f=0}^{u_\pm} \varphi(p^f)
=
(\chi_p(\pm r) + \overline{\chi_p(r)}) p^{u_\pm}.$$ When $s_p\leq \min\{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, u_{\pm}\}$, i.e. $s_p\leq u_{\pm}$, we get $$\chi_p(\pm r^2) = \chi_p(\pm 1) \chi_p(r)^2 = 1.$$ We get $$f_{\chi, p}^{\pm}(r)
=
(\chi_p(\pm r) + \overline{\chi_p(r)})
\bigg(p^{\min\{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}, u_\pm\}}+ \frac{1}{2}\delta_{2\mid e_p, \frac{e_p}{2} \leq u_\pm } \varphi(p^{\frac{e_p}{2}})\bigg)$$ So we have $$f_{\chi, p}^{\pm}(r)
=
(\chi_p(\pm r) + \overline{\chi_p(r)})
\begin{cases}
\frac{1}{2}\big(p^{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}} + p^{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}}\big)
& \text{if } w_\pm(r) \leq{\left\lfloor}\frac{e_p}{2}{\right\rfloor}, \\
p^{e_p-w_\pm(r)}
& \text{if } w_\pm(r) > {\left\lfloor}\frac{e_p}{2}{\right\rfloor}.
\end{cases}$$
The proposition follows from $$\begin{aligned}
\operatorname{Eis}(N,\chi;n)=\operatorname{Eis}(\Gamma, \chi^{(0)})+n\operatorname{Eis}(\Gamma, \chi^{(1)}),\end{aligned}$$ by evaluating $\operatorname{Eis}(\Gamma, \chi^{(0)})$ and $\operatorname{Eis}(\Gamma, \chi^{(1)})$ using Lemmas \[lem:Eis\_integral\] and \[lem:TrPhiGamma\_1/2\] from the previous section together with Lemmas \[lem:|Fm\_ve|log\]–\[lem:chi\_ve\] in the present section.
Cuspidal and Continuous contributions $(\operatorname{C}+\operatorname{Eis})(\Gamma, \chi)$
-------------------------------------------------------------------------------------------
Finally we conclude the proof of Theorem \[thm:STF\_N\] by computing the sum of $\operatorname{C}(N, \chi; n)$ from Proposition \[prop:Cu\] and $\operatorname{Eis}(\Gamma, \chi; n)$ from Proposition \[prop:Eis\_N\].
For $n=1$, we combine and , and note that $$\begin{gathered}
(\Psi_4 + \Psi_5+ \Psi_6)(p^{e_p}, p^{s_p})
\\
=
e_p\!\left(p^{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}}+p^{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}}\right)
-
\frac{p^{{\left\lfloor}\frac{e_p}{2}{\right\rfloor}} +p^{{\left\lfloor}\frac{e_p-1}{2}{\right\rfloor}}-2}{p-1}
+
p^{s_p-1}
+
2\frac{p^{s_p-1}-1}{p-1}
=
\Psi_3(p^{e_p}, p^{s_p}).\end{gathered}$$ Here $\Psi_3$ is given in . Hence we obtain .
For $n=-1$, we combine and . Here one notes that, if $I_\chi=I_{q,4}=1$, $$\chi(\sqrt{-1})\min\{e_2+1, 4\} \prod_{p\mid N, p\equiv_41} \Psi_2 (p^{e_p}, p^{s_p}) \prod_{p\mid N, p\equiv_4-1} 2
=
{\tilde{\Psi}}_2(N, \chi),$$ with ${\tilde{\Psi}}_2$ as in . Recall also that if $I_\chi=1$, i.e. if $\chi$ is pure, then $s_2=0$ or $s_2\geq 3$. Therefore, $$I_\chi 2^{\omega(N)} \frac{1}{2}
\Big(
\Omega_1(N, q) (-e_2+2) + {\varepsilon}_N -2 -\Omega_1(N, q) \Psi_7(2^{e_2}, 2^{s_2})
\Big)
=
I_\chi 2^{\omega(N)}\Omega_2(N, q),$$ where $\Omega_2$ is given in . Finally for $p\mid N$ odd, recalling , $$\frac{1}{2}\big(\Psi_7(p^{e_p}, p^{s_p}) + e_p\big)
=
\max\left\{\frac{1}{2}, s_p\right\}.$$ Hence we obtain .
Recall also that the formulas for $\operatorname{I}(\Gamma, \chi; n)$ and $(\operatorname{NEll}+\operatorname{Ell})(\Gamma, \chi; n)$ stated in Theorem \[thm:STF\_N\] were proved in Lemma \[lem:I\_N\] and Proposition \[prop:NEll+Ell\_N\], respectively. Hence the proof of Theorem \[thm:STF\_N\] is now complete. $\square$
Sieving {#sec:sieve}
=======
Our goal in this section is to prove Theorem \[THMmintf\], by sieving out the contribution from the twist-minimal Hecke eigenforms in the trace formula in Theorem \[thm:STF\_N\].
The sieving is carried out in two steps: We first sieve for newforms, and then sieve for twist-minimal forms among the newforms. The first step is as in [@BS07 §2.3]: For any Dirichlet character $\chi$ modulo $N$ of conductor $\operatorname{cond}(\chi)$, any $\lambda>0$ and $n\in\{\pm1\}$, we have $$\begin{aligned}
\label{newformsieve}
\operatorname{Tr}T_n|_{{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)}
=
\sum_{M \mid \frac N{\operatorname{cond}(\chi)}} \beta\left(\frac{N/\operatorname{cond}(\chi)}{M}\right)
\operatorname{Tr}T_n|_{{\mathcal{A}}_\lambda(\chi|_{M\operatorname{cond}(\chi)})},\end{aligned}$$ where $\beta(m)$ is the multiplicative function given by $\zeta(s)^{-2} = \sum_{m=1}^\infty \beta(m) m^{-s}$.
For the second step, sieving down to ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)$, we assume that $\chi$ is minimal as in Definition \[def:minimal\]. Let $S_\chi$ be the set of pairs $\langle M,\psi\rangle$ where $M$ is a (positive) divisor of $N$ and $\psi$ is a primitive Dirichlet character such that $\operatorname{lcm}(M,\operatorname{cond}(\psi)\operatorname{cond}(\chi\psi))=N$, $\operatorname{cond}(\chi\psi^2)\mid M$ and $\chi\psi^2|_M$ is minimal. Also let $\sim$ be the equivalence relation on $S_\chi$ defined by $\langle M,\psi\rangle\sim\langle M',\psi'\rangle$ if and only if $M'=M$ and there is some Dirichlet character $\upsilon$ satisfying $\operatorname{cond}(\upsilon)\operatorname{cond}(\chi\psi^2\upsilon)\mid M$ and $\operatorname{cond}(\psi\upsilon\overline{\psi'})=1$. It then follows from Lemma \[lem:twistconductor\] and Lemma \[lem:twistminimal\_isom\] that, for any $\lambda>0$ and $n\in\{\pm1\}$, we have the direct sum decomposition $$\begin{aligned}
\label{newmindecomp}
{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)=\bigoplus_{\langle M,\psi\rangle\in S_\chi/\sim}\Bigl({\mathcal{A}^{\mathrm{min}}}_\lambda(\chi\psi^2|_M)\otimes{\overline{\psi}}\Bigr),\end{aligned}$$ where $S_\chi/\!\!\sim$ denotes any set of representatives for $S_\chi$ modulo $\sim$. Hence also, using $\overline{\psi(n)}=\psi(n)\in\{\pm1\}$: $$\begin{aligned}
\label{newmindecomp2}
\operatorname{Tr}T_n|_{{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)}=\sum_{\langle M,\psi\rangle\in S_\chi/\sim}\psi(n)\operatorname{Tr}T_n|_{{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi\psi^2|_M)}.\end{aligned}$$ Note that the pairs $\langle M,\psi\rangle$ in $S_\chi$ with $M=N$ form a single equivalence class, and this class contributes via the term $\operatorname{Tr}T_n|_{{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)}$ in the direct sum in .
In order to invert the formula , we first note that the set $S_\chi$ and the relation $\chi$ can be fully described by local conditions: Writing $\chi=\prod_{p\mid N} \chi_p$ as usual, and setting $N_p=p^{e_p}$ ($e_p=\operatorname{ord}_p(N)$) and $M_p=p^{\operatorname{ord}_p(M)}$, we have that a pair $\langle M,\psi\rangle$ lies in $S_\chi$ if and only if $\langle M_p,\psi_p\rangle$ lies in $S_{\chi_p}$ for each prime $p$, and furthermore $\langle M,\psi\rangle\sim\langle M',\psi'\rangle$ holds if and only if $\langle M_p,\psi_p\rangle\sim\langle M'_p,\psi'_p\rangle$ for all primes $p\mid N$. Also the equivalence classes in $S_{\chi_p}$ are easily classified: If $p=2$ or $e_p\leq1$ or $2\nmid e_p$ or $s_p>1$ then *all* elements in $S_{\chi_p}$ are equivalent with $\langle N_p,1\rangle$. In the remaining case when $p>2$, $e_p\geq2$, $2\mid e_p$ and $s_p\in\{0,1\}$, then the set of elements in $S_{\chi_p}$ outside the equivalence class of $\langle N_p,1\rangle$ equals $$\begin{aligned}
\label{PAIRS1}
\{\langle p^{e_p/2},\psi_p\rangle\::\:\operatorname{cond}(\psi_p)=p^{e_p/2},\:\psi_p\neq{\overline{\chi}}_p\}
\qquad\text{if $e_p\geq4$ or $s_p=1$,}\end{aligned}$$ and $$\begin{aligned}
\label{PAIRS2}
\{\langle p,\psi_p\rangle\: :\:\operatorname{cond}(\psi_p)=p\}
\:\cup\:\{\langle 1,(\tfrac{\cdot}p)\rangle\}
\qquad\text{if $e_p=2$ and $s_p=0$.}\end{aligned}$$ In the case of , each $\sim$ equivalence class in that set has exactly two elements. In the case of , the elements $\langle p,(\frac{\cdot}p)\rangle$ and $\langle 1,(\frac{\cdot}p)\rangle$ form singleton equivalence classes, while the remaining $p-3$ elements group together into equivalence classes with exactly two elements each.
In particular it follows from the above description that can be rewritten as $$\begin{aligned}
\operatorname{Tr}T_n|_{{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi)}=
\sum_{\langle M,\psi\rangle\in S_\chi'}
2^{-k(N,M,\psi)}\,\psi(n)\operatorname{Tr}T_n|_{{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi\psi^2|_M)},\end{aligned}$$ where $S_\chi'$ is the subset of all $\langle M,\psi\rangle\in S_\chi$ satisfying \[$M_p<N_p$ or $\psi_p=1$\] for each prime $p\mid N$, and $k(N,M,\psi)$ is the number of primes $p\mid N$ for which $M_p<N_p$ and \[$s_p=1$ or $\psi_p\neq(\frac{\cdot}p)$\]. This formula can now be inverted as follows: $$\begin{aligned}
\label{minsieveformula}
\operatorname{Tr}T_n|_{{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)}=
\sum_{\langle M,\psi\rangle\in S_\chi'}
(-1)^{k'(N,M)}
2^{-k(N,M,\psi)}\,
\psi(n)\operatorname{Tr}T_n|_{{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi\psi^2|_M)},\end{aligned}$$ where $k'(N,M)$ is the number of primes $p\mid N$ for which $M_p<N_p$.
We will apply the sieving in and then to the trace formula in Theorem \[thm:STF\_N\]. Note that the right hand side of that formula is a sum of terms $f(N,\chi)$ each of which is multiplicative with respect to $\chi$, in the sense that $f(N, \chi) = \prod_{p\mid N} f(p^{e_p}, \chi_p)$ for any Dirichlet character $\chi$ modulo $N$. One verifies that this multiplicativity property is preserved by the sieving, i.e., for any multiplicative function $f(N,\chi)$, if we define $f^{{\mathrm{new}}}$ and $f^{\min}$ via $$\begin{aligned}
\label{newformsieve2}
f^{{\mathrm{new}}}(N,\chi)=
\sum_{M \mid \frac N{\operatorname{cond}(\chi)}} \beta\left(\frac{N/\operatorname{cond}(\chi)}{M}\right)
f(M\operatorname{cond}(\chi),\chi)\end{aligned}$$ and $$\begin{aligned}
\label{minsieveformula2}
f^{\min}(N,\chi)=\sum_{\langle M,\psi\rangle\in S_\chi'}
(-1)^{k'(N,M)}
2^{-k(N,M,\psi)}\,
\psi(n) f(M,\chi\psi^2),\end{aligned}$$ then also $f^{{\mathrm{new}}}$ and $f^{\min}$ are multiplicative. Hence our task is reduced to computing $f^{\min}(p^{e_p},\chi_p)$ for each term $f(N,\chi)$ appearing in the trace formula in Theorem \[thm:STF\_N\]. Writing $e=e_p$ and $s=s_p$, we note that implies $$\begin{aligned}
\label{fnewp}
f^{{\mathrm{new}}}(p^{e}, \chi_p) = \sum_{j=0}^{e-s} \beta(p^{j}) f(p^{e-j}, \chi_p),\end{aligned}$$ and we have, for each $j\geq0$, $$\beta(p^j) = \begin{cases}
1 & \text{if } j\in \{0, 2\}, \\
-2 & \text{if } j=1, \\
0 & \text{otherwise.}
\end{cases}$$ Next assume again that $\chi=\prod_p\chi_p$ is minimal as in Definition \[def:minimal\]. It now follows from and the description of the equivalence classes of $S_{\chi_p}$ given around and that, for each odd prime $p\mid N$: $$\begin{gathered}
\label{e:fmin_odd}
f^{\min}(p^{e}, \chi_p)
=
f^{\mathrm{new}}(p^{e}, \chi_p)-\delta_{e=2,s=0}\bigl(\tfrac{n}{p}\bigr)
\big(f^{\mathrm{new}}(1, 1)+\tfrac12f^{\mathrm{new}}(p, 1)\big)
\\
-\delta_{2\mid e,s\leq1}\cdot\frac12\sum_{\substack{\mathrm{cond}(\psi)=p^{\frac{e}{2}}\\\psi\neq\overline{\chi}_p}}
\psi(n)f^{\mathrm{new}}(p^{\frac{e}{2}},\chi_p\psi^2).\end{gathered}$$ On the other hand for $p=2$ we have simply $$\label{e:fmin_p=2}
f^{\min}(2^{e}, \chi_2)
=
f^{{\mathrm{new}}} (2^{e}, \chi_2).$$
$\operatorname{I}(\Gamma, \chi)$ {#IMINSIEVE}
--------------------------------
Recalling Lemma \[lem:I\_N\], by and , we have $$\operatorname{I}^{\min}(\Gamma, \chi; n)
=
\frac{1+n}{2}
\frac{M(\chi)}{12}
\int_{\mathbb{R}}rh(r) \tanh(\pi r) \; dr.$$ Here $M(\chi)$ is given in . By [@BS07 p.141], $$\int_{\mathbb{R}}rh(r) \tanh(\pi r) \; dr = -\int_{-\infty}^\infty \frac{g'(u)}{\sinh(u/2)} \; du,$$ so we get $$\operatorname{I}^{\min}(\Gamma, \chi; n)
=
-
\frac{1+n}{2}
\frac{M(\chi)}{12}
\int_{-\infty}^\infty \frac{g'(u)}{\sinh(u/2)} \; du.$$
${\bold{S}}_p(p^{e}, \chi; t, n)$
---------------------------------
Using the Dirichlet class number formula and the formula for ${\bold{S}}_p(1,1;t,n)$ in , the elliptic and hyperbolic terms in Theorem \[thm:STF\_N\] can be rewritten as $$\begin{aligned}
(\operatorname{NEll}+\operatorname{Ell})(\Gamma, \chi; n)
=
\sum_{\substack{t\in {\mathbb{Z}}\\ D=t^2-4n\\\sqrt D\notin{\mathbb{Q}}}}
\biggl(\prod_{p\mid N}\frac{{\bold{S}}_p(p^e,\chi_p;t,n)}{1+(p-(\frac dp))\frac{p^{\operatorname{ord}_p\ell}-1}{p-1}}\biggr)
\:L(1,\psi_{D})
\hspace{40pt}
\\
\cdot\begin{cases}
g\!\left(2\log\frac{|t|+\sqrt{D}}2\right)
&\text{if }D>0,\\
\frac{\sqrt{|D|}}{\pi}\int_{\mathbb{R}}\frac{g(u)\cosh(u/2)}{4\sinh^2(u/2)+|D|}\,du
&\text{if }D<0,
\end{cases}\end{aligned}$$ where the character $\psi_D$ is as in Section \[ss:twist-minimal\_TF\] (cf. also [@BL17 Sec. 2]). Hence to show that we obtain the corresponding sum in Theorem \[THMmintf\], it remains to prove that for each prime $p$ with $e=e_p>0$, $$\begin{aligned}
\label{HtnchiWTP}
\frac{{\bold{S}}_p^{\min}(p^e,\chi_p;t,n)}{1+(p-(\frac dp))\frac{p^{\operatorname{ord}_p\ell}-1}{p-1}}
=H_{t,n}(\chi_p).\end{aligned}$$
From now on until further notice, let us assume that *$p$ is odd.* If $e=s>0$ then ${\bold{S}}_p^{\min}={\bold{S}}_p^{{\mathrm{new}}}={\bold{S}}_p$ by and , and one then verifies directly from and (with $h=\max\{2e-1, e\} = 2e-1$) that holds.
Next assume $s\in\{0, 1\}$ and $e>2$. Then by (where now $h=\max\{2s-1, e\} =e$) and , we get $$\begin{gathered}
\label{e:Spnew}
{\bold{S}}_p^{{\mathrm{new}}}(p^e, \chi; t, n)
=
{\bold{S}}_p(p^e, \chi; t, n) - 2{\bold{S}}_p(p^{e-1}, \chi; t, n) + {\bold{S}}_p(p^{e-2}, \chi; t, n)
\\
=
\chi\!\left(\frac{t+\delta_{2\nmid t} p^s}{2}\right)
p^{e-3}
\begin{cases}
(p-1)
\bigg( \delta_{2\mid e} \left(p-\left(\frac{d}{p}\right)\right) p^{f-\frac{e}{2}+1}
+
\left(\left(\frac{d}{p}\right)-1\right)(p+1)
\bigg)
& \text{if } g\geq e, \\
\left(\left(\frac{d}{p}\right)-1\right) (p-1) (1+\delta_{2\nmid e} p),
& \text{if } g=e-1, \\
\left(\left(\frac{d}{p}\right)-1\right) (p-1) + \left(\frac{d}{p}\right),
& \text{if } g=e-2, \\
0 & \text{otherwise.}
\end{cases}\end{gathered}$$ Note that for $\alpha\geq 2$, $$\sum_{\substack{\psi\text{ primitive}\\ \operatorname{cond}(\psi)=p^\alpha}}
\psi(x)
=
\varphi(p^{\alpha-1})
\begin{cases}
p-1 & \text{if } \alpha \leq \operatorname{ord}_p(x-1), \\
-1 & \text{if } \alpha = \operatorname{ord}_p(x-1)+1, \\
0 & \text{otherwise.}
\end{cases}$$ To evaluate the last sum in , we need the following lemma.
When $\left(\frac{d}{p}\right)=1$, $n\left(\frac{t}{2}\pm \frac{\sqrt{d}\ell}{2}\right)^2 \equiv_{p^{\alpha}} 1$ if and only if $\ell\equiv_{p^{\alpha}} 0$.
Set $x \equiv_{p^{\alpha}} \frac{t}{2}\pm \frac{\sqrt{d}\ell}{2}$ and $y\equiv_{p^{\alpha}} \frac{t}{2}\mp \frac{\sqrt{d}\ell}{2}$. Then since $t^2-4n=d\ell^2$, we get $xy \equiv_{p^{\alpha}} n$ and $x^2\equiv_{p^{\alpha}} tx-n$. Assume that $n(tx-n)=nx^2 \equiv_{p^{\alpha}}1$. Then $ntx \equiv_{p^{\alpha}} 2$. Multiplying by $y$ on both sides, we get $$t\equiv 2y \equiv t-\sqrt{d}\ell \pmod*{p^{\alpha}}.$$ So $p^{\alpha}\mid \ell$. Conversely, assume that $\ell\equiv_{p^{\alpha}}0$. Then $$nx^2 \equiv ntx-1 \equiv n\frac{t^2}{2} -1 = n\frac{t^2-4n}{2}+1
\equiv 1\pmod*{p^{\alpha}}.$$
By the above lemma, for $e>2$, $2\mid e$, we have $$\begin{gathered}
\sum_{\operatorname{cond}(\psi)=p^{\frac{e}{2}}}
\psi(n)\,{\bold{S}}_p^{{\mathrm{new}}}(p^{\frac{e}{2}}, \chi\psi^2; t, n)
=
\chi\!\left(\tfrac{t+\delta_{2\nmid t} p^s}{2}\right)
(p-1) p^{e-3}
\\
\times
\begin{cases}
2(p-1)p^{f-\frac{e}{2}+1} + \left(\left(\frac{d}{p}\right)-1\right)\big(-2p^{f-\frac{e}{2}+1} + p +1\big)
& \text{if } g\geq e-1, \\
-2 & \text{if } g=e-2 \text{ and } \left(\frac{d}{p}\right)=1, \\
0 & \text{otherwise.}
\end{cases}\end{gathered}$$ By and we now obtain, when $s\in\{0,1\}$ and $e>2$, $${\bold{S}}_p^{\min}(p^e, \chi; t, n)
=
\left(\left(\tfrac{d}{p}\right)-1\right)
\frac{1+\delta_{2\nmid e}}{2}
\chi\!\left(\tfrac{t+\delta_{2\nmid t} p^s}{2}\right)
p^{e-3}
\begin{cases}
(p-1)(p+1) & \text{if } g\geq e-1, \\
-\delta_{2\mid e} p - 1 & \text{if } g=e-2, \\
0 & \text{otherwise.}
\end{cases}$$ Hence again holds; cf. .
We next turn to the case $e\in\{1, 2\}$, $s<e$. By , when $e=1$ and $s=0$, we have $${\bold{S}}_p^{\min}(p, 1; t, n)= {\bold{S}}_p^{{\mathrm{new}}}(p, 1; t, n) = {\bold{S}}_p(p, 1; t, n) - 2{\bold{S}}_p(1, 1; t, n) = \left(\tfrac{d}{p}\right)-1.$$ For $e=2$, $$\begin{gathered}
{\bold{S}}_p^{{\mathrm{new}}}(p^2, \chi; t, n)
\\
=
\chi\!\left(\tfrac{t+\delta_{2\nmid t}p^s}{2}\right)
\begin{cases}
p^f (p-2-s) + \left(\left(\frac{d}{p}\right)-1\right) \frac{-p^f(p-2-s)+p^2-p-1-s}{p-1}
& \text{if } g\geq 2, \\
-1-s & \text{if } g=1, \\
0 & \text{if } g=0
\end{cases}
\\
+
\delta_{g=0}
\bigg\{
(1-s)\frac{1-\left(\frac{d}{p}\right)}{2}
-
\frac{1}{2-s}
\frac{1+\left(\frac{d}{p}\right)}{2}
\left(\chi\!\left(\tfrac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\tfrac{t-\sqrt{d}\ell}{2}\right)\right)
\bigg\}.\end{gathered}$$ Assuming first $s=1$, $e=2$, we compute: $$\begin{gathered}
\sum_{\substack{\psi\pmod*{p}\\\psi\notin\{{\overline{\chi}}_p,1\}}}
{\bold{S}}_p^{{\mathrm{new}}}(p, \chi\psi^2; t, n) \psi(n)
\\
=
\begin{cases}
\chi\!\left(\frac{t+\delta_{2\nmid t} p}{2}\right)
(p-3)
\Bigl(2p^f + \bigl(\bigl(\frac{d}{p}\bigr)-1\bigr) \frac{-2p^f + p+1}{p-1}\Bigr)
& \text{if } g\geq 1, \\
-2\Bigl(\chi\!\bigl(\frac{t+\sqrt{d}\ell}{2}\bigr) + \chi\!\bigl(\frac{t-\sqrt{d}\ell}{2}\bigr)\Bigr)
& \text{if } g=0 \text{ and } \bigl(\frac{d}{p}\bigr)=1, \\
0 & \text{otherwise.}
\end{cases}\end{gathered}$$ Hence by , $$\begin{gathered}
{\bold{S}}_p^{\min}(p^2, \chi; t, n)
=
{\bold{S}}_p^{{\mathrm{new}}}(p^2, \chi; t, n) -
\frac{1}{2}\sum_{\substack{\psi\pmod*{p}\\\psi\notin\{{\overline{\chi}}_p,1\}}}
{\bold{S}}_p^{{\mathrm{new}}}(p^2, \chi\psi^2; t, n) \psi(n)
\\
=
\begin{cases}
\chi\!\left(\frac{t+\delta_{2\nmid t} p^s}{2}\right) \left(\left(\frac{d}{p}\right)-1\right)
\frac{p+1}{2}
& \text{if } g\geq 1, \\
0
& \text{if } g=0.
\end{cases}\end{gathered}$$ Finally for $s=0$, $e=2$, we have $$\begin{gathered}
\left(\tfrac{n}{p}\right)
\big( {\bold{S}}_p(1, 1; t, n) + \tfrac12{\bold{S}}_p^{{\mathrm{new}}}(p, 1; t, n)\big)
+
\frac{1}{2} \sum_{\substack{\psi\pmod*{p}\\\psi\neq1}}
{\bold{S}}_p^{{\mathrm{new}}}(p, \psi^2; t, n) \psi(n)
\\
=
\begin{cases}
(p-2)p^f + \left(\left(\frac{d}{p}\right)-1\right) \frac{\frac{1}{2}(p^2-3) - p^f(p-2)}{p-1}
& \text{if } g\geq 1, \\
-1+\left(\left(\frac{d}{p}\right)-1\right) \frac{\left(\frac{n}{p}\right)-1}{2}
& \text{if } g=0, \\
0 & \text{otherwise,}
\end{cases}\end{gathered}$$ and so, by , $${\bold{S}}_p^{\min}(p^2, 1; t, n)
=
\frac{\left(\tfrac{d}{p}\right)-1}{2}
\begin{cases}
p-1 & \text{if } g\geq 1, \\
-\left(\frac{n}{p}\right)-1 & \text{if } g=0, \\
0 & \text{otherwise.}
\end{cases}$$ In all these cases we again see that holds; cf. .
Finally we turn to the case $p=2$. Since $\chi$ is minimal, we have to consider the following subcases (cf. Definition \[def:minimal\] and recall that $s=1$ is impossible when $p=2$): $$\begin{aligned}
\label{def:minimalp2expl}
s=
\begin{cases}
e & \text{if } e\geq 2, \\
{\left\lfloor}\frac{e}{2}{\right\rfloor}& \text{if } e\geq 4, \\
2 & \text{if } e\geq 5\text{ and } 2\nmid e, \\
0 & \text{if } 2\nmid e \text{ or } e=2.
\end{cases}\end{aligned}$$ Recalling , for $s\geq 2$ and $s\leq e\leq 2s$, we have $$\begin{gathered}
{\bold{S}}_2(2^e, \chi; t, n)
\\
=
\chi\!\left(\tfrac{t}{2}\right) 2^{e-1}
\begin{cases}
\delta_{2\nmid d} 2^{\frac{g}{2}-s} (4-\delta_{e=2s})
+ \left(\left(\frac{d}{2}\right)-1\right) (3-2^{{\left\lfloor}\frac{g}{2}{\right\rfloor}-s}(4-\delta_{e=2s}))
& \text{if } g\geq 2s+1, \\
-\delta_{2\nmid d} (4-\delta_{e=2s})
+ \delta_{e\leq 2s-1}\left(\left(\frac{d}{2}\right)-1\right)
& \text{if } g=2s, \\
0 & \text{otherwise}
\end{cases}
\\
+
\delta_{\left(\frac{d}{2}\right)=1, g\leq 2s-2}
\left(\chi\!\left(\tfrac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\tfrac{t-\sqrt{d}\ell}{2}\right)\right)
2^{f+\min\{e-s, f\}},\end{gathered}$$ For $e=2s+1$, we have $$\begin{gathered}
{\bold{S}}_2(2^{e}, \chi; t, n)
=
\chi\!\left(\tfrac{t}{2}\right) 2^{e-1}
\begin{cases}
\delta_{2\nmid d} 2^{\frac{g}{2}-s+1} + \left(\left(\frac{d}{2}\right) -1\right) (3-2^{{\left\lfloor}\frac{g}{2}{\right\rfloor}-s+1})
& \text{if } g\geq 2s+2, \\
0 & \text{otherwise}
\end{cases}
\\
+
\delta_{\left(\frac{d}{2}\right)=1, g\leq 2s}
\left(\chi\!\left(\tfrac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\tfrac{t-\sqrt{d}\ell}{2}\right)\right)
2^{2f}.\end{gathered}$$ Here $f=\operatorname{ord}_2(\ell)$. Then by , for $s={\left\lfloor}\frac{e}{2}{\right\rfloor}$, $e\geq 4$, we have $$\begin{aligned}
{\bold{S}}_2^{\min}(2^e, \chi; t, n)
=
{\bold{S}}_2^{{\mathrm{new}}}(2^{e}, \chi; t, n)
=
\chi\!\left(\tfrac{t}{2}\right)&\left(\left(\tfrac{d}{2}\right)-1\right)
2^{e-3}
\begin{cases}
3 & \text{if } g\geq e+1, \\
-1-\delta_{2\mid e} 2 & \text{if } g=e, \\
1-\delta_{2\nmid d}4 & \text{if } g=e-1, 2\nmid e, \\
0 & \text{otherwise}
\end{cases}
\\
&-
\delta_{\left(\frac{d}{2}\right)=1, g=e-2}
\left(\chi\!\left(\tfrac{t+\sqrt{d}\ell}{2}\right)+ \chi\!\left(\tfrac{t-\sqrt{d}\ell}{2}\right)\right)
2^{g-1}.\end{aligned}$$ For $e=2s$ and $g=2s-2$, we claim that $\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right)+ \chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)=0$. Note that $g=4$ cannot occur, so $s\geq 4$. Moreover, if $a$ is odd, then $$\chi\!\left(\tfrac{t}{2}+a2^{s-2}\right)^2
=
\chi\!\left(\tfrac{t^2}{4} + a 2^{s-1}\right)
=
-\chi\!\left(\tfrac{t^2}{4}\right) = -\chi(1) =-1,$$ since $t^2-4n=d\ell^2$ and $g=\operatorname{ord}_2(d\ell^2) = 2s-2>4$, which forces $n=1$. This implies that $$\chi\!\left(\tfrac{t}{2}+a 2^{s-2}\right) = \pm i.$$ Since $\left(\frac{t+\sqrt{d}\ell}{2}\right)\left(\frac{t-\sqrt{d}\ell}{2}\right) = 1$, we get $\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right)\chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)=1$, so $\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right)=\overline{\chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)}$. Then $\chi\!\left(\frac{t+\sqrt{d}\ell}{2}\right)+\chi\!\left(\frac{t-\sqrt{d}\ell}{2}\right)=0$ as claimed. Now consider the cases for $s\in\{0, 2\}$. For $e\geq \max\{1, 2s-1\}$, we have $$\begin{gathered}
{\bold{S}}_2(2^e, \chi; t, n)
\\
=
\chi\!\left(\tfrac{t}{2}\right)
\begin{cases}
\delta_{2\nmid d}
2^{\frac{g}{2}+{\left\lfloor}\frac{e}{2}{\right\rfloor}}(1+2^{{\left\lceil}\frac{e}{2}{\right\rceil}-{\left\lfloor}\frac{e}{2}{\right\rfloor}-1})
\\
+
\left(\left(\frac{d}{2}\right)-1\right)
(3\cdot 2^{e-1} - 2^{{\left\lfloor}\frac{g}{2}{\right\rfloor}+{\left\lfloor}\frac{e}{2}{\right\rfloor}}(1+2^{{\left\lceil}\frac{e}{2}{\right\rceil}-{\left\lfloor}\frac{e}{2}{\right\rfloor}-1}))
& \text{if } g\geq e+1 \text{ and } 2\nmid g, \\
& \text{or } g\geq \max\{e, 2s+2\} \text{ and } 2\mid g, \\
0 & \text{otherwise}
\end{cases}
\\
+
\delta_{\left(\frac{d}{2}\right)=1, g\leq e-1}
\left(\chi\!\left(\tfrac{t+\sqrt{d}\ell}{2}\right) + \chi\!\left(\tfrac{t-\sqrt{d}\ell}{2}\right)\right) 2^g.\end{gathered}$$ For $e=s=0$, we have $${\bold{S}}_2(1, 1; t, n)
=
\delta_{2\nmid d} 2^{\frac{g}{2}} + \left(\left(\tfrac{d}{2}\right)-1\right)(1-2^{{\left\lfloor}\frac{g}{2}{\right\rfloor}}).$$ For $e=4$ and $s=2$, we have $${\bold{S}}_2^{\min}(2^4, \chi_{-4}; t, n)
=
\chi_{-4}\!\left(\tfrac{t}{2}\right) \left(\left(\tfrac{d}{2}\right)-1\right)
\begin{cases}
6 & \text{if } g\geq 5, \\
0 & \text{otherwise.}
\end{cases}$$ For $s=0$ and $e\in\{1, 2\}$, we have $${\bold{S}}_2^{\min}(2^e, 1; t,n)
=
\begin{cases}
\left(\frac{d}{2}\right)-1
& \text{if } g\geq 2, \\
-2 & \text{if } e=1 \text{ and } g=0, \\
1 & \text{if } e=2 \text{ and } g=0.
\end{cases}$$ For $s\in\{0, 2\}$ and $e \geq\max\{3, 2s+1\}$ and $2\nmid e$, we have $${\bold{S}}_2^{\min}(2^e, \chi; t, n)
=
\chi\!\left(\tfrac{t}{2}\right)
\left(\left(\tfrac{d}{2}\right)-1\right) 2^{e-3}
\begin{cases}
3 & \text{if } g\geq e+1, \\
\delta_{2\nmid d} 4 -1 & \text{if } g\in\{e, e-1\}, \\
0 & \text{otherwise.}
\end{cases}$$ In all these cases we again see that holds; cf. .
$\Psi_1$, $\Psi_2$, ${\tilde{\Psi}}_2$ and $\Psi_3$
---------------------------------------------------
Note that $\Psi_1$ appears only when $n=1$.
\[lem:Psi1\_min\] We have $$\label{e:Psi1_min}
\Psi_1^{\min}(p^e, p^s)
=
\delta_{e=s} 2.$$ So $$\label{e:Psi1_min_N}
\Psi_1^{\min}(N, q) = \delta_{N=q} 2^{\omega(N)}.$$
Recalling , when $p$ is odd and $e=s\geq 1$ or $e\geq\max\{1, 2s\}$ for $s\in\{0, 1\}$, we have $$\Psi_1^{{\mathrm{new}}}(p^e, p^s)
=
\begin{cases}
p^{\frac{e}{2}-2} (p-1)^2 & \text{if } e\geq \max\{3, 2s+2\} \text{ and } 2\mid e, \\
p-2-s & \text{if } e=2\text{ and } s\in\{0, 1\}, \\
2 & \text{if } e=s\geq 1, \\
0 & \text{otherwise.}
\end{cases}$$ Then by , we get . When $p=2$ and $s$ is as in , we have by $$\Psi_1^{\min}(2^e, 2^s) = \Psi_1^{{\mathrm{new}}}(2^e, 2^s)
= \begin{cases}
2 & \text{if } e=s\geq 2, \\
0 & \text{otherwise.}
\end{cases}$$
Recalling , let $$\Psi_2(p^e, \chi) = \frac{\chi(-1)+1}{2} \Psi_2(p^e, p^s).$$ Note that $\Psi_2$ appears only when $n=1$. But it also occurs in the definition of ${\tilde{\Psi}}_2$ when $n=-1$.
\[lem:Psi2\_min\] For $p\equiv_41$, we have $$\label{e:Psi2_min}
\Psi_2^{\min}(p^e, \chi)
=
0,$$ so that $\Psi_2^{\min}(N,\chi)=\delta_{N=1}$.
Recall . When $p$ is odd, $e\geq \max\{1, 2s\}$, $s\in\{0, 1\}$ or $e=s\geq 1$, we have $$\Psi_2^{{\mathrm{new}}}(p^e, \chi)
=
\delta_{e=2, s=0}.$$ Then by and , we get .
Note that ${\tilde{\Psi}}_2$ appears only when $n=-1$.
\[lem:tPsi2\_min\] We have $${\tilde{\Psi}}_2^{\min}(p^e, \chi) = 0,$$ so that ${\tilde{\Psi}}_2^{\min}(N,\chi)=\delta_{N=1}$.
Recalling , for $p\not\equiv_41$ and $s=0$, we have $${\tilde{\Psi}}_2^{{\mathrm{new}}}(p^e, 1)
=
\begin{cases}
-1 & \text{if } p\equiv_4-1 \text{ and } e=2, \\
0 & \text{otherwise.}
\end{cases}$$ For $p\equiv_41$, by , and , we have ${\tilde{\Psi}}_2^{\min}(p^e, \chi)=0$.
Note that $\Psi_3$ appears only for $n=1$.
\[lem:Psi3\_min\] When $p$ is odd, we have $$\Psi_3^{\min}(p^e, p^s)
=
\begin{cases}
4e-1 & \text{if } e=s>0, \\
2 & \text{if } e=1 \text{ and } s=0, \\
\frac{1}{2}(p-1+2s) & \text{if } e=2 \text{ and } s\in\{0, 1\}, \\
p^{{\left\lfloor}\frac{e-3}{2}{\right\rfloor}} (p-1) \frac{\delta_{2\mid e} p+ \delta_{2\nmid e} 3+1 }{2}
& \text{if } e\geq 3 \text{ and } s\in \{0, 1\}.
\end{cases}$$ For $p=2$, we have $$\Psi_3^{\min}(2^e, 2^s)
=
\begin{cases}
4e-1 & \text{if } e=s\geq 2, \\
2 & \text{if } e\in\{1, 3\} \text{ and } s=0, \\
1 & \text{if } e=2 \text{ and } s=0, \\
3\cdot 2^{\frac{e}{2}-2} & \text{if } e=2s\geq 4, \\
2^{\frac{e-1}{2}} & \text{if } e\geq 5, 2\nmid e \text{ and } s\in\{0, 2, \frac{e-1}{2}\}.
\end{cases}$$ Thus, combining with , $$\begin{gathered}
\sum_{p\mid N} \Psi_3^{\min}(p^{e_p}, p^{s_p})
\prod_{p'\mid N, p'\neq p} \Psi_1^{\min}({p'}^{e_{p'}}, {p'}^{s_{p'}}) \log p
\\
=
\begin{cases}
2^{\omega(N)} \big(\log N^2 - \frac{1}{2} \sum_{p\mid N} \log p\big)
& \text{if } N=q,\\
2^{\omega(N)-1} \Psi_3^{\min}(p^{e_p}, p^{s_p}) \log p
& \text{if } \frac{N}{q} = p^{e_p-s_p}>1, \\
0 & \text{otherwise.}
\end{cases}\end{gathered}$$
Recalling , for $e=s>0$ or $e\geq \max\{1, 2s\}$ for $s\in\{0, 1, 2\}$, we have $$\Psi_3^{{\mathrm{new}}}(p^e, p^s)
=
\begin{cases}
4e-1 & \text{if } e=s>0, \\
p^{\frac{e}{2}-2} (p-1) (e(p-1)+1)
& \text{if } e\geq \max\{3, 2s+2\} \text{ and } 2\mid e, \\
2p^{\frac{e-3}{2}} (p-1)
& \text{if } e\geq \max\{3, 2s+2\} \text{ and } 2\nmid e, \\
2p-3 & \text{if } e=2 \text{ and } s=0, \\
2 & \text{if } e=1 \text{ and } s=0, \\
2(p-1) & \text{if } e=3 \text{ and } s=1, \\
2(p-2) & \text{if } e=2 \text{ and } s=1, \\
e-1 & \text{if } p=2, e\in\{4, 5\} \text{ and } s=2, \\
\end{cases}$$ For $n=1$, by applying and , we prove the lemma.
$\Phi_\pm$
----------
Let us define $$\Phi_\pm (p^{e}, \chi; m)
=\begin{cases}
1&\text{if }s=0,\: p\mid m
\\[3pt]
\frac12\bigl(\overline{\chi(m)} + \chi(\pm m)\bigr)\Phi_\pm(p^e, m)&\text{otherwise.}
\end{cases}$$ Then in the trace formula in Theorem \[thm:STF\_N\], the sums over $m$ appearing in the last lines of (for $n=1$) and (for $n=-1$), can be expressed as $$\begin{aligned}
2\sum_{m=2}^\infty \frac{\Lambda(m)}m\biggl(\prod_{p\mid N}\Phi_n(p^e,\chi_p;m)\biggr)\,g(2\log m).\end{aligned}$$ Hence to show that we obtain the corresponding sum in Theorem \[THMmintf\], we have to prove that for each prime $p\mid N$, writing now $\chi$ in place of $\chi_p$, $$\begin{aligned}
\label{e:Phi_min}
\Phi_n^{\min}(p^e,\chi;m)=\Phi_{m,n}(\chi)
=\begin{cases}
\overline{\chi(m)}+\chi(nm)&\text{if }s=e,\\
-1&\text{if }p\mid m,\: e=1\text{ and }s=0,\\
0&\text{otherwise}
\end{cases}\end{aligned}$$ (cf. ).
By we have, for $e>0$, $$\Phi_\pm(p^e, \chi;m)
=
\begin{cases}
\delta_{s=0} & \text{if } p\mid m, \\[3pt]
\bigl(\overline{\chi(m)} + \chi(\pm m)\bigr)\,p^{e-w_\pm} & \text{if } p\nmid m,\: e<2w_\pm, \\[3pt]
\bigl(\overline{\chi(m)} + \chi(\pm m)\bigr)\,\frac12(p^{\lfloor e/2\rfloor}+p^{\lfloor (e-1)/2\rfloor})
& \text{if } p\nmid m,\: e\geq2w_\pm,
\end{cases}$$ where $w_\pm = w_\pm (m) = \max\{s, e-\operatorname{ord}_p(\pm m^2-1)\}$. Let $f_\pm = \operatorname{ord}_p(\pm m^2-1)$. For $s=0$, we compute $$\Phi_\pm^{{\mathrm{new}}}(p^e, \chi; m)
=
\begin{cases}
-1 & \text{if } p\mid m \text{ and } e=1, \\
\begin{cases}
-1 & \text{if } e=2 \text{ and } f_\pm = 0, \\
-p^{f_\pm}+p^{f_\pm -1} & \text{if } e=2f_\pm +2 \text{ and } f_\pm \geq 1, \\
p^{\frac{e}{2}-2}(p-1)^2 & \text{if } 3\leq e \leq 2f_\pm \text{ and } 2\mid e, \\
p-2 & \text{if } e=2 \text{ and } f_\pm \geq 1,
\end{cases}
& \text{if } p\nmid m, \\
0 & \text{otherwise.}
\end{cases}$$ For $s=1$, $p$ odd and $e\geq 2$, and assuming $p\nmid m$, we compute $$\begin{gathered}
\Phi_\pm^{{\mathrm{new}}}(p^e, \chi; m)
\\
=
\frac{1}{2}\big(\overline{\chi(m)}+\chi(\pm m)\big)
\begin{cases}
-2 & \text{if } e=2 \text{ and } f_\pm =0, \\
\begin{cases}
p-3 & \text{if } e=2, \\
p^{\frac{e}{2}-2} (p-1)^2 & \text{if } 4\leq e \leq 2f_\pm \text{ and } 2\mid e, \\
-p^{f_\pm}+p^{f_\pm -1} & \text{if } e=2f_\pm +2,
\end{cases}
& \text{if } f_\pm \geq 1, \\
0 & \text{otherwise. }
\end{cases}\end{gathered}$$ For $p$ odd, using the above formulas together with and the fact that $\Phi_\pm^{{\mathrm{new}}}=\Phi_\pm$ when $s=e>0$, the desired result now follows by a direct computation in each case.
Next assume $p=2$. Recalling and , we consider the following cases. For $e\geq 5$, $2\nmid e$ and $s\in \{0, 2\}$, we get $$\Phi_\pm^{\min}(2^e, \chi ;m) = 0,$$ and for $s=e\geq2$ we get $$\Phi_\pm^{\min}(2^e, \chi ;m) = \overline{\chi(m)}+\chi(nm).$$ For $s\geq 2$ and $e\in\{2s, 2s+1\}$, we obtain $$\Phi_{\pm}^{\min}(2^e, \chi; m)
=
\frac{1}{2}\big(\overline{\chi(m)}+\chi(\pm m)\big)
\begin{cases}
-2^{f_\pm} & \text{if } 2\nmid m, e=2s \text{ and } f_\pm = s-1, \\
0 & \text{otherwise.}
\end{cases}$$ For $f_\pm = s-1$, we have $m^2 = \pm 1+a \cdot 2^{s-1}$ for an odd integer $a$. Then $$\chi(m)^2 = \chi(\pm 1+a\cdot 2^{s-1}) =-\chi(\pm 1).$$ For $\Phi_+$, we get $\chi(m)=\pm i$ and $\overline{\chi(m)}+\chi(m)=0$. For $\Phi_-$, if $\chi$ is even then $\chi(m)=\pm i$. If $\chi$ is odd then $\chi(m) = \pm 1$. For either case we get $\overline{\chi(m)}+\chi(-m)=0$. So finally we conclude that $\Phi_\pm^{\min}(2^e, \chi; m)=0$ for $e\in\{2s, 2s+1\}$. For $s=0$ and $e\in\{1, 2, 3\}$, we get $$\Phi_\pm^{\min}(2^e, \chi; m) =
\begin{cases}
-1 & \text{if } 2\mid m \text{ and } e=1, \\
& \text{or } 2\nmid m, e=2 \text{ and } f_\pm=0, \\
0 & \text{otherwise.}
\end{cases}$$ This formula agrees with , since in fact $f_{\pm}=\operatorname{ord}_p(\pm m^2-1)\geq1$ must hold when $p=2$ and $2\nmid m$.
Now has been proved in all cases.
$\Omega_1$ and $\Omega_2$ {#Omega12MINSIEVE}
-------------------------
For $j\in \{1, 2\}$, define $${\tilde{\Omega}}_j(p^e, \chi)
=
\begin{cases}
2\Omega_j(N, q) & \text{if } p=2, e\geq 1 \text{ and } \chi \text{ even}, \\
\Omega_j(N, q) & \text{if } p=2 \text{ and } e=0, \\
2 & \text{if } p \text{ odd}, e\geq 1 \text{ and } \chi \text{ even}, \\
1 & \text{if } p \text{ odd } \text{ and } e=0, \\
0 & \text{otherwise.}
\end{cases}$$ Here $\Omega_j$ is given in and . Then $${\tilde{\Omega}}(N, \chi) = \prod_{p\mid N} {\tilde{\Omega}}_j(p^{e_p}, \chi_p) = I_\chi 2^{\omega(N)} \Omega_j(N, q).$$
\[lem:tOmega\_min\] When $p$ is odd and $n=-1$, we have $$\label{e:tOmega_min_odd}
{\tilde{\Omega}}_j^{\min}(p^e, \chi) =
\delta_{e=s\geq 1} (1+\chi(-1)).$$ When $p=2$ and $n=-1$, we have $$\label{e:tOmega1_min_p=2}
{\tilde{\Omega}}_1^{\min}(2^e, \chi) = \delta_{e=s\geq 2} (1+\chi(-1))$$ and $$\label{e:tOmega2_min_p=2}
{\tilde{\Omega}}_2^{\min}(2^e, \chi)
=
\begin{cases}
\frac{1}{2} & \text{if } e=0, \\
-2e & \text{if } e=s\geq 3 \text{ and } \chi\text{ even}, \\
-\frac{3}{2} & \text{if } e=1 \text{ and }s=0, \\
-\frac{1}{2} & \text{if } e\in \{2, 3\} \text{ and } s=0, \\
0 & \text{otherwise.}
\end{cases}$$
When $p$ is odd, for $e=s\geq 1$ or $e\geq \max\{1, 2s\}$, we have $${\tilde{\Omega}}_j^{{\mathrm{new}}}(p^e, \chi) =
\begin{cases}
2 & \text{if } e=s>0 \text{ and } \chi \text{ even}, \\
-2 & \text{if } e=2, s=1 \text{ and } \chi \text{ even}, \\
-1 & \text{if } e=2 \text{ and } s=0, \\
0 & \text{otherwise.}
\end{cases}$$ By , for $j\in \{1, 2\}$ and $n=-1$, we get . When $p=2$, for $j\in\{1, 2\}$, by , since ${\tilde{\Omega}}_j^{\min}(2^e, \chi) = {\tilde{\Omega}}_j^{{\mathrm{new}}}(2^e, \chi)$, we get and .
Concluding the proof of Theorem \[THMmintf\]
--------------------------------------------
It follows from and that for any minimal character $\chi\pmod*{N}$, any $n\in\{\pm1\}$ and any pair of test functions of trace class $(g,h)$, we have $$\begin{aligned}
\label{THMmintfpf2}
\sum_{\lambda>0}&\operatorname{tr}T_n|_{{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)}
h\Bigl(\sqrt{\lambda-\tfrac14}\Bigr)=
\sum_{\langle M,\psi\rangle\in S_\chi'}
(-1)^{k'(N,M)}
2^{-k(N,M,\psi)}\,
\psi(n)\operatorname{Tr}^{{\mathrm{new}}}\bigl(M,\chi\psi^2\bigr),\end{aligned}$$ where $$\begin{aligned}
\label{THMmintfpf1}
\operatorname{Tr}^{{\mathrm{new}}}(N',\chi')
=
\sum_{M \mid \frac {N'}{\operatorname{cond}(\chi')}} \beta\left(\frac{N'/\operatorname{cond}(\chi')}{M}\right)
\sum_{\lambda>0}\operatorname{Tr}T_n|_{{\mathcal{A}}_\lambda(\chi'|_{M\operatorname{cond}(\chi')})} h\Bigl(\sqrt{\lambda-\tfrac{1}{4}}\Bigr).\end{aligned}$$ Now the inner sum in can be evaluated by using Theorem \[thm:STF\_N\] and compensating for the possible contribution from the Laplace eigenvalue $\lambda=0$. Specifically, we have ${\mathcal{A}}_0(\chi)={\mathbb{C}}$ if $\chi$ is the trivial character, otherwise ${\mathcal{A}}_0(\chi)=\{0\}$; therefore $\lambda=0$ contributes with a term $h(\frac i2)$ in the left hand side of if and only if $n=1$ and $\chi$ is trivial. This means that for $n=1$ and $\chi'$ trivial, when using the right hand side of to evaluate , we need to compensate by subtracting a term $(\sum_{M\mid N'}\beta(M))h(\frac i2)=\mu(N')h(\frac i2)$ from the resulting formula. This compensation has an effect in the right hand side of only if $n=1$, $\chi$ is trivial and $\operatorname{ord}_p(N)\leq2$ for all $p$; and if $\operatorname{ord}_p(N)=2$ for at least one $p$ then the compensations cancel each other out; hence it is only for $N$ squarefree that the net effect is nonzero. Using $h(\frac i2)=\int_{{\mathbb{R}}}g(u)\cosh(u/2)\,du$ we now see that the total contribution from these compensations is exactly the term $-\delta_{n=1}\mu(\chi)\int_{{\mathbb{R}}}g(u)\cosh(u/2)\,du$ appearing in the formula of Theorem \[THMmintf\].
Taking the results of Sections \[IMINSIEVE\]–\[Omega12MINSIEVE\] into account, we now see that the formula in Theorem \[THMmintf\] follows by using and to evaluate . Hence the proof of Theorem \[THMmintf\] is now complete. $\square$
Artin representations and $\Gamma(N)$ {#sec:galois}
=====================================
In this section we prove two lemmas that restrict the computations of twist-minimial spaces needed to prove Theorem \[thm:SEC\] for $\Gamma(N)$ and Theorem \[thm:Artin\].
\[lem:Gammaadmissible\] For any $N\in{\mathbb{Z}}_{>0}$, the Selberg eigenvalue conjecture holds for $\Gamma(N)$ if and only if ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)=\{0\}$ for all $\lambda\in(0,\tfrac14)$ and $\chi\pmod*{M}$ with $\operatorname{lcm}(M,\operatorname{cond}(\chi)^2)\mid N^2$.
Given $\lambda>0$, let ${\mathcal{A}}_\lambda(N)$ denote the space of Maass cusp forms of eigenvalue $\lambda$ that are invariant under the action of $\Gamma(N)$. Then, by the same type of argument as in [@Miy89 Lemma 4.3.1], we have the isomorphism $${\mathcal{A}}_\lambda(N)\cong\bigoplus_{\chi\pmod*{N}}{\mathcal{A}}_\lambda(\chi|_{N^2}),$$ where to a given $f\in{\mathcal{A}}_\lambda(\chi|_{N^2})$ we associate the function $z\mapsto f(N^{-1}z)$, which is an element of ${\mathcal{A}}_\lambda(N)$. In turn, for a given pair $\lambda>0$ and $\chi\pmod*{N}$, we have $${\mathcal{A}}_\lambda(\chi|_{N^2})=\sum_{\substack{M,d\in{\mathbb{Z}}_{>0}\\
\operatorname{cond}(\chi)\mid M\mid N^2\\d\mid\frac{N^2}{M}}}
\ell_{M,d}\,{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi|_M),$$ in the notation of §\[sec:prelim\].
By Lemma \[lem:twistconductor\], the space ${\mathcal{A}^{\mathrm{new}}}_\lambda(\chi|_M)$ is spanned by the Hecke eigenforms $f\otimes\psi$, where $f$ is twist minimial and $M=\operatorname{lcm}(\operatorname{cond}(f),\operatorname{cond}(\psi)\operatorname{cond}(\chi\overline{\psi}))$. Put $M'=\operatorname{cond}(f)$ and $\chi'=\chi\overline{\psi}^2|_{M'}$, so that $f\in{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi')$. Since $M\mid N^2$ and $\operatorname{cond}(\chi)\mid N$, we have $M'\mid N^2$ and $\operatorname{cond}(\chi')\mid N$.
Thus, if ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi')=\{0\}$ whenever $\lambda<\frac14$ and $\chi'\pmod*{M'}$ satisfies $M'\mid N^2$ and $\operatorname{cond}(\chi')\mid
N$ then the Selberg eigenvalue conjecture holds for $\Gamma(N)$. Conversely, we have ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi')\subseteq{\mathcal{A}^{\mathrm{new}}}_\lambda(\chi')$ so the Selberg conjecture for $\Gamma(N)$ implies that ${\mathcal{A}^{\mathrm{min}}}_\lambda(\chi')=\{0\}$ for $\lambda<\frac14$.
\[lem:Galoisadmissible\] Let $\rho:\operatorname{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})\to\operatorname{GL}_2({\mathbb{C}})$ be a nondihedral, irreducible Artin representation of conductor $N$, and let $\chi=\prod_{p\mid N}\chi_p$ be the Dirichlet character associated to $\det\rho$ via class field theory. If $p\mid N$ is a prime such that $\operatorname{ord}_p{N}\in\{1,\operatorname{ord}_p\operatorname{cond}(\chi)\}$, then $\chi_p$ has order $2$, $3$, $4$ or $5$. Further, if $p$ and $q$ are two such primes then $\chi_p\chi_q$ cannot have order $20$.
Let $\rho_p$ denote the restriction of $\rho$ to $\operatorname{Gal}\bigl(\overline{{\mathbb{Q}}}_p/{\mathbb{Q}}_p\bigr)$. Then $\rho_p$ factors through $G=\operatorname{Gal}(L/{\mathbb{Q}}_p)$ for some finite extension $L/{\mathbb{Q}}_p$. Let $G_i$, $i=0,1,2,\ldots$, denote the ramification subgroups of $G$, with $G_0$ the inertia group. Then $\rho_p$ and $\det\rho_p$ have conductor exponents $$e=\frac1{\#G_0}\sum_{i\ge0}\sum_{g\in G_i}\bigl(2-\operatorname{tr}\rho_p(g)\bigr)
\quad\text{and}\quad
s=\frac1{\#G_0}\sum_{i\ge0}\sum_{g\in G_i}\bigl(1-\det\rho_p(g)\bigr),$$ respectively. Note that the average of $\operatorname{tr}\rho_p$ over $G_i$ is the number of copies of the trivial representation in $\rho_p|G_i$. If this is nonzero then $\rho_p|G_i\cong\det\rho_p|G_i\oplus1$, from which it follows that the $i$th terms of the two sums above are the same. If $\rho_p|G_i$ does not contain the trivial representation then $$\frac1{\#G_i}\sum_{g\in G_i}\bigl(2-\operatorname{tr}\rho_p(g)\bigr)=2
>1\ge\frac1{\#G_i}\sum_{g\in G_i}\bigl(1-\det\rho_p(g)\bigr).$$ Thus, the $i$th term of the formula for $e$ is always $\ge$ the $i$th term of the formula for $s$, with equality if and only if $\rho_p|G_i$ contains the trivial representation. If $e\in\{1,s\}$ then equality must hold for every term; in particular, $\rho_p|G_0$ contains the trivial representation, so that $\rho_p|G_0\cong\det\rho_p|G_0\oplus1$. When $e=1$, this in turn implies that $s=1$, so $\det\rho_p|G_0$ is nontrivial.
Let $\bar\rho_p$ denote the composition of $\rho_p$ with the canonical projection $\operatorname{GL}(2,{\mathbb{C}})\to\operatorname{PGL}(2,{\mathbb{C}})$. Then when $e\in\{1,s\}$, the natural maps $\rho_p(G_0)\to\det\rho_p(G_0)$ and $\rho_p(G_0)\to\bar\rho_p(G_0)$ are isomorphisms. Since $\det\rho_p(G_0)$ is a nontrivial cyclic subgroup of ${\mathbb{C}}^\times$ and $\bar\rho_p(G_0)$ is a subgroup of $A_4$, $S_4$ or $A_5$, it follows that $\det\rho_p(G_0)\cong\bar\rho_p(G_0)$ is cyclic of order $2$, $3$, $4$ or $5$. Since the Dirichlet character $\chi_p$ associated to $\det\rho_p$ is determined by $\det\rho_p|G_0$, they have the same order, which implies the first claim.
Finally, order $4$ (resp. $5$) can only occur when $\rho$ is octahedral (resp. icosahedral). These possibilities are mutually exclusive, whence the second claim.
In the $A_5$ case, we may also take advantage of the fact that icosahedral representations occur in Galois-conjugate pairs that are not twist equivalent. Thus, assuming Artin’s conjecture, we can still rule out the existence of an icosahedral representation of a given conductor when our computation accommodates one representation (in total over all characters, modulo twist equivalence) but not two. We used this trick to rule out icosahedral representations with conductor $N\in\{2221,2341,2381,2529,2799\}$.
Numerical remarks {#sec:numerics}
=================
To prove Theorems \[thm:SEC\] and \[thm:Artin\], we applied the numerical method described in [@BS07 §4]. Briefly, we consider test functions of the form $$h(r)=\left(\operatorname{sinc}^2\!\left(\frac{\delta{r}}2\right)
\sum_{j=0}^{M-1}x_j\cos(j\delta{r})\right)^2,$$ where $$\operatorname{sinc}{r}:=\begin{cases}
\sin(r)/r&\text{if }r\ne0,\\
1&\text{if }r=0,
\end{cases}$$ $\delta=X/2M$ for $X,M\in{\mathbb{Z}}_{>0}$ and $x_0,\ldots,x_{M-1}\in{\mathbb{R}}$ are arbitrary. For each $\chi\pmod*{N}$ and $\epsilon\in\{0,1\}$, set $$m_\chi=\#\{\psi\pmod*{N}:\psi(-1)=1,\;\psi^2=1,\;
\operatorname{cond}(\psi)\operatorname{cond}(\chi\psi)\mid N\}$$ and let $n_{\chi,\epsilon}$ be a lower bound for the number of twist-minimal forms of character $\chi$ and parity $\epsilon$ arising from Artin representations; we use `PARI/GP` [@PARI] to compute the contribution from dihedral representations, as described in [@BS07 §3.2] (see also the source code at [@code]), and the data from Table \[tab:Artin\] for the rest. Then the quantity $$Q_{\chi,\epsilon}(x_0,\ldots,x_{M-1})
:=\frac1{m_\chi}\left[
\sum_{\lambda>0}\operatorname{tr}\tfrac12(T_1+(-1)^\epsilon T_{-1})|_{{\mathcal{A}^{\mathrm{min}}}_\lambda(\chi)}
h\Bigl(\sqrt{\lambda-\tfrac14}\Bigr)-n_{\chi,\epsilon}h(0)\right],$$ is a positive-definite quadratic form in the $x_j$. By standard trigonometric identities, the matrix of $Q_{\chi,\epsilon}$ is determined from the traces of $\operatorname{sinc}^4(\delta{r}/2)\cos(j\delta{r})$ for $0\le j\le2M-2$. We apply the trace formula to compute these, and then minimize $Q_{\chi,\epsilon}$ with respect to the constraint $\sum_{j=0}^{M-1}x_j=1$.
In every case, it turned out that the criterion from [@BS07 §4.3] applied, so that the optimal test function $h$ satisfied $h(r)\ge1$ for $r\in i{\mathbb{R}}$. As explained in [@BS07 §3.4], every non-CM form occurs with multiplicity $m_\chi$. Thus, since the CM forms satisfy Selberg’s conjecture,[^4] whenever the resulting minimal value of $Q_{\chi,\epsilon}$ is less than $1$, we deduce both the Selberg conjecture and the completeness of the list of nondihedral Artin representations for twist-minimal forms of character $\chi$.
To ensure the accuracy of our numerical computations, we used the interval arithmetic library [`Arb`](http://arblib.org/) [@Joh17] throughout. To handle the integral terms of the trace formula, for each basis function we first computed $\int_0^\infty g'(u)\log{u}\,du$ symbolically, which allowed us to replace $\log(\sinh(u/2))$ and $\log(\tanh(u/4))$ by the real-analytic functions $\log(\sinh(u/2)/u)$ and $\log(\tanh(u/4)/u)$, respectively. Thus, in every integral term, the integrand agrees with an analytic function on each interval $[j\delta,(j+1)\delta]$. After applying a suitable affine transformation to replace the interval by $[-1,1]$, we use the following rigorous numerical quadrature estimate of Molin [@Mol10]:
Let $f$ be an analytic function on an open neighborhood of $D=\{z\in{\mathbb{C}}:|z|\le 2\}$. Then for any $n\ge1$ we have $$\left|\int_{-1}^1 f(x)\,dx-\sum_{k=-n}^na_kf(x_k)\right|
\le\exp(4-5/h)\sup_{z\in\partial{D}}|f(z)|,$$ where $h=\log(5n)/n$, $a_k=\frac{h\cosh(kh)}{\cosh^2(\sinh(kh))}$ and $x_k=\tanh(\sinh(kh))$.
Note that the error term decays exponentially in $n/\log{n}$. To obtain a bound for $|f|$ on $\partial D$, we write $\partial D=\bigcup_{j=0}^{n-1}\{2e(\theta):\theta\in[j/n,(j+1)/n)\}$ and use interval arithmetic to bound $|f(2e(\theta))|$ on each segment.
Using the algorithm from [@BBJ], we computed the class numbers of ${\mathbb{Q}}(\sqrt{t^2\pm4})$ for all $t\le e^{20}$, which enables us to take $X$ as large as $40$ in the above. Taking $M=200$, various $X\le 40$ and $\chi$ as indicated by Lemmas \[lem:Gammaadmissible\] and \[lem:Galoisadmissible\] sufficed to prove Theorems \[thm:SEC\] and \[thm:Artin\].
[^1]: A. R. B. and M. L. were partially supported by EPSRC Grant `EP/K034383/1`. M. L. was partially supported by Royal Society University Research Fellowship “Automorphic forms, L-functions and trace formulas”. A. S. was supported by the Swedish Research Council Grant 2016-03360.
[^2]: The existence of this representation was first shown by Doud and Moore [@DM06], who also proved that $1951$ is minimal among *prime* conductors of even icosahedral representations.
[^3]: We correct for a misprint in [@Miy89 Theorem 6.6.6]: $\Omega'/p^{\nu+\rho+1}$ should be replaced by $\Omega'/p^{\nu+\rho}$
[^4]: For squarefree $N$, we actually remove the contribution from CM forms, following [@BS07 §3.3]. With more work one could generalize that approach to arbitrary $N$, but it turns out not to be necessary for our applications.
|
---
abstract: 'Electrocardiogram (ECG) is the most widely used diagnostic tool to monitor the condition of the cardiovascular system. Deep neural networks (DNNs), have been developed in many research labs for automatic interpretation of ECG signals to identify potential abnormalities in patient hearts. Studies have shown that given a sufficiently large amount of data, the classification accuracy of DNNs could reach human-expert cardiologist level. A DNN-based automated ECG diagnostic system would be an affordable solution for patients in developing countries where human-expert cardiologist are lacking. However, despite of the excellent performance in classification accuracy, it has been shown that DNNs are highly vulnerable to adversarial attacks: subtle changes in input of a DNN can lead to a wrong classification output with high confidence. Thus, it is challenging and essential to improve adversarial robustness of DNNs for ECG signal classification – a life-critical application. In this work, we proposed to improve DNN robustness from the perspective of noise-to-signal ratio (NSR) and developed two methods to minimize NSR during training process. We evaluated the proposed methods on PhysionNet’s MIT-BIH dataset, and the results show that our proposed methods lead to an enhancement in robustness against PGD adversarial attack and SPSA attack, with a minimal change in accuracy on clean data.'
author:
- |
Linhai Ma & Liang Liang\
Department of Computer Science\
University of Miami\
Coral Gables, FL 33146, USA\
`{l.ma, liang.liang}@miami.edu`\
bibliography:
- 'iclr2020\_conference.bib'
title: |
Improve robustness of DNN for ECG signal classification:\
a noise-to-signal ratio perspective
---
Introduction
============
Electrocardiogram (ECG) is widely used for monitoring the condition of cardiovascular system. After many years of residency training, a cardiologist becomes experienced in reading ECG graphs, detect abnormalities and classify signals into different disease categories. This is tedious and time-consuming. Researchers found out that deep neural networks (DNNs), especially convolutional neural networks (CNNs) can be trained for ECG signal analysis with excellent classification accuracy [@Kachuee2018] [@hannun2019cardiologist]. Therefore, with a sufficiently large amount of data and a carefully-designed network structure, DNN models could reach human expert cardiologist level for ECG signal classification, and the analysis for a patient can be done in a fraction of a second. For patients in developing countries where human-expert cardiologists are lacking, a DNN-based automated ECG diagnostic system would be an affordable solution to improve health outcomes.
However, recent studies have shown that despite the high classification accuracy of DNNs, they are susceptible to adversarial attacks in the form of small perturbations to input of the networks, and the perturbation is even imperceptible to human eyes and not expected (by humans) to change the prediction of DNNs [@Akhtar2018]. Adversarial attacks can be classified into two types based on whether the whole structure of the network is known by the attacker. It is a white box attack if the attacker knows the inner structure of the network, such as Fast Gradient Signed Method (FGSM) [@Goodfellow2015] and Projected Gradient Descent (PGD)[@madry2017towards]. It is a black box attack if the attacker has almost no knowledge of the inner structure of the network, such as transfer-based attack [@papernot2017practical] and SPSA [@uesato2018adversarial]. These attacks [@Akhtar2018] pose significant threats to the deep learning systems in sensitive and life-critical application fields such as ECG classification.
To improve DNN robustness, lots of effort has been made by researchers to develop defense methods. Currently, the most popular defense strategy is adversarial training. The basic idea of adversarial training is to added noise to the training samples and the noise is from specific adversarial attacks (e.g. PGD). Through adversarial training, the network can learn some features of adversarial noises and its decision boundary is modified so that it will become difficult to push the input across the decision boundary by adding a small amount of noise. Adversarial training is straightforward but has many problems. For example, generating adversarial samples is very time-consuming, and low-quality adversarial samples can be misleading and even reduce the classification accuracy of networks. Therefore, different adversarial training based defense methods were proposed [@Akhtar2018], which share the same basic idea and vary in how the adversarial samples are generated. Parallel to adversarial training, regularization terms can be added to loss function to reduce the sensitivity of network output with respect to the input. The regularization terms could be the gradient magnitude of loss with respect to input [@Ros2017], or Jacobian regularization [@jakubovitz2018improving].
Methodology
===========
In this paper, we proposed two new loss functions with regularization terms to improve robustness of neural networks for ECG signal classification. Our methods aim to reduce the effect of noises added to input, from the perspective of noise-to-signal ratio (NSR), which may make the network more robust. We evaluated our methods on PhysioNet MIT-BIH Arrhythmia ECG dataset [@moody2001impact] [@ECG]. The results from our experiment show that our proposed methods can achieve a significant improvement in network robustness against PGD white-box attack and SPSA black-box attack, and outperforms the standard adversarial training.
ECG Dataset
-----------
The PhysioNet MIT-BIH Arrhythmia ECG Dataset contains 109446 ECG heartbeat records, and there are 5 categories: N (Normal, Left/Right bundle branch block, Atrial escape and Nodal escape), S (Atrial premature, Aberrant atrial premature, Nodal premature and Supra-ventricular premature), V (Premature ventricular contraction and Ventricular escape), F (Fusion of ventricular and normal) and Q (Paced, Fusion of paced and normal and Unclassifiable). The dataset has been divided into a training set (87554 samples) and a testing set (21892 samples), which is publically available [@ECG]. We further divided the training set into a ’pure’ training set (70043 samples, 80%) and a validation set (17511 samples, 20%). The dataset has a large imbalance between classes, and we performed up-sampling to ensure that there are roughly the same number of samples in each class in the training set and testing set.
New loss functions to reduce noise-to-signal ratio
--------------------------------------------------
In this section, we will introduce two new loss functions to improve network robustness by reducing noise-to-signal ratio (NSR). The noise refers to the adversarial noise generated from an adversarial attack. The signal refers to the output of the classification network (the logits before the softmax layer). Our methods assume that the nonlinear activation function is ReLU or its variants, and the network may have convolution layers, fully-connected layers, pooling layers, batch-normalization layers, dropout layers, and skip-connections. Since a convolution layer is equivalent to a fully-connected layer with shared weights, we can convert a CNN into a multiple layer perceptron (MLP) in theory. Therefore, we will introduce our method using MLP only in this section. Given an input sample $x$ (a vector), the output of a classification network can be exactly expressed by a “linear” equation [@Ding2018]: $$z=W^Tx+b$$ where the weight matrix $W$ will be different for different $x$, and the bias vector $b$ will different for different $x$. The output is a vector$ z=\left[z_1,\ldots,z_5\right]^T $where $z_i$ is the output logit of Class $i$ and the total number of classes is 5 in this application. Let $w_i$ be the $i^{th}$ column of $W$ and $b_i$ be the $ i^{th}$ element of $ b$, then we have $$z_i=w_i^T x +b_i$$ The two new loss functions will be introduced in the following sub-sections using the above notations and equations.
### Loss1
Often, the bias terms of a DNN are very small, and they can be ignored. Therefore, equation (2) is simplified to $$z_y = w_y^T x$$ where $y$ is the true class label of $x$. $z_y$ is the dot product of two vectors, $w_y$ and $x$. Given the magnitude of $w_y$, the dot product reaches the maximum when the two vectors are aligned in the same direction. In other words, we can train a classification network such that $x$ is “memorized” by $w_y$, similar to a RBF network in some sense (deep RBF network is difficult to train [@Goodfellow2015]). Thus, we define a regularization term to encourage the alignment of the two vectors, which is given by $$R_1= || w_y - \gamma \frac {x}{x^T x}||_2^2$$ where $\gamma$ is a scalar coefficient of the unit vector in the direction of $x$. We combine the regularization term with mean square error (MSE) loss and margin loss for classification, given by $$loss_1(x,y)=(z_y-1)^2+\sum_{i\neq y} (z_i-0)^2+\sum_{i} max(0,1-z_y+z_i) + \beta_1 R_1$$ where $\beta_1$ is a scalar parameter to be determined on validation set. In this way, $\left|z_y\right|=\left|w_y^Tx\right|\rightarrow$1. As a result, $\gamma$ can be fixed to 1. The margin loss and the regularization term $R_1$ are only used for correctly-classified samples; and for wrongly-classified samples, $loss_1$ only contains MSE loss.
This new loss may improve robustness, which can be explained intuitively by Figure 3. Given a small magnitude of adversarial noise $\epsilon$ (a vector), to maximally change the output $z_y$, the direction of $\epsilon$ should be aligned with the direction of $w_y$. Assuming $\epsilon$ and $w_y$ are almost in the same direction, then $\epsilon$ and $x$ will be almost in the same direction because of $loss_1$, i.e., the noise may look like the input. Therefore, if the magnitude of $\epsilon$ is small, it may not alter $z_y$ significantly; and if the magnitude of $\epsilon$ is large enough to cause miss-classification, then $\epsilon$ becomes visually noticeable.
### Loss2
Here, we introduce the second loss which directly minimizes noise-to-signal (NSR) ratio. During an adversarial attack, a noisy vector $\epsilon$ is generated and added to the input x, and then the output becomes $$z_{y,\epsilon}=w_{y,\epsilon}^T(x+\epsilon)+b_{y,\epsilon}$$ If the noise $\epsilon$ is small enough, we can assume that: $w_y\approx w_{y,\epsilon}$ and $b_y\approx b_{y,\epsilon}$. This assumption is valid as long as the “on/off” states of ReLU units do not change significantly and pooling masks do not change significantly when adversarial noise is added. Therefore, $$z_{y,\epsilon} \approx w_y^T x+b_y+ w_y^T \epsilon = z_y+w_y^T \epsilon$$ Then, we define NSR and apply Hölder’s inequality: $$NSR_y=\frac{|w_y^T \epsilon|}{|z_y|} \le \frac{||w_y||_q . ||\epsilon||_p}{|z_y|}$$ where $\frac{1}{p}+\frac{1}{q}=1$. In this work, we focus on infinite norm $||\epsilon||_\infty = \epsilon_{max}$, and therefore $$NSR_y \le \frac{||w_y||_1 . \epsilon_{max}}{|z_y|} = R_2$$ We combine the regularization term $R_2$ with mean square error (MSE) loss and margin loss for classification, given by $$loss_2 (x,y)=(z_y-1)^2+\sum_{i\neq y} (z_i-0)^2+\sum_{i}{max (0,1-z_y+z_i)}+\beta_2 log(1+R_2)$$ In the experiment, $\epsilon_{max}$ is set to 1, and $\beta_2$ is determined on validation set. The margin loss and the regularization term $R_2$ are only used for correctly-classified samples; and for wrongly-classified samples, $loss_2$ only contains MSE loss.
Experiment and Results
======================
In the experiment, we applied the two proposed losses on two DNNs to evaluate their robustness. One of the networks is the CNN proposed in [@Kachuee2018], which has 30 layers in total. The other network is an MLP designed by us, and it has 8 layers. The structure of this MLP is (187-128)-RELU-(128-128)-RELU-(128-128)-RELU-(128-32)-(32-5). Based on the performance on validation sets, we set $\beta_1$ as 0.2 and set $\beta_2$ as 0.5 to obtain a good balance between robustness and accuracy. Number of epochs is 50, optimizer is “Adamax” and learning rate is 0.001.
To study the contribution of each term in the loss functions, we also performed ablation experiment. To obtain the baseline performance, the MLP/CNN were trained with Cross-entropy loss for 50 epochs, with “Adamax” optimizer, and learning rate of 0.001. The two DNNs achieved good performance in classification of ECG signals but are vulnerable to adversarial attacks, as shown in figures from Figure 1 to Figure 2. We also evaluated the other two defense methods: (1) Jacobian regularization [@jakubovitz2018improving] and (2) standard adversarial training using 10-PGD [@shafahi2019adversarial].
To evaluate the performance, we compare our proposed methods with other methods, which includes: MLP/CNN with Jacobian regularization [@jakubovitz2018improving] and MLP/CNN under 10-PGD standard adversarial training [@shafahi2019adversarial] with noise 0.1, 0.2 and 0.3, where 10 is number of steps of PGD attack to generate adversarial samples. In this experiment, there are totally 9 methods to be compared. They are: MLP/CNN with Cross-entropy loss denoted as “ce” in plots, MLP/CNN with mean square error (MSE) loss denoted as “mse” in plots, MLP/CNN with loss1 denoted as “loss1” , MLP/CNN with loss2 denoted as “loss2”, MLP/CNN with Jacobian regularization denoted as “jacob”, MLP/CNN with MSE and margin loss denoted as “mseMargin” and MLP/CNN trained with adversarial samples generated by 10-PGD adversarial attack under noise level $\epsilon$ denoted as “adv $\epsilon$”. The loss function of this adversarial training is [@Goodfellow2015]: $$loss_{adv} = 0.5 L_{CE}(x, y) + 0.5 L_{CE} (x_\epsilon, y)$$ where $L_{CE}$ is Cross-entropy loss and $x_\epsilon$ is adversarial sample. Training batch size is 128 in this experiment. Because we performed up-sampling to ensure that there are roughly the same number of samples in each class in the testing data, average recall is the same as overall accuracy. So, we only report overall accuracy of classification (denoted as “ACC”) and average precision across the five classes (denoted as “PREC”) as metrics of performance in this experiment.
100-PGD evaluation
------------------
First, we used PGD adversarial attack to test these methods. Projected Gradient Descent (PGD)[@madry2017towards] is regarded as the strongest first-order attack. K-PGD attacks clean input $x$ with K steps: $$x^k=\prod\left(x^{k-1}+\alpha\cdot s i g n\left(\nabla_xLoss\left(x^{k-1}\right)\right)\right)$$ where $\alpha$ is the step size and $x^k$ is the adversarial example from the $k^{th}$ step. If the noise added to input $x$ is larger than the given noise level $\epsilon$, PGD will project it back to the noise level $\epsilon$.
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In this experiment, we used 100-PGD to evaluate the methods. Figure 1 presents the results under 100-PGD attack, comparing our proposed methods to Jacobian regulation and 10-PGD adversarial training for MLP. It can be seen that MLP with our proposed loss2 can achieve the best accuracy and precision under all noise levels except for 0.01 (loss1 is better). Under attack on noise level 0.1, MLP with loss2 has accuracy of 61% and precision of 64%, which is the highest among all methods. Figure 2 presents the results under 100-PGD attack, by comparing our proposed methods to Jacobian regulation and 10-PGD adversarial training for CNN. CNN with loss2 has the best performance. Under the PGD attack with the noise level 0.1, CNN with loss2 can maintain an accuracy of more than 71% and a precision of approximately 78%. In this experiment, “adv0.2” and “adv0.3” are not as good as loss2 on all of the noise levels, which means these adversarially-trained CNNs are not strong enough to resist against 100-PGD attack. The other tests can be seen in Appendix.
Conclusion
==========
In this study, we proposed two methods to improve the robustness of deep neural networks for classification of ECG signals. Our methods aim to reduce the proportion of introduced noises in the output of the neural network, namely, noise-to-signal ratio. In this way, our methods can help reduce the effect of noise on the prediction of the network and thus improve the robustness against adversarial attack. The results of the experiment have shown that our proposed loss2, outperforms all other methods under white-box and black-box attacks (PDG and SPSA), for the classification task. We hope that our approaches may faciliate the development of robust and affordable solutions for automated ECG diagnosis for developing countries.
Appendix
========
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Appendix
========
In this experiment, we used 20-PGD to evaluate all methods. Figure 4 presents the results comparing our proposed methods to Jacobian regulation and 10-PGD adversarial training for MLP under 20-PGD attack. First, MLP with our proposed loss2 achieved the best accuracy and precision in general. Under attack on noise level 0.1, MLP with loss2 can still have an accuracy 69% and precision 75%, which is the highest. Second, performance of this adversarial training is not as good as loss2. We can see that there is an intersection of “adv0.1” and “adv0.2” around noise level 0.1 and an intersection of “adv0.2” and “adv0.3” around noise level 0.2. This means that model trained from adversarial samples under noise $\epsilon$ only behaves well around noise level $\epsilon$. Then, we can notice that both our proposed losses can improve the robustness of MLP against PGD attack while keeping a high accuracy and precision in classification of clean data. MLP with loss1 have accuracy 92% and precision 92% and MLP with loss2 can achieve accuracy 90% and precision 92%. MLP with “ce” has accuracy 92% and precision 93%, and the performance dropped quickly as noise level increased
Figure 5 presents the results comparing our proposed methods to Jacobian regulation and 10-PGD adversarial training for CNN under 20-PGD attack. CNN with loss2 has the best performance as well. Under the PGD attack on noise level 0.1, CNN with loss2 can still maintain an accuracy of 75% and a precision of 81%. We can see that “adv0.2” outperforms “loss2” around only noise level 0.2 and “adv0.3”, and outperforms “loss2” around only noise level 0.3, while on lower noise level it has a very weak resist to PGD attack (shown in Figure 3 (b)). This also demonstrates that model trained with adversarial samples with noise level $\epsilon$ only behaves well around noise level $\epsilon$, which is a weakness of the standard adversarial training.
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Appendix
========
We evaluated the proposed methods with a black-box attack, SPSA adversarial attack[@uesato2018adversarial]. SPSA is a gradient-free method, which is useful when the model is non-differentiable, or more generally, the gradients do not point in useful directions. This attack is applied with almost no knowledge of the inner structure of the network. It randomly generates samples around a given input sample (e.g. an image) of the target network, uses these random samples to estimate the gradient of the target network, and then the estimated gradient can be used for attacks. SPSA runs for 100 iterations with perturbation size 0.01, Adamax learning rate 0.01, and 2048 samples for each gradient estimation. SPSA attack is very computationally expensive, and therefore we only selected the first 162 samples from each class for this experiment.
From Figure 6, we can see that loss2 has the best performance in general for the test of MLP. In Figure 6 (a), “jacob” is the best from noise level 0.03 to noise level 0.05, slightly better than that of loss2. However, “jacob” falls sharply when the noise level is larger than 0.05. When noise level is 0.1, loss2 can still have an accuracy of about 51% while “jacob” method can only achieve 42%. As the noise level becomes higher, “jacob” drops below 10%, while loss2 is always above 20%. In Figure6 (b), we can see that loss2 outperforms “jacob” method under all noise levels. So, loss2 has a better resist against SPSA black-box attack than Jacobian regularization for MLP. In Figure 6 (a), “adv0.1” only outperforms loss2 on noise levels from 0.05 to 0.1, which again confirms that model trained with adversarial samples with noise level $\epsilon$ only behaves well around noise level $\epsilon$. In general, our proposed loss2 has a better performance against SPSA attack than all the other methods in comparing.
From Figure 7, we can see that under SPSA black-box attack, loss2 has the best performance in general. In Figure 7 (a), “jacob” method is outperformed by loss2. When noise level is 0.1, loss2 can still have an accuracy of about 60%, better than “jacob” method that can achieve only 52%. In Figure 7 (b), we can see that loss2 outperforms “jacob” methods almost on all noise levels, with precision of 74% at noise level 0.1. So, in the test on CNN, loss2 has a better resist against SPSA black-box attack than Jacobian regularization as well. In Figure 7 (a), “adv0.1” outperforms loss2 at noise level only from 0.01 to 0.1, and “adv0.2” outperforms loss2 only on noise levels from 0.1 to 0.2, which again confirms that model trained with adversarial samples with noise level $\epsilon$ only behaves well around noise level $\epsilon$. In general, our proposed loss2 has a better performance against SPSA attack than the other methods.
The good performance of our proposed loss2, under the black-box adversarial attack, also suggests that loss2 is not doing gradient obfuscation to improve the robustness of the target neural network.
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Appendix
========
The tables corresponding to figures from Figure 2 to Figure 7 are shown in this section.
Appendix
========
In this section, ECG signals with different levels of noise are shown. The attack is PGD-100 and the target network is MLP, which has been trained with Cross-entropy loss for 50 epochs. As we can see, when the noise level reaches 0.1 (shown in Figure 8 (f)), the ECG signal is even hardly recognizable by human eye.
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Appendix
========
In the experiment, loss funtions $loss_1$ and $loss_2$ run in mini-batches. Assume there are $k$ classes,$loss_1$ should be:
$$R_1= \sum_i || w_i - \gamma \frac {X}{X^T X}||_2^2$$
We combine the regularization term $R_1$ with mean square error (MSE) loss and margin loss for classification, given by
$$loss_1(X,Y)=\sum (Z-Y)^2+multi\_margin\_loss(Y, Z) + \beta_1 R_1$$
$loss_2$ should be:
$$R_2 = \sum_i^k \frac{||w_i||_1 . \epsilon_{max}}{|z_i|}$$
We combine the regularization term $R_2$ with mean square error (MSE) loss and margin loss for classification, given by
$$loss_2 (X, Y)=(Z-Y)^2+multi\_margin\_loss(Y, Z)+\beta_2 log(1+R_2)$$
|
---
abstract: 'The timing method, using either stellar pulsations or eclipse timing of close binaries as a clock, is proving to be an efficient way to detect planets around stars that have evolved beyond the red giant branch. In this article we present a short review of the recent discoveries and we investigate the potential of the timing method using data both from ground-based facilities as well as from the Kepler and CoRoT space missions.'
author:
- Roberto Silvotti
- 'R[ó]{}bert Szab[ó]{}'
- Pieter Degroote
- 'Roy H. Østensen'
- Sonja Schuh
title: The Potential of the Timing Method to Detect Evolved Planetary Systems
---
[address=[INAF-Osservatorio Astronomico di Torino, strada dell’Osservatorio 20,\
10025 Pino Torinese, Italy]{}]{}
[address=[Konkoly Observatory of the Hungarian Academy of Sciences, Konkoly Thege Miklós út 15-17, 1121 Budapest, Hungary]{}]{}
[address=[Instituut voor Sterrenkunde, K.U. Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium]{}]{}
[address=[Instituut voor Sterrenkunde, K.U. Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium]{}]{}
[address=[Georg-August-Universität Göttingen, Institut für Astrophysik, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany]{}]{}
Introduction
============
The large majority of the almost 500 extrasolar planets detected so far[^1] orbit main sequence (MS) stars, and these planets were detected mostly using radial velocities (RVs) or transits. For various reasons, not only technical, current exoplanet research is strongly oriented towards MS stars, with the main goal of finding rocky analogs of our Earth, and towards understanding planetary system formation, while the late-stage evolution after the MS remains poorly understood. The first attempts to model the evolution of the planetary systems after the red giant phase are very recent (Villaver & Livio 2009 and ref. therein), and in particular lack observational constraints. The two main methods to detect exoplanets, RVs and transits, are not efficient for compact stars with high gravities and small radii, such as the extreme horizontal branch stars and the white dwarfs. An alternative method for these stars is given by the timing method, in analogy with the first pulsars’ planet discoveries [@Wolszczan+1992]. This method is based on the reflex motion of the parent star due to the companion, which changes periodically the star-observer distance, causing a delay (or advance) on the arrival time of the photons. In order to use the timing method, what we need is a precise astrophysical clock. For the pulsars’ planets, this is given by the pulsar’s radio signal modulated by the fast and stable rotation of the neutron star. But other potentially precise clocks exist in astrophysics and are given, for example, by stellar pulsation timing or the eclipse timing in close binary systems. An early example of application of the timing method is given by Paparó et al. (1988), who calculated the orbital elements of an unseen stellar companion to the Delta Scuti star SZ Lyncis using the O–C technique (Sterken 2005) on the pulsation timing. The method has also been discussed for pulsating white dwarfs by Kepler et al. (1991) and Provencal (1997).
The next sections contain an introduction to the field of post-RGB planetary systems and a short review of the recent discoveries, respectively. Then we investigate the potential of the timing method to find new planetary systems around evolved stars using pulsation or eclipse timing, with particular attention to the ongoing space missions CoRoT and Kepler. Finally, in the last section, a few conclusions are given and future perspectives are discussed.
Post-RGB evolution of planetary systems
=======================================
During the red giant (RG) expansion of a star, two main opposite phenomena influence the orbits of potential planetary companions: mass loss from the star, which lowers the binding energy and hence tends to cause an outward drift of the planet, and tidal effects, which tend to attract the planet close to the star and eventually inside the envelope. If the planet enters the stellar envelope, it is not clear in which conditions it can survive and what can be the influence on the subsequent evolution of the star. On the other hand, we know that during the RG phase a small fraction of stars, of the order of 1%, loose almost all their envelope and are left with a very thin H envelope during their core He burning phase. These are the so-called subdwarf B (or sdB) stars, that are located on the extreme horizontal branch (EHB), at high effective temperatures ( 20,000K). At least 50% of them are in binary systems and in this case the formation of an sdB star can be explained in terms of binary evolution (Han et al. 2002, 2003). However, for the remaining fraction of apparently single sdB stars, it is more difficult to explain why the huge mass loss during the RGB occurred (see review by Heber 2009). In this context planets might play a role [@Soker1998].
Similar phenomena of planetary migration, planet engulfment and planet accretion from stellar ejected material can happen also during the second large expansion of a star, when it enters the asymptotic giant branch and the thermal pulses phase, and during the planetary nebula (PN) ejection.
In order to achieve consistent modelling of all these complex phenomena, tests against observational constraints are essential. To detect planets around white dwarfs is (or more exactly will be) a key step to study the late-stage evolution of planetary systems. Equally important is to identify planetary systems in previous shorter and more rare evolutionary phases like the RGB or the horizontal branch. Only in this way can we try to disentangle the various phenomena that changed the configuration of the planets and their orbits, with the final goal of being able to trace back the entire history of a planetary system from the final white dwarf configuration to the original protoplanetary disk. What we need first is to find a sufficient number of planetary systems orbiting post-RGB stars and white dwarfs, covering a wide range of orbital parameters and planetary masses.
Concerning white dwarfs, what we know already is that some of these stars are surrounded by metal-rich debris disks: at least 17 white dwarfs show circumstellar dust at $\sim$ 1,000 K (Reach et al. 2005 and ref. therein, Farihi et al. 2010) and about 20% of them show also Ca[ii]{} emission from a hotter gaseous disk (Gänsicke et al. 2006, 2010). These disks are generally very close to the white dwarf, within about 1 [R$_{\odot}$]{}. Recently, also much more distant and cooler debris disks have been detected (Su et al. 2007, Chu et al. 2010 and Bilíková et al. 2010), which are probably the final evolution of the solar system’s Kuiper belt analogues. In other words, we don’t know yet how many white dwarfs host planets and at which distances from the star, but we know that at least the material to build planets is there.
Another interesting possibility is that second (or even third) generation planets can form from the stellar material ejected during stellar evolution (Perets et al. 2010). But again, to confirm these theoretical scenarios, we need observations and statistics.
Recent detections of post-RGB planetary systems
===============================================
From the observational side, the late-stage evolution of the planetary systems is a very recent field and only in the last three years a few first post-RGB planetary systems were detected. They belong mainly to two classes:
1. Extreme horizontal branch hot subdwarfs B:\
V391 Peg b [@Silvotti+2007],\
HW Vir b and c [@Lee+2009],\
HS0705+67003 b [@Qian+2009],\
HD149382 b (Geier et al. 2009, but see also Jacobs et al. 2010).
2. Cataclysmic variables:\
NN Ser b and c (Qian et al. 2009b, Beuermann et al. 2010a, Hessman et al. 2010),\
DP Leo b (Qian et al. 2010a, Beuermann et al. 2010b) and\
QS Vir b (Qian et al. 2010b).
We note that the issue of nomenclature for circumbinary planets, in contrast to the nomenclature for planets around components of visually resolved binaries, is unsolved. On recommendation by the *A&A* editor, Beuermann et al. (2010a, 2010b) follow the rule by which any unseen companions, whether stellar or sub-stellar, are designated by lowercase letters. They hence have introduced (ab) for the two components of the central binary and start numbering the third bodies with the letter c, resulting in the planet designations NN Ser (ab) c, NN Ser (ab) d, and DP Leo (ab) c.
Another interesting new planet candidate was presented during this meeting by Setiawan et al. 2010a (a 1.3 M$_{\rm Jup}$ planet at 0.116 AU from a red HB star of likely extragalactic origin).
In Table 1 we summarize the list of known post-RGB planets, including the pulsars’ planets and a few brown dwarfs (BDs) near the planet mass limit. We include also the brown dwarf WD0137$-$349b because its very small orbital distance, only 0.375 [R$_{\odot}$]{}, implies that the BD was engulfed by the red giant progenitor of the white dwarf (Maxted et al. 2006). As one can see, most of the detections reported in Table 1 were obtained through timing measurements.
----------------- ----------- ---- --- -------- ------ --- --------------------- ------ --- -------- -------- ------------------------- ----------------------- ----------------------------
PSR1257$+$12b $M\simeq$ 6e$-$5 0 . 19 25 . 262d 0 pulsar timing (radio signal) Wolszczan & Frail 1992
PSR1257$+$12c $M\simeq$ 0 . 014 0 . 36 66 . 5419 d 0.0186 timing (radio signal) Konacki & Wolszczan 2003
PSR1257$+$12d $M\simeq$ 0 . 012 0 . 46 98 . 2114 d 0.0252 timing (radio signal)
PSRB1620$-$26b 2 . 5 23 100 pulsar+WD timing (radio signal) Thorsett et al. 1993
in globular cluster Sigurdsson et al. 2003
WD0137$-$349b $M\simeq$ 55 0 . 375 [R$_{\odot}$]{} 1 . 927h WD RVs Maxted et al. 2006
GD66 b (?) $M<$7: 2 . 4 2 . 75 5 . 7 0 WD (DAV) timing (pulsations) Mullally et al. 2008, 2009
GD356 b (?) $M<$12 $>$2 . 7h WD, magnetic inferred from Wickramasinghe et al. 2010
Zeeman splitting
V391Pegb 3 . 2 1 . 7 3 . 2 0 EHB (sdBV$_{\!\rm rs}$) timing (pulsations) Silvotti et al. 2007
HWVirb 19 . 2 5 . 3 15 . 8 0.46 EHB (ecl. sdB+M) timing (eclipse) Lee et al. 2009
HWVirc 8 . 5 3 . 6 9 . 1 0.31 timing (eclipse)
HS0705+67003b 39 . 5 $<$3 . 6 7 . 15 EHB (ecl. sdB+M) timing (eclipse) Qian et al. 2009a
HD149382b (?) $M=$8– 23 5–6 . 1 [R$_{\odot}$]{} 2 . 391d EHB (sdB) RVs Geier et al. 2009
Jacobs et al. 2010
NNSerb 6 . 9 5 . 4 15 . 5 0 pre-CV timing (eclipse) Qian et al. 2009b
NNSerc$^{\ast}$ 2 . 3 3 . 4 7 . 7 0.2 timing (eclipse) Beuermann et al. 2010a
Hessman et al. 2010
DPLeob 6 . 05 8 . 2 28 . 0 0.39 CV (polar) timing (eclipse) Qian et al. 2010a
Beuermann et al. 2010b
QSVirb $M\simeq$ 6 . 65 4 . 2 7 . 86 0.37 CV, hibernating timing (eclipse) Qian et al. 2010b
HIP13044b 1 . 25 0 . 116 16 . 2d 0.25 RHB RVs Setiawan et al. 2010a
(extragal. orig.?) Setiawan et al. 2010b
----------------- ----------- ---- --- -------- ------ --- --------------------- ------ --- -------- -------- ------------------------- ----------------------- ----------------------------
: Post-RGB planets[]{data-label="tab:a"}
A circumbinary planet with a minimum mass of about 5 M$_ {\rm Jup}$ could be the reason of the periodic component in the O–C plot of the eclipsing polar HU Aquarii (Schwarz et al. 2009, not included in Table 1). O’Toole et al. (2010) also report early results from their on-going RV survey. We refer to their proceeding papers for more details.
For what concerns the white dwarfs, until now there are no clear detections of planets around single white dwarfs, with only two candidates: GD66 (Mullally et al. 2008, 2009) and GD356 (Wickramasinghe et al. 2010). We know at least three planet hosting MS stars having a wide binary WD companion: HD13445=Gl86, HD147513 and HD27442 (Desidera & Barbieri 2007 and ref. therein). In all these three systems the planet orbit the MS star at an orbital distance much smaller than the WD distance: 0.11 versus 19 AU for HD13445, 1.26 versus at least 100 AU for HD147513, and 1.18 versus 240 AU for HD27442. Therefore it is unlikely that the WD had an influence on the evolution of the planetary system and this is why these systems are not included in Table 1. More interesting (although quite exotic) is the case of the circumbinary planet orbiting the pulsar+WD system PSR 1620$-$26 (Thorsett et al. 1999, Sigurdsson et al. 2003) in the globular cluster M4. Following Sigurdsson et al. (2003), the most likely scenario is that the planet was initially in orbit around the MS star progenitor of what is now the WD companion. The system underwent an exchange interaction with a neutron star binary, with the MS star replacing the original high-mass WD companion to the neutron star. The end results were a new eccentric orbit for the MS star, and the displacement of the planet onto a wide circumbinary orbit.
Ongoing projects and the potential of CoRoT and Kepler data
===========================================================
Similarly to the astrometric method (and contrary to the RV and transit methods), the timing method is sensitive to large orbital separations. Equation (1) shows that the (maximum) change in the arrival time of the photons (or time shift) $\tau$ is indeed proportional to the orbital distance $a$ ($m_p$, $M_{\star}$, $c$ and $i$ are, respectively, planet mass, stellar mass, light speed and inclination of the orbit to the line of sight):
$$\tau~=~\frac{a}{c}~~\frac{{m_p\,\,{\rm sin}\,i}}{M_{\star}}$$
For $a$=1AU, $m_p$=1M$_{\rm Jup}$, $M_{\star}$=1[M$_{\odot}$]{}(and $i$=90$^\circ$), the time shift is about 0.5 seconds; while it is 1s for a stellar mass of 0.5[M$_{\odot}$]{}, as in the case of a typical sdB star or a slightly lower-than-average-mass white dwarf.
As shown by the recent discoveries in the previous section, it is possible to detect planets with the timing method using ground-based data. What is needed is a long-term photometric monitoring of the targets for many years with frequent runs, ideally each month or two. A few ground-based programs to search for post-RGB planets exist, e.g.the EXOTIME program (Schuh et al. 2009, Schuh 2010, Lutz et al. 2010[^2]) which is monitoring five sdB pulsators, a similar program at the McDonald Observatory on 15 ZZ Ceti pulsators [@Mullally+2008], and the long-term monitoring of one DBV pulsator by Dalessio et al. (2010). A secondary goal of these programs is the measurement of the secular variation of the pulsation periods, which is related to the change of the stellar structure because of its evolution (e.g. Kepler et al. 2005, Costa & Kepler 2008, Silvotti et al. 2007). Similar ground-based programs exist also for the eclipsing binaries.
However it is clear that, potentially, the maximum results can be achieved using continuous monitoring from space craft. The exceptional photometric precision and duty cycle of the CoRoT and Kepler satellite data represent a unique opportunity to use the timing method to detect low mass substellar companions to pulsating stars and eclipsing binaries. In particular, Kepler has a longer coverage, up to 3.5 years, as the satellite will observe the same $\sim$105 square degree field for the whole mission duration; while the CoRoT “long runs” have a maximum duration of about 6 months. For more details concerning the Kepler and CoRoT missions we refer the reader to Borucki et al. (2010), Gilliland et al. (2010) and Auvergne et al. (2009). The main limitation of Kepler is the relatively long sampling time of 58 s (short-cadence data). Although the photometer samples the field every 6.54 s, telemetry restrictions do not permit to save and transmit to the ground the original data. This is a severe limitation only for the very rapid pulsators like the short-period sdB stars of the sdBV$\!_{\rm r}$ class and the white dwarfs, which can have pulsation periods as short as $\approx$1–2 minutes. However, in the survey phase (1st year of observation) that has just ended, no pulsating white dwarfs were found and only one short-period sdB pulsator was observed (Kawaler et al. 2010) plus an hybrid pulsator with a very rich amplitude spectrum in a eclipsing binary system with a M dwarf (Østensen et al. 2010a, see also end of this section on [*5. Eclipse timing*]{}).
For all the other 13 long-period sdB pulsators (sdBV$\!_{\rm s}$) discovered by Kepler (Østensen et al. 2010b, 2010c), with typical periods between 30 min to few hours, the 58 s sampling is largely sufficient, even though longer pulsation periods mean higher minimum detectable mass of a substellar companion. This is shown by the following equation (2), in which $\sigma_\tau$ is the timing error, $N$ is the number of data points, $P$ the pulsation period, $\sigma_I$ the photometric (relative intensity) error, $A$ the pulsation amplitude and $\sigma_A$ the amplitude error, which decreases with the square root of the length of the run and can be estimated directly by measuring the mean noise of the Fourier amplitude spectrum. Equation (2) is valid under the assumption of uncorrelated errors and no cross terms between the different pulsation frequencies, and is obtained multiplying by $P$ equation (A15) from Breger et al. 1999 (see also Montgomery & O’Donoghue 1999 and Silvotti et al. 2006 for more details).
$$\sigma_\tau=\frac{P}{2\pi}~\sqrt{\frac{2}{N}}~\frac{\sigma_I}{A}
=\frac{P}{2\pi}~\frac{\sigma_A}{A}$$
A similar equation can be obtained also for the eclipse timing (Doyle & Deeg 2002), assuming a simple model of triangular eclipse (which implies that the error in eclipse minimum timing $\sigma_\tau$ is a lower limit to the real error). In the following equation (3), $T_{\rm ecl}$ is the duration of the eclipse from first to last contact, $\Delta I$ is the relative intensity of the eclipse (or (1-$\Delta I$) is the eclipse’s relative depth, with $I$=1 out of eclipse) and $N_{\rm ecl}$ is the number of data points taken during $T_{\rm ecl}$; $\sigma_I$ was defined above.
$$\sigma_\tau=\frac{T_{\rm ecl}}{2\sqrt{N_{\rm ecl}}}~\frac{\sigma_I}{(1-\Delta I)}$$
As we do not expect to find close-in planets with short orbital periods around post-RGB stars[^3], we will need many months/years of Kepler data to be able to set limits on the presence of planets around these stars. At the moment, with the available Kepler data, we can only start making some tests to see what the typical phase errors are that we can obtain and therefore what we can expect in terms of minimum detectable masses and orbital periods. The first goal of these preliminary tests is to verify the phase stability (or “phase coherence”) of different classes of pulsating stars and eclipsing binaries.
Regarding the pulsators, the most promising targets are those with stable pulsations (in terms of period, amplitude and phase), short periods (to reduce the minimum detectable mass), several independent periods and relatively large amplitudes. To have several independent periods is important as it permits to check whether the same periodic pattern produced by a planet is present in all the periods (see Dalessio et al. 2010 for an example where this is not the case!), helping to exclude false detections due to other periodic effects (e.g. beating of close frequencies). SdB pulsators fit well these constraints and, due to the relatively short periods, are the best candidates. Other potential targets are the Delta Scuti stars, $\beta$ Cephei and RR Lyrae. In the next pages we show the results of a few preliminary tests on these pulsating stars using Kepler and CoRoT data, including also an eclipsing binary with a pulsating sdB component. The Kepler data were retrieved from the KASC data base (KASC is the Kepler AsteroSeismic Consortium, Gilliland et al. 2010). The CoRoT public data were downloaded from the CoRoT archive at LAEFF (Laboratory for Space Astrophysics and Theoretical Physics) in Spain.
[**1. Subdwarf B stars**]{}
Before the Kepler observations, the oscillating sdB stars could be divided into two groups: the short-period (p-mode) pulsators (or sdBV$\!_{\rm r}$=rapid pulsators) of the V361 Hya class from the prototype (Kilkenny et al. 1997), with effective temperatures between about 27,000 and 35,000 K, and the longer period (g-mode) pulsators (or sdBV$\!_{\rm s}$=slow pulsators) of the V1093 Her class (Green et al. 2003), with effective temperatures between $\sim$21,000 and $\sim$28,000 K. Four hybrid pulsators (sdBV$\!_{\rm rs}$) were known, showing both p- and g-modes at the same time (Schuh et al. 2006, see also Baran & Fox-Machado 2010 for the complete list of hybrid pulsators before Kepler). This classification (compare also Kilkenny et al. 2010) may need some adjustment because the Kepler observations have shown that the temperature range of the two instability strips and the fraction of hybrid pulsators could be larger than previously believed.
In order to have a first feeling of what can be obtained with the timing technique on Kepler sdB data, we consider the star KIC010139564, the only rapid sdB pulsator observed by Kepler [@Kawaler+2010], and one of the best candidates to search for planets. Figure 1 shows the O–C plot of six pulsation frequencies: these frequencies are considered also in Figs.4 and 5 of Kawaler et al. (2010).
![O–C plots of 6 pulsation periods of the short-period sdB pulsator KIC10139564 (see Kawaler et al. 2010 for more details). Each point represents about 7.5 days of short cadence data. The phase stability of all these periods suggests that substellar companions can be detected. The small error bars of the two periods with higher amplitude (173.60 and 182.72 s) allow to detect a planet of about 7 M$_{\rm Jup}$ at 1 AU from the star. Using longer subsets, this limit can be reduced to about 2 M$_{\rm Jup}$ and even further (see text for more details).[]{data-label="f1"}](silvotti_talk_fig1.ps){width="\textwidth"}
First of all, we see from Fig.1 that the frequencies have in general stable phases. This is true not only for these six frequencies but also for almost all the other frequencies of this star, although they are not represented here. By comparing Fig.1 with Fig.4 of Kawaler et al. (2010), we see that, for most of these frequencies, the error bars are smaller by a factor of (7.5/2)$^{0.5}\simeq$1.9, which is exactly what we expect considering that each point in Fig.1 represents $\sim$7.5 days of data while in Kawaler’s plot each point corresponds to 2 days. However, at least one frequency at 5760.2 $\mu$Hz behaves differently: the error bar of Fig.1 is larger than the one obtained by Kawaler et al. (2010). The reason is in their Fig.5, where we see that both phase and amplitude are changing periodically with a phase difference of about 90 degrees, suggesting that beating between closely spaced stable oscillation frequencies is occuring. From the smaller error bars of Fig.1 we can estimate the minimum mass of a detectable planet: we obtain about 7 M$_{\rm Jup}$ for an orbital distance of 1 AU. However, potentially, the minimum mass can be reduced to about 2 M$_{\rm Jup}$ using longer subsets of about 3 months each. This limit could be further extended to about 0.7 M$_{\rm Jup}$ extrapolating from the small error bars in Fig.4 of Kawaler et al. (2010), provided that the new Kepler data that will be collected in the next months/years will actually allow to separate the close frequencies and resolve the amplitude spectrum of this star.
For the long-period sdB pulsators, the first observations have shown that in general the phases are stable allowing to use the timing method also for these stars. For them, the minimum detectable mass of a substellar companion should be $\approx$10–50 times higher than the numbers reported above considering that the periods are at least one order of magnitude longer. However, some of these stars show also a few peaks at high frequency, that are probably due to a few excited p-modes (Reed et al. 2010, Charpinet et al. 2011). If these short-period p-modes will be confirmed by the future Kepler observations, they could bring down again the minimum detectable masses of the potential companions.
[**2. Delta Scuti stars**]{}
Another interesting class of pulsators are the Delta Scuti stars. Although the periods are relatively long and comparable with those of the long-period sdB pulsators, their amplitudes can be larger, allowing to reduce the timing errors to a few seconds (see equation 2). An example of a Delta Scuti star with a relatively simple amplitude spectrum (not too many frequencies) and short periods is given in Fig.2, where we see that the phases are stable, at least during the 1-month Kepler run in short cadence.
![An example of O–C plots of a Delta Scuti star observed by Kepler. Each point represents about 7 days of short cadence data. The phase errors of the two main pulsation frequencies F1 and F2, of the order of $\pm$4 and $\pm$10 s respectively, suggest that Delta Scuti stars can be stable enough to use the timing technique. The fractional amplitudes of the two frequencies are 0.25% and 0.17% respectively.[]{data-label="f2"}](silvotti_talk_fig2.ps){width="\textwidth"}
[**3. $\beta$ Cephei stars**]{}
The timing technique can also be used for $\beta$ Cephei stars. Here we show an example on the $\beta$ Cephei star HD 180642 (V1449 Aquilae), illustrating that these stars also may have stable phases.
HD 180642 was observed by the CoRoT satellite during 156 days of the long run LRc01, with a sampling cadence of 32s. The CoRoT data were analyzed in detail by Degroote et al. (2009), who found a complex amplitude spectrum with several combination frequencies. We applied a least-square sinusoidal fit to the CoRoT data using 15 frequencies obtained from pre-whitening, including a few combinations. In Figure 3 (upper panel) a preliminary O–C plot shows an apparent phase variation of the order of 20s in 2 months, whose origin and nature is not clear. If this effect were due to a secular variation of the pulsation period, we would obtain [$\dot{\rm P}$]{} $\approx$ $4\times$10$^{-9}$ s/s. This number is significantly larger than the value found by Degroote et al. (2009) from direct measurement: about 1.3$\times$10$^{-9}$ s/s. This difference deserves further study.
![The upper and lower panels represent the phase and (fractional) amplitude variations, respectively, of the main pulsation frequency F=5.486889 c/d (P=4.374063 h) of the $\beta$ Cephei star HD 180642 (V1449 Aquilae). Each point represents about 17.4 days of the CoRoT long run LRc01 (156 days). The two panels suggest that phase and amplitude variations are not correlated and therefore these variations can not be attributed to beating of close frequencies. See Degroote et al. (2009) for more details on the pulsational behavior of this star.[]{data-label="f3"}](silvotti_talk_fig3.ps){width="\textwidth"}
[**4. RR Lyrae stars**]{}
From the KASC data base we selected two RR Lyrae stars with the longest observational coverage (about 6 months). The O–C diagram of these RR Lyrae stars, shown in Fig.4, indicates that the phase errors are generally large and, even more important, they are highly inhomogeneous. The reason of this behaviour is not clear. Although before the Kepler observation, both stars were considered as non-Blazhko stars, the non-uniform error distribution might be due to a low-level Blazhko modulation of phases and/or frequencies. Noise due to convection might also play a role. Although a deeper testing on a larger number of RR Lyrae stars is needed before drawing conclusions, this first example suggests that RR Lyrae might not be stable enough for the timing technique.
![O–C diagram of two RR Lyrae stars observed by Kepler in long cadence (a measurement each 30 min). Each point represents about 20 days of data. The vertical scale is different in each panel. The reason of the large and inhomogeneous phase errors is not clear (see text for more comments).[]{data-label="f4"}](silvotti_talk_fig4.ps){width="\textwidth"}
[**5. Eclipse timing**]{}
In this section we show an example of eclipse timing. We note that this method is related to the search for transit timing variations. The target, 2M1938+4603 (or KIC09472174), is of particular interest because it is a sdB+dM eclipsing binary system and the sdB component shows a very rich pulsational spectrum with periods ranging from $\sim$4 min to 5.5 hours. The orbital period of the system is 0.12576530 d. For more details on 2M1938+4603 we refer the reader to the recent article by Østensen et al. (2010a). The eclipsing binary systems with a pulsating component, like this one, have the advantage of having two independent clocks given by the eclipse timing and by the pulsations (provided that there is no coupling between them). For this reason they are an ideal test case to verify the potential of the timing method in finding low-mass substellar objects.
![O–C diagram of the eclipsing sdB+dM binary 2M1938+4603 (KIC09472174), based on eclipse timing. The times of the primary minima are taken from Østensen et al. (2010a). With a binary period of 0.12576530 d, the upper panel corresponds to 695 days of ground based eclipse timings, with the Kepler 9.7-day commissioning data around cycle 2500. The latter are shown in more detail in the lower panel. The noise in the lower panel (and the detectable planetary mass) can be reduced by fitting several eclipses at once. To have an idea of the realistic timing errors, if we consider all the 77 Kepler eclipses in the lower panel, we obtain an O–C standard deviation of 1.8 s. With a 2 years (or longer) observation, and using 2.5 months of data for each point in the O–C plot, the 4 $\sigma$ detection limit would be reduced to about 4 M$_{\rm Jup}$ considering an orbital distance of 1 AU from the binary. Potentially, the presence of a planet could be independently confirmed also from the pulsations of the sdB star.[]{data-label="f5"}](silvotti_talk_fig5.ps){width="\textwidth"}
Conclusions and future perspectives
===================================
In the last three years, the first discoveries of planets orbiting evolved stars after the red giant branch have proven that the timing method is an efficient way to detect substellar objects around compact stars, using either the stellar pulsation or the eclipse timing as a clock. This method, which is sensitive to Jupiter-mass planets in large orbits of $\approx$1 AU or more, requires long-time photometric monitoring with frequent, or, better still, continuous, runs. In order to validate the method, it would be important to confirm these discoveries with other independent techniques. This in general is not easy because the targets are generally faint and many have spectral lines broadened by high gravity.
The high quality data from the CoRoT and Kepler satellites represent a unique opportunity due to the exceptional duty cycle. Kepler has the additional advantage of longer runs, up to 3.5 years. Preliminary tests show that sdB stars are the best candidates, in particular the short-period pulsators. Other good candidates are the Delta Scuti stars, at least those with a relatively simple amplitude spectrum (not too many frequencies). An example on a $\beta$ Cephei star shows that also these stars may have sufficiently stable phases, although the higher stellar mass implies a higher minimum mass of the potential substellar companions. RR Lyrae stars are more problematic and our preliminary tests suggest that phases and/or frequencies may have significant variations on time scales of weeks. An example of an eclipsing binary is given and shows that also for these targets Jovian planets can be detected. Moreover, if the binary system has a pulsating component like 2M1938+4603, the pulsations can be used, potentially, as an independent clock to confirm the detections.
R.S. acknowledges financial support from the organizers, who waived the conference fees.
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[^1]: 494 from the Extrasolar Planets Encyclopaedia (<http://exoplanet.eu/>) on October 12th, 2010.
[^2]: see also [<http://www.na.astro.it/~silvotti/exotime/>]{}
[^3]: And in any case the timing method is not very sensitive to small orbital distances, see equation (1).
|
---
abstract: |
We present a study of analytic models of starless cores whose line profiles have “infall asymmetry,” or blue-skewed shapes indicative of contracting motions. We compare the ability of two types of analytical radiative transfer models to reproduce the line profiles and infall speeds of centrally condensed starless cores whose infall speeds are spatially constant and range between 0 and 0.2 km s$^{-1}$. The model line profiles of [HCO$^{+}$]{} () and [HCO$^{+}$]{} () are produced by a self-consistent Monte Carlo radiative transfer code. The analytic models assume that the excitation temperature in the front of the cloud is either constant (“two-layer” model) or increases inward as a linear function of optical depth (“hill” model). Each analytic model is matched to the line profile by rapid least-squares fitting.
The blue-asymmetric line profiles with two peaks, or with a blue shifted peak and a red shifted shoulder, can be well fit by one or both of the analytic models. For two-peak profiles is best matched by the “[hill5]{}” model (a five parameter version of the hill model), with an RMS error of 0.01 km s$^{-1}$, while the “[twolayer6]{}” model underestimates the infall speed by a factor of $\sim 2$. For red-shoulder profiles, the [hill5]{} and [twolayer6]{} fits reproduce infall speeds equally well, with an RMS error of 0.04 km s$^{-1}$. The fits are most accurate when the line has a brightness temperature greater than 3 K. Our most accurate models tend to reproduce not only the line profile shape, but also match the excitation conditions along the line of sight. A better match to the line profile shape does not necessarily imply a better match to the infall speed and provide guidance on how to minimize the risk of obtaining a poor infall speed fit.
A peak signal to noise ratio of at least 30 in the molecular line observations is required for performing these analytic radiative transfer fits to the line profiles. Moderate amounts of depletion and beam smoothing do not adversely affect the accuracy of the infall speeds obtained from these models.
author:
- 'Christopher H. De Vries and Philip C. Myers'
title: Molecular Line Profile Fitting with Analytic Radiative Transfer Models
---
Introduction
============
There are four distinct stages in isolated star formation: (1) formation of a gravitationally bound core withing a molecular cloud, (2) the gravitationally driven collapse of that core, (3) the formation of a central star-like object, (4) dispersal of the remaining cloud material [@sal]. The initial two stages are predominantly studied through millimeter- and submillimeter-wave-length observations. The molecular line rotational transition emission visible in these wavelength regimes allow one to probe both the physical and chemical composition of the molecular cloud cores out of which stars form. Spectral analysis of the molecular line emission also allows one to investigate the kinematic motions within the molecular cloud, particularly the phase of gravitational collapse [@meo] as well as the associated bipolar outflows which are ubiquitous around young stars [@bac].
In a dense cloud core, commonly used molecular line tracers of moderate optical depth, such as CS (), or [HCO$^{+}$]{} () tend to become self-absorbed. As the cloud contracts under the influence of gravity, the central regions tend to become more dense than the outer regions. This sets up a gradient in the excitation temperature of the molecular gas within the cloud such that the excitation temperatures in the central regions of the cloud are greater than those in the outer regions, even if the kinetic temperature is constant within the molecular cloud. If the cloud is spherical and static the spectral line emission from this cloud in these tracers will be symmetric about the line of sight velocity and self-absorbed. However, if there is radial motion around the molecular cloud core, a velocity gradient develops around the center of the core, precisely where the excitation temperature is the greatest. Much of the emission from the rear of the cloud is not absorbed as it travels through the core, due to the high excitation temperature within the core, and continues unabsorbed through the front section of the cloud because the velocity gradient between the rear and front sections of the cloud results in a Doppler shift which is significant compared to the broadening by thermal motions. Emission from the front of the cloud however is absorbed by nearby molecular gas with lower excitation temperature and only a small Doppler shift. In the case of inward radial motion the rear emission is blue-shifted and the front emission is red-shifted, yielding the “classic” blue-asymmetric infall line profile. This line profile has become the predominant tool for investigating infall motions within star-forming molecular cloud cores [@sl; @zhou2; @zhou; @mmtwbg; @gemm; @lmt2].
Although this blue-asymmetry results from infall, it is not unique to that phenomenon. The superposition of multiple clouds along the line of sight can also cause an asymmetric line profile. By observing an optically thin molecular line, which should remain symmetric around the line of sight velocity of the cloud core, it is possible to reduce the likelihood of mistaking separate clouds for an infalling cloud. Asymmetric line profiles can also arise in kinematically complex regions. Studies have shown that rotation and outflow can also produce asymmetric line profiles [@slp; @al; @nmwb]. Even the asymmetric line emission of B335, once thought to be the prototypical example of spectrally detected inside-out collapse [@zekw; @zekw2; @zhou], is now also thought to also contain an outflow which gives rise to the line wings previously modeled as part of the collapse [@wmmt]. In spite of the multiple molecular cloud configurations that can result in blue-asymmetric line profiles, they are still the best way to study infall motions available, provided we understand the context in which they occur and make clear the necessary caveats.
Understanding how blue-asymmetric line profiles arise from infall motion is easy, but creating a complete model matching the molecular emission from one or more tracers is a difficult and time-consuming task. Typically one constructs an infalling physical model and uses LVG, Monte Carlo [@cegw] or microturbulent [@zhou] radiative transfer to model the emission. By iteratively changing the model parameters one is able to match some or all of the observations reasonably well. This process can take several minutes per iteration, and becomes prohibitively slow as the number of physical parameters becomes large and as the number of molecular species increases. The derived cloud parameters are also highly dependent on the input physical model. Changing the physical model can lead to an entirely different set of solutions.
“Starless cores” are useful regions for observations of inward motion because they are numerous, nearby, relatively simple in their density structure, and their motions are not confused by outflows. They have been the subject of several studies [@lm; @lmt1; @lmt2; @bapabw; @mm; @tmcwc; @tmcw; @ts; @plm], some of which have used radiative transfer models to fit their line profiles. In this paper we present simple analytic radiative transfer models, based on the simplest available assumptions of possible excitation temperature trends within starless molecular cloud cores. These models can easily be fit to blue-asymmetric self-absorbed molecular line profiles, and produce an estimate of the infall velocity suggested by those line profiles. We also test the efficacy of the analytic radiative transfer models by comparing the infall velocities derived using those models with actual infall velocities in rigorously modeled condensing molecular clouds. Our main finding is that our analytic radiative transfer models can yield good estimates for the physical conditions of the starless cores with few assumptions and little computational investment.
Analytic Models
===============
We compare two analytic models in this paper. Although it is possible to construct more analytic models using the same techniques in this paper we feel that these two models are the most appropriate for modeling infall. It is possible to analytically integrate the equation of transfer $$\frac{dT_{\rm B}}{d\tau_{\nu}} = -T_{\rm B} + J(T),$$ where $T_{\rm B}$ is the brightness temperature, $T$ is the excitation temperature and $J(T) =
\left(h\nu/k\right)\left[\exp\left(h\nu/kT\right)-1\right]^{-1}$. The brightness temperature is defined as $T_{\rm B} = \left(c^{2}/2\nu^{2}k\right) I_{\nu}$ and is directly proportional to the specific intensity $I_{\nu}$. The general solution to the equation of transfer, assuming that optical depth increases away from the observer is $$\label{xfer}
T_{\rm B} = T_{i} e^{-\tau_{0}} + \int_{0}^{\tau_{0}} J(T) e^{-\tau}
d\tau,$$ where $T$ is the excitation temperature of a region which varies over the optical depth interval from 0 to $\tau_{0}$, and $T_{i}$ is the incident specific intensity of radiation on that region at $\tau_{0}$ in units of brightness temperature. This equation can easily be integrated over regions of constant excitation temperature. We assume regions of constant excitation temperature along the line of sight in our first model.
The first model, first discussed in @mmtww, we call the “two-layer” model. It is perhaps unfortunate that this name has been given to this model and has stuck, as it implies that this is a model of two plane-parallel layers along the line of sight. This is not true. The model actually applies to a line of sight in which two regions with differing excitation temperatures are moving toward each other. We assume the near region has a lower excitation temperature than the far region. We assume both regions have a velocity dispersion for the observed molecule $\sigma$ and a total optical depth of $\tau_{0}$ at line center. The excitation temperature of the front region is $T_{f}$, while that of the rear region is $T_{r}$, and the regions are approaching each other with a speed of $2v_{\rm in}$. The optical depth of each region at a velocity $v$ is $$\begin{aligned}
\tau_{f}(v) & = & \tau_{0}\exp\left[-(v-v_{\rm LSR}-v_{\rm in})^{2}/2\sigma^{2}\right], \\
\tau_{r}(v) & = & \tau_{0}\exp\left[-(v-v_{\rm LSR}+v_{\rm
in})^{2}/2\sigma^{2}\right],\end{aligned}$$ assuming that the average line of sight velocity of both regions is $v_{\rm
LSR}$. Figure \[models\] provides a graphical representation of the model. The brightness temperature of the spectral line, obtained from the equation of radiative transfer, is $$\Delta T_{\rm B}(v) = J(T_{f})\left[1-e^{-\tau_{f}(v)}\right] +
J(T_{r})\left[1-e^{-\tau_{r}(v)}\right]e^{-\tau_{f}(v)} -
J(T_{b})\left[1-e^{-\tau_{r}(v)-\tau_{f}(v)}\right],$$ where $T_{b}$ is the background temperature. The parameters $\tau_{0}$, $\sigma$, $T_{f}$, $T_{r}$, $v_{\rm LSR}$, and $v_{\rm in}$ are free parameters which can be adjusted to fit blue-asymmetric line profiles. @lmt2 has used this analytic model to derive infall velocity estimates for 29 starless cores. They find that infall velocities are typically on the order of a tenth of a kilometer per second, comparable to the velocity dispersions measured in these sources.
Equation \[xfer\] is also analytically integrable if $J(T)$ is a linear function of the optical depth $\tau$. Assuming a simple linear function $J(T) = J_{1} +
\left[\left(J_{2}-J_{1}\right)/\tau_{0}\right]\tau$ we can integrate the equation of transfer to obtain $$T_{\rm B} = T_{i} e^{-\tau_{0}} + (J_{2}-J_{1})\frac{1-e^{-\tau_{0}}}{\tau_{0}}
+ J_{1} - J_{2}e^{-\tau_{0}}.$$ In the above equation $\tau$ is a function of the Doppler velocity, so in order to calculate the intensity of emission across a line profile $J(T)$ must be a linear function of $\tau$ at every frequency. It is possible to construct a series of regions along the line of sight where $J(T)$ varies linearly with $\tau$ within each region and analytically calculate the brightness of radiation at a given frequency emitted from that line of sight using the above equation. We have constructed our next model using the above technique and equations which we feel approximates the excitation conditions seen along the line of sight in infalling clouds.
The second model, which we call the “hill” model, is introduced for the first time in this paper. This model consists of a core with a peak excitation temperature $T_{P}$ at the center and an excitation temperature of $T_{0}$ at the near and far edges of the core. The $J(T)$ drops linearly from $J(T_{P})$ at the center to $J(T_{0})$ at edges of the core, forming a hill in the $J(T)$ profile. The optical depth of the core is $\tau_{\rm C}$, and its infall velocity is $v_{\rm C}$, while the systematic velocity of the system is $v_{\rm LSR}$. A schematic representation of this model is shown in figure \[models\]. We introduce this new type of model because we believe it may be more analogous to the excitation profile we are likely to be observing in starless cores. In figure \[excivtau\] we compare the excitation profile as a function of optical depth at line center for a simulated cloud, based on a Monte Carlo radiative transfer model, to the best fit hill and two-layer models of the line profile derived from that cloud. The excitation profile of our modeled starless core has a significant slope, which the hill model can replicate, but the two-layer model, due to its use of constant excitation temperature zones, can not replicate. This improvement provides motivation for introducing a new analytic radiative transfer model of infall.
In order to solve the equation of radiative transfer we separate the hill model profile into two regions along the line of sight: (1) The portion of the cloud in which the excitation temperature is rising along the line of sight, whose optical depth is $\tau_{f}$; And (2) The portion of the cloud in which the excitation temperature is falling along the line of sight, whose optical depth is $\tau_{r}$. The optical depth as a function of line of sight velocity for each of these regions is $$\begin{aligned}
\tau_{f}(v) & = & \tau_{\rm C}\exp\left[-(v-v_{\rm LSR}-v_{\rm C})^{2}/2\sigma^{2}\right], \\
\tau_{r}(v) & = & \tau_{\rm C}\exp\left[-(v-v_{\rm LSR}+v_{\rm C})^{2}/2\sigma^{2}\right],\end{aligned}$$ assuming a velocity dispersion of $\sigma$ in the entire cloud. We can now solve the equation of transfer by integrating along the line of sight through each of these two regions to derive the brightness temperature $$\begin{aligned}
\Delta T_{\rm B}(v) & = &
\left(J(T_{P})-J(T_{0})\right)\left[\left(1-e^{-\tau_{f}(v)}\right)/\tau_{f}(v)-
e^{-\tau_{f}(v)}\left(1-e^{-\tau_{r}(v)}\right)/\tau_{r}(v)\right]
\nonumber \\
& & \mbox{ } + \left(J(T_{0})-J(T_{b})\right)\left[1-e^{-\tau_{r}(v)-\tau_{f}(v)}\right],\end{aligned}$$ where $T_{b}$ is the background temperature. We believe the excitation temperature gradient in this model might more closely approximate the excitation profiles in observed clouds. The six free parameters in the hill model are $\tau_{\rm C}$, $\sigma$, $T_{0}$, $T_{P}$, $v_{\rm LSR}$, and $v_{\rm C}$. It is also possible to add an envelope of constant excitation temperature moving with its own infall velocity. Assuming this envelope has an excitation temperature of $T_{0}$, the optical depth of the envelope and the velocity of the envelope adds two more free parameters, creating a maximum of eight free parameters possible in the hill models.
In order to reduce the total number of free parameters we consider several variants of each model by holding certain parameters fixed. We considered two variants of the two-layer model, which we call [twolayer5]{} and [twolayer6]{}, as well as 4 variants of the hill model, which we call [hill5]{}, [hill6]{}, [hill6core]{}, and [hill7]{}. The number is each variant name refers to the number of free parameters in that variant. The parameterization of the two-layer model variants are shown in table \[twolayervar\], while those of the hill model variants are shown in table \[hillvar\].
[lllllll]{} & Free & Free & Free & Free & $T_{b}$ & Free\
[twolayer6]{} & Free & Free & Free & Free & Free & Free\
[lllllllll]{} & 0 & Free & Free & 0 & Free & Free & $T_{b}$ & Free\
[hill6]{} & Free & Free & Free & Free & 0 & Free & $T_{b}$ & Free\
[hill6core]{} & Free & Free & Free & 0 & Free & Free & $T_{b}$ & Free\
[hill7]{} & Free & Free & Free & Free & 0 & Free & Free & Free\
Physical Models {#physicalmodels}
===============
The analytic radiative transfer models do an excellent job of fitting blue-asymmetric line profiles [@lmt2]. Their efficacy in determining physical parameters of the molecular cloud, such as infall velocity, has not yet been studied. In this paper we study how effectively we can recover the physical parameters of modeled clouds using both the two-layer and hill analytic radiative transfer models. We then use this information to determine the point at which the systematic errors of the model dominate over random observational errors.
In our first three simulations, we modeled molecular clouds with a density law of the form $$n(r) = \frac{n_{0}}{1+\left(r/r_{0}\right)^{\alpha}},$$ where $n_{0}$ is the central density, $r_{0}$ is the turnover radius, and $\alpha$ is the power law of the density profile at large radii [@tmcwc]. This model reproduces the flat density region (at $r<r_{0}$) and power law region (at $r>r_{0}$) required to model starless cores. It also is in good agreement with the density profile of a Bonnor-Ebert sphere [@tmcw] as well as the “Plummer-like” density model of @ww. The clouds modeled in this paper have peak densities ($n_{0}$) of $6\times
10^{4}$ cm$^{-3}$, a turnover radius ($r_{0}$) of $5\times 10^{16}$ cm, an outer radius of $2\times 10^{17}$ cm, and an $\alpha$ of 4. We chose to model [HCO$^{+}$]{} rotational line emission from these clouds, as it is often moderately optically thick in starless cores. We assumed that the kinetic temperature of the gas is 10 K throughout the cloud. We set the relative abundance of [HCO$^{+}$]{} at $2\times 10^{-9}$ [@dbjg] and the [HCO$^{+}$]{}/[H$^{13}$CO$^{+}$]{} abundance ratio to 64. Depletion is not included here, however we do investigate the effect of depletion in §\[depletionbeam\]. There are four velocity laws which we investigate in this paper. The first (Simulation A) is constant infall velocity for all radii in the cloud. The second (Simulation B) is a constant infall velocity for $r>r_{0}$ and 0 velocity for $r<r_{0}$. The third (Simulation C) is constant infall velocity for $r<r_{0}$ and 0 velocity for $r>r_{0}$. These simulations are summarized in Table \[sims\]. These velocity laws were chosen because they are simple velocity laws to investigate. Our analytic radiative transfer models define a characteristic single infall velocity which is analogous to Simulations A, B, and C. We set the velocity dispersion to 0.1 km s$^{-1}$ over the entire cloud and typically allow the velocities within the cloud to be up to twice the velocity dispersion. These results should all scale with the velocity dispersion and the peak line brightness. In addition to the simulations described above, we also simulated a different density profile based on the best fit Bonnor-Ebert model of B68 [@all2]. The parameters of this simulation (Simulation D) are discussed in §\[besphere\].
[llll]{} A & TMCWC & $v_{\rm in} = a$ & $a =$ 0.00, 0.02, …, 0.2 km s$^{-1}$\
B & TMCWC & $v_{\rm in} = \left\{ \begin{array}{ll} 0 & \mbox{if $r<r_{0}$} \\ a &
\mbox{if $r\geq r_{0}$} \end{array} \right. $ & $a =$ 0.00, 0.02, …, 0.2 km s$^{-1}$\
C & TMCWC & $v_{\rm in} = \left\{ \begin{array}{ll} a & \mbox{if $r<r_{0}$} \\ 0 &
\mbox{if $r\geq r_{0}$} \end{array} \right. $ & $a =$ 0.00, 0.02, …, 0.2 km s$^{-1}$\
D & BE & $v_{\rm in} = a$ & $a =$ 0.000, 0.024, …, 0.24 km s$^{-1}$\
We use these simulated starless cloud cores to synthesize [HCO$^{+}$]{} () and [HCO$^{+}$]{} () molecular line emission using the @hv 1-D Monte Carlo radiative transfer model. The simulated clouds were divided into 20 spherical shells, each of uniform density, kinetic temperature, radial velocity, and with a radial size of $10^{16}$ cm. The level populations within the first 21 rotational levels of the [HCO$^{+}$]{} molecule were calculated using the Einstein-A and [HCO$^{+}$]{} with H$_{2}$ collisional rate coefficients from @monteiro and @green. We then integrate along the central line of sight to find the and molecular line spectral profiles of [HCO$^{+}$]{} because they are frequently observed, and often have blue-asymmetric line profiles attributed to infall [@gezc; @nw; @nwb; @nmwb; @brc1]. We then fit these synthesized spectra with the two-layer and hill analytic radiative transfer models to check the ability of the analytic models to recover physical parameters within the modeled cloud.
Several algorithms exist for minimizing multidimensional functions. There is always the danger of minimizing to a “local” minimum (meaning the lowest value within some finite region of the function to be minimized) instead of the “global” minimum (meaning the lowest function value over the entire parameter space) and it becomes more difficult to avoid local minima as your function goes to higher dimensions. There are several strategies to avoid local minima, and we have chosen a hybrid minimization algorithm to do just that. We use the Differential Evolution (DE) algorithm of @devo initially to separate the local minima from the global minimum, then we use the Nelder-Mead simplex method [@nm] to optimize the fit. The DE minimization algorithm is applicable to many astronomical problems, and is similar in scope to simulated annealing. DE is an evolutionary algorithm in that several parameter sets are randomly selected and these are randomly modified over several “generations” until acceptable convergence has been achieved. The variability in each parameter is scaled by the variation of the solution sets in each generation, so as the algorithm converges the region of the search decreases without the need to include a decreasing “temperature” as in simulated annealing. DE is also superior to simulated annealing in that it is more likely to achieve convergence, is a simpler algorithm, and has fewer adjustable parameters [@devo].
Results
=======
We made four simulations of infalling clouds and used the @hv Monte Carlo radiative transfer model to extract [HCO$^{+}$]{} () and [HCO$^{+}$]{} () line profiles from those simulations. In this section we will discuss how effective our analytic models are at deriving the input velocities in those simulation, quantify the relative contributions of both random and systematic errors in model fits of simulated data sets, and apply this model to an observed infall profile in a source which has been rigorously modeled using Monte Carlo radiative transfer techniques.
Simulation A — Constant Infall
------------------------------
Simulation A, which has a constant infall velocity for all cloud radii, has asymmetric line profile typical of those seen in infalling clouds. The [HCO$^{+}$]{} () line profiles are shown in figure \[majo\], along with their optically thin [H$^{13}$CO$^{+}$]{} () counterparts. The spectra in the bottom on the left and right side are for a simulation with no infall. The middle spectra on the left and right are of a simulation with an infall velocity of 0.1 km s$^{-1}$. The top spectra on the left and right are from a simulation with an infall velocity of 0.2 km s$^{-1}$. Overlain on the [HCO$^{+}$]{} () spectra on the left panel are the [twolayer5]{} and [hill5]{} best fit models, while on the right the [twolayer6]{} and [hill6]{} spectra are shown. Figure \[majt\] depicts [HCO$^{+}$]{} () and [H$^{13}$CO$^{+}$]{} () spectra from the same simulations along with their [twolayer5]{}, [hill5]{}, [twolayer6]{}, and [hill6]{} best fit spectral line profiles. Two variants are not shown in these fits, the [hill6core]{} model best fit in most cases has a $\tau_{\rm E}$ which is very close to 0, yielding the same parameters and spectral line shape as the [hill5]{} model. The [hill7]{} model always manages a very good fit to the spectral line shape, however in some cases, especially in spectra with a small red component that is not well separated from the blue component (such as the middle and top spectra in figure \[majt\]), the parameters of the [hill7]{} model are not very well constrained. A wide range of infall velocities, optical depths, and excitation temperatures in the seven parameter [hill7]{} model can yield the same line profile.
There are several interesting features in these spectral line fits. When the infall velocity is greater than the velocity dispersion the [H$^{13}$CO$^{+}$]{} line profiles become double peaked. This is not due to self-absorption as the lines are optically thin. It is due to the fact that the molecular gas along the central line of sight is traveling at two velocities separated by more than the velocity dispersion, resulting in two peaks in the optically thin component. This feature is unlikely to be observed in [H$^{13}$CO$^{+}$]{} () lines as they are too weak to be easily detected. Also interesting is that while the dip between the red and blue peaks of the spectral line profile is deep, both the hill variants and the two-layer variants do a good job a reproducing the spectral line, but as the red peak drops in intensity relative to the blue peak the analytic model fits tend to produce a spectral line with a single blue peak and a red shoulder with no local maximum but only a flattening of the slope. The two-layer model variants tend to produce this type of spectrum at lower infall speeds than the hill model variants (Figure \[majo\]). We call the spectra with distinct blue and red peaks “dip” spectra and those with only a flattening of the slope, “shoulder” spectra. We also see that none of the models can reproduce a spectral line which has a minimal red peak superimposed on a broadly asymmetric line (as in the top spectra of figure \[majt\]). These models tend to produce spectra that are similar in shape to two overlapping gaussians of equal width. If we try to fit a spectral line that differs greatly from one that can be obtained by adding two gaussians of equal width, then we tend to not get good line fits with any of the analytic models.
In figure \[maf\] we show the quality of the infall velocity fit attained by the analytic model fits to the [HCO$^{+}$]{} () line profiles in this simulation (left side) as well as the model fits to the [HCO$^{+}$]{} () line profiles in this simulation (right side). In this simulation, for the entire infall velocity range up to twice the velocity dispersion, the [hill5]{} model tends to reproduce the modeled infall velocity with an RMS systematic error of less than 0.01 km s$^{-1}$ in the [HCO$^{+}$]{} () lines and 0.02 km s$^{-1}$ in the [HCO$^{+}$]{} () lines. The [twolayer5]{} model tends to consistently underestimate the infall velocity by over a factor of 2 in both transitions when it produces good line fits. The [twolayer6]{} model fits the infall velocity with an RMS error of 0.03 km s$^{-1}$ in both the [HCO$^{+}$]{} () lines and the [HCO$^{+}$]{} () lines. Generally the [twolayer6]{} model does a better job in cases where the spectrum is dominated by the blue component and the relative size or integrated intensity of the red component is small. This happens to be the true for cases with higher infall velocities in the [HCO$^{+}$]{} () and [HCO$^{+}$]{} () spectra in this simulation. The [twolayer6]{} model also performs well when the excitation temperature of the rear layer matches the simulated excitation temperature in the cloud. The [hill6]{} model does a good job at fitting the infall velocity for infall velocities greater than the velocity dispersion, but tends to greatly overestimate the infall velocity in cases where it is smaller than the velocity dispersion. The [hill7]{} model also tends to reproduce the infall velocity fairly well, but not as well as the [hill5]{} model.
It should be noted that a good fit to the line profile does not always translate to a good fit to the physical parameters, such as the infall velocity, in the simulated clouds. In the case of the [twolayer6]{} models, often the dip solution is a good fit to the line profile, but a worse fit to the infall velocity than the shoulder solution. The [hill7]{} model is able to fit just about any line profile, but often yields worse infall velocities than [hill5]{} model fits with larger minimized $\chi^{2}$. In summary, the infall velocities for Simulation A (constant infall) are most accurately reproduced by the [hill5]{} model with a RMS deviation of 0.016 km s$^{-1}$ over 22 cases. The [twolayer6]{} model is significantly worse with an overall RMS deviation of 0.030 km s$^{-1}$ over 22 cases.
Simulation B — Envelope Infall
------------------------------
Simulation B, which has a constant infall velocity for $r>r_{0}$ and no infall for $r<r_{0}$, has asymmetric line profiles typical of those seen in infalling clouds. Figure \[mbjo\] shows the [HCO$^{+}$]{} () line profiles for three different infall velocities. The self-absorbed solid line in the lowest left and right panels is the line profile for a cloud with no infall, as well as the optically thin [H$^{13}$CO$^{+}$]{} () line profile. In the middle left and right are the [HCO$^{+}$]{} and [H$^{13}$CO$^{+}$]{} () line profiles for a cloud with an infall velocity of 0.1 km s$^{-1}$, equal to the velocity dispersion in that cloud. The upper left and right panels are the [HCO$^{+}$]{} and [H$^{13}$CO$^{+}$]{} () line profiles for a cloud with an infall velocity of 0.2 km s$^{-1}$. These line profiles are slightly narrower than those seen in Simulation A (Constant Infall). This is due to the fact that the cores, where the most dense gas is located, are stationary and not producing any broad line wings. As in our previous simulation we have overlaid the [twolayer5]{}, [hill5]{}, [twolayer6]{}, and [hill6]{} best fit to the [HCO$^{+}$]{} spectra. Figure \[mbjt\] indicates these same results for the [HCO$^{+}$]{} () and [H$^{13}$CO$^{+}$]{} () spectra in Simulation B (Envelope Infall).
We investigate the efficacy of our models in matching the infall velocity of the Simulation B (Envelope Infall) clouds in figure \[mbf\]. The [twolayer5]{} model once again underestimates the infall velocity in all cases and in both transitions, although it monotonically increases throughout the simulation. The [twolayer6]{} model tends to yield the same result as the [twolayer5]{} model over many of the infall velocities, but jumps to a higher infall velocity which in many cases approximates the correct infall velocity (especially in the [HCO$^{+}$]{} () fits). We investigated these jumps in the [twolayer6]{} model further and found that there are two local minima in the $\chi^{2}$ surface for many spectrum shapes.
The [twolayer6]{} minima we found correspond to “dip” spectra, with two distinct peaks, and “shoulder” spectra with one peak and a flattening of the spectrum on the red side. These two spectral line fits tend to have very different parameters for the infall velocity which we show in figure \[dipshoulder\] for the [HCO$^{+}$]{} () lines generated in this simulation. The letter with the larger size indicates the fit with the lowest $\chi^{2}$, and the solid line indicates the expected value if the infall velocity modeled analytically matches the infall velocity of the cloud. In the [HCO$^{+}$]{} () profiles, the “dip” solution is a better fit to the line profile up to a modeled infall velocity of 0.12 km s$^{-1}$, however the “shoulder” solution yields a better fit to the infall velocity for modeled infall velocities above 0.04 km s$^{-1}$. A similar trend is seen in the [HCO$^{+}$]{} () results. The “shoulder” fits tend to have a higher infall velocity, and are often a much better match to the infall velocity. The “dip” spectra are typically the equivalent to the [twolayer5]{} model results. The best “dip” spectrum fit of the [twolayer6]{} model usually results in an excitation temperature of the front layer equal to the background temperature and an underestimate of the infall velocity. We therefore recommend that one should attempt to find the local minimum corresponding to a “shoulder” fit when using the [twolayer6]{} model, and that one can verify that one has found the “shoulder” fit by making sure that the excitation temperature of the front layer is significantly higher than the background temperature.
Another trend revealed in figure \[mbf\] is that the [twolayer6]{} model does not do a good job of fitting high velocity infall using the [HCO$^{+}$]{} () line profile in this modeled cloud. We can see in the upper-most line profiles of figure \[mbjt\] that the [HCO$^{+}$]{} () line profiles do not separate into multiple distinct components at high infall velocities in this model. As the lines begin to blend into a profile that is not easily separable into multiple components, the two-layer model begins to break down and does not accurately match the actual infall velocity.
The lower panels of figure \[mbf\] show the hill model variants infall velocities compared to the infall velocities in those same clouds. Once again we see that [hill5]{} does a good job of matching most of the infall velocities using the [HCO$^{+}$]{} () line profile, but tends to break down at high velocities. Overall the [hill5]{} model fits to the [HCO$^{+}$]{} ( match the simulated infall velocities with an RMS error of 0.08 km s$^{-1}$ over the full range of simulated infall velocities, however for infall velocities less than 1.5 times the velocity dispersion the RMS error is less than 0.01 km s$^{-1}$. The [hill5]{} infall velocity fits break down at an even lower infall velocity in the case of the [HCO$^{+}$]{} () line, however the overall RMS error in the infall velocity fits for that transition are only 0.07 km s$^{-1}$. We believe this is because the contribution of the red component is so low in these cases. When the magnitude of the red component drops [hill5]{} does not produce a good profile fit (see the topmost spectrum of figure \[mbjo\]). We also believe that the [hill5]{} model does a poor job when the shoulder starts to disappear yielding a sloping asymmetric line such as the uppermost spectrum of figure \[mbjt\]. Once again the [hill6]{} model does a poor job at fitting low infall velocities. The [hill7]{} does a good job at fitting all the infall velocities in both line transitions, typically within 0.03 km s$^{-1}$ of the modeled value. The error in the fit begins to increase, especially in the case of the [HCO$^{+}$]{} () fit, as the infall velocity increases. This is due to the diminishing magnitude of the red-shifted component. As the blue-shifted component begins to dominate the [hill7]{} model fit, the seven parameters of the model begin to overdetermine the line profile, resulting in a range of solutions that fit equally well. In practice it is possible to avoid this by reducing the number of parameters and using the [hill6]{} model, however in most observed infall profiles the magnitudes of both the red and blue components are not as large as seen in the top panel of figure \[mbjo\] [cf. @lmt2]. In typical line profiles, such as those seen in this spherically symmetric infalling cloud, with a uniform infall velocity in the outer layers, the [hill5]{} does an excellent job at extracting the infall velocity, while the [twolayer6]{} does an adequate job for lines with a small red peak or narrow shoulder component.
Simulation C — Core Infall
--------------------------
We confine infall to the core ($r<r_{0}$), and surround the infall region with a static envelope in Simulation C. The resulting [HCO$^{+}$]{} and [H$^{13}$CO$^{+}$]{} () line profiles are shown in figure \[mcjo\]. The [HCO$^{+}$]{} and [H$^{13}$CO$^{+}$]{} () line profiles are shown in figure \[mcjt\]. Once again modeled infall velocity, this time in the core, is increasing upwards. The bottom spectra are for no infall velocity, the middle spectra are for an infall velocity equal to the velocity dispersion of 0.1 km s$^{-1}$, and the top spectra are for an infall velocity of 0.2 km s$^{-1}$. The first thing we notice is that the [HCO$^{+}$]{} () lines are not very blue-asymmetric. Although the width increases as the infall velocity in the core increases, the infall asymmetry does not because the transition is mostly sensitive to motions in the envelope. The emission from the core is mostly absorbed by the envelope because the optical depth in the line is high. The [HCO$^{+}$]{} () transition, which is more optically thin, does a better job of probing the motions in the core, and shows a greater degree of asymmetry as the infall velocity increases.
The low level of asymmetry in the line results in a relatively poor fit to the infall velocity for both the two-layer model variants and the hill model variants. Figure \[mcf\] depicts the efficacy of the model variants at matching the simulated infall velocity in this simulation. The [twolayer5]{}, [twolayer6]{}, [hill5]{}, and [hill7]{} all significantly underestimate the infall velocity from the [HCO$^{+}$]{} () line profiles. This is not surprising as that transition is very thick and not very sensitive to the motions in the core, where the infall is occurring. The [hill5]{} fit tends to be consistently a factor of $\sim5$ lower than the simulated infall velocity. The situation is more encouraging when examining the thinner [HCO$^{+}$]{} () line profile fits. The [twolayer5]{} model continues to significantly underestimate the infall velocity, but the [twolayer6]{} model matches the simulated infall velocity well for velocities between 0.1 km s$^{-1}$ and to 0.15 km s$^{-1}$. Overall the RMS error on the infall velocity fits of the [twolayer6]{} model to the [HCO$^{+}$]{} () lines is 0.03 km s$^{-1}$. The [hill5]{} model continues to underestimate the infall velocity from the [HCO$^{+}$]{} () line, resulting in an RMS error on the fit infall velocity of 0.07 km s$^{-1}$. We believe this is due to the fact that the red peak quickly becomes a shoulder in this simulation, and the [hill5]{} model tends to do much better when fitting a distinct red peak. The [hill6]{} model overestimates the infall velocity, while the [hill7]{} model tends to do about as well as the [twolayer6]{} model at estimating the infall velocity when fitting the [HCO$^{+}$]{} () lines in this simulation. Both the [hill7]{} model and the [twolayer6]{} model begin to underestimate the infall velocity for core infall velocities greater that approximately 0.14 km s$^{-1}$. At this point we run into the problem that the line profile is not becoming more asymmetric, just wider, due to the increasing velocities in the core.
Simulation D — Dense Bonnor-Ebert Sphere {#besphere}
----------------------------------------
In our fourth simulation we choose a different density profile and a uniform velocity throughout the cloud. We specifically choose the Bonnor-Ebert fit to the density profile of B68 obtained by using extinction measurements of the background K giant population to produce an azimuthally averaged column density profile [@all2]. We calculate the Bonnor-Ebert density profile using the modified Lane-Emden equation $$\frac{1}{\xi^{2}}\frac{d}{d\xi} \left(\xi^{2}\frac{d\Psi}{d\xi}\right) = e^{-\Psi},$$ where $\xi = (r/a)\sqrt{4\pi G\rho_{c}}$ is the non-dimensional radial parameter, $a$ is the isothermal sound speed, $\rho_{c}$ is the volume density at the cloud center, and $\Psi(\xi) = -\ln(\rho/\rho_{c})$. We split this second order differential equation in to two first order equations by using the variable substitutions suggested by @bkv and integrate the equations numerically to derive the density profile. We use $\xi_{\rm max} = 6.9$, the free parameter derived by @all2 to be the best fit to the B68 cloud extinction observations. @dhwb find that the relative abundance of [HCO$^{+}$]{} to molecular hydrogen is less than $1.4\times
10^{-10}$, which is significantly lower than the value of $2\times 10^{-9}$ which we have been using. Under such low abundances the [HCO$^{+}$]{} lines we would derive from radiative transfer calculations within this model cloud would be optically thin. Since optically thick lines are required for our analysis, we choose to maintain the [HCO$^{+}$]{} to molecular hydrogen relative abundance at $2
\times 10^{-9}$ and only change the density profile in this simulation. We divide the spherical cloud into thirty equally spaced shells, the central shell has a molecular hydrogen density of $2.6\times 10^{5}$ cm$^{-3}$, while the outermost shell has a molecular hydrogen density of $1.6\times
10^{4}$ cm$^{-3}$ and a maximum radius of $1.87\times 10^{17}$ cm or 12,500 AU. We assume a constant kinetic temperature of 11 K throughout the cloud, as obtained by @lvlk using NH$_{3}$ (1,1) and (2,2) hyperfine line observations conducted at the DSN 34m telescope.
We assume a constant infall velocity throughout the cloud at all radii. We modeled 11 infall velocities ranging from 0 to twice the velocity dispersion, which we chose to be 0.12 km s$^{-1}$. We base our choice of the velocity dispersion on the [N$_{2}$H$^{+}$]{} () line width in B68 reported by @cbmt. [HCO$^{+}$]{} () and [H$^{13}$CO$^{+}$]{} () spectra from this simulation are shown in figure \[mdjo\]. As in previous simulations we choose a representative sample of three spectra in the direction of the cloud center at different infall velocities. In this case the infall velocities are 0 km s$^{-1}$, 0.12 km s$^{-1}$, and 0.24 km s$^{-1}$. On the left side of figure \[mdjo\] the best fit [twolayer5]{} and [hill5]{} models are shown, while on the right side the best fit [twolayer6]{} and [hill6]{} models are shown. The first thing to note is that the [H$^{13}$CO$^{+}$]{} lines are slightly asymmetric, similar to [H$^{13}$CO$^{+}$]{} (), HC$^{18}$O$^{+}$ (), and D$^{13}$CO$^{+}$ () observations of L1544 seen in figure 1 of @cwztdm1. This is as a result of self-absorption, because although the [H$^{13}$CO$^{+}$]{} molecule has an abundance 64 times lower than that of [HCO$^{+}$]{}, it is not so low that it is optically thin. It is important to be aware of that asymmetry may arise in less abundant species, however the degree of asymmetry is indeed less than the asymmetry of the more abundant [HCO$^{+}$]{} molecule. As in previous simulations, the line profiles generated by the analytic models reproduce the Monte Carlo generated profiles very well. Similar results for the [HCO$^{+}$]{} () and [H$^{13}$CO$^{+}$]{} () lines are shown in figure \[mdjt\]. The [H$^{13}$CO$^{+}$]{} lines are more symmetric in this case as there is less optical depth along the line of sight.
We compare the infall velocities in this simulation with those obtained by fitting the line profiles with our analytic models in figure \[mdf\]. As in previous sections we present the fits to the [HCO$^{+}$]{} () line profiles on the left panels of the figure, and those to the [HCO$^{+}$]{} () line profiles on the right panels of the figure. The upper panels refer to the two-layer model fits, while the lower panels refer to the hill model fits. Neither of the two-layer models reproduce the simulated infall velocity across the entire range of infall velocities we tested. The RMS systematic error in the [twolayer6]{} model fits of [HCO$^{+}$]{} () is 0.03 km s$^{-1}$, and 0.04 km s$^{-1}$ with respect to the [HCO$^{+}$]{} () fits. The [hill5]{} model does a better job at estimating the infall velocity in the simulated clouds. The RMS systematic error in the [hill5]{} estimates of the infall velocity based on the [HCO$^{+}$]{} () profile fits is 0.03 km s$^{-1}$, while the RMS systematic error based on the [HCO$^{+}$]{} () profile fits is 0.01 km s$^{-1}$. As we have seen in previous simulations the [twolayer6]{} model yields a closed fit to the infall velocity at higher infall velocities. The [hill5]{} model provides a fairly good fit for all infall velocities. These results are similar to the results obtained in Simulation A (Constant Infall), which is the other simulation in which infall was constant at all radii. This is reassuring as the central densities in these two simulations differ by a factor of 4 and they have different radial density profiles. This suggests that these analytic models may be able to predict infall velocities from optically thick line profiles for a wide range of densities and radial density profiles in star forming clouds.
Error Analysis
--------------
There are two components to the error in any fit of a model to data. The first, systematic error, is error which arise in the model’s description of the data. It is due to the fact that the model may not correctly describe the nature of the observed region. The second component is random error, which arises from noise in the observation. Although random error can be reduced over time or with better instrumentation, usually up to some limit, it is always a factor in any observation, and in fitting a model to those observations. In the preceding sections we examined how well our analytic radiative transfer models fit synthetic noiseless spectra. From this we found systematic errors due to the models shown in figures \[maf\] and \[mbf\]. In order to investigate the behavior of the fits in the presence of random error we decided to add noise to the spectra.
@bg have derived analytic expressions for the contribution of random error in a spectrum to the three parameters of a Gaussian fit based upon the work of @lrdt. They find that the random error in the line of sight velocity and in the velocity dispersion of a Gaussian fit are the same and equal to $$\label{gauss}
\sigma_{v_{\rm LSR}} = \sigma_{\sigma} = 1.06\left(\delta_{v}\sigma
\right)^{1/2} \frac{\sigma_{\rm RMS}}{T_A^{*}},$$ where $v_{\rm LSR}$ is the line of sight velocity of the Gaussian, $\sigma$ is the velocity dispersion of the Gaussian, $\delta_{v}$ is the width of the spectral channels, $T_A^{*}$ is the peak intensity of the Gaussian line profile, and $\sigma_{\rm RMS}$ is the RMS noise level in the spectrum. Unfortunately deriving expressions for the random error in the parameters of the two-layer or hill models is more difficult. We therefore use a bootstrap method to investigate the errors in the parameters of these models. We generate a noiseless synthetic spectrum from one of the above simulations, add random noise to it, and then perform the analytic radiative transfer model fits. We do this 100 times with different random noise of the same RMS value in order to record the variance of each model parameter. Assuming the errors are normally distributed, we find that the random error in each parameter is merely the square root of the variance in that parameter.
We initially tested the method on Gaussian line fits, whose random errors should follow the expression in equation \[gauss\]. Figure \[gaussboot\] shows our resulting error determinations for different RMS noise values (filled circles) versus the expected value (solid line). We express the RMS noise in terms of the signal to noise ratio on the lower axis, and also in terms of the integration time it would take to achieve a given RMS for a telescope with a system temperature of 200 K and a main beam efficiency of 0.5 when observing a Gaussian line with a peak brightness temperature of 5 K, along the top axis. We also use a channel width of 0.04 km s$^{-1}$ in this case, less than the input 0.1 km s$^{-1}$ velocity dispersion of the Gaussian line and typical of current spectrometers. The bootstrap estimates are a good fit to equation \[gauss\], with minimal scatter due to the random nature of the technique. This test indicates that the bootstrap method is a robust estimator of the random error in the parameters of the Gaussian fit and should be a robust estimator of the random error in the parameters of the analytic radiative transfer model fits.
We apply the bootstrap method to the [HCO$^{+}$]{} () line profiles of Simulation A (Constant Infall) shown in figure \[majo\]. Figure \[hillboot\] summarizes our findings for the random errors in the velocity dispersion, infall velocity, and line of sight velocity parameters of the [hill5]{} model. The closed circles indicate the error in parameters as a function of integration time for a spectral resolution of 0.04 km s$^{-1}$, while the open circles are those for a spectral resolution of 0.16 km s$^{-1}$. The error increases with the lower spectral resolution because the self-absorption feature starts to be resolved out at that resolution. The cross in the center of each panel indicates the average systematic error obtained from the fits on figure \[maf\]. The average systematic error in the velocity dispersion is 0.008 km s$^{-1}$, the average systematic error in the infall velocity is 0.009 km s$^{-1}$, while the average systematic error in the line of sight velocity is 0.005 km s$^{-1}$. The solid line in both the velocity dispersion panel and the line of sight velocity panel are the Gaussian errors from figure \[gaussboot\]. Unsurprisingly the errors in the hill model parameters are greater than those in the Gaussian fit, which has fewer free parameters of the [hill5]{} model, however they tend to follow a similar power law trend with increasing integration times. What we can see from this plot is that it is possible to be dominated by the systematic errors of the hill model after a reasonable amount of integration time on a typical millimeter radio telescope. For spectral resolutions adequate to observe the line shape, the velocity dispersion random error drops below the systematic level after less than 30 seconds of integration, while the line of sight velocity is well determined after less than two minutes. Getting to a point where the systematic errors in the infall velocity dominate over the random errors takes the longest amount of time, but only about 10 minutes for a line whose brightness temperature is approximately 5 K. This means that generally the infall velocity is as well determined as it can get by the hill model when the signal to noise ratio is greater than $\sim30$. We assume, based on our experience with gaussian line fitting that maintaining a similar signal to noise ratio in lines of differing peak intensities should yield similar errors. This implies an integration time of 30 minutes for a line whose brightness temperature is approximately 3 K, and 250 minutes for a line whose brightness temperature is approximately 1 K. This means the observations required to obtain the required signal to noise can take a prohibitively long time for weak asymmetric line profiles.
Depletion and Beam Smoothing {#depletionbeam}
----------------------------
Depletion in high density regions of a molecular cloud core is a significant effect in observed infall regions. Many observations of CO, CS, and [HCO$^{+}$]{} in dense cores show a ring distribution thought to result from an abundance drop in the gas phase due to freeze out of these molecules onto grains [@klv; @wlv; @kallsuw; @all; @cwtdm; @bclal; @jw; @blccsl; @tmcwc; @tmcw]. We have added depletion in some cases to perform an investigation of the efficacy of the analytic models at estimating the infall velocity as depletion increases. We adopt the formalism of @tmcwc and set the abundance of [HCO$^{+}$]{} to vary exponentially with the density, $$X(r) = X_{0} \exp\left[-n(r)/n_{d}\right],$$ where $X(r)$ is the abundance of [HCO$^{+}$]{} as a function of cloud radius, $X_{0}$ is the [HCO$^{+}$]{} abundance in the low density limit ($2\times 10^{-9}$), $n(r)$ is the density of molecular hydrogen as a function of cloud radius, and $n_{d}$ is the density at which the abundance drops by a factor of $e$. We included depletion in the Simulation A (Constant Infall) case where $v_{\rm in} = 0.1$ km s$^{-1}$ for $n_{d} = 10^{6},
10^{5.5}, 10^{5}, 10^{4.5}, 10^{4}, 10^{3.5},$ and $10^{3}$ cm$^{-3}$. We found for the [hill5]{} model the fit infall velocity changed by less that one hundredth of a kilometer per second down to an $n_{d}$ of $10^{4}$ cm$^{-3}$ for the [HCO$^{+}$]{} () line and $10^{5}$ for the [HCO$^{+}$]{} () line. Figure \[depletion\] illustrates that in each of these cases that level of depletion corresponds to a change in the line profile from a two-peaked self-absorbed profile to a “shoulder” profile. Severe depletion, beyond the levels reported in figure \[depletion\] tends to result in a line too weak to be observed. Based on this simulation, we infer that moderate levels of depletion which does not render the line profile symmetric or unobservable has only a small effect on the [hill5]{} model’s infall velocity determinations.
Another important observational effect is smoothing by a telescope beam. Our simulated line profiles were generated by integrating along a pencil beam through the center of a simulated cloud. We have generated 2 dimensional grids of pencil beam integrations of our simulated clouds and can convolve them to produce a simulated Gaussian telescope beam. We begin by projecting the cloud on the sky plane at a distance of 140 pc, so that a beam of 60 is equal to a distance of about 0.04 pc in the cloud. We then convolved two cases from Simulation A (Constant Infall), where $v_{\rm in} =
0.1$ and $0.2$ km s$^{-1}$, to generate beam sizes of 20 (0.014 pc), 40 (0.027 pc), and 60 (0.041 pc) in the [HCO$^{+}$]{} () and [H$^{13}$CO$^{+}$]{} () lines. The resulting beam-smoothed line profile seen toward the center of each simulated clouds is shown in figure \[beam\]. From the left column of figure \[beam\] we see that beam smoothing has a similar effect to depleting the core. This is not surprising as both depletion and beam smoothing tend to increase the relative contribution of emission from the envelope of the molecular cloud. Our investigation indicates that the infall speed is well fit in cases where the beam width is less than or equal to the core diameter ($10^{17}$ cm in our simulation), however this is dependent on both the density structure and velocity structure of the molecular cloud. In practical terms, a higher resolution probes a smaller region which may have a large infall velocity and therefore a more asymmetric line.
Applying the Analytic Models {#method}
============================
As a final exercise we have applied the [hill5]{} model to a high signal to noise line profile. We have learned from the simulations that the [hill5]{} model is the most robust model for determining the infall velocity of a molecular cloud from the line profile for a majority of line profile shapes. In particular the [hill5]{} model yields a good fit when the line profile has distinct blue-shifted and red-shifted peaks with a self-absorption minimum between them. For line profiles with this shape, in simulations A and D, which had the same infall speed at all radii, the RMS error of the [hill5]{} infall velocity determination is 0.01 km s$^{-1}$, better than the overall RMS error in these simulations of 0.02 km s$^{-1}$. We recommend using the [hill5]{} model to derive infall velocities from asymmetric molecular line profiles.
In some cases there is little or no red-shifted peak, but merely a shoulder on the red side of the line profile. These cases tend to occur when the infall velocity is greater than the velocity dispersion. In Simulation A (Constant Infall) and Simulation C (Core Infall) red-shoulder profiles occur in the [HCO$^{+}$]{} () transition for simulated infall velocities greater than 0.1 km s$^{-1}$, in Simulation B (Envelope Infall) red-shoulder profiles occur in both the transitions for infall velocities greater than 0.1 km s$^{-1}$, while in Simulation D (Dense Bonnor-Ebert Sphere) red-shoulder profiles never occur for all the infall velocities in our simulations. In these “shoulder” spectra the [twolayer6]{} model gives a comparable infall velocity estimate to the [hill5]{} model. We note that in Simulation C, where the [twolayer6]{} model performs better than the [hill5]{} model the red-shoulder model, lines tend to be weak ($<1$ K). When using the [twolayer6]{} model it is important to choose the local minimum (which should also be the global minimum in these cases) that corresponds to the “shoulder” solution set. One can verify that the “shoulder” solution has been found by examining the best fit excitation temperature of the front layer, which should be greater than the background temperature. If the front layer excitation temperature is equal to the background temperature then it is likely that the one has found the “dip” solution which is usually never a good fit to the infall velocity.
Why is the [hill5]{} model good at finding the infall velocity of most clouds, while the [twolayer6]{} model is successful only on spectra with an asymmetric shoulder? The key to understanding this lies in how well each model matches the excitation conditions along the line of sight at each Doppler velocity. In figure \[excivtau\] we presented cases in which both the [hill5]{} and [twolayer6]{} models match the excitation profile of the simulation quite well for $\tau<3$ at the line center velocity. Although the figure depicts only one Doppler velocity, the match continues to hold over the entire line profile. The resulting infall velocities derived from these models closely match the infall velocities in the simulated contracting cloud. In cases where the infall velocities derived from the analytic models do not match the infall velocities in the simulated clouds, the modeled excitation temperature profiles as a function of optical depth along the line of sight derived from the spectral line fits do not match the simulated excitation temperature profiles. Even in cases where the spectral line match is quite good, if the excitation conditions are not well matched then all the physical parameters, including the infall velocity, will not match the parameters of the simulated or observed cloud.
We find that the [hill5]{} model tends to reproduce the excitation temperature profile as a function of optical depth quite well for a wide range of infall velocities. The reason that this is true is that the excitation conditions in a cloud do tend to rise as a function of optical depth over the first couple of optical depths, until the excitation temperature reaches the kinetic temperature of the core or begins dropping again as the density drops. In all of our simulations a linear fit to the rising excitation temperature, starting with an excitation temperature equal to the background temperature, is a good approximation of the actual excitation temperature profile at all Doppler velocities. As a result the [hill5]{} model is usually sensitive to the physical conditions of the collapsing cloud and yields a good estimate of the infall velocity if the cloud radii which comprise the first few optical depths of the line profile are infalling. For this reason we recommend using the [hill5]{} model. Our simulations and error analysis suggest that the line profile should have a peak brightness temperature greater than 1 K and a signal to noise ratio greater than 30. The line profile must also have a well defined dip or shoulder which must be resolved by the spectrometer resolution. If these conditions are met the [hill5]{} model can be expected to provide an estimate of the infall velocity with an accuracy of 0.02 km s$^{-1}$. Our simulations suggest the accuracy for dip profiles alone is 0.01 km s$^{-1}$.
The [twolayer6]{} model tends to reproduce the excitation temperature profile as a function of optical depth well only for higher infall velocities which tend to produce spectral lines that are very asymmetric or have well defined red shoulders. In these cases it may be tempting to use the [twolayer6]{} method to derive the infall velocity, however there are several caveats which one must take in to account. In cases with low infall velocities or more symmetric line profiles the excitation temperature of the front layer is often low (at or near the cosmic microwave background temperature) while that of the rear layers is often much higher than the maximum excitation temperature in the cloud, even in cases where the [twolayer6]{} model produces a good fit to the emerging line profiles. In these cases the infall velocities as other physical conditions derived from the [twolayer6]{} model do not match the conditions in the simulated cloud. The reason this occurs is that there is a certain amount of degeneracy in reproducing a symmetric or nearly symmetric line profile with the [twolayer6]{} model. A very high excitation temperature region behind a very low excitation temperature region produces the same nearly symmetric profile as a moderately high excitation temperature region behind a moderately low excitation temperature region. At higher infall velocities, when the line profile is very asymmetric or has only a shoulder, the excitation temperature of the front region can be accurately determined from the red-shifted emission of that line profile. This breaks the degeneracy and creates an excitation temperature profile along the line of sight that more closely matches the simulation and is more well determined overall. As a result the physical parameters of the simulation, including the infall velocity, tend to be well determined by the [twolayer6]{} model in these cases.
In figure \[data\] we present a high signal to noise line profiles upon which we will carry out the procedure prescribed above. The line profile is the isolated component of an [N$_{2}$H$^{+}$]{} () line profile in L1544 observed by Bourke et al. (in preparation). Based on our results in the previous sections we choose to model the emission using the [hill5]{} model. That best fit is shown by the points in figure \[data\]. We see that the fit is fairly good, though it deviates from the line profile by producing a second peak instead of a shoulder. The [hill5]{} parameters that result in this fit are shown in table \[datahillfit\]. In this case our [hill5]{} infall estimate of 0.099 km s$^{-1}$ matches well with the best fit [twolayer6]{} estimate of 0.095 km s$^{-1}$. Unfortunately since we did not create this profile from a model we do not know the actual kinematic conditions in the cloud, however there are two other estimates obtained using detailed radiative transfer methods. @krbp have fit a model to [N$_{2}$H$^{+}$]{} () and [N$_{2}$H$^{+}$]{} () spectra at 4 positions around L1544. They find an infall speed profile that accelerates inwards from the cloud edge to 0.24 km s$^{-1}$. Although this peak infall speed is much more than our estimate, we note that their fits to L1544 include no velocity dispersion. We speculate that the infall speed may be artificially high in order to match the line width in the absence of any turbulent velocity component. Bourke et al. (in preparation) have observed this particular source in several molecular line transitions, include [N$_{2}$H$^{+}$]{} (), [N$_{2}$H$^{+}$]{} (), DCO$^{+}$ (), DCO$^{+}$ (), CS (), CS (), CS (), and C$^{34}$S (). They have used the density model of this cloud proposed by @tmcwc and assumed infall at the same velocity over the entire cloud along with a kinetic temperature of 10 K to create radiative transfer models of L1544 and match their observed line profiles. A simultaneous match of all the components in the [N$_{2}$H$^{+}$]{} spectral lines currently yields an infall velocity of 0.1 km s$^{-1}$, which is in agreement with our estimate to within a hundredth of a kilometer per second.
[lllll]{} 3.26 & 7.183 & 0.099 & 0.070 & 6.14\
Conclusion
==========
We have shown that analytic radiative transfer models are effective at estimating the infall velocities of contracting clouds from the blue-asymmetric line profiles of species of moderate optical depth in the region of collapse. The models provide good velocity estimates with few assumptions. The greatest advantage of the analytic models is that they allow for a quick and reasonable estimate of infall velocity with a minimum of computational effort. In cases where nothing else is known about the source, they provide the only possible estimate of infall, and in regions where a more robust model can be constructed, the analytic models provide an excellent starting point for further modeling [^1]. The main conclusions of this work are as follows:
1. Analytic radiative transfer models can reproduce the blue-asymmetric line profiles seen in infalling starless cores. In addition to the two-layer model first presented in @mmtww we have found a second class of radiative transfer models (the hill models) and explored 5 variants of these two classes of analytic radiative transfer models. All the model variations were able to produce blue-asymmetric line profiles similar to those we observe in starless cores.
2. Successful analytic radiative transfer models reproduce not only the line profile shape, but also match the initial excitation conditions along the line of sight up to or beyond an optical depth of 1. When this requirement is met, the modeled infall velocity successfully fits the simulated infall velocity. Those models which match the line profile shape, but do not match the line of sight excitation conditions cannot reveal the infall velocity and velocity dispersion in the cloud.
3. The [hill5]{} analytic radiative transfer model is the preferred model when fitting a blue-asymmetric line profile. In simulations with constant velocity as at all cloud radii the infall velocities attained by fitting the [hill5]{} analytic radiative transfer model to the simulated line profiles match the infall speed with an RMS error 0.02 km s$^{-1}$. The [hill5]{} model tends to perform especially well when there are two distinct peaks such that the intensity difference between the self-absorption trough and the redshifted peak is greater than 10% of the intensity of the blueshifted peak. The RMS error in the infall determination of [hill5]{} fits to simulated spectral lines with those characteristics from Simulations A and D is 0.01 km s$^{-1}$.
4. The physical parameters of simulated clouds that produce blue-asymmetric line profiles with no distinct shoulder or redshifted peak are generally not well determined by any of the model variations we have studied in this paper. We recommend not using these models to estimate infall velocities from line profiles that do not have a separate redshifted peak or a distinct redshifted shoulder.
5. A better match to the line profile shape does not necessarily imply a better match to the excitation profile or the physical parameters, such as the infall velocity, of the observed cloud. This can be seen most clearly in figure \[dipshoulder\] where local minimum of the “dip” solution often provides a lower $\chi^{2}$ fit to the line profile shape, but a poorer fit to the infall velocity than the “shoulder” solution. Following the instructions in §\[method\] when fitting a blue-asymmetric line profile will minimize the risk of obtaining an unusually poor estimate of the physical parameters from the analytic radiative transfer model fit to the line profile.
6. All the analytic models lose sensitivity to the infall velocity if there is a significantly optically thick static envelope between the observer and the infall region. In these cases a more optically thin line can provide a better estimate to the infall velocity in the obscured region. Conversely an optically thin tracer may underestimate the infall velocity of an infalling envelope around a static cloud core. These methods are sensitive to the velocities and conditions in the regions in which the line is becoming optically thick.
7. A peak signal to noise ratio greater than or equal to 30 is required for the systematic error on the infall velocity estimate to be greater than the random error. We recommend achieving a peak signal to noise ratio of 30 in order to minimize the random error of a line profile fit using our analytic radiative transfer models, however nothing is gained by achieving a higher signal to noise level. Achieving this signal to noise level can take a prohibitively long time for lines whose peak antenna temperature is less than or equal to 1 K.
8. Depletion and beam smoothing can greatly affect the observed line profile in infalling starless cores. We conclude that, even in the presence of these effects, if the line profile is asymmetric and a good candidate to be fit using these analytic radiative transfer models (as described in §\[method\]), then generally the radiative transfer models do yield an accurate estimate of the infall velocity. If the cloud is depleted to such an extent or viewed with such a large beam that the line profile becomes very weak or symmetric, then these models will not accurately probe the infall region of the starless core.
We are grateful to Tyler Bourke for contributing his L1544 data and some preliminary results from his radiative transfer modeling for use in this paper. We also acknowledge Henrik Beuther for his encouraging advice and an anonymous referee for his or her valuable comments and suggestions. This research was conducted with support from NASA Origins of Solar Systems Grant NAG5-13050.
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[^1]: Implementations of the models used for this paper are available from the authors at [http://cfa-www.harvard.edu/\~{}cdevries/analytic\_infall.html](http://cfa-www.harvard.edu/~{}cdevries/analytic_infall.html)
|
---
author:
- |
Lloyd Windrim, Rishi Ramakrishnan, Arman Melkumyan and Richard Murphy\
Australian Centre for Field Robotics\
University of Sydney\
[{ l.windrim,r.ramakrishnan,a.melkumyan,r.murphy } @acfr.usyd.edu.au]{}
bibliography:
- 'ICRA2017.bib'
title: Hyperspectral CNN Classification with Limited Training Samples
---
Previous Work {#sec: prevWork}
=============
Illumination Invariance in Hyperspectral Classification
-------------------------------------------------------
In the remote-sensing community, a well-known classifier is the Spectral Angle Mapper (SAM) which computes the inverse cosine of the normalised dot product between a spectra of interest and library spectra [@Yuhas1992]. SAM achieves some degree of illumination invariance as the classifier is robust to multiplications of the spectra that are constant across wavelength. Whilst this makes it invariant to different intensities, it does not cater for the effect of shadows which is a wavelength dependant multiplication of the spectra. An alternative approach is to transform the spectra into an illumination invariant form prior to classification. From computer vision, the method of [@Finlayson2006a] derives a transformation for RGB images to a 2D log-chromaticity space in which an axis exists where changes in illumination due to intensity and shadow are suppressed. This approach is extended to hyperspectral images in [@Salekdeh2011]. The problem with this approach is in the assumptions that are made in order to derive the transformation, particularly the assumption of Planckian illumination where the spectral power distribution of the incident light is modelled by Wein’s approximation to Plank’s law. This is because atmospheric absorption features are not accounted for by the approximation and have a large impact on the incident light spectra. Finally, multi-modal approaches [@Friman2011; @Ramakrishnan2015] which use additional sensors such as Light Detection and Ranging (LiDAR) and Global Positioning Systems (GPS) can form geometry-based illumination models of the scene and compensate for the variations in lighting, however, the additional sensors required in these approaches are not always available.
CNNs for Hyperspectral Classification
-------------------------------------
In recent years, CNNs have been utilised for pixel-wise classification of hyperspectral images. However, many of the proposed CNNs convolve in the spatial domain and not the spectral domain. The CNN in [@Makantasis2015] convolves a small window over spatial patches extracted from a dimensionality reduced hyperspectral cube. A similar approach is taken by [@Zhao2016] where a CNN learns spatial features from patches extracted from a dimensionality reduced hyperspectral cube, but these spatial features are combined with spectral features learnt using a method based on Local Discriminant Embedding (LDE). These features are used together to do classification with Logistic Regression and Support Vector Machines (SVMs). In [@Lee2016] spatial patches are extracted from the hyperspectral cube, but this time the cubes dimensionality has not been reduced such that the patches cover the whole spectrum. This network convolves spatial filters of different sizes over the patch, again with no convolutions occurring over the spectral channel. There are some simple spatial augmentations done to avoid overfitting, but nothing to cover the large variability that can occur in the shape of the spectra. A very simple CNN was proposed by [@Hu2015] which learnt features by convolving over the spectral channel. This approach was shown to perform favourably against other types of neural networks as well as SVMs. The architecture chosen consisted of only one convolutional layer and one fully-connected layer. In all of these works the datasets used for evaluation were almost exclusively captured from satellite or airborne platforms. Also, there is no analysis into the use of radiometric normalisation methods other than flat-field correction (requiring a calibration panel in the scene). Finally, all of the scenarios assume that the training examples are sufficient for capturing the variability in the data, which is not always possible in a robotics or autonomous application. Hence, it is unknown how robust these networks are when the training data does not capture all of the variability in the data.
Conclusion {#sec: conclusion}
==========
This work has proposed a method for training a CNN to be robust to several factors of variation in hyperspectral data with only a limited amount of training data. This was done through relighting augmentation using an approximation of the irradiance ratio found from the scene. This work also showed that it is possible to train a CNN to classify hyperspectral data without first radiometrically normalising it using a calibration panel in the scene. This has paved the way for utilisation of CNNs with hyperspectral cameras in robotics applications. Future work involves the incorporation of spatial information and developing large scale, multi-label classifiers.
Acknowledgments
===============
This work has been supported by the Rio Tinto Centre for Mine Automation and the Australian Centre for Field Robotics, University of Sydney.
|
---
---
[**[A Categorification of the Spin Representation of $U(\mf{so}(7,\C))$ via Projective Functors]{}** ]{}
Yongjun Xu, Shilin Yang$^{\dag}$
[*College of Applied Sciences, Beijing University of Technology\
*Beijing 100124, P. R. China** ]{}
> [[**Abstract.**]{} The purpose of this paper is to study a categorification of the $n$-th tensor power of the spin representation of $U(\mf{so}(7,\C))$ by using certain subcategories and projective functors of the BGG category of the complex Lie algebra $\mf{gl}_n$.]{}
[**Key Words:**]{}BGG category; Categorification; Lie algebra; Projective functor; Spin representation.
[**Mathematics Subject Classification:**]{}17B10.
Introduction
============
The general idea of categorification was introduced by Crane and Frenkel [@C; @CF]. In recent years categorifications of algebras and their representations have been studied by many mathematicians, see for example [@KMS479; @M; @MM; @R] and references therein.
Let $\Og{\mf{g}}$ be the BGG category associated to a triangular decomposition of a finite dimensional complex semisimple or reductive Lie algebra $\mf{g}$. The BGG category $\Og{\mf{g}}$ and its projective functors play an important role in many algebraic categorifications, which can be seen from the following two facets. On one hand, the projective functors of $\Og{\mf{g}}$ are extensively used in catetegorifications of group algebras of finite Weyl groups and their Hecke algebras. In [@KMS479], Khovanov, $et~al.$ presented several examples about categorifications of various representations of the symmetric group $S_n$ via projective functors acting on certain subcategories of the BGG category $\Og{\mf{sl}_n}$. Especially in [@KMS1163] they categorified integral Specht modules over $S_n$ and its Hecke algebra by some translation functors of $\Og{\mf{sl}_n}$. Mazorchuk and Stroppel [@MS1363] constructed a subcategory of $\Og{\mf{g}}$ on which the actions of translation functors categorify (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. Basing on the results in [@MS1363], they [@MS1] gave a categorification of Wedderburn¡¯s basis for $\C[S_n]$. Moreover, Mazorchuk and Miemietz [@MM] reproved and extended the result in [@MS1363] by studying 2-representations of abstract 2-categories from a more systematic and more abstract prospective. In addition, Mazorchuk and Stroppel [@MS2939] applied graded versions of translation functors and a subcategory of the principal block of $\Og{\mf{g}}$ to categorifications of a parabolic Hecke module (see also [@S]). On the other hand, the BGG category $\Og{\mf{g}}$ and its projective functors can be applied to categorificaitons of universal enveloping algebras of simple Lie algebras. In [@BFK] Bernstein, $et~al.$ investigated a categorification of the $n$-th tensor power of the fundamental representation of $U(\mf{sl}_2)$ via certain singular blocks and projective functors of $\Og{\mf{gl}_n}$ (see also [@KMS479]). Following [@BFK] Sussan [@SU] generalized the case of $\mf{sl}_2$ in [@BFK] to that of $\mf{sl}_k$ and studied $\mf{sl}_k$-link invariants.
In [@BFK] Bernstein, $et~al.$ raised a more difficult problem: categorifications of the representation theory of arbitrary simple Lie algebra $\mf{g}$. The main purpose of this article is to study a categorification of the $n$-th tensor power of the spin representation of $U(\mf{so}(7,\C))$. The main tools for our categorification are also the BGG category $\Og{\mf{gl}_n}$ and its projective functors. Our work can be considered as a part of categorifications of the representation theory of $U(\mf{g})$ for the simple Lie algebra $\mf{g}$ of type $B_3$. In other words, we categorify the image of $U(\mf{so}(7,\C))$ under the algebra homomorphism $\Phi:U(\mf{so}(7,\C))\to {\End}(V_{\sp}^{\otimes n})$ corresponding to the $n$-th tensor power of the spin representation $V_{\sp}$ of $U(\mf{so}(7,\C))$. In fact, as standard representations of the special orthogonal Lie algebras $\mf{so}(m,\C)$, the spin representations are especially important since they not only play a fundamental role in the realization of exceptional simple Lie algebras but also have many important applications in Lie group, geometry and physics (see [@LM] Chapters I.5 and IV.9, and [@FH] Chapter 20).
This paper is organized as follows. In Section \[sect-2\], we collect the background material that will be necessary in the sequel. In Section \[sect-3\], we obtain a categorification of the $n$-th tensor power of the spin representation of $U(\mf{so}(7,\C))$. First, we categorify the underlying space of the $n$-th tensor power $V_{\sp}^{\otm n}$ of the spin representation $V_{\sp}$ for $U(\mf{so}(7,\C))$ by using certain subcategories of $\O(\mf{gl}_n)$ (Theorem \[t1\]). Next we yield a categorification of the $U(\mf{so}(7,\C))$ action on $V_{\sp}^{\otm
n}$ by projective functors of $\O(\mf{gl}_n)$ (Theorem \[c1\]). Finally, we lift defining relations of $U(\mf{so}(7,\C))$ to natural isomorphisms between functors (Theorem \[t3\]).
Throughout, we denote by $\C,$ $\R$ and $\Z$ the complex number field, the real number field and the set of integers respectively.
Preliminaries {#sect-2}
=============
We start by reviewing some basic results about the universal enveloping algebra of Lie algebra $\mf{so}(7,\C)$ and the BGG category of a complex reductive Lie algebra.
As an associative algebra, the universal enveloping algebra $U(\mf{so}(7,\C))$ of the special orthogonal Lie algebra $\mf{so}(7,\C)$ is generated by $h_i, e_i, f_i (1\leq i\leq 3)$ over $\C$ which are subject to the following relations: $$\begin{aligned}
h_i h_j=h_j h_i,\quad e_i f_j-f_j e_i=\delta_{i,j}h_i,\end{aligned}$$ $$\begin{aligned}
h_i e_j-e_j h_i=a_{i,j}e_j,\quad h_i f_j-f_j h_i=-a_{i,j}f_j,\end{aligned}$$ $$\begin{aligned}
{\mathop{\sum}\limits_{k=0}^{1-a_{i,j}}}
(-1)^k \Big( {\begin{array}{*{20}c}
1-a_{i,j} \\
k \\
\end{array}} \Big)
e_i^{1-a_{i,j}-k}e_j e_i^k=0~for~i\neq j,\end{aligned}$$ $$\begin{aligned}
{\mathop{\sum}\limits_{k=0}^{1-a_{i,j}}}
(-1)^k \Big( {\begin{array}{*{20}c}
1-a_{i,j} \\
k \\
\end{array}} \Big)
f_i^{1-a_{i,j}-k}f_j f_i^k=0~for~i\neq j,\end{aligned}$$ where $a_{i,j}(1\leq i,j\leq 3)$ are the entries of the Cartan matrix $A=(a_{i,j})_{3\times 3}$ of $\mf{so}(7,\C)$.
Let $V_{\sp}={\mathop{\oplus}\limits_{i=0}^{7}}\C v_i$ be an 8-dimensional vector space over $\C$. Then $V_{\sp}$ is a $U(\mf{so}(7,\C))$-module in the following way: $$\begin{aligned}
&& h_1 v_7=0,~h_1 v_6=0,~h_1 v_5=v_5,~h_1 v_4=v_4,~h_1 v_3=-v_3,
~h_1 v_2=-v_2, ~h_1 v_1=0,~ h_1 v_0=0,\\
&&h_2 v_7=0,~ h_2 v_6=v_6,~ h_2 v_5=-v_5,~ h_2 v_4=0,~ h_2
v_3=0,~h_2
v_2=v_2, ~h_2 v_1=-v_1, ~ h_2 v_0=0,\\
&&h_3 v_7=v_7, ~ h_3 v_6=-v_6, ~ h_3 v_5=v_5, ~ h_3 v_4=-v_4, ~h_3
v_3=v_3, ~ h_3 v_2=-v_2, ~ h_3 v_1=v_1, ~ h_3 v_0=-v_0,\\
&&e_1 v_7=0, ~ e_1 v_6=0, ~ e_1 v_5=0, ~ e_1 v_4=0, ~ e_1 v_3=v_5, ~
e_1 v_2=v_4, ~ e_1 v_1=0, ~ e_1 v_0=0,\\
&&e_2 v_7=0, ~ e_2 v_6=0, ~ e_2 v_5=v_6, ~ e_2 v_4=0, ~
e_2 v_3=0, ~ e_2 v_2=0, ~ e_2 v_1=v_2, ~ e_2 v_0=0,\end{aligned}$$ $$\begin{aligned}
&&e_3 v_7=0, ~ e_3 v_6=v_7, ~ e_3 v_5=0, ~ e_3 v_4=v_5, ~ e_3 v_3=0,
~ e_3 v_2=v_3, ~ e_3 v_1=0, ~ e_3
v_0=v_1,\\
&&f_1 v_7=0, ~ f_1 v_6=0, ~ f_1 v_5=v_3, ~ f_1
v_4=v_2, ~ f_1 v_3=0, ~ f_1 v_2=0, ~ f_1 v_1=0, ~ f_1 v_0=0,\\
&&f_2 v_7=0, ~ f_2 v_6=v_5, ~ f_2 v_5=0, ~ f_2 v_4=0, ~
f_2 v_3=0, ~ f_2 v_2=v_1, ~ f_2 v_1=0, ~ f_2 v_0=0,\\
&&f_3 v_7=v_6, ~ f_3 v_6=0, ~ f_3 v_5=v_4, ~ f_3 v_4=0, ~ f_3
v_3=v_2, ~ f_3 v_2=0, ~ f_3 v_1=v_0, ~ f_3 v_0=0.\end{aligned}$$ The $U(\mf{so}(7,\C))$-module $V_{\sp}$ is called the spin representation of $U(\mf{so}(7,\C))$.
For convenience, we fix some notations which we need in the sequel. All Lie algebras and their representations are defined over $\C$. Let $\mf{g}$ be a finite dimensional reductive Lie algebra with a fixed triangular decomposition $\mf{g}=\mf{n}_+\oplus \mf{h}\oplus
\mf{n}_-.$ Denote by $U(\mf{g})$ the universal enveloping algebra of $\mf{g}$, $Z(U(\mf{g}))$ the center of $U(\mf{g})$ and $\Theta$ the set of the central characters. $W$ denotes the Weyl group of $\mf
g$. $\rho$ is the half-sum of positive roots. Define the dot-action of $W$ on $\ch$ as follows: $w\cdot \lambda=w(\lambda+\rho)-\rho$. For $\lambda\in \ch$, let $\theta_{\lambda}:Z(U(\mf{g}))\rightarrow\C$ be the corresponding central character and $M(\lambda)$ the Verma module with the highest weight $\lambda.$
Let $\ch_{dom}$ be the set of all elements in $\ch$ dominant with respect to the dot-action. Then there is a map $\eta:\ch\rightarrow\Theta$ which maps $\lambda$ to $\theta_{\lambda}$ sets up a bijection between $\ch_{dom}$ and $\Theta$ (see [@Di], Section 7.4). The notation $\Og{\mf{g}}$ denotes the BGG category of $\mf{g}$ associated to the triangular decomposition $\mf{g}=\mf{n}_+\oplus \mf{h}\oplus\mf{n}_-.$ For any $\theta\in \Theta$ denote by $\Ogc{\theta}{\mf{g}}$ the full subcategory of $\Og{\mf{g}}$ whose objects are the modules $M$ where $$M=\bigg\{m\in M\big|(z-\theta(z))^{n}\cdot m=0~{\rm for~some~}n\in
\N~{\rm for~each}~z\in Z(U(\mf{g}))\bigg\}.$$ The BGG category $\Og{\mf{g}}$ is the direct sum of the subcategories $\Ogc{\theta}{\mf{g}}$ as $\theta$ ranges over the central characters of the form $\theta_{\lambda}$ (see [@H], Section 1.12).
Now we give a brief introduction to projective functors.
Denote by $\textrm{proj}_{\theta}$ the functor from $\Og{\mf{g}}$ to $\Og{\mf{g}}$ that, to a module $M={{\mathop{\oplus}\limits_{\theta\in
\Theta}}}M(\theta),$ associates the $\theta$-component summand $M(\theta)$ of $M$. Let $F_V$ be the functor of tensoring with a finite-dimensional $\mf{g}$-module $V.$
$F:\Og{\mf{g}} \to \Og{\mf{g}}$ is a projective functor if it is isomorphic to a direct summand of the functor $F_V$ for some finite dimensional $\mf{g}$-module $V.$
Denote by $K(\A)$ the Grothendieck group of an abelian or triangulated category $\A.$ It is the free abelian group generated by the symbols $[M]$ where $M$ is an object of $\A.$ The only relations in this group are of the form $[N]=[M]+[P]$ when there is a short exact sequence or distinguished triangle of the form $0 \to
M \to N \to P \to 0.$ The image of an object $M$ and an exact functor $F$ in the Grothendieck group will be denoted by $[M]$ and $[F]$ respectively.
The following properties of projective functors can be found in Section 3.2 or 3.4 of [@BG].
\[p3\]
1. Projective functors are exact.
2. Any direct sum of projective functors is a projective functor.
3. Any composition of projective functors is a projective functor.
4. The functor $\textrm{\rm proj}_{\theta}:\Og{\mf{g}} \to \Og{\mf{g}}$ is a projective functor.
5. Let $F,G$ be projective functors. If $[F]=[G],$ then $F\cong G.$
Fix a central character $\theta,$ then $$\begin{aligned}
\label{base}
\bigg\{~[M(\lambda)]~\big|~\theta=\theta_{\lambda}~\bigg\}
=\bigg\{~[M(\mu)]~\big|~\mu\in W\cdot\lambda~\bigg\}\end{aligned}$$ forms a $\Z$-basis of the Grothendieck group $K(\Ogc{\theta}{\mf{g}})$ (see [@H], Section 1.10 and 1.12). The following proposition shows that this basis is handy for writing the action of projective functors on the Grothendieck group of $\O(\mf{g})$(see [@BFK], Section 2.3.2, and [@BG], Section 1.12).
\[p4\] Let $V$ be a finite-dimensional $\mf{g}$-module, $\mu_1,\cdots,\mu_m$ the multiset of weights of $V$, i.e., there is a basis $v_1,v_2,\cdots, v_m$ of $V$ such that the weight of the vector $v_i$ equals $\mu_i$, $M(\lambda)$ the Verma module with the highest weight $\lambda,$ then we have $[V\otimes
M(\lambda)]=\sum_{i=1}^m [M(\lambda+\mu_i)]$ in the Grothendieck group $K(\Og{\mf{g}}).$
For unexplained concepts and notations, we refer the reader to [@BG; @BGG; @FH; @HK; @H; @M].
Categorification of the Spin Representation of $U(\mf{so}(7,\C))$ {#sect-3}
=================================================================
This section is to obtain a categorification of the $n$-th tensor power $V_{\sp}^{\otm n}$ of the spin representation $V_{\sp}$ for $U(\mf{so}(7,\C))$ in the following three steps.
- Categorifying the underlying space of the $n$-th tensor power $V_{\sp}^{\otm n}$ of the spin representation $V_{\sp}$ of $U(\mf{so}(7,\C))$ by using certain subcategories of the BGG category $\O(\mf{gl}_n)$.
- Yielding a categorification of the $U(\mf{so}(7,\C))$ action on $V_{\sp}^{\otm
n}$ by projective functors of $\O(\mf{gl}_n)$.
- Lifting defining relations of $U(\mf{so}(7,\C))$ to natural isomorphisms between functors.
We fix once and for all a triangular decomposition $\mathfrak{n}_+\oplus \mathfrak{h}\oplus \mathfrak{n}_-$ of the Lie algebra $\gl.$ The Weyl group of $\mf{gl_n}$ is isomorphic to the symmetric group $S_n.$ Choose a standard orthogonal basis $\varepsilon_1,\cdots,\varepsilon_n$ in the Euclidean space $\R^n$ and identify the complexification $\C\otm_{\R}\R^n$ with the dual $\ch$ of Cartan subalgebra so that $R_+=\{~
\varepsilon_i-\varepsilon_j~|~1\leq i < j\leq n~\}$ is the set of positive roots and $\beta_i=\varepsilon_i-\varepsilon_{i+1},1\le
i\le n-1$ are simple roots. The generator $s_i$ of the Weyl group $W=S_n$ acts on $\ch$ by permuting $\varepsilon_i$ and $\varepsilon_{i+1}.$ Denote by $\rho$ the half-sum of positive roots $$\rho=\frac{n-1}{2}\varepsilon_1+\frac{n-3}{2}\varepsilon_2+\cdots+\frac{1-n}{2}\varepsilon_n.$$ We denote by $[0, 7]$ the integers $0\leq k\leq 7.$ For a sequence $(a_1,\cdots, a_n)\in[0, 7]^n$ we denote by $M(a_1,\cdots, a_n)$ the Verma module with the highest weight $a_1\varepsilon_1+\cdots+a_n\varepsilon_n-\rho.$
Let ${\bf D}$ be the set of all 8-tuples of nonnegative integers ${\bf d}=(d_0,d_1,d_2,d_3,d_4,d_5,d_6,d_7)$ such that ${\mathop{\sum}\limits_{k=0}^{7}}d_k=n$. We define the following equivalence relation $\thicksim$ on ${\bf D}$: $$\begin{aligned}
{\bf d}\thicksim {\bf d^{'}}&\Leftrightarrow&\left\{
\begin{array}{ll}
d_7+d_6+d_5+d_4-d_3-d_2-d_1-d_0=d_7^{'}+d_6^{'}+d_5^{'}+d_4^{'}-d_3^{'}-d_2^{'}-d_1^{'}-d_0^{'}, &\\
d_7+d_6-d_5-d_4+d_3+d_2-d_1-d_0=d_7^{'}+d_6^{'}-d_5^{'}-d_4^{'}+d_3^{'}+d_2^{'}-d_1^{'}-d_0^{'},& \\
d_7-d_6+d_5-d_4+d_3-d_2+d_1-d_0=d_7^{'}-d_6^{'}+d_5^{'}-d_4^{'}+d_3^{'}-d_2^{'}+d_1^{'}-d_0^{'},&
\end{array}
\right.\end{aligned}$$ for any ${\bf d}=(d_0,d_1,d_2,d_3,d_4,d_5,d_6,d_7),{\bf
d^{'}}=(d_0^{'},d_1^{'},d_2^{'},d_3^{'},d_4^{'},d_5^{'},d_6^{'},d_7^{'})\in
{\bf D}$. In the following $[{\bf d}]$ and $\widetilde{{\bf D}}$ represent the equivalent class of ${\bf d}$ and the set of all the the equivalent classes respectively.
The spin representation $V_{\sp}$ has the weight space decomposition $V_{\sp}={\mathop{\oplus}\limits_{k=0}^{7}}V_{k},$ where $V_{k}=\C
v_k$ for $0\leq k\leq 7$ (see [@HK] Chapter 2). For ${\bf d'}\in
{\bf D}$ and $(a_1,\cdots, a_n)\in[0, 7]^n$, we define the following condition: $$\begin{aligned}
\label{eqn3-1}\sharp~ \{~a_m~|~a_m=k, 1\leq m \leq n\}
=d_k^{'}~for~0\leq k\leq 7.\end{aligned}$$ It follows that $V_{\sp}^{\otm n}$ has the weight space decomposition $V_{\sp}^{\otm n}={\mathop{\oplus}\limits_{[{\bf
d}]\in \widetilde{{\bf D}}}}(V_{\sp}^{\otm n})_{[{\bf d}]},$ where $(V_{\sp}^{\otm n})_{[{\bf d}]}$ is the $\C$-linear space spanned by $$B^{'}_{[{\bf d}]}:=\bigg\{ v_{a_{1}}\otimes v_{a_{2}}\otimes\cdots
\otimes v_{a_{n}} \big{|} (a_1, \cdots, a_n)\in [0, 7]^n \hbox{
satisfying the conditon (\ref{eqn3-1}) for some}~{\bf d'}\in[{\bf
d}]\bigg\}.$$
From now on, we denote by $^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]}$ the $\Z$-module spanned by $B^{'}_{[{\bf d}]}$ and $^{\Z}V_{\sp}^{\otm
n}:={\mathop{\oplus}\limits_{[{\bf d}]\in \widetilde{{\bf
D}}}}{^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]}}.$ It is easy to see that $\C\otm_{\Z}{^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]}}=(V_{\sp}^{\otm
n})_{[{\bf d}]}$ and $\C\otm_{\Z}{^{\Z}V_{\sp}^{\otm
n}}=V_{\sp}^{\otm n}.$ For each ${\bf d}\in {\bf D}$, set $\lambda_{\bf d}={\mathop{\sum}\limits_{i=0}^{7}}
{\mathop{\sum}\limits_{j=1}^{d_i}} (7-i) \varepsilon_{d_0 + \cdots +
d_{i-1} + j}.$ Denote by $\theta_{\bf d}=\eta(\lambda_{\bf d}-\rho)$ the corresponding central character of $\mf{gl}_n$ under the map $\eta: \ch\rightarrow\Theta$. We define $\O_{\bf d}:=\O_{\theta_{\bf
d}}(\mf{gl}_n),$ $\O_{[{\bf d}]}:={\mathop{\oplus}\limits_{{\bf
d^{'}}\in[{\bf d}]}}\O_{\bf d^{'}}$ and $\O^{n}:={\mathop{\oplus}\limits_{[{\bf d}]\in \widetilde{{\bf
D}}}}\O_{[{\bf d}]}.$
Now we are prepared to realize $^{\Z}V_{\sp}^{\otm n}$ and its weight space $^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]}$ for any $[{\bf d}]\in
\widetilde{{\bf D}}$ as the Grothendieck groups of the categories $\O^n$ and $\O_{[{\bf d}]}$ respectively. Indeed, we have the following result.
\[t1\] There exists an isomorphism of abelian groups $\gamma_n:K(\O^n)\rightarrow ^{\Z}V_{\sp}^{\otm n}$ given by $$\gamma_n([M(a_1,\cdots, a_n)])=v_{a_{1}}\otimes v_{a_{2}}\otimes\cdots \otimes v_{a_{n}}$$ for any sequence $(a_1,\cdots, a_n)\in[0, 7]^n.$ Moreover, the restriction of $\gamma_n$ on $K(\O_{[{\bf d}]})$ is an abelian group isomorphism between $K(\O_{[{\bf d}]})$ and $^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]}$ for any $[{\bf d}]\in
\widetilde{{\bf D}}$.
To prove the theorem, it is sufficient to prove that $\gamma_n:K(\O_{[{\bf d}]})\otimes_{\Z} \C\rightarrow
^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]}$ is an abelian group isomorphism for any $[{\bf d}]\in \widetilde{{\bf D}}$. Indeed, the above abelian group isomorphism will be obvious if we note the following facts.
For any $[{\bf d}]\in \widetilde{{\bf D}}$ and ${\bf d^{'}}\in[{\bf
d}]$ it is seen from (\[base\]) that the set of all the symbols $[M(a_1,\cdots, a_n)]$ satisfying the condition (\[eqn3-1\]) is a $\Z$-basis of the Grothendieck group $K(\O_{\bf d^{'}}).$ We denote this $\Z$-basis by $B_{\bf d^{'}}$. It follows that $B_{[{\bf
d}]}={\mathop{\cup}\limits_{{\bf d^{'}}\in {\bf [d]}}}B_{\bf d^{'}}$ is a $\Z$-basis of the Grothendieck group $K(\O_{[{\bf d}]})$. On the other hand, if we denote by $B'_{\bf d^{'}}$ the set of $v_{a_{1}}\otimes v_{a_{2}}\otimes\cdots \otimes v_{a_{n}}$ such that the sequence $(a_1, \cdots, a_n)\in [0, 7]^n$ satisfies (\[eqn3-1\]), then $B^{'}_{[{\bf d}]}:={\mathop{\cup}\limits_{{\bf
d^{'}}\in [{\bf d}]}}B'_{\bf d^{'}}$ is a $\Z$-basis of the weight space $^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]}$.
Let $\Lo$ be the $n$-dimensional fundamental representation of $\gl$ with weights $\varepsilon_1,\varepsilon_2,\cdots ,\varepsilon_n$ and the corresponding weight vectors $u_1, u_2, \cdots, u_n$. Then its dual representation $\Ld$ has weights $-\varepsilon_1,-\varepsilon_2,\cdots, -\varepsilon_n.$ In addition, we recall some submodules of $\Lo^{\otimes 2}$ and $(\Ld)^{\otimes
2}$ which will be used to construct the functors for our categorification. Denote by $Sym^2(\Lo)$ the symmetric square of $\Lo$, i.e, the submodule of $\Lo^{\otimes 2}$ spanned by $u_i\otimes u_i(1\leq i\leq n)$ and $u_i\otimes u_j+u_j\otimes
u_i(1\leq i< j\leq n)$, and denote by $Alt^2(\Lo)$ the alternative square of $\Lo$, i.e, the submodule of $\Lo^{\otimes 2}$ spanned by $u_i\otimes u_j-u_j\otimes u_i(1\leq i< j\leq n)$. Similarly, we denote by $Sym^2(\Ld)$ and $Alt^2(\Ld)$ the symmetric square and the alternative square of $\Ld$ respectively. In the following, we define $\O_{\bf d}$ to be the trivial subcategory of $\O(\gl)$ for ${\bf d}\notin {\bf D}.$ For ${\bf d}\in {\bf D}$ let ${\bf d}_i$ denote the fact that one subtracts 1 from the coefficient at place $i,$ and ${\bf d}^i$ the fact that one adds 1 to the coefficient at place $i.$ Then ${\bf d}_i^j$ means that one subtracts 1 from the coefficient at place $i$ and adds 1 to the coefficient at place $j.$
To categorify the action of $U(\mf{so}(7,\C))$ on $V_{\sp}^{\otimes
n}$, we introduce a series of projective functors of $\O(\gl).$
For ${\bf d}=(d_0,d_1,d_2,d_3,d_4,d_5,d_6,d_7)\in {\bf D}$, define $$\begin{aligned}
&&c_1({\bf d}):=d_5+d_4-d_3-d_2,\\
&&c_2({\bf d}):=d_6-d_5+d_2-d_1,\\
&&c_3({\bf d}):=d_7-d_6+d_5-d_4+d_3-d_2+d_1-d_0, \end{aligned}$$ and for $1\leq i\leq 3$, denote by $\sgn(c_i({\bf d}))$ the sign function of $c_i({\bf d})$, i.e., $$\sgn(c_i({\bf d}))=\left\{
\begin{array}{ll}
1,~{\rm if}~c_i({\bf d})>0,&\\
0,~{\rm if}~c_i({\bf d})=0,& \\
-1,~{\rm if}~c_i({\bf d})<0.&
\end{array}
\right.$$ Then set $$\H_i^{\sgn(c_i({\bf d}))}([{\bf d}])=(\Id_{\O_{[{\bf d}]}})^{\oplus \sgn(c_i({\bf d}))c_i({\bf d})}:
\O_{[{\bf d}]}\to\O_{[{\bf d}]},$$ where $\Id_{\O_{[{\bf d}]}}$ is the identity functor of $\O_{[{\bf
d}]}$. From the definition of the equivalence relation $\thicksim$ on ${\bf D}$ we can see that the functors $\H_i^{\sgn(c_i({\bf
d}))}([{\bf d}])(1\leq i\leq 3)$ are independent on the choice of the representative ${\bf d}$ of \[[**d**]{}\].
For ${\bf d}\in {\bf D}$ denote $$\E_1^{+2}({\bf d}):=\textrm{proj}_{\theta_{{\bf d}_2^4}}
\circ F_{Sym^2(\Lo)}: \O_{\bf d}\to\O_{{\bf d}_2^4},$$ $$\E_1^{-2}({\bf d}):=\textrm{proj}_{\theta_{{\bf d}_2^4}}
\circ F_{Alt^2(\Lo)}: \O_{\bf d}\to \O_{{\bf d}_2^4},$$ $$\E_1^{+3}({\bf
d}):=\textrm{proj}_{\theta_{{\bf d}_3^5}} \circ F_{Sym^2(\Lo)}:
\O_{\bf d}\to \O_{{\bf d}_3^5},$$ $$\E_1^{-3}({\bf d}):=\textrm{proj}_{\theta_{{\bf d}_3^5}}
\circ F_{Alt^2(\Lo)}: \O_{\bf d}\to \O_{{\bf d}_3^5}.$$ For any ${\bf [d]}\in \widetilde{{\bf D}}$, we set $$\E_{1}^+([{\bf d}])={\mathop{\oplus}\limits_{{\bf d^{'}}\in[{\bf
d}]}}(\E_1^{+2}({\bf d^{'}})\oplus\E_1^{+3}({\bf d^{'}})):
\O_{[{\bf d}]}\to \O_{[\overleftarrow{{\bf d}_1}],}$$ $$\E_{1}^-([{\bf d}])={\mathop{\oplus}\limits_{{\bf d^{'}}\in[{\bf
d}]}}(\E_1^{-2}({\bf d^{'}})\oplus\E_1^{-3}({\bf d^{'}})):
\O_{[{\bf d}]}\to \O_{[{\bf \overleftarrow{\bf d_1}}]},$$ where $[{\bf \overleftarrow{\bf d_1}}]=[{{\bf d}_3^5}] =[{{\bf
d}_2^4}].$
For ${\bf d}\in {\bf D}$ denote $$\E_2^1({\bf
d}):=\textrm{proj}_{\theta_{{\bf d}_1^2}} \circ F_{\Lo}: \O_{\bf
d}\to \O_{{\bf d}_1^2},$$ $$\E_2^5({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_5^6}}} \circ F_{\Lo}: \O_{\bf
d}\to \O_{{\bf d}_5^6}.$$ For any $[{\bf d}]\in \widetilde{{\bf
D}}$, set $$\E_{2}([{\bf d}])={\mathop{\oplus}\limits_{{\bf d^{'}}\in[{\bf
d}]}}(\E_2^1({\bf d^{'}})\oplus\E_2^5({\bf d^{'}})): \O_{[{\bf
d}]}\to \O_{[\bf \overleftarrow{\bf d_2}]},$$ where ${\bf [\overleftarrow{\bf d_2}]}=[{{\bf d}_1^2}] =[{{\bf
d}_5^6}].$
For ${\bf d}\in {\bf D}$ denote $$\E_3^0({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_0^1}}} \circ F_{\Lo}: \O_{\bf
d}\to \O_{{\bf d}_0^1},$$ $$\E_3^2({\bf d}):=\textrm{proj}_{\theta_{{{\bf d}_2^3}}}
\circ F_{\Lo}: \O_{\bf d}\to \O_{{\bf d}_2^3},$$ $$\E_3^4({\bf d}):=\textrm{proj}_{\theta_{{{\bf d}_4^5}}} \circ F_{\Lo}:
\O_{\bf d}\to \O_{{\bf d}_4^5},$$ $$\E_3^6({\bf d}):=\textrm{proj}_{\theta_{{{\bf d}_6^7}}}
\circ F_{\Lo}: \O_{\bf d}\to \O_{{\bf d}_6^7}.$$ For any $[{\bf
d}]\in \widetilde{{\bf D}}$, set $$\E_{3}([{\bf d}])={\mathop{\oplus}\limits_{{\bf d^{'}}\in[{\bf
d}]}}(\E_3^0({\bf d^{'}})\oplus\E_3^2({\bf d^{'}})\oplus\E_3^4({\bf
d^{'}})\oplus\E_3^6({\bf d^{'}})): \O_{[{\bf d}]}\to \O_{[\bf
\overleftarrow{\bf d_3}]},$$ where ${\bf [\overleftarrow{\bf d_3}]} =[{{\bf d}_0^1}] =[{{\bf
d}_2^3}]=[{{\bf d}_4^5}]=[{{\bf d}_6^7}].$
For ${\bf d}\in {\bf D}$ denote $$\F_1^{+4}({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_4^2}}} \circ F_{Sym^2(\Ld)}:
\O_{\bf d}\to \O_{{\bf d}_4^2},$$ $$\F_1^{-4}({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_4^2}}} \circ F_{Alt^2(\Ld)}:
\O_{\bf d}\to \O_{{\bf d}_4^2},$$ $$\F_1^{+5}({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_5^3}}} \circ F_{Sym^2(\Ld)}:
\O_{\bf d}\to \O_{{\bf d}_5^3},$$ $$\F_1^{-5}({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_5^3}}}\circ F_{Alt^2(\Ld)}:
\O_{\bf d}\to \O_{{\bf d}_5^3}$$ For any ${\bf [d]}\in
\widetilde{{\bf D}}$, set $$\F^+_{1}([{\bf d}])={\mathop{\oplus}\limits_{{\bf
d^{'}}\in[{\bf d}]}}(\F_1^{+4}({\bf d^{'}})\oplus\F_1^{+5}({\bf
d^{'}})): \O_{[{\bf d}]}\to \O_{\bf [\overrightarrow{\bf d_1}]},$$ $$\F^-_{1}([{\bf d}])={\mathop{\oplus}\limits_{{\bf
d^{'}}\in[{\bf d}]}}(\F_1^{-4}({\bf d^{'}})\oplus\F_1^{-5}({\bf
d^{'}})): \O_{[{\bf d}]}\to \O_{[\bf \overrightarrow{\bf d_1}]},$$ where ${\bf [\overrightarrow{\bf d_1}]}=[{{\bf d}_4^2}] =[{{\bf
d}_5^3}].$
For ${\bf d}\in {\bf D}$ denote $$\F_2^2({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_2^1}}} \circ F_{\Ld}: \O_{\bf
d}\to \O_{{\bf d}_2^1},$$ $$\F_2^6({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_6^5}}} \circ F_{\Ld}: \O_{\bf
d}\to \O_{{\bf d}_6^5}.$$ For any $[{\bf d}]\in \widetilde{{\bf
D}}$, set $$\F_{2}([{\bf d}])={\mathop{\oplus}\limits_{{\bf
d^{'}}\in[{\bf d}]}}(\F_2^2({\bf d^{'}})\oplus\F_2^6({\bf d^{'}})):
\O_{[{\bf d}]}\to \O_{\bf [\overrightarrow{\bf d_2}]},$$ where ${\bf
[\overrightarrow{\bf d_2}]}=[{{\bf d}_2^1}] =[{{\bf d}_6^5}].$
For ${\bf d}\in {\bf D}$ denote $$\F_3^1({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_1^0}}} \circ F_{\Ld}: \O_{\bf
d}\to \O_{{\bf d}_1^0},$$ $$\F_3^3({\bf d}):=\textrm{proj}_{\theta_{{{\bf d}_3^2}}}
\circ F_{\Ld}: \O_{\bf d}\to \O_{{\bf d}_3^2},$$ $$\F_3^5({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_5^4}}} \circ F_{\Ld}: \O_{\bf
d}\to \O_{{\bf d}_5^4},$$ $$\F_3^7({\bf
d}):=\textrm{proj}_{\theta_{{{\bf d}_7^6}}} \circ F_{\Ld}: \O_{\bf
d}\to \O_{{\bf d}_7^6}.$$ For any $[{\bf d}]\in \widetilde{{\bf
D}}$, set $$\F_{3}([{\bf d}])={\mathop{\oplus}\limits_{{\bf
d^{'}}\in[{\bf d}]}}(\F_3^1({\bf d^{'}})\oplus\F_3^3({\bf
d^{'}})\oplus\F_3^5({\bf d^{'}})\oplus\F_3^7({\bf d^{'}})):
\O_{[{\bf d}]}\to \O_{\bf [\overrightarrow{\bf d_3}]},$$ where ${\bf
[\overrightarrow{\bf d_3}]}=[{{\bf d}_1^0}] =[{{\bf d}_3^2}] =[{{\bf
d}_5^4}] =[{{\bf d}_7^6}].$
It can be seen from Proposition \[p3\] (1), (2), (3) and (4) that the above functors we introduce are exact and projective functors. Therefore, they can induce abelian group homomorphisms of the corresponding Grothendieck groups. By Proposition \[p4\] and direct calculations we can obtain the following explicit formulas of their induced homomorphisms on the basis element $[M(a_1,\cdots,
a_n)]\in B_{\bf d}$, which will be used in checking the commutativity of the diagrams in Proposition \[t2\]. $$\begin{aligned}
\label{E1}
&\quad&[\E_1^{+2}({\bf
d})]([M(a_1,\cdots,a_n)])\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}[M(a_1,\cdots,a_{m-1},a_{m}+2,a_{m+1},\cdots,a_n)]\nonumber\\
&&+{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(2,3)~{\rm
or}~(3,2)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)].\nonumber\end{aligned}$$ $$\begin{aligned}
\label{E2}
&\quad&[\E_1^{-2}({\bf d})]([M(a_1,\cdots,a_n)])\\
&=&{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(2,3)~{\rm
or}~(3,2)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)].\nonumber\end{aligned}$$ $$\begin{aligned}
\label{E3}
&\quad&[\E_1^{+3}({\bf d})]([M(a_1,\cdots,
a_n)])\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=3}^n}[M(a_1,\cdots,a_{m-1},a_{m}+2,a_{m+1},\cdots,a_n)]\nonumber\\
&&+{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(3,4)~{\rm
or}~(4,3)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)].\nonumber\end{aligned}$$ $$\begin{aligned}
\label{E4}
&\quad&[\E_1^{-3}({\bf d})]([M(a_1,\cdots,
a_n)])\\
&=&{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(3,4)~{\rm
or}~(4,3)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)].\nonumber\end{aligned}$$ $$\begin{aligned}
\label{E5}
[\E_2^1({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=1}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E6}
[\E_2^5({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=5}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E7}
[\E_3^0({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=0}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E8}
[\E_3^2({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E9}
[\E_3^4({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=4}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E10}
[\E_3^6({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=6}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E11}
&\quad&[\F_1^{+4}({\bf d})]([M(a_1,\cdots,
a_n)])\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=4}^n}[M(a_1,\cdots,a_{m-1},a_{m}-2,a_{m+1},\cdots,a_n)]\nonumber\\
&&+{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(3,4)~{\rm
or}~(4,3)}}[M(a_1,\cdots,a_{i-1},a_{i}-1,a_{i+1},\cdots,
a_{j-1},a_{j}-1,a_{j+1},\cdots,a_n)].\nonumber\end{aligned}$$ $$\begin{aligned}
\label{E12}
&\quad&[\F_1^{-4}({\bf d})]([M(a_1,\cdots,
a_n)])\\
&=&{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(3,4)~{\rm
or}~(4,3)}}[M(a_1,\cdots,a_{i-1},a_{i}-1,a_{i+1},\cdots,
a_{j-1},a_{j}-1,a_{j+1},\cdots,a_n)].\nonumber\end{aligned}$$ $$\begin{aligned}
\label{E13}
&\quad&[\F_1^{+5}({\bf d})]([M(a_1,\cdots,
a_n)])\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=5}^n}[M(a_1,\cdots,a_{m-1},a_{m}-2,a_{m+1},\cdots,a_n)]\nonumber\\
&&+{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(4,5)~{\rm
or}~(4,5)}}[M(a_1,\cdots,a_{i-1},a_{i}-1,a_{i+1},\cdots,
a_{j-1},a_{j}-1,a_{j+1},\cdots,a_n)].\nonumber\end{aligned}$$ $$\begin{aligned}
\label{E14}
&\quad&[\F_1^{-5}({\bf d})]([M(a_1,\cdots,
a_n)])\\
&=&{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(4,5)~{\rm
or}~(5,4)}}[M(a_1,\cdots,a_{i-1},a_{i}-1,a_{i+1},\cdots,
a_{j-1},a_{j}-1,a_{j+1},\cdots,a_n)].\nonumber\end{aligned}$$ $$\begin{aligned}
\label{E15}
[\F_2^2({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}[M(a_1,\cdots,a_{m-1},a_{m}-1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E16}
[\F_2^6({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=6}^n}[M(a_1,\cdots,a_{m-1},a_{m}-1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E17}
[\F_3^1({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=1}^n}[M(a_1,\cdots,a_{m-1},a_{m}-1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E18}
[\F_3^3({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=3}^n}[M(a_1,\cdots,a_{m-1},a_{m}-1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E19}
[\F_3^5({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=5}^n}[M(a_1,\cdots,a_{m-1},a_{m}-1,a_{m+1},\cdots,a_n)].\end{aligned}$$ $$\begin{aligned}
\label{E20}
[\F_3^7({\bf d})]([M(a_1,\cdots,
a_n)])={\mathop{\sum}\limits_{m=1,\atop
a_{m}=7}^n}[M(a_1,\cdots,a_{m-1},a_{m}-1,a_{m+1},\cdots,a_n)].\end{aligned}$$
Indeed, we check the formula (\[E1\]) as follows: $$\begin{aligned}
&&\quad[\E_1^{+2}({\bf d})]([M(a_1,\cdots, a_n)])\\
&&=[\textrm{proj}_{\theta_{{{\bf d}_2^4}}}(Sym^2(\Lo)\otimes
M(a_1,\cdots, a_n))]\\
&&=[\textrm{proj}_{\theta_{{{\bf d}_2^4}}}]
({\mathop{\sum}\limits_{1\leq i\leq j\leq
n}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},
\cdots,a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)])\\
&&={\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}[M(a_1,\cdots,a_{m-1},a_{m}+2,a_{m+1},\cdots,a_n)]\\
&&\quad+{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(2,3)~{\rm
or}~(3,2)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)].\end{aligned}$$ Note that the second equality is obtained by Proposition \[p4\], while others are obvious by the definitions of the projective functors $\E_1^{+2}({\bf d})$, $F_{Sym^2(\Lo)}$ and $\textrm{proj}_{\theta}$.
Similarly, we can get the formulas from (\[E2\]) to (\[E20\]).
Now, by the formulas (\[E1\])–(\[E20\]), a categorification of the action of $U(\mf{so}(7,\C))$ on the $n$-th tensor power of its spin representation can be obtained as follows.
\[t2\]
1. For any $1\leq i\leq 3$ and $[{\bf d}]\in \widetilde{{\bf D}}$, the action of $h_i$ on $(V_{\sp}^{\otm n})_{[{\bf d}]}$ can be categorified by the exact functor $\H_i^{\sgn(c_i({\bf d}))}([{\bf
d}])$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O_{[{\bf d}]}) &
\stackrel{}{\xrightarrow[\quad\quad\quad]{\gamma_n}}
& ^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]} \\
\Big\downarrow\vcenter{ \llap{$[\H_i^{\sgn(c_i({\bf d}))}({[{\bf
d}]})]~$}} & &
\Big\downarrow\vcenter{ \rlap{$\sgn(c_i({\bf d}))h_i$}} \\
K(\O_{[\bf d]}) &
\stackrel{}{\xrightarrow[\quad\quad\quad]{\gamma_n}} &
^{\Z}(V_{\sp}^{\otm n})_{[\bf d]}.
\end{array}$$
2. 1. For any $[{\bf d}]\in \widetilde{{\bf D}},$ the restriction of $e_1$ from $(V_{\sp}^{\otm n})_{[{\bf d}]}$ to $(V_{\sp}^{\otm n})_{\bf
[\overleftarrow{{\bf d_1}}]}$ can be categorified by a pair of exact functors $(\E^+_{1}([{\bf d}]),\E^-_{1}([{\bf d}]))$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O_{[{\bf d}]}) & \stackrel{}
{\xrightarrow[\quad\quad\quad\quad\quad\quad]{\gamma_n}} & ^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]} \\
\Big\downarrow\vcenter{ \llap{$[\E^+_{1}([{\bf d}])]-[\E^-_{1}({\bf
[d]})]~$}} & &
\Big\downarrow\vcenter{ \rlap{$e_1$}} \\
K(\O_{\bf [\overleftarrow{{\bf d_1}}]}) &
\stackrel{}{\xrightarrow[\quad\quad\quad\quad\quad\quad]{\gamma_n}}
& ^{\Z}(V_{\sp}^{\otm n})_{\bf [\overleftarrow{{\bf d_1}}]}.
\end{array}$$
2. For any $2\leq i\leq 3$ and $[{\bf d}]\in \widetilde{{\bf D}}$, the restriction of $e_i$ from $(V_{\sp}^{\otm n})_{[{\bf d}]}$ to $(V_{\sp}^{\otm n})_{\bf [\overleftarrow{{\bf d_i}}]}$ can be categorified by the exact functor $\E_{i}([{\bf d}])$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O_{[{\bf d}]}) & \stackrel{\gamma_n}{\longrightarrow} & ^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]} \\
\Big\downarrow\vcenter{ \llap{$[\E_{i}([{\bf d}])]~$}} & &
\Big\downarrow\vcenter{ \rlap{$e_i$}} \\
K(\O_{\bf [\overleftarrow{{\bf d_i}}]}) &
\stackrel{\gamma_n}{\longrightarrow} & ^{\Z}(V_{\sp}^{\otm n})_{\bf
[\overleftarrow{{\bf d_i}}]}.
\end{array}$$
3. 1. For any $[{\bf d}]\in \widetilde{{\bf D}},$ the restriction of $f_1$ from $(V_{\sp}^{\otm n})_{[{\bf d}]}$ to $(V_{\sp}^{\otm n})_{\bf
[\overrightarrow{{\bf d_1}}]}$ can be categorified by a pair of exact functors $(\F^+_{1}([{\bf d}]),\F^-_{1}([{\bf d}]))$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O_{[{\bf d}]}) & \stackrel{}
{\xrightarrow[\quad\quad\quad\quad\quad\quad]{\gamma_n}} & ^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]} \\
\Big\downarrow\vcenter{ \llap{$[\F^+_{1}([{\bf d}])]-[\F^-_{1}({\bf
[d]})]~$}} & &
\Big\downarrow\vcenter{ \rlap{$f_1$}} \\
K(\O_{\bf [\overrightarrow{{\bf d_1}}]}) &
\stackrel{}{\xrightarrow[\quad\quad\quad\quad\quad\quad]{\gamma_n}}
& ^{\Z}(V_{\sp}^{\otm n})_{\bf [\overrightarrow{{\bf d_1}}]}.
\end{array}$$
2. For any $2\leq i\leq 3$ and $[{\bf d}]\in \widetilde{{\bf D}}$, the restriction of $f_i$ from $(V_{\sp}^{\otm n})_{[{\bf d}]}$ to $(V_{\sp}^{\otm
n})_{\bf
[\overrightarrow{{\bf d_i}}]}$ can be categorified by the exact functor $\F_{i}([{\bf d}])$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O_{[{\bf d}]}) & \stackrel{\gamma_n}{\longrightarrow} & ^{\Z}(V_{\sp}^{\otm n})_{[{\bf d}]} \\
\Big\downarrow\vcenter{ \llap{$[\F_{i}([{\bf d}])]~$}} & &
\Big\downarrow\vcenter{ \rlap{$f_i$}} \\
K(\O_{\bf [\overrightarrow{{\bf d_i}}]}) &
\stackrel{\gamma_n}{\longrightarrow} & ^{\Z}(V_{\sp}^{\otm n})_{\bf
[\overrightarrow{{\bf d_i}}]}.
\end{array}$$
Here we check $(1)$ and $(2)$ in some cases. Other cases can be verified similarly.
\(1) To check $\gamma_n\circ[\H_i^{\sgn(c_i({\bf d}))}({[{\bf
d}]})]=\sgn(c_i({\bf d}))h_i\circ\gamma_n$ is equivalent to check $$\gamma_n\circ[\H_i^{\sgn(c_i({\bf d}))}({[{\bf
d}]})]([M(a_1,\cdots, a_n)])=\sgn(c_i({\bf
d}))h_i\circ\gamma_n([M(a_1,\cdots, a_n)])$$ for any $[M(a_1,\cdots,
a_n)]\in B_{[{\bf d}]}={\mathop{\cup}\limits_{{\bf d^{'}}\in [{\bf
d}]}}B_{\bf d^{'}}.$ In fact, we have $$\begin{aligned}
&\quad&\gamma_n\circ[\H_i^{\sgn(c_i({\bf d}))}({[{\bf
d}]})]([M(a_1,\cdots,
a_n)])\\
&=&|c_i({\bf d})|\gamma_n([M(a_1,\cdots, a_n)])\\
&=&|c_i({\bf d})|(v_{a_1}\otimes v_{a_2}\otimes\cdots \otimes
v_{a_n})\\
&=&\sgn(c_i({\bf d}))c_i({\bf d})(v_{a_1}\otimes
v_{a_2}\otimes\cdots
\otimes v_{a_n})\\
&=&\sgn(c_i({\bf d}))h_i\circ\gamma_n([M(a_1,\cdots, a_n)]).\end{aligned}$$
\(2) (a) To verify the commutativity of the diagram in (a), it suffices to check $\gamma_{n}\circ([\E^+_{1}([{\bf
d}])]-[\E^-_{1}({\bf [d]})])=e_1\circ\gamma_{n}.$ Indeed, for any $[{\bf d}]\in \widetilde{{\bf D}}$ and $[M(a_1,\cdots, a_n)]\in
B_{[{\bf d}]}={\mathop{\cup}\limits_{{\bf d^{'}}\in [{\bf
d}]}}B_{\bf d^{'}}$, we assume $[M(a_1,\cdots, a_n)]\in B_{{\bf
d}_{0}}$ for some ${\bf d}_{0}\in [{\bf d}]$, by (\[E1\])–(\[E4\]), we have the following formulas: $$\begin{aligned}
&\quad&[\E^+_{1}([{\bf d}])]([M(a_1,\cdots, a_n)])\\
&=&[{\mathop{\oplus}\limits_{{\bf
d^{'}}\in[{\bf d}]}}(\E_1^{+2}({\bf d^{'}})\oplus\E_1^{+3}({\bf d^{'}}))M(a_1,\cdots, a_n)]\\
&=&[(\E_1^{+2}({\bf d}_{0})\oplus\E_1^{+3}({\bf d}_{0}))M(a_1,\cdots,a_n)]\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}[M(a_1,\cdots,a_{m-1},a_{m}+2,a_{m+1},\cdots,a_n)]\\
&&+{\mathop{\sum}\limits_{m=1,\atop a_{m}=3}^n}[M(a_1,\cdots,a_{m-1},a_{m}+2,a_{m+1},\cdots,a_n)]\\
&&+{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(2,3)~{\rm
or}~(3,2)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)]\\
&&+{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(3,4)~{\rm
or}~(4,3)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)],\end{aligned}$$ and $$\begin{aligned}
&\quad&[\E^-_{1}([{\bf d}])]([M(a_1,\cdots, a_n)])\\
&=&[{\mathop{\oplus}\limits_{{\bf
d^{'}}\in[{\bf d}]}}(\E_1^{-2}({\bf d^{'}})\oplus\E_1^{-3}({\bf d^{'}}))M(a_1,\cdots, a_n)]\\
&=&[(\E_1^{-2}({\bf d}_{0})\oplus\E_1^{-3}({\bf d}_{0}))M(a_1,\cdots,a_n)]\\
&=&{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(2,3)~{\rm
or}~(3,2)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)]\\
&&+{\mathop{\sum}\limits_{1\leq i<j\leq n,\atop
(a_{i},a_{j})=(3,4)~{\rm
or}~(4,3)}}[M(a_1,\cdots,a_{i-1},a_{i}+1,a_{i+1},\cdots,
a_{j-1},a_{j}+1,a_{j+1},\cdots,a_n)].\end{aligned}$$
It follows that $$\begin{aligned}
&\quad&\gamma_{n}\circ([\E^+_{1}([{\bf d}])]-[\E^-_{1}({\bf
[d]})])([M(a_1,\cdots, a_n)])\\
&=&\gamma_{n}({\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}[M(a_1,\cdots,a_{m-1},a_{m}+2,a_{m+1},\cdots,a_n)])
+\gamma_{n}({\mathop{\sum}\limits_{m=1,\atop
a_{m}=3}^n}[M(a_1,\cdots,a_{m-1},a_{m}+2,a_{m+1},\cdots,a_n)])\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+2}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})
+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=3}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+2}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}}).\end{aligned}$$
Moreover, we have $$\begin{aligned}
&\quad&e_1\circ\gamma_{n}([M(a_1,\cdots, a_n)])\\
&=&e_1(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}}\otimes v_{a_{m+1}}\otimes\cdots\otimes
v_{a_{n}})\\
&=&{\mathop{\sum}\limits_{m=1}^{n}}(v_{a_1}\otimes\cdots\otimes
v_{a_{m-1}}\otimes e_1v_{a_{m}}\otimes
v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+2}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})\\
&&+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=3}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+2}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}}).\end{aligned}$$ Therefore $\gamma_{n}\circ([\E^+_{1}([{\bf d}])]-[\E^-_{1}({\bf
[d]})])=e_1\circ\gamma_{n}.$
\(b) We give the proof of the case $i=3.$
In the following we will prove $\gamma_{n}\circ[\E_{3}([{\bf
d}])]=e_3\circ\gamma_{n}$ which means the diagram in this case commutes. In fact, for any $[{\bf d}]\in \widetilde{{\bf D}}$ and $[M(a_1,\cdots, a_n)]\in B_{[{\bf d}]}={\mathop{\cup}\limits_{{\bf
d^{'}}\in [{\bf d}]}}B_{\bf d^{'}}$, we only need to check $$\gamma_{n}\circ[\E_{3}([{\bf d}])]([M(a_1,\cdots,
a_n)])=e_3\circ\gamma_{n}([M(a_1,\cdots, a_n)]).$$ Assume that $[M(a_1,\cdots, a_n)]\in B_{{\bf d}_{0}}$ for some ${\bf d}_{0}\in
[{\bf d}]$, by (\[E7\])–(\[E10\]), we have
$$\begin{aligned}
&\quad&[\E_{3}([{\bf d}])]([M(a_1,\cdots, a_n)])\\
&=&[{\mathop{\oplus}\limits_{{\bf d^{'}}\in[{\bf d}]}}(\E_3^{0}({\bf
d^{'}})\oplus\E_3^{2}({\bf d^{'}})
\oplus\E_3^{4}({\bf d^{'}})\oplus\E_3^{6}({\bf d^{'}}))M(a_1,\cdots, a_n)]\\
&=&[(\E_3^{0}({\bf d}_{0})\oplus\E_3^{2}({\bf d}_{0})
\oplus\E_3^{4}({\bf d}_{0})\oplus\E_3^{6}({\bf d}_{0}))M(a_1,\cdots, a_n)]\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=0}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)]
+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)]\\
&&+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=4}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)]
+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=6}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)].\\\end{aligned}$$
It follows that $$\begin{aligned}
&\quad&\gamma_{n}\circ[\E_{3}([{\bf d}])]([M(a_1,\cdots, a_n)])\\
&=&\gamma_{n}({\mathop{\sum}\limits_{m=1,\atop
a_{m}=0}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)])+\gamma_{n}({\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)])\\
&&+\gamma_{n}({\mathop{\sum}\limits_{m=1,\atop
a_{m}=4}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)])+\gamma_{n}({\mathop{\sum}\limits_{m=1,\atop
a_{m}=6}^n}[M(a_1,\cdots,a_{m-1},a_{m}+1,a_{m+1},\cdots,a_n)])\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=0}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+2}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})
+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+2}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})\\
&&+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=4}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+2}\otimes v_{a_{m+1}}\otimes\cdots\otimes
v_{a_{n}})+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=6}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+2}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}}).\end{aligned}$$
Moreover, we have $$\begin{aligned}
&\quad&e_3\circ\gamma_{n}([M(a_1,\cdots, a_n)])\\
&=&e_3(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}}\otimes v_{a_{m+1}}\otimes\cdots\otimes
v_{a_{n}})\\
&=&{\mathop{\sum}\limits_{m=1}^{n}}(v_{a_1}\otimes\cdots\otimes
v_{a_{m-1}}\otimes e_3v_{a_{m}}\otimes
v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})\\
&=&{\mathop{\sum}\limits_{m=1,\atop
a_{m}=0}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+1}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})
+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=2}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+1}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})\\
&&+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=4}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+1}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}})
+{\mathop{\sum}\limits_{m=1,\atop
a_{m}=6}^n}(v_{a_1}\otimes\cdots\otimes v_{a_{m-1}}\otimes
v_{a_{m}+1}\otimes v_{a_{m+1}}\otimes\cdots\otimes v_{a_{n}}).\end{aligned}$$ Hence $\gamma_{n}\circ[\E_{3}([{\bf d}])]([M(a_1,\cdots,
a_n)])=e_3\circ\gamma_{n}([M(a_1,\cdots, a_n)]).$
\[c1\] For any $1\leq i\leq 3$ and $2\leq j\leq 3,$ let $$\H^+_i={\mathop{\oplus}\limits_{[{\bf d}]\in
\widetilde{{\bf D}},\atop \sgn(c_i({\bf d}))=1~{\rm
or}~0}}\H_i^{\sgn(c_i({\bf d}))}([{\bf d}]):\O^{n}\to\O^{n},\quad
\H^-_i={\mathop{\oplus}\limits_{[{\bf d}]\in \widetilde{{\bf
D}},\atop \sgn(c_i({\bf d}))=-1}}\H_i^{\sgn(c_i({\bf d}))}([{\bf
d}]):\O^{n}\to\O^{n},$$ $$\E^+_{1}={\mathop{\oplus}\limits_{[{\bf d}]\in \widetilde{{\bf
D}}}}\E^+_{1}([{\bf d}]):\O^{n}\to\O^{n},\quad
\E^-_{1}={\mathop{\oplus}\limits_{[{\bf d}]\in \widetilde{{\bf
D}}}}\E^-_{1}([{\bf d}]):\O^{n}\to\O^{n},$$ $$\F^+_{1}={\mathop{\oplus}\limits_{[{\bf d}]\in
\widetilde{{\bf D}}}}\F^+_{1}([{\bf d}]):\O^{n}\to\O^{n},\quad
\F^-_{1}={\mathop{\oplus}\limits_{[{\bf d}]\in
\widetilde{{\bf D}}}}\F^-_{1}([{\bf d}]):\O^{n}\to\O^{n},$$ $$\E_{j}={\mathop{\oplus}\limits_{[{\bf d}]\in
\widetilde{{\bf D}}}}\E_{j}([{\bf d}]):\O^{n}\to\O^{n},\quad
\F_{j}={\mathop{\oplus}\limits_{[{\bf d}]\in \widetilde{{\bf
D}}}}\F_{j}([{\bf d}]):\O^{n}\to\O^{n}.$$ Then we have the following results.
1. For any $1\leq i\leq 3,$ the action of $h_i$ on $V_{\sp}^{\otm n}$ can be categorified by a pair of exact functors $(\H^+_{i},\H^-_{i}),$ which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O^{n}) & \stackrel{}{\xrightarrow[\quad\quad\quad]{\gamma_n}} & ^{\Z}V_{\sp}^{\otm n} \\
\Big\downarrow\vcenter{ \llap{$[\H^+_{i}]-[\H^-_{i}]~$}} & &
\Big\downarrow\vcenter{ \rlap{$h_i$}} \\
K(\O^{n}) & \stackrel{}{\xrightarrow[\quad\quad\quad]{\gamma_n}} &
^{\Z}V_{\sp}^{\otm n}.
\end{array}$$
2. 1. The action of $e_1$ on $V_{\sp}^{\otm n}$ can be categorified by a pair of exact functors $(\E^+_{1},\E^-_{1})$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O^{n}) & \stackrel{}{\xrightarrow[\quad\quad\quad]{\gamma_n}} & ^{\Z}V_{\sp}^{\otm n} \\
\Big\downarrow\vcenter{ \llap{$[\E^+_{1}]-[\E^-_{1}]~$}} & &
\Big\downarrow\vcenter{ \rlap{$e_1$}} \\
K(\O^{n}) & \stackrel{}{\xrightarrow[\quad\quad\quad]{\gamma_n}} &
^{\Z}V_{\sp}^{\otm n}.
\end{array}$$
2. For any $2\leq i\leq 3$, the action of $e_i$ on $V_{\sp}^{\otm n}$ can be categorified by the exact functor $\E_{i}$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O^{n}) & \stackrel{\gamma_n}{\longrightarrow} & ^{\Z}V_{\sp}^{\otm n} \\
\Big\downarrow\vcenter{ \llap{$[\E_{i}]~$}} & &
\Big\downarrow\vcenter{ \rlap{$e_i$}} \\
K(\O^{n}) & \stackrel{\gamma_n}{\longrightarrow} &
^{\Z}V_{\sp}^{\otm n}.
\end{array}$$
3. 1. The action of $f_1$ on $V_{\sp}^{\otm n}$ can be categorified by a pair of exact functors $(\F^+_{1},\F^-_{1})$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O^{n}) & \stackrel{}{\xrightarrow[\quad\quad\quad]{\gamma_n}} & ^{\Z}V_{\sp}^{\otm n} \\
\Big\downarrow\vcenter{ \llap{$[\F^+_{1}]-[\F^-_{1}]~$}} & &
\Big\downarrow\vcenter{ \rlap{$f_1$}} \\
K(\O^{n}) & \stackrel{}{\xrightarrow[\quad\quad\quad]{\gamma_n}} &
^{\Z}V_{\sp}^{\otm n}.
\end{array}$$
2. For any $2\leq i\leq 3$, the action of $f_i$ on $V_{\sp}^{\otm n}$ can be categorified by the exact functor $\F_{i}$, which means that the following diagram commutes$:$ $$\begin{array}{ccc}
K(\O^{n}) & \stackrel{\gamma_n}{\longrightarrow} & ^{\Z}V_{\sp}^{\otm n} \\
\Big\downarrow\vcenter{ \llap{$[\F_{i}]~$}} & &
\Big\downarrow\vcenter{ \rlap{$f_i$}} \\
K(\O^{n}) & \stackrel{\gamma_n}{\longrightarrow} &
^{\Z}V_{\sp}^{\otm n}.
\end{array}$$
It is not difficult to check the diagrams are commutative by Proposition \[t2\].
Now we categorify defining relations of $U(\mf{so}(7,\C))$ as natural isomorphisms between some projective functors of $\O(\mf{gl}_n)$. Proposition \[p3\] (5) plays an important role in the proof of the following results.
\[t3\]
1. $\H^+_i\circ\H^+_j\oplus\H^-_i\circ\H^-_j
\oplus\H^+_j\circ\H^-_i\oplus\H^-_j\circ\H^+_i
\cong
\H^+_i\circ\H^-_j\oplus\H^-_i\circ\H^+_j
\oplus\H^+_j\circ\H^+_i\oplus\H^-_j\circ\H^-_i$ for $1\leq i,j\leq 3.$
2. 1. $\E^+_1\circ\F^+_1\oplus\E^-_1\circ\F^-_1
\oplus\F^+_1\circ\E^-_1\oplus\F^-_1\circ\E^+_1\oplus\H^-_1
\cong\E^+_1\circ\F^-_1\oplus\E^-_1\circ\F^+_1
\oplus\F^+_1\circ\E^+_1\oplus\F^-_1\circ\E^-_1\oplus
\H^+_1;$
2. $\E^+_1\circ\F_j\oplus\F_j\circ\E^-_1
\cong\E^-_1\circ\F_j\oplus\F_j\circ\E^+_1$ for $j=2,3;$
3. $\E_i\circ\F^+_1\oplus\F^-_1\circ\E_i
\cong\E_i\circ\F^-_1\oplus\F^+_1\circ\E_i$ for $i=2,3;$
4. $\E_i\circ\F_j\oplus\delta_{i,j}\H^-_i
\cong\F_j\circ\E_i\oplus\delta_{i,j}\H^+_i$ for $(i,j)=(2,2), (2,3), (3,2)$ or $(3,3).$
3. 1. $\H^+_1\circ\E^+_1\oplus\H^-_1\circ\E^-_1
\oplus\E^+_1\circ\H^-_1\oplus\E^-_1\circ\H^+_1\oplus(\E^-_1)^{\oplus 2}
\cong
\H^+_1\circ\E^-_1\oplus\H^-_1\circ\E^+_1
\oplus\E^+_1\circ\H^+_1\oplus\E^-_1\circ\H^-_1\oplus(\E^+_1)^{\oplus
2};$
2. $\H^+_i\circ\E^+_1\oplus\H^-_i\circ\E^-_1\oplus\E^+_1\circ\H^-_i\oplus\E^-_1\circ\H^+_i\oplus(\E^+_1)^{\oplus
(-a_{i,1})}
\cong
\H^+_i\circ\E^-_1\oplus\H^-_i\circ\E^+_1\oplus\E^+_1\circ\H^+_i\oplus\E^-_1\circ\H^-_i\oplus(\E^-_1)^{\oplus
(-a_{i,1})}$ for $i=2, 3;$
3. $\H^+_i\circ\E_j\oplus
\E_j\circ\H^-_i\oplus(\E_j)^{\oplus(-a_{i,j})} \cong
\H^-_i\circ\E_j\oplus \E_j\circ\H^+_i$ for $(i,j)=(1,2), (1,3), (2,3)$ or $(3,2);$
4. $\H^+_i\circ\E_i\oplus
\E_i\circ\H^-_i\cong \H^-_i\circ\E_i\oplus
\E_i\circ\H^+_i\oplus\E_i^{\oplus 2}$ for $i=2, 3.$
4. 1. $\H^+_1\circ\F^+_1\oplus\H^-_1\circ\F^-_1\oplus\F^+_1\circ\H^-_1\oplus\F^-_1
\circ\H^+_1\oplus(\F^+_1)^{\oplus 2}
\cong
\H^+_1\circ\F^-_1\oplus\H^-_1\circ\F^+_1\oplus\F^+_1\circ\H^+_1\oplus\F^-_1
\circ\H^-_1\oplus(\F^-_1)^{\oplus 2};$
2. $\H^+_i\circ\F^+_1\oplus\H^-_i\circ\F^-_1\oplus\F^+_1\circ\H^-_i\oplus\F^-_1\circ\H^+_i\oplus(\F^-_1)^{\oplus
(-a_{i,1})}
\cong
\H^+_i\circ\F^-_1\oplus\H^-_i\circ\F^+_1\oplus\F^+_1\circ\H^+_i\oplus\F^-_1\circ\H^-_i\oplus(\F^+_1)^{\oplus
(-a_{i,1})}$ for $i=2,3;$
3. $\H^+_i\circ\F_j\oplus
\F_j\circ\H^-_i \cong \H^-_i\circ\F_j\oplus
\F_j\circ\H^+_i\oplus(\F_j)^{\oplus(-a_{i,j})}$ for $(i,j)=(1,2), (1,3), (2,3)$ or $(3,2);$
4. $\H^+_i\circ\F_i\oplus
\F_i\circ\H^-_i\oplus\F_i^{\oplus 2}\cong \H^-_i\circ\F_i\oplus
\F_i\circ\H^+_i$ for $i=2, 3.$
5. 1. $\E^+_{1}\circ\E^+_{1}\circ\E_{2}\oplus\E^-_{1}\circ\E^-_{1}\circ\E_{2}\oplus
\E_{2}\circ\E^+_{1}\circ\E^+_{1}\oplus
\E_{2}\circ\E^-_{1}\circ\E^-_{1}\oplus(\E^+_{1}\circ\E_{2}\circ\E^-_{1})^{\oplus
2}\oplus(\E^-_{1}\circ\E_{2}\circ\E^+_{1})^{\oplus 2}\\
\cong\E^+_{1}\circ\E^-_{1}\circ\E_{2}\oplus\E^-_{1}\circ\E^+_{1}\circ\E_{2}
\oplus \E_{2}\circ\E^+_{1}\circ\E^-_{1}\oplus
\E_{2}\circ\E^-_{1}\circ\E^+_{1}\oplus(\E^+_{1}\circ\E_{2}\circ\E^+_{1})^{\oplus
2}\oplus(\E^-_{1}\circ\E_{2}\circ\E^-_{1})^{\oplus 2};$
2. $\E^+_{1}\circ\E_{3}\oplus\E_{3}\circ\E^-_{1}\cong\E^-_{1}\circ\E_{3}\oplus\E_{3}\circ\E^+_{1};$
3. $\E_{2}\circ\E_{2}\circ\E^+_{1}\oplus
\E^+_{1}\circ\E_{2}\circ\E_{2}\oplus(\E_{2}\circ\E^-_{1}\circ\E_{2})^{\oplus
2} \cong\E_{2}\circ\E_{2}\circ\E^-_{1}\oplus
\E^-_{1}\circ\E_{2}\circ\E_{2}\oplus(\E_{2}\circ\E^+_{1}\circ\E_{2})^{\oplus
2};$
4. $\E_{2}\circ\E_{2}\circ\E_{3}\oplus
\E_{3}\circ\E_{2}\circ\E_{2}\cong(\E_{2}\circ\E_{3}\circ\E_{2})^{\oplus
2};$
5. $\E_{3}\circ\E_{3}\circ\E_{3}\circ\E_{2}\oplus(\E_{3}\circ\E_{2}\circ\E_{3}\circ\E_{3})^{\oplus
3} \cong(\E_{3}\circ\E_{3}\circ\E_{2}\circ\E_{3})^{\oplus
3}\oplus\E_{2}\circ\E_{3}\circ\E_{3}\circ\E_{3}$.
6. 1. $\F^+_{1}\circ\F^+_{1}\circ\F_{2}\oplus\F^-_{1}\circ\F^-_{1}\circ\F_{2}\oplus
\F_{2}\circ\F^+_{1}\circ\F^+_{1}\oplus
\F_{2}\circ\F^-_{1}\circ\F^-_{1}\oplus(\F^+_{1}\circ\F_{2}\circ\F^-_{1})^{\oplus
2}\oplus(\F^-_{1}\circ\F_{2}\circ\F^+_{1})^{\oplus 2}\\
\cong\F^+_{1}\circ\F^-_{1}\circ\F_{2}\oplus\F^-_{1}\circ\F^+_{1}\circ\F_{2}
\oplus \F_{2}\circ\F^+_{1}\circ\F^-_{1}\oplus
\F_{2}\circ\F^-_{1}\circ\F^+_{1}\oplus(\F^+_{1}\circ\F_{2}\circ\F^+_{1})^{\oplus
2}\oplus(\F^-_{1}\circ\F_{2}\circ\F^-_{1})^{\oplus 2};$
2. $\F^+_{1}\circ\F_{3}\oplus\F_{3}\circ\F^-_{1}\cong\F^-_{1}\circ\F_{3}\oplus\F_{3}\circ\F^+_{1};$
3. $\F_{2}\circ\F_{2}\circ\F^+_{1}\oplus
\F^+_{1}\circ\F_{2}\circ\F_{2}\oplus(\F_{2}\circ\F^-_{1}\circ\F_{2})^{\oplus
2} \cong\F_{2}\circ\F_{2}\circ\F^-_{1}\oplus
\F^-_{1}\circ\F_{2}\circ\F_{2}\oplus(\F_{2}\circ\F^+_{1}\circ\F_{2})^{\oplus
2};$
4. $\F_{2}\circ\F_{2}\circ\F_{3}\oplus
\F_{3}\circ\F_{2}\circ\F_{2}\cong(\F_{2}\circ\F_{3}\circ\F_{2})^{\oplus
2};$
5. $\F_{3}\circ\F_{3}\circ\F_{3}\circ\F_{2}\oplus(\F_{3}\circ\F_{2}\circ\F_{3}\circ\F_{3})^{\oplus
3} \cong(\F_{3}\circ\F_{3}\circ\F_{2}\circ\F_{3})^{\oplus
3}\oplus\F_{2}\circ\F_{3}\circ\F_{3}\circ\F_{3}.$
We check Theorem \[t3\] in some cases, while the remaining cases can be verified similarly.
\(1) By Proposition \[p3\] (5), we only need to check that $$\begin{aligned}
\label{equ3.2}
&\quad&[\H^+_i]\circ[\H^+_j]+[\H^-_i]\circ[\H^-_j]+[\H^+_j]\circ[\H^-_i]+[\H^-_j]\circ[\H^+_i]\\
&=&[\H^+_i]\circ[\H^-_j]+[\H^-_i]\circ[\H^+_j]+[\H^+_j]\circ[\H^+_i]+[\H^-_j]
\circ[\H^-_i].\nonumber\end{aligned}$$ Indeed, it is easy to see that $h_ih_j\gamma_n=h_jh_i\gamma_n.$ By Theorem \[c1\] (1), we have $$h_ih_j\gamma_n=\gamma_n\circ([\H^+_i]-[\H^-_i])\circ([\H^+_j]-[\H^-_j]),$$ $$h_jh_i\gamma_n=\gamma_n\circ([\H^+_j]-[\H^-_j])\circ([\H^+_i]-[\H^-_i]).$$ Therefore, $$\begin{aligned}
\label{equ3.3}
([\H^+_i]-[\H^-_i])\circ([\H^+_j]-[\H^-_j])=([\H^+_j]-[\H^-_j])
\circ([\H^+_i]-[\H^-_i]).\end{aligned}$$ We immediately get (\[equ3.2\]) by expanding (\[equ3.3\]).
\(2) (a) By Proposition \[p3\] (5), it is enough to check that $$\begin{aligned}
\label{equ3.4}
&&[\E^+_1]\circ[\F^+_1]+[\E^-_1]\circ[\F^-_1]+[\F^+_1]\circ[\E^-_1]+[\F^-_1]\circ[\E^+_1]+[\H^-_1]\\
&=&[\E^+_1]\circ[\F^-_1]+[\E^-_1]\circ[\F^+_1]+[\F^+_1]\circ[\E^+_1]+[\F^-_1]\circ[\E^-_1]+[\H^+_1].\nonumber\end{aligned}$$ Indeed, noting that\
(i)$(e_1f_1-f_1e_1)\gamma_n=h_1\gamma_n,$\
(ii)$(e_1f_1-f_1e_1)\gamma_n=\gamma_n\circ\big[([\E^+_1]-[\E^-_1])\circ([\F^+_1]-[\F^-_1])
-([\F^+_1]-[\F^-_1])\circ([\E^+_1]-[\E^-_1])\big],$\
(iii)$h_1\gamma_n=\gamma_n\circ([\H^+_1]-[\H^-_1]),$\
where (ii) and (iii) follow from Theorem \[c1\] (1), (2)(a) and (3)(a), we have $$\begin{aligned}
\label{equ3.5}
([\E^+_1]-[\E^-_1])\circ([\F^+_1]-[\F^-_1])-([\F^+_1]-[\F^-_1])\circ([\E^+_1]-[\E^-_1])=[\H^+_1]-[\H^-_1],\end{aligned}$$ (\[equ3.4\]) is obtained by expanding (\[equ3.5\]).
\(3) (a) By Proposition \[p3\] (5), it suffices to check that $$\begin{aligned}
\label{equ3.6}
&\quad&[\H^+_1]\circ[\E^+_1]+[\H^-_1]\circ[\E^-_1]+[\E^+_1]\circ[\H^-_1]+[\E^-_1]\circ[\H^+_1]+2[\E^-_1]\\
&=&
[\H^+_1]\circ[\E^-_1]+[\H^-_1]\circ[\E^+_1]+[\E^+_1]\circ[\H^+_1]+[\E^-_1]\circ[\H^-_1]+2[\E^+_1].\nonumber\end{aligned}$$ Indeed, noting that\
(i)$(h_1e_1-e_1h_1)\gamma_n=2e_1\gamma_n,$\
(ii)$(h_1e_1-e_1h_1)\gamma_n=\gamma_n\circ\big[([\H^+_1]-[\H^-_1])\circ([\E^+_1]-[\E^-_1])
-([\E^+_1]-[\E^-_1])\circ([\H^+_1]-[\H^-_1])\big],$\
(iii)$2e_1\gamma_n=2\gamma_n\circ([\E^+_1]-[\E^-_1]),$\
where (ii) and (iii) can be seen from Theorem \[c1\] (1) and (2)(a), we have $$\begin{aligned}
\label{equ3.7}
([\H^+_1]-[\H^-_1])\circ([\E^+_1]-[\E^-_1])
-([\E^+_1]-[\E^-_1])\circ([\H^+_1]-[\H^-_1])=2([\E^+_1]-[\E^-_1]),\end{aligned}$$ (\[equ3.6\]) follows from expanding (\[equ3.7\]).
\(4) (b) We only give the proof of the case $(i,j)=(2,1).$ By Proposition \[p3\] (5), it is sufficient to check that $$\begin{aligned}
\label{equ3.8}
&\quad&[\H^+_2]\circ[\F^+_1]+[\H^-_2]\circ[\F^-_1]+[\F^+_1]\circ[\H^-_2]+[\F^-_1]\circ[\H^+_2]+[\F^-_1]\\
&=&
[\H^+_2]\circ[\F^-_1]+[\H^-_2]\circ[\F^+_1]+[\F^+_1]\circ[\H^+_2]+[\F^-_1]\circ[\H^-_2]+[\F^+_1].\nonumber\end{aligned}$$ Indeed, noting that\
(i)$(h_2f_1-f_1h_2)\gamma_n=f_1\gamma_n,$\
(ii)$(h_2f_1-f_1h_2)\gamma_n=\gamma_n\circ\big[([\H^+_2]-[\H^-_2])\circ([\F^+_1]-[\F^-_1])
-([\F^+_1]-[\F^-_1])\circ([\H^+_2]-[\H^-_2])\big],$\
(iii)$f_1\gamma_n=\gamma_n\circ([\F^+_1]-[\F^-_1]),$\
where (ii) and (iii) follow from Theorem \[c1\] (1) and (3)(a), we have $$\begin{aligned}
\label{equ3.9}
([\H^+_2]-[\H^-_2])\circ([\F^+_1]-[\F^-_1])
-([\F^+_1]-[\F^-_1])\circ([\H^+_2]-[\H^-_2])=[\F^+_1]-[\F^-_1],\end{aligned}$$ We obtain (\[equ3.8\]) by expanding (\[equ3.9\]).
\(5) (a) By Proposition \[p3\] (5), it is equivalent to check $$\begin{aligned}
\label{equ3.10}
&\quad&[\E^+_{1}]\circ[\E^+_{1}]\circ[\E_{2}]+[\E^-_{1}]\circ[\E^-_{1}]\circ[\E_{2}]+
[\E_{2}]\circ[\E^+_{1}]\circ[\E^+_{1}]\\
&\quad&+[\E_{2}]\circ[\E^-_{1}]\circ[\E^-_{1}]+2[\E^+_{1}]\circ[\E_{2}]\circ[\E^-_{1}]
+2[\E^-_{1}]\circ[\E_{2}]\circ[\E^+_{1}]\nonumber\\
&=&[\E^+_{1}]\circ[\E^-_{1}]\circ[\E_{2}]+[\E^-_{1}]\circ[\E^+_{1}]\circ[\E_{2}]
+[\E_{2}]\circ[\E^+_{1}]\circ[\E^-_{1}]\nonumber\\
&\quad&+[\E_{2}]\circ[\E^-_{1}]\circ[\E^+_{1}]+2[\E^+_{1}]\circ[\E_{2}]\circ[\E^+_{1}]
+2[(\E^-_{1}]\circ[\E_{2}]\circ[\E^-_{1}]).\nonumber\end{aligned}$$ Indeed, noting that\
(i)$(e^2_1e_2-2e_1e_2e_1+e_2e^2_1)\gamma_n=0,$\
(ii)by Theorem \[c1\] (2) (a) and (b), $(e^2_1e_2-2e_1e_2e_1+e_2e^2_1)\gamma_n
=\gamma_n\circ\big[([\E^+_1]-[\E^-_1])
\circ([\E^+_1]-[\E^-_1])\circ[\E_2]
-2([\E^+_1]-[\E^-_1])\circ[\E_2]\circ([\E^+_1]-[\E^-_1])
+[\E_2]\circ([\E^+_1]-[\E^-_1])\circ([\E^+_1]-[\E^-_1])\big],$\
we have $(3.31)$ $$\begin{aligned}
([\E^+_1]-[\E^-_1])\circ([\E^+_1]-[\E^-_1])\circ[\E_2]
-2([\E^+_1]-[\E^-_1])\circ[\E_2]\circ([\E^+_1]-[\E^-_1])
+[\E_2]\circ([\E^+_1]-[\E^-_1])\circ([\E^+_1]-[\E^-_1])=0,\end{aligned}$$ which is (\[equ3.10\]) after expanding.
\(6) (e) By Proposition \[p3\] (5), it is equivalent to verify $$[\F_{3}]\circ[\F_{3}]\circ[\F_{3}]\circ[\F_{2}]+3[\F_{3}]\circ[\F_{2}]\circ[\F_{3}]\circ[\F_{3}]
=3[\F_{3}]\circ[\F_{3}]\circ[\F_{2}]\circ[\F_{3}]+[\F_{2}]\circ[\F_{3}]
\circ[\F_{3}]\circ[\F_{3}].$$ By Theorem \[c1\] (3) (b), we have $$\begin{aligned}
&&\gamma_n\circ([\F_{3}]\circ[\F_{3}]\circ[\F_{3}]\circ[\F_{2}]
+3[\F_{3}]\circ[\F_{2}]\circ[\F_{3}]\circ[\F_{3}])\\
&&=(f^3_{3}f_{2}+3f_{3}f_{2}f^2_{3})\circ\gamma_n=(3f^2_{3}f_{2}f_{3}+f_{2}f^3_{3})\circ\gamma_n\\
&&=\gamma_n\circ(3[\F_{3}]\circ[\F_{3}]\circ[\F_{2}]\circ[\F_{3}]
+[\F_{2}]\circ[\F_{3}]\circ[\F_{3}]\circ[\F_{3}]).\end{aligned}$$ Thus $[\F_{3}]\circ[\F_{3}]\circ[\F_{3}]\circ[\F_{2}]+3[\F_{3}]\circ[\F_{2}]\circ[\F_{3}]\circ[\F_{3}]
=3[\F_{3}]\circ[\F_{3}]\circ[\F_{2}]\circ[\F_{3}]+[\F_{2}]\circ[\F_{3}]
\circ[\F_{3}]\circ[\F_{3}]$ since $\gamma_n$ is an abelian group isomorphism.
Acknowledgement {#acknowledgement .unnumbered}
===============
We are grateful to the referee for many useful comments, suggestions and information about relevant references.
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\[SU\] J. Sussan, Category $\O$ and $\mf sl_k$-link invariants, arXiv:math.QA/0701045.
|
---
abstract: 'In this paper, we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, we prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. We extend the result of [@AW] on sequence matching to $\alpha$-mixing processes with exponential decay.'
address:
- |
Vanessa Barros and Jérôme Rousseau, Departamento de Matemática, Universidade Federal da Bahia\
Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
- 'Lingmin Liao, LAMA (Laboratoire d’Analyse et de Mathématiques Appliquées), Université Paris-Est Créteil Val de Marne, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France '
author:
- 'Vanessa Barros, Lingmin Liao, Jerome Rousseau'
title: On the shortest distance between orbits and the longest common substring problem
---
[^1]
Introduction
============
Motivations to study sequence matching or sequence alignment can be found in various fields of research (e.g. computer science, biology, bioinformatics, geology and linguistics, etc). For instance, to compare two DNA strands, one can be interested in finding the longest common substring, i.e. the longest string of DNA which appears in both strands. Thus, one can measure the level of relationship of the two strands by studying the length of this common substring. For example, for the following two strands $$ACAATGAGAGGATGACCTTG$$ $$TGACTGTAACTGACACAAGC$$ a longest common substring is ACAA (TGAC is also a longest common substring) and is of length 4 when the total length of the strands is 20.
Other quantities may be of interest in DNA comparison or more generally in sequence alignment and we refer to [@RSW; @W-book] for more information on the subject. Here we will concentrate on the behaviour of the length of the longest common substring when the length of the strings grows, more precisely, for two sequences $X$ and $Y$, the behaviour, when $n$ goes to infinity, of $$M_n(X,Y)=\max\{m:X_{i+k}=Y_{j+k}\textrm{ for $k=1,\dots,m$ and for some $0\leq i,j\leq n-m$}\}.$$ This problem was studied by Arratia and Waterman [@AW], who proved that if $X_1,X_2,\dots$,\
$Y_1,Y_2,\dots$ are i.i.d. such that $\P(X_1=Y_1)=p\in (0,1)$ then $$\P\left(\lim_{n\rightarrow\infty}\frac{M_n}{\log n}=\frac{2}{-\log p}\right)=1.$$ The same result was also proved for independent irreducible and aperiodic Markov chains on a finite alphabet, and in this case $p$ is the largest eigenvalue of the matrix $[(p_{ij})^2]$ (where $[p_{ij}]$ is the transition matrix).
In this paper, we generalize Arratia and Waterman’s result to $\alpha$-mixing process with exponential decay (or $\psi$-mixing with polynomial decay) and prove that if the Rényi entropy $H_2$ exists then $$\P\left(\lim_{n\rightarrow\infty}\frac{M_n}{\log n}=\frac{2}{H_2}\right)=1.$$
Our theorem applies to both cases of [@AW] which are $\alpha$-mixing with exponential decay. Other examples of $\alpha$-mixing process with exponential decay include Gibbs states of a Hölder-continuous potential [@Bowen; @Ruelle]. One can see [@Bradley] for a nice introduction on strong mixing conditions of stochastic processes (or [@brad1] for a more complete version).
Further developments of the work [@AW] (e.g. sequences of different lengths, different distributions, more than two sequences, extreme value theory for sequence matching) can be found in [@AW1; @AGW; @AW2; @AW3; @KO; @DKZ; @Neu]. We also refer the reader to [@pittel; @konto; @lambert] for related sequence matching problems.
A generalization of the longest common substring problem for dynamical systems is to study the behaviour of the shortest distance between two orbits, that is, for a dynamical system $(X,T,\mu)$, the behaviour, when $n$ goes to infinity, of $$m_n(x,y)=\min_{i,j=0,\dots,n-1}\left(d(T^ix,T^jy)\right).$$ Indeed, when $X=\mathcal{A}^\N$ for some alphabet $\mathcal{A}$ and $T$ is the shift on $X$, we can consider the distance between two sequences $x,y\in X$ defined by $d(x,y)=e^{-k}$ where $k=\inf\{i\geq0, x_i\neq y_i\}$.
Then, assuming that $m_n$ is not too small, that is $-\log m_n(x,y)\leq n$ (we will see in Theorem \[thineq\] that this condition is satisfied for almost all couples $(x,y)$ if $n$ is large enough), one can observe that almost surely $$M_n(x,y)\leq -\log m_n(x,y)\leq M_{2n}(x,y).$$ Thus ${M}_n(x,y)$ and $-\log m_n(x,y)$ have the same asymptotic behaviour.
Even if the shortest distance between two orbits seems to be something natural to define and study, to the best of our knowledge, it has not been done in the literature before. One can observe that this quantity shares some similarities with the correlation sum and the correlation integral of the Grassberger-Procaccia algorithm [@gp1; @gp2] and the nearest neighbour analysis [@cutler], with the synchronization of coupled map lattices [@vaienti1], with dynamical extremal index [@faranda-vaienti], with the connectivity, proximality and recurrence gauges defined by Boshernitzan and Chaika [@BC] and also with logarithm laws and shrinking target properties (see e.g. the survey [@athreya]). One can also remark that information on the hitting time (see e.g. [@saussol]) can give information on the shortest distance. Indeed, if we define the hitting time of a point $x$ in the ball $B(y,r)$ as $W_r(x,y)=\inf\{k\geq1,T^kx\in B(y,r)\}$ then if $W_r(x,y)\leq n$, we have $m_n(x,y)<r$.
In this paper, we show that the behaviour of the shortest distance $m_n$ is linked to the correlation dimension of the invariant measure $\mu$, defined (when the limit exists) by $${C}_\mu=\underset{r\rightarrow0}{\lim}\frac{\log\int_X\mu\left(B\left(x,r\right)\right)d\mu(x)}{\log r}.$$ More precisely, if the correlation dimension exists, then under some rapid mixing conditions of the system $(X,T,\mu)$, we deduce that for $\mu\otimes\mu$-almost every $(x,y)\in X\times X$, $$\underset{n\rightarrow+\infty}{\lim}\frac{\log m_n(x,y)}{-\log n}=\frac{2}{{C}_\mu}.$$ For irrational rotations, we prove that this result does not hold and that the previous limit depends on the irrationality exponent of the angle of the rotation. In the proof, the duality between hitting times and the shortest distance and the result of Kim and Seo [@KimSeo] on hitting times for irrational rotations are useful.
Our main results on the shortest distance between orbits and its relation with the correlation dimension are stated in Section \[secshort\] and proved in Section \[sec-proof\]. In Section \[sec-lcs\] we state an equivalent formulation of our main theorem (Theorem \[thprinc\]) for random processes. More precisely, we establish a relation between the longest common substring and the Rényi entropy. This result is proved in Section \[sec-discrete\]. The case of irrational rotations is treated in Section \[sec-rot\]. We apply our results to multidimensional expanding maps in Section \[secexp\].
Shortest distance between orbits {#secshort}
================================
Let $(X,d)$ be a finite dimensional metric space and $\mathcal{A}$ its Borel $\sigma$-algebra. Let $(X,\mathcal{A},\mu,T)$ be a measure preserving system (m.p.s.) which means that $T:X\rightarrow X$ is a transformation on $X$ and $\mu$ is a probability measure on $(X,\mathcal{A})$ such that $\mu$ is invariant by $T$, i.e., $\mu(T^{-1}A)=\mu(A)$ for all $A\in\mathcal{A}$.
We would like to study the behaviour of the shortest distance between two orbits: $$m_n(x,y)=\min_{i,j=0,\dots,n-1}\left(d(T^ix,T^jy)\right).$$ We will show that the behaviour of $m_n$ as $n\rightarrow\infty$ is linked with the correlation dimension. Before stating the next theorem, we recall the definition of the lower and upper correlation dimensions of $\mu$: $$\underline{C}_\mu=\underset{r\rightarrow0}{\underline\lim}\frac{\log\int_X\mu\left(B\left(x,r\right)\right)d\mu(x)}{\log r}\qquad\textrm{and}\qquad\overline{C}_\mu=\underset{r\rightarrow0}{\overline\lim}\frac{\log\int_X\mu\left(B\left(x,r\right)\right)d\mu(x)}{\log r}.$$ When the limit exists we will denote the common value of $\underline{C}_\mu$ and $\overline{C}_\mu$ by $C_\mu$. The existence of the correlation dimension and its relation with other dimensions can be found in [@Pes; @PW; @barbaroux].
\[thineq\]
Let $(X,\mathcal{A},\mu,T)$ be a measure preserving system (m.p.s.) such that $\underline{C}_\mu>0$. Then for $\mu\otimes\mu$-almost every $(x,y)\in X\times X$, $$\underset{n\rightarrow+\infty}{\overline\lim}\frac{\log m_n(x,y)}{-\log n}\leq\frac{2}{\underline{C}_\mu}.$$
Theorem \[thineq\] is a general result which can be applied to any dynamical system such that $\underline{C}_\mu>0$ and shows us that $m_n$ cannot be too small. If $\underline{C}_\mu=0$, one cannot expect to obtain such information since one can have $m_n(x,y)=0$ on a set of positive measure (for example if the measure $\mu$ is a finite linear combination of Dirac measures). We can also observe that the inequality in Theorem \[thineq\] can be strict (noting for example the trivial case when $T$ is the identity; a more interesting example, irrational rotations, will be treated in Section \[sec-rot\]) but under some natural rapidly mixing conditions we will prove an equality.
We need the following hypotheses.\
(H1) There exists a Banach space $\mathcal{C}$, such that for all $\psi,\ \phi\in \mathcal{C}$ and for all $n\in\N^*$, we have $$\left|\int_X\psi.\phi\circ T^n\, d\mu-\int_X \psi d\mu\int_X\phi d \mu\right|\leq\|\psi\|_\mathcal{C}\|\phi\|_\mathcal{C}\theta_n,$$ with $\theta_n =a^{n}$ ($0\leq a<1$) and where $\|\cdot\|_\mathcal{C}$ is the norm in the Banach space $\mathcal{C}$.
There exist $0<r_0<1$, $c\geq0$ and $\xi \geq0$ such that
(H2) For any $0<r<r_0$, the function $\psi_1:x\mapsto \mu(B(x,r))$ belongs to the Banach space $\mathcal{C}$ and $$\|\psi_1\|_\mathcal{C}\leq cr^{-\xi}.$$ (H3) For $\mu$-almost every $y\in X$ and any $0<r<r_0$, the function $\psi_2:x\mapsto \mathbbm{1}_{B(y,r)}(x)$ belongs to the Banach space $\mathcal{C}$ and $$\|\psi_2\|_\mathcal{C}\leq cr^{-\xi}.$$ We observe that the hypothesis (H3) cannot be satisfied when the Banach space $\mathcal{C}$ is the space of Hölder functions since the characteristic functions are not continuous. We will treat this case separately in Theorem \[thprinc2\].
We will also need some topological information on the space $X$.
\[deftight\] A separable metric space $(X,d)$ is called tight if there exist $r_0>0$ and $N_0\in \N$, such that for any $0 < r < r_0$ and any $x\in X$ one can cover $B(x, 2r)$ by at most $N_0$ balls of radius $r$.
We observe that this is not a very restrictive condition. Indeed, any subset of $\R^n$ with the Euclidian metric is tight and any subset of a Riemannian manifold of bounded curvature is tight (see [@GY]). In [@GY] it was also proved that if $(X,d)$ admits a doubling measure then it is tight and some examples of spaces which are not tight were given.
Now we can state our main result.
\[thprinc\] Let $(X,\mathcal{A},\mu,T)$ be a measure preserving system, such that $(X,d)$ is tight, satisfying (H1), (H2), (H3) and such that ${C}_\mu$ exists and is strictly positive. Then for $\mu\otimes\mu$-almost every $(x,y)\in X\times X$, $$\underset{n\rightarrow+\infty}{\lim}\frac{\log m_n(x,y)}{-\log n}=\frac{2}{{C}_\mu}.$$
Since our hypotheses are similar to the ones in [@FFT2; @AFV], it is natural to apply our theorem to the same family of examples. Here, we give a short list of simple examples. In Section \[secexp\], we apply our results to a more interesting family of examples: multidimensional piecewise expanding maps.
Denote by $Leb$ the Lebesgue measure.
Theorem \[thprinc\] can be applied to the following systems:
1. For $m\in\{2,3,\dots\}$, let $T: [0,1] \rightarrow [0,1]$ be such that $x\mapsto mx \mod 1$ and $\mu=Leb$.
2. Let $T:(0,1]\rightarrow(0,1]$ be such that $T(x)=2^k(x-2^{-k})$ for $x\in(2^{-k},2^{-k+1}]$ and $\mu=Leb$.
3. ($\beta$-transformations) For $\beta>1$, let $T: [0,1] \rightarrow [0,1]$ be such that $x\mapsto \beta x \mod 1$ and $\mu$ be the Parry measure (see [@parry]), which is an absolutely continuous probability measure with density $\rho$ satisfying $1-\frac{1}{\beta}\leq\rho(x)\leq(1-\frac{1}{\beta})^{-1}$ for all $x\in[0,1]$.
4. (Gauss map) Let $T:(0,1]\rightarrow(0,1]$ be such that $T(x)=\left\{\frac{1}{x}\right\}$ and $d\mu=\frac{1}{\log 2}\frac{dx}{1+x}$.
In these examples it is easy to see that $C_\mu=1$. Moreover, (H1), (H2) and (H3) are satisfied with the Banach space $\mathcal{C}=BV$ of functions having bounded variations (see e.g. [@FFT] Section 4.1 and [@philipp; @philipp2; @hof]).
One can observe that Theorem \[thprinc\] is an immediate consequence of Theorem \[thineq\] and the next theorem.
\[dsup\]
Let $(X,\mathcal{A},\mu,T)$ be a measure preserving system, such that $\underline{C}_\mu>0$ and such that $(X,d)$ is tight, satisfying (H1), (H2) and (H3). Then for $\mu\otimes\mu$-almost every $(x,y)\in X\times X$, $$\underset{n\rightarrow+\infty}{\underline\lim}\frac{\log m_n(x,y)}{-\log n}\geq\frac{2}{\overline{C}_\mu}.$$
When the Banach space $\mathcal{C}$ is the space of Hölder functions $\mathcal{H}^\alpha(X,\R)$ we can adapt our proof and do not need to assume (H3). Moreover, (H2) can be replaced by a condition on the measure of an annulus:
(HA) There exist $r_0>0$, $\xi\geq0$ and $\beta>0$ such that for every $x\in X$ and any $r_0>r>\rho>0$, $$\mu(B(x,r+\rho)\backslash B(x,r-\rho))\leq r^{-\xi}\rho^\beta.$$ In fact, we will show in the proof of the next theorem that (HA) implies (H2). Analogous conditions to (HA) have already appeared in the literature (e.g. [@GHN; @saussol; @chazottes; @haydn-w]) but in a local version. Here, we need a stronger global version. Nevertheless, one can easily observe that this assumption is still satisfied if the measure is Lebesgue or absolutely continuous with respect to Lebesgue with a bounded density.
\[thprinc2\]
Let $(X,\mathcal{A},\mu,T)$ be a measure preserving system, such that $\underline{C}_\mu>0$ and such that $(X,d)$ is tight, satisfying (H1) with $\mathcal{C}=\mathcal{H}^\alpha(X,\R)$ and (HA) or (H2). Then for $\mu\otimes\mu$-almost every $(x,y)\in X\times X$, $$\underset{n\rightarrow+\infty}{\underline\lim}\frac{\log m_n(x,y)}{-\log n}\geq\frac{2}{\overline{C}_\mu}.$$
Here are some interesting examples where Theorem \[thprinc2\] applies: planar dispersing billiard maps (with finite and infinite horizon) and Lorenz maps (see [@GHN] Section 4 and the references therein), expanding maps of the interval with a Gibbs measure associated to a Hölder potential (see [@saussol] where (HA) is proved in Lemma 44) and $C^2$ endomorphism (of a $d$-dimensional compact Riemannian manifold) admitting a Young tower with exponential tail (see [@FFT10] Section 6 and [@collet]).
Longest common substring problem {#sec-lcs}
================================
As explained in the introduction, finding the shortest distance between two orbits corresponds, when working with symbolic dynamical systems, to a sequence matching problem: finding the size of the longest common substrings between two sequences.
We will consider the symbolic dynamical system $(\Omega,\P,\sigma)$ where $\Omega=\mathcal{A}^\N$ for some alphabet $\mathcal{A}$, $\sigma$ is the (left) shift on $\Omega$ and $\P$ is an invariant probability measure. For two sequences $x,y\in\Omega$, we are interested in the behaviour of $$M_n(x,y)=\max\{m:x_{i+k}=y_{j+k}\textrm{ for $k=1,\dots,m$ and for some $1\leq i,j\leq n-m$}\}.$$ We will show that the behaviour of $M_n$ is linked with the Rényi entropy of the system.
For $y\in \Omega$ we denote by $C_n(y)=\{z \in \Omega :z_i = y_i\text{ for all }
0\le i\le n-1\} $ the *$n$-cylinder* containing $y$. Set $\F_0^n$ as the sigma-algebra over $\Omega$ generated by all $n$-cylinders.
We define the lower and upper Rényi entropy as the following limits: $$\underline{H}_2=\underset{k\rightarrow+\infty}{\underline\lim}\frac{\log\sum \P(C_k)^2}{-k}\qquad\textrm{and}\qquad\overline{H}_2=\underset{k\rightarrow+\infty}{\overline\lim}\frac{\log\sum \P(C_k)^2}{-k},$$ where the sums are taken over all $k$-cylinders. When the limit exists, we will denote it $H_2$.
The existence of the Rényi entropy has been proved for Bernoulli and Markov measures, Gibbs states of a Hölder-continuous potential, weakly $\psi$-mixing processes [@haydn-vaienti] and recently for $\psi_g$-regular processes [@abadi-car].
We say that our system is [*$\alpha$-mixing*]{} if there exists a function $\alpha:\N \rightarrow\R$ satisfying $\alpha(g)\to0$ when $g\to+\infty$ and such that for all $m,n \in \N$, $A\in\F_0^n$ and $B\in \F_0^{m}$: $$\left|\P(A\cap\sigma^{-g-n}B) -\P(A)\P(B)\right|\le \alpha(g).$$ It is said to be [*$\alpha$-mixing with an exponential decay*]{} if the function $\alpha(g)$ decreases exponentially fast to $0$.
We say that our system is [*$\psi$-mixing*]{} if there exists a function $\psi:\N \rightarrow\R$ satisfying $\psi(g)\to 0$ when $g\to+\infty$ and such that for all $m,n \in \N$, $A\in\F_0^n$ and $B\in \F_0^{m}$: $$\left|\P(A\cap\sigma^{-g-n}B) -\P(A)\P(B)\right|\le \psi(g)\P(A)\P(B).$$ Now we are ready to state our next result.
\[seqmat\]
If $\underline{H}_2>0$, then for $\P\otimes\P$-almost every $(x,y)\in \Omega\times \Omega$, $$\label{eqren1}
\underset{n\rightarrow+\infty}{\overline\lim}\frac{M_n(x,y)}{\log n}\leq\frac{2}{\underline{H}_2}.$$ Moreover, if the system is $\alpha$-mixing with an exponential decay or if the system is $\psi$-mixing with $\psi(g)=g^{-a}$ for some $a>0$ then, for $\P\otimes\P$-almost every $(x,y)\in \Omega\times \Omega$, $$\label{eqren2} \underset{n\rightarrow+\infty}{\underline\lim}\frac{M_n(x,y)}{\log n}\geq\frac{2}{\overline{H}_2}.$$ Furthermore, if the Rényi entropy exists, then for $\P\otimes\P$-almost every $(x,y)\in \Omega\times \Omega$, $$\underset{n\rightarrow+\infty}{\lim}\frac{M_n(x,y)}{\log n}=\frac{2}{H_2}.$$
Remark that Theorem \[seqmat\] generalizes the results in [@AW] since the processes treated there (i.i.d. and independent irreducible and aperiodic Markov chains on a finite alphabet) are $\alpha$-mixing with an exponential decay and their Rényi entropies exist. Moreover, in [@AW] the authors used a different proof for each case, while here we present a single and simpler proof. Our proof which will be presented in Section \[sec-discrete\] is an adaptation to symbolic dynamical systems of the results presented in Section \[secshort\].
One can apply our results to the following examples (which cannot be obtained from [@AW]).
Gibbs states of a Hölder-continuous potential $\phi$ are $\psi$-mixing with an exponential decay [@Bowen; @Ruelle]. Moreover, the Rényi entropy exists and $H_2=2P(\phi)-P(2\phi)$ where $P(\phi)$ is the pressure of the potential $\phi$ [@haydn-vaienti].
Let $0<q_i<1$ for any $i\in\N$. Consider the Markov chain $(Y_n)_n$ with the following transition probabilities $$Q_Y(i,j)=\left\{\begin{array}{ll} q_i & \textrm{if }j=0\\
1-q_i & \textrm{if }j=i+1\\
0 & \textrm{otherwise}\end{array}\right.$$ for any $i\in\N , j\in\N$.
Let $(X_n)_n$ be the process defined by $X_n=1$ when $Y_n=0$ and $X_n=0$ when $Y_n\neq 0$ and let $\P$ be its stationary measure. This process is called a binary renewal process.
Assuming that there exists $a\in(0,1/2)$ such that $a\leq q_i\leq 1-a$ for any $i\in\N$, then this process is $\alpha$-mixing with exponential decay [@abadi-car-gallo], thus we have the inequalities and . However, we observe that the existence of the Rényi entropy in this case is not known.
Irrational rotations {#sec-rot}
====================
In this section we consider the irrational rotations. For $\theta\in \mathbb{R}\setminus \mathbb{Q}$, let $T_\theta$ be the irrational rotation on the unit circle $\mathbb{T}= \mathbb{R}/\mathbb{Z}$ defined by $$T_\theta x= x+ \theta. $$ Then for any $n\in\mathbb{Z}$, we have $T_\theta^n x=x+n\theta$ and the shortest distance becomes $$m_n(x,y)=\min_{-n\leq j \leq n} \|(x-y)+j\theta\|.$$ The limit behavior of $m_n(x,y)$ is thus linked to the inhomogeneous Diophantine approximation.
Let $$\eta=\eta(\theta):=\sup\{\beta\geq 1: \liminf_{j\to\infty} j^\beta\|j\theta\|=0\}$$ be the irrationality exponent of $\theta$. Now we will show that the result of Theorem \[thprinc\] does not hold for $T_\theta$.
For $n\in \mathbb{N}$ and $(x,y)\in \mathbb{T}^2$, let $m_n(x,y)$ be the shortest distance between the orbits of $x$ and $y$ defined as above. Then for Lebesgue almost all $(x,y)\in \mathbb{T}^2$, we have $$\begin{aligned}
\liminf_{n\to\infty} {\log m_n(x,y) \over -\log n } = {1 \over \eta} \quad \text{and} \quad
\limsup_{n\to\infty} {\log m_n(x,y) \over -\log n }= 1.\end{aligned}$$
Let $$W_{B(y,r)} (x):=\inf\{n\geq 1: T_\theta^n(x)\in B(y,r)\}$$ be the waiting time for $x\in \mathbb{T}$ entering the ball $B(y,r)$ of center $y\in \mathbb{T}$ and radius $r>0$. Kim and Seo ([@KimSeo]) proved that for almost all $x$ and $y$ in $\mathbb{T}$, $$\label{KimS}
\liminf_{r\to 0} {\log W_{B(y,r)} (x) \over -\log r}=1 \quad \text{and} \quad \limsup_{r\to 0} {\log W_{B(y,r)} (x) \over -\log r}=\eta.$$ Let us denote $\widetilde W_{B(y,r)} (x)$ the waiting time under the action of the irrational rotation of angle $-\theta$. Since $\eta(\theta)=\eta(-\theta)$, is also satisfied for $\widetilde W_{B(y,r)} (x)$.
By definition, for the time $k=W_{B(y,r)} (x)$, $T_\theta^kx$ firstly enters the ball $B(y,r)$. So, we have $m_n(x,y)< r$, when $W_{B(y,r)} (x)\leq n$. On the other hand, when $ W_{B(y,r)} (x)>n$ and $\widetilde W_{B(y,r)} (x)>n$, we have $m_n(x,y)> r$.
By the first equality of (\[KimS\]), there is a sequence $(r_k)_k$ tending to $0$ such that $$\lim_{k\to \infty} {\log W_{B(y,r_k)} (x) \over -\log r_k}=1.$$ Let $n_k=W_{B(y,r_k)} (x)$. Then $m_{n_k}(x,y)< r_k$. Thus, $$\label{eta}
\limsup_{n\to\infty} {\log m_n(x,y) \over -\log n } \geq \limsup_{k\to\infty} {\log m_{n_k}(x,y) \over -\log n_k } \geq \lim_{k\to\infty}{\log r_k \over -\log W_{B(y,r_k)} (x)}=1.$$
Using again the first equality of (\[KimS\]), for any $0<\epsilon<1$, we have $$W_{B(y,r)} (x)> (1/r)^{1-\epsilon} \qquad\textrm{and}\qquad \widetilde W_{B(y,r)} (x)> (1/r)^{1-\epsilon},$$ provided $r>0$ is small enough.
Therefore, taking $0<\epsilon<1$ and defining $r=n^{-1/(1-\epsilon)}$ for $n\gg1$, we get
$$W_{B(y,r)} (x)> n \qquad\textrm{and}\qquad \widetilde W_{B(y,r)} (x)> n,$$ which implies $$m_n(x,y) > r= n^{-1/(1-\epsilon)}.$$ Thus $$\limsup_{n\to\infty} {\log m_n(x,y) \over -\log n } < {1 \over 1-\epsilon}.$$ By the arbitrariness of $\epsilon>0$, we have $$\label{eta1}
\limsup_{n\to\infty} {\log m_n(x,y) \over -\log n } \leq 1.$$
From the inequalities and we get the second part of the theorem.
By the same arguments and the second equality of (\[KimS\]), we deduce that for any $\epsilon>0$ $$W_{B(y,r)} (x)\leq (1/r)^{\eta+\epsilon},$$ provided $r>0$ small enough.
Hence, defining $r=n^{-1/(\eta+\epsilon)},\ n\gg1$, we have $
W_{B(y,r)} (x)\leq n,
$ which implies $$m_n(x,y) \leq r= n^{-1/(\eta+\epsilon)}.$$ Thus $$\liminf_{n\to\infty} {\log m_n(x,y) \over -\log n } \geq {1 \over \eta+\epsilon}.$$ By the arbitrariness of $\epsilon>0$, we then obtain $$\label{eta2}
\liminf_{n\to\infty} {\log m_n(x,y) \over -\log n } \geq {1 \over \eta}.$$
From the inequalities and we see that for $\eta=1$ the following result holds $$\lim_{n\to\infty} {\log m_n(x,y) \over -\log n } =1.$$ Thus, from now on we can suppose $\eta >1$ and it only remains to show $$\begin{aligned}
\label{upper_liminf}
\liminf_{n\to\infty} {\log m_n(x,y) \over -\log n } \leq {1 \over \eta}.\end{aligned}$$
We remark that the results of Kim and Seo ([@KimSeo]) are not applicable for proving (\[upper\_liminf\]), so we will give a direct proof.
Let $q_k=q_k(\theta)$ be the denominators of the $k$-th convergent of the continued fraction of $\theta$. Then (see for example Khintchine’s book [@Khinchin]) $$\label{qn-estimate}
{1\over 2q_{n+1}} < {1\over q_{n+1}+q_n}< \|q_{n}\theta\| \leq {1 \over q_{n+1}}.$$ By the theorem of best approximation (e.g. [@RS]), we have $$\eta(\theta) =\limsup_{n\to\infty} {\log q_{n+1}\over \log q_n}. \label{def-w}$$
First note that since $$m_n(x,y)=m_n(0,x-y),$$ it is enough to show that for almost all $y\in [0,1]$ $$\begin{aligned}
\liminf_{n\to\infty} {\log m_n(0,y) \over -\log n } \leq {1 \over \eta}.\end{aligned}$$
Second, we consider the following function $$f:y\in [0,1]\mapsto \liminf_{n\to\infty} {\log m_n(0,y) \over -\log n }.$$ Observing $$m_{n+1}(0,y)\leq m_n(0,T_\theta y)=\min_{-n+1\leq j \leq n+1} \|y+j\theta\|\leq m_{n-1}(0,y),$$ we see that $f$ is a $T_\theta$-invariant function. By the ergodicity of the Lebesgue measure with respect to the irrational rotations, we conclude that $f$ is a constant for almost all $y\in[0,1]$. Since we have already proved that $$\begin{aligned}
\liminf_{n\to\infty} {\log m_n(0,y) \over -\log n } \geq {1 \over \eta},\end{aligned}$$ we only need to show that for any $\delta>0$, the set $$E_\delta:=\left\{y\in [0,1]: \text{there exists a sequence } \{N_k\}_k \ \text{such that $\forall k$, } {\log m_{N_k}(0,y) \over \log N_k} \leq {1 \over \eta}+\delta\right\}$$ has positive Lebesgue measure. By the definition of $m_n(x,y)$, we can rewrite $E_\delta$ as $$E_\delta=\left\{y\in [0,1]: \exists \ \{ N_k \}_k \ \text{s.t. } \forall\ k, \ \forall j=-N_k+1,...,N_k-1, \ \| j\theta-y \|\geq {1 \over N_k^{{1 \over \eta}+\delta}}\right\}.$$
Let $\tau={1 \over \eta}+\delta$ and take $0<\epsilon<{\delta \eta^2 \over 1+\delta\eta}$. From the identity there exists a subsequence $\{n_k\}_k$ such that $$q_{n_k+1} \geq q_{n_k}^{\eta-{\epsilon\over 2}} \geq q_{n_k}^{\eta-\epsilon}>q_{n_k}^{\eta \over 1+\delta\eta}, \ \text{ since $q_k \geq 1$. }$$ Without loss of generality, we still write $\{k\}$ this subsequence.
Take $N_k=\lceil q_k^{\eta-\epsilon} \rceil$ and define the following decreasing sequence of sets $$E_{\delta,k}:=\left\{y\in [0,1]: \forall j=-N_k+1,...,N_k-1, \ \| j\theta-y \|\geq {1 \over N_k^{{\tau}}}\right\}.$$ Since $E_{\delta}=\cap_{k\geq k_0} E_{\delta,k}$ for any $k_0\in \mathbb{N}$, we only need to show that $\cup_{k\geq k_0} E_{\delta,k}^c$ has Lebesgue measure strictly less than $1$ for some $k_0\in \mathbb{N}$.
Now observing that $
E_{\delta,k}^c=\bigcup_{-N_k< j < N_k} B\big(j\theta, {1\over N_k^\tau}\big),
$ we only have to estimate the Lebesgue measure of the following set $$\bigcup_{k \geq k_0} \bigcup_{-N_k< j < N_k} B\big(j\theta, {1\over N_k^\tau}\big).$$
Let us first consider the union $
\bigcup_{0\leq j < N_k} B\big(j\theta, {1\over N_k^\tau}\big).
$
By the definition of $N_k$ and the facts $\eta-\epsilon>1$ and that $\{q_k \}\subset \N$ is increasing we have $$q_k\leq N_k\leq q_{k+1}.$$ Therefore $$\begin{aligned}
\label{eq-union}
\bigcup_{0\leq j < N_k} B\big(j\theta, {1\over N_k^\tau}\big) \subset \bigcup_{i=0}^{q_k-1} \bigcup_{j=0}^{\lceil N_k/q_k\rceil} B\Big((i+jq_k)\theta, {1\over N_k^\tau}\Big).\end{aligned}$$ Moreover, we have $$\begin{aligned}
\label{N_k_tau_upper}
{1\over N_k^\tau} \leq {1 \over q_k^{(\eta-\epsilon)\tau}}= {1 \over q_k^{(\eta-\epsilon)({1 \over \eta}+\delta)}} = {1 \over q_k^{1+\delta\eta-\epsilon({1 \over \eta}+\delta)}} ={1 \over q_k^{1+\epsilon_1}}. $$ where $\epsilon_1:=\delta\eta-\epsilon({1 \over \eta}+\delta)>0$.
To estimate the measure of the following set $$\begin{aligned}
\bigcup_{j=0}^{\lceil N_k/q_k\rceil} B\Big((i+jq_k)\theta, {1\over N_k^\tau}\Big),\end{aligned}$$ one can observe that the distance between two consecutive centers of the balls in the union is $$\| (i+(j+1)q_k)\theta-(i+jq_k)\theta\|=\|q_k\theta\|.$$ Thus $$Leb\left(\bigcup_{j=0}^{\lceil N_k/q_k\rceil} B\Big((i+jq_k)\theta, {1\over N_k^\tau}\Big)\right)\leq \left({N_k \over q_k}+2\right)\cdot \|q_k\theta\|+{2 \over N_k^\tau} .$$
From the inequalities and we have $$\begin{aligned}
\label{}
\left({N_k \over q_k}+2\right)\cdot \|q_k\theta\|+{2 \over N_k^\tau} \leq & \left({q_k^{\eta-\epsilon}+1 \over q_k}+2\right)\cdot {1 \over q_{k+1}} +{2 \over N_k^\tau}
\leq {3q_k^{\eta-\epsilon} \over q_k} \cdot {1 \over q_{k}^{\eta-{\epsilon \over 2}}} + {2\over N_k^\tau}\\
\leq & {3 \over q_{k}^{1+{\epsilon \over 2}}} + {2\over q_k^{1+\epsilon_1}}
\leq {5 \over q_k^{1+\epsilon_2}}, $$ where $0<\epsilon_2\leq \min\{\epsilon/2,\epsilon_1\}$.
Let $\beta>1$. For $k$ large enough we obtain $$\begin{aligned}
Leb\left(\bigcup_{0\leq j < N_k} B\big(j\theta, {1\over N_k^\tau}\big) \right)\leq q_k \cdot {5 \over q_k^{1+\epsilon_2}}={5 \over q_k^{\epsilon_2}} \leq 5 \beta^{-\epsilon_2 k},\end{aligned}$$ where the last inequality comes from the assumption $\eta(\theta)>1$.
By symmetry, we also deduce that $$\begin{aligned}
Leb\left(\bigcup_{-N_k< j \leq 0} B\big(j\theta, {1\over N_k^\tau}\big) \right)\leq 5 \beta^{-\epsilon_2 k}.\end{aligned}$$
Therefore $$\begin{aligned}
Leb\left(\bigcup_{-N_k< j < N_k} B\big(j\theta, {1\over N_k^\tau}\big) \right)\leq 10 \beta^{-\epsilon_2 k}.\end{aligned}$$
Note that there exists $k_0\geq 1$, such that $$10\sum_{k=k_0}^\infty \beta^{-\epsilon_2 k} = {10\beta^{-\epsilon_2k_0} \over 1- \beta^{-\epsilon_2}}<1.$$ Thus $$\bigcup_{k\geq k_0} E_{\delta k}^c = \bigcup_{k\geq k_0}\bigcup_{-N_k< j < N_k} B\big(j\theta, {1\over N_k^\tau}\big)$$ has Lebesgue measure strictly less than $1$. Therefore, (\[upper\_liminf\]) follows.
Hence, finally, we conclude that for almost all $x$ and $y$, $$\liminf_{n\to\infty} {\log m_n(x,y) \over -\log n } = {1 \over \eta}, \quad \text{and} \quad
\limsup_{n\to\infty} {\log m_n(x,y) \over -\log n }= 1.$$
Multidimensional piecewise expanding maps {#secexp}
=========================================
In this section, we will apply our main result to a family of maps defined by Saussol [@Saussol1]: multidimensional piecewise uniformly expanding maps. It was observed in [@AFLV11] that these maps generalize Markov maps which also contain one-dimensional piecewise uniformly expanding maps.
Let $N\geq 1$ be an integer. We will work in the Euclidean space $\mathbb{R}^N$. We denote by $B_\epsilon(x)$ the ball with center $x$ and radius $\epsilon$. For a set $E\subset\mathbb{R}^N$, we write $$B_\epsilon(E):=\{y\in\mathbb{R}^N: \sup_{x\in E} |x-y|\leq \epsilon\}.$$
Let $X$ be a compact subset of $\R^N$ with $\overline{X^\circ}=X$ and $T:X \rightarrow X$. The system $(X,T)$ is a multidimensional piecewise expanding system if there exists a family of at most countably many disjoint open sets $U_i \subset X$ and $V_i$ such that $\overline{U_i} \subset V_i$ and maps $T_i:V_i \rightarrow \R^N$ satisfying for some $0< \alpha \leq1$, for some small enough $ \epsilon_0>0$, and for all $i$:
1. $T\vert_{U_i}=T_i\vert_{ U_i} $ and $B_{\epsilon_0}(TU_i)\subset T_i(V_i) $;
2. $T_i \in C^1(V_i),T_i \text{ is injective and } T_i^{-1}\in C^1(T_iV_i) .$ Moreover, there exists a constant $c$, such that for all $\epsilon \leq \epsilon_0, z \in T_iV_i$ and $x,y \in B_{\epsilon}(z)\cap T_iV_i$ we have $$|\det D_xT_i^{-1}-\det D_yT_i^{-1}| \leq c \epsilon^{\alpha} |\det D_zT_i^{-1}|;$$
3. $Leb(X\setminus \bigcup_i U_i)=0$;
4. there exists $s=s(T)<1$ such that for all $u,v \in TV_i$ with $d(u,v) \leq \epsilon_0$ we have $d(T_i^{-1}u,T_i^{-1}v)\leq s d(u,v)$;
5. let $G(\epsilon,\epsilon_0):=\sup_x G(x,\epsilon,\epsilon_0)$ where $$G(x,\epsilon,\epsilon_0)=\sum_i\frac{Leb(T_i^{-1}B_{\epsilon}(\partial TU_i)\cap B_{(1-s)\epsilon_0}(x))}{m(B_{(1-s)\epsilon_0}(x))},$$ then the number $\eta=\eta(\delta):=s^\alpha +2 \sup_{\epsilon \leq \delta} \frac{G(\epsilon)}{\epsilon^\alpha}\delta^\alpha$ satisfies $\sup_{\delta \leq \epsilon_0} \eta(\delta)<1.$
We will prove that the multidimensional piecewise expanding systems satisfy the conditions of Theorem \[thprinc\].
Let $(X,T)$ be a topologically mixing multidimensional piecewise expanding map and $\mu$ be its absolutely continuous invariant probability measure. If the density of $\mu$ is bounded away from zero, then for $\mu\otimes\mu$-almost every $(x,y)\in X\times X$, $$\underset{n\rightarrow+\infty}{\lim}\frac{\log m_n(x,y)}{-\log n}=\frac{2}{N}.$$
First of all, we define the Banach space involved in the mixing conditions. Let $\Gamma \subset X$ be a Borel set. We define the oscillation of $\varphi \in L^1(Leb)$ over $\Gamma$ as
$$osc(\varphi,\Gamma)= \underset{ \Gamma}{ \textrm{ess-sup} } (\varphi)- \underset{ \ \Gamma}{ \textrm{ess-inf} } (\varphi).$$
Now, given real numbers $0<\alpha \leq 1$ and $0<\epsilon_0<1$ consider the following $\alpha$-seminorm $$|\varphi|_\alpha=\underset{ 0<\epsilon\leq\epsilon_0}{ \sup \ } \epsilon^{ -\alpha} \int_{ X}osc(\varphi,B_\epsilon(x))dx.$$ We observe (see [@Saussol1]) that $X\ni x \mapsto osc(\varphi,B_\epsilon(x))$ is a measurable function and $$\text{supp} (\text{osc}(\varphi,B_\epsilon(x)))\subset B_\epsilon(\text{supp } \varphi).$$ Let $V_{\alpha}$ be the space of $L^1(Leb)-$functions such that $|\varphi|_\alpha<\infty$ endowed with the norm $$\|\varphi\|_{\alpha}=\|\varphi\|_{L^1(Leb)}+|\varphi|_\alpha.$$ Then $(V_{\alpha},\|\cdot\|_{\alpha})$ is a Banach space which does not depend on the choice of $\epsilon_0 $ and $V_{\alpha} \subset L^\infty $ (see [@Saussol1]).
Saussol ([@Saussol1]) proved that for a piecewise expanding map $T:X \longrightarrow X$, where $X\subset \R^N$ is a compact set, there exists an absolutely continuous invariant probability measure $\mu$ which enjoys exponential decay of correlations against $L^1$ observables on $V_{\alpha}$. More precisely, for all $\psi\in V_{\alpha}$, for all $\phi\in L^1(\mu)$ and for all $n\in\N^*$, we have $$\left|\int_X\psi.\phi\circ T^n\, d\mu-\int_X \psi d\mu\int_X\phi d \mu\right|\leq\|\psi\|_{\alpha}\|\phi\|_1\theta_n,$$ with $\theta_n =a^{n}$ ($0\leq a<1$). This means that the system $(X,T,\mu)$ satisfies the condition (H1) with $\mathcal{C}=V_\alpha$.
It remains to show that the system also satisfies the conditions (H2) and (H3) (with $r_0=\epsilon_0$). To this end, we need to estimate the norms $\|\psi_1\|_{\alpha}$ and $\|\psi_2\|_{\alpha}$, where $\psi_1$ and $\psi_2$ are the functions defined in (H2) and (H3). Since $\psi_1$ and $\psi_2$ are both in $L^1(Leb)$ we just need to estimate their $\alpha$-seminorms.
From the above observation we notice that $$\begin{aligned}
\textrm{supp} \ osc(\psi_j,B_\epsilon(\cdot)) \subset X_\epsilon,\ j=1,2,
\end{aligned}$$ where $X_\epsilon=\{ x\in \R^N, d(x,X)\leq \epsilon\} $ is a compact set. Therefore $$|\psi_j|_\alpha=\underset{ 0<\epsilon\leq\epsilon_0}{ \sup \ } \epsilon^{ -\alpha} \int_{ X_\epsilon}osc(\psi_j,B_\epsilon(x))dx,\ j=1,2.$$
To estimate $|\psi_j|_\alpha,\ j=1,2$ we define $$S_j^\epsilon:= \epsilon^{ -\alpha} \int_{ X_\epsilon}osc(\psi_j,B_\epsilon(x))dx,$$ and prove that $S_j^\epsilon$ is bounded from above by $C_j\epsilon_0^{ 1-\alpha}$, for some $ C_j>0,\ j=1,2. $
Let us start with $S_1^\epsilon=\epsilon^{ -\alpha} \int_{ X_\epsilon}osc(\mu(B(\cdot,r),B_\epsilon(x))dx$. Suppose $r\leq \epsilon$. Since $\mu$ is absolutely continuous and its density is bounded away from zero, we can write $$\begin{aligned}
\label{ACpsi1}
\psi_1(y)=\mu(B(y,r))=\int_{B(y,r)} h(z)dz,\end{aligned}$$ where the density $h$ belongs to $V_{\alpha}\subset L^\infty$. It means that $0< c_1\leq h \leq c_2$ for some constants $c_1$ and $c_2.$
By we have $$\begin{aligned}
osc(\psi_1,B_\epsilon(x))= \underset{ y\in B(x,\epsilon)\cap X }{ \textrm{ess-sup} } \int_{B(y,r)} h(z)dz- \underset{ \tilde y\in B(x,\epsilon) \cap X}{ \textrm{ess-inf} } \int_{B(\tilde y,r)} h(z)dz.
\end{aligned}$$ Therefore $S^1_ \epsilon$ becomes $$S^1_ \epsilon= \epsilon^{ -\alpha} \int_{ X_\epsilon}\left(\underset{ y\in B(x,\epsilon) \cap X}{ \textrm{ess-sup} } \int_{B(y,r)} h(z)dz- \underset{ \tilde y\in B(x,\epsilon) \cap X}{ \textrm{ess-inf} } \int_{B(\tilde y,r)} h(z)dz
\right)dx.$$ Since $c_1 \leq h \leq c_2$, $$\begin{aligned}
S_1^\epsilon& \leq \epsilon^{ -\alpha} \int_{ X_\epsilon}\left(\underset{ y\in B(x,\epsilon) \cap X}{ \textrm{ess-sup} } \int_{B(y,r)} c_2dz- \underset{ \tilde y\in B(x,\epsilon) \cap X}{ \textrm{ess-inf} } \int_{B(\tilde y,r)} c_1dz
\right)dx\\
&\leq C_0\epsilon^{ -\alpha} \int_{ X_\epsilon} (c_2-c_1) r^Ndx \leq C_0(c_2-c_1)\epsilon^{ -\alpha+N} Leb(X_{\epsilon}),
\end{aligned}$$ where $C_0$ is the Lebesgue measure of the unit ball in $\mathbb{R}^N$. Using the facts that $X_\epsilon\subset X_{\epsilon_0}$ and that $X_{\epsilon_0}$ is compact, we have
$$\label{sup1}
S_1^\epsilon \leq (c_2-c_1)\epsilon^{ -\alpha+N} Leb(X_{\epsilon_0})\leq C \epsilon_0^{ -\alpha+N}.$$
Now suppose $r>\epsilon$. Then for each $y\in B(x,\epsilon)$ we have $$B(x,r-\epsilon)\subset B(y,r)\subset B(x,r+\epsilon).$$ Therefore $$\begin{aligned}
\label{a}
\ osc(\psi_1,B_\epsilon(x))\leq \int_{B(x,r+\epsilon)} h(z)dz-\int_{B(x,r-\epsilon)} h(z)dz= \int_{D} h(z)dz
\leq \|h\|_{\infty}Leb(D),\end{aligned}$$ where $D=B(x,r+\epsilon)\setminus B(x,r-\epsilon)$. It is easy to see that $$\begin{aligned}
\label{b}
Leb(D)\leq 2C_0\epsilon \sum_{k=0}^{N-1}\binom{N}{k}
\leq 2^{N+1} C_0\epsilon .\end{aligned}$$ From the inequalities and we deduce that $$\begin{aligned}
\label{c}
S_1^\epsilon\leq\epsilon^{-\alpha}\|h\|_{\infty}Leb(D)Leb(X_\epsilon)
\leq 2^N \epsilon_0^{1-\alpha}\|h\|_{\infty}Leb(X_{\epsilon_0}) \leq C \epsilon_0^{ 1-\alpha}.\end{aligned}$$
Combining and , we obtain $$\label{g}
|\psi_1|_\alpha \leq C_1\epsilon_0^{ 1-\alpha}$$ for some constant $C_1$.
It remains to estimate $S_2^\epsilon$. First, let us estimate the oscilation of $\psi_2(z)=\mathbbm{1}_{B(y,r)}(z)$ over the set $B(x,\epsilon)$ when $r\leq\epsilon$: $$\begin{aligned}
osc(\psi_2,B(x,\epsilon))&=& \underset{ z\in B(x,\epsilon)\cap X}{ \textrm{ess-sup} } \mathbbm{1}_{B(y,r)}(z)- \underset{ \tilde{z}\in B(x,\epsilon)\cap X}{ \textrm{ess-inf} } \mathbbm{1}_{B(y,r)}(\tilde{z})\\
&\leq &\mathbbm{1}_{B(y,r+\epsilon)}(x).\end{aligned}$$ Thus, $$\label{f}
S_2^\epsilon:=\epsilon^{-\alpha}\int_{ X_\epsilon}osc(\psi_2,B_\epsilon(x))dx
\leq C_0\epsilon^{-\alpha}(r+\epsilon)^N
\leq 2^N \epsilon_0^{N-\alpha}.$$
When $r>\epsilon$, we have $$\begin{aligned}
osc(\psi_2,B(x,\epsilon))\leq \mathbbm{1}_{B(y,r+\epsilon)\setminus B(y,r-\epsilon)}(x).\end{aligned}$$ Using the same ideas as in the estimation of $\|\psi_1\|_\alpha$ and the last inequality, we obtain $$S_2^\epsilon
\leq2^N \epsilon_0^{1-\alpha}Leb(X_{\epsilon_0}). \label{sup2-psi2}$$
Thus, and give us $$\label{psi2}
|\psi_2|_\alpha \leq C_2\epsilon_0^{ 1-\alpha}$$ for some constant $C_2$.
From the inequalities and , we get (H2) and (H3).
Finally, a straightforward calculation leads to ${C}_\mu=N$.
Proofs of the main results {#sec-proof}
==========================
For $\eps>0$, let us define $$k_n=\frac{1}{\underline{C}_\mu-\eps}(2\log n+\log\log n) \quad \text{and} \quad r_n=e^{-k_n}.$$ We also define $$A_{ij}(y)=T^{-i}B(T^jy,e^{-k_n})$$ and $$S_n(x,y)=\sum_{i,j=0,\dots,n-1}\mathbbm{1}_{A_{ij}(y)}(x).$$ Observe that $$\label{eqMnSn}
\left\{(x,y): m_n(x,y)< r_n\right\}= \left\{(x,y):S_n(x,y)>0\right\}.$$ Thus, we have $$\mu\otimes\mu \left((x,y):m_n(x,y)< r_n\right)=\mu\otimes\mu \left((x,y): S_n(x,y)>0\right)=\mu\otimes\mu \left((x,y): S_n(x,y)\geq1\right).$$ Then, using Markov’s inequality, we obtain $$\begin{aligned}
\mu\otimes\mu \left((x,y):m_n(x,y)< r_n\right)&\leq& \E(S_n)=\iint\sum_{i,j=0,\dots,n-1}\mathbbm{1}_{A_{ij}(y)}(x)d\mu\otimes\mu(x,y)\\
&=&\sum_{i,j=0,\dots,n-1}\int\left(\int\mathbbm{1}_{A_{ij}(y)}(x)d\mu(x)\right)d\mu(y)\\
&=&\sum_{i,j=0,\dots,n-1}\int\mu\left(B(T^jy,r_n)\right)d\mu(y),\end{aligned}$$ since $\mu$ is invariant.
Using again the invariance of $\mu$, we get $$\mu\otimes\mu \left((x,y):m_n(x,y)< r_n\right)\leq n^2\int\mu(B(y,r_n)d\mu(y).$$ By the definition of the lower correlation dimension and the definition of $k_n$, for $n$ large enough, we have $$\mu\otimes\mu \left((x,y):m_n(x,y)< r_n\right)\leq n^2r_n^{\underline{C}_\mu-\eps}=\frac{1}{\log n}.$$ Finally, choosing a subsequence $n_\ell=\lceil e^{\ell^2}\rceil$, we have $$\mu\otimes\mu \left((x,y):m_{n_\ell}(x,y)< r_{n_\ell}\right)\leq \frac{1}{\log n_\ell}\leq \frac{1}{\ell^2}.$$ Thus $\sum_\ell \mu\otimes\mu \left((x,y):m_{n_\ell}(x,y)< r_{n_\ell}\right)<+\infty$. By the Borel-Cantelli Lemma, for $\mu\otimes\mu$-almost every $(x,y)\in X\times X$, if $\ell$ is large enough then $$m_{n_\ell}(x,y)\geq r_{n_\ell}$$ and $$\frac{\log m_{n_\ell}(x,y)}{-\log n_\ell}\leq \frac{1}{\underline{C}_\mu-\eps}\left(2+\frac{\log\log n_\ell}{\log n_\ell}\right).$$ Finally, taking the limit superior in the previous equation and observing that $(n_\ell)_\ell$ is increasing, $(m_n)_n$ is decreasing and $\underset{\ell\rightarrow+\infty}\lim\frac{\log n_\ell}{\log n_{\ell+1}}=1$, we have $$\underset{n\rightarrow+\infty}{\overline\lim}\frac{\log m_n(x,y)}{-\log n}=\underset{\ell\rightarrow+\infty}{\overline\lim}\frac{\log m_{n_\ell}(x,y)}{-\log n_\ell}\leq \frac{2}{\underline{C}_\mu-\eps}.$$ Then the theorem is proved since $\eps$ can be chosen arbitrarily small.
Before proving Theorem \[dsup\] we state a few facts in order to simplify the calculations. At first let us recall the notion of $(\lambda,r)$-grid partition.
Let $0<\lambda<1$ and $r>0$. A partition $\{Q_i\}_{i=1}^{\infty}$ of $X$ is called a $(\lambda,r)$-grid partition if there exists a sequence $\{x_i\}_{i=1}^{\infty}$ such that for any $i\in\N$ $$B(x_i,\lambda r)\subset Q_i\subset B(x_i, r).$$
Now we prove a technical lemma.
\[lema1\] Under the hypotheses of Theorem \[dsup\], there exists a constant $K>0$ such that $$\int_X\mu\left(B(y,r_n)\right)^2d\mu(y) \leq K\left(\int_X \mu \left(B(y,r_n)\right)d\mu(y)\right)^{3/2},\ \ \ \text{for } n\ \text{large enough}.$$
Since $X$ is a metric space, there exist $0<\lambda<\frac12$ and $R>0$ such that for any $0<r<R$ there exists a $(\lambda,r)$-grid partition (see Proposition 2.1 in [@GY]).
Let us choose $n$ large enough so that $r_n<\min\{R,r_0/2\}$ ($r_0$ as in Definition \[deftight\]). Let $\{Q_i\}_{i=1}^{\infty}$ be a $(\lambda,\frac{r_n}{2})$-grid partition and $\{x_i\}_{i=1}^\infty$ be such that $$B\left(x_i,\lambda \frac{r_n}{2}\right)\subset Q_i
\subset B\left(x_i, \frac{r_n}{2}\right).$$ Then we have $$\begin{aligned}
\label{a1}
\int_X\mu\left(B(y,r_n)\right)^2d\mu(y)&=&\sum_{i}\int_{Q_i}\mu\left(B(y,r_n)\right)^2d\mu(y).\end{aligned}$$
Now, fix a ball $B(x_i, {2r_n})$ and consider the set $$D_i=\{x_j: Q_j \cap B(x_i,2r_n) \neq \emptyset\}.$$ Since the space is tight, one can conclude that (see the proof of Theorem 4.1 in [@GY] ) there exists a constant $K_0$ depending only on $N_0$ such that the cardinality card($D_i$)$\leq K_0.$ Therefore $$\bigcup_{y \in Q_i}B(y,r_n)\subset B(x_i,2r_n) \subset\bigcup_{j=1}^{K_0}Q_{i,j},$$ where $Q_{i,j}$ are elements of the partition.
By we have $$\begin{aligned}
& &\int_X\mu\left(B(y,r_n)\right)^2d\mu(y)
\leq\sum_{i}\int_{Q_i}\left(\sum_{j=1}^{K_0}\mu\left(Q_{i,j}\right)\right)^2d\mu(y)\\
&=&\sum_{i}\mu(Q_i)\left(\sum_{j=1}^{K_0}\mu\left(Q_{i,j}\right)\right)^2
\leq\sum_{i}\left(\sum_{j=1}^{K_0}\mu\left(Q_{i,j}\right)\right)^3
\leq K_0^2\sum_{i}\sum_{j=1}^{K_0}\mu\left(Q_{i,j}\right)^3,\end{aligned}$$ where the last inequality is deduced from Jensen’s inequality. Now, since the elements $Q_{i,j}$ cannot participate in more than $K_0$ different sums (one can see the arguments leading to (12) in [@GY]) and since $x\mapsto x^{2/3}$ is a countably subbadditive function, we have $$\begin{aligned}
& &\int_X\mu\left(B(y,r_n)\right)^2d\mu(y)
\leq K_0^3\sum_{i}\mu\left(Q_{i}\right)^3\\
&\leq& K_0^3\left(\sum_{i}\mu\left(Q_{i})\right)^2\right)^{3/2}
=K_0^3\left(\sum_{i}\int_{Q_i}\mu\left(Q_i\right)d\mu(y)\right)^{3/2}.\end{aligned}$$ Finally, note that for any $y\in Q_i$, we have $Q_i\subset B(y,r_n)$. Thus $$\begin{aligned}
\int_X\mu\left(B(y,r_n)\right)^2d\mu(y)
\leq K_0^3\left(\sum_{i}\int_{Q_i}\mu\left(B(y,r_n\right)d\mu(y)\right)^{3/2},\end{aligned}$$ and the result follows with $K=K_0^3$.
We are now ready to prove Theorem \[dsup\].
Without loss of generality, we will assume in the proof that $\theta_n=e^{-n}$.
For $\eps>0$, let us define $$k_n=\frac{1}{\overline{C}_\mu+\eps}(2\log n+b\log\log n) \quad \text{and} \quad r_n=e^{-k_n}.$$ Using the same notation as in the proof of Theorem \[thineq\], we recall that $$\label{eqexp}\E(S_n)=n^2\int\mu(B(y,r_n))d\mu(y).$$ Moreover, using and Chebyshev’s inequality, we obtain $$\begin{aligned}
&\mu\otimes\mu \left((x,y):m_n(x,y)\geq r_n\right)\leq\mu\otimes\mu \left((x,y):S_n(x,y)=0\right) \nonumber \\
\leq&\mu\otimes\mu \left((x,y):|S_n(x,y)-\E(S_n)|\geq |\E(S_n)|\right)
\leq\frac{\operatorname{var}(S_n)}{\E(S_n)^2}. \label{eqmnvarsn}\end{aligned}$$ Thus, we need to control the variance of $S_n$. First of all, we have $$\begin{aligned}
\operatorname{var}(S_n)
=&\sum_{1\leq i,i',j,j'\leq n}cov(\mathbbm{1}_{A_{ij}},\mathbbm{1}_{A_{i'j'}})\\
=&\sum_{1\leq i,i',j,j'\leq n}\iint \mathbbm{1}_{A_{ij}}\mathbbm{1}_{A_{i'j'}}-\iint\mathbbm{1}_{A_{ij}}\iint\mathbbm{1}_{A_{i'j'}}\\
=&\sum_{1\leq i,i',j,j'\leq n}\iint \mathbbm{1}_{B(T^jy,r_n)}(T^ix)\mathbbm{1}_{B(T^{j'}y,r_n)}(T^{i'}x)-n^4\left(\int\mu(B(y,r_n)d\mu(y)\right)^2.\end{aligned}$$ Let $g=g(n)=\log(n^{4+4\xi/(\overline{C}_\mu+\eps)})$. We will split the last sum into the following four parts: $$\begin{aligned}
\sum_{1\leq i,i',j,j'\leq n}=&\sum_{|i-i'|>g,|j-j'|> g}+\sum_{|i-i'|>g,|j-j'|\leq g}+\sum_{|i-i'|\leq g,|j-j'|> g}+\sum_{|i-i'|\leq g,|j-j'|\leq g}\\
=:& I+II+III+IV.\end{aligned}$$
At first we observe that if $|i-i'|>g$, then by (H1) and (H3), $$\begin{aligned}
& &\iint \mathbbm{1}_{B(T^jy,r_n)}(T^{i-i'}x)\mathbbm{1}_{B(T^{j'}y,r_n)}(x)d\mu(x)d\mu(y)\nonumber\\
&\leq&\int\left(\int\mathbbm{1}_{B(T^{j}y,r_n)}(x)d\mu(x)\int\mathbbm{1}_{B(T^{j'}y,r_n)}(x)d\mu(x)\right.\nonumber\\
& &+\left.\theta_g \cdot\|\mathbbm{1}_{B(T^{j}y,r_n)}\|_\mathcal{C} \cdot \|\mathbbm{1}_{B(T^{j'}y,r_n)}\|_\mathcal{C}\right)d\mu(y)\nonumber\\
&\leq&c^2r_n^{-2\xi}\theta_g+\int\mu\left(B(T^{j}y,r_n)\right)\mu\left(B(T^{j'}y,r_n)\right)d\mu(y).\label{eqmix1}\end{aligned}$$ Therefore $$\begin{aligned}
I+II\leq& n^4c^2r_n^{-2\xi}\theta_g+n^2\sum_{|j-j'|> g}\int\mu\left(B(T^{j}y,r_n)\right)\mu\left(B(T^{j'}y,r_n)\right)d\mu(y)\\
&+n^2\sum_{|j-j'|\leq g}\int\mu\left(B(T^{j}y,r_n)\right)\mu\left(B(T^{j'}y,r_n)\right)d\mu(y).\end{aligned}$$
Now, in the case where $|j-j'|>g$, we use (H1) and (H2) to get $$\begin{aligned}
& &\int\mu\left(B(T^{j}y,r_n)\right)\mu\left(B(T^{j'}y,r_n)\right)d\mu(y)\nonumber\\
&\leq&\left(\int\mu\left(B(y,r_n)\right)d\mu(y)\right)^2+\theta_g \cdot \|\mu\left(B(\cdot,r_n)\right)\|_\mathcal{C}\cdot \|\mu\left(B(\cdot,r_n)\right)\|_\mathcal{C}\nonumber\\
&\leq&\left(\int\mu\left(B(y,r_n)\right)d\mu(y)\right)^2+c^2 r_n^{-2\xi}\theta_g.\label{eqmix2}\end{aligned}$$ Otherwise we use Hölder’s inequality and the invariance of the measure to obtain $$\begin{aligned}
& &\int\mu\left(B(T^{j}y,r_n)\right)\mu\left(B(T^{j'}y,r_n)\right)d\mu(y)\nonumber\\
&\leq&\left(\int\mu\left(B(T^{j}y,r_n)\right)^2d\mu(y)\right)^{1/2}\left(\int\mu\left(B(T^{j'}y,r_n)\right)^2d\mu(y)\right)^{1/2}\nonumber\\
&=&\int\mu\left(B(y,r_n)\right)^2d\mu(y).\label{eqhold}\end{aligned}$$ So the first two terms can be estimated as below $$\begin{aligned}
\label{c1}
I+II\leq& 2n^4c^2r_n^{-2\xi}\theta_g+n^4\left(\int\mu\left(B(y,r_n)\right)d\mu(y)\right)^2\nonumber \\
&+2n^3{g}\int\mu\left(B(y,r_n)\right)^2d\mu(y).\end{aligned}$$
The third term can be treated exactly as the second one using the following symmetry on $x$ and $y$: $$\begin{aligned}
\iint \mathbbm{1}_{B(T^jy,r_n)}(T^ix)\mathbbm{1}_{B(T^{j'}y,r_n)}(T^{i'}x)d\mu(x)d\mu(y)\\=\iint \mathbbm{1}_{B(T^ix,r_n)}(T^jy)\mathbbm{1}_{B(T^{i'}x,r_n)}(T^{j'}y)d\mu(y)d\mu(x).\end{aligned}$$ Finally, for the last term we use the boundedness of the indicator function and the invariance of the measure to obtain $$\begin{aligned}
& &\iint \mathbbm{1}_{B(T^jy,r_n)}(T^ix)\mathbbm{1}_{B(T^{j'}y,r_n)}(T^{i'}x)d\mu(x)d\mu(y)\nonumber\\
&\leq&\iint \mathbbm{1}_{B(T^jy,r_n)}(T^ix)d\mu(x)d\mu(y)\nonumber\\
&\leq&\int\mu\left(B(y,r_n)\right)d\mu(y).\label{eqdiag}\end{aligned}$$
Therefore, $$\begin{aligned}
\label{d}
III+IV\leq& n^4c^2r_n^{-2\xi}\theta_g+2 g n^3\int\mu\left(B(y,r_n)\right)^2d\mu(y)\nonumber \\
&+4n^2 g^2\int\mu\left(B(y,r_n)\right)d\mu(y).\end{aligned}$$
Combining together the estimates , , and , we obtain $$\begin{aligned}
\frac{\operatorname{var}(S_n)}{(\E(S_n))^2}&\leq& \frac{3n^4c^2r_n^{-2\xi}\theta_g+4n^2 g^2\int\mu\left(B(y,r_n)\right)d\mu(y)}{\left(n^2\int\mu(B(y,r_n)d\mu(y)\right)^2}\\
& &+\frac{4n^3 g\int\mu\left(B(y,r_n)\right)^2d\mu(y)}{\left(n^2\int\mu(B(y,r_n)d\mu(y)\right)^2}.\end{aligned}$$
To estimate the first term, we use the information on the decay of correlations (H1) and the choice of $g$ to obtain $$\begin{aligned}
\frac{3n^4c^2r_n^{-2\xi}\theta_g}{\left(n^2\int\mu(B(y,r_n))d\mu(y)\right)^2}
&\leq&3c^2\theta_gr_n^{-2(\overline{C}_\mu+\eps)}r_n^{-2\xi}\nonumber\\
&\leq&3c^2\theta_gn^{4}(\log n)^{2b}n^{4\xi/(\overline{C}_\mu+\eps)}(\log n)^{{2\xi b \over \overline{C}_\mu+\eps}}\nonumber\\
&\leq&3c^2(\log n)^{2b\left(1+{\xi \over\overline{C}_\mu+\eps}\right)}. \label{est1}\end{aligned}$$
For the second term, we use again our choice of $g$ to get $$\begin{aligned}
\frac{4n^2 g^2\int\mu\left(B(y,r_n)\right)d\mu(y)}{\left(n^2\int\mu(B(y,r_n))d\mu(y)\right)^2}
&\leq &4 g^2(\log n)^b
=4\left(4+\frac{4\xi}{\overline{C}_\mu+\eps}\right)^2(\log n)^{2+b}.\label{est3}\end{aligned}$$
For the last estimate, we use Lemma \[lema1\] to obtain $$\begin{aligned}
\frac{4n^3 g\int\mu\left(B(y,r_n)\right)^2d\mu(y)}{\left(n^2\int\mu(B(y,r_n))d\mu(y)\right)^2}
&\leq&\frac{4K g }{n\left(\int\mu(B(y,r_n)d\mu(y)\right)^{1/2}}\nonumber\\
&\leq& 4K\frac{ g}{n}r_n^{-\frac{\overline{C}_\mu+\eps}{2}}\nonumber\\
&=&4K g\cdot(\log n)^{{b\over 2}}\nonumber \\
&=&4K\left(4+{4\xi \over \overline{C}_\mu+\eps}\right)(\log n)^{1+{b\over 2}}.\label{est2}\end{aligned}$$
Now, taking $b<-2$, and combining , , and , we have $$\mu\otimes\mu \left((x,y),m_n(x,y)\geq r_n\right)\leq\frac{\operatorname{var}(S_n)}{(\E(S_n))^2}\leq K_1(\log n)^{1+{b\over 2}},$$ for large enough $n$, and for some constant $K_1$.
Thus, choosing $b<-4$ and the subsequence $n_\ell=\lceil e^{\ell^2}\rceil$, we can apply the Borel-Cantelli Lemma as in the proof of Theorem \[thineq\]. Moreover, since by Theorem \[thineq\] $m_n(x,y)>0$ a.e., we can consider the quantity $\log m_n(x,y)$ and we obtain $$\underset{n\rightarrow+\infty}{\underline\lim}\frac{\log m_n(x,y)}{-\log n}=\underset{\ell \rightarrow+\infty}{\underline\lim}\frac{\log m_{n_\ell}(x,y)}{-\log n_\ell}\geq \frac{2}{\overline{C}_\mu+\eps}.$$ Then the theorem is proved since $\eps$ can be chosen arbitrarily small.
Now we explain how to modify the proof of Theorem \[dsup\] to prove Theorem \[thprinc2\].
When our Banach space $\mathcal{C}$ is the space of Hölder functions, (H3) cannot be satisfied since the characteristic functions are not continuous. Thus the only difference in the proof will be in and where we use the mixing hypothesis.
First, one can easily adapt in this setting approximating the characteristic functions by Lipschitz functions exactly as in the proof of Lemma 9 in [@rapid].
Then, to obtain we just need to prove that if (HA) is satisfied then the function $x\mapsto \mu(B(x,r))$ is Hölder, i.e. (H2) is satisfied.
In fact, let $x,y\in X$ and $0<r<r_0$. If $\|x-y\|<r $ then by (HA), $$\begin{aligned}
\| \mu(B(x,r))- \mu(B(y,r))\|&\leq& \mu\left(B(x,r+\|x-y\|)\backslash B(x,r-\|x-y\|)\right)\\
&\leq&r^{-\xi}\|x-y\|^{\beta}.\end{aligned}$$ If $\|x-y\|\geq r $ then $$\begin{aligned}
\| \mu(B(x,r))- \mu(B(y,r))\|\leq 2
\leq\frac{2}{r}\|x-y\|.\end{aligned}$$ Thus, the function $x\mapsto \mu(B(x,r))$ is Hölder, one can applied (H1) and (H2) to obtain and the theorem is proved.
Proof of the symbolic case {#sec-discrete}
==========================
The proof of the first part of Theorem \[seqmat\] is a simple adaptation of the proof of Theorem \[thineq\] and a simpler version was also proved in [@AW]. To do this adaptation, one must substitute $-\log m_n(x,y)$ by $M_n(x,y)$ and the balls $B(x, e^{-k})$ must be substituted by cylinders $C_k(x)$.
We will just focus on the second part of the theorem and explain the main differences with Theorem \[thprinc\]. We will assume that the system is $\alpha$-mixing with an exponential decay, the $\psi$-mixing case can be easily deduced using the same ideas.
For $\eps>0$, let us define $$k_n=\frac{1}{\overline{H}_2+\eps}(2\log n+b\log\log n).$$ We also define $$A_{ij}(y)=\sigma^{-i}C_{k_n}(\sigma^jy)$$ and $$S_n(x,y)=\sum_{i,j=1,\dots,n}\mathbbm{1}_{A_{ij}(y)}(x).$$ We observe that $$\label{expsh}\E(S_n)=n^2\sum_{C_{k_n}} \P(C_{k_n})^2,$$ where the sum is taken over all the cylinders of size $k_n$.
Following the lines of the proof of Theorem \[thprinc\], we have $$\label{eqvarsh}
\P\otimes\P \left((x,y): M_n(x,y)\leq k_n\right)\leq\frac{\operatorname{var}(S_n)}{\E(S_n)^2}.$$ Again, we will estimate the variance dividing the sum of $\operatorname{var}(S_n)$ into $4$ terms. Let $g=\log(n^4).$ For $i-i'>g+k_n$, we have the equivalent of equation : $$\begin{aligned}
& &\iint \mathbbm{1}_{C_{k_n}(\sigma^jy)}(\sigma^ix)\mathbbm{1}_{C_{k_n}(\sigma^{j'}y)}(\sigma^{i'}x)d\P(x)d\P(y)\nonumber\\
&\leq&\alpha(g)+\int\P\left(C_{k_n}(\sigma^{j}y)\right)\P\left(C_{k_n}(\sigma^{j'}y)\right)d\P(y).\nonumber$$ If, moreover, $j-j'>g+k_n$, we have the equivalent of , that is $$\begin{aligned}
& &\int\P\left(C_{k_n}(\sigma^{j}y)\right)\P\left(C_{k_n}(\sigma^{j'}y)\right)d\P(y)\nonumber\\
&=&\int\P\left(C_{k_n}(\sigma^{j-j'}y)\right)\P\left(C_{k_n}(y)\right)d\P(y)\nonumber\\
&=&\sum_{C_{k_n},C'_{k_n}}\P(C_{k_n})\P(C'_{k_n})\P\left(C_{k_n}\cap \sigma^{-(j-j')}C'_{k_n}\right)\nonumber\\
&\leq&\sum_{C_{k_n},C'_{k_n}}\P(C_{k_n})\P(C'_{k_n})\left(\P(C_{k_n})\P(C'_{k_n})+\alpha(g)\right)\nonumber\\
&\leq&\alpha(g)+\left(\sum_{C_{k_n}}\P(C_{k_n})^2\right)^2.\nonumber$$ However, if $j-j'\leq g+k_n$, we obtain the following inequality, equivalent of : $$\int\P\left(C_{k_n}(\sigma^{j}y)\right)\P\left(C_{k_n}(\sigma^{j'}y)\right)d\P(y)\leq
\sum\P(C_{k_n})^3.$$ As in the proof of Lemma \[lema1\], using the subbaditivity of $x\mapsto x^{2/3}$, we have $$\sum\P(C_{k_n})^3\leq\left(\sum\P(C_{k_n})^2\right)^{3/2}.$$ Finally, when $|i-i'|\leq g+k_n$ and $|j-j'|\leq g+k_n$, we have the equivalent of : $$\iint \mathbbm{1}_{C_{k_n}(\sigma^jy)}(\sigma^ix)\mathbbm{1}_{C_{k_n}(\sigma^{j'}y)}(\sigma^{i'}x)d\P(x)d\P(y)\leq\sum\P(C_{k_n})^2.$$
Then, one can gather these estimates to obtain $$\begin{aligned}
&\P\otimes\P \left((x,y):M_n(x,y)\leq k_n\right)\\
\leq &\frac{2n^4\alpha(g)+2n^3(g+k_n)\left(\sum\P(C_{k_n})^2\right)^{3/2}+n^2(g+k_n)^2\sum\P(C_{k_n})^2}{\left(n^2\sum \P(C_{k_n})^2\right)^2}.\end{aligned}$$ Thus, for $b<-2$, $$\P\otimes\P \left((x,y),M_n(x,y)\leq k_n\right)=\mathcal{O}((\log n)^{1+{b\over 2}}).$$ To conclude the proof, we use the Borel-Cantelli Lemma, exactly as in the proof of Theorem \[thprinc\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors would like to thank Rodrigo Lambert and Mike Todd for various comments on a first draft of this article.
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[^1]: This work was partially supported by CNPq and FAPESB
|
\#1[[$\backslash$\#1]{}]{}
Introduction
============
In the study of YBa$_2$Cu$_3$O$_7$ (YBCO) and its family cuprates, the physical properties of the CuO$_3$ chain itself have been a subject of interest as well as those of the CuO$_2$ plane. In particular, evidence for charge instability has been observed in the hole-doped CuO$_3$ chain of YBCO and its relation with superconductivity has been discussed. [@CDW] Among the YBCO family compounds, PrBa$_2$Cu$_3$O$_7$ (PBCO) is unique in that it does not show superconductivity while the other rare-earth substituted YBCO compounds are superconducting. [@Pr123] Optical studies [@Takenaka] have revealed that the CuO$_2$ plane is not doped with holes and consequently that the superconductivity is suppressed in PBCO. In order to explain the hole depletion in the CuO$_2$ plane, several models have been proposed. Among them, the model proposed by Fehrenbacher and Rice (FR), [@FR] in which the Pr 4$f$-O 2$p_{\pi}$ states trap holes from the Cu 3$d$-O 2$p_{\sigma}$ band, has been most successful. It was also argued that hole transfer between the CuO$_2$ plane and the CuO$_3$ chain may play an important role. [@Khomskii] Experimental studies of the band structure of the CuO$_2$ plane, the CuO$_3$ chain and the Pr 4$f$ state should further reveal the difference between the superconducting YBCO and non-superconducting PBCO. [@ETL] Another important and potentially even more interesting point is that, if the CuO$_2$ plane is undoped and the CuO$_3$ chain is doped as suggested by the various experimental and theoretical studies, PBCO may give us an opportunity to study a hole doped CuO$_3$ chain with good one-dimensionality compared to YBCO. Recently, an angle-resolved photoemission spectroscopy (ARPES) study of SrCuO$_2$ by Kim [*et al.*]{} [@Kim] has shown the spinon and holon bands of the undoped CuO$_3$ chain, which is a manifestation of spin-charge separation in the one-dimensional system, and has attracted much interest. However, no ARPES study of a hole-doped CuO$_3$ chain has been made so far. Although YBCO has been studied by ARPES, [@YBCO] the band dispersion from the CuO$_3$ chain is far from being one dimensional because of the strong interaction between the chain and the plane.
In this paper, we report an ARPES study of untwinned PBCO samples which have relatively low resistivity but is not superconducting. As we show below, we observe at least two kinds of dispersive features: one from the undoped antiferromagnetic CuO$_2$ plane and the other which is highly one-dimensional band from the doped CuO$_3$ chain.
Experimental
============
Single crystals of PBCO were grown in a MgO crucible by a pulling technique. The composition was determined to be Pr$_{1.02}$Ba$_{1.98}$Cu$_{2.92}$Mg$_{0.08}$O$_7$ by inductively coupled plasma analysis. [@ICP] Because a small amount of Mg-impurities originating from the crucible are substituted preferentially for the plane Cu sites, [@Mg] it is expected that the CuO$_3$ chain is not affected by these impurities. Rectangular shaped samples were cut out from the as-grown crystal and annealed at 500 $^\circ$C in oxygen atmosphere under uniaxial pressure. The resistivity parallel to the chain direction increases with cooling from 8 m$\Omega$cm at 300 K up to 10 $\Omega$cm at 10K and that perpendicular to the chain direction increases from 400 m$\Omega$cm at 300 K to 70 $\Omega$cm at 10K. The ARPES measurements were performed using the Vacuum Science Workshop chamber attached to the undulator beamline 5-3 of Stanford Synchrotron Radiation Laboratory (SSRL). Incident photons were linearly polarized and had energy of 29 eV. The total energy resolution including the monochromator and the analyzer was approximately 40 meV. The angular resolution was $\pm 1$ degree, which gives the momentum resolution of $\pm$0.05$\pi$ at $h\nu$ = 29 eV. \[In the text, momenta along the $a$- and $b$-axes are given in units of $1/a$ and $1/b$, respectively. Here, $a$ (= 3.87 $\stackrel{\circ}{\rm A}$) and $b$ (= 3.93 $\stackrel{\circ}{\rm A}$) are the lattice constants of PBCO perpendicular and parallel to the chain direction, respectively.\] The chamber pressure during the measurements was less than $5\times 10^{-11}$ Torr. The samples were cooled to 10 K and cleaved [*in situ*]{}. The cleaved surface was the $ab$ plane. Orientation of the $a$- and $b$-axes was done by Laue diffraction before and after the measurement. The cleanliness of the surfaces was checked by the absence of a hump at energy $\sim -9.5$ eV. [@YBCO] All the spectra presented here were taken within 30 hours of cleaving. We cleaved the samples three times and checked the reproducibility. As shown in Fig. \[arrange\], ARPES data were taken in two arrangements: an $E \parallel ac$ arrangement in which the photon polarization $E$ is in the $ac$ plane and is perpendicular to the chain and an $E \parallel bc$ arrangement in which $E$ is in the $bc$ plane and has a component parallel to the chain. The position of the Fermi level ($E_{\rm F}$) was calibrated with gold spectra for every measurement.
Results and discussion
======================
The entire valence band data measured at ($k_a$,$k_b$) = (0,0) are shown in Fig. \[val\]. Here, $k_a$ and $k_b$ are momenta along the $a$- and $b$-axes, respectively. The Ba 5$p$ core levels are split into surface and bulk components as observed in YBCO, [@YBCO] indicating that the cleaved surface is as good as that of YBCO. The peak labeled as A is intense for $E \parallel bc$, while it loses its weight for $E \parallel ac$. This strong polarization dependence indicates that peak A is derived from the CuO$_3$ chain and that the CuO$_3$ chain is well aligned at the surface. Peak A can be attributed to the non-bonding O 2$p$ states of the CuO$_3$ chain. [@Kim] On the other hand, the intensity of peak B is insensitive to the photon polarization, indicating that it is derived from the CuO$_2$ plane. Peak B has energy of $\sim$ -2.3 eV and is similar to that found in Sr$_2$CuO$_2$Cl$_2$. [@Sr2CuO2Cl2] As pointed out by Pothuizen [*et al.*]{}, [@Sr2CuO2Cl2] peak B corresponds to the non-bonding O 2$p$ states of the CuO$_2$ plane.
Figure \[pp1\] shows ARPES spectra along the (0,0)$\rightarrow$($\pi$,$\pi$) direction and those along the (0,$\pi$/2)$\rightarrow$($\pi$,$\pi$/2) direction taken with $E \parallel ac$. The uppermost and second uppermost spectra in the left panel were taken at (0,0) and ($\pi$/4,$\pi$/4), respectively. The intensity from -0.2 to -0.4 eV is enhanced at ($\pi$/4,$\pi$/4) compared to that at (0,0). Let us denote this feature as $\alpha$ and discuss it in following paragraphs. The spectra taken at ($\pi$/2,$\pi$/2) are shown by the thicker solid curves and marked by the closed circles in the two panels of Fig. \[pp1\]. Figure \[pp2\] shows ARPES spectra nearly along the (0,0)$\rightarrow$($\pi$,$\pi$) direction for $E \parallel bc$. In Fig. \[pp2\], the spectrum taken at ($\pi$/2,$\pi$/2) is shown by the thicker solid curve and marked by the closed circle. In these spectra taken around ($\pi$/2,$\pi$/2), a dispersive feature with a band maximum at $\sim$ -0.4 eV is clearly seen and is labeled as $\beta$. This structure $\beta$ is very similar to that found at ($\pi$/2,$\pi$/2) in Sr$_2$CuO$_2$Cl$_2$ and can be interpreted as the Zhang-Rice (ZR) singlet state of the undoped antiferromagnetic CuO$_2$ plane. [@Sr2CuO2Cl2] The energy difference between the ZR state at ($\pi$/2,$\pi$/2) (structure $\beta$) and the non-bonding O 2$p$ state at (0,0) (peak B) is $\sim$ 2 eV in PBCO, in agreement with the observation in Sr$_2$CuO$_2$Cl$_2$. [@Sr2CuO2Cl2] In addition to the ZR state, a weak structure $\gamma$ at -0.2 eV was observed around ($\pi$/2,$\pi$/2).
By comparing Figs. \[pp1\] and \[pp2\], one can notice that the intensity of structure $\alpha$ at ($\pi$/4,$\pi$/4) is weak for $E \parallel bc$ compared to that for $E \parallel ac$. In addition, while the relative intensity of structure $\alpha$ to structure $\beta$ has strong polarization dependence, that of structure $\gamma$ to structure $\beta$ has only small polarization dependence. This suggests that these two structures $\alpha$ and $\gamma$ have different origins. As discussed in the next paragraph, structure $\alpha$ at ($\pi$/4,$\pi$/4), which shows strong polarization dependence, is part of a one-dimensional band from the CuO$_3$ chain. On the other hand, since structure $\gamma$ shows weak polarization dependence and probably has two-dimensional character, it is tempting to interpret structure $\gamma$ as a Pr 4$f$-O 2$p_{\pi}$ hybridized state or a so-called FR state. If this is the FR state, the present spectra are consistent with the FR scenario because the FR state is closer to $E_{\rm F}$ than the ZR state of the CuO$_2$ plane. [@XAS] The situation that the FR state is partially occupied and is observable in photoemission spectroscopy is consistent with the optical result, which has shown that the formal valence of the Pr ion is +3.5 and the FR state is occupied by 0.5 electrons on the average if it exists. [@Takenaka] This argument on structure $\gamma$ should be confirmed by using Pr 4$d$-4$f$ resonant photoemission in the future.
In Figs. \[chain1\] and \[chain2\], we have plotted ARPES data along the chain direction taken with $E \parallel ac$. In the spectra taken at ($k_a$,0) ($k_a$ is 0, $\pi$/10, $\pi$/5, 3$\pi$/10, or $\pi$), which are shown at the uppermost position in each panel, the broad feature labeled as $\alpha'$ is located at -0.7 $\pm$ 0.2 eV. This feature moves toward $E_{\rm F}$ as $k_b$ increases and reaches a band maximum with energy of $\sim$ -0.2 eV around ($k_a$,$\pi$/4). The spectra taken at ($k_a$,$\pi$/4) are shown by the thicker solid curves and marked by the closed circles in each panel of Figs. \[chain1\] and \[chain2\]. This feature loses its weight for $k_b > \pi/4$ and the spectrum at ($k_a$,$\pi$/2) is almost featureless. This behavior does not depend on $k_a$, i.e., the momentum perpendicular to the chain, meaning that this dispersive feature is highly one dimensional. In order to demonstrate the dispersion more clearly, the contour plots of the difference spectra, which are obtained by subtracting the featureless spectrum at ($\pi$,$\pi$/2) from the spectra at each $k$-point, are shown for $k_a$ = 0, 3$\pi$/10, and $\pi$ in Fig. \[chain3\]. Indeed, the overall dispersion of structure $\alpha'$ does not depend on $k_a$ so much. Therefore, we can conclude that the band is derived from the CuO$_3$ chain. Structure $\alpha$ at ($\pi$/4,$\pi$/4) \[see Fig. \[pp1\]\] is also part of this one-dimensional band. The fact that the one-dimensional band reaches a band maximum at $k_b$ $\sim$ $\pi$/4 and disappears for $k_b > \pi/4$ indicates that the filling of the Cu 3$d_{x^{2}-y^{2}}$-O 2$p_\sigma$ band in the CuO$_3$ chain is close to 1/4, namely, the formal valence of Cu in the chain is $\sim$ +2.5. This is consistent with the optical study. [@Takenaka]
There are two interesting points in this one-dimensional band. The first point is that the other structure ($\alpha''$), which has higher energy than structure $\alpha'$, is observed around ($\pi$,0) \[Compare the uppermost spectra shown by the thicker solid curves in the two panels of Fig. \[chain1\].\]. In order to show the dispersion clearly, the density plots of the second derivatives of the ARPES spectra along the chain direction are diplayed for $k_a$ = 0, 3$\pi$/10, and $\pi$ in Fig. \[chain4\]. The dispersion of $\alpha''$ is visible in the spectra for $k_a$ = $\pi$ which is shown in the right panel of Fig. \[chain4\]. It is possible to attribute $\alpha'$ and $\alpha''$ to the holon and spinon bands of the Tomonaga-Luttinger (TL) liquid. [@TL] While the total width of the holon band is predicted to be $\sim$ 4$t$, that of the spinon band has an energy scale of $J$. [@Maekawa] In Figs. \[chain3\] and \[chain4\], model holon and spinon dispersions of -2$t$cos($k_b$+$\pi$/4) and -$\pi J/2$cos(2$k_b$) [@TL; @Maekawa] are shown by solid curves. The two curves with $t$ of 0.5 eV and $J$ of 0.16 eV, which are reasonable values for the cuprates, [@Maekawa] roughly follow the dispersions in Fig. \[chain4\]. The second point is that the spectral weight near $E_{\rm F}$ is considerably suppressed in the experimental data in disagreement with the theoretical prediction on the TL liquid. [@TL; @Maekawa] Although the lengths of the CuO$_3$ chain at the surface are finite because of the surface termination, [@YBCO] the observation of the nice dispersive behavior indicates that the lengths of the CuO$_3$ chains are long enough to allow us to compare the data with the theory for the doped CuO$_3$ chain. A possible origin of the intensity suppression near $E_{\rm F}$ is the instability of the nearly-1/4-filled CuO$_3$ chain leading to charge density waves (CDW). [@Lee] Actually, charge instability in the hole-doped CuO$_3$ chain of PBCO has been observed by NMR and NQR measurements. [@Grevin] In addition, it has recently been pointed out that the spectral function of one-dimensional CDW insulators can have the holon and spinon dispersions. [@Voit]
Here, it should be remarked how we would observe the feature with $E \parallel ac$. In PBCO, the CuO$_4$ square plane of the CuO$_3$ chain is perpendicular to the $ab$ plane, i.e., the sample surface. Therefore, the photon polarization has a component perpendicular to the sample surface, namely, parallel to the CuO$_4$ square plane of the CuO$_3$ chain as shown in Fig. \[arrange\]. It is this additional component of the polarization that gives a finite transition matrix element to the ZR state in the CuO$_3$ chain.
With $E \parallel bc$, the contribution from the chain is very weak when $k_a$ is small. As $k_a$ becomes larger, the intensity of the one-dimensional band for $E \parallel bc$ becomes larger and, at $k_a$ = $\pi$, is comparable to that for $E \parallel ac$. Figure \[chain5\] shows ARPES spectra along the ($\pi$,$\pi$/2)$\rightarrow$($\pi$,-$\pi$/2) direction, namely, along the chain direction taken with $E \parallel bc$. The two dispersive features, which can be interpreted as the holon and spinon dispersions, are also observed for $E \parallel bc$. In order to show the dispersion obtained for $E \parallel bc$ clearly, the density plot of the second derivatives of the ARPES spectra is displayed in Fig. \[chain6\]. The dispersions of the holon and spinon bands are almost symmetric with respect to $k_b$ = 0 and reach maxima around $k_b$ = $\pi$/4 and -$\pi$/4. These dispersions obtained for $E \parallel bc$ are consistent with those obtained for $E \parallel ac$.
In the present ARPES data taken with $E \parallel ac$ and $E \parallel bc$, the relative intensity of the spinon band to the holon band strongly depends on $k_a$. The spinon band becomes more intense as $k_a$ becomes larger. The same behavior was found in the undoped CuO$_3$ chain by Kim [*et al.*]{} [@Kim] Further experimental and theoretical investigation is required to reveal this peculiar $k_a$ dependence of the spectral function.
Conclusion
==========
In conclusion, we have observed the band dispersions from the CuO$_2$ plane and the hole-doped CuO$_3$ chain of non-superconducting PBCO. The band dispersion from the CuO$_2$ plane clearly shows that PBCO has an undoped insulating CuO$_2$ plane. On the other hand, the one-dimensional dispersive feature from the doped CuO$_3$ chain consists of two structures which can be interpreted as holon and spinon bands. These bands lose intensity beyond $k_b$ $\sim$ $\pi$/4, indicating that the CuO$_3$ chain is nearly 1/4-filled. Further experimental and theoretical studies are desirable to reveal the nature of the one-dimensional band including its momentum and polarization dependence.
Acknowledgment {#acknowledgment .unnumbered}
==============
We would like to thank M. Schabel, I. Terasaki, T. Thoyama, S. Maekawa, D. D. Sarma, and K. Penc for valuable comments. We are grateful to P. J. White, A. Y. Matsuura and the staff of SSRL for technical support. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan, Special Coordination Funds of the Science and Technology Agency of Japan, the New Energy and Industrial Technology Development Organization (NEDO), the U. S. DOE, Office of Basic Energy Science and Division of Material Science. SSRL is operated by the U. S. DOE, Office of Basic Energy Sciences, Division of Chemical Sciences.
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abstract: 'Large and well defined variations exist between the distribution of mass and the light of stars on extragalactic scales. Mass concentrations in the range $10^{12} - 10^{13} M_{\odot}$ manifest the most light per unit mass. Group halos in this range are typically the hosts of spiral and irregular galaxies with ongoing star formation. On average $M/L_B \sim 90 M_{\odot}/L_{\odot}$ in these groups . More massive halos have less light per unit mass. Within a given mass range, halos that are dynamically old as measured by crossing times and galaxy morphologies have distinctly less light per unit mass. At the other end of the mass spectrum, below $10^{12} M_{\odot}$, there is a cutoff in the manifestation of light. Group halos in the range $10^{11} - 10^{12} M_{\odot}$ can host dwarf galaxies but with such low luminosities that $M/L_B$ values can range from several hundred to several thousand. It is suspected that there must be completely dark halos at lower masses. Given the form of the halo mass function, it is the low relative luminosities of the high mass halos that has the greatest cosmological implications. Of order half the clustered mass may reside in halos with greater than $10^{14} M_{\odot}$. By contrast, only $5-10\%$ of clustered mass would lie in entities with less than $10^{12} M_{\odot}$.'
author:
- 'R. Brent Tully'
title: Light to Mass Variations with Environment
---
Introduction
============
It would be remarkable if the naive assumption that light linearly traces mass is correct. Indeed, on scales of tens of kiloparsecs around galaxies it is known from rotation curve information that dark matter is more dispersed than starlight. It has long been entertained that dark matter might be more dispersed than light on megaparsec scales [@kai84]. If so, we would say there is a ‘bias’ between what we see and the physically more fundamental parameter, the mass. Semi-analytic models [@bla99; @som01; @ost03] anticipate variations in the conversion of baryons into stars and the manifestation as starlight at both very low and very high densities.
Theoretical considerations prepare us for the possibility of a complex relationship between the distribution of mass and of the lighthouses sitting atop that mass. In this article we present observational evidence for such complexity. Three environmental regimes will be considered. At home in the Local Group we find ourselves in an intermediate regime, common to the majority of spiral galaxies. Most elliptical galaxies find themselves in a high density regime. Then there is a low density regime where galaxies are few or absent. What is the relationship between light and mass in each of these three kinds of environments?
Groups of Galaxies
==================
Visible galaxies are embedded in halos that extend beyond the radii of observable gas and stars. We need to go to the scale of groups and clusters to obtain a measure of the total mass of galaxies. The current investigation relies on the analysis of groups by @tul87; hereafter T87. The individual galaxies associated with the groups in T87 are identified in Table II of @tul88. This particular group compendium has the attractive features that (a) it was constructed with quantitative, well defined rules, (b) the groups have been demonstrated to contain few interlopers, and (c) a subset has a high level of volume completeness.
The group catalog was constructed through a tree or ‘dendogram’ procedure. The construction begins by looking for the linkage between galaxies that gives the largest value of the product $L_B/R_{ij}^2$ where two candidates $i$ and $j$ are considered from the ensemble of $N$ objects, $L_B$ is the blue luminosity of the brighter candidate, and $R_{ij}$ is the linear separation between the two. For details on the calculation of $R_{ij}$ from angular and radial velocity information see T87 but essentially, for linkages relevant to group scales, the $R_{ij}$ are based on angular separations only. The dominant pair that is selected is now inserted back into the catalog as a single unit with the sum of the component luminosities and with position and velocity constructed from the luminosity weighted contributions of the components. There are now $N-1$ objects to consider and the process is repeated, over and over, until $N=1$.
In a dendogram constructed in this manner, ultimately all galaxies under consideration are linked. Obviously there is a transition at some point from collections of galaxies that are bound to those that are unbound. The issue of how to distinguish the bound and unbound domains was the subject of extensive discussion in T87. A crossing time argument was used. At any point in the dendogram, the galaxies that are linked together can be characterized by a separation dimension, the inertial radius $R_I$, and a velocity dispersion, $V_p$. The crossing time $t_{\rm x} \sim R_I / V_p$ (see T87 for details including corrections for projection) can be compared with the free expansion timescale, the inverse of the Hubble Constant, H$_0$. Units with $t_{\rm x} {\rm H}_0 << 1$ have a high probability of being bound. Units with $t_{\rm x} {\rm H}_0 > 1$ can be inferred to be unbound. With a judicious choice of level, the dendogram could be split between mostly bound and mostly unbound. It would follow from this choice, allowing for measurement and projection errors, that essentially all the merged units that make the dendogram cut on the high density side would have $t_{\rm x} {\rm H}_0 < 1$.
An interesting aspect of this analysis is that it provides an inventory not only of all the galaxies in a volume that are in groups but also [*all that are not*]{} in groups. In the sample considered by T87 there was the usual incompletion with distance. However a high Galactic latitude volume with the distance limit of $25 h_{75}^{-1}$ Mpc ($h_{75} = {\rm H}_0/75$) has completion in the sense that all galaxies brighter than $M_B^{\star}$ are included. Within this volume, T87 identified 179 groups of 2 or more galaxies and 49 groups of 5 or more. For statistical robustness, we will restrict most of the following discussion to the groups of 5 or more galaxies (one of the small groups in T87 has been boosted to above 5 members with our current inventory, giving us 50 groups to consider).
We should note a subtle point that arises because the dendogram linkages involve luminosities. Linkages that involve bright galaxies are made more easily, as might well be reasonable if brighter galaxies are more massive. However could we miss linkages in cases where mass is underrepresented by light? This possibility would potentially undermine the basis of this investigation. It is to be noted that T87 provides a safety net to guard against this concern. In addition to the dendogram cut that resulted in the definition of ‘groups’, a second cut was made an order of magnitude lower in luminosity density that resulted in what were called ‘associations’. We will first continue with a discussion of the groups but later we will revisit the entities called associations.
Light to Mass Variations within Groups
======================================
The analysis in T87 missed an amazing correlation! It is seen in Figure 1, which draws on data extracted from T87 and presented here in Table 1. The data in the left panel represent all the 50 groups in the volume with completion above $M_B^{\star}$ . In the right panel, symbols distinguish between groups with majority early morphological types (E-S0-Sa: filled squares) and groups with majority late types (Sab-Irr: open circles). There are fewer points in this panel because, in order to make a statistically meaningful distinction in morphological class, we require there be at least 6 galaxies brighter than $M_B=-17$. These groups are distinguished by an entry in the third column of Table 1 where the percentage type Sa or earlier among galaxies brighter than $M_B=-17$ is given. The numbers of early type groups are small in this restricted local volume, so the slightly more distant early type groups Virgo W, Antlia, and NGC 5846 are also considered (last 3 entries in Table 1). Groups with crossing times much less than a Hubble time have relatively high mass to light ($M/L_B$) ratios, values of several hundred in solar units (masses from T87 are unweighted virial masses following the definition given in Section 6 of this paper). Groups with crossing times that approach a Hubble time have $M/L_B$ ratios that are much more modest, a few tens in solar units. It is also seen that there is a strong correlation between crossing time and the morphological classification. If the sample is split at a crossing time as a fraction of the Hubble timescale of $0.2 {\rm H}_0$ then almost all the groups with shorter crossing times are dominated by early types (8 of 9) and almost all the groups with longer crossing times are dominated by late types (11 of 14).
{width="8.7cm"}
Could the correlation seen in Fig. 1 be induced by parameter coupling? The vertical axis has dependencies on $V_p^2 R$ while the horizontal axis has dependencies $R/V_p$, where $R$ is a characteristic of the group dimension and $V_p$ is the velocity dispersion. Changes in $V_p$ cause shifts in the sense of the observed correlation. Changes in $R$ cause shifts orthogonal to the correlation. The concern would have to be with $V_p$ values but the typical measurement uncertainties in these values are 20% for the groups in question, much smaller than the factor ten range in crossing time values. Most clearly, though, the correlation cannot be an artifact of parameter errors and coupling between the axes because there is a strong correlation between the parameter of either axis in Fig. 1 and the morphology of the galaxies. The qualitative morphological information is decoupled from the measured parameters. In the right panel of Fig. 1, the early type groups have a median crossing time about a factor 3 less and a median $M/L_B$ about a factor of 3 higher than the late type groups.
Could the correlation be an artifact caused by systematic departures from virial equilibrium? Presumably the groups with the shortest crossing times have most closely approached the virial equipartition of energy. There could be a systematic mis-estimation of mass with the virial approximation that depends on the dynamical age of a system. Pacheco (private communication) has studied the collapse of a group with the expression: $${1 \over 2} {{\rm d}^2 I \over {\rm d} t^2} = {2 (T+U) + W }$$ where $I$ is the moment of inertia of the system, $T$ is the kinetic energy in the randomized motions of the constituents, $U$ is the kinetic energy in the ordered motions of infall, and $W$ is the potential energy. Pacheco made analytic assumptions about what happens when, during the collapse, violent relaxation [@lyn67; @fun92] causes the transform of energy from bulk motion to a state approximated by the virial condition. Dynamical friction induces the transfer of energy from ordered to random motions. The approach to equilibrium, with ${\rm d}^2 I /{\rm d} t^2 \sim 0$, occurs on the order of one dynamical timescale ($t_{dyn} = (G\rho)^{-1/2} \sim t_x$ where $\rho$ is the average density of the group within the virial radius). Measurements of groups caught before relaxation would tend to result in underestimates of mass by typically 20% and a factor 2 in the extreme (theoretically assuming full 3-D information is available).
The situation is complex. It was suggested by @bar85 that virial mass estimates might be biased low because of segregation as the visible components sink toward the center relative to dark matter. This possible systematic would increase as the group ages, consequently it would run counter to the correlation seen in Fig. 1. There is room for further analysis of the properties of groups extracted from cosmological simulations. High resolution is required to avoid the concern of artificial 2-body relaxation from massive numerical particles.
There is firmer ground on the observational side. There is general agreement that groups and clusters with high densities and short dynamical times have $M/L_B$ values of several hundred. These circumstances are confirmed by X-ray and gravitational lensing studies. We suggest that the trend to substantially lower $M/L_B$ values for systems with longer dynamical times seen in Fig. 1 is consistent with what is now known from the few detailed studies of low density groups.
The situation is clearest in the Local Group and has received confirmation from recent studies of the nearest neighboring groups. These groups are represented in the left panel of Fig. 1 but are not sufficiently populated to enter the right panel. They lie in the lower right corner, with large crossing times and low $M/L_B$.
The Local Group encompasses two dynamical regimes. On a 100 kpc scale, each of the two major galaxies host a school of minor systems that approximate virial conditions. On a 1 Mpc scale, the two major galaxies and ten or so other small galaxies are falling together, most on first approach. This latter assertion is based on the observation that almost all the Local Group systems that are not tightly associated with either of the two major galaxies have negative velocities in the Local Group standard of rest (NGC 6822 is an exception).
The masses of the Milky Way and Andromeda sub-groups can be estimated through the virial theorem since these sub-systems have densities and crossing times that suggest they are dynamically evolved [@eva00]. The stellar stream on a scale of 125 kpc around M31 provides a compatible mass estimate [@iba04]. Each sub-group is determined to have $0.7-1 \times 10^{12} M_{\odot}$. The mass of the larger infall region can be estimated separately. One approach follows the @kah59 timing argument. A variation on this approach is provided by the identification of the current turnaround radius or zero-velocity surface for the Local Group [@san86; @kar02c]. There is the important result that the sum of the masses of the Andromeda and Milky Way sub-systems add to approximately the mass determined to lie within the zero-velocity surface of $1.2 \times 10^{12} M_{\odot}$. A similar mass is found from orbit reconstructions [@pee95]. Our naive virial estimation (Table 1: group 14-12) gives a Local Group mass of $1 \times 10^{12} M_{\odot}$ and $M/L_B=17 M_{\odot}/L_{\odot}$, in factor 2 agreement with other results.
Recent observations of the resolved stars in nearby galaxies with Hubble Space Telescope have dramatically improved our knowledge of distances to many nearby galaxies and hence of the structure of the nearest groups. We now know that the best studied neighboring groups, those about M81 (Table 1: group 14-10) and Centaurus A (Table 1: group 14-15), have dumbbell structures like the Local Group. In both cases the two largest galaxies are surrounded by swarms of small galaxies which transit their hosts in times much less than the age of the universe. Almost certainly in the case of the M81 Group and plausibly in the case of the Centaurus Group, the substructures are falling together. As with the Local Group, the masses inferred for the entire bound entities (2 and 3 times $10^{12} M_{\odot}$ respectively) are close to the sum of the masses of the dynamically evolved subcomponents [@kar02a; @kar02b]. The halos of the giant galaxies must not extend much beyond the domain of the immediate companions and the group $M/L_B$ values are a few tens. These groups that are overall spiral rich and low density evidently have $M/L_B$ values well below 100. The virial approximation for these groups, while crude, gives mass estimates that are not strongly deviant.
While theoretical expectations might cause one to anticipate systematic underestimations of mass assuming the virial approximation, studies of the nearest structures reveal that a crude application of the virial theorem may frequently result in [*overestimations*]{} of mass. Frequently, unbound objects will be erroneously included as group members. The availability of good distances to individual galaxies shows that the very nearby Sculptor Group (Table 1: 14-13) and CVn I Group (Table 1: 14-07) are composites of regions that are unlikely to be bound in the ensemble. The inclusion of expansion velocities and the inflation of group scale cause overestimates of mass. These effects are pernicious for the extended, low dispersion systems that inhabit the lower right corner of Fig. 1.
The limited information available on morphological variations from studies of nearby groups is giving a hint of higher mass in early type systems. The kinematics of the sub-group around Centaurus A suggest a mass of $4 \times 10^{12} M_{\odot}$ and $M/L_B \sim 100 M_{\odot}/L_{\odot}$ (albeit including the large spiral NGC 4945) while the M83 sub-group is found to have a mass of $1 \times 10^{12} M_{\odot}$ and $M/L_B \sim 50 M_{\odot}/L_{\odot}$. The elliptical Cen A may have a higher $M/L_B$ by a factor 2–3 relative to the large spirals in our vicinity. The situation in the nearest groups is reviewed by @kar04.
Figure 2 shows the correlation between mass and light for the TF87 sample of nearby groups. In the left panel, all the 50 groups in the volume-limited sample with at least 5 members are represented. The groups with $t_{\rm x} {\rm H}_0 < 0.2$ are plotted with filled squares and the groups with larger crossing times are plotted with open circles. The 3 more distant early type groups, all with $t_{\rm x} {\rm H}_0 < 0.2$, are plotted as inverted triangles. The solid line at $45^{\circ}$ indicates the mean $M/L_B$ value determined by T87 for this sample. Again we see that the groups with short crossing times have less blue luminosity per unit of mass. In addition, it can be said that groups with more mass manifest less blue light per unit mass.
We consider the same data in the right panel but now the differentiation is made on the basis of the morphology of the groups. It is seen again that the qualitative morphological description and the quantitative crossing time measure provide an equivalent basis for distinguishing between high and low $M/L_B$ systems.
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The Virgo Cluster (group 11-1 in TF87) and three other specific groups are identified in the left panel of Fig. 2. It is instructive to consider the pairwise properties of these groups. It can be seen that the NGC 1407 (51-8) Group has roughly the same mass as the Ursa Major Cluster (group 12-1) but much lower luminosity. The NGC 1407 Group has the same luminosity as the NGC 1023 (17-1) Group but much more mass. These similarities and differences are not in observational doubt. There is a summary of the properties of these groups in Table 1 of @tre02; see also Fig. 1 in that reference for histograms of the group velocities. The unusual nature of the NGC 1407 Group has been noted by @gou93 and @qui94. The group contains only two $L^{\star}$ galaxies but has a velocity dispersion of 385 km s$^{-1}$. The Ursa Major Cluster, by contrast, has almost enough luminous galaxies to qualify as an Abell richness class 0 cluster! The velocity dispersion, though, is only 148 km s$^{-1}$. The virial masses of the two groups are almost the same. In the NGC 1407 Group motions are large in a small volume while in the Ursa Major Cluster motions are low in a large volume. Yet the luminosities differ by a factor 5 and $M/L_B$ values differ accordingly. The NGC 1023 Group is a scaled down version of the Ursa Major Cluster. There are three $L^{\star}$ galaxies and the group velocity dispersion is only 57 km s$^{-1}$. The NGC 1023 and NGC 1407 groups have similar galaxy content in terms of numbers and luminosity (the type content is late and early, respectively). However the motions in the NGC 1407 Group are a factor of 7 higher. This difference can not be questioned observationally, and the implied mass of the NGC 1407 Group is higher by a factor 30.
In summary of this section, evidence is provided of an order of magnitude variation of $M/L_B$ that depends on mass and/or stage of dynamical evolution. The derivation of mass from the virial theorem might have biases but there are effects that both raise and lower estimates. The cases that are particularly well studied provide confirmation of the trends shown in Figs. 1 and 2.
Numerical Action Models of Large Scale Flows
============================================
There is a strong inference from Numerical Action orbit reconstructions that the Virgo Cluster is very underluminous for its mass compared with the general field. From a first analysis of the Local Supercluster [@sha95] with 300 distance estimates as constraints and a later analysis [@tul98] with 900 distance estimates, it was determined that the mean density of the Universe is $\Omega_m \sim 0.2$, consistent with an overall mean $M/L_B \sim 200 M_{\odot}/L{_\odot}$ assignment. However, as emphasized in the latter paper, it is not possible with a [*single*]{} assignment of $M/L_B$ for all galaxies to obtain a satisfactory model that simultaneously gives a good description of flows in the general vicinity of the Local Supercluster and a good description of the infall region around the Virgo Cluster.
Evidence for this claim is provided in Figures 3 and 4. The first of these figures show velocity-distance data for galaxies in a group along a specific line-of-sight with respect to the Virgo Cluster. The two panels illustrate attempts to fit this data with two distinct Numerical Action models. The solid curves represent the run of velocity with distance expected by the separate models. In the left panel, a value $M/L_B=200 M_{\odot}/L_{\odot}$ is assigned to all groups and galaxies. The location of the galaxy data points in velocity and distance cannot be understood in the context of the model in this panel. In the right panel, $M/L_B=1000 M_{\odot}/L_{\odot}$ is assigned to the Virgo Cluster, and incidentally to all E/S0 knots. In compensation for this increase in the mass of some of the objects, $M/L_B=150 M_{\odot}/L_{\odot}$ is given all the rest. Now the infall motions toward Virgo are greatly enhanced and the model gives a physical basis for the location of this particular group in velocity-distance space. We see that the group must be slightly nearer than the cluster and falling away from us into the cluster.
The swing in amplitude of the wave in velocity-distance seen in Fig. 3 (the ‘triple-value’ characteristic [@ton81]) depends on the mass assigned to the cluster. It can be seen that the curve in the right panel of Fig. 3 only minimally reaches the location of the data points, hence represents a [*minimum*]{} required mass. A larger mass would cause a larger swing and would not be in conflict with the data but is not required. Fig. 4 shows the constraints provided by many lines-of-sight through the Virgo infall region. The individual points record velocity and angular distance from Virgo, as open circles for galaxies in the cluster proper (defined by the ‘caustic’ radius established by galaxies that have fallen into the cluster and back out to a second turnaround), and as filled circles for galaxies identified to be within the first infall region. To keep the plot clean, galaxies at foreground and background ‘triple-value’ locations have been rejected based on distance information. The swings in amplitude of the triple-value waves along various lines-of-sight are indicated by the brackets in the two panels. It is clear that the swing generated by the Numerical Action model of the left panel, with $M/L_B=200 M_{\odot}/L_{\odot}$ for all points, fails by a wide margin to explain the infall motions into the Virgo Cluster. The model used in the right panel, with $M/L_B=1000 M_{\odot}/L_{\odot}$ given to the Virgo Cluster and $M/L_B=150 M_{\odot}/L_{\odot}$ assigned to the field, provides a satisfactory description to the observed data.
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[*Almost a factor 7 higher $M/L_B$ value is required for the Virgo Cluster than for the general field.*]{} The evidence for a lot of mass in the cluster which comes from the extended nature of the infall region (extending to $\sim25^{\circ}$ from the center of the cluster or 8 Mpc) and high infall velocities (galaxies hit the caustic surface of the cluster at $6^{\circ}$ with a velocity of 1500 km s$^{-1}$) requires a remarkable $1.3 \times 10^{15} M_{\odot}$ within the 2 Mpc caustic surface of second turnaround. This mass is almost twice the virial mass estimate but it pertains to a radius three times larger than the virial radius. The infall mass estimate pertains to the cluster on it’s largest scale. It can be noted that a similar conclusion was reached much earlier [@tul84] with a non-linear but spherically symmetric analytic model of Virgo Cluster infall. The very same conclusions were reached that there had to be a lot of mass for the amount of light in the cluster but much less mass with respect to light in the supercluster environs surrounding the cluster.
The preferred model in the @tul98 analysis used two discrete $M/L_B$ assignments: $M/L_B=1000 M_{\odot}/L_{\odot}$ was assigned to 12 E-S0 dominated groups including the Virgo Cluster and $M/L_B=150 M_{\odot}/L_{\odot}$ was assigned otherwise. (The action models do not provide constraints on the masses of the E-S0 groups smaller than Virgo because the triple value regions are too restricted to map infall. The large $M/L_B$ assignments for these groups were based on the virial information discussed earlier.) It was concluded that $\Omega_m \sim 0.2$ associated with galaxies in the mean but that the Virgo Cluster in particular must have a much higher $M/L_B$ value than the mean. The Numerical Action mean result for the field ($M/L_B=150 M_{\odot}/L_{\odot}$) and the large mass for Virgo ($M \sim 1.3 \times 10^{13} M_{\odot}$) are recorded in the right panel of Fig. 2.
As an aside, it should be noted that what is meant by the terms ‘group’ and ‘cluster’ are not well defined, other than that there is the inference that the structures are bound. In the case of the Local Group, it was discussed that there is a zero-velocity surface at $\sim 1$ Mpc radius [@san86]. This surface defines a bound domain and provides a logical limit to the group. However, the Local Group is certainly not virialized over this domain. At most, there may be limited quasi-virialized regimes within $\sim 150$ kpc about each of the two dominant galaxies. By contrast, when we talk about the Virgo Cluster we certainly do not consider it to extend to the current zero-velocity surface which lies on the near side at about half the distance from us to the cluster core. Rather, a good working definition of the volume of the cluster is the domain within the first caustic, the outer limit of objects that have completed a single passage. The cluster is not virialized to this radius but the approach toward virialization will be greater within a cluster defined this way than in the Local Group analog.
The discussion to this point, involving the properties of groups and the inferences drawn from modeling of flow patterns in the Local Supercluster, has dealt with comparisons between intermediate and high density regimes. Attention will now turn to lower density environments.
The Low Density Regime
======================
The algorithm for the selection of groups used by T87 contained luminosity. Could we loose real groups because their constituents are underluminous? We have just been arguing that groups with short crossing times and dominated by early type galaxies are environments that are underluminous compared with the environments of groups with crossing times approaching a Hubble time and dominated by spirals and irregulars. Fortunately, no groups of the former type would have been missed in the construction of the group catalog because the E/S0 groups with short crossing times have considerably higher spatial densities than the spiral groups. Consequently they comfortably cross the luminosity density threshold set for the group catalog even though they turn out to be underluminous. In the years that have passed since the catalog was developed we have never become aware of a high spatial density but low luminosity group in the volume of our sample that escaped identification.
However we cannot be so sanguine in the limit of low spatial densities. Indeed, among the entities called ‘associations’ that came out of the T87 study there were some that look suspiciously like the entities identified as groups. The dimensions of these units are only several hundred kiloparsecs, like the groups. They can have [*very low*]{} velocity dispersions, comparable with the lowest values in the established groups. [*Their members are all dwarfs.*]{} They were not picked up in the groups analysis because they have very low luminosities. If they are bound groups then the associated $M/L_B$ is much larger than is familiar to us from our experience with the intermediate regime spiral groups.
Our interest in these entities was rekindled by work from a very different direction. We were interested in possible environmental dependencies at the faint end of the luminosity function of galaxies. It seems that there are unexpectedly few dwarf galaxies compared with the expectations of the standard hierarchical model for galaxy formation [@kly99][@moo99]. We suggested that the deficiency with respect to the standard model might be greater at lower densities [@tul02], hereafter TSTV. There might be an expectation that there are many low mass dark matter halos floating about with little or no gas and stars. We speculated that there might be a possibility to identify the dynamical manifestations of such halos in a certain special environment. These low mass dark halos would not be of sufficient consequence to be noticed in our standard groups, even those as modest as the Local Group. At a mass and density regime well below that of the Local Group, according to the hypothesis that we are pursuing that low mass halos are voided of gas and stars, then there would be nothing to be seen. However, in the interval [*between*]{} the normal groups and putative much smaller structures it might be expected that [*there are groups that are mostly, but not entirely dark.*]{}
As discussed in TSTV, it occurred to us that the ‘mostly dark’ groups that could be anticipated from theoretical grounds might be found among the subset of associations in the T87 study with group-like dimensions and low velocity dispersions but high implied $M/L_B$ values. We drew attention to four such groups thought to lie within 5 Mpc.
The key issue regarding these ‘groups of dwarfs’ that we identify is whether they are bound. [*If they are bound then they must have high $M/L_B$ values.*]{} In order to illuminate this issue, we have been involved in programs to identify other potential group members and to get good distances to individual objects. These galaxies are all near enough that they are easily resolved with Hubble Space Telescope (HST) and distances can be established from the luminosities of stars at the tip of the Red Giant Branch [@lee93], the ‘TRGB’ distance method. We first review the present status of these programs.
New Data on Candidate Groups of Dwarfs
======================================
At the time of the paper by TSTV, good distances were only known for members of one of the four putative groups of dwarfs. Now there are reasonable distances from the TRGB method based on HST images for [*most of the galaxies in all of the groups*]{}. The new distances are reported in the following publications: @doh98, @free01, @kar02c, @kar03a, @kar03b, [@mai02], and @men02.
The updated results are summarized in Table 2 for the four candidate groups of dwarfs and one additional comparison group. No weighting is applied in the calculation of the dimension, velocity, and mass parameters since it is assumed that the individual galaxies are inconsequential test particles in the potential of the group. The following information is provided in the table. (1) Group name from @tul88. (2) Prominent galaxy in group. (3) Number of candidate group members. (4) Mean group distance. (5) Inertial radius $R_I^{3D}=(\sum_i^N r_i^2/N)^{1/2}$ where $r_i$ is the 3-dimensional distance of a galaxy from the group centroid. (6) Radial velocity dispersion $V_r=(\sum_i^N v_i^2/(N-1))^{1/2}$ where $v_i$ is the radial velocity difference between a galaxy and the group mean (7) Blue Luminosity. (8) “Projected mass estimate” $M_{pm} = {f_{pm}^{3D} \over G (N-\alpha)} \sum_i^N r_i v_i^2$ [@hei85] where $f_{pm}^{3D}=5$ (TSTV) and $\alpha=1.5$. (9) Virial mass estimate $M_v = 3 {(N-1)\over N} V_r^2 R_G / G$ where $R_G = N / \sum_{pairs} (1/r_{ij})$ and $r_{ij}$ is the separation between pairs in the group counted only once. The factor 3 gives the statistical conversion from the observed radial velocity to 3-dimensions. Radii are already provided in 3-dimensions because of the availability of distances. (10) Mass to light ratio based on the projected mass estimator. (11) Mass to light ratio based on the virial analysis. (12) Mass to light ratio given by TSTV. (13) Crossing time as a fraction of the Hubble time $t_{\rm x} H_0 = 38 R_I^{3D}/V_r$ where $R_I^{3D}$ is in Mpc and $V_r$ is in km s$^{-1}$ and $H_0 = 75$ km s$^{-1}$ Mpc$^{-1}$.
In TSTV, only the projected mass estimator of @hei85 was considered. Here we also provide the unweighted virial mass estimator. There can be circumstances with small numbers where the virial mass estimator gives ambiguous results; see the case of the NGC 3109 Group discussed below. However the use of the two mass estimators provides a better feel for the considerable uncertainties.
The $M/L_B$ values determined by TSTV are carried for convenience into Table 2. It can be seen that $M/L_B$ values have increased slightly overall with the current analysis. A principal cause of this small increase was a mistake in corrections to luminosities of galaxies in TSTV. In that paper the luminosities had corrections for obscuration based on the prescription by @dev91. However, the obscuration in dwarf galaxies is sufficiently low as to be unmeasurable [@tul98b]. In this paper, no correction is made for obscuration within galaxies with $M_B>-16$. The result is that luminosities are systematically lowered and $M/L_B$ values are increased. Corrections are made for obscuration due to lines of sight through our Galaxy [@sch98]. In any event, the luminosities of dwarf galaxies are often poorly known. Errors in the group luminosities of up to 30% are possible, but errors at this level do not bring the conclusions of this study into question. Here are comments on the situation in individual groups.
[*14+12 = NGC 3109 Group:*]{} The existence of this nearby entity as a distinct group was suggested by @vdb99. In our earlier analysis we entertained that GR8 = DDO 155 might be a member but this galaxy is significantly farther away than was assumed by TSTV, at 2.24 Mpc rather than 1.51 Mpc [@doh98]. This galaxy is also, by a substantial amount, the farthest removed from the other group candidates on the plane of the sky. Consequently, we eliminate this object from consideration as a group member. Moreover the object LSBC D634-03 is now revealed to be a background galaxy coincident with an HI high velocity cloud. The other 4 candidates are still considered at the distances given in TSTV. In the case of this group, the virial mass estimator is particularly untrustworthy because two of the candidates are very close neighbors; NGC 3109 and Antlia are separated by 29 kpc. Since $R_G \sim 1/\sum (1/r_{ij})$, the single pair dominates by an order of magnitude over the next pair in the calculation of the scale of the potential well. If the separation of this pair is arbitrarily set to 100 kpc, still smaller than any other separation in the group, then the virial mass estimate doubles. There is a chronic ambiguity in the virial mass estimator with small numbers and instances of close pairs. However, only this one group suffers this problem among the cases we are considering. Overall, the viability of the 14+12 candidate group remains as it was with the discussion by TSTV, with an implied $M/L_B \sim 400$ if the entity is bound.
[*14+8 = UGC 8760 Group:*]{} Distance measurements have revealed that this group is much closer than the $\sim 5$ Mpc that we expected. There are now TRGB distances for all the candidates, including a new and provisionary one for UGC 8760 itself. Being closer, the linear separations are smaller than we appreciated. Also we have become aware of a fourth system, UGC 9240, which also is probably associated. The three well measured distances are very similar (UGC 8651 at 3.01 Mpc; UGC 8833 at 3.19 Mpc; UGC 9240 at 2.79 Mpc). The first two were associated with the group by TSTV and it is a powerful claim for the reality of the group that they both have unexpectedly low and similar distances. A nice test for the hypothesis being pursued is that the distance of UGC 8760 be also low and an HST observation returned as this paper goes to press reveals UGC 8760 to be at $\sim 3.2$ Mpc. It is seen in Table 2 that the new $M/L_B$ values are consistent between methods and considerably larger than found previously. The difference from the TSTV results is most marked in this case. Part of this increase is a direct consequence of the downward revision in distance; $M/L_B$ is inversely dependent on distance. The other causes are the addition of the fourth candidate member and the changes to luminosities discussed above. A value $M/L_B \sim 1000$ is required if the entity is bound.
[*14+19 = UGC 3974 Group:*]{} All four candidate members now have TRGB distance measures and they all turn out to have distances similar to our expectation of $\sim 5$ Mpc (UGC 3755 at 5.0 Mpc; UGC 3974 at 5.2 Mpc; UGC 4115 at 5.5 Mpc; KK98 65 at 4.5 Mpc). These galaxies are rather at the limit of the TRGB method with the HST WFPC detector in one orbit so the uncertainties are larger in these distances. Even so, contributions to the scale of the group from distances and projected separations are comparable. Again $M/L_B$ values are larger than found before. The earlier result was based on the projected mass estimator with 2-D positions and particularly uncertain luminosities. The current analysis has revised luminosities and 3-D positions. Giving consideration to both mass estimators, the current information suggests $M/L_B \sim 2000$ if the group is bound.
[*17+6 = NGC 784 Group:*]{} All 4 candidates have TRGB distances now although those for NGC 784 and UGC 1281 are unpublished and poor quality. The agreement is satisfactory and give a mean distance of 5.0 Mpc, as was initially anticipated. (UGC 1281 at 4.8 Mpc, NGC 784 at 4.6 Mpc, KK98 16 at 5.7 Mpc; KK98 17 at 5.0 Mpc). The factor 3 larger $M/L_B$ than previously is attributable to the correction to luminosities and the distance differentials that reveal this group is extended more in the line-of-sight than in projection. If the group is bound then $M/L_B \sim 1000$.
[*14+13 = Foreground Sculptor Group, a comparison case:*]{} This structure was included in the discussion by TSTV. In the past, several of the galaxies have been considered as part of the Sculptor Group but we consider this historical group to consist of two distinct entities. We give attention to the nearer part. One good new candidate is added to the group specified by TSTV: ESO 294-010. All but the group’s most luminous galaxy, NGC 55, have measured distances now (ESO 294-010 at 1.92 Mpc; NGC 300 at 2.00 Mpc; UGCA 438 at 2.23 Mpc; IC 5152 at 2.07 Mpc). Results are little different than previously. The mass estimates are as low as for any of the groups discussed above because of the very low velocity dispersion. However there are two moderate galaxies in the group so there is substantial luminosity. IC 5152 is removed by 770 kpc from the group centroid, mostly in projection, so the status of this object is in doubt. Whether or not it is included, if the entity is assumed to be bound then $M/L_B \sim 17$.
Comparison of Dwarf Group Properties
====================================
The candidate groups have mass estimates in the modest range $1-9 \times 10^{11} M_{\odot}$. Characteristic dimensions range 290 kpc $ < R_I <~$ 540 kpc with a mean of 420 kpc, typical of more familiar groups (T87). Velocity dispersions range from 36 km s$^{-1}$ down to an incredibly low 13 km s$^{-1}$ with a mean of 22 km s$^{-1}$. These values are all very low compared to familiar luminous groups and account for the low mass estimates. Luminosities are all in the decade $1-10 \times 10^8 L_{\odot}$ except for the comparison case of the 14+13 or Foreground Sculptor Group which has almost an order of magnitude more luminosity than the most luminous of the others.
Crossing times as a fraction of the Hubble time are given in the last column of Table 2. Typically values are $\sim 0.8 {\rm H}_0$. Of course, the crossing times are long because the velocity dispersions are so low. They are a sufficiently large fraction of the age of the Universe that one can worry that velocities are simply attributable to the expansion of the Universe; ie, that these are not bound groups. Only in the case of the tight 17+6 (NGC 784) Group is the crossing time substantially less than a Hubble time, H$_0^{-1}$. If IC 5152 is accepted as part of the 14+13 (Foreground Sculptor) Group then the crossing time for this group is 50% longer than H$_0^{-1}$, but this is the comparison case: a group with low $M/L_B$.
To this point there has been almost no discussion of uncertainties. First, it is to be emphasized that the measurement errors in the observed parameters are of almost no consequence. The radial velocities are obtained by observations of the 21cm line of HI and for these dwarfs are determined with an accuracy of $\pm 5$ km s$^{-1}$. The distances are determined with the TRGB method. There might be a distance scale zero point issue at the level of 10% that would shift the entire sample in a similar fashion and not affect this discussion. Relatively, the distances should be accurate to 5% for galaxies within 3 Mpc. There is a degradation to $10-15\%$ at 5 Mpc. However in none of the groups discussed here is the depth of the group dominant over the projected dimensions. Hence errors in measured dimensions do not make a significant contribution to uncertainties. The most poorly known direct observables are the luminosities. The global group luminosities attributable to the identified candidates can have errors as large as 30%. The luminosities come from heterogeneous sources, some with large errors. The situation could be improved with dedicated observations. Still, the real uncertainties lie elsewhere.
The greatest uncertainty of all is whether or not the groups are bound. Assuming for the moment that they are, then there are four dominant uncertainties in the calculation of a group mass: (a) small number statistics, (b) the statistics of velocity deprojection from one to three dimensions, (c) potentially poor coverage of the group gravitational well, and (d) the nature of the galaxy orbits. The first two of these problems lead to uncertainties that are statistical in nature. The groups under consideration have 4 or 5 candidate members each. The number of pairs grows as ${N(N-1)/2}$ so each additional candidate gives a big improvement. As for the velocity deprojection problem, recall that mass calculation involves $V^2$ and the deprojection correction is a factor 3. The latter two problems raise possibilities of systematic errors in the mass calculations. With regard to the question of coverage of the group potential well, one aspect is simply the limited coverage with only 4 or 5 test particles and another aspect is the uncertainty in the limits of the putative bounded regions. Do the candidates at larger radii represent those limits, or are they frequently beyond those limits and escaping? Finally, there is the nature of the galaxy orbits. Are they isotropized to a degree as would occur as a system approaches virialization or are they largely radial as would be expected of a group still in formation, with orbits dominated by infall? The alternatives have different implications for the projected velocities. Following the assumption of the virial theorem: $2T/|W| = 1$, where the kinetic energy per unit mass is $T = {3(N-1) \over 2N} V_r^2$ and the potential energy per unit mass is $W = -G M_v / R_G$ (see section 6). The possibility of systematic departures from the virial condition was discussed in section 3. Both from analytic and N-body simulations [@mer03] it is found that mean values of $2T/|W|$ evolve from small values at turnaround to $\sim 1$ at first collapse, to ratios up to 20% above unity, whence there is a slow approach to virialization from values above 1. There can be large excursions in individual cases. The results of this paper could require modification due to this probable systematic effect. Dense groups that are well advanced toward virialization will tend to have $2T/|W|$ greater than unity by $0-20\%$ so mass estimates will be above true values while groups that are still collapsing will tend to have $2T/|W|$ less than unity by up to a factor 2 so mass estimates will be below true values.
Given the other large sources of errors, the uncertainty regarding the dynamical state is not dominant so the virial condition will be assumed. Mass estimates have been calculated two ways, following the discussion by @hei85. Except in the special case of the 14+12 (NGC 3109) Group that received discussion, the two mass estimates for a group differ from the mean by no more than 35%. If it is simply assumed that the group is bound then one gets a [*lower mass limit*]{} which is one-half the virial estimate. In the plots that will be shown further along, factor 3 uncertainties in mass have been assumed for groups with 4 candidates.
Even the very large uncertainties that have been discussed do not bring into question the fundamental claim being made pertaining to the low density regime. [*If the candidate dwarf groups are bound then high $M/L_B$ values are required for the groups.*]{} We can think of two arguments that favor the point of view that the groups are bound. The first notes the continuity in properties with well established groups in terms of dimensions and velocity dispersions. These properties are illustrated in Figure 5. To a reasonable degree, the E/S0 dominated groups and spiral dominated groups separate on this plot. The early-type groups tend to be restricted spatially but can manifest large velocity dispersions while the late-type groups inevitably have modest velocity dispersions but can be dispersed spatially (there are a few groups with majority early types with the properties of the spiral groups). The candidate dwarf groups can be viewed as an extension of the distribution of the groups dominated by spirals. There is nothing to suggest from these parameters that characterize the dynamical state that the dwarf groups are other than a continuation of the family of bound groups to lower masses. It is only when attention is given to the luminosities of the systems that one appreciates that the situation is quite different from the familiar.
The second argument draws on the highly correlated nature of the distribution of dwarf galaxies. Within 5 Mpc, most known dwarfs are members of the few well established groups (14-7: CVn I Group; 14-10: M81 Group; 14-11: Maffei/IC 342 Group; 14-12: Local Group; 14-13: Sculptor Group; 14-15: Centaurus Group) and most of the rest are associated with a few relatively isolated bright galaxies or are members of the dwarf groups that have been identified. The current census of the sky for dwarf galaxies is mainly due to the hard work extracting candidates from the Second Palomar Observatory Sky Survey in the northern sky and the ESO/SERC survey of the southern sky by Karachentseva and collaborators [@kaa98], [@kaa99], [@kar00], [@kaa00], with HI follow up reported in @huc01 and earlier papers. The HI Parkes All Sky Survey (HIPASS) is providing an independent search of the southern part of the sky [@bar01].
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@kar03a provide a tabulation of 156 galaxies, big and small, identified to exist between 1 and 5.5 Mpc of us (hence excludes the Local Group). Fully 101 of the objects are associated with the well established groups identified in the previous paragraph. Another 13 are associated with relatively isolated luminous galaxies and 6 are associated with somewhat more distant groups or are at very low galactic latitude. Then there are the 21 galaxies we associate with the groups of dwarfs or the comparison 14+13 (NGC 55) Group. That leaves only 15 low luminosity galaxies remaining through the rest of the volume. Several of these are in close proximity to one another.
The strong correlation in position of the nearby galaxies is illustrated in Figure 6. The top panel shows the two-point correlation function for all known galaxies at $\vert b \vert > 28$ within 5 Mpc, excluding the Local Group within 1.1 Mpc. Good distances exist for a majority of these galaxies, and rough distances exist for the rest, so what is shown here is the three-dimensional two-point function. Normalization was achieved by comparison with 1000 monte carlo random distributions within the volume. The strong positive correlation at separations less than 1 Mpc is dominated by the contributions from the well known nearby galaxy groups.
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The bottom panel in Fig. 6 examines the two-point correlation when contributions from the dominant groups have been excised. In this case, spheres of radius 1.4 Mpc have been defined centered on the 5 dominant nearby high latitude groups: those associated with M81, CVn I, Centaurus, Sculptor, and NGC 1313 (the Maffei Group lies at low latitude as does a majority of the Centaurus Group). The 80% of the volume that remains (220 Mpc$^3$) contains 30% of the original sample (42 galaxies). The Foreground Sculpter and three of four of our candidate dwarf groups are in the remaining volume, contributing 17 of the 42 galaxies (the 14+19 Group lies at low Galactic latitude; in fact the fussy $\vert b \vert > 28$ limit was chosen to allow inclusion of the entire 17+6 Group).
It is seen that even when the historically known groups are eliminated there is still a strong correlation signal at separations less than 1 Mpc. The signal at separations less than 0.5 Mpc is 63% of the signal in the top panel. This signal is dominated by the correlations within the four low mass candidate groups.
The positive correlation signal at separations greater than 9 Mpc has an easy explanation. The monte carlo normalization assumes objects are distributed randomly throughout the available volume but in fact most nearby galaxies lie in the supergalactic equatorial plane. The signal at large separations comes from cross-correlations from opposite ends of the equatorial plane, between the 17+6 members and galaxies in the region of the Centaurus Group on the one hand and between galaxies in the vicinity of the Sculptor Group with those near CVn I and near Centaurus. In this small sample, only 3-4 galaxies in each of these regions is enough to create the spurious cross-correlation signal. The weaker signal at large separations in the top panel arises from the same source: the concentration of local galaxies to a plane.
The present analysis is preliminary because the census of dwarfs is still uncertain. It is still not clear that even the high latitude sky has been uniformly surveyed. It is not clear how rapidly candidates are lost with Galactic latitude. HI signals for objects with velocities near zero can be lost in the confusion of Galactic emission. Our knowledge of the distances of candidates is improving rapidly with HST imaging but is still very incomplete. On the basis of present information the dwarf groups that have been identified are manifest enhancements over a random distribution. They correlate comparably well as the galaxies in previously established groups.
Everything Together
===================
The small dimensions of the dwarf groups and the highly significant 2-point correlation signal on scales $<1$ Mpc suggest that the groups of dwarfs are bound. Given that proposition, we can add the information from the low density regime to the data from the intermediate and high density regime shown in Fig. 2. The combined data is seen in Figure 7. T87 groups of 2-4 members are added to provide information about the situation at low luminosities, though mass estimates are very uncertain for these entities. The four groups of dwarfs are plotted with error bars in mass. They lie at luminosities below $10^9 L_{\odot}$ and well below the dotted $M/L_B$ constant line. The comparison 14+13 (NGC 55) Group lies in the same mass range but at higher luminosity, well above the dotted line. The solid curve was fit to the ensemble of data. The curve corresponds to the expression: $$L_B = \phi M^{\gamma} e^{-M^{\dag}/M}$$ This equation relates luminosity and mass with 3 constraints: a logarithmic slope at the high mass end, $\gamma$, a low mass exponential cutoff set by $M^{\dag}$, and a normalization, $\phi$. The curve was fit by minimizing a $\chi^2$ with uncertainties taken in mass only: $$\chi^2 = \sum_i^N ({{\rm log}M_i - {\rm log}M_{fit} \over {\rm log}\sigma})^2$$ Here, $M_{fit}$ is the mass in the relationship described by eq. (2) at the luminosity $L_i$ of group $i$, with measured mass $M_i$. The $\chi^2$ normalization is given the dependence on the number of group members $N$ according to the formulation ${\rm log}\sigma = A/N^{0.75}$. A weighting that favors groups with large $N$ more than the statistical $N^{-0.5}$ is justified by the expected better approximation to a virialized state for the larger groups and to compensate for the small numbers of large $N$ systems. The absolute normalization of the $\chi^2$ evaluator is arbitrary: we simply search for the lowest value. We set $A=2$ for the traditional groups and $A=1$ for the groups of dwarfs; ie, give double weight to the groups of dwarfs because they are drawn from a restricted region and are few in number.
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The curve superimposed on the data in Fig. 7 has a slope at the high mass end of $\gamma = 0.70$. It can be appreciated that taking errors only in mass leads to a steeper slope than if errors are distributed into luminosity. At the low mass end the curve demonstrates an exponential cutoff characterized by $M^{\dag} = 6 \times 10^{11} M_{\odot}$. The normal luminous groups with 5 or more members only provide information at masses $>10^{12} M_{\odot}$. Groups of 2–4 members provide information at lower luminosities and masses but at the cost of large uncertainties in mass. The evidence for the cutoff comes from the four candidate dwarf groups. In the absence of the dwarf groups we might be left to suppose that there is simply a lower limit to the mass range of groups. However, from Fig. 7 we instead infer that smaller mass groups do exist but groups are not manifested by the light of stars at very low masses. The break is more dramatically represented in Figure 8. This plot shows the same data but the vertical axis gives the departure from a $45^{\circ}$ line in the previous plot. Large errors in masses in small groups create diagonal scatter in this plot. The inference we draw from the Foreground Sculptor Group, with very low $M_B/L$, is that there is tremendous real scatter in the baryon content of halos in the proximity of the mass cutoff.
Interestingly, both @mar02 and @vdb03 have deduced a similar dependence between mass and light from observations of the luminosity function of galaxies derived from redshift surveys and the assumption that the underlying halo mass function follows the modified Press-Schechter description of @she99. The deduction follows from the observation that, compared with the mass function, the luminosity function is shallower at the faint end and cuts off more abruptly at the bright end, implying $M/L$ increases at the two extremes relative to the intermediate range.
{width="8.7cm"}
To conclude, we take a look at the mass function in the volume with $\vert b \vert >30$ and a distance limit of $25 h_{75}^{-1}$ Mpc (distances from a numerical action velocity field model) and the fraction of the total mass lying in specific environments. This information is summarized in Figure 9. The mass function incorporates all the groups involved in the earlier discussion, plus contributions from galaxies that lie outside any of these groups. We have no information on the extended mass around individual galaxies but for the purpose of constructing this mass function it is assumed that $M/L_B^{single}=100 M_{\odot}/L_{\odot}$. The contribution of the individual galaxies by number is given by the figures in brackets in Fig. 9. It is seen that the individual galaxies only become dominant in their contribution below $10^{12} M_{\odot}$ and with our $M/L_B$ assumption for the individual galaxies their contribution to the overall mass budget is small.
The mass function shown in Fig. 9 becomes incomplete below $10^{12} M_{\odot}$. The error bars only reflect the statistical uncertainty associated with the observed contributions. To get an estimate of the contribution to the total mass budget of low mass groups we consider the restricted volume within 5 Mpc. In this volume there are 5 groups of luminous galaxies, excluding the low latitude 14-11 (Maffei/IC 342) Group, and these groups cumulatively contain about $12 \times 10^{12} M_{\odot}$. Three high latitude luminous galaxies unassociated with significant groups (NGC 1313, NGC 3621, and NGC 6503), if assumed to have an associated $M/L_B=100$, would contain about $2 \times 10^{12} M_{\odot}$ in their vicinity. The five low mass groups discussed here would cumulatively contain about $2.0 \times 10^{12} M_{\odot}$. Roughly $3/4$ of the mass related to observable galaxies within 5 Mpc would lie in the major groups, almost 90% would be associated in some way with luminous galaxies, and only about 12% would be in the regions of the groups of dwarfs. Although the number of groups associated with low mass halos are comparable to the number of familiar groups in the local region, they contribute only a small fraction of the mass. Hence the inventory of the mass of the Universe tied up in bound structures represented by Fig. 9 is unlikely to be seriously in error at the low mass end.
{width="8.7cm"}
The histogram of mass fractions by mass interval seen in Fig. 9 can be summarized as follows. Fully 40% of the mass in this local volume is in the single object, the Virgo Cluster (clearly, the high mass end of the mass function is poorly determined in such a restricted volume). The groups that have short crossing times and are dominated by E/S0/Sa galaxies (including Virgo) contain 60% of the mass. Looking at groups of all types, 90% of the mass is in groups with ${\rm log} M > 12.5$. The luminous groups within 5 Mpc all lie essentially within the mass bin $12.0 < {\rm log} M < 12.5$, on the tail of the histogram seen in Fig. 9. Yet even with respect to such groups, the low mass groups within 5 Mpc make only a minor contribution to the mass budget. It can be concluded that they do not contain an important fraction of the mass of the Universe.
Summary
=======
There is surprisingly strong evidence for variations in the relationship between blue light and mass as a function of environment. On the scale of groups, the lowest $M/L_B$ values are found on mass scales of $10^{12} - 10^{13} M_{\odot}$ where typical values are $M/L_B \sim 90 M_{\odot}/L_{\odot}$. Groups in this mass range are almost always composed of late type galaxies with ongoing star formation. Groups with higher mass produce less blue light. The trend is enhanced by the local density as identified by the crossing time. High density regions, those with short crossing times, are darker. The morphology of the group members is highly correlated with these trends: groups with predominantly E/S0/Sa systems have short crossing times and large $M/L_B$. This pattern is consistent with a picture in which dense regions formed earlier and today star formation is largely exhausted in these places. The stellar populations are fainter and redder. Possibly in these environments there have been multiple collisions between galaxies that have scattered many stars into the intracluster environment where their light goes undetected in our light inventory. Then it is certainly known that the dense environments of E/S0 groups and clusters glow with thermal X-ray emission, to the degree that most of the baryons in these environments are in the hot intracluster gas and not in stars. Together, these astrophysical processes could explain the observed factor of 7 increase in $M/L_B$ in proceeding from spiral dominated groups in the mass range $10^{12} - 10^{13} M_{\odot}$ to E/S0 dominated groups in the mass range $10^{14} - 10^{15} M_{\odot}$.
At the other end of the mass spectrum, below $10^{12} M_{\odot}$, there is a cutoff in the visible manifestations of groups. One unlikely possibility is that there simply are few groups with less mass than $10^{12} M_{\odot}$. A more likely possibility is that group halos of lower mass exist but are difficult to identify because of a deficiency of light. We have presented evidence that such low mass groups might be common. The number of such groups that can be identified within 5 Mpc are comparable to the number of luminous groups. The deficiency of light is very great in these cases, assuming that the groups are bound. Groups in the mass range $10^{11} - 10^{12} M_{\odot}$ can have $M/L_B$ values 5 to 20 times higher than the groups of spiral galaxies in the range $10^{12} - 10^{13} M_{\odot}$. The possibility is open that there may be halos in the dwarf group mass range and lower that are totally invisible.
As an aside, if the hypothesis of this paper is correct that groups of dwarfs are bound then the two decade old controversy between the alternatives of dark matter versus non-Newtonian gravity (Milgrom 1983) is definitively resolved. It is implausible that the groups of dwarfs are bound by stars or gas. The relationship between the observable constituents and the gravitational field is quite unlike that of more luminous groups.
The transform between light and mass is being shown to be complex. Of particular cosmological importance is the sharp trend toward darkness at high mass. The Virgo Cluster has 40% of the mass attributed to groups in the volume of our study but it only contributes 15% of the blue light. The rich clusters have a dynamic importance much greater than would be expected from their light. By contrast, though there is evidence that low mass groups may be rendered almost or completely invisible, their cumulative contribution to the clustered mass of the Universe seems to be small. To the limit that it can be traced, the mass function is too flat for low mass structures to make a substantial contribution to the inventory of mass.
The work over the years involving the Numerical Action modeling has involved a happy collaboration with Jim Peebles and, especially, Ed Shaya. My principal collaborator on work at the faint end of the galaxy luminosity function is Neil Trentham. Jose Pacheco has provided valuable insight regarding the dynamic state of unrelaxed structures. This research is being supported by JPL Contract 1243647 and STScI awards HST-GO-09162 and HST-GO-10210.
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[lcccccccccc]{}
11-01 & 174 & 61 & 16.8 & 1042 & 715 & 1.02 & 0.08 & 12.20 & 14.95 & 562\
11-04 & 7 & & 13.8 & 1576 & 124 & 0.30 & 0.13 & 10.59 & 12.77 & 151\
11-10 & 9 & 17 & 23.9 & 1221 & 79 & 0.54 & 0.36 & 11.36 & 12.79 & 27\
11-14 & 5 & & 19.2 & 838 & 106 & 1.10 & 0.55 & 11.50 & 13.21 & 51\
12-01 & 57 & 22 & 17.2 & 967 & 148 & 1.30 & 0.47 & 11.70 & 13.68 & 95\
12-03 & 9 & 22 & 22.9 & 1352 & 112 & 0.71 & 0.34 & 11.01 & 13.28 & 185\
12-05 & 6 & & 23.0 & 1384 & 83 & 0.61 & 0.39 & 10.95 & 12.62 & 47\
12-06 & 9 & 29 & 20.1 & 1020 & 125 & 0.64 & 0.27 & 10.91 & 13.01 & 126\
14-01 & 25 & 52 & 9.7 & 911 & 266 & 0.50 & 0.10 & 11.00 & 13.72 & 523\
14-04 & 22 & 14 & 7.6 & 596 & 58 & 0.62 & 0.56 & 10.78 & 12.44 & 46\
14-05 & 8 & & 7.3 & 571 & 129 & 0.46 & 0.19 & 10.77 & 12.90 & 134\
14-06 & 5 & & 7.8 & 698 & 78 & 0.29 & 0.20 & 10.44 & 12.50 & 116\
14-07 & 22 & & 3.5 & 309 & 51 & 0.39 & 0.41 & 10.24 & 12.21 & 93\
14-09 & 9 & & 5.0 & 367 & 82 & 0.33 & 0.21 & 10.46 & 12.53 & 117\
14-10 & 12 & & 3.3 & 242 & 108 & 0.35 & 0.17 & 10.46 & 12.70 & 174\
14-11 & 8 & & 3.0 & 188 & 75 & 0.47 & 0.33 & 11.17 & 12.47 & 20\
14-12 & 10 & & 0.0 & -18 & 57 & 0.43 & 0.40 & 10.78 & 12.01 & 17\
14-13 & 11 & & 2.1 & 197 & 118 & 0.43 & 0.19 & 10.38 & 12.90 & 28\
14-15 & 12 & & 4.3 & 304 & 68 & 0.66 & 0.52 & 10.96 & 12.66 & 50\
15-01 & 9 & 88 & 7.2 & 626 & 112 & 0.22 & 0.10 & 10.62 & 12.58 & 92\
17-01 & 13 & 9 & 10.0 & 752 & 36 & 0.58 & 0.85 & 10.78 & 12.18 & 25\
17-04 & 6 & & 9.8 & 813 & 114 & 0.53 & 0.25 & 10.57 & 12.99 & 266\
19-01 & 8 & & 10.6 & 735 & 87 & 0.65 & 0.39 & 10.86 & 12.76 & 79\
21-01 & 11 & 36 & 22.9 & 1087 & 220 & 0.61 & 0.15 & 11.28 & 13.50 & 165\
21-03 & 10 & 0 & 20.9 & 1009 & 98 & 0.51 & 0.28 & 10.82 & 12.70 & 75\
21-06 & 12 & 55 & 22.3 & 1207 & 124 & 0.86 & 0.37 & 11.15 & 13.00 & 71\
21-10 & 5 & & 20.1 & 1082 & 61 & 0.57 & 0.49 & 10.77 & 12.20 & 27\
21-12 & 10 & 30 & 24.5 & 1453 & 88 & 0.65 & 0.39 & 11.02 & 12.81 & 62\
41-07 & 5 & & 24.0 & 1219 & 132 & 0.39 & 0.16 & 11.07 & 12.86 & 61\
43-01 & 13 & 11 & 19.2 & 970 & 128 & 0.75 & 0.31 & 11.15 & 13.24 & 124\
44-01 & 7 & & 15.9 & 922 & 76 & 0.50 & 0.35 & 10.79 & 12.71 & 83\
51-01 & 31 & 90 & 16.9 & 1344 & 434 & 0.56 & 0.07 & 11.48 & 14.11 & 431\
51-04 & 17 & 62 & 17.9 & 1427 & 110 & 0.99 & 0.48 & 11.18 & 13.12 & 87\
51-05 & 6 & & 19.9 & 1601 & 85 & 0.53 & 0.33 & 10.87 & 12.87 & 100\
51-07 & 6 & & 20.2 & 1626 & 112 & 0.67 & 0.32 & 10.99 & 12.80 & 65\
51-08 & 23 & 94 & 25.0 & 1552 & 385 & 0.35 & 0.06 & 11.10 & 13.66 & 364\
52-01 & 11 & 29 & 17.1 & 1405 & 99 & 0.53 & 0.28 & 10.95 & 12.98 & 107\
52-02 & 8 & & 13.8 & 1125 & 75 & 0.64 & 0.45 & 10.91 & 12.77 & 73\
52-03 & 5 & & 19.1 & 1567 & 105 & 0.45 & 0.23 & 10.58 & 12.86 & 190\
52-06 & 6 & & 18.6 & 1531 & 79 & 0.44 & 0.29 & 10.65 & 12.28 & 43\
52-07 & 6 & & 23.7 & 1954 & 48 & 0.44 & 0.49 & 11.07 & 12.25 & 15\
53-01 & 13 & 82 & 13.4 & 1006 & 236 & 0.34 & 0.08 & 11.04 & 13.54 & 319\
53-03 & 5 & & 15.0 & 1109 & 121 & 0.45 & 0.20 & 10.42 & 13.08 & 460\
53-07 & 15 & 50 & 10.8 & 833 & 93 & 0.69 & 0.39 & 10.75 & 12.96 & 161\
53-10 & 5 & & 11.4 & 831 & 103 & 0.45 & 0.23 & 10.51 & 12.76 & 179\
54-01 & 7 & & 13.7 & 802 & 111 & 0.55 & 0.26 & 10.72 & 13.06 & 219\
54-03 & 5 & & 8.4 & 519 & 68 & 0.29 & 0.22 & 10.28 & 12.48 & 160\
61-11 & 6 & & 23.2 & 1852 & 106 & 0.45 & 0.22 & 10.71 & 12.51 & 63\
61-16 & 12 & 40 & 19.3 & 1572 & 93 & 0.84 & 0.48 & 11.25 & 12.88 & 43\
65-01 & 6 & & 14.6 & 1132 & 74 & 0.49 & 0.35 & 10.72 & 12.69 & 94\
11-24 & 12 & 92 & 35.2 & 2148 & 311 & 0.69 & 0.12 & 11.74 & 14.00 & 184\
31-02 & 10 & 70 & 37.5 & 2188 & 479 & 0.13 & 0.01 & 11.12 & 13.76 & 434\
41-01 & 11 & 73 & 28.4 & 1807 & 344 & 0.49 & 0.08 & 11.51 & 13.84 & 214\
[llccccccccccc]{}
14+12 & NGC 3109 & 4 &1.4& 0.34 & 18 & 3.6 & 1.4 & 0.6& 400 & 200& 1220 & 0.83\
14 +8 & UGC 8760 & 4 &3.0& 0.29 & 16 & 1.1 & 1.0 & 1.2 & 860 &1030 & 250 & 0.76\
14+19 & UGC 3974 & 4 &5.0& 0.54 & 28 & 3.3 & 5.1 & 8.5 & 1520 &2560 & 1060 & 0.84\
17 +6 & NGC 784& 4 &5.0& 0.45 & 36 & 8.5 & 8.3 &10.5 & 980 &1240 & 330 & 0.54\
14+13 & NGC 55& 4 &2.1& 0.30 & 15 &57.6 & 0.9 & 1.1 & 16 & 18 & 13 & 0.85\
& & 5 & & 0.47 & 13 &61.5 & 0.8 & 1.1 & 14 & 18 & & 1.5 \
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abstract: '[Spectrally-resolved observations of three pure rotational lines of H$_2$, conducted with the EXES instrument on SOFIA toward the classic bow shock HH7, reveal systematic velocity shifts between the S(5) line of ortho-H$_2$ and the two para-H$_2$ lines \[S(4) and S(6)\] lying immediately above and below it on the rotational ladder. These shifts, reported here for the first time, imply that we are witnessing the conversion of para-H$_2$ to ortho-H$_2$ within a shock wave driven by an outflow from a young stellar object. The observations are in good agreement with the predictions of models for non-dissociative, C-type molecular shocks. They provide a [ clear demonstration]{} of the chemical changes wrought by interstellar shock waves, in this case the conversion of para-H$_2$ to ortho-H$_2$ in reactive collisions with atomic hydrogen, and [ provide among the most compelling evidence yet obtained for C-type shocks in which the flow velocity changes continuously.]{}]{}'
author:
- |
David A. Neufeld, Curtis DeWitt, Pierre Lesaffre, Sylvie Cabrit, Antoine Gusdorf,\
[Le Ngoc Tram]{}, and Matthew Richter
title: 'SOFIA/EXES observations of warm H$_2$ at high spectral resolution: witnessing para-to-ortho conversion behind a molecular shock wave in HH7'
---
Introduction
============
Shock waves are a widespread phenomenon in the interstellar medium; they may be driven by supersonic motions associated with protostellar outflows, supernova explosions, and cloud-cloud collisions. In addition to heating and compressing the medium through which they propagate, shock waves can alter the chemical state of the interstellar gas. Even in slower shocks that are non-dissociative, the passage of a shock wave through a molecular cloud can cause significant changes in its composition, either as a result of grain mantle sputtering, or because of large enhancements in the rates of chemical reactions that are endothermic or possess an activation energy barrier [(Godard et al. 2019, and references therein.)]{} One theorized example of such a change is the conversion of para-H$_2$ to ortho-H$_2$ behind a [molecular]{} shock wave (e.g. Timmermann 1998; Wilgenbus et al. 2000).
Because their interconversion by means of radiative processes or in non-reactive collisions is negligibly slow, ortho- and para-H$_2$ may be regarded as two distinct chemical species. On long timescales ([$\sim 0.1$ Myr in dark clouds; e.g. Harju et al. 2017]{}), reactions with H$_3^+$, H$^+$, and H – involving the breaking of chemical bonds – can lead to ortho-para conversion and thereby establish an equilibrium ortho-to-para ratio (OPR) that is determined by the gas temperature. The OPR in local thermodynamic equilibrium (LTE) is temperature-dependent, because of the intimate connection between the rotational state of the molecule and the spin state of the nuclei. If the spin wavefunction is anti-symmetric (para-H$_2$ [with total nuclear spin 0]{}), then the rotational wavefunction must be symmetric and the rotational quantum number, $J$, must be even. Conversely, rotational states of ortho-H$_2$ ([total nuclear spin 1]{}) all have odd $J$. In the limit of high temperature, multiple states of both ortho- and para-H$_2$ are populated, and the OPR approaches 3, the ratio of the nuclear spin degeneracies. In the opposite limit of low temperature [(i.e. for $T \ll E_1/k = 170\,\rm K$, where $E_1$ is the energy of the $J=1$ rotational state)]{}, only the lowest $J=0$ state is significantly populated and the OPR approaches zero. In LTE, OPRs of 1, 2, and 2.9 are achieved at gas kinetic temperatures of 78, 118, and 216 K, respectively.
Observations of the H$_2$ OPR within shock-heated molecular gas became possible with the [*Infrared Space Observatory*]{} ([*ISO*]{}). Through observations of the H$_2$ S(1) – S(5) pure rotational lines toward shocked gas in HH54, [*ISO*]{} revealed an OPR $\sim 1.2$ in gas of kinetic temperature $\sim 650$ K (Neufeld et al. 1998). This measurement of a non-equilibrium OPR in warm gas for which the OPR would be 3 in LTE implied that para-to-ortho conversion, if it occurred at all, did not proceed to completion within the time period for which the emitting gas was warm. Thus, the warm H$_2$ observed by [*ISO*]{} retained an OPR that was a relic of an earlier phase in which the gas had equilibrated at a lower temperature. [Non-equilibrium OPRs were also measured toward HH2 by [*ISO*]{} (Lefloch et al. 2003), and were in good agreement with the predictions of Wilgenbus et al. (2000).]{}
Later observations with the IRS instrument on [*Spitzer*]{} allowed the H$_2$ S(0) – S(7) line intensities to be measured toward multiple sources with better sensitivity than [*ISO*]{} (e.g. Neufeld et al. 2006, 2007, 2009). Such measurements are conveniently represented in rotational diagrams, obtained by plotting ${\rm log}(N_J/g_J)$ versus $E_J$, where $N_J$ is the column density in rotational state $J$, $g_J$ is the degeneracy ($2J+1$ for even $J$ and $3(2J+1)$ for odd $J$) and $E_J$ is the rotational energy. The rotational diagrams obtained with [*Spitzer*]{} typically exhibited three key behaviors. First, positive curvatures in the rotational diagrams indicated the presence of multiple gas temperatures, suggesting that shocks with a range of velocities were present within the beam. Second, zigzag patterns in the rotational diagrams – with ${\rm log}(N_J/g_J)$ systematically higher for states of even $J$ – implied an OPR below the value expected in LTE. And third, the departure from OPR (i.e. the degree of the zigzag) was typically greater for the lower rotational states than for the higher ones. This third feature of the rotational diagrams suggested indirectly that para-to-ortho conversion was taking place within the shocks that had been observed, and that the faster shocks responsible for the higher-$J$ line emission were more efficient in converting para-H$_2$ to ortho-H$_2$. The greater efficiency of faster shocks could be readily understood in the context of shock models, which indicated that para-to-ortho conversion is dominated by reactive collisions with atomic H that possess an activation energy barrier corresponding to a temperature of $\sim 4000$ K (Schofield 1967; Lique 2015). In this picture, faster shocks – which can be partially dissociative – produce more atomic hydrogen which can react more quickly with H$_2$ at the higher postshock temperatures.
Despite the success of shock models in explaining the H$_2$ rotational diagrams obtained with [*Spitzer*]{}, direct evidence for para-to-ortho conversion has been elusive. Even in the nearest sources, para-to-ortho conversion occurs on a length scale that is too small[^1] to be resolved with [*Spitzer*]{} (although future JWST observations may provide the necessary angular resolution). But in addition to spatial offsets between the ortho- and para-H$_2$ emissions from shocks in which para-to-ortho conversion is occurring, shock models also predict small velocity shifts of a few km s$^{-1}$. [ These shifts occur because the gas is being decelerated while the ortho-to-para ratio is increasing (Wilgenbus et al. 2000); thus, the ortho-H$_2$ emissions emerge preferentially from a region where the gas has been more significantly decelerated with respect to the preshock material.]{} While the spectral resolution of [*Spitzer*]{}/IRS ($\lambda/\Delta \lambda \sim 60$ at the wavelengths of the S(3) – S(5) lines) was far too poor to permit the detection of such small velocity shifts, ground-based observations of the para-H$_2$ S(4) line toward shocked gas in HH54 have revealed an intriguing shift relative to the ortho-H$_2$ S(9) pure rotational line and the $v = 1-0$ S(1) line (Santangelo et al. 2014). However, because the latter two lines are both of much higher excitation than the S(4) line, the velocity difference might be related to [gradients in]{} excitation rather than to spin-symmetry; [moreover, the slit position used for the H$_2$ S(4) observation was different from that used for the ortho-H$_2$ observations, making a direct comparison difficult.]{}
With the advent of the EXES spectrometer (Richter et al. 2018) on SOFIA, it is now possible to observe the S(4), S(5), S(6), and S(7) pure rotational lines of H$_2$ [in the same observational configuration and]{} at a spectral resolution sufficient to resolve the expected ortho-para velocity shifts. In this paper, we discuss the first unequivocal detection of such shifts, obtained toward HH7, a classic bow shock [located in the NGC1333 cloud in the Perseus complex, where it is]{} driven by a jet from the young stellar object (YSO) SVS 13 [(Bachiller et al. 1990)]{}. The observations and data reduction are described in Section 2, and the results presented in Section 3. A discussion follows in Section 4.
Observations and data reduction
===============================
We observed the S(4), S(5), S(6), and S(7) pure rotational transitions of H$_2$ toward two positions in HH7. The observations were carried out using the EXES instrument in High-Medium mode, with a slit of width $1.9^{\prime\prime}$ centered on $[\alpha,\delta]=\rm 3^h29^m08\fs 46$,+31$^0$15$^\prime 29\farcs 2$ [(J2000)]{} and $\rm 3^h29^m08 \fs 38$,+31$^0$15$^\prime 26\farcs 3$ [(J2000)]{}. These positions, which we denote P1 and P2 respectively, are located [close to the peak of HH7A (i.e. near the apex of the bow shock, as shown in Figure 1). The offsets relative to the putative driving source, SVS 13A/VLA4B, were $(60\farcs 3, -34\farcs 8)$ and $(59\farcs 3, -37\farcs 7)$ respectively.]{} [The SOFIA image size at these wavelengths is typically $\sim 3.8^{\prime\prime}$ (50$\%$ encircled power; Temi et al. 2018)]{}.
![Slit locations overlaid on the H$_2$ v$= 1 - 0$ [S(1)]{} map of Khanzadyan et al. (2003). The slit position angle is shown at the average value during the observations.](fig1.ps)
This configuration of EXES provides a spectral resolving power $\lambda/\Delta \lambda$ of 86,000, corresponding to a Doppler velocity of 3.5 km s$^{-1}$. In Table 1, we list for each spectral line the rest wavelength, slit length, date(s) of observation, integration time, and slit rotation angle (degrees East of North). The latter varied during the observations, so a range is given. Figure 1 shows the average slit locations overlaid on the H$_2$ v$= 1 - 0$ [S(1)]{} map of Khanzadyan et al. (2003). The telescope was nodded after every [52 – 68]{} s of integration to a reference position devoid of H$_2$ emission, enabling the subtraction of telluric emission features.
0.1 true in
----------------------------------- ----------------- ----------------- ----------------- ---------------------
Line S(4) S(5) S(6) S(7)
Rest wavelength ($\mu$m) 8.02504108 6.90950858 6.10856384 5.51118327
Slit length (arcsec) 12.4 9.8 8.4 7.6
Dates of observations (P1)$^a$ Oct 19, 20 Oct 19, 20 Oct 20 Oct 19
Dates of observations (P2)$^a$ Oct 24 Oct 24, 25 Oct 24 Oct 25
Integration time$^b$ (s) (P1) 1792 2150 1800 1250
Integration time$^b$ (s) (P2) 1024 2316 1280 640
Slit position angle$^c$ (P1) 14.6 – 25.3 22.4 – 47.8 35.0 – 47.1 29.2 – 35.3
Slit position angle$^c$ (P2) 30.3 – 36.7 36.6 – 46.7 12.6 – 27.4 15.9 – 17.3
Line centroid$^{d,e}$ (P1) $5.81 \pm 0.18$ $2.19 \pm 0.22$ $5.26 \pm 0.28$ $1.74 \pm 0.57$
Line centroid$^{d,e}$ (P2) $7.39 \pm 0.18$ $3.51 \pm 0.25$ $6.35 \pm 0.43$ $2.50 \pm 0.72$
Line width $^{d,f}$ (P1) $10.1 \pm 0.4$ $14.4 \pm 0.6$ $11.6 \pm 0.7$ $19.3 \pm 1.5$
Line width $^{d,f}$ (P2) $8.9 \pm 0.4$ $14.3 \pm 0.6$ $13.0 \pm 1.1$ $19.1 \pm 1.9$
Integrated intensity$^{d,g}$ (P1) $4.30 \pm 0.19$ $4.19 \pm 0.17$ $1.84 \pm 0.11$ [$3.63 \pm 0.31$]{}
Integrated intensity$^{d,g}$ (P2) $3.51 \pm 0.17$ $3.57 \pm 0.16$ $1.99 \pm 0.17$ [$4.52 \pm 0.31$]{}
----------------------------------- ----------------- ----------------- ----------------- ---------------------
: [Observational details]{}
The data were reduced using the Redux pipeline (Clarke et al. 2015) with the fspextool software package – a modification of the Spextool package (Cushing et al. 2004) – which performs wavelength calibration, and a custom python script for aperture extraction. [The absolute flux calibration is accurate to]{} $\sim \rm 25\%$ and [the relative flux calibration to]{} $\sim \rm 12.5\%$. [The wavelength calibration was obtained from observations of multiple atmospheric lines of water and methane within each bandpass: we conservatively estimate its accuracy as $0.3\, \rm km \, s^{-1}$.]{} The atmospheric transmission is expected to exceed [90]{}$\%$ for all observed transitions.
Results
=======
The observations led to clear detections of all four target lines toward both positions. In the bottom two panels of Figure 2, we show the resultant spectra, after subtraction of a linear baseline, each normalized with respect to the peak intensity. Here, the S(4) – S(6) lines were extracted over a region of length $8.4^{\prime\prime}$, the largest extraction region common to all three lines. The extraction region for the S(7) line had a length of 6$^{\prime\prime}$. [ For the S(4) – S(6) lines, we also present spectral extractions for three equal subregions of length $2.8^{\prime\prime}$ along the slit, which we designate P1A/P2A, P1B/P2B, and P1C/P2C from southwest to northeast; these are shown in the top six panels of Figure 2.]{}
![$\rm H_2$ pure rotational spectra obtained with SOFIA/EXES toward HH7. The slit positions, extraction regions, and slit position angles are given in Table 1 or the text. The solid curves show Gaussian fits to the data plotted in the histogram. [ Vertical colored lines indicate the velocity centroids for each fit, with horizontal bars at the bottom indicating the standard errors (i.e. 68 % confidence limits); dashed black vertical lines show for comparison the centroids obtained for the full slit extractions in bottom panels.]{}](fig2.ps)
Solid lines show the best Gaussian fits to the observed spectra; the velocity centroids, line widths (FWHM) and integrated line fluxes for those fits are presented in Table 1, along with the standard errors on each parameter. Both the fits and the observed data are normalized relative to the peaks of the Gaussian fits. Vertical colored lines in Figure 2 indicate the velocity centroids for the Gaussian fits, [with horizontal bars at the bottom indicating the standard errors (i.e. 68% confidence limits).]{} As is immediately evident from even a cursory inspection of the bottom panels in Figure 2, both positions P1 and P2 exhibit statistically-significant shifts between the S(5) ortho-H$_2$ line and two para-H$_2$ lines, S(4) and S(6). Because, the latter two lines “sandwich” the former in excitation level, $E_J$, the shift must be associated with the different spin symmetries rather than the excitation. The measured magnitude of the ortho-para shift is [ $3.62 \pm 0.28 $ km s$^{-1}$ and $3.88 \pm 0.31 $ km s$^{-1}$]{}, respectively, at positions P1 and P2. The errors given here are 1 $\sigma$ statistical uncertainties; the systematic uncertainties for each line position are [less than]{} 0.3 km s$^{-1}$.
The upper panels show that statistically significant velocity shifts between the S(4) and S(5) lines are detected within every subregion as well. Here, the dashed black vertical lines show the velocity centroids obtained for the full $8.4^{\prime\prime}$ spectral extractions. [ The location of these lines – relative to the horizontal bars that show the standard errors on the velocity centroids for the subregions – indicate]{} that there is no statistically-significant evidence for a velocity gradient along the slit at either position P1 or P2; thus, [ the observed ortho-para velocity shifts cannot be attributed to a chance combination of gradients in velocity and ambient (preshock) OPR along the slit.]{}
Unfortunately, at the time of the observations, the S(7) line was Doppler-shifted close to an order edge where the sensitivity is significantly reduced; here, the signal-to-noise ratio is insufficient to constrain the centroid velocity nearly as well as was possible for S(4), S(5) and S(6).[^2] For the S(7) line, the shorter slit length prevented us from using the same extraction region as for the other lines.
Discussion
==========
Variation of the OPR behind the shock
-------------------------------------
Our observations of HH7 provide the first definitive evidence for the conversion of para-H$_2$ to ortho-H$_2$ behind shock wave propagating in molecular gas. Given a triad of observed line intensities for the H$_2$ S(4), S(5), and S(6) lines, we may determine uniquely the excitation temperature, $T_{86}$, implied by the S(6)/S(4) line ratio (which is proportional to the population ratio in $J=8$ and $J=6$), and the ortho-to-para, $\rm OPR_{678}$, that is needed to account for the S(5) line strength ($J=7$ level population) based on a linear interpolation of the rotational diagram between $J = 6$ and $J = 8$. The results for both parameters are shown in Figure 3 (crosses) in velocity bins of width 1 km s$^{-1}$. The results are consistent with a steady increase of the OPR with increasing blueshift [ (see the top panel of Fig. 4 below)]{}; this is the expected sense of the variation if the shock is propagating towards us into ambient molecular gas.
Comparison with shock models
----------------------------
We have compared the observational results presented here with predictions from the Paris-Durham shock model ([Lesaffre et al. 2013; Flower & Pineau des Forêts 2015; Godard et al. 2019]{}). The upper panel of Figure 4 shows the gas kinetic temperature, the flow velocities for the ionized and neutral species, and the OPR predicted for a plane-parallel, steady-state C-type shock[^3] of preshock H nucleus density $n_0=10^4\,\rm cm^{-3}$, velocity $v_s=$ 25 km s$^{-1}$, perpendicular preshock magnetic field $B_0= 130\,\rm \mu G$, and initial ortho-to-para ratio OPR$_0=0.01$. Here, the shock velocity and preshock magnetic field [have been adjusted to provide the optimal fit to the observed S(6)/S(4) line ratio, and the preshock density lies within the range of values derived in previous studies (Smith et al. 2003; Molinari et al. 2000; Yuan et al 2012). The initial OPR is the value that would be expected if the preshock gas had achieved equilibrium at a temperature of 25 K; significantly larger values for OPR$_0$ worsen the agreement with the measurements of OPR$_{678}$ plotted in Figure 3. The shock was assumed to be irradiated by an ultraviolet radiation field with an intensity equal to the mean interstellar value.]{}
![Top panel: neutral gas temperature ($T_{3} = T/[10^3$ K\]), velocities for the neutral and ionized fluids ($v_{n6,i6} = v_{n,i} / [10^6 \rm \, cm \,s^{-1}]$), and H$_2$ OPR for the simple C-type shock model described in the text. Bottom panels: predicted H$_2$ line profiles for the simple model described in the text, overlaid on the observed spectra for positions P1 and P2.](fig4.ps)
In the middle and lower panels of Figure 4, we show the predicted line profiles, after convolution with the instrumental profile [ (which is only of minor importance)]{}. These were obtained using the methods described by Tram et al. (2018), and include the effects of gradients in velocity and OPR within the shock (upper panel) as well as thermal broadening. Here, we adopted a LSR velocity of $8.4\, \rm km\,s^{-1}$ for the ambient material [(Lefloch et al. 1998; Lefèvre et al. 2017)]{} and assumed the flow was inclined towards us at an angle of [60$^0$]{} to the line-of-sight, consistent with the inclination angle of 30 - 60$^0$ estimated previously (Davis et al. 2000, and references therein). [ For this inclination angle, the ortho-para velocity shift in the predicted line profiles is $1.5\,\rm km\,s^{-1}$ (shift between the centroid of a Gaussian fit to the S(5) line and the mean of the centroids for S(4) and S(6)); and thermal broadening and the velocity gradient make comparable contributions to the overall line width.]{}
Given the simplicity of this model, in which a single plane-parallel shock is responsible for the observed emission, the fit to the data is reasonable, particularly at position P1. Clearly, the S(5) – S(7) lines tend to show a blue excess relative to the predicted profile. Most likely, this discrepancy reflects a shortcoming of the simple model that we have adopted; if, as seems probable, the slit contains a mixture of shocks of varying speed, then the faster shocks could create a blue excess (characterized by a larger temperature and OPR). [ These blue excesses, which are most pronounced for the S(5) and S(7) lines, account for part of the ortho-para shifts that are apparent in Figure 2. Nevertheless, an examination of the right half of each observed line profile in Position P1 clearly indicates that the ortho-para shifts in the predicted line profiles are needed to fit the data.]{} A more complete analysis, which we defer to a future study, will take account of the three-dimensional structure of the bow shock in HH7, using methods similar those introduced by Tram et al. (2018), [ and will attempt to model the different observed line profiles at the two positions.]{}
To enable a comparison with the results presented in Figure 3 for OPR$_{678}$ and $T_{86}$, we have treated the predicted line profiles in Figure 4 in exactly the same way as we did the observed line profiles, i.e.we computed the line ratios in $1\,\rm km\,s^{-1}$ velocity bins and derived OPR$_{678}$ and $T_{86}$ from those ratios. The result is shown by the solid curves [in Figure 3]{}. The observed data points do not extend into the region of the blue excess ($v_{\rm LSR} \le -5 \, \rm km\,s^{-1}$), because the S(4) line intensity there is too small to permit a meaningful determination of OPR$_{678}$ and $T_{86}.$ Given the limitations of our simple model, the agreement is quite good. Thus the changing OPR, witnessed here for the first time, is broadly consistent with the predictions of interstellar shocks models. [ In particular, the data provide among the most compelling evidence yet obtained for C-type shocks in which the flow velocity changes continuously.]{}
Based on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA). SOFIA is jointly operated by the Universities Space Research Association, Inc. (USRA), under NASA contract NAS2-97001, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK 0901 to the University of Stuttgart. D.A.N gratefully acknowledges the support of an USRA grant, SOF06-0022; and S.C., A.G., P.L. and L.N.T. that of the Programme National “Physique et Chimie du Milieu Interstellaire" (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES.
Bachiller, R., Gueth, F., Guilloteau, S., Tafalla, M., & Dutrey, A. 2000, , 362, L33
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[^1]: [The length scale is $\sim 10^{-3} (10^4 \, {\rm cm}^{-3} / n_0)\,\rm pc,$ where $n_0$ is the preshock density of H nuclei. For HH7, which is among the most favorable sources ($n_0 = 10^4 \, \rm cm^{-3}$, distance $\sim 250$ pc), this corresponds to an angular scale of $\simlt 1^{\prime\prime}$.]{}
[^2]: Owing to the US government shutdown, follow-up observations of H$_2$ S(7) – scheduled for late January 2019 when the Doppler shift was considerably more favorable – could not be performed.
[^3]: Models for J-type (i.e. “Jump"-type) shocks are unsuccessful in accounting for these data. Because the gas is rapidly decelerated at the shock front within a J-type shock, the predicted line profiles are very narrow; hence, ortho-para velocity shifts are not expected from a single J-shock. While a collection of multiple J-type shocks of varying [shock]{} velocity might yield an ortho-para [velocity]{} shift in their combined emission, this would be accompanied by a shift between the S(4) and S(6) lines because the postshock temperature (which determines the S(6)/S(4) line ratio) and degree of para-to-ortho conversion both increase with shock velocity. [These shock-velocity-dependent effects for J-shocks were considered quantitatively by Wilgenbus et al. (2000; their Figure 5b). Their analysis showed that an increase in the OPR from 0.1 to $ > 2$ – resulting from an increase in the shock velocity – should go hand in hand with a sharp increase in H$_2$ rotational temperature, leading to temperatures greater than 1500 K at shock velocities where the OPR $ > 2$. No such increase in the rotational temperature with OPR was observed (Figure 3).]{}
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abstract: 'The unit sphere $\mathbb S^3$ can be identified with the unitary group $SU(2)$. Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.'
address:
- 'Department of Mathematics, Georgetown University, Washington D.C. 20057, USA'
- 'Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway'
- 'Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen 5008, Norway'
author:
- 'Der-Chen Chang'
- Irina Markina
- 'Alexander Vasil’ev'
title: 'Sub-Riemannian geodesics on the 3-D sphere'
---
[^1]
[^2]
Introduction
============
The unit $3$-sphere centered on the origin is the set of $\mathbb
R^4$ defined by $$\mathbb S^3=\{(x_1,x_2,x_3,x_4)\in\mathbb R^4:\ x_1^2+x_2^2+x_3^2+x_4^2=1)\}.$$ It is often convenient to regard $\mathbb R^4$ as the two complex dimensional space $\mathbb C^2$ or the space of quaternions $\mathbb H$. The unit 3-sphere is then given by $$\mathbb S^3=\{(z_1,z_2)\in\mathbb C^2:\ |z_1|^2+|z_2|^2=1)\}\quad\text{or}\quad
\mathbb S^3=\{q\in\mathbb H:\ |q|^2=1)\}.$$ The latter description represents the sphere $\mathbb S^3$ as a set of unit quaternions and it can be considered as a group $Sp(1)$, where the group operation is just a multiplication of quaternions. The group $Sp(1)$ is a three-dimensional Lie group, isomorphic to $SU(2)$ by the isomorphism $\mathbb C^2\ni (z_1,z_2)\to q\in\mathbb H$. The unitary group $SU(2)$ is the group of matrices $$\begin{pmatrix}
z_1 & z_2 \\
-\bar z_2 & \bar z_1
\end{pmatrix}, \qquad z_1, z_2 \in \mathbb C,\quad |z_1|^2+|z_2|^2=1,$$ where the group law is given by the multiplication of matrices. Let us identify $\mathbb R^3$ with pure imaginary quaternions. The conjugation $qh\bar q$ of a pure imaginary quaternion $h$ by a unit quaternion $q$ defines rotation in $\mathbb R^3$, and since $|qh\bar q|=|h|$, the map $h\mapsto qh\bar q$ defines a two-to-one homomorphism $Sp(1)\to
SO(3)$. The Hopf map $\pi:\mathbb S^3\to\mathbb S^2$ can be defined by $$\mathbb S^3\ni q\mapsto qi\bar q=\pi(q)\in\mathbb
S^2.$$ The Hopf map defines a principle circle bundle also known as the Hopf bundle. Topologically $\mathbb S^3$ is a compact, simply-connected, 3-dimensional manifold without boundary.
Even a small part of properties of the unit 3-sphere finds numerous applications in complex geometry, topology, group theory, mathematical physics and others fields of mathematics. In the present paper we give a new look at the unit 3-sphere, considering it as a sub-Riemannian manifold. The sub-Riemannian structure comes naturally from the non-commutative group structure of the sphere in a sense that two vector fields span the smoothly varying distribution of the tangent bundle and their commutator generates the missing direction. The sub-Riemannian metric is defined as the restriction of the euclidean inner product from $\mathbb R^4$ to the distribution. The present paper devoted to the description of sub-Riemannian geodesics on the sphere. The sub-Riemannian geodesics are defined as a projection of the solution to the corresponding Hamiltonian system onto the manifold. We give explicit formulas using different parametrizations and discuss the number of geodesics starting from the unity of the group. While working on this paper the authors became aware on the results in [@Hurtado] where the Lagrangian approach was developed and the minimizers were found (in our terminology geodesics are solutions to a Hamiltonian system so a minimizer is one of them).
Left-invariant vector fields and the horizontal distribution
============================================================
In order to calculate left-invariant vector fields we use the definition of $\mathbb S^3$ as a set of unit quaternions equipped with the following noncommutative multiplication “$\circ$”: if $x=(x_1,x_2,x_3,x_4)$ and $y=(y_1,y_2,y_3,y_4)$, then $$\begin{aligned}
\label{grlaw2}
x\circ y=(x_1,x_2,x_3,x_4)\circ(y_1,y_2,y_3,y_4) & = & \Big((x_1y_1-x_2y_2-x_3y_3-x_4y_4),
\nonumber \\ & \ \ & (x_2y_1+x_1y_2-x_4y_3+x_3y_4),
\nonumber
\\& \ & (x_3y_1+x_4y_2+x_1y_3-x_2y_4),\\ & \ & (x_4y_1-x_3y_2+x_2y_3+x_1y_4)\Big).\nonumber\end{aligned}$$ The rule gives us the left translation $L_x(y)$ of an element $y=(y_1,y_2,y_3,y_4)$ by the element $x=(x_1,x_2,x_3,x_4)$. The left-invariant basis vector fields are defined as $X(x)=(L_x(y))_*X(0)$, where $X(0)$ are basis vectors at the unity of the group. The matrix corresponding to the tangent map $(L_x(y))_*$ calculated by becomes $$(L_x(y))_* =\begin{pmatrix}
x_1 & -x_2 & -x_3 & -x_4\\
x_2 & x_1 & -x_4 & x_3\\
x_3 & x_4 & x_1 & -x_2\\
x_4 & -x_3 &x_2 & x_1
\end{pmatrix}.$$ Calculating the action of $(L_x(y))_*$ in the basis of unit vectors of $\mathbb R^4$ we get four vector fields $$\begin{aligned}
\label{vf}
X_1(x) & = & x_1\partial_{x_1}+x_2\partial_{x_2}+x_3\partial_{x_3}+x_4\partial_{x_4},\nonumber \\
X_2(x) & = & -x_2\partial_{x_1}+x_1\partial_{x_2}+x_4\partial_{x_3}-x_3\partial_{x_4},\\
X_3(x) & = & -x_3\partial_{x_1}-x_4\partial_{x_2}+x_1\partial_{x_3}+x_2\partial_{x_4},\nonumber \\
X_4(x) & = & -x_4\partial_{x_1}+x_3\partial_{x_2}-x_2\partial_{x_3}+x_1\partial_{x_4}.\nonumber\end{aligned}$$ It is easy to see that the vector $X_1(x)$ is the unit normal to $\mathbb S^3$ at $x$ with respect to the usual inner product $\langle\cdot,\cdot\rangle$ in $\mathbb R^4$, hence, we denote $X_1(x)$ by $N$. Moreover, $$\langle N, X_2(x)\rangle=\langle N, X_3(x)\rangle=\langle N, X_4(x)\rangle=0,\quad\text{and}\quad|X_k(x)|^2=\langle X_k(x), X_k(x)\rangle=1,$$ for $k=2,3,4$, and for any $x\in\mathbb S^3$. The matrix $$\begin{pmatrix}
-x_2 & x_1 & x_4 & -x_3\\
-x_3 & -x_4 & x_1 & x_2\\
-x_4 & x_3 &-x_2 & x_1
\end{pmatrix}$$ has rank 3, and we conclude that the vector fields $X_2(x)$, $X_3(x)$, $X_4(x)$ form an orthonormal basis of the tangent space $T_x\mathbb S^3$ with respect to $\langle\cdot,\cdot\rangle$ at any point $x\in\mathbb S^3$. Let us denote the vector fields by $$X_3=X,\quad X_4=Y,\quad X_2=Z.$$
The vector fields possess the following commutation relations $$[X,Y]=XY-YX=2Z,\quad [Z,X]=2Y,\quad [Y,Z]=2X.$$ Let $\mathcal{D} = \operatorname{span}\{ X, Y\}$ be the distribution generated by the vector fields $X$ and $Y$. Since $[X, Y] = 2Z \notin \mathcal{D} $, it follows that $\mathcal{D}$ is not involutive. The distribution $\mathcal{D}$ will be called [*horizontal*]{}. Any curve on the sphere with the velocity vector contained in the distribution $\mathcal{D}$ will be called a [*horizontal curve*]{}. Since $T_x\mathbb
S^3=\operatorname{span}\{X,Y,Z=1/2[X,Y]\}$, the distribution is bracket generating. We define the metric on the distribution $\mathcal D$ as the restriction of the metric $\langle\cdot,\cdot\rangle$ onto $\mathcal D$, and the same notation $\langle\cdot,\cdot\rangle$ will be used. The manifold $(\mathbb S^3,\mathcal D,\langle\cdot,\cdot\rangle)$ is a step two sub-Riemannian manifold.
Notice that the choice of the horizontal distribution is not unique. The relations $[Z,X]=2Y$ and $[Y,Z]=2X$ imply possible choices $\mathcal{D} =
\operatorname{span}\{ X, Z\}$ or $\mathcal{D} = \operatorname{span}\{ Y, Z\}$. The geometries defined by different horizontal distributions are cyclically symmetric, so we restrict our attention to the $\mathcal{D} = \operatorname{span}\{ X, Y\}$.
We also can define the distribution as a kernel of the following one form $$\omega = -x_2 dx_1 + x_1 dx_2 + x_4 dx_3 - x_3 dx_4$$ on $\mathbb{R}^4$. One can easily check that $$\omega(X) =0,\quad \omega(Y)=0,\quad \omega (Z)=1 \not=0,\quad \omega (N)=0.$$ Hence, $\ker
\omega = \operatorname{span}\{ X, Y, N \}$, and the horizontal distribution can be written as $$\mathbb{S}^3 \ni x \rightarrow \mathcal{D}_x = \ker \omega \cap T_x \mathbb{S}^3.$$
Let $\gamma(s)=(x_1(s),x_2(s),x_3(s),x_4(s))$ be a curve on $\mathcal S^3$. Then the velocity vector, written in the left-invariant basis, is $$\dot\gamma(s)=a(s)X(\gamma(s))+b(s)Y(\gamma(s))+c(s)Z(\gamma(s)),$$ where $$\begin{aligned}
\label{coord}
a & = & \langle\dot\gamma, X\rangle=-x_3\dot x_1-x_4\dot x_2+x_1\dot x_3+x_2\dot x_4,\nonumber \\
b & = & \langle\dot\gamma, Y\rangle=-x_4\dot x_1+x_3\dot x_2-x_2\dot x_3+x_1\dot x_4, \\
c & = & \langle\dot\gamma, Y\rangle=-x_2\dot x_1 +x_1\dot x_2+x_4\dot x_3-x_3\dot x_4.\nonumber\end{aligned}$$ The following proposition holds.
\[prop1\] Let $\gamma (s) = (x_1(s), x_2(s), y_1(s),
y_2(s))$ be a curve on $\mathbb{S}^3$. The curve $\gamma$ is horizontal, if and only if, $$\label{hc}
c=\langle \dot\gamma, Z \rangle = \langle\dot\gamma, X\rangle=-x_2\dot x_1 +x_1\dot x_2+x_4\dot x_3-x_3\dot x_4=0.$$
The manifold $\mathbb S^3$ is connected and it satisfies the bracket generating condition. By the Chow theorem [@Chow], there exists piecewise $C^1$ horizontal curves connecting two arbitrary points on $\mathbb S^3$. In fact, smooth horizontal curves connecting two arbitrary points on $\mathbb S^3$ were constructed in [@CChM].
The horizontality property is invariant under the left translation.
It can be shown that does not change under the left translation. This implies the conclusion of the proposition.
Hamiltonian system
==================
Ones we have a system of curves, in our case the system of horizontal curves, we can define the length as in the Riemannian geometry. Let $\gamma:[0,1]\to \mathbb S^3$ be a horizontal curve such that $\gamma(0)=x$, $\gamma(1)=y$, then the length $l(\gamma)$ of $\gamma$ is defined as the following $$\label{length}
l(\gamma)=\int_0^1\langle\dot\gamma, \dot\gamma\rangle^{1/2}\,dt=\int_0^1\big(a^2(t)+b^2(t)\big)^{1/2}\,dt.$$ Now we are able to define the distance between the points $x$ and $y$ by minimizing the integral or the corresponding energy integral $\int_0^1\big(a^2(t)+b^2(t)\big)\,dt$ under the non-holonomic constraint . This is a Lagrangian approach. The Lagrangian formalism was applied to study the sub-Riemannian geometry of $\mathbb S^3$ in [@CChM; @Hurtado]. In the Riemannian geometry the minimizing curve locally coinsides with the geodesic, but it is not the case for the sub-Riemannian manifolds. Interesting examples and discussions can be found, for instance in [@LS; @Mon1; @Mon2; @Mon; @Str]. Given the sub-Riemannian metric we can form a Hamiltonian function defined on the cotangent bundle of $\mathbb S^3$. The geodesics in the sub-Riemannian manifolds are defined as a projection of the solution to the corresponding Hamiltonian system onto the manifold. It is a good generalization of the Riemannian case in the following sense. The Riemannian geodesics (that are defined as curves with vanishing acceleration) can be lifted to the solutions of the Hamilton system on the cotangent bundle.
In the present paper we are interested in the construction of sub-Riemannian geodesics on $(\mathbb S^3,\mathcal D,\langle\cdot,\cdot\rangle)$. Let us write the left-invariant vector fields $X,Y,Z$, using the matrices $$\label{eq:matrix}
I_1=\left[\array{rrrr}0 & 0 & -1 & 0
\\ 0 & 0 & 0 & -1
\\
1 & 0 & 0 & 0
\\ 0 & 1 & 0 & 0\endarray\right],\quad
I_2=\left[\array{rrrr}0 & 0 & 0 & -1
\\ 0 & 0 & 1 & 0
\\
0 & -1 & 0 & 0
\\ 1 & 0 & 0 & 0\endarray\right],\quad
I_3=\left[\array{rrrr}0 & -1 & 0 & 0
\\ 1 & 0 & 0 & 0
\\
0 & 0 & 0 & 1
\\ 0 & 0 & -1 & 0\endarray\right].$$ Then $$X=\langle I_1x,\nabla x\rangle,\quad Y=\langle I_2x,\nabla x\rangle,
\quad Z=\langle I_3x,\nabla x\rangle.$$ The Hamiltonian function is defined as $$H=\frac{1}{2}(X^2+Y^2)=\frac{1}{2}\Big(\langle I_1x,\xi\rangle^2
+\langle I_2x,\xi\rangle^2\Big),$$ where $\xi=\nabla x$. Then the Hamiltonian system follows as $$\label{hs}
\begin{split}
&\dot x=\frac{\partial H}{\partial \xi}\ \ \Rightarrow \ \ \dot
x=\langle I_1x,\xi\rangle\cdot (I_1x)+\langle I_2x,\xi\rangle\cdot (I_2x)\\
&\dot \xi=-\frac{\partial H}{\partial x}\ \ \Rightarrow \ \
\dot\xi=\langle I_1x,\xi\rangle\cdot (I_1\xi)+\langle
I_2x,\xi\rangle\cdot (I_2\xi).
\end{split}$$ As it was mentioned, a geodesic is the projection of a solution to the Hamiltonian system onto the $x$-space. We obtain the following properties.
- [ Since $\langle I_1x,x\rangle=\langle I_2x,
x\rangle=\langle I_3x,x\rangle=0$, multiplying the first equation of by $x$ we get $$\langle\dot x,x\rangle=0\ \ \Rightarrow \ \ |x|^2=const.$$ We conclude that [*any solution to the Hamiltonian system belongs to the sphere.*]{} Taking the constant equal to $1$ we get geodesics on $\mathbb S^3$.]{}
- [Multiplying the first equation of by $I_3x$, we get $$\label{hc1}
\langle\dot x,I_3x\rangle=0,$$ by the rule of multiplication for $I_1$, $I_2$, and $I_3$. The reader easily recognizes the horizontality condition $\langle\dot
x,Z\rangle=0$ in . It means that [*any solution to the Hamiltonian system is a horizontal curve*]{}]{}.
- [Multiplying the first equation of by $I_1x$, and then by $I_2x$, we get $$\langle \xi,I_1x\rangle=\langle\dot x,I_1x\rangle,
\qquad \langle \xi,xI_2\rangle=\langle\dot x,xI_2\rangle.$$ From the other side, we know that $\langle\dot x,I_1x\rangle=a$ and $\langle\dot x,xI_2\rangle=b$. The Hamiltonian function can be written in the form $$H=\frac{1}{2}\Big(\langle I_1x,\xi\rangle^2+\langle I_2x,\xi\rangle^2\Big)
=\frac{1}{2}\Big(\langle I_1x,\dot x\rangle^2+\langle I_2x,\dot
x\rangle^2\Big)=\frac{1}{2}\Big(a^2+b^2\Big).$$ Thus, [*the Hamiltonian function gives the kinetic energy $H=\frac{|\dot
q|^2}{2}$ and it is a constant along the geodesics.*]{}]{}
- [If we multiply the first equation of by $\dot x$, then we get $$|\dot x|^2=\langle I_1x,\xi\rangle^2+\langle I_2x,\xi\rangle^2
=\langle I_1x,\dot x\rangle^2+\langle I_2x,\dot
x\rangle^2=a^2+b^2=2H.$$ Therefore $$\label{eq1}|\dot
x|^2=a^2+b^2.$$]{}
Velocity vector with constant coordinates
=========================================
We know that the length of the velocity vector is constant along geodesics. Let us start from the simplest case, when the coordinates of the velocity vector are constant. Suppose that $\dot a=\dot b=0$. The first line of system can be written as $$\begin{aligned}
\label{hs1}
\dot x_1 & = -ax_3-bx_4 \qquad \qquad \dot x_3 & = +ax_1-bx_2 \\
\dot x_2 & = -ax_4+bx_3 \qquad \qquad \dot x_4 & = +ax_2+bx_1 \nonumber.\end{aligned}$$
Differentiation of system yields $$\begin{aligned}
\label{hs2}
\ddot x_1 & = -a\dot x_3-b\dot x_4 \qquad \qquad \ddot x_3 & = +a\dot x_1-b\dot x_2 \\
\ddot x_2 & = -a\dot x_4+b\dot x_3 \qquad\qquad \ddot x_4 & = +a\dot x_2+b\dot x_1 \nonumber.\end{aligned}$$ We substitute the first derivatives from in , and get $$\label{hs3}
\ddot x_k=-r^2 x_k,\quad r^2=a^2+b^2,\quad k=1,2,3,4.$$
\[th1\] The set of geodesics with constant velocity coordinates form a unit sphere $\mathbb S^2$ in $\mathbb R^3$
We are looking for horizontal geodesics parametrized by the arc length and starting from the point $x(0)=x_0$. So, we set $r=1$ and $a=\cos\psi$, $b=\sin\psi$, where $\psi$ is a constant from $[0,2\pi)$. Solving the equation we get the general solution $x(s)=A\cos s+B\sin s$. We conclude that $A=x_0$ from the initial data. To find $B$ let us substitute the general solution in equations and get $B=(aI_1+bI_2)x_0$. Thus, the horizontal geodesics with constant horizontal coordinates are $$x(s)=x_0\cos s + (\cos\psi I_1+\sin \psi
I_2)x_0\sin s$$ Since the geodesics are invariant under the left translation it is sufficient to describe the situation at the unity element, e.g., $x_0=(1,0,0,0)$ of $\mathbb S^3$. In this case the geodesics are $$\begin{aligned}
\label{constgeod}
x_1 & = \cos s,\qquad\qquad x_3 & = \cos\psi\sin s, \\
x_2 & = 0, \qquad\qquad\ \ \ \ x_4 & = \sin\psi\sin s. \nonumber\end{aligned}$$ We see that the set of geodesics with constant velocity coordinates form the unit sphere $\mathbb S^2$ in $\mathbb
R^3=\{(x_1,0,x_3,x_4)\}$. The parameter $\psi\in [0,2\pi)$ corresponds to the initial velocity.
The sphere is a direct analogue of the horizontal plane in the Heisenberg group $\mathbb H^1$ at the unity. We remark that this result was obtained independently in [@Hurtado].
Let us calculate the analogue of the vertical axis in $\mathbb H^1$. We wish to find an integral curve for the vector field $Z$. In other words, we solve the system $$\begin{aligned}
\label{vertical}
a & = & \langle\dot\gamma, X\rangle=-x_3\dot x_1-x_4\dot x_2+x_1\dot x_3+x_2\dot x_4=0,\nonumber \\
b & = & \langle\dot\gamma, Y\rangle=-x_4\dot x_1+x_3\dot x_2-x_2\dot x_3+x_1\dot x_4=0, \\
c & = & \langle\dot\gamma, Z\rangle=-x_2\dot x_1 +x_1\dot x_2+x_4\dot x_3-x_3\dot x_4=1,\nonumber \\
n & = & \langle\dot\gamma, N\rangle=+x_1\dot x_1 +x_2\dot
x_2+x_3\dot x_3+x_4\dot x_4=0.\nonumber\end{aligned}$$ The determinant of the system is $1$ and it is redused to $$\begin{aligned}
\dot x_1 & = -x_2,\qquad\qquad \dot x_3 & = +x_4,\\
\dot x_2 & = +x_1,\qquad\qquad \dot x_4 & = -x_3.\end{aligned}$$ Differentiating again, we get the equation $\ddot x=-x$. The initial point is $x(0)=x_0$. System gives the value of the initial velocity $\dot x(0)=I_3x_0$. Taking into account this initial data we get the equation of the [*vertical line*]{} as $$x(s)=x_0\cos s +
I_3x_0\sin s.$$ In particular, at the point $(1,0,0,0)$ the equation of the vertical line is $$\label{vcircl}
x_1 = \cos s,\quad x_2 = \sin s,\quad
x_3 = 0,\quad
x_4 = 0,\qquad s\in[0,2\pi].$$
Velocity vector with non-constant coordinates
=============================================
[*Cartesian coordinates*]{}
Fix the initial point $x^{(0)}=(1,0,0,0)$. It is convenient to introduce complex coordinates $z=x_1+ix_2$, $w=x_3+ix_4$, $\varphi=\xi_1+i\xi_2$, and $\psi=\xi_3+i\xi_4$. Hence, the Hamiltonian admits the form $2H=|\bar w\varphi -z\bar\psi|^2$. The corresponding Hamiltonian system becomes $$\begin{array}{lllll}
\dot z & = & w(\bar{w} \varphi- z\bar{\psi}),\quad & &z(0)=1, \\
\dot w & = & -z(w\bar{\varphi}-\bar{z}\psi), \quad & &w(0)=0,\\
\dot {\bar{\varphi}} & = & \bar{\psi}(w\bar{\varphi}-\bar{z}\psi), \quad & &\bar{\varphi}(0)=A-iB,\\
\dot {\bar{\psi}} & = & -\bar{\varphi}(\bar{w} \varphi-z\bar{\psi}), \quad & &\bar{\psi}(0)=C-iD.
\end{array}$$ Here the constants $B,C$, and $D$ have the following dynamical meaning: $\dot w(0)=C+iD$, and $B=-i \ddot w(0)/2\dot w(0)$. So $C,D$ is the velocity and $B/\sqrt{C^2+D^2}$ is the curvature of a geodesic at the initial point. This complex Hamiltonian system has the first integrals $$\begin{array}{rcl}
z\psi -w \varphi & = & C+iD, \\
z\bar{\varphi}+w \bar{\psi} & = & A-iB,\\
\end{array}$$ and we have $|z|^2+|w|^2=1$ and $2H=C^2+D^2=1$ as an additional normalization. The latter means that we parametrize geodesics by the natural parameter. Therefore, $$\begin{array}{rcl}
\varphi & = & z(A+iB)-\bar w ( C+iD), \\
\psi & = & \bar z (C+iD) +w (A+iB).\\
\end{array}$$ Let us introduce an auxiliary function $p=\bar w/{z}$. Then substituting $\varphi$ and $\psi$ in the Hamiltonian system we get the equation for $p$ as $$\dot{p}=(C+iD)p^2-2iBp+(C-iD),\quad p(0)=0.$$ The solution is $$p(s)=\frac{(C-iD)\sin(s\sqrt{1+B^2})}{\sqrt{1+B^2}\cos(s\sqrt{1+B^2})+iB \sin(s\sqrt{1+B^2})}.$$ Taking into account that $\dot z \bar z= -w \dot{\bar w}$, we get the solution $$\label{geodz}
z(s)=\left(\cos(s\sqrt{1+B^2})+i\frac{B}{\sqrt{1+B^2}}\sin(s\sqrt{1+B^2})\right)e^{-iBs},$$ and $$\label{geodw}
w(s)=\frac{C+iD}{\sqrt{1+B^2}} \sin(s\sqrt{1+B^2})e^{iBs}.$$ If $B=0$ we get the solutions with constant horizontal velocity coordinates $$z(s)=\cos s,
\qquad w(s)=(\dot x_3(0)+i\dot x_4(0))\sin s$$ from the previous section.
\[numbergeod1\] Let $A$ be a point of the vertical line, i. e. $A=(\cos \omega,\sin \omega, 0,0)$, $\omega\in[0,2\pi)$, then there are countably many geometrically different geodesics $\gamma_n$ connecting $O=(1,0,0,0)$ with $A$. They have the following parametric equations $$\begin{aligned}
\label{geod1}
z_n(s) & = & \Big(\cos\big(s\frac{\pi n}{\sqrt{\pi^2 n^2-\omega^2}}\big)-
i\frac{\omega}{\pi n}\sin\big(s\frac{\pi n}{\sqrt{\pi^2 n^2-\omega^2}}\big)\Big)e^{\frac{is\omega}{\sqrt{\pi^2n^2-\omega^2}}},\\
w_n(s) & = & \big(\dot x_3(0)+i\dot x_4(0)\big)\frac{\sqrt{\pi^2n^2-\omega^2}}{\pi n}
\sin\big(s\frac{\pi n}{\sqrt{\pi^2 n^2-\omega^2}}\big)e^{\frac{-is\omega}{\sqrt{\pi^2n^2-\omega^2}}},\nonumber\end{aligned}$$ $n\in\mathbb Z\setminus\{0,\pm 1\}$, $s\in[0,s_n]$, where $l_n=\frac{1}{\sqrt{2}}s_n=\frac{1}{\sqrt{2}}\sqrt{\pi^2 n^2-\omega^2}$ is the length of the geodesic $\gamma_n$.
Since we use the condition $2H=|\dot z|^2+|\dot w|^2=1$ we conclude that the geodesics are parametrized by the arc length, and the length of a geodesic at the value of the parameter $s=l \sqrt{2}$ is equal to $l$. If the point $A=(z(s),w(s))$ belongs to the vertical line starting at $O=(1,0,0,0)$, then $|z(s)|=1$ and $|w(s)|=0$ provided that $-Bs=\omega$. It implies $$\cos^2
(s\sqrt{1+B^2})+\frac{B^2}{1+B^2}\sin^2(s\sqrt{1+B^2})=1,\qquad
\sin(s\sqrt{1+B^2})=0,\qquad -Bs=\omega.$$ These equations are satisfied when $$s_n=\sqrt{\pi^2 n^2-\omega^2},\quad B_n=-\frac{\omega}{\sqrt{\pi^2
n^2-\omega^2}},\quad n\in\mathbb Z\setminus\{0\}.$$ We conclude, that for $n(s)\in\mathbb Z\setminus\{0\}$ there is a constant $B_n(s)=-\frac{\omega}{\sqrt{\pi^2 n^2-\omega^2}}$, such that the corresponding geodesic $\gamma_n(s)$, $s\in[0,s_n]$, satisfying equation joins the points $O$ and $A$ and the length of the geodesic is equal to $s_n=\sqrt{\pi^2 n^2-\omega^2}$.
In the formulation of the theorem the words ‘geometrically different’ mean that due to the rotation of the argument of $C+iD$ in $w(s)$, there exist uncountably many geodesics.
So far we have had a clear picture of trivial geodesics whose velocity has constant coordinates. They are essentially unique (up to periodicity). The situation with geodesics joining the point $(1,0,0,0)$ with the points of the vertical line $A$ has been described in the preceding theorem. Let us consider the general position of points on $\mathbb S^3$.
\[numbergeod2\] Given an arbitrary point $(z_1,w_1)\in \mathbb S^3$ which neither belongs to the vertical line $A$ nor to the horizontal sphere $\mathbb S^2$, there is a finite number of geometrically different geodesics joining the initial point $(z_0,w_0)\in \mathbb S^3$ with $(z_1,w_1)$, $z_0=1$, $w_0=0$.
Let us denote $$w_1=\rho e^{i\varphi},\quad z_1=re^{i\alpha},\quad C+iD=e^{i\theta}.$$ Then from and we have that $$\label{ugol}
r^2=1-\frac{1}{1+B^2}\sin^2(s\sqrt{1+B^2}),\quad \mbox{and\ \ }\varphi=Bs+\theta,$$ where $s$ is the value of the length arc parameter when the point $(z_1,w_1)$ is reached. We want to exclude the parameter $s$ and rewrite the equations and in terms of the parameter $B$ and the given dates $r,\rho\neq 0,\alpha\neq 0$, and $\varphi$. We suppose for the moment that the angles $s\sqrt{1+B^2}$ and $sB$ are from the first quadrant. Other cases are treated similarly. Then we have $$z=(\sqrt{1-(1+B^2) \rho^2}+iB\rho)e^{i(\theta-\varphi)},$$ and $$\theta=\theta(B)=\alpha+\varphi-\arctan\frac{B\rho}{\sqrt{1-(1+B^2)\rho^2}}.$$ The first expression in leads to the value of the length parameter $s$ at $(z_1,w_1)$ $$s=\frac{1}{\sqrt{1+B^2}}\arcsin (\rho\sqrt{1+B^2}),$$ and the second to $$\varphi=\theta+\frac{B}{\sqrt{1+B^2}}\arcsin (\rho\sqrt{1+B^2}).$$ Substituting $\theta(B)$ in the latter equation we obtain $$\label{equa}
\sin\left((\alpha-\arctan\Big(\frac{B\rho}{r^2-B\rho^2}\Big))\sqrt{1+\frac{1}{B^2}}\right)=\rho\sqrt{1+B^2},$$ as an equation for the parameter $B$. Observe that $\varphi-\theta(B)=\alpha-\arctan\Big(\frac{B\rho}{r^2-B\rho^2}\Big)$ is a bounded function and $\lim_{B\to 0}\theta(B)\neq 0$. Indeed, if the latter limit were vanishing, then the value of given $\varphi$ would be zero and the solution of the problem would be only $B=0$ which is the trivial case excluded from the theorem. So the left-hand side of equation is a function of $B$ which is bounded by 1 in absolute value and fast oscillating about the point $B=0$. Observe, that $\alpha=0$ corresponds to the horizontal sphere which also was excluded from the theorem. The right-hand side of is an even function increasing for $B>0$, see Figure . Therefore, there exists a countable number of non-vanishing different solutions $\{B_n\}$ of the equation within the interval $|B|\leq \sqrt{\frac{1}{\rho^2}-1}=\frac{|z|}{|w|}$ with a limit point at the origin.
However, in order do define the parameters $B$, we need to solve the equations (\[ugol\]), (\[equa\]), and not all $B_n$ satisfy all three equations. Let us consider positive $B_n$. We calculate the argument of $z$ as $$\alpha=-B_ns+\arctan\Bigg[\frac{B_n}{\sqrt{B_n^2+1}}\tan\left(s\sqrt{B_n^2+1}\right)\Bigg]$$ $$=-B_ns+\arctan\Bigg[\frac{B_n\rho}{\sqrt{1-(1+B_n^2)\rho^2}}\Bigg]$$ $$<-B_n s+\frac{B_n\rho}{\sqrt{1-(1+B_n^2)\rho^2}}.$$
On the other hand, we have $$1=\frac{\arcsin(\rho\sqrt{1+B_n^2})}{s\sqrt{1+B_n^2}}>\frac{\rho}{s}.$$ Observe that due the remark before this theorem, $\alpha>0$ and $0<\rho<1$. Therefore, we deduce the inequality $$\alpha<B_n\rho\frac{1-\sqrt{1-\rho^2(1+B_n^2)}}{\sqrt{1-\rho^2(1+B_n^2)}},$$ or $$\label{equ}
B_n\rho>\alpha \frac{\sqrt{1-\rho^2(1+B_n^2)}}{1-\sqrt{1-\rho^2(1+B_n^2)}}.$$ The right-hand side of the inequality decreases with respect to $B_n>0$.
Set $\varepsilon=\frac{1+\rho^2}{2}$. If $\varepsilon <\rho^2(1+B_n^2)<1$, then immediately we have the inequality $B_n^2>\frac{1}{2}(\frac{1}{\rho^2}-1)>0$. If $0 <\rho^2(1+B_n^2)\leq \varepsilon$, then the inequality implies that $$B_n>\alpha\frac{\sqrt{1-\varepsilon}}{\rho(1-\sqrt{1-\varepsilon})}=\alpha\frac{\sqrt{1-\rho^2}}{\rho(\sqrt{2}-\sqrt{1-\rho^2})}>0.$$ Finally, we obtain $$B_n>\min\left\{\alpha\frac{\sqrt{1-\rho^2}}{\rho(\sqrt{2}-\sqrt{1-\rho^2})}, \sqrt{\frac{1}{2}(\frac{1}{\rho^2}-1)}\right\}\equiv b(\xi_1,\rho)>0.$$ This proves that all positive solutions to the equation must belong to the interval $(b(\xi_1,\rho), \sqrt{\frac{1}{\rho^2}-1})$, hence there are only finite number of such $B_n$. The same arguments are applied for negative values of $B_n$.
Let us discuss the limiting cases. If $\rho\to 0$ then the endpoint aproaches the vertical line. In this case the graph of the right hand side function in approaches the horizontal axis and the range of $|B|$ increases. If $\alpha\to 0$ then the end point aproaches the horizontal sphere $\mathbb S^2$, the number of geodesics is finite for any value $\alpha\neq 0$, but decreases.
This theorem reveals similarity of sub-Riemannian geodesics on the sphere with those for the Heisenberg group. The number of geodesics joining the origin with a point neither from the vertical axis nor from the horizontal plane is finite and approaching the vertical line becomes infinite.
[*Hyperspherical coordinates*]{}
Let us use the hyperspherical coordinates to find geodesics with non-constant velocity coordinates. $$\begin{aligned}
\label{hspc}
x_1+ix_2 & = & e^{i\xi_1}\cos\eta,\\
x_3+ix_4 & = & e^{i\xi_2}\sin\eta,\qquad \eta\in[0,\pi/2),\quad\xi_1,\xi_2\in[-\pi,\pi)\nonumber\end{aligned}$$
The horizontal coordinates are written as $$\begin{aligned}
a & = & \dot\eta\cos(\xi_1-\xi_2)+(\dot\xi_1+\dot\xi_2)\sin(\xi_1-\xi_2)\frac{\sin 2\eta}{2},\\
b & = & -\dot\eta\sin(\xi_1-\xi_2)+(\dot\xi_1+\dot\xi_2)\cos(\xi_1-\xi_2)\frac{\sin 2\eta}{2},\\
c & = & \dot\xi_1\cos^2\eta-\dot\xi_2\sin^2\eta.\end{aligned}$$
The horizontality condition in hyperspherical coordinates becomes $$\dot\xi_1\cos^2\eta-\dot\xi_2\sin^2\eta=0.$$
The horizontal sphere is obtained from the parametrization , if we set $\xi_1=0$, $\xi_2=\psi$, $\eta=s$. We get $$a^2+b^2=1=\dot\eta^2\quad\Longrightarrow\quad a=\cos\psi,\quad
b=\sin\psi.$$ The vertical line is obtained from the parametrization setting $\eta=0$, $\xi_1=s$.
Writing the vector fields $N,Z,X,Y$ in the hyperspherical coordinates we get $$N=-2\operatorname{cotan}2\eta\partial_{\eta},\quad
Z=\partial_{\xi_1}-\partial_{\xi_2},$$ $$X=\sin(\xi_1-\xi_2)\tan\eta\partial_{\xi_1}+\sin(\xi_1-\xi_2)\operatorname{cotan}\eta\partial_{\xi_2}
+2\cos(\xi_1-\xi_2)\partial_{\eta},$$ $$Y=\cos(\xi_1-\xi_2)\tan\eta\partial_{\xi_1}
+\cos(\xi_1-\xi_2)\operatorname{cotan}\eta\partial_{\xi_2}-2\sin(\xi_1-\xi_2)\partial_{\eta}.$$ In this parametrization the similarity with the Heisenberg group can be shown. The commutator of two horizontal vector fields $X,Y$ gives the constant vector field $Z$ which is orthogonal to the horizontal vector fields at each point of the manifold. In hyperspherical coordinates it is easy to see that the form $\omega=\cos^2\eta d\xi_1-\sin^2\eta d\xi_2$, that defines the horizontal distribution is contact because $$\omega\wedge d\omega=\sin (2\eta)\,d\eta\wedge d\xi_1\wedge
d\xi_2=2dV,$$ where $dV$ is the volume form. The sub-Laplacian is defined as $$\frac{1}{2}(X^2+Y^2)=\frac{1}{2}(\tan^2\eta\partial^2_{\xi_1}+\operatorname{cotan}^2\eta\partial^2_{\xi_2}
+4\partial^2_{\eta}+2\partial_{\xi_1}\partial_{\xi_2}).$$ The Hamiltonian becomes $$H(\xi_1,\xi_2,\eta,\psi_1,\psi_2,\theta)=\frac{1}{2}(\tan^2\eta\psi^2_1+\operatorname{cotan}^2\eta\psi^2_2
+4\theta^2+2\psi_1\psi_2),$$ and the corresponding Hamiltonian system is given as $$\begin{aligned}
\dot\xi_1 & = & \frac{\partial H}{\partial \psi_1}=\psi_1\tan^2\eta+\psi_2\\
\dot\xi_2 & = & \frac{\partial H}{\partial \psi_2}=\psi_2\operatorname{cotan}^2\eta+\psi_1\\
\dot\eta & = & \frac{\partial H}{\partial \theta}=4\theta\\
\dot\psi_1 & = & -\frac{\partial H}{\partial \xi_1}=0\\
\dot\psi_2 & = & -\frac{\partial H}{\partial \xi_2}=0\\
\dot\theta & = & -\frac{\partial H}{\partial \eta}=-\psi_1^2\frac{\tan\eta}{\cos^2\eta}
+\psi_2^2\frac{\operatorname{cotan}\eta}{\sin^2\eta}\\\end{aligned}$$ Let us solve this Hamiltonian system for the following initial data: $\eta(0)=0$, $\xi_1(0)=0$, $\xi_2(0)=0$, $\psi_1(0)=\psi_1$, $\psi_2(0)=\psi_2$, $\theta(0)=\frac{\dot\eta(0)}{4}=\theta_0$.
We see that $\psi_1$ and $\psi_2$ are constant. The horizontality condition at $(0,0,0)$ gives $\dot\xi(0)=0$ and the first equation of Hamiltonian system implies that $\psi_2=0$. If $\psi_1=0$, then $\dot\xi_1=\dot\xi_2=0$, $\dot\eta=4\theta_0$ and we get the variety of trivial geodesics up to the reparametrization $s\mapsto 4\theta_0 s$. To find other geodesics we suppose that $\psi_1\neq 0$ and $\dot\eta(0)>0$. This condition ensures us that the trajectory starting at the point $(0,0,0,)$ remains in the domain of parametrizaition locally in time. From the third and from the last equations of the Hamiltonian system we have $$\ddot\eta=-4\psi_1^2\frac{\sin\eta}{\cos^3\eta}\qquad\Rightarrow\qquad\dot\eta\,d\dot\eta=-4\psi_1^2\frac{\sin\eta}{\cos^3\eta}d\eta
\qquad\Rightarrow$$ $$\dot\eta^2=C-4\frac{\psi_1^2}{\cos^2\eta},$$ We observe that $C=\dot\eta^2(0)+4\psi_1^2>0$.
Continue to solve the Hamiltonian system finding $\eta(s)$. $$\frac{\cos\eta\,d\eta}{\sqrt{C\cos^2\eta-4\psi_1^2}}=ds.$$ Denote by $\sin\eta=p$. Then, $$\label{eq10}
\frac{dp}{\sqrt{-Cp^2+C-4\psi_1^2}}=ds.$$ Integrating from $0$ to $s$ we get $$s+K=\frac{1}{\sqrt{|C|}}\arcsin\Big(\sqrt{\frac{C}{C-4\psi_1^2}}\sin\eta(s)\Big),$$ where $K$ is found setting $s=0$ as $K=\frac{1}{\sqrt{|C|}}\arcsin 0=0$. We calculate $$\label{eta}
\sin^2\eta(s)=\frac{C-4\psi^2_1}{C}\sin^2(\sqrt C s).$$ From the Hamiltonian system we find $$\label{xi2}\xi_2(s)=\psi_1s,$$ and $$\dot\xi_1=\psi_1\frac{\sin^2\eta(s)}{1-\sin^2\eta(s)}
=\psi_1\frac{\sin^2(\sqrt C s)}{a+\sin^2(\sqrt C s)},\quad
a=\frac{C}{C-4\psi_1^2}.$$ It gives $$\label{xi1}\xi_1(s)=-\psi_1s+\frac{\psi_1}{2|\psi_1|}\arctan\Big[\frac{2|\psi_1|}{\sqrt
C}\tan(\sqrt C s)\Big].$$
Let us suppose for the moment that the geodesics are parametrized on the interval $[0,1]$. If the initial point and the finite point are on the vertical line: $\eta(0)=\eta(1)=0$, then $$0=\sin^2\eta(1)=\frac{C-4\psi^2_1}{C}\sin^2(\sqrt C)\quad\Rightarrow\quad
C=\pi^2n^2.$$ Since the value of $\xi_2$ on the vertical line is arbitrary, the values of $C$ and $\xi_1(1)$ give us the value of $\xi_2(1)$. Setting $C=\pi^2n^2$ in the equation for $\xi_1(1)$, we find $\psi_1=-\xi_1(1)$. Then $$\xi_2(1)=-\xi_1(1).$$ The finite point on the vertical line corresponds to the value of $\xi_1(1), \eta(1)=0$, and $\xi_2=-\xi_1(1)$.
We also note that the square of the velocity $|v|^2=\dot\eta^2(s)+(\dot\xi_1(s)+\dot\xi_2(s))^2\frac{\sin^2
2\eta(s)}{2}$ is constant along geodesics. Applying the initial condition $\eta(0)=0$, we get $$|v|^2=\dot\eta^2(0)=C-4\psi_1^2.$$ In the case when a geodesic ends at the vertical line at $\xi_1(1)$, its lengths is expressed as $$\sqrt{2}l_n=\sqrt{C-4\psi_1^2}=\sqrt{\pi^2n^2-4\xi_1^2(1)}.$$ We see that this result coincides with the result given by Theorem \[numbergeod1\] and we state it as follows.
\[numbergeod4\] Let $A$ be a point of the vertical line, i. e. $\xi_1$ is given and $\eta=0$. There are countably many geodesics $\gamma_n$ connecting $O=(0,0,0)$ and $A$, given parametrically as $$\begin{aligned}
\xi_1(s) &= & \xi_1s+\frac{1}{2}\arctan\Big(\frac{2\xi_1}{\pi n}\tan(\pi n s)\Big),\\
\xi_2(s) & = & -\xi_1 s,\\
\sin\eta(s) & = & \frac{s_n}{\pi n}\sin(\pi n s),\end{aligned}$$ where $\frac{1}{\sqrt{2}}s_n=\frac{1}{\sqrt{2}}\sqrt{\pi^2 n^2-4\xi_1^2}$ is the length of the geodesic $\gamma_n$, $n\in\mathbb N$.
Hopf fibration
==============
There is a close relation between the sub-Riemannian sphere $S^3$ and the Hopf fibration. Let $\mathbb S^2$ and $\mathbb S^3$ be unit 2-dimensional and 3-dimensional sphere respectively. We remind that the Hopf fibration is a principal circle bundle over two-sphere given by the map $h: \mathbb S^3\to \mathbb S^2$: $$h(x_1,x_2,x_3,x_4)=((x_1^2+x_2^2)-(x_3^2+x_4^2),2(x_1x_4+x_2x_3),2(x_2x_4-x_1x_3)).$$ Another way to define the Hopf fibration is to write $$h(q)=qiq^*\in \mathbb S^2,\quad q\in \mathbb S^3,\quad i=(0,1,0,0).$$ The fiber passing through the unity of the group $(1,0,0,0)$ has equation $(\cos\theta,\sin\theta,0,0)$, which as we see, coincides with the equation of the vertical line at this point. The sphere $\mathbb
S^2$ represents the horizontal “plane” sweep out by the geodesics with constant horizontal coordinates.
Let $Q\to M$ be a principle $G$-bundle with the horizontal distribution $\mathcal D$ on $Q$. A sub-Riemannian metric on $Q$ that has distribution $\mathcal D$ and it is invariant under the action of $G$ is called a metric of bundle type.
In our situation $\mathbb S^3\to \mathbb S^2$ is a principle $\mathbb S^1$-bundle given by the Hopf map. The sub-Riemannian metric on the distribution $\mathcal D=\operatorname{span}\{X,Y\}$ was defined as the restriction of the euclidean metric $\langle\cdot,\cdot\rangle$ from $\mathbb R^4$ and we used the same notation $\langle\cdot,\cdot\rangle$ for sub-Riemannian metric.
The sub-Riemannian metric $\langle\cdot,\cdot\rangle$ on $\mathbb S^3$ is a metric of bundle type.
The action of the group $\mathbb S^1$ on $q=(x_1,x_2,x_3,x_4)\in\mathbb S^3$ can be written as $q\circ e^{it}$, $e^{it}=(\cos t+i\sin t)\in \mathbb S^1$, $t\in [0,2\pi)$, where $\circ$ is the quaternion multiplication. If we write $q=(e^{i\xi_1}\cos\eta,e^{i\xi_2}\sin\eta)$, $\tilde q=q\circ e^{it}$, then $\tilde q=(e^{i(\xi_1+t)}\cos\eta,e^{i(\xi_2-t)}\sin\eta)$. In order to show that the metric $\langle\cdot,\cdot\rangle$ is of bundle type, we have to prove that the metric is invariant under the action of the group $\mathbb S^1$. The metric $\langle\cdot,\cdot\rangle$ at any $q$ is given by the matrix $$\left[\array{rrr}1 & 0 & 0
\\ 0 & \cos^2\eta & 0
\\
0 & 0 & \sin^2\eta
\endarray\right]$$ and it is easy to see that it is invariant under the action $\tilde q=q\circ e^{it}$.
We can formulate the results of Theorems \[numbergeod1\] and \[numbergeod4\] as an isoholonomic problem. First let us give some definitions. Let $c:[0,1]\to\mathbb S^2$ be a curve in $\mathbb S^2$ For a given point $x$ of the fiber $Q_{c(0)}$ at $c(0)$, let $\gamma$ be a horizontal lift of $c$ that starts at $q$ (it means that the projection of $\gamma$ under the Hopf map $h:\mathbb S^3\to\mathbb S^2$ coincides with $c$). The map $\Phi(c):Q_{c(0)}\to Q_{c(1)}$ sending $x=\gamma(0)$ to the endpoint $\gamma(1)$ of the horizontal lift is called [*parallel transport*]{} along $c$. The action of $\mathbb S^1$ takes horizontal curves to horizontal curves, so any two horizontal curves $\gamma_1$ and $\gamma_2$ of $c$ are related by $\gamma_1=\gamma_2g$ for some $g\in\mathbb S^1$. It follows that the action of $\mathbb S^1$ commutes with $\Phi(c)$, that is, $\Phi(c)(xg)=\big(\Phi(c)(x)\big)g$.
If $c$ is a closed loop, parallel transport $\Phi(c)$ maps the fiber $Q_{c(0)}$ onto itself. Fix a point $x_0\in Q_{c(0)}$. Since $\mathbb S^1$ acts transitively on $Q_{c(0)}$ we have $\phi(c)(x_0)=x_0l$ for some $l\in\mathbb S^1$. If we choose another point $y_0=x_0g\in Q_{c(0)}$, we get $$\Phi(c)(y_0)=\big(\Phi(c)(x_0)\big)(g)=x_0lg=y_0(g^{-1}lg).$$ The curve $c$ therefore, determines a conjugacy class in $\mathbb S^1$, called the holonomy class of $c$. The element $l\in\mathbb S^1$ for which $\Phi(c)(x_0)=x_0l$ is called the [*representative holonomy*]{} of $c$ with respect to $x_0$. The set of all such $l\in\mathcal S^1$ for $c$ running over all closed loops with $c(0)=c(1)=h(x_0)$ is a subgroup of $\mathbb S^1$ called the [*holonomy group*]{} of the distribution $\mathcal D$ at $x_0$.
Let us fix a representative holonomy $l\in\mathbb S^1$ and a point $x_0\in\mathbb S^3$, or equivalently the initial and the finite points $x_0$ and $x_1=x_0l$. The set of horizontal curves that join $x_0$ to $x_1$ is in one-to-one correspondence with the set of all closed loops on $\mathbb S^2$, based at $h(x_0)$, whose holonomy with respect to $x_0$ is $l$. Recall that the Riemannian length of a loop on $\mathbb S^2$ equals the sub-Riemannian length of its horizontal lift. Thus, the sub-Riemannian geodesic problem for geodesics with endpoints at the same fiber is equivalent to the following isoholonomic problem: [*Among all loops with a given holonomy, find the shortest*]{}.
If we take $x_0=(1,0,0,0)$, $l=e^{i\omega}$, $x_1=x_0l=(\cos \omega,\sin \omega,0,0)$, then Theorem \[numbergeod1\] says $$\begin{aligned}
\text{if}\quad \omega\in [0,\pi) & \quad\text{then the shortest loop has length}\quad s_1=\sqrt{\pi^2n^2-\omega^2},\\
\text{if}\quad \omega\in [\pi,2\pi) & \quad\text{then the shortest loop has length}\quad s_2=\sqrt{4\pi^2n^2-\omega^2}.\end{aligned}$$
[123]{}
O. Calin, D.-Ch. Chang, I. Markina, [*Sub-Riemannian geometry of the sphere $S^3$*]{}, to appear in Canadian J. Math.
W. L. Chow. [*Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung*]{}, Math. Ann., [**117**]{} (1939), 98-105.
A. Hurtado, C. Rosales, [*Area-stationary surfaces inside the sub-Riemannian three-sphere*]{}, Math. Ann. [**340**]{} (2008), 675–708.
W. Liu, H. J. Sussman, [*Shortest paths for sub-Riemannian metrics on rank-two distributions.*]{} Mem. Amer. Math. Soc. [**118**]{} (1995), no. 564, 104 pp.
D. W. Lyons. [*An elementary introduction to the Hopf fibration.*]{} Math. Mag. [**76**]{} (2003), no. 2, 87-98.
R. Montgomery, [Abnormal minimizers.]{} SIAM J. Control Optim. [**32**]{} (1994), no. 6, 1605–1620.
R. Montgomery, [*Survey of singular geodesics.*]{} Sub-Riemannian geometry, 325–339, Progr. Math., 144, Birkhäuser, Basel, 1996.
R. Montgomery, [*A tour of subriemannian geometries, their geodesics and applications.*]{} Mathematical Surveys and Monographs, [**91**]{}. American Mathematical Society, Providence, RI, 2002. 259 pp.
R.S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. [**24**]{} (1986) 221–263; Correction, ibid. [**30**]{} (1989) 595-596.
[^1]:
[^2]: The first author is partially supported by a research grant from the United State Army Research Office and a Hong Kong RGC competitive earmarked research grant $\#$600607. The second and the third authors have been supported by the grant of the Norwegian Research Council \#177355/V30, and by the European Science Foundation Research Networking Programme HCAA
|
---
abstract: 'The theory of controlled quantum open systems describes quantum systems interacting with quantum environments and influenced by external forces varying according to given algorithms. It is aimed, for instance, to model quantum devices which can find applications in the future technology based on quantum information processing. One of the main problems making difficult the practical implementations of quantum information theory is the fragility of quantum states under external perturbations. The aim of this note is to present the relevant results concerning ergodic properties of open quantum systems which are useful for the optimization of quantum devices and noise (errors) reduction. In particular we present mathematical characterization of the so-called “decoherence-free subspaces” for discrete and continuous-time quantum dynamical semigroups in terms of $C^*$-algebras and group representations. We analyze the non-Markovian models also, presenting the formulas for errors in the Born approximation. The obtained results are used to discuss the proposed different strategies of error reduction.'
address: 'Institute of Theoretical Physics and Astrophysics, University of Gdańsk, Wita Stwosza 57, PL 80-952 Gdańsk, Poland'
author:
- Robert Alicki
title: Controlled Quantum Open Systems
---
Introduction
============
There exists a widespread believe that sooner or later the technology of the future will employ typical quantum features of microscopic or mesoscopic systems like entanglement or superpositions of quantum states. The most prominent examples are the rapid development of the quantum information and computation theory \[1,2\] and the corresponding progress in experimental techniques. The later includes realization of quantum teleportation \[3\], monitoring of dissipation and decoherence for mesoscopic quantum systems \[4\], construction of quantum gates \[5\], etc.
The theory and experiment show that entanglement of separated systems and superpositions of well distinguishable states are very sensitive to perturbations by environment. The interaction with environment introduces noise (errors) which should be taken into account by designing any quantum devices and their operations. The appropriate theory has been developed in the 70-ties and 80-ties as the Quantum Theory of Open Systems (see \[6-11\] for the survey of this theory). The basic idea of this theory is the decomposition of the Universe into the open system $S$ and the environment (reservoir , heat bath) $R$ with their corresponding Hilbert spaces ${\cal H}_S$ and ${\cal H}_R$ respectively. Assuming that the interaction between $S$ and $R$ is weak we can always write, within a reasonable approximation, that for the initial moment $t_0 \equiv 0$ the initial state of the total system is given by the product formula $$\rho_{SR}(0) = \rho\otimes \omega_R
\eqno(1)$$ where $\rho$ is an arbitrarily prepared initial state of $S$ and $\omega_R$ is a stable reference state of $R$. Our interest is concentrated on the time evolution of $S$ for times $t \geq 0$ given by the reduced density matrix (we omit subscript $S$) $$\rho_t = {\rm Tr}_R\bigl( U_t \rho\otimes\omega_R U_t^*\bigr) = \Lambda_t(\rho)
\eqno(2)$$ where $\{U_t ; t\in{\bf R}\}$ is a family of unitary operators acting on ${\cal H}_{SR} =
{\cal H}_S \otimes {\cal H}_R$ and describing the reversible (Hamiltonian) evolution of the total system. ${\rm Tr}_R$ is a partial trace over ${\cal H}_R$. The dynamical maps $\{\Lambda_t ; t\geq 0\}$ , their properties and different approximation schemes for them (e.g. Markovian approximations) are the main topics of the Quantum Theory of Open Systems.
In order to deal with the problems posed by quantum information processing and quantum computing one has to extend the standard approach introducing, beside the open system $S$ and the reservoir $R$, the third element – the controler $C$. The role of $C$ is to prepare an initial state , to measure a final one and to control, as far as it is possible, the time evolution of $S$ by means of the time-dependent Hamiltonian $H_S(t)$. The presence of the time-dependent Hamiltonian makes, for example, the application of Markovian approximation often impossible and demands new mathematical techniques. Moreover, the physical structure of $C$, the resources used for state preparation and measurements must be taken into account in the estimation of efficiency of quantum devices. We shall call such systems Controlled Quantum Open Systems (CQOP).
One of the main problems in the theory of CQOP’s is to reduce errors due to the interaction of a quantum device with an environment. At least three elements of the presented scheme can be optimized: the choice of the subspace in ${\cal H}_S$ which supports the initial states of $S$, the trajectory $\{H_S (s) ; 0\leq s
\leq t\}$ which leads to a given unitary operation on ${\cal H}_S$ and the structure of the coupling of $S$ to $R$. In the recent literature one can find a number of proposals of such optimization schemes \[12-14\]. However, some of these ideas are essentially based on the independent derivations of the earlier results, some other employ oversimplified phenomenological models which should be scrutinized from the point of view of a more fundamental theory. Therefore, it seems to be useful to collect, improve and generalize the relevant results concerning mainly the ergodic properties of quantum open systems and discuss their consequences for the optimization schemes mentioned above.
In the Section 2 we present mathematical characterization of the subalgebras of observables which evolve according to reversible dynamics for a given irreversible quantum dynamical map or quantum dynamical semigroup. Such subalgebras correspond to “decoherence-free subspaces” in the language of modern quantum information theory. The general ideas are illustrated by examples of collective dissipation models in Section3. Section 4 is devoted to the derivation and discussion of the formula describing errors in CQOP in the non-Markovian weak coupling Born approximation.
Ergodic properties of quantum dynamical maps and semigroups
============================================================
In this Section we restrict ourselves in most cases to quantum open systems with finite-dimensional Hilbert space ${\cal H}_S = {\bf C}^n$. We discuss the irreversible dynamics either in terms of the single dynamical map $\Gamma $ and its iterations ${\Gamma^k , k=0,1,2,...}$ or the dynamical semigroup $\{T_t ; t\geq 0\}$ with the composition law $$T_t T_s = T_{t+s}\ , t,s \geq 0\ ,\ T_0\equiv 1\ .
\eqno(3)$$ Both $\Gamma$ and $T_t$ describe the dynamics in Heisenberg picture i.e. they act on a matrix algebra ($C^*$-algebra) ${\cal M}_n$ representing quantum observables of the system. In the following all results independent on the semigroup property (3) are formulated in terms of a single dynamical map $\Gamma$. The Schrödinger picture evolution is given by the dual map $\Gamma^*$ acting on ${\cal M}_n$ which is treated now as the smallest complex linear space containing density matrices (quantum states) of the system and satisfying $${\rm Tr}(A \Gamma(B))= {\rm Tr}(\Gamma^*(A)B)\ .
\eqno(4)$$ Dynamical maps must preserve positivity and normalization i.e. $$\Gamma (A^* A) \geq 0\ ,\ \Gamma^*(B^* B)\geq 0\ ,$$ $$\Gamma ({\bf 1})= {\bf 1}\ , \ {\rm Tr}\Gamma^*(B) = {\rm Tr}(B)
\eqno(5)$$ for all $A,B\in {\cal M}_n$.
Complete positivity
-------------------
Positivity preserving condition (5) is generally to weak for a mathematically and physically consistent theory. The minimal condition which must be imposed on dynamical maps to allow meaningful construction of joint dynamics for noninteracting composed systems is [*complete positivity*]{}. It means that for any $d = 1,2,3,...$, $\Gamma\otimes I_d$ is positive,where $I_d$ is an identity map acting on $d\times d$ matrices (i.e. trivial dynamical map on $d$-level quantum system). Any reduced dynamical map obtained from the formula (2) is completely positive and any completely positive dynamical map can be obtained from the reduced dynamics scheme. On the mathematical side completely positive maps on operator algebras were studied already in the 50-ties and the celebrated Stinespring representation \[15\] leads to a general form of completely positive dynamical map called often Kraus decomposition \[16\] $$\Gamma (A) = \sum_{\alpha}W_{\alpha}^* A W_{\alpha}
\eqno(6)$$ where $W_{\alpha}$ are bounded operators on ${\cal H}$ satisfying $\sum_{\alpha}W_{\alpha}^*W_{\alpha} = {\bf 1}$. The decomposition (6) is highly nonunique, in particular the sum over $\alpha$ can be replaced by an integral. If ${\cal H}$ is $n$-dimensional then complete positivity is equivalent to $n$-positivity and one can always find Kraus decomposition in terms of at most $n^2$ terms.
From now on by the [*quantum dynamical semigroup*]{} (in the Heisenberg picture) we mean a a one-parameter strongly continuous semigroup ${T_t
, t\geq 0}$ of completely positive unity preserving maps. For finite dimensional case we have always $$A_t = T_t(A)= e^{tL} (A)\ ,\ {d\over dt} A_t = L (A_t)
\eqno(7)$$ with the generator of the standard Lindblad, Gorini, Kossakowski and Sudarshan form \[17\] $$L(A) = i[H, A] + \sum_{j=1}^p V_j^* A V_j -{1\over 2}
\bigl\{\sum_{j=1}^p V_j^*V_j , A \bigr\}$$ $$= i[H, A] + {1\over 2}\sum_{j=1}^p\bigl( V_j^* [A , V_j] + [V_j^*, A]V_j \bigr)\ .
\eqno(8)$$ The choice of the Hamiltonian $H=H^*$ and the operators (matrices) $V_j$ is again not unique and the sum over $j$ can be replaced by an integral. In our finite dimensional case one can always find at most $p=n^2-1$ such $V_j$-s. To simplify the notation we put always $\hbar\equiv 1$ and $k_B \equiv 1$ to have the same units for energy, frequency and temperature.
The irreversible character of a dynamical map or semigroup is expressed in terms of the following inequalities valid for any $A\in {\cal M}_n$ $$\Gamma (A^* A)\geq \Gamma(A^*)\Gamma(A)\ --\ {\rm Kadison\ inequality}
\eqno(9)$$ $$L(A^* A)\geq L(A^*) A + A^* L(A)\ --\ {\rm dissipativity} .
\eqno(10)$$ Indeed, puting equality in (9) we obtain in our case the reversible dynamical map $A\mapsto U^* A U$ with a unitary $U$ while the equality in (10) implies the Hamiltonian derivation structure $L(A)= i[H,A]$. Both inequalities are the consequences of 2-positivity condition obviously satisfied for any completely positive dynamical map and proved very useful in the analysis of ergodic properties.
$C^*$-algebras and group representations
----------------------------------------
In this Section we briefly review the basic mathematical facts about finite dimensional $C^*$-algebras and representation theory of compact groups which will be used later on \[18\].
The full matrix algebra is an example of $C^*$- algebra i.e. a complex linear space with adjoint operation (hermitian conjugation) $A\mapsto A^*$, multiplication $AB$ and complete with respect to the norm satisfying $\|AA^*\|= \|A\|^2$. We are interested in finite-dimensional $C^*$-algebras which can always be seen as $C^*$-subalgebras of full matrix algebras i.e. linear subspaces of ${\cal M}_n$, closed with respect to multiplication of matrices and hermitian conjugation and containing the unit matrix [**1**]{}.
Any such $C^*$-subalgebra ${\cal N}$ can be represented as the algebra of linear maps on the finite dimensional Hilbert space in the following way. The Hilbert space is decomposed as $${\bf C}^n = \bigoplus_{j=1}^m {\bf C}^{n_j}\otimes {\bf C}^{d_j}\ ,\ \ \sum_{j=1}^m n_jd_j = n
\eqno(11)$$ where $n_j$ is the dimension of the irreducible representation of ${\cal N}$ and $d_j$ its multiplicity. Then any element $A\in {\cal N}$ can be written in the form $$A = \bigoplus_{j=1}^m A_j\otimes {\bf 1}_{d_j}
\eqno(12)$$ where $A_j \in {\cal M}_{n_j}$ and ${\bf 1}_{d_j}$ is a unit matrix acting on ${\bf C}^{d_j}$.
In a general algebraic framework the physical symmetry or reversible dynamical map described in the Heisenberg picture is given in terms of an [*automorphism*]{} of the corresponding $C^*$-algebra, i.e. a linear map $A\mapsto{\cal U}(A)$ satisfying ${\cal U}(A^*) = {\cal U}(A)^*$, ${\cal U}(AB) = {\cal U}(A){\cal U}(B)$ , ${\cal U}({\bf 1})= {\bf 1}$ and reversible. For an automorphism of a finite dimensional algebra ${\cal N}\subset {\cal M}_n$ reversibility follows from the former conditions and moreover there exists a unitary matrix $U\in {\cal M}_n$ such that ${\cal U}(A) = U^* A U$. The group of all automorphisms of ${\cal N}$ is denoted by $Aut({\cal N})$. A subgroup $Aut_0({\cal N})\subset
Aut({\cal N})$ contains all [*inner automorphisms*]{} given in terms of unitaries $U\in {\cal N}$. For our finite dimensional case $Aut_0({\cal N})$ coincides with the [*path connected component of identity*]{} of $Aut({\cal N})$. In order to illustrate the fact, that even in the finite dimensional case not all automorphisms are inner, consider the abelian subalgebra of ${\cal M}_2$ of diagonal matrices and the “flip” map $
\pmatrix{a & 0 \cr
0 & b \cr} \mapsto \pmatrix{b & 0 \cr
0 & a \cr}\ .
$ This automorphism can be writen as $A\mapsto \sigma_x A \sigma_x$, where the (unitary) Pauli matrix $\sigma_x$ is neither diagonal nor cannot be continuosly transformed into identity.
We denote by ${\cal A}'= \{X\in {\cal M}_n ; [X,{\cal A}]=0\}$ the [*commutant*]{} of the subset ${\cal A}\subset{\cal M}_n$. If ${\cal A}$ is self-adjoint, i.e. $A\in{\cal A}\Rightarrow
A^*\in{\cal A}$, then the commutant ${\cal A}'$ is a unital $C^*$-subalgebra of ${\cal M}_n$. For a subalgebra ${\cal N}$ we have $({\cal N}')'={\cal N}$. Obviously if ${\cal A}\subset
{\cal B}$ then ${\cal B}'\subset {\cal A}'$. Denoting by $Alg({\cal A})$ the smallest $C^*$- algebra containing ${\cal A}$ we have ${\cal A}' = Alg({\cal A})'$. For an algebra ${\cal N} = \bigoplus_{j=1}^p {\cal M}_{n_j}\otimes {\bf 1}_{dj}$ (see (12)) its commutant ${\cal N}' = \bigoplus_{j=1}^p {\bf 1}_{n_j}\otimes{\cal M}_{d_j}$.
The theory of group representations provides natural examples of finite dimensional $C^*$-algebras. Take a finite-dimensional unitary representation ${\cal R}: G\mapsto
{\cal M}_n$; $g\mapsto R(g)$ of the compact group $G$ (discrete or Lie group). Such representation can be always decomposed into irreducible representations ${\cal R}_j$ according to (11). Any subspace ${\bf C}^{n_j}$ is a carrier of an irreducible representation and appears $d_j$-times in the decomposition. Hence any unitary $R(g)\in {\cal R}(G)$ is decomposed as $R(g) = \bigoplus_{j=1}^p R_j(g)\otimes {\bf 1}_{d_j}$. The $C^*$-algebra generated by ${\cal R}(G)$ will be denoted by $Alg({\cal R}(G))$ and consists of elements of the form $A = \sum_g \alpha_g R(g)\ ,\ \alpha_g\in{\bf C}$ which can be written as (12).
Decoherence-free subalgebras
-----------------------------
For an open system with a time-independent Hamiltonian interacting with a stable environment being at the equilibrium state (heat bath) we expect that for an arbitrary initial state the system returns to its thermal equilibrium state determined by the temperature of environment. In mathematical terms it means that $\Gamma^*$ (or $T^*_t$) possesses a unique stationary state $\rho_{eq}$ and $\lim_{n\to\infty}\Gamma^n(A) = {\rm Tr}(\rho_{eq} A){\bf 1}$ ($\lim_{t\to\infty}T_t(A) = {\rm Tr}(\rho_{eq} A){\bf 1})$. We say that the dynamics $\Gamma$ ($T_t$) is [*ergodic*]{} in this case. The relaxation process can be very complex and involving many time scales of very different magnitude, some of them being much longer than the time scale relevant for the operation of our quantum device. Therefore, it is often very useful to study simplified approximative nonergodic dynamics which take into account only the fastest relaxation processes. For such a dynamics we are interested in observables and states which evolve according to the reversible (unitary , Hamiltonian) evolution i.e. in a “decoherence-free” way. We shall prove that such observables are hermitian elements of a certain unital $C^*$- subalgebra of ${\cal M}_n$. In practice, the manifest form of this subalgebra is difficult to find and quite often we must be satisfied with a detailed knowledge of its certain nontrivial subalgebras. It turns out that the symmetries of the system can be very useful in this context.
The mathematical techniques used here are based on the earlier results on the ergodic theory of quantum dynamical semigroups \[19,20\]. Some partial results of this type can be found in the recent literature also \[21\].
A minimal condition which should be satisfied by a decoherence-free observable $A$ with respect to the dynamical map $\Gamma$ is the [*lack of dissipation*]{} expressed by $$\Gamma(A^2) = \bigl(\Gamma(A)\bigr)^2\ ,\ A=A^*\ .
\eqno(13)$$ We formulate this condition in terms of a [*dissipation function*]{} $D_{\Gamma}(.,.)$ from ${\cal M}_n \times {\cal M}_n$ to ${\cal M}_n$ defined by $$D_{\Gamma}(A,B) = \Gamma (A^* B) - \Gamma (A^*)\Gamma (B)
\eqno(14)$$ Denote by ${\cal N}_{\bf R}(\Gamma)$ the following subset of hermitian matrices (observables) in ${\cal M}_n$ and by ${\cal N}(\Gamma)$ its complexification $${\cal N}_{\bf R}(\Gamma) = \{ A=A^*\in {\cal M}_n; D_{\Gamma} (A,A) =0
\}\ ,$$ $${\cal N}(\Gamma)=\{ A+iB ; A,B\in{\cal N}_{\bf R}(\Gamma)\}\ .
\eqno(15)$$ We shall call ${\cal N}(\Gamma)$ the [*decoherence free subalgebra*]{} of $\Gamma$ and this name is justified by the following theorem
[**Theorem 1**]{}
1\) ${\cal N}(\Gamma)$ is a unital $C^*$-subalgebra of ${\cal M}_n$
2)$\Gamma$ restricted to ${\cal N}(\Gamma)$ is a unity preserving homomorhism, i.e. for any $A , B\in {\cal N}(\Gamma)$ $$\Gamma (AB) = \Gamma(A)\Gamma(B) \ ,\ \Gamma ({\bf 1})={\bf 1}
\eqno(16)$$ and moreover $\Gamma\ {\rm is \ invertible\ on}$ $\Gamma\bigl({\cal N}(\Gamma)\bigr)$.
3\) There exists a unitary $U_{\Gamma}\in {\cal M}_n$ such that $$\Gamma (A) = U^*_{\Gamma} A U_{\Gamma}\ ,\ {\rm for\ all}\ A \in {\cal N}(\Gamma) .
\eqno(17)$$
[**Proof**]{} We show first that for any $A\in {\cal N}_{\bf R}(\Gamma)$ and arbitrary $B\in{\cal M}_n$ $$\Gamma (AB)= \Gamma(A)\Gamma(B)\ ,\ \Gamma (BA)= \Gamma(B)\Gamma(A)\ .
\eqno(18)$$ The relations (18) follow from the Kadison inequality (9) applied to $A +\lambda B$ with an arbitrary $\lambda\in{\bf C}$. Namely, $$0\leq D_{\Gamma}(A+\lambda B,A+\lambda B)$$ $$= \Gamma (A^2) + |\lambda|^2 \Gamma(B^*B)+\lambda \Gamma (AB)
+ {\bar\lambda} \Gamma (B^*A)$$ $$- \Gamma(A)^2- |\lambda|^2 \Gamma(B^*)\Gamma(B)
- \lambda \Gamma (A)\Gamma(B) - {\bar\lambda} \Gamma (B^*)\Gamma(A)$$ $$= |\lambda|^2\bigl(\Gamma(B^* B)-\Gamma(B^*)\Gamma(B)\bigr)$$ $$+ \lambda\bigl(\Gamma(AB)-\Gamma(A)\Gamma(B)\bigr)
+ {\bar\lambda}\bigl(\Gamma(B^*A)-\Gamma(B^*)\Gamma(A)\bigr)\ .
\eqno(19)$$ Dividing both sides of the inequality (19) by $|\lambda|$ and taking $|\lambda|\to 0$ we have for any $z\in {\bf C}, |z|=1$ $$z\bigl(\Gamma(AB)-\Gamma(A)\Gamma(B)\bigr)
+ {\bar z}\bigl(\Gamma(B^*A)-\Gamma(B^*)\Gamma(A)\bigr)\geq 0
\eqno(20)$$ what implies (18).
It is now a simple exercise to prove using (18) that if $\ A,B\in {\cal N}(\Gamma)$, ${\rm then}\ A^*$,$ B^*$ , $A+B$, $AB\ \in {\cal N}(\Gamma)$. Hence ${\cal N}(\Gamma)$ is a unital $C^*$-subalgebra and from the general results on finite dimensional $C^*$-algebras reviewed in Section 2.2 it follows that (17) holds also.
Theorem 1 gives us the general mathematical structure of observables evolving in a reversible way under a single step dynamics. The physical meaning of the algebra ${\cal N}(\Gamma)$ is illustrated by the most elementary physical process consisting of the following stages:
1\) preparation of the initial state $\rho$,
2\) irreversible evolution governed by the dynamical map $\Gamma$,
3\) measurement of a given observable with a spectral resolution $$A=\sum_k a_k P_k\ .$$
The probability of obtaining the value $a_k$ is equal to $ p_k = {\rm Tr}(\rho \Gamma(P_k))$ and if $A\in{\cal N}(\Gamma)$ then $p_k = {\rm Tr}(\rho U_{\Gamma}^*P_k U_{\Gamma})$. In the language of quantum information theory it means that if we choose decoherence-free [*decoding observables*]{} then the noisy quantum channel is equivalent to a certain noiseless one.
Analogical structures can be defined for the discrete (continuous) time dynamical semigroups $\{\Gamma^k , k= 0,1,2,...\}$ ($\{T_t ,t\geq 0\}$) to obtain the global decoherence-free subalgebras which contain observables evolving in a reversible way for all times $${\cal N}(\Gamma^{(\cdot)})= \bigcap_{k=1}^{\infty} {\cal N}(\Gamma^k)\ ,\
{\cal N}(T_{(\cdot)})= \bigcap_{t\geq 0} {\cal N}(T_t)
\eqno(21)$$ respectively.
As a direct consequence of this definition, the Theorem 1 and the general results of Section 2.2 we obtain
[**Collorary**]{} ${\cal N}(\Gamma^{(\cdot)})$ and ${\cal N}(T_{(\cdot)})$ are globaly invariant under the action of $\Gamma^k $ and $T_t $ respectively. The both semigroups restricted to decoherence-free algebras produce groups of unitary automorhisms $\{U_{\Gamma}^k ,
k\in {\bf Z}\}$ and $\{U_t ; t\in{\bf R}\}$. Moreover if for the discrete case $U_{\Gamma}$ can be continuosly transformed into identity then $U_{\Gamma}
\in {\cal N}(\Gamma^{(\cdot)})$. Analogical result holds automatically for continuous time semigroup, i.e. $U_t\in {\cal N}(T^{(\cdot)})$.
For the future applications it is useful to define a set of [*fixed points*]{} of $\Gamma$ as ${\cal F}(\Gamma)=\{A\in {\cal M}_n ; \Gamma(A) = A\}$. In general, ${\cal F}(\Gamma)$ is a self-adjoint linear subspace of ${\cal M}_n$ but under additional assumptions it becomes $C^*$-algebra. For the dynamical semigroup ${\cal F}(T_{(\cdot)}) = Ker (L)=
\{A\in{\cal M}_n ; L(A)=0\}$.
The algebras ${\cal N}(\Gamma)$, ${\cal N}(\Gamma^{(\cdot)})$ and ${\cal N}(T_{(\cdot)})$ are the largest ones with the decoherence-free property but their definition is not constructive and often difficult to apply. Therefore, it is sometimes convenient to consider smaller subalgebras which are easier to construct explicitely. Theorems 2,3 provide us with the appropriate examples.
[**Theorem 2**]{}
Assume that $\Gamma = {\cal U}\Gamma_D $ where ${\cal U}(X) = U^*XU$ is a unitary automorphism and $$\Gamma_D(X) = \sum_{\alpha =1}^r W_{\alpha}^* X W_{\alpha}\ .
\eqno(22)$$ Then the commutants defined below are $C^*$-subalgebras and $${\cal W}_1 = \{W_{\alpha}, W_{\alpha}^*; \alpha = 1,2,...,r\}' \subset {\cal W}_2 =
\{W_{\alpha}W_{\beta}^*;\alpha,\beta = 1,2,...,r\}'\subset {\cal N}(\Gamma)\ .
\eqno(23)$$ If additionally ${\cal U}$ commutes with $\Gamma_D$ then $${\cal W}_1\subset {\cal W}_3 = \{W_{\alpha}W_{\beta} , W_{\alpha}W_{\beta}^* ,
W_{\alpha}^*W_{\beta}^*;\alpha,\beta=1,2,...,r\}'\subset {\cal N}(\Gamma^{(\cdot)})\ .
\eqno(24)$$
For continuous time dynamical semigroups we have:
[**Theorem 3**]{}
Assume that the semigroup generator (8) decomposed as $L = L_H +L_D , L_H = i[H,\cdot]$ satisfies $$L_H L_D = L_D L_H\ .$$ Then the commutant defined below is a $C^*$-subalgebra and $$\{V_j , V_j^*;j=1,2,...,p\}'\subset {\cal N}(T_{(\cdot)})\ .$$
[**Proof of Theorems 2 and 3**]{} From the definition of the commutant ${\cal W}_1
\subset {\cal W}_2 \subset {\cal N}(\Gamma_D)= {\cal N}(\Gamma)$.
Due to the condition ${\cal U}\Gamma_D = \Gamma_D {\cal U}$, ${\cal N}(\Gamma_D^{(\cdot)}) = {\cal N}(\Gamma^{(\cdot)})$. Finally, if $X=X^*$ commutes with all $W_{\alpha}W_{\beta}$ , $ W_{\alpha}W_{\beta}^*$ , $W_{\alpha}^*W_{\beta}^*, \alpha,\beta = 1,2,...,r$ then $\Gamma^k(X^2) = \bigl(\Gamma^k (X)\bigr)^2 $ for all $k=0,1,2,...$
Similarly, for continuous time semigroups the commutation of the Hamiltonian part $L_H$ with $L_D$ implies that the elements of the commutant $\{V_j ,V_j;j=1,2,...,p\}'$, which lie in a kernel of $L_D$, are decoherence-free.
[**Remark**]{} The decomposition of the dynamical map into reversible and dissipative parts in Theorems 2,3 is not unique and can be optimized to obtain the largest commutants.
To apply Theorems 2, 3 it is often convenient to use the results of the theory of group representations. Assume that the matrices $W_{\alpha}$ belong to $Alg({\cal R}(G))\subset {\cal M}_n$ of a certain unitary representation of a finite or compact Lie group $G$. In this case the Hilbert space is decomposed into carriers of irreducible representations of $G$ as in (11) where $n_j$ are dimensions of irreducible representations and $d_j$ their multiplicities. The commutant $$\{{\cal R}(G)\}'= \bigoplus_{j=1}^m {\bf 1}_{n_j}\otimes{\cal M}_{d_j}\subset {\cal W}_1
\eqno(25)$$ in this case.
Similarly, for quantum dynamical semigroup satisfying the assumptions of the Theorem 3 the matrices $V_j , V_j^*, j=1,2,...,p$ generate a certain Lie algebra in ${\cal M}_n$ and hence the representation ${\cal R}(G)$ of the corresponding group $G$. In this case the commutant (25) coincides with $\{V_j , V_j^*;j=1,2,...,p\}'$.
Another application of the group theory is possible when the finite or compact Lie group $G$ is a [*symmetry group for the dynamical map*]{} $\Gamma$ i.e. $${\cal R}_g \Gamma {\cal R}_g^{-1} = \Gamma\ ,\ {\cal R}_g (X) = R(g)^* X R(g)
\eqno(26)$$ for any unitary $R(g)\in{\cal R}(G), g\in G$. We call the condition (26) [*global invariance*]{} of the dynamical map $\Gamma$ with respect to the group $G$. Using the definition of ${\cal N}(\Gamma^{(\cdot)})$ and then applying (26) we can prove that from $A\in {\cal N}(\Gamma^{(\cdot)})$ it follows that ${\cal R}_g (A)\in {\cal N}(\Gamma^{(\cdot)})$ too. Therefore, the invariance holds $${\cal R}_g \bigl({\cal N}(\Gamma^{(\cdot)})\bigr) = {\cal N}(\Gamma^{(\cdot)})\ .
\eqno(27)$$ If $G$ is a connected compact Lie group then any $R(g)$ can be continuosly connected to identity and hence by general theorems from Section 2.2 ${\cal R}_g$ is an inner automorhism of ${\cal N}(\Gamma^{(\cdot)})$, i.e. $R(g) \in {\cal N}(\Gamma^{(\cdot)})$. Hence the $C^*$-algebra generated by ${\cal R}(G)$ is included in in the decoherence-free algebra ${\cal N}(\Gamma^{(\cdot)})$. The situation for discrete symmetry may be quite different due to “incidental degeneracies” of $\Gamma$ which admit larger group algebras. All that is true for the dynamical semigroup $\{T_t ; t\geq0\}$ also and will be illustrated by examples in Section 3.
The stronger invariance conditions with respect to the symmetry group called [*local invariance*]{} can be formulated for the dynamical map $\Gamma ={\cal U}\Gamma_D
= \Gamma_D {\cal U}$ as in the Theorem 2. It reads for all $\alpha$ and $g\in G$ $$R(g)^*W_{\alpha} R(g) = W_{\alpha}
\eqno(28)$$ Therefore the algebra generated by the group representation $Alg ({\cal R}(G))\subset
{\cal N}(\Gamma^{(\cdot)})$. Similarly, for quantum dynamical semigroup as in the Theorem 3 the condition $$R(g)^*V_{\alpha} R(g) = V_{\alpha}
\eqno(29)$$ implies $Alg ({\cal R}(G))\subset {\cal N}(T_{(\cdot)})= Ker (L_D)$.
Summarizing, global invariance (26) for connected compact Lie group implies $Alg ({\cal R}(G))\subset {\cal N}(\Gamma^{(\cdot)})$ while local invariance (28) or (29) implies for any compact group $G$ that $Alg ({\cal R}(G))\subset {\cal N}(\Gamma^{(\cdot)})$ or $Alg ({\cal R}(G))\subset {\cal N}(T_{(\cdot)})$ respectively.
The concrete examples of semigroups and their decoherence-free observables obtained using the methods proposed above will be discussed in Section 3.
Limited relaxation
-------------------
We expect that the open system governed by a non-ergodic dynamics shows a behaviour which can be called [*limited relaxation*]{}. It means that in the Heisenberg picture any initial observable becomes for long times decoherence-free and the set of decoherence-free observables is nontrivial. We are able to prove this property under additional assumptions which are typically fulfilled for open quantum systems weakly coupled to heat baths.
We begin with the dynamical map $\Gamma$ satisfying the following conditions:
1\) There exist a [*faithful*]{} (i.e. given by a strictly positive density matrix) stationary state $\sigma$ , $\Gamma^*(\sigma) =\sigma$.
2\) [*Detailed balance*]{} condition with respect to $\sigma$ holds, i.e. $\Gamma = {\cal U}\Gamma_D = \Gamma_D {\cal U}$ where ${\cal U}(X) = U^*XU$ is a unitary automorphism , ${\cal U}(\sigma) = \sigma$ and the dynamical map $\Gamma_D$ is a hermitian operator on the Hilbert space $({\cal M}_n; <\cdot , \cdot>_{\sigma})$ equipped with the scalar product $$<A,B>_{\sigma} = {\rm Tr}(\sigma A^* B)\ ,\ A,B\in {\cal M}_n
\eqno(30)$$ and the norm $\|A\|_{\sigma}= <A,A>^{1/2}$ . This Hilbert space is often called [*Liouville space*]{}. More explicitely ${\cal U}$ is unitary, $\Gamma_D$ is hermitian and ${\cal U}\Gamma_D =
\Gamma_D{\cal U}$ is a [*normal contraction*]{} on the Liouville space$({\cal M}_n; <\cdot , \cdot>_{\sigma})$.
The existence of a stationary state , generally non unique, is quaranteed by the finite-dimensionality of the system’s Hilbert space but its faithfulness is a nontrivial assumption.
[**Lemma 1**]{}
If the dynamical map $\Gamma$ possesses a faithful stationary state $\sigma$ then the fixed point set ${\cal F}(\Gamma) $ is a $C^*$-subalgebra of ${\cal M}_n$ and ${\cal F}(\Gamma)\subset
{\cal N}(\Gamma^{(\cdot)})$.
[**Proof**]{} Take $A=A^*\in {\cal F}(\Gamma)$ then, $ A^2 = \bigl(\Gamma(A)\bigr)^2$ and hence $$0= {\rm Tr}\bigl( \sigma [A^2 - \Gamma(A)\Gamma(A)]\bigr) =
{\rm Tr}\bigl( \sigma (\Gamma(A^2) - \Gamma(A)\Gamma(A))\bigr) \ .
\eqno(31)$$ As $\Gamma(A^2) - \Gamma(A)\Gamma(A)\geq 0$ and $\sigma$ is faithful then (31) is equivalent to $\Gamma(A^2)= \bigl(\Gamma(A)\bigr)^2$. It follows that $A\in{\cal N}(\Gamma)$. If $A=A^*\in {\cal N}(\Gamma)$ then we can reverse the reasoning to prove that $A\in {\cal F}(\Gamma)$. For a non-hermitian $A$ we can use the decomposition $A= A_1 +iA_2$ into hermitian ones.
We shall denote by $P_{\Gamma}$ the orthogonal projector acting on the Liouville space $({\cal M}_n; <\cdot,\cdot>_{\sigma})$ and projecting on the subspace ${\cal N}(\Gamma^{(\cdot)})$.
Limited relaxation process is characterized by the following theorem.
[**Theorem 4**]{}
Consider a dynamical map $\Gamma$ satisfying the detailed balance condition of above. Then ${\cal N}(\Gamma^{(\cdot)}) = {\cal F}(\Gamma_D)$ and $$\lim_{k\to\infty}\|\Gamma^k (A) - {\cal U}^k(P_{\Gamma}A)\| = 0\ ,\ {\rm for\ all}
\ A\in{\cal M}_n\ .
\eqno(32)$$
[**Proof**]{} From $\Gamma^k = {\cal U}^k\Gamma_D^k$ it follows that ${\cal N}(\Gamma^{(\cdot)})
={\cal N}(\Gamma_D^{(\cdot)})$. For any $A\in{\cal N}(\Gamma^{(\cdot)})$ we have $\|\Gamma_D^k(A)\|_{\sigma} = \|A\|_{\sigma}$.Therefore, because $\Gamma_D$ is a hermitian contraction on the Liouville space, ${\cal N}(\Gamma_D^{(\cdot)})={\cal F}(\Gamma_D)$.
For the dynamical semigroup with the generator $L= i[H,\cdot]+L_D$ the [*detailed balance condition*]{} means that the Hamiltonian part $L_H$ commutes with $L_D$, $[H,\sigma] = 0$ and $L_D$ is a hermitian operator on the Liouville space $({\cal M}_n; <\cdot,\cdot>_{\sigma})$ \[7\]. Introducing a kernel ${\rm Ker}(L_D) = \{A\in{\cal M}_n; L_D(A)=0\}$, the orthogonal projection on ${\rm Ker}(L_D)$ denoted by $P_{L_D}$ and applying the same idea of the proof we obtain the following analog of the Theorem 4.
[**Theorem 5**]{}
Consider the dynamical semigroup $T_t$ with the generator satisfying detailed balance condition of above. Then ${\cal N}(T_{(\cdot)}) = {\rm Ker}(L_D)$ and $$\lim_{t\to\infty}\|T_t(A) - e^{itH}P_{L_D}(A)e^{-itH}\| = 0\ ,\ {\rm for\ all}
\ A\in{\cal M}_n\ .$$
[**Example**]{}
The particular case of detailed balance generator with the stationary Gibbs state $$\sigma = Z^{-1}e^{-H/T}
\eqno(33)$$ leads to a very specific structure of $L_D$ \[7\] $$L_D(A) =\sum_{j;\omega_j\geq 0}\Bigl( V_j^* A V_j -{1\over 2}
\{V_j^*V_j , A \}+ e^{-\omega_j/T}\bigl(V_j A V_j^* -{1\over 2}
\{V_jV_j^* , A \}\bigr)\Bigr)
\eqno(34)$$ with $[H,V_j] = \omega_j V_j$ . Simple calculation involving the dissipativity condition (10) and hermicity of $L_D$ on the Liouville space shows that in this case $$0\leq {\rm Tr}\bigl(\sigma[L(A^*A) - L(A^*)A - A^*L(A)]\bigr)= - <A ,L_D(A)>_{\sigma}$$ $$= \sum_{j;\omega_j\geq 0}\Bigl(< [V_j, A], [ V_j, A]>_{\sigma}+
e^{-\omega_j/T} < [V_j^* , A], [V_j^* , A ]>_{\sigma}\Bigr)\ .
\eqno(35)$$ Therefore, the commutant $\{V_j,V_j^*;\omega_j \geq 0\}'= {\rm Ker}(L_D) ={\cal N}(T_{(\cdot)})$ .
Examples
========
In this Section we give examples of physical systems governed by nonergodic quantum dynamical semigroups and illustrate the different methods of searching for decoherence-free observables introduced in Section 2.
N-particle systems with permutation invariance
----------------------------------------------
Consider a system which consists of $N$ identical particles each of them is described by a finite dimensional Hilbert space ${\cal H}^{(j)}\equiv {\bf C}^d ; j=1,2,...,N$. There are two possibilities of constructing a quantum dynamical semigroup generator invariant with respect to the permutation group $S_N$. The first one corresponds to particles interacting with identical “private” reservoirs and is given by the Master equation in the Schrödinger picture for the $N$-particle density matrix $\rho_t$ $${d\over dt}\rho_t = -i [H ,\rho_t] + \sum_{m=1}^N L^*_m (\rho_t)
\eqno(36)$$ where $H$ is a $N$- particle permutation invariant Hamiltonian and $L^*_m$ are identical copies of a single-particle dynamical semigroup generator.
The second example corresponds to the collective coupling of $N$-particle system to a single “common” reservoir $${d\over dt}\rho_t = -i [H ,\rho_t] + {1\over 2}\sum_{\alpha}\bigl([V_{\alpha},\rho_t V_{\alpha}]
+ [V_{\alpha}\rho_t, V_{\alpha}]\bigr)
\eqno(37)$$ where $V_{\alpha}= \sum_{m=1}^N v^{(m)}_{\alpha}$ with identical copies of a single particle operator $v_{\alpha}$.
The Hilbert space of the system possesses the structure of $N$-fold tensor product $\bigotimes_N {\bf C}^d$ with the basis $\{e_{j_1,j_2,...j_N} = e_{j_1}\otimes e_{j_2}\otimes
\cdots\otimes e_{j_N}\}$. Any permutation $\pi\in S_N$ is represented by the operator $R(\pi)$ defined in terms of the basis as $$R(\pi)e_{j_1,j_2,...j_N} = e_{\pi(j_1),\pi(j_2),...\pi(j_N)}\ .
\eqno(38)$$ It is well-known that this representation is highly reducible and therefore $ Alg({\cal R}(S_N))$ is nontrivial. The explicite construction of the decomposition (11)(12) is given in terms of Young tables and described in the texbooks on group theory \[22\].
The semigroup generated by the master equation(36) is globally invariant with respect to $S_N$ but because $S_N$ is a discrete group according to the results of Section 2.3 we can have no nontrivial decoherence-free observables i.e. ${\cal N}(T_{(\cdot)})= {\bf C}$. Indeed, if a single-particle semigroup generated by $L^*_m$ is ergodic and the Hamiltonian $H=0$, then the whole $N$-particle dynamics governed by (36) is ergodic too.
For the case of the dynamics goverened by the master equation (37) the situation is different. The corresponding semigroup is locally invariant with respect to $S_N$ and therefore according to the results of Section 2.3 $ Alg({\cal R}(S_N))\subset {\cal N}(T_{(\cdot)})$.
Superradiance model with $SU(2)$ symmetry
-----------------------------------------
The superradiance \[23\] and subradiance phenomena \[24\] are the first examples of collective dissipative phenomena in quantum systems which leads to a nonergodic behaviour (on a certain time scale) due to (approximative) permutation symmetry of the interaction of atoms with the electromagnetic field. In the simplest case of 2-level atoms in open space and for the temperature $T=0$, we obtain the master equation of the form (37) with $$H= {1\over 2}\omega \sum_{m=1}^N \sigma_z^{(m)}\ ,\ v_{\alpha} =
{1\over 2}\sqrt{\gamma}(\sigma_x + i\sigma_y)
\eqno(39)$$ where $\omega$ is an atomic frequency, $\gamma$ is a radiation damping constant and $\sigma_j^{(m)}, j= x,y,z,$ are Pauli matrices for the $m$-th atom.
In order to find decoherence-free observables we can use either the local symmetry of the master equation with respect to the permutation group $S_N$ or the fact that the collective operators $\sum_{m=1}^N \sigma_j^{(m)}$ which appear in the semigroup generator define a reducible $N$-fold product representation of the group $SU(2)$ on $\bigotimes_N {\bf C}^2$.Then we obtain $$Alg ({\cal R}(S_N))\subset Alg({\cal R}(SU(2)))\subset {\cal N}(T_{(\cdot)})
\eqno(40)$$ where ${\cal R}(S_N)$ is a standard representation of $S_N$ defined by (38) and ${\cal R}(SU(2))$ is the mentiond above product representation of $SU(2)$. For our case of $T=0$ the explicite structure of ${\cal N}(T_{(\cdot)})$ is unknown while for $T>0$ we can use detailed balance condition and the Theorem 5 to characterize ${\cal N}(T_{(\cdot)})$ \[20\].
Non-markovian controlled open systems
=====================================
The analysis of the previous Section has a rather mathematical and phenomenological character and its most developed part is applicable to dynamical semigroups (discrete or continuous) and hence involve Markovian assumption or at least stationary external conditions. For a generic controlled system we expect that the external conditions like, for instance, external electromagnetic fields used to control the quantum device vary in time. Therefore, we need other approximation schemes which are based on the assumption of the [*weak influence of the environment on the system*]{} and express the dynamics in terms of the fundamental ingredients (controlled Hamiltonian, correlation functions of the bath)\[25\].
Errors in CQOP
---------------
We would like to describe a quantum device operating during the time interval $[0,t]$. We assume that in the initial moment the prepared state can be given by a product formula (1) with a pure state $\rho = |\psi><\psi|$ while in the final moment the state of our open system is described by the reduced density matrix (2). The actual dynamics is compared with the ideal one given in terms of the controlled unitary evolution $U_S(s)$ which solves the Schrödinger equation governed by the controlled Hamiltonian $H_S(s) ; 0\leq s\leq t$. The error $\epsilon$ due to the interaction with environment can be defined as $$\epsilon = 1 - <U_S(t)\psi ,\Lambda_t (|\psi><\psi|)U_S(t)\psi>\ .
\eqno(41)$$ The choice of the Hamiltonian $H_S(t)$ is a tricky problem. Typically, one takes a sum of the time-independent Hamiltonian $H_S^0$ of the bare system $S$ and a time-dependent contributions $H_S^1(s)$ describing the influence of controlled external fields on our system. However the interaction with environment induces generally time-dependent Hamiltonian corrections. For example, vacuum fluctuations of the electromagnetic field produce Lamb shift of the atomic energy levels and van der Waals type interactions between neutral atoms \[24\]. Similar phenomena appear in solid state physics due to interactions with phonons and other quasi-particles. In the following we adopt the optimistic point of view assuming that we can include these corrections in the process of designing quantum algorithms. In practice one should expect the presence of the Hamiltonian errors due to an approximative knowledge of the true physical Hamiltonian $H_S(s)$.
Reduced dynamics in Born approximation
--------------------------------------
We consider the system which consists of the quantum device $S$ driven by the time-dependent Hamiltonian $H_S(t)$ and the bath with its Hamiltonian $H_R$. Then the total Hamiltonian reads $$H(t)=H_S(t) +H_R + H_{int}
\eqno(42)$$ with the interaction Hamiltonian of the form $$H_{int}=\sum_\alpha S_\alpha\otimes R_\alpha
\eqno(43)$$ where $S_\alpha,R_\alpha$ are self-adjoint operators.
In the following we investigate the dynamical map $\Gamma^*$ in the Schrödinger picture which describes the evolution of $S$ from the initial moment $t_{in}= -\tau$ to to the final one $t_{fin} = \tau$ in Born approximation. The exact dynamical map can be expressed in terms of $H_{int}$, the unitary propagator for the total systems $U(t,s)$, the free propagator for the total system $U_0(t,s)$ and the propagator $U_S(t,s)$ defined by $$U(t,s)={\bf T} e^{-i\int^t_sH(u)\; du}\ ,\
U_0(t,s)={\bf T} e^{-i\int^t_s(H_S(u)+H_R)\;du}\ ,$$ $$U_S(t,s)={\bf T} e^{-i\int^t_sH_S(u) \; du}
\eqno(44)$$ where ${\bf T}$ is the time ordering operator. Introducing the "superoperator’ notation: ${\hat U}\rho = U \rho U^*$ and ${\hat H}_{int}\rho = [H_{int},\rho]$, choosing the initial state similarly to (1) $$\rho_{SR}(-\tau)= \rho\otimes\omega_R\ ,\ [H_R,\omega_R]=0\ ,
{\rm Tr}(\omega_R R_{\alpha}) = 0
\eqno(45)$$ and applying the second order integral identity for ${\hat U}(t,s)$ we obtain the exact equation $$\Gamma^*(\rho)= {\hat U}_S(\tau,-\tau)\rho$$ $$- {\rm Tr}\Bigl\{\int_{-\tau}^{\tau}
ds\int_{-\tau}^s du {\hat U}_0(\tau,s){\hat H}_{int} {\hat U}_0(s,u){\hat H}_{int}
{\hat U}(u,-\tau)\rho\otimes\omega_R\Bigr\}\ .
\eqno(46)$$ The Born approximation consists in replacing ${\hat U}(u,-\tau)\rho\otimes\omega_R$ by ${\hat U}_S(u,-\tau)\rho\otimes\omega_R$ in eq.(46). Our main assumption is that for a given initial state $\rho$ and a given trajectory $\{H_S(t); -\tau \leq t\leq \tau\}$ of the controlling Hamiltonian the influence of the reservoir on the state of our system $S$ given by the second term on the RHS of eq.(46) is small, say of the order $\epsilon<<1$. Then up to the higher order terms $\sim \epsilon^2$ we can use the proposed approximation. One should notice that we do not necessarily impose weak coupling or/and short time $\tau$ regime but we can search for the optimal choice of $\rho$ and $H_S(t)$ to fulfill the Born approximation beyond this regime.
The final formula for $\Gamma^*(\rho)$ in Born approximation reads $$\Gamma^*(\rho) = {\hat U}_S\Bigl(\rho + \Phi^* (\rho) - {1\over 2}\{K,\rho\}\Bigr)
\eqno(47)$$ where ${\hat U}_S \equiv {\hat U}_S(\tau,-\tau)$. The superoperator $\Phi^*$ called [*error map*]{} is a completely positive map given by $$\Phi^*(\rho) =\int_{-\tau}^{\tau}ds\int_{-\tau}^{\tau} du {\rm Tr}\bigl(\omega_R R_{\alpha}
R_{\beta}(s-u)\bigr) S_{\beta}(s,-\tau) \rho S_{\alpha}(u,-\tau)
\eqno(48)$$ where $S_{\alpha}(s,u)= {\hat U}_S(u,s)S_{\alpha}$ , $R_{\alpha}(t)= e^{it{\hat H}_R}
R_{\alpha}$. The operator $K\geq 0$ is given by $K = \Phi ({\bf 1})$. Strictly speaking one has also a contribution of the form $-i[h,\rho]$ which contains Hamiltonian corrections due to the interaction with $R$ like for example Lamb shift, collective Lamb shift, etc. but we put $h\equiv 0$. This can be justified by starting with a bare Hamiltonian $H^0_S(t)$ containing appropriate counterterms and performing renormalization procedure. It is , however, equivalent to treating $H_S (t)$ as a full physical Hamiltonian and puting $h \equiv 0$.
The equation (47) resembles in its structure the short time approximation to quantum dynamical semigroup and indeed, if the conditions of Markovian approximation are fulfilled we can derive from (47) different types of the Markovian master equations \[7,9,11\]. The approximative map (47) is trace preserving but complete positivity is satisfied up to the higher order terms in $\epsilon$.
Finally, we pass to the frequency (energy) domain introducing spectral density of the reservoir $R_{\alpha \beta}(\omega)$ by $${\rm Tr}\bigl(\omega_R R_{\alpha}R_{\beta}(t)\bigr)=
\int_{-\infty}^{\infty} R_{\alpha\beta}(\omega)e^{-i\omega t}d\omega
\eqno(49)$$ and defining $$Y_{\alpha}(\omega) = \int_{-\tau}^{\tau} S_{\alpha}(s,-\tau) e^{-i\omega s}ds\ .
\eqno(50)$$ We obtain the following formula for the error map $$\Phi^*(\rho)= \sum_{\alpha,\beta}\int_{-\infty}^{\infty}d\omega\; R_{\alpha\beta}(\omega)\;
Y_{\beta}(\omega)\rho Y_{\alpha}^*(\omega)
\eqno(51)$$ which is a kind of quantum [*fluctuation-dissipation theorem*]{} in the linear response regime \[9,10\].
Error formula
-------------
The general definition of the error due to the interaction with the environment (41) can be applied to our approximative evolution $$\epsilon = 1- <U_S\psi , \Gamma^*(|\psi><\psi|)U_S\psi>\ .
\eqno(52)$$ Putting the expression (47) we obtain the error formula in Born approximation $$\epsilon = <\psi, K \psi> - <\psi ,\Phi^*(|\psi><\psi|)\psi>
\eqno(53)$$ which can be the starting point for the analysis of efficiency of quantum devices. Writing it in terms of (49-51) we obtain $$\epsilon =\sum_{\alpha,\beta}\int_{-\infty}^{\infty}d\omega\; R_{\alpha\beta}(\omega)\;
\Bigl( <\psi ,Y_{\alpha}^*(\omega) Y_{\beta}(\omega)\psi>$$ $$-
<\psi ,Y_{\alpha}^*(\omega)\psi><\psi, Y_{\beta}(\omega)\psi>\Bigr)\ .
\eqno(54)$$ The interpretation of (54) becomes simple and elegant if we introduce the following notation $$<\psi ,Y_{\alpha}^*(\omega) Y_{\beta}(\omega)\psi> -
<\psi ,Y_{\alpha}^*(\omega)\psi><\psi, Y_{\beta}(\omega)\psi>
= 2\tau S_{\alpha\beta}(\omega)\ .
\eqno(55)$$ Now we can write the error formula in a compact form as an overlap of two “correlators” ${\cal R}(\omega) = [R_{\alpha \beta}(\omega)]$ and ${\cal S}(\omega) = [S_{\alpha \beta}(\omega)]$ multiplied by the duration of the operation time of the quantum device $2\tau$ $$\epsilon = 2\tau \int_{-\infty}^{\infty}d\omega\;{\rm Tr}\bigl({\cal R}(\omega)
{\cal S}(\omega)\bigr)\ .
\eqno(56)$$ Both correlators encode the information about the dynamics and the initial state of the baths and the quantum device, respectively. Moreover, one can see that their mathematical structure is essentially the same. To show this we consider a sequence of random (quantum or classical) time-dependent real variables $f_{\alpha}(t)$ and denote by $<\cdot>_F$ the appropriate average. Define a correlator ${\cal F}(\omega) = [F_{\alpha\beta}(\omega)]$ by the following formula $$\lim_{\tau\to\infty}{1\over 2\tau} \int_{-\tau}^{\tau} ds\Bigl( <f_{\alpha}(t+s)
f_{\beta}(s)>_F - <f_{\alpha}(t+s)>_F <f_{\beta}(s)>_F \Bigr)$$ $$=\int_{-\infty}^{\infty} F_{\alpha\beta}(\omega) e^{-i\omega t}d\omega\ .
\eqno(57)$$ Then puting $f_{\alpha}\equiv R_{\alpha} , <\cdot>_F \equiv {\rm Tr}(\omega_R \cdot)$ we obtain ${\cal F}(\omega)\equiv {\cal R}(\omega)$. Similarly, ${\cal F}(\omega)\equiv {\cal S}(\omega)$ for $f_{\alpha}
\equiv S_{\alpha} , <\cdot>_F \equiv <\psi ,\cdot \psi>$ if only $\tau$ is long enough such that the limit $\tau\to\infty$ in (57) can be used.
Having the formula (56) one can try to design different strategies reducing errors. Although very roughly the error seems to be proportional to the operation time $2\tau$ the situation is much more complicated. For example, if the leading error source is described by the correlator satisfying for small $\omega$ $${\cal R}(\omega)\sim \omega^{\kappa} \ ,\ \kappa >1
\eqno(58)$$ the optimal strategy is to slow down the time evolution of $S$ in order to move the correlator ${\cal S}(\omega)$ into the low frequency domain. On the contrary, if ${\cal R}(\omega)$ is flat “fast gates” are optimal. Generally, one can try to shift the support of ${\cal S}(\omega)$ into the frequency domain where ${\cal R}(\omega)$ is small. However, because both correlators are positively defined matrices we cannot expect dramatic reduction of errors by some “cancelation effects”. Moreover, different type of reservoirs and interactions produce different shapes of ${\cal R}(\omega)$, e.g. scattering with gas particles gives flat shapes while the linear coupling to quantum bosonic field produces the relations of type (58) with some cut-offs for large $\omega$ (see \[25\]and the authors contribution to \[11\]).
Another different strategy is to select an optimal subspace of initial states $\psi$ which are “decoherence-free” with respect to the leading error source. The following simple criterion is a generalization of the idea of quantum correcting codes \[12\].
Assume that the matrics ${\cal R}(\omega)$ is strictly positive for $\omega\in\Omega$. Than any $\psi$ satisfying the eigenvector condition for all $\alpha$ and $\omega\in\Omega$ $$Y_{\alpha}(\omega)\psi = \lambda_{\alpha}(\omega)\psi\ ,\ \lambda_{\alpha}(\omega)\in {\bf C}
\eqno(59)$$ evolves without errors.
Another partial results and examples of applications can be found in \[25\].
Conclusions
===========
The author believes that the presented results, although often formulated in a rather abstract mathematical form, can serve as a guide in the discussion concerning the optimal strategies in controlling quantum devices. These strategies can be roughly divided into passive and active ones. At present, it seems that the passive ones can be realized in three apparently different ways:
I\) using the symmetries of the coupling to the baths and systems’ Hamiltonian to obtain large enough decoherence-free subalgebras as presented in Section 2.3,
II\) slowing down the action of quantum algorithm (slow gates) for the environments which satisfy the low frequency scaling (58),
III\) Speeding up the quantum algorithm (fast gates) for the environments with flat ${\cal R}(\omega)\simeq {\cal R}$.
The best known models of the type I) are given by the “superradiance” systems governed by the equations (37). In principle, they can be realized in different settings using collective coupling to bosonic fields (photons, phonons, etc.,). Such reservoirs satisfy also the condition (58) what suggests that both I) and II) demand the similar technology. Indeed, the collective coupling of qubits to the bosonic field, which is a necessary condition for the validity of (37) is possible if the wave-length of the relevant bosonic modes is long in comparison with the diameter of the device \[24\]. It means that only the low frequences can appear in the correlator ${\cal S}(\omega)$ (55),(56). This is equivalent to the slow gates strategy II). Moreover, in both cases I) and II) the time scale of gates must grow with the size of the device (proportional to the number of qubits) in order to keep the total error constant.
The main obstacle for the strategy I) is the approximative character of the symmetry which becomes less accurate with the growing size of the device. Moreover, the Hamiltonian corrections due to the renormalization effects discussed in Section 4.2 do not possess this symmetry except of the cases with the very special geometry \[24\].
The strategies II) and III) are obviously contradictory and therefore for real systems , for which both types of reservoirs are at work, the optimization is needed.
The most elaborated active strategy is the idea of quantum error corrections for quantum computers \[12\]. The existing models of error corrections possess, however, several drawbacks. First of all, the noise is assumed to be independent of the algorithm what is equivalent to the assumption of the strategy III) (${\cal R}(\omega)\simeq {\cal R}$). Moreover, the entropy produced in the computer must be very precisely taken away by “fresh” qubits prepared in pure states. Finally, initial errors for all qubits used in this scheme are not taken into account seriously.
Another active approach, called dynamical decoupling \[13\], involves the existence of the frequency domain for which the correlator ${\cal R}(\omega)\simeq 0$. Then applying a suitable fast modulation of the system’s dynamics we can, in principle, move the support of the correlator ${\cal S}(\omega)$ into this domain. Although for the specific environments the correlator ${\cal R}(\omega)$ may have some “valleys”, the overall tendency for ${\cal R}(\omega)$ is to grow with $\omega$. This is related to the fact that the number of accesible quantum states of the environment typically grows with increasing energy scale. In any case, dynamical decoupling strategy demands fast gates technology as for III).
In authors’ opinion the optimized combination of the passive strategies I,II and III may be the most realistic attempt to deal with quantum noise and associated errors. It would be interesting to apply the formulas of the type (56) to estimate the minimal error per a single gate within the different proposed physical implementations of quantum computers (e.g. trapped ions , NMR , quantum dots , etc).
A part of this paper (Sections 4,5) is based on the collaboration with Michał ,Paweł and Ryszard Horodecki.
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|
---
abstract: 'Deep neural networks (DNNs) can easily fit a random labeling of the training data with zero training error. What is the difference between DNNs trained with random labels and the ones trained with true labels? Our paper answers this question with two contributions. First, we study the memorization properties of DNNs. Our empirical experiments shed light on how DNNs prioritize the learning of simple input patterns. In the second part, we propose to measure the similarity between what different DNNs have learned and memorized. With the proposed approach, we analyze and compare DNNs trained on data with true labels and random labels. The analysis shows that DNNs have *One way to Learn* and *N ways to Memorize*. We also use gradient information to gain an understanding of the analysis results.'
author:
- |
Jindong Gu\
The University of Munich\
Siemens AG, Corporate Technology\
`[email protected]`\
Volker Tresp\
The University of Munich\
Siemens AG, Corporate Technology\
`[email protected]`\
bibliography:
- 'neurips\_2019.bib'
title: Neural Network Memorization Dissection
---
Introduction
============
From a traditional perspective of machine learning, a model with sufficient capacity can overfit the training data, e.g., by memorizing each training sample. The overfitted model will yield poor performance on unseen test data [@GoodfellowDL]. However, over-parametrized deep neural networks (DNNs) with high expressiveness often show excellent generalization ability, even without explicit regularization techniques. [@Zhang2016UnderstandingDL] showed that the generalization of DNNs cannot be explained by traditional approaches, such as *Rademacher complexity* [@Bartlett2001RademacherAG], *VC-dimension* [@Vapnik1998], and *Uniform stability* [@bousquet2002stability]).
[@Zhang2016UnderstandingDL] demonstrated that DNNs are able to fit data with random labels and pure noise inputs. They raised the interesting question whether similar memorization tactics are used when DNNs are trained on data with true labels. [@arpit2017closer] showed that DNNs learn simple patterns first before memorizing, based on observations in extensive experiments. How DNNs prioritize the learning of simple patterns in early training has not been well-understood.
Paradigmatic examples of memorization algorithms are Lookup Tables and $k$-nearest neighbor approaches [@fix1951discriminatory]. [@chatterjee18a] built a network of Lookup Table Network that memorizes training examples. Although they memorize training examples, the built network also showed competitive generalization ability. Moreover, their built network shows little variation from run to run. [@cohen2018dnn] experimentally demonstrated the similarity between DNNs and $k$-NN.
It is evident that, given training examples and corresponding hyperparameters, the built $k$-NN model is unique. However, it is not clear whether the learned functions are also similar in the case of DNNs, when trained with different initializations, network sizes, and architectures. To explore this question, we propose an approach to measure the similarity and dissimilarity between what individual DNNs have learned and what they have memorized. By using our proposed approach, we find that DNNs have *One way to Learn* and *N ways to Memorize* training data. A deeper analysis shows that there is a qualitative difference between neural networks and $k$-NN.
Our contributions can be summarized as follows: 1) With extensive experiments, we show that how DNNs initially learn or prioritize simple patterns; 2) We propose an approach to measure the similarity between what different DNNs have learned and memorized given the same training data; 3) With the proposed approach, we analyze DNNs trained on data with true labels and random labels, respectively. We discuss our understanding of the dissection results.
[0.45]{} ![The training and test accuracy, and the gradient magnitude $\overline{G}$ in the training processes.[]{data-label="fig:vgg11"}](VGG11_RandL "fig:")
[0.45]{} ![The training and test accuracy, and the gradient magnitude $\overline{G}$ in the training processes.[]{data-label="fig:vgg11"}](VGG11_TrueL "fig:")
How DNNs Prioritize the Learning of Simple Patterns? {#sec:prio}
====================================================
We first train VGG11 on CIFAR10 with true labels for 100 epochs using SGD with a constant learning rate of 0.01, moment 0.9 and a batch size of 128, using weight decay ($\lambda=$1e-4) as a regularizer. The model achieves 100% accuracy on training data and 83,73% accuracy on test data. Under the same setting, we train another VGG11 model with random labels without the regularizer, which achieves 100% accuracy on training data and $\approx$10% accuracy on test data.
Given the training loss $\mathcal{L}$ and a training data point $\pmb{x}=\{x_1, x_2, \cdots, x_M\}$, the gradient of the loss with respect to each input dimension is [$\frac{\partial \mathcal{L}}{\partial x_j}$]{}. The gradient magnitude of the sample is defined as the sum of absolute gradient values of an input [$G(\pmb{x}) = \sum^M_{j=1} |\frac{\partial \mathcal{L}}{\partial x_j}|$]{}. We record [$G(\pmb{x})$]{} during training and average them over all training examples [$\overline{G} = \frac{1}{N} \sum^N_{i=1} G(\pmb{x}^{(i)})$]{}.
In the computational graph of a DNN, inputs and parameters of the DNN are leaf nodes. Their received gradients depend on the gradients of shared intermediate nodes due to the chain rule. In a backpropagation, gradients become smaller and smaller when propagated back to the input layer. To a great extent, a big gradient magnitude $\overline{G}$ corresponds to large gradients of the parameters.
Figure \[fig:vgg11\] shows the accuracy and the gradient magnitude $\overline{G}$ of each epoch in the training process. In the model trained with random labels, the training process started with a hunting stage. Once it starts to fit, it converges quickly. This phenomenon is also observed in [@Zhang2016UnderstandingDL]. In the model trained with true labels, the model fits the data directly without a hunting stage. What we observed and summarised is not specific for the VGG11 model, but also for other models, e.g., a similar model with different network size (VGG13) and a model with a different architecture (Resnet32) (see Supplementary material).
We use a constant learning rate in our experiments. In the hunting stage, the model updates itself slowly since gradients are small. In the fitting stage, the big gradients make the quick update of the model possible to achieve high accuracy. An interesting question is how the gradients of parameters relate to the complexity of patterns in the training data.
In the training data with true labels, simple patterns (i.e., easily recognizable patterns shared by the images with of the same class) exist. Since visually similar images with similar intermediate feature representations have the same labels, the gradients of a parameter inside a batch are consistent, which leads to a quick update of the parameter. In contrast, in the training process on random labels, in the beginning, the corresponding gradients are partly canceled due to the inconsistent gradient information. After fine-tuning for several epochs, the neural network is able to map inputs of the same class (i.e., visually different images) to similar intermediate feature representations, which makes the gradients consistent again. The neural network starts the fitting stage.
![The images generated by the GAN: the first row shows the images generated by GAN with different random seeds; other rows show the ones after the optimization of Equation \[equ:disect\_p\] for different output classes of VGG11, such as *Ship*, *Hourse* and *Bird*.[]{data-label="fig:gen_img"}](Sim_models)
![The images generated by the GAN: the first row shows the images generated by GAN with different random seeds; other rows show the ones after the optimization of Equation \[equ:disect\_p\] for different output classes of VGG11, such as *Ship*, *Hourse* and *Bird*.[]{data-label="fig:gen_img"}](gen_img)
Dissection Approach
===================
One path to understand DNNs is to visualize what each neuron has learned or detected. Activation Maximization synthesizes an input that highly activates a specific neuron using the gradients of the neural activation with respect to the input [@erhan2009visualizing]. Different regularizations are applied [@yosinski2015understanding; @mahendran2016visualizing; @nguyen2016synthesizing] to make the synthesized inputs (i.e., images) look realistic. In this work, we aim to study the similarity between what different DNNs have learned and memorized given the same training data.
The two DNNs are trained on the same training data $\mathbb{X} = \{\pmb{x}^{(1)}, \pmb{x}^{(2)}, \cdots, \pmb{x}^{(n)}\}$ assumed to be sampled from a data distribution $\chi$. Their learned functions are $f^1$ and $f^2$, respectively. We can find the input pattern that activates $j$-th output class of $f^1$ using Activation Maximization [@erhan2009visualizing], namely, $\pmb{x}^{j*} =\mathop{\operatorname*{argmax}}_{\tiny \pmb{x}} f^1_j(\pmb{x})$. We could test whether $f^2$ also has a high response on the pattern at the $j$-th class $f^2_j(\pmb{x}^{j*})$. By dissecting $f^2$ with the patterns learned by $f^1$, we can compare the two functions.
However, the challenge is that $\pmb{x}^{j*}$ could be very unlikely considering the data distribution $\chi$. The $f^2$ then could show unexpected outputs on the out-of-distribution inputs, which makes the comparison not accurate. To overcome the challenge, we apply a Generative Adversary Network (GAN) [@goodfellow2014generative] to regularize the Activation Maximization process.
A GAN consists of two components: a Discriminator to distinguish the real data from generated data and a Generator to fool the Discriminator by generating data. In the case of convergence, the Generator is able to map latent variables to samples in data space. We train a GAN on the same training data $\mathbb{X}$ as above until convergence. The Generator therefore maps latent variables $\pmb{z} \sim \mathcal{N}(0,\, 1)$ to a sample $G(\pmb{z}) \sim \chi$. The pattern learned by the $j$-th output class of the DNN $f^1$ is $$\footnotesize
\pmb{z}^{j*} = \mathop{\operatorname*{argmax}}_{\pmb{z}} f^1_j(G(\pmb{z})) \qquad \qquad \pmb{x}^{j*} = G(\pmb{z}^{j*})
\label{equ:disect_p}$$ where pre-trained $f^1$ and $G(\cdot)$ are fixed. The created $\pmb{x}^{j*}$ subject to $\chi$ can be used to dissect $f^2$.
Two strategies DNNs use are the lerning of simple patterns or the memorizing training samples. The pattern $\pmb{x}^{j*}$ in Equation \[equ:disect\_p\] is what the DNN $f^1$ has learned or memorized in the $j$-th class. By dissecting $f^2$ using the pattern, we can identify whether the two functions are similar. The similarity between $f^1$ and $f^2$ can be measured with the following metric. $$\footnotesize
Dist(f^1, f^2) = \frac{1}{C} \sum_j^C D_{{\tiny KL}}(f^1(\pmb{x}^{j*})\| f^2(\pmb{x}^{j*}))
\label{equ:metric}$$ The lower the dissimilarity value [$Dist(f^1, f^2)$]{} is, the more similar the two functions are. Another closely related work also uses a similar metric by replacing $\pmb{x}^{j*}$ with the images in test dataset [@hacohen2019all]. Different from their work, we focus on the patterns that DNNs has learned or memorized, not the random ones on the data distribution.
Dissecting DNNs via Experiments
===============================
The CIFAR10 training dataset with true labels is notated as $\mathbb{X}^T$. By replacing the true labels with random labels, we get another training dataset $\mathbb{X}^R$. We train four DNNs on $\mathbb{X}^T$ and $\mathbb{X}^R$, respectively. The first two models VGG11(s1), VGG11(s2) are VGG11 trained with different random initializations (i.e., different seeds). The other two models have a different network size or architecture, namely, VGG13 and ResNet32. The setting of training processes is the same as in Section \[sec:prio\]. They all achieve 100% accuracy on the training datasets.
We train a deep convolutional GAN on the $\mathbb{X}^T$. The architectures and hyperparameters are the same as in [@Radford2015UnsupervisedRL]. The generator maps 100-dimensional gaussian noises $\pmb{z}$ to images $G(\pmb{z})$ with shape (32, 32, 3) using four transposed convolutional layers; the discriminator maps the images to a single output using four convolutional layers. They are trained on $\mathbb{X}^T$ for 30 epochs to converge.
![The similarity between models trained on data with different noise levels. With less noise, models have a greater similarity.[]{data-label="fig:noise"}](N_ways_M)
![The similarity between models trained on data with different noise levels. With less noise, models have a greater similarity.[]{data-label="fig:noise"}](Noise_level)
Given the four models pre-trained on $\mathbb{X}^T$ or $\mathbb{X}^R$, we set VGG11(s1) model as $f^1$ and the rest of models as $f^2$, respectively. To compute $Dis(f^1, f^2)$ in Equation \[equ:metric\], for each class, we initialize $\pmb{z} \sim \mathcal{N}(0,\, 1)$ with 100 different seeds. The Equation \[equ:disect\_p\] is optimized by Gradient Ascent with a learning rate of 0.05 for 1000 iterations.
**One way to Learn:** For well-trained models, the dissimilarity values between them are computed with Equation \[equ:metric\] and shown in Figure \[fig:dissim\]. Each bar corresponds to a averaged dissimilarity value between two models. The black line on each bar corresponds its variance.
The blue bars with very low values correspond to the dissimilarity between models trained on $\mathbb{X}^T$. What different models have learned are extremely similar. Given the training data $\mathbb{X}^T$, what neural networks have learned are not affected by different random initializations, network sizes, and neural architectures. In other words, DNNs have one way to learn.
We visualize the patterns what VGG11(s1) has learned. Given the $j$-th class, $\pmb{z}$ is initialized with different seeds. We show the images $G(\pmb{z})$ generated by GAN before the optimization of Equation \[equ:disect\_p\] and also the ones after the optimization $\pmb{x}^{j*}$. In Figure \[fig:gen\_img\], each column corresponds to a seed, and the images in the 2nd-4th row correspond to the patterns learned for the respective classes.
**N ways to Memorize:** As visible in Fig \[fig:dissim\], models trained on $\mathbb{X}^R$ memorize all the training samples. The yellow bars with high values mean that what different models have memorized from $\mathbb{X}^R$ are different, even between the models from run to run. DNNs have different ways to memorize samples.
In our experiments, as in previous works [@Zhang2016UnderstandingDL; @arpit2017closer], weight decay and data augmentations are turned off to reach zero error on the training data. We train VGG11 on $\mathbb{X}^R$ with the weight decay ($\lambda=$1e-4) and the standard data augmentation (4-pixels paddings, 32$\times$32 random croppings, and horizontal flips), respectively. The experiment results show that the weight decay and the data augmentation do not affect the similarity between what different models have memorized.
To further confirm our hypothesis about the qualitative difference between models trained on $\mathbb{X}^T$ and $\mathbb{X}^R$, we train VGG11 with 10 random seeds on the two datasets, respectively. We visualize the training accuracy and the gradient magnitude $\overline{G}$ of the 10 training processes in Figure \[fig:seeds\]. For the models trained on $\mathbb{X}^T$, the optimization always follows similar paths, as shown in the left subplot. Contrarily, the right subplot shows that the training process on $\mathbb{X}^R$ follows obviously different optimization paths. We conjecture that the optimization on $\mathbb{X}^R$ encounters a large number of saddle points [@dauphin2014identifying; @choromanska2015loss] and that the number of total saddle points encountered during learning depends on how the optimizer choose path directions on each saddle point, which might be quite sensitive to data.
**Learn and Memorize:** When we train models on the training data with different noise levels (i.e., a certain percentage of training data are specified with random labels), it is not clear what DNNs learn and/or memorize. Our analysis in Section \[sec:prio\] supports the hypotheses: instead of only learning on the simple patterns or being totally misled by the examples with random labels, DNNs start to learn simple patterns and then memorize the rest of samples. The learned functions become gradually far away from each other when the noise level increases (see Figure \[fig:noise\]).
Conclusion
==========
Our work builds a connection between the learning process and the complexity of patterns in training data by analyzing their gradients. We further propose an approach to compare what different DNNs learned and memorized. The experimental results reveal that DNNs have one way to learn and N ways to memorize.
|
---
author:
- |
Sho Tanaka[^1]\
Kurodani 33-4, Sakyo-ku, Kyoto 606-8331, Japan
title: ' An attempt to resolve the cosmological constant problem in the modified Yang’s noncommutative quantized space-time '
---
addtoreset[equation]{}[section]{}
Introduction
============
In our preceding papers, “Where does black-hole entropy lie?" \[1\] and “A short essay on quantum black holes and underlying noncommutative quantized space-time"\[2\] hereafter referred as I and II, respectively, we emphasized the importance of the underlying noncommutative geometry such as Snyder’s and Yang‘s noncommutative quantized space-time \[3-6\] towards the ultimate theory of quantum gravity or Planck scale physics. In the present paper, we more specifically notice the [*modified*]{} Yang’s noncommutative quantized space-time.
In fact, at the end of [**5. Concluding arguments and further outlook**]{} in II, we stated: “Before closing this short essay, let us note another interesting possibility of Yang’s quantized space-time algebra (YSTA, see Appendix A). Indeed, one should notice that YSTA is intrinsically equipped with the long scale parameter [*R*]{}, together with the short scale parameter [*a*]{}[^2] which has been identified with Planck length $ l_P $ in our present research so far. On the other hand, as was preliminarily pointed out in \[25\], ${\it R}$ might be promisingly related to a [*fundamental cosmological constant*]{} in connection with the recent dark-energy problem, under the further idea that YSTA subject to the [*SO(D+1,1)*]{} algebra might be understood in terms of some kind of local reference frame in the ultimate theory of quantum gravity, on the analogy of the familiar local Lorentz frame in Einstein’s General Theory of Relativity."
The above statement must be just the subject of the present paper. In this connection, it might be appropriate here in advance to mention the [*modified version*]{} of Yang’s original quantized space-time (see Appendix A ).
Indeed, as will be shown in Appendix A, we intend in its modified version (MYST) to replace the long scale parameter $R$ in the original version (YST) to be some kind of ${\it time}\ \tau -$dependent: let us say it the [*cosmological time*]{} dependent, $R(\tau)$ ( see the beginning of section [**3**]{} ). In fact, by this treatment, we intend that the series of MYST characterized by $(\lambda, R(\tau))$ becomes able to describe the [*historical development*]{} of our universe beginning with the so-called big bang.
The essence of cosmological constant problem
============================================
Today, we know that the observed value of cosmological constant $\Lambda_{obs.}$ is over one hundred twenty orders of magnitude smaller than the Planck energy density. That is,
$$\begin{aligned}
\Lambda_{\rm obs.} \sim 10^{-122} G^{-2},\end{aligned}$$
where $G$ denotes the Newton constant and $G^{-2}$ is known to be the so-called Planck energy density, see for instance \[7\].
At this point, we are almost uniquely led to the following [*theoretical*]{} expression $\Lambda (\tau)_{\rm theor.}$ at the present cosmological time $\tau$
$$\begin{aligned}
\Lambda (\tau)_{\rm theor.} = {n_{\rm dof} ( V_3^{R(\tau)}) }^{-1} G^{-2}\end{aligned}$$
on the basis of $ n_{\rm dof} ( V_3^{R(\tau)}),$ which is the key concept in the modified Yang’s quantized space-time (see Appendix A), and most importantly describes the quantized number of spatial degrees of freedom inside $V_3^{R(\tau)}$, i.e., $d=3$ - dimensional volume with radius $R(\tau)$: $$\begin{aligned}
{n_{\rm dof}} ( V_3^{R(\tau)} ) = ( [R(\tau)/\lambda] +1)^2, \end{aligned}$$ ( see (3.1) in II ), where $ [R(\tau)/\lambda] $ means the nearest integer of $ R(\tau)/\lambda. $
In fact, one can easily confirm that the factor ${n_{\rm dof} ( V_3^{R(\tau)}) }^{-1} ( \sim(\lambda/R(\tau))^2 $ in (2.2) just presents the extremely small factor $10^{-122}$ in (2.1) under $R(\tau) \sim 10^{28} {\rm cm}$ and $\lambda (= l_P) \sim 10^{-33} {\rm cm}.$
That is, the essence of cosmological constant problem is now clearly understood or resolved in the fundamental structure of $\Lambda (\tau)_{theor.} $ derived in (2.2) where the most puzzling factor $10^{-122}$ in the $\Lambda_{\rm obs.} $ in (2.1) is [*theoretically*]{} understood in terms of the key concept ${n_{\rm dof}( V_3^{R(\tau)}) }.$
By the way, $n_{\rm dof} ( V_3^{R(\tau)} )$ and the relation (2.3) itself as a whole were derived according to the general consideration of “Kinematical Holographic Relation" [**KHR**]{} given by (3.1) in II, through the proper irreducible representation of Yang’s quantized space-time algebra YSTA (see Appendix A in II ), while it is now developed on the modified Yang’s quantized space-time algebra (MYSTA) (see Appendix A).
Furthermore, one should notice the fact that $n_{\rm dof} ( V_3^{R(\tau)} )$ is nothing but the whole number of quantized spatial degrees of freedom inside of $ V_3^{R(\tau)},$ that is, the whole spatial volume of our universe at the cosmological time $\tau$, as seen from the consideration given in I, II and originally in \[8\].
Concluding arguments and further outlook
=========================================
First, le us notice that the theoretical cosmological constant $\Lambda (\tau)_{\rm theor.} = {n_{\rm dof} ( V_3^{R(\tau)}) }^{-1} G^{-2}$ derived in Eq. (2.2) becomes the cosmological time ${\tau}$ dependent through the term ${n_{\rm dof} ( V_3^{R(\tau)})}$, as noted in Appendix A in general. In this connection, it is quite important to notice that the term ${n_{\rm dof} ( V_3^{R(\tau)})}$ is directly related to the [*entropy*]{} of the static and equilibrium system in the following way
$$\begin{aligned}
S(V_3^{R(\tau)}) ( = - {\rm Tr} [{\bf W} (V_3^{R(\tau)}){\rm ln} {\bf W} (V_3^{R(\tau)})] )
= n_{\rm dof} (V_3^{R(\tau)}) S[site]
\nonumber\\
= ([R(\tau)/\lambda]+1)^2 S[site],\end{aligned}$$
as seen in Eq.(3.10) in II with [*L*]{} replaced by $R(\tau)$. In the above expression $S[site]$ denotes the entropy assumed to be commonly realized in every \[site\], that is, every basis vector in Hilbert space II (see, section [**2**]{} in II). This fact guarantees that the cosmological time $ {\tau}$ acquires the so-called arrow of time, beyond a simple parameter (see Appendix [**A**]{} ).
Second, let us briefly consider the big bang stage. We assume that it starts at $\tau \sim 0$ with the uncertainty $\lambda.$ Then, one finds in Eq.(2.2) ${\rm n_{dof}} ( V_3^{\sim \lambda}) \sim 4 $ and $$\begin{aligned}
\Lambda (\lambda)_{\rm theor.} \sim {1\over 4} \ G^{-2}\end{aligned}$$ at the very beginning stage of the big bang, in a sharp contrast to Eq. (2.1). It reminds us the extremely microscopic black hole system considered in the subsection [**4.3**]{} in II. It is our great interest further to investigate the later various stages of the big bang in accordance with the consideration of the black holes ranging over from macroscopic to extremely microscopic scales given also in [**4.3**]{} in II.
It should be furthermore noted here that the above consideration of the big bang might be fulfilled without suffering from the so-called “Hawking-Penrose singularity theorems," because of the fact that our present research based on the noncommutative geometry or quantized space-time is well associated with the Heisenberg’s Uncertainty Principle, so as to be free from singularities (see Appendix B in II, “Historical background of noncommutative quantized space and time").
In this connection, it might be possibly claimed that the above derivation of $\Lambda (\tau)_{\rm theor.} = {n_{\rm dof} ( V_3^{R(\tau)}) }^{-1} G^{-2}$ in Eq.(2.2) seems a bit [*ad hoc*]{} and a certain numerical fine tuning to explain the puzzling factor $10^{-122}$ in the Eq.(2.1) [*accidentally*]{} through $ n_{\rm dof} ( V_3^{R(\tau)}) .$ However, the present author should like to notice that on the back of the key concept $ n_{\rm dof}(V_3^{R(\tau)}),$ clearly lies the idea of H. Yukawa’s Atomism of quantized space-time or Elementary Domain beginning from the preceding idea of P.A.M. Dirac’s Generalized transformation function (‘g.t.f.’ see Appendix B in II), which governs the whole space of universe, in assosiation with the present cosmological constant problem or the dark energy problem. We anticipate that our $ \Lambda (\tau)_{\rm theor.} $ given by Eq. (2.2) will show widely the [*predictive*]{} power in the later various stages of the big bang over the present single prediction: $\Lambda (\lambda)_{\rm theor.} \sim {1\over 4} \ G^{-2} $ derived in Eq.(3.2) for its very beginning stage.
In this article we have omitted the important arguments related to the so-called standard model of high energy physics. It is our important task to reconstruct the M -theory \[9\] in terms of the present Modified Yang’s Lorentz covariant quantized space-time towards the ultimate theory of quantum gravity and quantum cosmology.
1.5cm
**Acknowledgments**
0.8cm
The author would like to thank Taichiro Kugo for valuable discussions from the different point of view. The author is grateful to Hideaki Aoyama for giving constant encouragement to the present research.
1.5cm
**Appendix**
Modified Yang’s Lorentz covariant quantized space-time (MYST) {#appendixa}
==============================================================
Even in the MYST, as in the original Yang’s quantized space-time (YST) (see Appendix A in II) it is equipped with the so-called Inonu-Wigner’s two contraction parameters, long $R$ and short $\lambda$. The MYST is simply given by replacing the long scale parameter $R$ in YST to be the [*cosmological time*]{} $\tau-$dependent $R(\tau) $, as noted at the end of [**Introduction**]{}.
In addition, we intend now to introduce this cosmological time $\tau$ so as to describe the observation time of $\Lambda_{obs.} ( \sim 10^{-122} G^{-2}$ ) in (2.1) on the one hand and on the other hand, the set of long and short scale parameters $ ( R(\tau), \lambda)$ at $\tau$ to specify the modified Yang’s quantized space-time as the [*local reference frame*]{} at the cosmological time $\tau$, as was mentioned in [**Introduction**]{}, according to the idea of the local Lorentz frame in General Theory of Relativity.
First, (A.1) holds as it stands $$\begin{aligned}
\hat{\Sigma}_{MN} \equiv i (q_M \partial /{\partial{q_N}}-q_N\partial/{\partial{q_M}})
\label{sec_a: stp}\end{aligned}$$ and (A.2) clearly tends into $$\begin{aligned}
- q_0^2 + q_1^2 + \cdots + q_{D-1}^2 + q_a^2 + q_b^2 = R (\tau)^2.\end{aligned}$$
Now, $D$-dimensional space-time and momentum operators, $\hat{X}_\mu$ and $\hat{P}_\mu$, with $\mu =1,2,\cdots,D,$ are defined by $$\begin{aligned}
&&\hat{X}_\mu \equiv \lambda\ \hat{\Sigma}_{\mu a}
\\
&&\hat{P}_\mu (\tau) \equiv \hbar /R (\tau) \hat{\Sigma}_{\mu b}, \end{aligned}$$ together with $D-$dimensional angular momentum $\hat{M}_{\mu \nu}$ $$\begin{aligned}
\hat{M}_{\mu \nu} \equiv \hbar \hat{\Sigma}_{\mu \nu}\end{aligned}$$ and the so-called reciprocity operator $$\begin{aligned}
\hat{N} (\tau) \equiv \lambda /R (\tau) \hat{\Sigma}_{ab}.\end{aligned}$$
Here and hereafter, one should notice that in MYST the momentum operator or the reciprocity operator and so on become the cosmological constant $\tau-$dependent in general.
Finally, one finds that the following relations are as a whole to hold
$$\begin{aligned}
&&[{\hat X}_\mu, {\hat X}_\nu ] = - i (\lambda^2/\hbar) {\hat M}_{\mu \nu}\end{aligned}$$
$$\begin{aligned}
&&[{\hat P} _\mu, (\tau) , {\hat P}_\nu (\tau)] = - i {\hbar} / R(\tau)^2 \ {\hat M}_{\mu \nu}\end{aligned}$$
$$\begin{aligned}
&&[{\hat X}_\mu, {\hat P}_\nu (\tau)] = - i \hbar {\hat N} (\tau) {\hat M}_{\mu \nu}\end{aligned}$$
.$$\begin{aligned}
[{\hat N} (\tau), {\hat X}_\mu ] = - i \lambda^2 /\hbar {\hat P}_\mu (\tau)\end{aligned}$$ $$\begin{aligned}
[{\hat N} (\tau), {\hat P}_\mu ] = i \hbar / R(\tau)^2\ {\hat X}_\mu,\end{aligned}$$ with other familiar relations concerning ${\hat M}_{\mu \nu}$’s omitted.
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[^1]: Em. Professor of Kyoto University, E-mail: [email protected]
[^2]: In our present researches, we have described so far the short scale parameter in YSTA by $\lambda$ instead of [*a*]{} which C.N. Yang \[5-6\] used in accord with S. Snyder \[\[3-4\], as early footnoted inside Appendix A in II.
|
---
abstract: 'The late time accelerated expansion of the Universe demands that even in local galactic-scales it is desirable to study astrophysical phenomena, particularly relativistic accretion related phenomena in massive galaxies or in galaxy mergers and the dynamics of the kiloparsecs-scale structure and beyond, in the local-galaxies in Schwarzschild-de Sitter (SDS) background, rather than in Schwarzschild or Newtonian paradigm. Owing to the complex and nonlinear character of the underlying magnetohydrodynamical equations in general relativistic (GR) regime, it is quite useful to have an Newtonian analogous potential containing all the important GR features that allows to treat the problem in Newtonian framework for study of accretion and its related processes. From the principle of conserved Hamiltonian of the test particle motion, here, a three dimensional Newtonian analogous potential has been obtained in spherical geometry corresponding to SDS/Schwarzschild anti-de Sitter (SADS) spacetime, that reproduces almost all of the GR features in its entirety with remarkable accuracy. The derived potential contains an explicit velocity dependent term of the test particle that renders an approximate relativistic modification of Newtonian like potential. The complete orbital dynamics around SDS geometry and the epicyclic frequency corresponding to SDS metric have been extensively studied in the Newtonian framework using the derived potential. Applying the derived analogous potential it is found that the current accepted value of $\Lambda \sim 10^{-56} \rm cm^{-2}$ moderately influences both sonic radius as well as Bondi accretion rate, especially for spherical accretion with smaller values of adiabatic constant and temperature, which might have interesting consequences on the stability of accretion disk in AGNs/radio galaxies.'
author:
- 'Tamal Sarkar [^1], Shubhrangshu Ghosh [^2], Arunava Bhadra [^3]'
title: 'Newtonian analogue of Schwarzschild de-Sitter spacetime: Influence on the local kinematics in galaxies'
---
Introduction
============
The simplest and most attractive general relativistic model that explains the late time accelerated expansion of the Universe \[1\] is the $\Lambda$CDM model where $\Lambda$ denotes the positive cosmological constant with a value of nearly $10^{-56} \; {\rm cm}^{-2}$ \[2\] and CDM refers to cold dark matter. The $\Lambda$CDM model is more or less consistent with all the current cosmological observations \[3\] though the origin of cosmological constant still remains elusive \[4\] and is one of the most pressing problems in modern physics.
In the presence of a repulsive cosmological constant (positive) ‘$\Lambda$’ the spacetime geometry exterior to a static spherically symmetric metric is Schwarzschild-de Sitter (SDS), which describes an isolated black hole (BH) in a spatially inflated Universe, rather than Schwarzschild metric. Therefore, the cosmological constant may affect any local gravitational phenomenon like perihelion shift of the orbits of gravitationally bound systems \[5\], gravitational bending of light \[6\], geodetic precession \[5\] etc., but the general perception is that owing to its tiny value, cosmological constant does not lead to any significant observable effects in a local gravitational phenomenon. In solar system the influence of cosmological constant is known to be maximum in the case of perihelion shift of mercury orbit where the $\Lambda$ contribution is about $10^{-15}$ of the total shift \[5\]. However, the contribution of repulsive $\Lambda$ could be significant (larger than the second order term) even in a local gravitational phenomenon when kiloparsecs to megaparsecs-scale distances are involved, such as the gravitational bending of light by cluster of galaxies \[7\]. The study of particle dynamics in SDS spacetime shows a significant contribution of $\Lambda$ when kiloparsecs-scale distance is involved \[8,9\]. The trajectories of both Small and Large Magellanic Clouds in the gravitational field of the Milky Way are affected significantly ($\sim 10 \%$ level or higher) by $\Lambda$ \[10\]. Cosmological constant also leads to non-spherical effects with non-negligible contributions on the local dynamics of clusters and superclusters \[11\]. Some authors claimed \[12\] that the repulsive $\Lambda$ is responsible for significantly lower value ($\sim 60 \; {\rm Km} \; {\rm s}^{-1}/{\rm Mpc}$) of Hubble parameter in our close neighborhood than its large-scale value ($\sim 70 \; {\rm Km} \; {\rm s}^{-1}/{\rm Mpc}$). The cosmological constant also influences gravitational equilibrium \[13\].
A plausible local scenario where cosmological constant may contribute significantly is the relativistic accretion phenomena around massive BHs which involve distance-scale of the order of hundreds of parsecs or even more. This is because the asymptotic character of the BH spacetime changes significantly in the presence of cosmological constant \[7\]. Note that the accretion of matter on to BH is believed to act as the engine for the active galactic nuclei (AGNs). A few studies have been carried out so far to investigate the effect of $\Lambda$ in astrophysical jet/accretion flow paradigm \[14-19\]. It was found that the presence of cosmological constant leads to suppression of black hole evaporation \[14\]; instead of evaporation black hole will accrete energy. The effect of $\Lambda$ on the dynamics of kiloparsecs to megaparsecs-scale astrophysical jets would be worth intriguing to investigate as suggested by Stuchlík, Slańy and Hledík \[15\]. Their work indicated that positive $\Lambda$ can have strong collimation effect on jets. Rezzolla et al. \[16\] have analyzed the effect of $\Lambda$ on the dynamical stability of a geometrically thick accretion disk with constant angular momentum around a SDS BH. Their work showed that $\Lambda$ introduces an outflow of matter through the outer cusp of disk causing a considerable impact on the runway instability. Slobodov et al. \[17\] demonstrated that characteristic peaks in the iron line profile generated in the BH accretion disk in a SDS background are less noticeable and distinguishable with the increase in the value of $\Lambda$. In an interesting scenario, Karkowski and Malec \[18\] and Mach, Malec and Karkowski \[19\] have recently explored the possible dependence of $\Lambda$ on relativistic Bondi-type accretion flow around a nonrotating BH. They defined a quasilocal mass accretion rate and found that $\Lambda$ suppresses the mass accretion rate of the flow and dramatically impacts the transonic nature of the accretion flow.
All the works on the effect of $\Lambda$ on accreting systems are carried out under some restricted conditions/situations. This is because the study of accreting BH systems involves solving general relativistic (GR) hydrodynamic/magnetohydrodynamic (MHD) equations in a strong gravitational field regime. Owing to the complex and nonlinear character of the underlying equations in GR regime, analytical/quasi numerical treatment of the problem is virtually discarded. Even numerical simulation is complicated by several issues such as different characteristic time scales for propagating modes of general relativity and relativistic hydrodynamics. Several early works on these accretion related phenomena were based on pure Newtonian gravity. A few GR effects were incorporated ad hocly. After the seminal work of Paczyński and Witta \[20\], most of the authors treated accretion and its related processes around BHs using hydrodynamical/MHD equations in the Newtonian framework by using some PNPs which are essentially modification of Newtonian gravitational potential developed with the objective to reproduce (certain) features of relativistic gravitation. This is to avoid GR gas dynamical equations, which in most occasions become inconceivable in practice in describing a complex physical system like accreting plasma. Consequently, adopting PNPs, one can comprehensively construct more realistic accretion flow models in simple Newtonian paradigm, while the corresponding PNP would capture the essential GR effects in the vicinity of the compact objects. Instead of PNPs, some authors simply use parameterized post Newtonian (PPN) expansion up to a certain order \[21\]. Since the PPN expansion converges very slowly, the latter option is valid only for orbits at large distances, but not for particle trajectories in the vicinity of the BHs.
Several PNPs exist in the astrophysical literature which are mostly prescribed either in an ad hoc manner or are devised employing certain explicit method \[20,22-33\], mostly developed for Schwarzschild and Kerr BHs. A PNP corresponding to SDS metric has also been prescribed \[34\] based on the method in \[30-32\] for a Keplerian rotation flow which can be more appropriately used to study gas dynamical properties of accretion disk, and thereby concentrating mostly in its use in accretion tori \[35\], though it has also been recently used to investigate the influence of the repulsive cosmological constant on the kinematics of Small and Large Magellanic cloud \[10\]. Although notionally PNPs are aimed at mimicking GR geometries, however, they, in general, are devoid of their uniqueness to effectively describe the GR features in its entirety. To replicate GR features, PNPs lay emphasize mostly to reproduce marginally stable and marginally bound orbits. Moreover, in general, they are unable to reproduce observationally verified tests in general relativity like geodetic precession, gravitational bending of light or gravitational time delay.
If the effects of cosmological constant need to be properly revealed in different astrophysical phenomena in local galactic-scales circumventing the complex GR treatment, it is desirable to have a correct Newtonian analogous like potential corresponding to SDS geometry that will reproduce all the salient features of the SDS metric with reasonable accuracy and extensively mimic a wide spectrum of GR behavior. Recently, Tejeda and Rosswog \[36\] derived a generalized effective potential for a Schwarzschild BH based on a proper axiomatic procedure. Their generalized potential, which has an explicit dependence on radial velocity and orbital angular velocity of test particle, reproduces exactly several relativistic features of corresponding Schwarzschild geometry. Later, the work has been extended for the Kerr BH \[37\]. In the present work, we would derive a modified Newtonian like potential from the conserved Hamiltonian of the test particle motion \[36\], which would then be an approximate relativistic potential analogue corresponding to the SDS metric. We would show from a detailed investigation that the derived potential, which depends on the velocity of the test particle, reproduces almost all of the corresponding GR features with remarkable accuracy. It would be worthy enough to explore the feasibility of measurement of cosmological constant in local-scales or in galactic-scales, and further to explore its influence on the dynamics on kiloparsecs-scale structure and beyond in the local-galaxies, or the accretion related scenarios/conditions where cosmological constant may contribute significantly, in which case, the proposed modified Newtonian like potential analogue to SDS geometry considering it as a background metric, would be of remarkable use.
The plan of the paper is the following. In the next section, we would formulate the approximate relativistic potential analogue in the Newtonian framework corresponding to both SDS and Schwarzschild anti-de Sitter (SADS) spacetimes simultaneously, starting from the conserved Hamiltonian of the system in the low energy limit of the test particle motion \[36\]. Subsequently in §, we evaluate the geodesic equations of motion and extensively analyze their solutions corresponding to SDS geometry, with both our analogous potential and general relativity, for the current accepted value of $\Lambda \sim 10^{-56} \rm cm^{-2}$. In § we explicitly investigate the influence of repulsive $\Lambda$ on Bondi accretion rate. Finally, we end in § with comments and discussion.
Formulation of a modified Newtonian analogous potential corresponding to SDS/SADS geometry
==========================================================================================
For a general class of static spherically symmetric spacetimes of the form (in the standard coordinates system)
$$\begin{aligned}
ds^2 =-f(r)\, c^2 \, dt^2 + \frac{1}{f(r)} \, dr^2 + r^2 d\Omega^2 \, ,
\label{1}\end{aligned}$$
where, $d\Omega^2 = d\theta^2 + \sin^2 \theta \, d\phi^2 $ and f(r) is the generic metric function, the Lagrangian density of a particle of mass $m$ is given by
$$\begin{aligned}
2 {\cal L}=-f(r) \, c^2 \, \left(\frac{dt}{d\tau}\right) ^2 + \frac{1}{f(r)} \, \left(\frac{dr}{d\tau}\right) ^2 +
r^2 \, \left(\frac{d\Omega}{d\tau}\right) ^2\, .
\label{2}\end{aligned}$$
From the symmetries, one obtains two constants of motion corresponding to two ignorable coordinates $t$ and $\Omega$ as given by
$$\begin{aligned}
{\cal P}_t =\frac{\partial \cal L}{\partial \tilde{t}}= - c^2 \, f(r)\frac{dt}{d\tau}={\rm constant}=-\epsilon
\label{3}\end{aligned}$$
and
$$\begin{aligned}
{\cal P}_{\Omega}=\frac{\partial \cal L}{\partial \tilde{\Omega}} = r^2 \frac{d\Omega}{d\tau}={\rm constant} = \lambda \, ,
\label{4}\end{aligned}$$
where, $\epsilon$ and $\lambda$ are specific energy and generalized specific angular momentum of the orbiting particle, respectively. Here, $\tilde{t}$ and $\tilde{\phi}$ represent the derivatives of ‘$t$’ and ‘$\phi$’ with respect to proper time $\tau$. It needs to be mentioned that from now onwards, throughout the paper, the terms related to momentum, energy/Hamiltonian, potential and frequency, all of which are in fact their specific quantities, would be addressed without the using of word ‘specific’. Using Eq. (3) we then can write
$$\begin{aligned}
\frac{dt}{d\tau} = \frac{\epsilon}{c^2} \frac{1}{f(r)} \, .
\label{5}\end{aligned}$$
Using $2{\cal L} =g_{\rm {\alpha \beta}} \, p^{\alpha} \, p^{\beta} =-m^2 c^2$ and substituting Eqs. (3) and (4) in Eq. (2), we obtain
$$\begin{aligned}
\left (\frac{dr}{d\tau} \right)^{2} = \left(\frac{\epsilon^{2}}{c^{2}}- c^{2} \right)
-c^{2} \, \left[f(r)-1 \right] - f(r) \, \frac{\lambda^2}{r^2} \, .
\label{6}\end{aligned}$$
By considering a locally inertial frame for a test particle motion, we write $E_{\rm GN} = (\epsilon^2 - c^4)/{2c^2}$ (‘$\rm GN$’ symbolizes ‘GR-Newtonian’). Second term in the above definition of $E_{\rm GN}$ is the rest mass energy of the particle which is subtracted from relativistic energy owing to low energy limit.
From Eqs. (5) and (6) and using Eq. (4), we get
$$\begin{aligned}
\frac{dr}{dt} = \frac{c^2}{\epsilon} \, f(r) \sqrt{2 \, E_{\rm GN}
-c^2 \, \left[f(r)-1 \right] - f(r) \,{\dot \Omega}^2 \frac{r^2}{f(r)} }\, ,
\label{7}\end{aligned}$$
where, $\dot \Omega$ is the derivative with respect to coordinate time ‘$t$’. Using the condition for low energy limit $\epsilon/c^2 \sim 1$, as we prefer to, $E_{\rm GN}$ is given by
$$\begin{aligned}
E_{\rm GN} = \frac{1}{2} \left(\frac{dr}{dt} \right)^2 \frac{1}{f(r)^2} + \frac{r^2 \, {\dot \Omega}^2}{2f(r)} +
\frac{c^2}{2} \, [f(r)-1] \, .
\label{8}\end{aligned}$$
In the asymptotic non-relativistic limit $E_{\rm GN}$ reduces to the Newtonian mechanical energy (= Hamiltonian of the motion). The generalized Hamiltonian $E_{\rm GN}$ in the low energy limit should then be equivalent to the Hamiltonian in Newtonian regime. The Hamiltonian in the Newtonian regime with the generalized analogous potential in spherical polar geometry will then be equivalent to $E_{\rm GN}$ in Eq. (8). Thus
$$\begin{aligned}
E_{\rm GN} \equiv \frac{1}{2} \left({\dot r}^2 + r^2 {\dot \Omega}^2 \right)
+ V_{\rm GN} - {\dot r} \frac{\partial V_{\rm GN}}{\partial {\dot r}}
- {\dot \Omega} \frac{\partial V_{\rm GN}}{\partial {\dot \Omega}} \, ,
\label{9}\end{aligned}$$
where, $T = 1/2 \, \left({\dot r}^2 + r^2 {\dot \Omega}^2 \right)$ is the non-relativistic kinetic energy of the test particle. $\dot r$ is the derivative with respect to coordinate time ‘$t$’. $V_{\rm GN}$ is the analogous potential which would then be given by
$$\begin{aligned}
V_{\rm GN} = \frac{c^2 [f(r)-1]}{2}- \frac{[1-f(r)}{2f(r)}
\left[ \frac{1+f(r)}{f(r)} \dot r^{2} + r^{2} \, {\dot \Omega}^2 \right] \, .
\label{10}\end{aligned}$$
$V_{\rm GN}$, thus, is the generalized three dimensional potential in spherical geometry in Newtonian analogue, corresponding to any generalized static GR metric given in Eq. (1), with test particle motion in the low energy limit. Note that, the first term on the right hand side of the potential contains the explicit information of the source. For a purely spherically symmetric gravitational mass with zero charge and without any external effects, the classical Newtonian gravitational potential $-GM/r$ will be recovered from this term. The second term is the explicit velocity dependent and contains the information of the test particle motion, thus contributing to the modification of Newtonian gravity.
For SDS/SADS metric the metric function $f(r)= 1- \frac{2r_s}{r} - \frac{\Lambda r^2}{3}$, where $\Lambda$ is the cosmological constant. $\Lambda > 0$ represents the SDS metric with spatially inflated Universe, where $\Lambda < 0$ represents a SADS metric corresponding to negative vacuum energy density. $r_s = GM/c^2$. From Eq. (10) we then obtain the three dimensional generalized Newtonian analogous potential in spherical geometry, corresponding to SDS/SADS geometry in the low energy limit, which is given by
$$\begin{aligned}
V_{\rm DS/ADS} = -\left( \frac{GM}{r} + \frac{\Lambda \, c^{2} \, r^{2}}{6} \right)
- \left(\frac{2r_s + \frac{\Lambda \, r^3}{3}}{r-2r_s - \frac{\Lambda \, r^3}{3}} \right) \left( \frac{r-r_s - \frac{\Lambda \, r^3}{6}}{r-2r_s - \frac{\Lambda \, r^3}{3}} \dot r^2 + \frac{r^2 \, {\dot \Omega}^2}{2} \right) \, , \nonumber \\
\label{11}\end{aligned}$$
where, subscript ‘DS/ADS’ denotes Schwarzschild de-Sitter/anti-de Sitter. With $\Lambda = 0$, the above potential reduces to the potential corresponding to the simplest static Schwarzschild geometry, given in \[36\]. $M \equiv M_{\rm BH}$ is the mass of the BH/central object. The denominator in the second term of the potential contains the exact metric function $f(r)$ of SDS/SADS geometry, and hence the potential $V_{\rm DS/ADS}$ would reproduce the exact location of the event horizon and cosmological horizon and other horizon properties, as that in full general relativity. Introducing a dimensional parameter or cosmological parameter ${\zeta} = {\Lambda r^2_s}/3$, the vanishing cubic polynomial $f(r)$ with repulsive cosmological constant ($\Lambda > 0$) would give two real positive roots representing the locations of two horizons, namely the BH horizon and the cosmological horizon. The locations of these two horizons are then given by $r_H = \frac{2}{\sqrt{3 {\zeta}}} \, \cos \left[\frac{\pi}{3} + \frac{\cos^{-1} \left(3 \, \sqrt{3 {\zeta}} \right) }{3} \right]$ and $r_{\rm CM} = \frac{2}{\sqrt{3 {\zeta}}} \, \cos \left[\frac{\pi}{3} - \frac{\cos^{-1} \left(3 \, \sqrt{3 {\zeta}} \right) }{3} \right]$, respectively.
In the next section, we will analyze different aspects of particle dynamics in the gravitational field of SDS geometry in the modified Newtonian framework.
Orbital dynamics around SDS spacetime
=====================================
In the Newtonian framework, the Lagrangian of a particle in the presence of the SDS analogous potential $V_{\rm DS}$ per unit mass is given by,
$$\begin{aligned}
{\cal L}_{\rm DS} = \frac{1}{2} \left[\frac{r^2 \, {\dot r}^2}{\left(r-2r_s - \frac{\Lambda \, r^3}{3}\right)^2} + \frac{r^3 \, {\dot \Omega}^2}{r-2r_s - \frac{\Lambda \, r^3}{3}} \right] \, + \, \frac{GM}{r} + \frac{\Lambda \, c^2 r^2}{6} \, ,
\label{12}\end{aligned}$$
where, ${\dot \Omega}^2= {\dot \theta}^2 + \sin^2 \theta \, {\dot \phi}^2$. Here overdots denote the derivative with respect to coordinate time ‘$t$’. We then compute the conserved angular momentum and Hamiltonian using $V_{\rm DS}$, given by
$$\begin{aligned}
\lambda_{\rm DS} = \frac{r^3 \, \dot \Omega}{r-2r_s - \frac{\Lambda \, r^3}{3}}
\label{13}\end{aligned}$$
and
$$\begin{aligned}
E_{\rm DS} = \frac{1}{2} \left[\frac{r^2 \, {\dot r}^2}{\left(r-2r_s - \frac{\Lambda \, r^3}{3}\right)^2} + \frac{r^3 \, {\dot \Omega}^2}{r-2r_s - \frac{\Lambda \, r^3}{3}} \right] \, - \, \frac{GM}{r} - \frac{\Lambda \, c^2 r^2}{6} \, ,
\label{14}\end{aligned}$$
respectively. Using Eqs. (13) and (14), we obtain $\dot r$ that uniquely describes the test particle motion, which is given by
$$\begin{aligned}
\frac{dr}{dt} = \frac{r-2r_s - \frac{\Lambda \, r^3}{3}}{r}
\sqrt{2 E_{\rm DS} + \frac{2GM}{r} + \frac{\Lambda \, c^{2} r^2}{3} -
\left(r-2r_s - \frac{\Lambda \, r^3}{3} \right) \frac{\lambda^2_{\rm DS}}{r^3} }\, , \nonumber \\
\label{15}\end{aligned}$$
which is exactly equivalent to $\dot r$ in general relativity in low energy limit. Replacing $\dot \Omega$ and $\dot r$ in Eq. (11) using Eqs. (13) and (15) respectively, SDS analogous potential can be written in terms of conserved Hamiltonian $E_{\rm DS}$ and angular momentum $\lambda_{\rm DS}$, given by
$$\begin{aligned}
V_{\rm DS} = -\left( \frac{GM}{r} + \frac{\Lambda \, c^{2} \, r^{2}}{6} \right)
- \left(2r_s + \frac{\Lambda \, r^3}{3}\right) \left[\left(r-2r_s - \frac{\Lambda \, r^3}{3}\right) \frac{\lambda^2_{\rm DS/ADS}}{r^4} \left(\frac{1}{2} - \frac{r - r_s - \frac{\Lambda \, r^3}{6}}{r} \right)\right] \nonumber \\
- \left(2r_s + \frac{\Lambda \, r^3}{3}\right) \left[\frac{1}{r^2} \left(r - r_s - \frac{\Lambda \, r^3}{6}\right) \left(2 E_{\rm DS/ADS} + \frac{2GM}{r} + \frac{\Lambda \, c^{2} r^2}{3} \right) \right]
\label{16}\end{aligned}$$
In Fig. 1a we show the variation of $V_{\rm DS}$ in the form given in Eq. (16) with the radial distance $r$, corresponding to $\Lambda > 0$ with $\Lambda r^2_s = 1 \times 10^{-27}$ in the low energy limit of test particle motion. The stated value of $\Lambda r^2_s$ corresponds to $\Lambda = 10^{-56} {\rm cm^{-2}}$ for $M_{\rm BH} \sim 10^9 \, M_{\odot}$. Unless stated explicitly, we shall use such a combination of $\Lambda$ and $M_{\rm BH}$ throughout the paper. The profile of $V_{\rm DS}$ clearly shows both the BH event horizon as well as the cosmological horizon. With $\Lambda r^2_s \sim 1 \times 10^{-27}$, cosmological horizon extends up to $\sim 5.5 \times 10^{13} \, r_s$. For a BH of $\sim 10^9 M_{\odot}$, it gives a radius of $\sim 5.5 \times 10^3 \, {\rm megaparsec}$. Figure 1b resembles that of 1a but for semi-relativistic test particle energy ($E_{\rm DS} \sim 0.5$). Although the locations of BH event horizon as well as cosmological horizon remain unaltered with the increase in test particle energy, nevertheless, there is a noticeable change in the magnitude of $V_{\rm DS}$ at both horizon radii. It is found that $\lambda_{\rm DS}$ has no effect on the nature of potential just beyond $\sim 100 \, r_s$. These figures indicate that with the increase in the value $\lambda_{\rm DS}$, inner horizon shifts to larger radii.
Figures 1c and 1d show the variation of effective potential $V^{\rm Eff}_{\rm DS} = V_{\rm DS} + \frac{\lambda^2_{\rm DS}} {2 r^4} \left(r-2r_s - \frac{\Lambda \, r^3}{3}\right)^2$ for the same value of $\Lambda$ as in figures 1a,b, corresponding to SDS spacetime. The nature of the profiles resemble that in figures 1a,b, however $V^{\rm Eff}_{\rm DS}$ attains a higher peak as compared to $V_{\rm DS}$ in the vicinity of inner horizon for values of $\lambda_{\rm DS} \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 3.5 $. The figure 1c shows the variation of ‘GR effective potential’ \[see Eq. (30)\] for the same values of angular momentum.
![Variation of potential $V_{\rm DS}$ \[Eq. (16)\] with radial distance $r$. The solid and long-dashed lines in figure 1a correspond to $V_{\rm DS}$ in low energy limit $E_{\rm DS} \sim 0$ for SDS geometry with $\Lambda r^2_s= 1 \times 10^{-27} $ with $\lambda_{\rm DS} = (0, 9.5)$, respectively. The short-dashed curve represents Schwarzschild spacetime ($\Lambda = 0$). Figure 1b is for semi-relativistic energy $E_{\rm DS} \sim 0.5$, otherwise resemble 1a. Figures 1c and 1d correspond to effective potential $V^{\rm Eff}_{\rm DS}$, with other parameters same to those in figures 1a and 1b, respectively. Figure 1e corresponds to ‘GR effective potential’ for SDS geometry with the same physical parameters as those in figures 1c,d, but independent of conserved energy. $E_{\rm DS}$ and $\lambda_{\rm DS}$ are expressed respectively in units of $c^2$ and $GM/c$. []{data-label="Fig1"}](f1.eps){width="1.0\columnwidth"}
In Fig. 2a, we depict the difference in the magnitude between $V_{\rm DS}$ in Eq. (16) and the analogous potential for Schwarzschild metric in \[36\] $\left(V_{\rm SW}\right)$, with radial distance $r$. For semi-relativistic energy of test particle, the magnitude of $V_{\rm DS}$ may even exceed the corresponding Schwarzschild value by ${\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 1.6 c^2$ at the outer radii near the cosmological horizon. In Fig. 2b, we show the relative deviation (in percentage) between the $V_{\rm DS}$ and $V_{\rm SW}$, as a function of radial distance $r$. For any arbritary physical quantity $\mathcal F$, the relative deviation $(\xi_{\imath})$ is defined as $\xi_{\imath} = 2 \lvert \frac{{\mathcal F}_{\rm DS} - {\mathcal F}_{\rm SW} }{ {\mathcal F}_{\rm DS} + {\mathcal F}_{\rm SW} } \rvert$ which essentially implies deviation between the analogous potential for SDS and Schwarzschild geometry relative to their average value. The subscript ‘SW’ represents corresponding quantities in ‘Schwarzschild’ geometry.
It is found from the Fig. 2b that $\xi_{\imath}$ sharply increases with $r$, especially, at the outer radii. At smaller distances $\xi_{\imath}$ is very small, and detecting such small deviations experimentally does not seem to be possible at present or in near future. At what distance $\xi_{\imath}$ can be said as non-negligible depends entirely on experimental accuracy. The solar system experiments presently are capable to detect any deviation from general relativity at $10^{-5}$ level. If we demand a similar experimental accuracy, we can then say that when $\xi_{\imath}$ attains a value of $\sim 10^{-4}$ (or equivalently $\sim 0.01 \%)$ the relative deviation may be considered to be non-negligible with the corresponding distance $r \sim 8.4 \times 10^7 \, r_s$. With this consideration, this then implies that beyond such radius the influence of $\Lambda$ cannot be neglected. At $r \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 8.5 \times 10^8 \, r_s$, the relative deviation becomes substantial with $\left(\xi_{\imath} \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 10 \% \right)$. Figure 2b also shows that $\xi_{\imath}$ increases rapidly up to a certain radius $r \sim 9 \times 10^9 \, r_s$, beyond which it mostly remains constant. Around this particular radius, the $\Lambda$ effects become too dominating, which renders $V_{\rm SW}$ to become negligible in compare to that of $V_{\rm DS}$. We describe this radius $r \sim 9 \times 10^9 \, r_s$ as some upper bound $\left(\equiv x_{\rm max} \right)$, where $\xi_{\imath}$ attains a values of $\sim 200 \%$. For $M_{\rm BH} \sim 10^9 \, M_{\odot}$, in accordance with the BH mass in many AGNs/quasars or in massive galaxies, the lower bound in $r \left(\equiv x_{\rm min} \right)$, gives a radius of $\sim 8$ kiloparsec, whereas the upper bound $x_{\rm max}$ gives a radius of $\sim 900$ kiloparsec. Thus, the region between $x_{\rm min}$ and $x_{\rm max}$ or a region approximately existing between few kiloparsecs to a few $100$ kiloparsecs would be strongly affected by cosmological constant $\Lambda$, thereby, directly influencing the kiloparsecs-scale structure in massive galaxies, in the local observable Universe. With a similar BH mass, the region beyond $80$ kiloparsec would have most significant $\Lambda$ effects, where $\xi_{\imath}$ would be greater than $\sim 10 \%$.
![Variation of the difference between $V_{\rm DS}$ ($\Lambda r^2_s = 1 \times 10^{-27}$) and $V_{\rm SW}$ with $r$. Solid and short-dashed curves in Fig. 2a show the difference between $V_{\rm DS}$ and $V_{\rm SW}$, corresponding to $E_{\rm DS} \sim 0$ and $E_{\rm DS} \sim 0.5$, respectively. Solid curve in Fig. 2b shows the variation of the relative deviation (in percentage) $\xi_{\imath}$ with $r$, for both $E_{\rm DS} \sim 0$ and $E{\rm DS} \sim 0.5$ (which coincide). The two extreme vertical dashed lines, denoted as $x_{\rm min}$ and $x_{\rm max}$ (in units of $r_s$, correspond to the value of $\xi_{\imath}$ $\sim 0.01 \%$ and $ \sim 200 \%$, respectively). The vertical dashed line in the middle represents $r$ at which $\xi_{\imath} \sim 10 \%$. $E_{\rm DS}$ is expressed in units of $c^2$. Note that $\lambda_{\rm DS}$ has no effect on the nature of potentials when $r > 100 \, r_s$. []{data-label="Fig2"}](f2.eps){width="1.0\columnwidth"}
Next we obtain the equation of the orbital trajectory using Eqs. (13) and (15), as given by
$$\begin{aligned}
\left(\frac{dr}{d\Omega}\right)^2 = \frac{r^4}{\lambda^2_{\rm DS/ADS}} \left[2 E_{\rm DS/ADS} + \frac{2GM}{r} + \frac{\Lambda \, c^{2} r^2}{3}
- \left(r-2r_s - \frac{\Lambda \, r^3}{3} \right) \frac{\lambda^2_{\rm DS/ADS}}{r^3} \right] \, . \nonumber \\
\label{17}\end{aligned}$$
Although Eq. (17) is derived with the condition of low energy limit, yet it is exactly the same as that in full general relativity which one can easily obtain using Eqs. (4) and (7) with relevant $f(r)$. To furnish a complete behavior of the particle dynamics in SDS background, we obtain the equations of motion of the test particle in the presence of $V_{\rm DS}$ from the Euler-Lagrange equations in spherical geometry which are given by
$$\begin{aligned}
\ddot r = \left( -\frac{GM}{r^2} + \frac{\Lambda \, c^2 r}{3} \right)
\left(\frac{r-2r_s - \frac{\Lambda \, r^3}{3}}{r} \right)^{2}
\, + \, \frac{2 \left(r_s - \frac{\Lambda \, r^3}{3} \right)}{r \left(r-2r_s - \frac{\Lambda \, r^3}{3} \right)} \, {\dot r}^2 + \left(r-3 r_s \right)
\left({\dot \theta}^2 + \sin^2 \theta \, {\dot \phi}^2 \right) \, ,
\label{18}\end{aligned}$$
$$\begin{aligned}
\ddot \phi = -\frac{2 \, \dot r \dot \phi}{r} \left(\frac{r-3r_s}{r - 2 r_s - \frac{\Lambda \, r^3}{3}} \right) \, - \, 2 \, \cot \theta \, \dot \phi \, \dot \theta
\label{19}\end{aligned}$$
and
$$\begin{aligned}
\ddot \theta = -\frac{2 \, \dot r \dot \theta}{r} \left(\frac{r-3r_s}{r - 2 r_s - \frac{\Lambda \, r^3}{3}} \right) \, + \, \sin \theta \, \cos \theta \,
{\dot \phi}^2 \, ,
\label{20}\end{aligned}$$
respectively. $\ddot \phi$ and $\ddot \theta$ equations are exactly the same to that in general relativity, whereas $\ddot r$ equation in (18) corresponds to that in general relativity in the low energy limit. The corresponding $\ddot r$ equation in general relativity is given by
$$\begin{aligned}
\ddot r = \left( -\frac{GM}{r^2} + \frac{\Lambda \, c^2 r}{3} \right)
\left(\frac{r-2r_s - \frac{\Lambda \, r^3}{3}}{r} \right)^{2} \,
\frac{c^4}{\epsilon^2}
\, + \, \frac{2 \left(r_s - \frac{\Lambda \, r^3}{3} \right)}{r \left(r-2r_s - \frac{\Lambda \, r^3}{3} \right)} \, {\dot r}^2 + \left(r-3 r_s \right)
\left({\dot \theta}^2 + \sin^2 \theta \, {\dot \phi}^2 \right) \, .
\label{21}\end{aligned}$$
It needs to be mentioned that previously a PNP has been prescribed in \[34\] corresponding to SDS geometry based on a method adopted by \[30-32\], which has been formulated by considering a Keplerian rotation profile of the test particle motion. The form of the PNP is given by
$$\begin{aligned}
\Phi (r) = \frac{r^3 \frac{\Lambda}{3} - 3 r \left(\frac{\Lambda}{3}\right)^{1/3} + 2}{2 \left[1-3\left(\frac{\Lambda}{3}\right)^{1/3} \right] \left(2-r+ r^3 \frac{\Lambda}{3}\right) } \, ,
\label{22}\end{aligned}$$
which is derived on the premise that $\Phi (r) = 0$ at the static radius, preserving the analogy that the gravitational potential tends to zero in asymptotically flat spacetime. This potential does not have any dependence on test particle velocity. With $\Lambda = 0$, the PNP in Eq. (22) reduces to that of Paczyński-Witta potential corresponding to Schwarzschild geometry. The corresponding GR behavior is mimicked through this PNP by intending only to reproduce the marginally stable and bound orbits for circular orbital trajectory. This is in sharp contrast with the velocity dependent potential $V_{\rm DS}$, which, a priori, focused on to replicate the general relativity by resembling the geodesic equations of motion. This ensures that most of the GR features could be reproduced accurately. The equation of the orbital trajectory and the equations of motion for the corresponding PNP in Eq. (22) in spherical geometry are then given by
$$\begin{aligned}
\left(\frac{dr}{d\Omega}\right)^2 = \frac{r^4}{\lambda^2} \left[2 E + \frac{r^3 \frac{\Lambda}{3} - 3 r \left(\frac{\Lambda}{3}\right)^{1/3} + 2}{2 \left[1-3\left(\frac{\Lambda}{3}\right)^{1/3} \right]\left(2-r+ r^3 \frac{\Lambda}{3}\right)}
- \frac{\lambda^2}{r^3} \right] \, ,
\label{23}\end{aligned}$$
$$\begin{aligned}
\ddot r = - \frac{\left(1- r^3 \frac{\Lambda}{3}\right)}{\left(2-r+ r^3 \frac{\Lambda}{3}\right)^2 } +
r \left({\dot \theta}^2 + \sin^2 \theta \, {\dot \phi}^2 \right) \, ,
\label{24}\end{aligned}$$
$$\begin{aligned}
\ddot \phi = -\frac{2 \, \dot r \dot \phi}{r} - \, 2 \, \cot \theta \, \dot \phi \, \dot \theta
\label{25}\end{aligned}$$
and
$$\begin{aligned}
\ddot \theta = -\frac{2 \, \dot r \dot \theta}{r} + \, \sin \theta \, \cos \theta \,
{\dot \phi}^2 \, ,
\label{26}\end{aligned}$$
It is seen that the trajectory equation as well as the equations of motion corresponding to the PNP in Eq. (22) do not at all resemble the equivalent relations in general relativity and hence this PNP cannot reproduce the features of SDS spacetime accurately. Notwithstanding, in the next few sections we would compare the dynamical behavior of the test particle motion obtained with $V_{\rm DS}$ and the PNP in Eq. (22) as well as with full general relativity, over the entire spatial regime relevant for SDS background.
Particle dynamics along circular orbit
--------------------------------------
In order to compare the behavior of the particle motion in presence of $V_{\rm DS}$ and those in general relativity, we compute the dynamical variables for the simplest circular orbit trajectory. With the conditions for the circular orbits $\dot r = 0$ and $\ddot r=0$, we obtain corresponding angular momentum $\lambda^C_{\rm DS}$, Hamiltonian $E^C_{\rm DS}$ and the orbital angular velocity $\dot \Omega^{C}_{\rm DS}$ with $V_{\rm DS}$ using Eqs. (13), (15) and (18), given by
$$\begin{aligned}
\lambda^C_{\rm DS} = r \sqrt{\frac{GM - \frac{\Lambda \, c^{2} r^3}{3}}{r-3r_s} } \, ,
\label{27}\end{aligned}$$
$$\begin{aligned}
E^C_{\rm DS} = - \frac{GM}{2r} \left( \frac{r-4r_s}{r-3r_s} \right) +
\left[\frac{\Lambda \, c^2 r^3}{6(r-3r_s)} \right] \, \left(\frac{\Lambda r^2}{3} + \frac{4r_s}{r} - 2 \right) \,
\label{28}\end{aligned}$$
and
$$\begin{aligned}
\dot \Omega^{C}_{\rm DS} = \frac{r - 2r_s - \frac{\Lambda \, r^3}{3} }{r^2}
\sqrt{\frac{GM - \frac{\Lambda \, c^2 r^3}{3}}{r-3r_s} } \, ,
\label{29}\end{aligned}$$
respectively. With $\Lambda = 0$, all the above equations get reduced to that obtained in Schwarzschild geometry. To compute these variables in exact SDS geometry we use the corresponding ‘GR effective potential’, given by
$$\begin{aligned}
V^{\rm GR}_{\rm eff} \, (r) = \left(1 - \frac{2 r_s}{r} - \frac{\Lambda \, r^2}{3}\right) \left(c^2 + \frac{\lambda^2}{r^2} \right) \, .
\label{30}\end{aligned}$$
As usual, circular orbits occur in general relativity when ${dr}/{d\tau} = 0 $ and ${\partial V^{\rm GR}_{\rm eff}}/{\partial r} = 0$. We obtain energy $\epsilon$ for particle motion in circular orbit, given by
$$\begin{aligned}
\frac{\epsilon}{c^2} = {\frac{\left(r - 2r_s - \frac{\Lambda \, r^3}{3}\right)}{\sqrt{r (r-3r_s)}}} \, .
\label{31}\end{aligned}$$
Angular momentum ($\lambda^{C}$) and the equivalent Hamiltonian $E^C$ $\left[= \left(\epsilon^2 - c^4 \right)/{2c^2} \right]$ for circular orbits in general relativity then exactly resemble the corresponding values obtained with $V_{\rm DS}$, given by Eqs. (27) and (28) respectively. The orbital angular velocity in general relativity is then given by
$$\begin{aligned}
\dot \Omega^{C} = \sqrt{\frac{GM}{r^3} - \frac{\Lambda \, c^2}{3} } \, ,
\label{32}\end{aligned}$$
whose analytical expression is not exactly equivalent to $\dot \Omega^{C}_{\rm DS}$ in Eq. (29). Note that the circular orbit corresponding to SDS metric exists down to $3 \, r_s$ in similarity to that in Schwarzschild geometry which represents the null hypersurface. The photon orbit is thus independent of the cosmological constant. For PNP in Eq. (22), the corresponding expressions of $\lambda^C$, $E^C$ and $\dot \Omega^C$ are given in \[34\]. The said PNP does not reproduce the correct photon orbit.
In Fig. 3 we show the appropriate comparison of the nature of potential $V_{\rm DS}$ in Eq. (11) and that of PNP in Eq. (22), corresponding to test particle motion in circular orbit. In the very inner and outer regions near to both the BH and cosmological horizons, the behavior of these potentials differ significantly (figures 3c,d). One of the major distinctive features of the PNP in Eq. (22) is that it becomes zero at the static radius $\left(\sim 1.4 \times 10^9 \, r_s \right)$, whereas $V_{\rm DS}$ attains a value of $\, \sim - 10^{-9} \, c^2$ (Fig. 3b). In the intermediate region, the nature of the potentials remain mostly similar. It is seen that at $r \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 10^8 \, r_s$, where the effect of $\Lambda$ is prominent, the PNP in Eq. (22) differs significantly as compared to $V_{\rm DS}$. This radius approximately resembles $x_{\rm min}$ in Fig. 2b.
![Variation of potential $V_{\rm DS}$ and the PNP in Eq. (22) with $r$ corresponding to circular motion trajectory for $\Lambda r^2_s= 1 \times 10^{-27}$. The solid and short-dashed lines correspond to $V_{\rm DS}$ in Eq. (11) and PNP in Eq. (22), respectively. Figure 3a covers the entire distance, figures 3b,c focus only on the outer radii; near to the static radius and the cosmological horizon, respectively, whereas figure 3d focuses on the inner radii near the BH event horizon. []{data-label="Fig3"}](f3.eps){width="1.0\columnwidth"}
In Fig. 4, we furnish the variation of $\lambda^C_{\rm DS}$ corresponding to $V_{\rm DS}$ which exactly coincides with that in general relativity, for the entire spatial regime. In Fig. 4b, we compare the nature of $\lambda^C_{\rm DS}$ with that corresponding to the PNP in Eq. (22) at the outer radii. The figures show that for SDS spacetime, there is a clear static radius at the outer radii where angular momentum abruptly falls to zero value. For BH of $\sim 10^9 M_{\odot}$, with $\Lambda = 10^{-56} {\rm cm^{-2}}$, the static radius will be located at $\sim 140 \, {\rm kiloparsec}$. Figure 4b shows that the angular momentum for circular orbit trajectory corresponding to $V_{\rm DS}$ as well as for the PNP in Eq. (22) behaves quite similarly at the outer radii, and coincides near the static radius. However, due to the profound effect of $\Lambda$ on the outer radii, the nature of $\lambda^C_{\rm DS}$ at the outer radii differs significantly from that corresponding to $V_{\rm SW}$, especially beyond $4 \times 10^8 \, r_s$. In the inner radii, where the effect of $\Lambda$ is negligible, $\lambda^C_{\rm DS}$ show appreciable deviation from the angular momentum profile corresponding to the PNP in Eq. (22). For a comparison of the profiles of different PNPs in the inner radii, the readers should see \[36\]. Resembling Fig. 2b, in Fig. 5, we show the relative deviation $\left(\xi_{\imath}\right)$ in angular momentum as a function of radial distance $r$, showing approximately similar behavior as that in Fig. 2b. For this profile (Fig. 5), we found that the value of $x_{\rm min} \sim 8.4 \times 10^7 $ and the value of $x_{\rm max} \sim 1.4 \times 10^9 $, corresponding to the value of $\xi_{\imath}$ $\sim 0.01\%$ and $\sim 200 \%$, respectively. At $x_{\rm max}$, which is also the static radius, the curve gets truncated. At $r \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 8 \times 10^8 \, r_s$, where the effect of $\Lambda$ is significant, large relative deviation (${\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 10 \%$) in angular momentum profile is noticed.
![Variation of angular momentum with $r$ for circular orbit trajectory. The short-dashed curve in Fig. 4a corresponds to $V_{\rm DS}$ with $\Lambda r^2_s = 1 \times 10^{-27}$ that coincides with the corresponding GR results. The solid curves in figures 4a and 4b are due to $V_{\rm SW}$. The long-dashed line in Fig. 4b corresponds to $V_{\rm DS}$ which coincides with the angular momentum profile corresponding to the PNP given in Eq. (22). []{data-label="Fig4"}](f4.eps){width="1.0\columnwidth"}
![Radial variation of the relative deviation (in percentage) in angular momentum. The two extreme vertical dashed lines, represented by $x_{\rm min}$ and $x_{\rm max}$, correspond to value of $\xi_{\imath}$ $\sim 0.01 \%$ and $\sim 200 \%$, respectively. The curve is truncated at $x_{\rm max}$ which also corresponds to the static radius. The vertical dashed line in the middle represents $r$ at which $\xi_{\imath}$ $\sim 10 \%$. []{data-label="Fig5"}](f5.eps){width="1.0\columnwidth"}
Figure 6 show the variation of conserved Hamiltonian $E^C_{\rm DS}$ with $r$, which is compared with the Hamiltonian corresponding to the PNP in Eq. (22) and the Hamiltonian corresponding to $V_{\rm SW}$. It needs to be noted that $E^C_{\rm DS}$ corresponding to $V_{\rm DS}$ exactly resembles the corresponding expression in general relativity. In the outer region of the test particle motion in circular orbit, where the effect of repulsive $\Lambda$ is prominent (figures 6b,c), there is a marked difference between the profile of conserved Hamiltonian corresponding to $V_{\rm DS}$ and that corresponding to the PNP in Eq. (22), especially at $r \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 5 \times 10^7 \, r_s$. This radius $r \sim 5 \times 10^7 \, r_s$ approximately resembles the location $x_{\rm min}$, the lower bound in $r$, at which the relative deviation $\left(\xi_{\imath}\right)$ in the Hamiltonian attains the value of $\sim 0.01 \%$ (see Fig. 6d). In fact, the value of $x_{\rm min}$ in this case is $\sim 5.3 \times 10^7 $. Around static radius $\left( \sim 10^9 \, r_s \right)$, the Hamiltonian corresponding to the PNP in Eq. (22) attains zero value (Fig. 6b), unlike the case of $E^C_{\rm DS}$. In the vicinity of cosmological horizon $\left(\sim 5.5 \times 10^{13} \, r_s \right)$, the Hamiltonian corresponding to the PNP in Eq. (22) diverges, showing a contrasting behavior with respect to the profile of $E^C_{\rm DS}$. $E^C_{\rm DS}$ after attaining a certain negative value becomes zero at a large outer radius, which is the outermost bound orbit (Fig. 6c). Due to the substantial effect of $\Lambda$, $E^C_{\rm DS}$ deviates significantly from that corresponding to $V_{\rm SW}$ at $r \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 5.5 \times 10^8 \, r_s$ (Fig. 6b), which precisely resembles the radius at which $\xi_{\imath}$ attains the value of $\sim 10 \%$ (Fig. 6d). Resembling Fig. 2b, here too, $x_{\rm max}$ is located at that radius where $\xi_{\imath}$ attains a value of $\sim 200 \%$. The value of $x_{\rm max}$ is then $\sim 7.5 \times 10^9$.
In the inner radii of the test particle in the circular orbit, where the effect of $\Lambda$ is negligible, $E^C_{\rm DS}$ shows appreciable deviation from the Hamiltonian profile corresponding to the PNP in Eq. (22). We do not show the corresponding profiles separately in this region, as already been furnished in \[36\].
![Variation of Hamiltonian with $r$ for test particle in circular orbit corresponding to $\Lambda r^2_s = 1 \times 10^{-27}$. The solid and long-dashed lines in Fig. 6a are for the Hamiltonian corresponding to $V_{\rm SW}$ and $V_{\rm DS}$, respectively. The profile of $E^C_{\rm DS}$ corresponding to $V_{\rm DS}$ coincides with that in general relativity. The solid, long-dashed and short-dashed lines in figures 6b,c are for the Hamiltonian corresponding to $V_{\rm SW}$, $V_{\rm DS}$ and the PNP in Eq. (22), respectively. Figures 6b,c focus on the outer radii; near to the static radius and the cosmological horizon, respectively. Fig. 6d resembles Fig. 2b and Fig. 5, however, showing the the relative deviation in Hamiltonian (in percentage), as a function of radial distance $r$. []{data-label="Fig6"}](f6.eps){width="1.0\columnwidth"}
In Fig. 7a, instead of the profile of orbital angular velocity $\dot \Omega^{C}_{\rm DS}$, we depict the profile of corresponding angular frequency $\omega^{C}_{\rm DS} \left(= \dot \Omega^{C}_{\rm DS}/{2\pi} \right)$ in $r$, which truncates at the static radius at $\sim 1.4 \times 10^9 \, r_s$. At outer radii where the effect of $\Lambda$ is quite substantial, the value of angular frequency of the particle in circular orbit is negligible. Owing to which, $\omega^{C}_{\rm DS}$, its corresponding GR counterpart, angular frequency corresponding to $V_{\rm SW}$ and the angular frequency corresponding to the PNP in Eq. (22), all coincide. Hence, we do not show the angular frequency profile of the test particle motion in the circular orbit separately, at the outer radii, as well the corresponding relative deviation profile. However, in the inner radii, where the effect of $\Lambda$ is negligible, $\omega^{C}_{\rm DS}$ deviates from the corresponding GR result by a maximum of $6 \%$. Moreover, the angular frequency profile of the particle in the circular orbit corresponding to the PNP in Eq. (22) shows a huge deviation from the profile of $\omega^{C}_{\rm DS}$, as well as from the corresponding GR result, with an error margin of $\sim 75 \%$. The corresponding orbital angular velocity profiles in the inner radii have already been shown in \[36\].
![The variation of the angular frequency and the epicyclic frequency with $r$ for the test particle in circular orbit. Solid, long-dashed and short-dashed curves in Fig. 7a depict angular frequency profiles corresponding to $V_{\rm DS}$ with $\Lambda r^2_s= 1 \times 10^{-27}$, the GR counterpart, and the PNP in Eq. (22) respectively. Note that $\omega^{C}_{\rm DS}$ coincides with the angular frequency profile corresponding to $V_{\rm SW}$, both in the inner as well as at the outer radii. Figure 7b depicts the variation of epicyclic frequency with $r$, corresponding to $V_{\rm DS}$ []{data-label="Fig7"}](f7.eps){width="1.0\columnwidth"}
Orbital perturbation and apsidal precession
-------------------------------------------
As discussed in the previous subsection about the importance of the repulsive cosmological constant on the dynamical behavior of the particle orbits and their stability at the outer radii near the static radius, it is necessary for us to investigate the perturbative effects on the orbital dynamics, which would indeed be affected by the positive cosmological constant around the cosmological horizon. The perturbative effects would plausibly have substantial influence on the accretion flow stability in the outer regions in SDS background, in the expanding Universe. We follow our analysis by computing the epicyclic frequency for small perturbation of the particle orbit in circular trajectory. It should be reminded that till date there is no exclusive analytical relation of epicyclic frequency in full general relativity corresponding to SDS spacetime, unlike that in the case of Schwarzschild and Kerr BHs \[38\]. Henceforth, we evaluate the relation for epicyclic frequency using our correct SDS analogous potential $V_{\rm DS}$. We restrict ourselves in the equatorial plane of test particle orbit in a circular trajectory. We then perturb $r$ and $\phi$ and their derivatives accordingly: $r \to r + \delta r, \, \dot r \to \delta {\dot r}, \, \ddot{r} \to \delta \ddot{r}$ and $\phi \to \phi + \delta \phi, \, {\dot \phi} \to {\dot \phi} \, {\vert}_C + \delta {\dot \phi}, \, \ddot{\phi} \to \delta {\ddot{\phi}} $ respectively. Using Eqs. (18), (19) and (20), the linearized perturbed equations of motion are then obtained as
$$\begin{aligned}
\delta {\ddot{r}} = \delta r \, \left[{\dot \phi}^2 \, {\vert}_C + \left(r-2r_s - {\Lambda r^3}{3} \right) \left[\frac{2GM}{r^5} (r-4r_s) +
\frac{\Lambda c^2}{3r} \left(1+\frac{4r_s}{r} - \frac{5 \Lambda r^2}{3} \right) \right] \right] \,
+ 2 {\dot \phi} \, {\vert}_C \, (r - 3r_s) \delta {\dot{\phi}}\, ,
\label{33}\end{aligned}$$
$$\begin{aligned}
\delta {\ddot{\phi}} = -\frac{2 {\dot \phi} \, {\vert}_C }{r} \, \left(\frac{r - 3r_s}{r - 2 r_s - \frac{\Lambda r^3}{3} } \right) \delta{\dot r}
\label{34}\end{aligned}$$
and
$$\begin{aligned}
\delta {\ddot{\theta}} = - {\dot \phi}^2 \, {\vert}_C \, {\delta \theta} \, ,
\label{35}\end{aligned}$$
respectively. Here, ${\dot \phi} \, {\vert}_C$ is the orbital angular velocity of the particle motion in the circular orbit in equatorial plane. We concentrate ourselves in computing only the radial epicyclic frequency as it is more significant. Using Eqs. (33) and (34), with the following relations of perturbed quantities $\delta r = \delta r_0 \exp^{\imath \kappa t}$ and $\delta \phi = \delta \phi_0 \exp^{\imath \kappa t}$ for harmonic oscillations, where $\kappa$ is the radial epicyclic frequency and $\delta r_0$ and $\delta \phi_0$ are amplitudes (see \[38\]), we eventually obtain the expression for radial epicyclic frequency $\kappa$ corresponding to $V_{\rm DS}$ after rigorous algebra, given by
$$\begin{aligned}
\kappa = \left(\frac{r-2r_s -\frac{\Lambda r^3}{3} }{r - 3r_s} \right)^{1/2} \, \sqrt{ \left[ \frac{GM}{r^5} (r-6r_s)(r-2r_s)
- \frac{\Lambda c^2}{3r^2} \left[2 (r-4r_s)(2r-3r_s) - \frac{\Lambda r^3}{3} (4r-15r_s) \right] \right] } \, .
\label{36}\end{aligned}$$
The relation in Eq. (36) reduces to that in Schwarzschild geometry with $\Lambda = 0$ whose value lies within a maximum error of just $\sim 6 \%$, as compared to the GR result. Although the radial epicyclic frequency $\kappa$ has been derived with SDS analogous potential, we anticipate that this relation could also be accurately used in full GR framework, as the effect of $\Lambda$ can only be realized at large outer radii from the central object, where the relativistic effect will itself be diminishing. As no other effective expressions for epicyclic frequency corresponding to SDS geometry is known, in Fig. 7b, we only show the variation of epicyclic frequency corresponding to $V_{\rm DS}$ with $r$. Resembling $\omega^{C}_{\rm DS}$, the value of $\kappa$ at the outer radii is negligible. Consequently, the profile of epicyclic frequency corresponding to $V_{\rm DS}$ coincides with that corresponding to $V_{\rm SW}$, both in the inner as well as at the outer radii, and hence the corresponding relative deviation would also be negligible. In the inner radii, where the effect of $\Lambda$ is negligible, the PNP in Eq. (22) reduces to Paczyński-Witta potential. The epicyclic frequency profile corresponding to this potential shows a huge deviation from the profile corresponding to $V_{\rm DS}$ by over an error margin of $\sim 75 \%$ (see \[36\]).
Before analyzing the dynamics of orbital trajectory, we display the profile of the radial velocity $v_r$ (see Fig. 8) of the test particle falling radially towards the central BH in the SDS background. The expression of $v_r$, here, solely depends on the energy $E_{\rm DS}$ of the test particle motion. In figures 8a,b we show the profiles of $v_r$ corresponding to $V_{\rm DS}$, $V_{\rm SW}$, and the PNP in Eq. (22), for the test particle with $E_{\rm DS} \sim 0$ and $E_{\rm DS} \sim 0.5$, respectively. Expression for $v_r$ corresponding to $V_{\rm DS}$ coincides with the GR expression. In figures 8c,d, we depict the detailed nature of the corresponding profiles at the outer radii which is our region of interest, where the effect of $\Lambda$ is significant. The figures show that $v_r$ profiles corresponding to the PNP in Eq. (22) distinctly differ; rather show opposite behavior from $v_r$ corresponding to $V_{\rm DS}$ at the outer radii, at $r \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 6 \times 10^{12} \, r_s$. For semi-relativistic/relativistic test particle energy, the radial velocity corresponding to the PNP in Eq. (22) even surpasses velocity of light at that radii, thus showing physically inconsistent behavior (figures 8b,d). In figures 8e,f, we show the profiles of relative deviation $\left(\xi_{\imath} \right)$ in $v_r$, corresponding to $E_{\rm DS} \sim 0$ and $E_{\rm DS} \sim 0.5$, respectively. $x_{\rm min}$ and $x_{\rm max}$ corresponding to $v_r$ for $E_{\rm DS} \sim 0$ are located at $r \sim 1 \times 10^{8} \, r_s$ and $r \sim 5.8 \times 10^{10} \, r_s$, at which $\xi_{\imath}$ attains the value of $\sim 0.01 \%$ and $\sim 200 \%$, respectively. However, the relative deviation corresponding to semi-relativistic/relativistic test particle energy (Fig. 8f) show marked difference as compared to test particle motion in the low energy limit, unlike that in Fig. 2b. For the semi-relativistic/relativistic energy, $v_r$ corresponding to $V_{\rm DS}$ deviates from that corresponding to $V_{\rm SW}$ at much outer radii. $x_{\rm min}$ and $x_{\rm max}$ thus move further outward attaining the value of $\sim 7 \times 10^{11}$ and $\sim 5.5 \times 10^{13}$, where $\xi_{\imath}$ is $\sim 0.01 \%$ and $\sim 200 \%$, respectively. In Fig. 8f, the curve gets truncated at $x_{\rm max}$ which is also the cosmological horizon.
The dynamics of orbital trajectory can be obtained from Eq. (17), which is identical to that obtained in general relativity, implying that the nature of trajectory of particle orbit obtained with $V_{\rm DS}$ would be similar to that in general relativity. In Fig. 9 we show the profile of ${dr}/{d\Omega}$ with $r$ corresponding to $\Lambda r^2_s = 1 \times 10^{-27}$. The profiles indicate that for semi-relativistic/relativistic test particle energy, ${dr}/{d\Omega}$ corresponding to $V_{\rm DS}$ deviate from ${dr}/{d\Omega}$ corresponding to $V_{\rm SW}$ at much outer radii (figures 9b,d), as compared to test particle energy in the low energy limit (figures 9a,c). ${dr}/{d\Omega}$ profiles corresponding to PNP in Eq. (22) also show different behavior, from that corresponding to $V_{\rm DS}$.
![Variation of radial velocity of the test particle falling radially towards the central BH. Solid, long-dashed and short-dashed curves in Fig. 8a are for $v_r$ corresponding to $V_{\rm DS}$ with $\Lambda r^2_s = 1 \times 10^{-27}$, $V_{\rm SW}$ and PNP in Eq. (22), respectively, for test particle with $E_{\rm DS} \sim 0$. Figure 8b is similar to that in 8a, however, for $E_{\rm DS} \sim 0.5$. In figures 8c,d which resemble figures 8a,b, the detailed nature of the corresponding profiles at the outer radii is depicted. In figures 8e,f we show the relative deviation (in percentage) in $v_r$, corresponding to $E_{\rm DS} \sim 0$ and $E_{\rm DS} \sim 0.5$, respectively. The vertical lines in the figures are same to those in Fig. 2b. $E_{\rm DS}$ is expressed in units of $c^2$. []{data-label="Fig8"}](f8.eps){width="1.0\columnwidth"}
![Variation of ${dr}/{d\Omega}$ in radial direction $r$ corresponding to $\Lambda r^2_s = 1 \times 10^{-27}$. The solid, long-dashed and short-dashed curves in all the figures correspond to $V_{\rm DS}$, PNP in Eq. (22), and $V_{\rm SW}$. Figures 9a and 9c are for $E_{\rm DS} \sim 0$ corresponding to angular momentum $\lambda_{\rm DS} =(1.1, 9.5)$, respectively. Figures 9b and 9d resemble figures 9a,c, however for $E_{\rm DS} \sim 0.5$. $E_{\rm DS}$ and $\lambda_{\rm DS}$ are expressed in units of $c^2$ and $GM/c$, respectively. []{data-label="Fig9"}](f9.eps){width="1.0\columnwidth"}
![The profile of elliptical like trajectory of particle orbit in equatorial plane due to $V_{\rm DS}$ with $\Lambda r^2_s = 1 \times 10^{-27}$. In Fig. 10a, the solid line is due to $V_{\rm SW}$ and the dotted line for $V_{\rm DS}$. The particle trajectory coincides with that in general relativity. Figure 10b shows the profile of perihelion advancement $\Psi$ as a function of eccentricity corresponding to $V_{\rm DS}$ for $r_a = 8 \times 10^8 \, r_s$. The solid and long-dashed curves correspond to $V_{\rm SW}$ and $V_{\rm DS}$ respectively. []{data-label="Fig10"}](f10.eps){width="1.0\columnwidth"}
In Fig. 10a, the elliptical-like trajectory of particle orbit in the $x-y$ plane due to $V_{\rm DS}$, along with that due to $V_{\rm SW}$, obtained from the equations of motion are shown. The particle starts with a tangential initial velocity of $1.58114 \times 10^{-5} \, c$ from an apoapsis $r_a = 8 \times 10^8 \, r_s$ attaining a constant eccentricity of magnitude $0.8$. We obtained the plot of elliptical trajectory using Cartesian transformation adopting the method of Euler-Cromer algorithm which preserves energy conservation. The distinct precessional effect can be seen around $\sim 80 \, {\rm kiloparsec}$. This is precisely the radius at which the $\xi_{\imath}$ corresponding to the potential (Fig. 2b) becomes greater than $\sim 10 \%$. In Fig. 10b, we show the apsidal precession or the perihelion advancement $\Psi$ \[given by Eq. (37)\] as a function of eccentricity which is identical to the GR results. This also guarantees that the photon orbit trajectory obtained with $V_{\rm DS}$ would resemble that in general relativity, which is independent of the cosmological constant.
$$\begin{aligned}
\Psi = \Pi - \pi \equiv \int_{r_p}^{r_a} \frac{d \phi}{dr} \, dr - \pi \, ,
\label{37}\end{aligned}$$
where $\Pi$ is the usual half orbital period of the test particle. $r_p$ and $r_a$ are periapsis and apoapsis of the orbit respectively.
Stability and boundedness of circular orbit
-------------------------------------------
We can obtain the last stable circular orbit using potential $V_{\rm DS}$ with the condition ${d\lambda^{C}_{\rm DS}}/{dr} = 0$ or equivalently
$$\begin{aligned}
3 r_s r - 18 r^2_s - 4 \Lambda r^4 + 15 \Lambda r^3 r_s = 0 \, ,
\label{38}\end{aligned}$$
which is exactly similar to that obtained in full general relativity. Thus, $V_{\rm DS}$ would accurately reproduce the last stable circular orbit in general relativity. Also the last bound circular orbit is obtained from Eq. (28) with $E^C_{\rm DS} =0$, and by virtue, is similar to that in full general relativity. The equivalent relation is
$$\begin{aligned}
9 r r_s - 36 r^2_s + 6 \Lambda r^4 - 12 \Lambda r^3 r_s - \Lambda^2 r^6 = 0 \, .
\label{39}\end{aligned}$$
For cosmological parameter ${\zeta} = 0$, the familiar circular orbit stability limit and the last bound circular orbit for the Schwarzschild metric is recovered. Unlike the case in the Schwarzschild geometry, circular orbits here become unstable at both inner and outer radii. Equation (38) gives two real positive roots. One of the root is the last stable circular orbit in the inner radii resembling the usual Schwarzschild case. The other root is the maximum possible circular stable orbit $r^{\rm SC}_{\rm max}$ at the outer radii which is located near to the static radius. The location of $r^{\rm SC}_{\rm max}$ is $\sim 0.88 \times 10^9 \, r_s$, whereas the corresponding static radius is located at $\sim 1.4 \times 10^9 \, r_s$. Resembling the case for stable circular orbit, Eq. (39) gives two real positive roots, corresponding to the marginally bound circular orbit in the inner radii and the maximum possible bound circular orbit $r^{\rm BC}_{\rm max}$ at the outer radii. Inner marginally bound circular orbit is the usual limit corresponding to Schwarzschild geometry. PNP in Eq. (22), too, accurately reproduces both marginally stable and bound orbits corresponding to SDS geometry. However, it needs to be noted that the test particle in SADS background with negative $\Lambda$ does not yield any outermost stable or outermost bound circular orbit.
Although we have studied the entire test particle dynamics with $\Lambda r^2_s = 1\times 10^{-27}$ for BH of $\sim 10^9 M_{\odot}$, however, in Fig. 11, we will depict the variation of $r^{\rm SC}_{\rm max}$ and $r^{\rm BC}_{\rm max}$ with $\Lambda r^2_s$, and the corresponding values of the dynamical variables for the test particle motion in circular orbit, in order to see the effect of $\Lambda$ on the stability of the orbits. With the decrease in value of $\Lambda r^2_s$, both $r^{\rm SC}_{\rm max}$ and $r^{\rm BC}_{\rm max}$ move further outwards. For all values of $\Lambda r^2_s$, $r^{\rm BC}_{\rm max}$ is located at much outer radii as compared to that of $r^{\rm SC}_{\rm max}$. We show the variation of $\lambda^{C}_{\rm DS}, E^C_{\rm DS}$ and $\omega^{C}_{\rm DS}$ along $r^{\rm SC}_{\rm max}$ corresponding to various values of $\Lambda r^2_s$ in figures 11b,c,d, respectively, which are identical to the GR results, as well as corresponding to the PNP in Eq. (22). On the other hand, the corresponding dynamical variable do not furnish any real values along $r^{\rm BC}_{\rm max}$. On the contrary, the values of the dynamical variables corresponding to $V_{\rm DS}$ in the innermost stable and innermost bound orbits where the effect of $\Lambda$ is negligible, differ considerably from that corresponding to the PNP in Eq. (22), as discussed in \[36\].
![Variation of the location of outermost stable circular and outermost bound circular orbit with repulsive $\Lambda$, obtained with $ V_{\rm DS}$. The solid and dashed lines in Fig. 11a show the variation of $r^{\rm SC}_{\rm max}$ and $r^{\rm BC}_{\rm max}$ with $\Lambda$, respectively. Figures 11b,c,d show the variation of Hamiltonian, angular momentum and angular frequency along $r^{\rm SC}_{\rm max}$ with corresponding values of $\Lambda$, obtained with $V_{\rm DS}$. The y-axis in Fig. 11a is in logarithmic-scale. []{data-label="Fig11"}](f11.eps){width="1.0\columnwidth"}
For supermassive BHs (SMBHs) in AGNs or quasars with $\left(10^6 M_{\odot} \, {\raisebox{-.4ex}{$\stackrel{<}{\scriptstyle \sim}$}}\, M_{\rm BH} \, {\raisebox{-.4ex}{$\stackrel{<}{\scriptstyle \sim}$}}\, 10^9 M_{\odot} \right)$, the current accepted value of $\Lambda$ yields a maximum possible radius of the outermost stable circular orbit in the range of \[$\sim (0.9 - 88)$\] kiloparsec. This also implies that the outer edge of the largest Keplerian accretion disk in AGNs/Quasars will never exceed length-scale of kiloparsecs. The instability of circular orbits at large outer radii can have serious astrophysical consequences. In the next section, we analyze the influence of $\Lambda$ on a simplistic astrophysical phenomenon, Bondi accretion rate.
A comparative analysis
----------------------
In the last section, we have exemplified that the test particle dynamics corresponding to $V_{\rm DS}$ are mostly identical to the corresponding GR results. This is due to the fact that geodesic equations obtained from the Lagrangian corresponding to $V_{\rm DS}$ are identical to the GR results, in the low energy limit. The derived potential exactly reproduces the GR results like horizon properties, orbits including apsidal precession, angular momentum, energy, radial velocity of the test particle and also the temporal properties.
In the inner radii, where the effect of $\Lambda$ is negligible, the profile of $V_{\rm DS}$ and particle dynamics corresponding to it essentially coincide with those of $V_{\rm SW}$. However, as seen from the analysis in the earlier subsections, the effect of $\Lambda$ starts to become prominent in the outer region from an approximate radius $x_{\rm min}$, at which $\xi_{\imath}$ corresponding to most of the quantities attain a value of $\sim 0.01 \%$. With $\Lambda r^2_s= 1 \times 10^{-27}$, for most of the quantities, the corresponding values of $x_{\rm min}$ is found out to be approximately in the range of $\sim \left(5 \times 10^7 - 1 \times 10^8 \right) \, r_s$. This is approximately the region from where $V_{\rm DS}$ and most of its corresponding dynamical quantities starts to deviate effectively, from that corresponding to $V_{\rm SW}$. For $M_{\rm BH} \sim 10^9 \, M_{\odot}$, this gives a radius of $\sim (5-10)$ kiloparsec. This implies that in massive galaxies, beyond a scale of few kiloparsecs, the influence of repulsive cosmological constant may not be neglected. Nevertheless, the most significant effect of $\Lambda$ occurs from around a radius of ${\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, \left(5 \times 10^8 - 1 \times 10^9 \right) \, r_s$, where $\xi_{\imath}$ corresponding to most of the dynamical quantities attain a value of ${\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 10 \%$. With BH of $\sim 10^9 \, M_{\odot}$, this gives a radius of $\sim (50-100)$ kiloparsec. This is approximately the region where $V_{\rm DS}$ and its corresponding dynamical quantities largely deviate from that corresponding to Schwarzschild case. This is also the radius where strong orbital precessional effect in the SDS background can be seen (Fig. 10).
On the other hand, the results obtained with $V_{\rm DS}$ differ significantly from the corresponding results of the PNP in Eq. (22), both in the inner radii as well as at the outer radii. In the inner radii $({\raisebox{-.4ex}{$\stackrel{<}{\scriptstyle \sim}$}}100 \, r_s)$, where the effect of $\Lambda$ is negligible, the PNP in Eq. (22) reduces to that of Paczyński-Witta potential, and the particle dynamics corresponding to this PNP deviates largely from that obtained with $V_{\rm DS}$, or in general relativity. The details of which have been elucidated in \[36\]. However, in the outer region where the effect of $\Lambda$ is predominant, especially around the static radius and beyond $\left({\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 10^9 \, r_s \right)$, the PNP in Eq. (22) as well as the profile of conserved Hamiltonian show prominent deviations from that of $V_{\rm DS}$ or that in general relativity, for test particle motion in circular orbit; sometimes showing contrary behavior near to the cosmological horizon (Fig. 6). In fact, the Hamiltonian corresponding to the PNP in Eq. (22) starts deviating from that corresponding to $V_{\rm DS}$ or GR result, from an approximate location of $x_{\rm min}$. Moreover, the PNP in Eq. (22) gives opposite trend for the test particle radial velocity (figures. 8c,d), as compared to $V_{\rm DS}$ or in general relativity, at $r \, {\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, 6 \times 10^{12} \, r_s$. However, in general, for circular motion, the velocity dependent part of $V_{\rm DS}$ vanishes and it gives results close to that predicted by Eq. (22). Therefore, major differences are expected for non-circular motions and in such situations the PNP proposed in \[34\] does not yield consistent results with the GR (with $\Lambda$) and hence with potential $V_{\rm DS}$.
Thus the PNP in Eq. (22) does not reproduce GR features with modest accuracy, unlike that with $V_{\rm DS}$. This occurs owing to the fact that $V_{\rm DS}$ has been derived based on the prerequisite, that, any analogous potential of corresponding GR geometry should reproduce identical or nearly-identical geodesic equations of motion. However, the only GR notion that the PNP in Eq. (22) tries to mimic is to reproduce last stable circular orbits and marginally bound circular orbits (both innermost and outermost).
Effect of $\Lambda$ on Bondi accretion rate
===========================================
Bondi accretion is a spherically symmetric steady gaseous accretion onto a compact star \[39\]. Although Bondi accretion flow is more hypothetical, yet it has serious astrophysical relevance. It is generally accepted that low luminous AGNs or low excitation radio galaxies (LERGs) are presumably powered by radiatively inefficient hot mode accretion, accreting gaseous plasma from hot X-ray emitting phase of interstellar medium (ISM) or intergalactic medium (IGM) \[40,41\]. Consequently the accretion paradigm is quasi spherical in nature, accreting at near ‘Bondi accretion rate’ from ISM/IGM. In the expanding Universe, Bondi accretion rate itself would be plausibly influenced by repulsive $\Lambda$, thus would explicitly govern the accretion dynamics \[19\].
Dynamically Bondi solution is transonic in nature with two critical radii or sonic radii ($r_c$) at which the radial velocity $v_r$ exactly equals to sound speed $a$ in the limits $r \rightarrow {\infty}$ and $r \rightarrow 0$ respectively. The limit $r \rightarrow {\infty}$ in astrophysical sense represents the location of the ambient medium (ISM/IGM). As usual, the flow is considered to be locally adiabatic, with $P \propto \rho^\gamma$ and $a = \sqrt{\gamma P/\rho}$. $P$ is the gas pressure, $\rho$ is the density of the accreting gas and $\gamma$ the adiabatic index. The mass accretion rate here is then defined by the relation
$$\begin{aligned}
\vert \dot M \vert = 4 \pi r^2 \rho \, v_r \, .
\label{40}\end{aligned}$$
Without going into the detailed physics of Bondi flow which has been extensively discussed in literature (for e.g. \[39,42\]), we simply furnish the relations at critical point $r_c$, obtained from Eq. (40) and radial momentum conservation equation. Using the condition at critical radius, i.e., at $r = r_c$, $v_r \vert_c = a_c$ and using $V_{\rm DS}$, instead of usual Newtonian gravity, the relations at critical point $r_c$ are then given by
$$\begin{aligned}
\frac{v^2_r \vert_c}{2} + \frac{a^2_c}{\gamma -1} + V_{\rm DS} \vert_c =
\frac{a^2_{\infty}}{\gamma-1}
\label{41}\end{aligned}$$
and
$$\begin{aligned}
\frac{2 a^2_c}{r_c} - \left. \frac{d V_{\rm DS/ADS}}{dr} \right\vert_c = 0 \, ,
\label{42}\end{aligned}$$
respectively. Solving these two equations we obtain $r_c$ and corresponding $a_c$ in terms of $a_{\infty}$, where $a_{\infty}$ is the sound speed at $r \rightarrow {\infty}$. Using this, the density at $r_c$ is given by $\rho_c = \rho_{\infty} \left(\frac{a^2_c}{a^2_{\infty}} \right)^{1/(\gamma-1)}$, where, $\rho_{\infty}$ is the density at $r \rightarrow {\infty}$. The transonic accretion rate is then given by
$$\begin{aligned}
\vert \dot M_t \vert = 4 \pi r^2_c \, \rho_c \, a_c \, .
\label{43}\end{aligned}$$
$\vert \dot M_t \vert$ can then be expressed in terms of $a_{\infty}$ and $\rho_{\infty}$, which is defined as ‘Bondi accretion rate’ $(\vert \dot M_B \vert )$ in SDS background with repulsive cosmological constant in presence of $V_{\rm DS}$. Assuming the accreting plasma to be of purely ionized hydrogen, we define $a_{\infty} =\sqrt{\gamma k_B T_{\infty}/m_p}$ and $\rho_{\infty} = n_p \vert_{\infty} \, m_p$, where $T_{\infty}$ and $n_p \vert_{\infty}$ are the temperature and the number density of the accreting gas at $r \rightarrow {\infty}$. $m_p$ and $k_B$ are the usual proton mass and Boltzmann constant, respectively. With these relations $\vert \dot M_B \vert$ in presence of $V_{\rm DS}$ can then be explicitly expressed in terms of $T_{\infty}, \rho_{\infty}, \Lambda, \gamma$
For Newtonian gravitational potential the usual relation of $\vert \dot M_B \vert = 4 \pi \, q(\gamma) \, G^2 M^2 \rho_{\infty} /{a^3_{\infty}} $, where $q(\gamma) = \frac{1}{4} \left(\frac{2}{5-3\gamma} \right)^{(5-3\gamma)/(2\gamma-2)}$. $T_{\infty}$ and $\rho_{\infty}$ in astrophysical sense are the temperature and density of the ambient medium, like hot X-ray emitting phase of ISM/IGM. In Fig. 12, we study the variation of $\vert \dot M_B \vert$ and sonic radius $r_c$ with $T_{\infty}$ and $\gamma$ for $\Lambda \sim 10^{-56} \rm cm^{-2}$. It is found that the current accepted value of cosmological constant does not have much impact on the sonic radius as well as the Bondi accretion rate for large values of ambient temperature. However, it is found that there is a moderate increase in the value of $\vert \dot M_B \vert$ as compared to the usual value of $\vert \dot M_B \vert$ with $\Lambda = 0$, for an ambient temperature $T_{\infty} < 10^4 K$, and for smaller values in $\gamma$ (${\raisebox{-.4ex}{$\stackrel{<}{\scriptstyle \sim}$}}\, 1.5$). The sonic radius of the flow, too, shift little inwards in SDS back ground as that compared to $\Lambda = 0$, corresponding to $T_{\infty} < 10^4 K$ and $\gamma \, {\raisebox{-.4ex}{$\stackrel{<}{\scriptstyle \sim}$}}\, 1.5$. This impact of $\Lambda$ on Bondi accretion rate might have interesting consequences on the advective accretion dynamics, especially in low luminous AGNs or in LERGs.
![Variation of sonic radius and Bondi accretion rate with temperature $T_{\infty}$ obtained with $V_{\rm DS}$, for $\Lambda \sim 10^{-56} \rm cm^{-2}$. Figure 12a shows the variation of the ratio of sonic radius in the SDS background to the sonic radius with $\Lambda = 0$, in temperature. Figure 12b is similar to 12a, but for the ratio of Bondi accretion rate in SDS background to Bondi accretion rate with $\Lambda = 0$. Solid, long-dashed and dotted curves in the figures 12a,b are for adiabatic constant $\gamma = (4/3, 1.5, 1.66)$, respectively. BH mass is chosen to be $\sim 10^9 M_{\odot}$. []{data-label="Fig12"}](f12.eps){width="1.0\columnwidth"}
Discussion
==========
The influence of positive cosmological constant is important in understanding the kinematics of the kiloparsecs-scale regions and beyond, in the local-galaxies. Any observational inference on the possible dependence of the local-scale structure of the galaxies on $\Lambda$ would be very intriguing, for which, Newtonian analogous potential would be of remarkable use. Here, we have obtained modified Newtonian analogous potential corresponding to SDS/SADS geometry which reproduces almost all of the GR features with remarkable accuracy. We have investigated the dynamical behavior of the test particle motion in SDS background for a typical BH mass $M_{\rm BH} \sim 10^9 M_{\odot} $ in accordance with the BH mass in many AGNs/quasars. For such massive black hole system, the non-negligible effects of cosmological constant $\Lambda$ start to occur at radius $\, x_{\rm min} \sim \, (5-10) \, {\rm kiloparsec}$ where the relative deviation between the quantities of SDS and Schwarzschild geometries attain a value of $\sim 0.01 \%$. The significant effects (at the level of $10 \%$ and above over those due to pure Schwarzschild geometry) of $\Lambda$ occur at $\, r{\raisebox{-.4ex}{$\stackrel{>}{\scriptstyle \sim}$}}\, (50-100) \, {\rm kiloparsec}$ and consequently the local astrophysical kinematics in many massive AGNs/quasars would be strongly influenced by $\Lambda$ beyond a distance of few kiloparsecs.
‘Low excitation radio galaxies’, which are often massive elliptical galaxies \[43\] are powered by the gaseous accretion directly from the X-ray emitting phase of ISM or either from the hot X-ray halos surrounding the galaxy or from the hot phase of the IGM \[40,41\], in which case the accretion flow region may well exceed parsecs-scale, extending even to kiloparsecs-scale. Quasar accretion disk may be susceptible to gravitational instability if the disk is self-gravitating and massive, which may trigger massive star formation in the outer regions extending beyond parsecs-scale \[44\]. Effects of $\Lambda$ in the outer regions of these kiloparsecs-scale accretion disk, especially on their dynamical instabilities, would then have interesting consequences in the expanding Universe. As remarked by \[15\] that positive cosmological constant could also influence the accretion process onto primordial BHs in the early epoch of the expanding Universe and can also induce strong collimation effects in astrophysical jets. Over kiloparsecs to megaparsecs-scale jets are, in fact, observed \[45,46\] in quasars and radio galaxies whose dynamics and structure would likely to be influenced by the effect of repulsive $\Lambda$ in the expanding Universe.
Seemingly, those are not the only plausible ways in which a positive cosmological constant can, in principle, influence the astrophysical processes on the kiloparsecs-scale regions and beyond, in the local-galaxies in SDS background. Recently \[47,48\], it has been suggested that nuclear spirals can be one of the mechanisms to power central SMBHs by transferring gas from galactic-scales (over kiloparsec-scales) to the central parsec region. In such circumstances $\Lambda$ can influence the feeding process where the mass inflow to nuclear SMBHs occur from kiloparsec-scales, and thus controlling the mass accretion rate of the accreting BH. Another interesting paradigm to explore the possible effects of $\Lambda$, could be in the area of AGN feedback, where outflows, jets and radiation from the nuclear region of the active galaxies harboring SMBHs interact with the ISM and beyond, with length-scales ranging over several kiloparsecs to even megaparsecs. The feedback might considerably effect the star formation rates as well as can curtail accretion onto host SMBH, and thus may influence the formation and evolution of galaxies \[49,50\]. There would be significant effects of $\Lambda$ at these length-scales in the expanding Universe.
To have a glimpse on how repulsive $\Lambda$ could influence the realistic astrophysical phenomena as well as to examine the effectiveness of the potential derived here, as a toy model, we investigated the impact of $\Lambda$ on Bondi accretion rate, which approximately governs the accretion dynamics plausibly in low luminous AGNs or in LERGs. Using SDS analogous potential, it is found that the current accepted value of cosmological constant impact the sonic radius as well as the Bondi accretion rate moderately, for an ambient temperature $T_{\infty} < 10^4 K$ and for smaller values in $\gamma$ (${\raisebox{-.4ex}{$\stackrel{<}{\scriptstyle \sim}$}}\, 1.5$). Thus, it appears that the said value of repulsive cosmological constant might influence the Bondi accretion phenomenon, with measurable effect in the local galactic-scales. This might have interesting consequences, especially on the dynamics of advective accretion flow; the detailed study of which would be pursued in the near future. Note that, \[19\] also analyzed the influence of $\Lambda$ on Bondi accretion rate, however with significantly large values of $\Lambda$. Interestingly the authors of \[19\] adopted unrealistic high value of $\Lambda$, however because for small $\Lambda$ their adopted technique becomes too difficult under pure GR framework.
The next goal is to test the SDS analogous potential in regard to the realistic astrophysical processes in local galactic-scales in order to explore the influence of $\Lambda$ on kiloparsecs-scale structure and beyond, in the local-galaxies.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors are thankful to an anonymous reviewer for insightful comments and suggestions that help us to improve the manuscript.
Here we furnish the Cartesian transformation of the acceleration terms of the test particle motion using potential $V_{\rm DS}$ in SDS back ground, which are useful to depict the elliptical like trajectory of the particle orbit. The acceleration of the test particle in $x^i$ direction, where $x^i = x, y, z$ is given by
$$\begin{aligned}
{\ddot{x}}^i = \frac{x^i}{r} fs(r) + fs1(r) \, {\dot x}^i \, \sum_i x^i {\dot x}^i \, ,
\label{A1}\end{aligned}$$
where,
$$\begin{aligned}
fs(r) = \left(-\frac{r_s}{r^2} + \frac{\Lambda r}{3} \right) \left[\frac{f(r)}{r}\right]^2 - \frac{3}{r^4} \, \sum_i \left(\sum_{jk} \epsilon_{ijk} x^j {\dot x}^k \right)
\label{A2}\end{aligned}$$
and
$$\begin{aligned}
fs1(r) = 2 \, \frac{r_s-\frac{\Lambda r^3}{3} }{r^2 f(r)} \, ,
\label{A3}\end{aligned}$$
respectively. $f(r) = r - 2r_s- \frac{\Lambda r^3}{3}$ and $r^2 = \sum_i x^i x^i$.
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[^1]: Email address: [email protected]
[^2]: Email address: [email protected]
[^3]: Email address: aru\[email protected]
|
---
abstract: 'In this letter we show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two-dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the $[k_x,k_y,t]$ space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the $[k_x,k_y,t]$ space, and the linking number of these two trajectories exactly equals to the Chern number of the static Hamiltonian. Applying this result to a recent experiment from the Hamburg group, we show that the linking number of the trajectories of the phase vortices determines the phase boundary of the static Hamiltonian.'
author:
- Ce Wang
- Pengfei Zhang
- Xin Chen
- Jinlong Yu
- Hui Zhai
date:
-
-
title: 'Measuring Topological Number of a Chern-Insulator from Quench Dynamics'
---
Recently cold atom experiments have realized a number of topological models including the Hofstadter model [@Bloch_Hofstadter; @Bloch_Hofstadter_Chern; @Ketterle_Hofstadter], the Haldane (and the Haldane-type) model [@Esslinger; @USTC], the Su-Schrieffer-Heeger model [@Bloch_SSH] and its Thouless charge pumping [@Bloch_pumping; @Japan_pumping; @Bloch_spin]. One major advantage of studying topological models in the context of cold atom systems, in comparison with its condensed matter counterpart, is that the experimental investigation of the dynamic processes can be more easily accessible. For example, considering non-interacting fermions initially in a topologically trivial insulator state of the initial Hamiltonian $\mathcal{H}^\text{i}$, we shall focus on a sudden quench to a final Hamiltonian $\mathcal{H}^\text{f}$, whose ground state is a topologically nontrivial insulator (e.g. a Chern insulator) at the same filling, and the question is whether the change of the topological number can be revealed from measuring the dynamical process after the quench. In fact, such a quench experiment has been performed recently in a Haldane-type model with cold atoms by the Hamburg group [@Sengstock]. Using a momentum resolved quantum state tomography method [@Sengstock2; @Tomo1; @Tomo2], they can map out the evolution of the wave function as time evolves after the quench.
At equilibrium, for a Chern insulator, it is known that the bulk Chern number, the number of edge states and the quantization value of the Hall conductance are equal, which is termed as “ the bulk-edge correspondence ". While for the non-equilibrium process after the quench, there is no such clear relations between them. First of all, because the time evolution after the quench is unitary, the Chern number of the quantum state does not change and does not reflect the topological number of the final Hamiltonian [@Rigol]. Nevertheless, the edge state gradually emerges [@Rigol; @Cooper]. Second, without dephasing, the Hall response will not be well quantized for either a slow or a sudden quench [@Hu; @Mueller]. While it is also found that the Hall conductance can become finite even after quenching to a topologically trivial final Hamiltonian [@Refael]. Therefore, it is desirable to know whether there is a way to rigorously map out the topology of the band structure of $\mathcal{H}^\text{f}$ through the quench dynamics.
In this letter we present a scheme to extract a *quantized* value from the dynamical process after the quench, and this quantized value is exactly the same as the topological Chern number of the final Hamiltonian $\mathcal{H}^\text{f}$. This scheme can be directly applied to analyze the recent experimental data from the Hamburg group (Ref. [@Sengstock]), as well as other similar systems (such as the ETH [@Esslinger] and the USTC experiments [@USTC]), to determine the topological phase diagram.
![(a) Schematic of hopping in the Haldane model in a honeycomb lattice; (b) The phase diagram of the Haldane model. The arrow indicates a quench from topologically trivial regime to a topologically nontrivial regime. []{data-label="Haldane_quench"}](Haldane_quench){width="3.2"}
*Summary of the Scheme.* Before proceeding to details, let us briefly summarize our scheme as follows:
Let us consider a general two-band tight-binding model in two-dimension, and at each momentum, the Hamiltonian can be written as $$\mathcal{H}({\bf k})=\frac{1}{2}{\bf h}({\bf k})\cdot{{\bm \sigma}}, \label{matHk}$$ where ${\bm \sigma}=(\sigma_x,\sigma_y,\sigma_z)$ is a vector of the Pauli matrices. Thus, the eigen-energies of the Hamiltonian are $\pm|{\bf h}({\bf k})|/2$, corresponding to the upper- and the lower-bands, respectively. We further consider at each ${\bf k}$, $|{\bf h}({\bf k})|$ is always non-zero, and the system is an insulator at half filling. The two-component wave function is denoted by $\zeta({\bf k})$.
Here we consider the quench process that corresponds to a sudden change of ${\bf h}({\bf k})$ from a topologically trivial ${\bf h}^\text{i}({\bf k})$ to ${\bf h}^\text{f}({\bf k})$. The initial wave function $\zeta^\text{i}({\bf k})$ is taken as the lower-band eigenstate of the initial Hamiltonian. After the quench, the wave function will involve according to the final Hamiltonian as $$\zeta({\bf k}, t)=\exp\left\{-\frac{i}{2}{\bf h}^\text{f}({\bf k})\cdot{\bm \sigma}t\right\}\zeta^\text{i}({\bf k}), \label{evolution}$$ and by introducing a Bloch vector $${\bf n}=\zeta^\dag({\bf k}, t){\bm \sigma}\zeta({\bf k}, t), \label{nzeta}$$ Eq. \[evolution\] and Eq. \[nzeta\] together define a mapping $f$ from $[k_x,k_y,t]$ to the Bloch sphere ${\bf n}$.
**The Scheme:** Taking any two constant vectors ${\bf n}_1$ and ${\bf n}_2$ on the Bloch sphere, their inverse images $f^{-1}({\bf n}_1)$ and $f^{-1}({\bf n}_2)$ are two trajectories in the $[k_x,k_y,t]$ space. The linking number of these two trajectories within the first Brillouin zone equals to the Chern number of the ground state for the final Hamiltonian at the same filling [@footnote_inverse].
![(a-b) The inverse images of two vectors ${\bf n}$ and $-{\bf n}$ on the equator, when the Hamiltonian is quenched from ${\bf h}^\text{i}({\bf k})$ with $M=-\infty$ (topologically trivial regime) to ${\bf h}^\text{f}({\bf k})$ with $\phi=0.1$ and $M=1$ (topologically trivial regime)(a), and to ${\bf h}^\text{f}({\bf k})$ with $\phi=\pi/2$ and $M=0$ (topologically nontrivial regime)(b), respectively; (c-d) The inverse images of the north and the south poles, when the Hamiltonian is quenched from ${\bf h}^\text{i}({\bf k})$ with $M=-1$ and $\phi=\pi/2$ to ${\bf h}^\text{f}({\bf k})$ with $M=0.33\sqrt{3}$ and $\phi=\pi/2$ (topologically trivial regime) (c), and to ${\bf h}^\text{f}({\bf k})$ with $M=0.27\sqrt{3}$ and $\phi=\pi/2$ (topologically nontrivial regime) (d). For all plots we have taken $J_0=1$ and $J_1=0.1$. []{data-label="Linking"}](linking){width="3.4"}
*Example to Illustrate the Scheme.* As a concrete example to illustrate our proposal, we consider the Haldane model in a honeycomb lattice \[see Fig. \[Haldane\_quench\](a)\]. The particle annihilation operators at two sublattices of the honeycomb lattices are denoted by $\hat{a}_{{\bf r}_i}$ and $\hat{b}_{{\bf r}_i}$. The tight-binding model is written as $$\begin{aligned}
\hat{H}=&-J_0\sum\limits_{{\bf r}_i,j}\left(\hat{a}^\dag_{{\bf r}_i}\hat{b}_{{\bf r}_i+{\bf d}_j}+\text{h.c.}\right)+M\sum\limits_{{\bf r}_i}\left(\hat{a}^\dag_{{\bf r}_i}\hat{a}_{{\bf r}_i}-\hat{b}^\dag_{{\bf r}_i}\hat{b}_{{\bf r}_i}\right)\nonumber\\
&+J_1\sum\limits_{{\bf r}_i,j}\left(e^{-i\phi}\hat{a}^\dag_{{\bf r}_i}\hat{a}_{{\bf r}_i+{\bf a}_j}+e^{i\phi}\hat{b}^\dag_{{\bf r}_i}\hat{b}_{{\bf r}_i+{\bf a}_j}+\text{h.c.}\right), \label{Ham}\end{aligned}$$ where ${\bf d}_{1,2}=(\pm\sqrt{3}/2,1/2)a_0$, ${\bf d}_3=(0,-1)a_0$ are the three vectors connecting the nearest neighboring sites; and ${\bf a}_{1,2}=(-\sqrt{3}/2,\pm 3/2)a_0$ and ${\bf a}_3=(\sqrt{3},0)a_0$ are the three vectors connecting the next nearest neighboring sites, with $a_0$ being the lattice spacing. The next nearest hopping has phase factor that is opposite between $A$ and $B$ sublattices. In the momentum space, Eq. \[Ham\] becomes $$\hat{H}=\sum\limits_{{\bf k}}(\hat{a}^\dag_{{\bf k}},\hat{b}^\dag_{{\bf k}})\mathcal{H}({\bf k})\left(\begin{array}{c}\hat{ a}_{{\bf k}} \\ \hat{b}_{{\bf k}}\end{array}\right),$$ and aside from a term proportional to the identity matrix, $\mathcal{H}({\bf k})$ takes the same form as Eq. \[matHk\] with $$\begin{aligned}
&h_x({\bf k})=-2J_0\sum_i\cos(\mathbf{k}\cdot\mathbf{d}_i), \label{hx}\\
&h_y({\bf k})=-2J_0\sum_i\sin(\mathbf{k}\cdot\mathbf{d}_i), \label{hy}\\
&h_z({\bf k})=2M+4J_1\sin\phi\sum_i\sin(\mathbf{k}\cdot\mathbf{a}_i). \label{hz}\end{aligned}$$ The phase diagram of this Haldane model at half filling (with the lower-band filled) is shown in Fig. \[Haldane\_quench\](b), where two topologically nontrivial regimes have the Chern numbers $+1$ and $-1$, respectively. Here we consider a sudden change of $M$ and $\phi$ starting from the topologically trivial regime, as indicated by the arrow in Fig. \[Haldane\_quench\](b).
In Fig. \[Linking\] we show two sets of examples. In Fig. \[Linking\](a-b), we consider the inverse image of two vectors ${\bf n}$ and $-{\bf n}$ on the equator. One can see that if $\mathcal{H}^\text{f}$ is in the topologically trivial regime, as shown in Fig. \[Linking\](a), $f^{-1}({\bf n})$ sets inside the trajectory of $f^{-1}(-{\bf n})$, and the linking number is zero; while if $\mathcal{H}^\text{f}$ is in the topologically nontrivial regime, as shown in Fig. \[Linking\](b), these two trajectories link three times. This is because, to avoid the discontinuity of the trajectory across the boundary of the first Brillouin zone, our plot spans the momentum regime including three replicas of the first Brillouin zone. Within the first Brillouin zone, the linking number is unity that equals to the Chern number of $\mathcal{H}^\text{f}$. Similarly, we consider the inverse images of the north and the south pole. As shown in Fig. \[Linking\](c-d), the inverse image of the north pole is a straight line in the $K$ and $K^\prime$ points. While the inverse image of the south pole does not enclose the $K$ or $K^\prime$ point if $\mathcal{H}^\text{f}$ is in the topologically trivial regime \[Fig. \[Linking\](c)\], giving rise to linking number zero; and it encloses three equivalent $K$ or $K^\prime$ point when $\mathcal{H}^\text{f}$ is topologically nontrivial \[Fig. \[Linking\](d)\], giving rise to linking number unity within the first Brillouin zone.
Here we should put a remark of how to determine the sign of the linking number. First of all, we note that each trajectory actually has a direction, defining as the direction of ${\bf \mathcal{J}}$ with [@Zee] $$\mathcal{J}^\mu=\frac{1}{8\pi}\epsilon^{\mu\nu\lambda}{\bf n}\cdot(\partial_\nu{\bf n}\times\partial_\lambda{\bf n}), \label{Jmu}$$ with the indices taking $k_x$, $k_y$ and $t$. Thus, when two trajectories link and one trajectory crosses through a surface enclosed by another trajectory, if the direction of the trajectory is the same as the normal direction of the surface (determined by the right hand rule), we denote the linking number as $+1$ [@math], otherwise we denote the linking number as $-1$. Hence, both cases shown in Fig. \[Linking\](b) and (d) have a linking number $+1$, consistent with the Chern number of $\mathcal{H}^\text{f}$ in the phase diagram Fig. \[Haldane\_quench\](b). We have also checked that, if the Chern number of the $\mathcal{H}^\text{f}$ changes sign, the linking number defined in this way also changes sign.
![Schematic of the mapping $f_1$ from $[k_x,k_y,t]$ ($[k_x,k_y]$ forms a torus) to $[{\bf h},t]$, and mapping $f_2$ from $[{\bf h},t]$ to ${\bf n}$ on a Bloch sphere. []{data-label="mapping"}](mapping){width="3.2"}
*Mathematical Proof of the Results.* The general proof follows following three steps:
Step 1: From Eq. \[evolution\] and Eq. \[nzeta\], one can see that at each ${\bf k}$, for $t=0$ and $t=2\pi/|{\bf h}^\text{f}({\bf k})|$, ${\bf n}$ always points to the north pole. Hence, they can be identified as one point. The mapping $f$ from $[k_x,k_y,t]$ to the Bloch sphere is thus a mapping from $T_3$ (three dimensional torus) to $S^2$, which is classified as $\mathcal{Z}$ for the situation considered here [@footnote]. The topological number is the Hopf invariant, and mathematically, it also equals to the linking number of two inverse images $f^{-1}$ [@math]. This number is invariant under continuous deformation of the mapping $f$.
Step 2: Now we decompose the mapping $f$ as $f_1\circ f_2$, as shown in Fig. \[mapping\]. $f_1$ maps $[k_x,k_y]$ to ${\bf h}^\text{f}$ with the definition of ${\bf h}^\text{f}({\bf k})$ (e.g. Eq. \[hx\]-\[hz\]), which is classified by the Chern number $C$. $f_2$ maps $[{\bf h}^\text{f},t]$ to ${\bf n}$ with Eq. \[evolution\] and Eq. \[nzeta\].
If $f_1$ is a topologically trivial map, it can be continuously deformed into a mapping that all ${\bf k}$ points are mapped to the same vector. Thus, $[{\bf h}^\text{f}, t]$ forms an $S^1$. Then $f_2$ becomes a mapping from $S^1$ to $S^2$ that is always topologically trivial. Thus, $f$ is a topologically trivial map.
In this step we focus on the case that $f_1$ is a topologically nontrivial map with non-zero Chern number $C$, we will show that one can construct a class of mapping $f$ such that the linking number of two inverse images of $f^{-1}$ equals $C$. For this construction, (i) we assume $|{\bf h}^\text{f}({\bf k})|=1$ for all momenta, thus, ${\bf h}^\text{f}$ itself is a Bloch sphere $S^2$ and the compact $[{\bf h}^\text{f},t]$ forms a $S^3$ ; (ii) we consider the initial wave function $\zeta^{i}({\bf k})=\zeta^0=\left(\begin{array}{c}1 \\0\end{array}\right)$ for all momenta, which corresponds to choosing ${\bf h}^i({\bf k})=(0,0,-1)$ or $M=-\infty$ in the example of the Haldane model; and (iii) we divide the first Brillouin zone into $|C|$ patches and we construct a mapping $f_1$ such that the boundary of each patch is mapped to the north pole of the Bloch sphere for ${\bf h}^\text{f}$, and therefore, different patches can be smoothly connected.
We can stretch each patch into a compactified two-dimensional plane and parametrize this plane with polar coordinate $r$ and $\varphi$. For the convenience of computing the Hopf invariant, we can construct ${\bf h}^\text{f}$ as $${\bf h}^\text{f}=(\cos\varphi\sin(f(r)),\sin\varphi\sin(f(r)),\cos(f(r))),$$ where $\varphi$ changes between zero and $2\pi$, and $r$ changes between zero and infinity, with $f(0)=\pi$ and $f(\infty)=0$.
Writing ${\cal J}^\mu=\epsilon^{\mu\nu\lambda}\partial_\nu B_\lambda$, the Hopf invariant is given by [@Zee] $$H=\int rdrd\varphi dt\ B_\mu {\cal J}^\mu. \label{Hopf}$$ With Eq. \[evolution\], \[nzeta\], \[Jmu\] and this parametrization, it is straightforward to show that $$\begin{aligned}
{\cal J}^0&=\frac{1}{2\pi r}\sin^2\left(\frac{t}{2}\right)\sin(2f)f^\prime, \label{J0}\\
{\cal J}^i&=-\frac{1}{4\pi}\sin f\left(\frac{r_i}{r^2}\sin t\sin f-2\epsilon_{ij}\frac{r_j}{r}\sin^2\left(\frac{t}{2}\right)f^\prime\right) \label{Ji}\end{aligned}$$ with $i$ being the two directions in the two-dimensional plane. Eq. and Eq. \[Ji\] further give $$\begin{aligned}
&B_0=-\frac{1}{2\pi}\sin^2\left(\frac{t}{2}\right)\cos f, \label{B0} \\
&B_i=-\frac{\epsilon_{ij}r_j}{2\pi r^2}\sin^2\left(\frac{t}{2}\right)\sin^2f. \label{Bi}\end{aligned}$$ Substituting Eq. \[J0\]-\[Bi\] into Eq. \[Hopf\], one can obtain $H=1$ for this case. Thus, $f_2$ is a topologically nontrivial Hopf map. It is proved mathematically that the Hopf invariant equals to the linking number of the inverse images in $S^3$ of two different ${\bf n}$’s on $S^2$ [@math], thus, the linking number of $f_2^{-1}$ equals unity. Furthermore, since there is a one-to-one correspondence between each patch in the first Brillouin zone and ${\bf h}^\text{f}$, thus, for the mapping constructed in this way, the linking number of the inverse mapping $f^{-1}$ equals $C$. Moreover, if the Chern number changes the sign, it corresponds to change $f(0)=0$ and $f(\infty)=\pi$ in the parametrization, and one obtains $H=-1$ for $f_2$. Consequently, the linking number also changes sign.
Step 3: The three conditions used in the Step 2 can be released by continuously deforming the mapping, which does not change the linking number. For (i), as long as the system is gapped, i.e. $|{\bf h}^\text{f}|$ is finite everywhere, the length of $|{\bf h}^\text{f}|$ can always be continuously adjusted to $|{\bf h}^\text{f}|=1$. For (ii), as long as the initial wave function is in the same topologically nontrivial regime, it can be obtained by a continuous transformation of a uniform state as $\zeta^\text{i}({\bf k})=\mathcal{U}({\bf k})\zeta^0$, where $\mathcal{U}({\bf k})$ is smooth function of ${\bf k}$ and smoothly connects to the identity matrix. Hence, it does not change the linking number. Finally, continuously deforming $f_1$ also does not change the linking number, which releases the condition (iii). Therefore, we prove our results for a general situation.
*Application to Cold Atom Experiments.* Finally, we discuss applying our theory to recent cold atom experiments, such as the one from the Hamburg group [@Sengstock]. The initial state is prepared with all ${\bf n}$ pointing around the north pole, and they quench the system by turning on a periodic shaking which can induce gauge field and topological Haldane model [@Aoki; @Zheng; @Neupert; @Esslinger; @Sengstock; @Sengstock2; @Sengstock_gauge1; @Sengstock_gauge2]. Using the method of momentum resolved quantum state tomography [@Sengstock2], they are able to measure the wave function in the pseudo-spin bases as $\zeta({\bf k})=\left(\begin{array}{c}\sin(\theta_{{\bf k}}/2) \\ -\cos(\theta_{{\bf k}}/2)e^{i\varphi_{{\bf k}}}\end{array}\right)$. Two types vortices of the phase $\varphi_{{\bf k}}$ are found in the momentum space. The first type of the phase vortices are naturally located in the $K$ and $\Gamma$ points and their locations do not evolve with time, where ${\bf n}$ always points to the north pole; while the second type of phase vortices locate at certain momenta, at which ${\bf n}$ rotates to the south pole. The second type of vortices can be pair-wisely created and annihilated in the momentum space, tracing a trajectory in the $[k_x,k_y,t]$ space.
In Ref. [@Sengstock], they use the appearance of the second type of phase vortex as criterion to determine what they call a “dynamical phase transition". And they find that the phase diagram for the “dynamical phase transition" is much wider than the expected topological regime for the static Hamiltonian. With our results, the topological regime of the static Hamiltonian is determined by whether the trajectories of the second type of vortices wind around the trajectories of the first type of vortices, because these two trajectories are the inverse images of the south and the north poles, respectively. Thus, the linking number of these trajectories, as we shown in Fig. \[Linking\](c-d), determines the Chern number. For the Haldane model, it can be shown that, if the initial state uniformly points to the north pole for all momenta, once the second type vortices appear, their trajectories always wind around $K$ or $K^\prime$ points [@Yu]. However, if the initial state spreads a finite regime around the north pole, as in the case of the real experiment, there exists certain regime where the second type of vortices appear but their trajectory enclose neither $K$ nor $K^\prime$ [@Yu]. In these regime, the final Hamiltonians are still topological trivial ones.
To summary, our result establishes a unique relation between quench dynamics and equilibrium property regarding the topological band structure. Our result can be apply to the Haldane model realized by the ETH [@Esslinger] and the Hamburg group [@Sengstock], with measurements of the momentum resolved quantum state tomography measurement, and can also be applied to the Haldane-like model realized by the USTC group [@USTC], where the real spin is used instead of the pseudo-spin, and momentum and spin resolved measurement will be sufficient. On the theory side, future generalizations include quench from topological nontrivial initial state, and quench in other classes of topological model, such as the time-reversal invariant $Z_2$ topological insulator.
*Acknowledgment.* This work is supported by MOST under Grant No. 2016YFA0301600 and NSFC Grant No. 11325418 and Tsinghua University Initiative Scientific Research Program.
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|
---
address: 'Department of Mathematics and Statistics, McGill University, 805 Sherbrooke West, Montreal H3A 2K6, Quebec, Canada'
author:
- 'Eyal Z. Goren & Payman L Kassaei'
date: 'January 7, 2005 '
title: 'The canonical subgroup: a “subgroup-free" approach'
---
Introduction
============
Canonical subgroups are essential to the theory of overconvergent modular forms. An elliptic curve $E$ with an ordinary reduction modulo a prime $p$ has a distinguished subgroup of rank $p$, which is the kernel of multiplication by $p$ on its formal group. This subgroup is a canonical lift of the kernel of ${{\operatorname{Fr }}}_p$ on $E$ modulo $p$. The overconvergence of the canonical subgroup, i.e. the fact that it can also be defined for elliptic curves with a “not too supersingular” reduction modulo $p$, allows one to define and study the ${\mathbf U}_p$ operator for overconvergent modular forms (See [@Katz §3.11].). Recently, in [@Buzzard; @Kassaei3], this theory has found new applications to the problem of analytic continuation of overconvergent modular forms. In these applications it is essential to understand the *precise extent* of overconvergence of the canonical subgroup, and to determine the “measure of supersingularity" of a quotient of an elliptic curve by a subgroup of order $p$ (including the canonical subgroup). These results appear in [@Katz Thms. 3.1, 3.10.7], where they are attributed to Lubin. A slightly more general version can be found in [@Buzzard].
Classically, the canonical subgroup of an elliptic curve (when it exists) is constructed by a close study of the power series of multiplication by $p$ in its formal group. In [@Kassaei1; @Kassaei2] this approach was used to develop a similar theory over certain PEL Shimura curves. Generalizing this approach to higher dimensions seems to pose a serious challenge, because it uses the one-dimensionality of the formal group in an essential way, including the existence of Newton polygon for power series in one variable.
The problem of constructing a canonical subgroup for each elliptic curve belonging to a certain region of a modular curve $X(\Gamma)$ can be rephrased as finding a partial section to the forgetful morphism of rigid analytic curves $\pi\colon X(\Gamma_0(p) \cap
\Gamma) \rightarrow X(\Gamma)$ whose moduli-theoretic description is $(E,\gamma,H) \mapsto (E,\gamma)$ where $(E,\gamma)$ is an elliptic curve with level $\Gamma$-structure and $H \subset E[p]$ is a finite flat subgroup of order $p$. Our approach ignores this moduli-theoretic description and just takes into account the geometry of the morphism $\pi$. This is what we call the “subgroup-free" approach. It has been known for a while that one can prove, using a general principle of rigid geometry due to Berthelot [@Berthelot], that such a section defined over the ordinary locus overconverges (to an a priori non-tractable extent) beyond the ordinary locus. This approach, which is expected to work in other situations, was used in [@KisinLai] to prove the overconvergence of canonical subgroups in the case of Hilbert modular varieties. However, other aspects of the theory, which were discussed in the opening paragraph, remain unsettled even in the case of Hilbert modular varieties. These aspects are also not fully covered by other recent approaches [@AbbesMokrane; @AndreattaGasbarri; @Nevens].
The purpose of this article is to derive all aspects of the theory of canonical subgroups via the“subgroup-free" approach. Our thesis is that the rigid geometric (or formal schematic) picture that arises in the familiar setting of the relevant Shimura varieties suffices by itself to guarantee the existence of the canonical subgroup and many of its properties. In this manuscript we demonstrate that for Shimura varieties of dimension one, even if they do not possess a natural modular interpretation. In fact, this lack of a moduli interpretation can be taken as a further motivation for our approach. Notice that our approach is such that inspires generalization to higher dimensional settings. More specifically, one constructs a section over the ordinary locus by lifting a section from characteristic $p$. One separately studies sections over the non-ordinary locus by using the theory of local models for the special fibre of the Shimura variety in question, and finally these two sections are glued together by using the above-mentioned principle of rigid geometry along with a certain uniqueness result. The authors hope to pursue this subject in a future publication.
Let $p$ be a prime. Let ${{\mathcal{O}}}={{\mathcal{O}}}_K$ be the ring of integers of a finite extension $K$ of $\QQ_p$, $\varpi$ a uniformizer of ${{\mathcal{O}}}$, $\kappa = {{\mathcal{O}}}/(\varpi)$ the residue field, and ${{\operatorname{val}}}={{\operatorname{val}}}_K$ be the valuation normalized so that ${{\operatorname{val}}}(\varpi)=1$. By a “curve" $X$ over ${{\mathcal{O}}}$ we mean a flat finite-type morphism $f\colon X \rightarrow {{\mathcal{O}}}$ of relative dimension $1$ of a reduced separated scheme $X$, such that the geometric fibres of $f$ are connected; $f$ need not be proper.
Let $X, Y$ be curves over ${{\mathcal{O}}}$. We assume that $X, Y$ are regular schemes, $X \rightarrow {{\operatorname{Spec }}}({{\mathcal{O}}})$ is smooth and $\pi\colon Y {{\; \longrightarrow \;}}X$ is a finite flat morphism of degree $e+1$. Moreover, we assume that (i) there exists a section $s\colon X\otimes \kappa {{\; \longrightarrow \;}}Y\otimes \kappa$ to $\pi
\otimes \kappa$, that (ii) the special fibre $Y\otimes \kappa$ is a reduced normal crossing divisor with two components, and (iii) the set theoretic preimage $(\pi\otimes \kappa)^{-1}(\pi\otimes
\kappa)(Q)$ is equal to $Q$ for any singular point $Q\in Y\otimes
\kappa$. To remove any doubt, we assume that $Y \otimes \kappa$ singular and by a normal crossing divisor we mean that each intersection is defined over $\kappa$ and its completed local ring is isomorphic to $\kappa[\![s, t]\!]/(st)$. We define $(Y \otimes \kappa)^\infty=s(X \otimes \kappa)\setminus (Y
\otimes \kappa)^{\rm sing}$, and $(Y \otimes \kappa)^0=(Y \otimes
\kappa) \setminus s(X \otimes \kappa)$.
From the point of view of a general theory this is a very specific situation, nonetheless it (and its appropriate generalization) is the one that occurs for Shimura curves (respectively, higher-dimensional PEL Shimura varieties); see §\[section:applications\]. In fact, condition (iii) is only put to have “cleaner statements"; it holds in the case of Shimura curves. Under these conditions, we prove in §\[section:main theorem\] the following result.
Let ${{\frak{X}}}, {{\frak{Y}}}$ be the formal schemes obtained, respectively, by completing $X, Y$ along their special fibres. The induced morphism ${{\frak{Y}}}\rightarrow {{\frak{X}}}$ is still denoted by $\pi$. Let ${{\pi_\text{\rm rig}}}\colon {{{{\frak{Y}}}_\text{\rm rig}}}\rightarrow {{{{\frak{X}}}_\text{\rm rig}}}$ be the induced morphism of rigid $K$-spaces á la Raynaud; cf. §\[subsection:recall\]. In §\[section:measure\] we define a “measure of singularity" $\nu_{{\frak{X}}}(P)\in \QQ^{\geq 0}$ (respectively, $\nu_{{\frak{Y}}}(Q)\in \QQ^{\geq 0}$) of a point $P$ of ${{{{\frak{X}}}_\text{\rm rig}}}$ (respectively, ${{{{\frak{Y}}}_\text{\rm rig}}}$); the definition is modelled over the notion of measure of supersingularity for modular curves. For every interval $I \subset \RR$ we have an admissible open set ${{\frak{Y}}}_{{\operatorname{rig}}}I$, whose closed point are $\{Q\in {{{{\frak{Y}}}_\text{\rm rig}}}:
\nu_{{\frak{Y}}}(Q) \in I\}$. The set ${{\frak{X}}}_{{\operatorname{rig}}}I$ is defined similarly. The following theorem is proven in §\[section:main theorem\].
[**Theorem A.**]{} *Assume $e>1$. The morphism ${{\pi_\text{\rm rig}}}\colon {{{{\frak{Y}}}_\text{\rm rig}}}\rightarrow {{{{\frak{X}}}_\text{\rm rig}}}$ admits a section $${{\frak{s}}}_{{\operatorname{rig}}}\colon {{\frak{X}}}_{{\operatorname{rig}}}[0,e/(e+1)) \rightarrow {{{{\frak{Y}}}_\text{\rm rig}}}.$$ This section is maximal, namely, it can not be extended to any connected admissible open properly containing ${{\frak{X}}}_{{\operatorname{rig}}}[0,e/(e+1))$.*
The reader acquainted with the theory of canonical subgroups will recognize that this theorem implies the classical existence theorem for canonical subgroups over modular curves, including the further statement (that to the best of our knowledge is not recorded in the literature) that the region over which one defines the canonical subgroup is the maximal possible, even from the point of view of maps of rigid spaces. The following theorem, proven in § \[section:involution\], will also be familiar to that reader as giving the behavior of the measure of supersingularity upon passing to a quotient by a subgroup of order $p$. We introduce the following terminology: Let $Q \in
{{{{\frak{Y}}}_\text{\rm rig}}}$. We say that $Q$ is (i) [*canonical*]{} if $\nu_{{\frak{Y}}}(Q) <
e/(e+1)$; (ii) [*anti-canonical*]{} if $\nu_{{\frak{Y}}}(Q) > e/(e+1)$; (iii) [*too singular*]{} if $\nu_{{\frak{Y}}}(Q)=e/(e+1)$.
[**Theorem B.**]{}
*Acknowledgments:* The authors benefited from an example of R. Coleman that inspired the proof of Proposition \[prop:Coleman\]. The first name author was partially supported by an NSERC grant no. 227040. The second-named author would like to thank CICMA and the department of mathematics at McGill university for their support and hospitality.
Background material
===================
Rigid analytic varieties and formal schemes {#subsection:recall}
-------------------------------------------
We recall here the connection between rigid analytic varieties and formal schemes as developed by Raynaud and Berthelot. Our exposition follows [@BoschLutkebohmertI; @BoschLutkebohmertII; @Berthelot; @deJong].
Let $R$ be a valuation ring of Krull dimension $1$, complete and separated with respect to the ${{\frak{I}}}$-adic topology, where ${{\frak{I}}}= (\varpi)$ is contained in the maximal ideal of $R$. Let $K$ be the field of fractions of $R$. For free variables $\xi =
(\xi_1, \dots, \xi_n)$ we let $R\langle \xi\rangle = \{ \sum_\nu
c_\nu\xi^\nu \in R[\![\xi]\!]: \lim c_\nu = 0\}$ be the strictly convergent powerseries, i.e. precisely those that converge on the polydisc $\{(a_1, \dots, a_n) : \vert a_i \vert \leq 1\; \forall
i\}$.
Recall that for a general commutative ring $B$ and an ideal $J$ of $B$ one defines the *$J$-torsion* of $B$ as the ideal $\{
b\in B: J^n b = 0 \; \text{\rm for some } n\in \NN\}$. If $J =
(g_1, \dots, g_r)$, the $J$-torsion is the kernel of the canonical homomorphism $R \rightarrow \prod_{i = 1}^r R[g_i^{-1}]$. If this ideal is $\{ 0 \}$ we say that $B$ has no $J$-torsion.
An *admissible $R$-algebra* is an $R$-algebra with no ${{\frak{I}}}$-torsion (equivalently, flat over $R$) that is isomorphic to $R\langle \xi\rangle/{{\frak{a}}}$, where $\xi = (\xi_1, \dots,
\xi_n)$ for some integer $n$; it implies that ${{\frak{a}}}$ is a finitely generated ideal. For us, the admissible $R$-algebras are the building blocks of two different categories - a category of rigid spaces and a category of formal schemes.
An affine formal $R$-scheme ${{\frak{X}}}$ is called admissible if it is of the type ${{\frak{X}}}= {{\operatorname{Spf }}}(A)$, where $A$ is an admissible $R$-algebra. We may then write ${{\frak{X}}}= \underset{\lambda
\rightarrow \infty}{\lim} {{\frak{X}}}_\lambda$, where ${{\frak{X}}}_\lambda :=
{{\frak{X}}}\otimes (R/(\varpi^\lambda))$, $\lambda \in \NN$, can be identified with the scheme ${{\operatorname{Spec }}}(A\otimes R/(\varpi^\lambda))$. Being admissible is a local property and so one gets a natural definition of an *admissible formal $R$-scheme*.
The notion of admissible blow-up is needed to define an equivalence of categories between a category of formal schemes and a category of rigid spaces. The definition of admissible formal blow-up is designed to be local on the base. We review, thus, only the affine case. Let ${{\frak{X}}}= {{\operatorname{Spf }}}(A)$ be an affine admissible $R$-formal scheme, $A = R\langle \xi\rangle/{{\frak{a}}}$. Let ${{\mathscr{A}}}$ be an open ideal, i.e., containing $(\varpi^\lambda)$ for some $\lambda>0$. The *admissible formal blow-up* of ${{\frak{X}}}$ at ${{\mathscr{A}}}$ is ${{\frak{X}}}^\prime = \underset{\lambda \rightarrow
\infty}{\lim}\; {\rm Proj} \oplus_{n=0}^\infty\left( {{\mathscr{A}}}^n
\otimes {{\mathcal{O}}}_{{\frak{X}}}/(\varpi^\lambda)\right)$ with the canonical map $\varphi\colon {{\frak{X}}}^\prime \rightarrow {{\frak{X}}}$. Then ${{\frak{X}}}^\prime$ is an admissible formal $R$-scheme over which ${{\mathscr{A}}}{{\mathcal{O}}}_{{{\frak{X}}}^\prime}$ is invertible.
Suppose that ${{\mathscr{A}}}= (f_0, \dots, f_m)$ and let $\widetilde{\varphi}\colon \widetilde{{{\frak{X}}}^\prime} \rightarrow
\widetilde{{{\frak{X}}}}={{\operatorname{Spec }}}(A)$ be the usual scheme theory blow-up of $A$ at the ideal ${{\mathscr{A}}}$. Then, upon taking $(\varpi)$-completion of $\widetilde{\varphi}\colon
\widetilde{{{\frak{X}}}^\prime} \rightarrow \widetilde{{{\frak{X}}}}$ we get $\varphi\colon {{\frak{X}}}^\prime \rightarrow {{\frak{X}}}$. On the other hand, $\widetilde{\varphi}\colon \widetilde{{{\frak{X}}}^\prime} \rightarrow
\widetilde{{{\frak{X}}}}$ admits a local description. The scheme $\widetilde{{{\frak{X}}}^\prime}$ has an affine cover by $\{
{{\operatorname{Spec }}}(A_i^\prime): i = 0, 1, \dots, m\}$, where $A_i^\prime = A_i^{\prime\prime}/(f_i-{\rm
torsion})$ and $A_i^{\prime\prime} = A\left[\frac{f_0}{f_i}, \dots,
\frac{f_m}{f_i} \right] = A\left[\frac{\zeta_0}{\zeta_i}, \dots,
\frac{\zeta_m}{\zeta_i} \right]/\left( f_i\frac{\zeta_j}{\zeta_i}
- f_j\right)$. Then the $(\varpi)$-completions of $A_i^\prime, A_i^{\prime\prime}$ are given by $\hat{A}_i^\prime = \hat{A}_i^{\prime\prime}/(f_i-{\rm torsion})$ and $\hat{A}_i^{\prime\prime} = A\left\langle\frac{f_0}{f_i}, \dots,
\frac{f_m}{f_i}\right\rangle = A\left\langle
\frac{\zeta_0}{\zeta_i}, \dots, \frac{\zeta_m}{\zeta_i}
\right\rangle/\left( f_i\frac{\zeta_j}{\zeta_i} - f_j\right)$; they give rise to an affine covering $\{ {{\operatorname{Spf }}}(\hat{A}_i^\prime):i = 0,
1, \dots, m\}$ of ${{\frak{X}}}^\prime$.
For an admissible $R$-algebra $A = R\langle \xi\rangle/{{\frak{a}}}$, let $A_{\rm rig}: = A\otimes_R K = K\langle \xi\rangle/{{\frak{a}}}K\langle \xi\rangle$ – this is an affinoid $K$-algebra. This construction extends to provide a functor $${\bf rig}\colon
\{\text{admissible formal~$R$-schemes}\} \;\rightarrow\;
\{\text{rigid~$K$-spaces}\},\qquad{{\frak{X}}}\mapsto {{\frak{X}}}_{{\operatorname{rig}}}.$$ One calls ${{\frak{X}}}_{{\operatorname{rig}}}~$ the *generic fibre* of the formal $R$-scheme ${{\frak{X}}}$.
*(Raynaud)* The functor ${\bf rig}$ is an equivalence of categories between (i) the category of quasi-compact admissible formal $R$-schemes, localized by admissible formal blow-ups, and (ii) the category of quasi-compact and quasi-separated rigid $K$-spaces.
It is easy to see from the construction that a flat morphism of formal schemes induces a flat morphism of rigid spaces. The converse is also true [@BoschLutkebohmertII Thm. 5.2]: every flat morphism of rigid $K$-spaces comes from a flat morphism of suitable formal schemes yielding the given rigid spaces. A flat morphism in the category of rigid spaces has image which is a finite union of affinoids, in particular, it is open [@BoschLutkebohmertII Cor. 5.11].
We will need to use the *specialization map*. In the affine case, the points of ${{\frak{X}}}_{{\operatorname{rig}}}$ are the maximal ideals of the algebra $A\otimes_R K$; these are in bijection with quotients of $A$ that are integral, finite and flat over $R$. If $T$ is such a quotient (it is the valuation ring of a finite extension of $K$), corresponding to a point $t\in {{\frak{X}}}_{{\operatorname{rig}}}$, we get a closed immersion of formal $R$-schemes ${{\operatorname{Spf }}}(T) \rightarrow {{\operatorname{Spf }}}(A)$, whose image is supported on a closed point of ${{\frak{X}}}$ that we denote by ${{\operatorname{sp }}}(t)$. The definition can be extended to any formal $R$-scheme. We get a morphism of ringed spaces ${{\operatorname{sp }}}\colon
{{\frak{X}}}_{{\operatorname{rig}}}\rightarrow {{\frak{X}}}$ [@SGA4 IV 4.9]. For every affine open $U = {{\operatorname{Spf }}}(B) \subset {{\frak{X}}}$, we have ${{\operatorname{sp }}}^{-1}(U) =
U_{{\operatorname{rig}}}$.
Assume that $R$ is a discrete valuation ring with residue field $\kappa$. In [@Berthelot] Berthelot generalizes the above construction to associate a generic fibre to any locally noetherian formal scheme ${{\frak{X}}}$ flat over $R$ that satisfies a condition weaker than admissibility: that the special fibre of ${{\frak{X}}}$, denoted by ${{\frak{X}}}_0$ and defined by the ideal of definition ${{\mathscr{I}}}$, is a scheme locally of finite type over $\kappa$. This condition is independent of the choice of ${{\mathscr{I}}}$ and coincides with admissibility if $\varpi{{\mathcal{O}}}_{{\frak{X}}}$ is an ideal of definition for ${{\frak{X}}}$. We will describe the construction in the affine case. Let ${{\frak{X}}}={{\operatorname{Spf }}}(A)$ and $I=H^0({{\frak{X}}},{{\mathscr{I}}})$ with generators $g_1,\dots,g_r$. For $n
\geq 1$ define $$A_n=A\langle T_1,\dots, T_r\rangle/(g_1^n-\varpi
T _1,\dots,g_r^n-\varpi T_r).$$ The condition on ${{\frak{X}}}$ implies that $A_n/\varpi A_n$ is finitely generated over $\kappa$, and hence ${{\frak{X}}}^n={{\operatorname{Spf }}}(A_n)$ is an admissible formal scheme over $R$. Applying Raynaud’s construction we obtain a rigid analytic space ${{\frak{X}}}^n_{{{\operatorname{rig}}}}$. For $m>n$ we have a homomorphism $A_m
\rightarrow A_n$, defined by sending $T_i$ to $g_i^{m-n}T_i$, inducing a morphism of rigid spaces ${{\frak{X}}}^n_{{{\operatorname{rig}}}} \rightarrow
{{\frak{X}}}^m_{{{\operatorname{rig}}}}$. It is easy to see that this morphism is an open immersion and identifies ${{\frak{X}}}^n_{{{\operatorname{rig}}}}$ with the subdomain of ${{\frak{X}}}^m_{{{\operatorname{rig}}}}$ over which $|g_i(x)|\leq |\varpi|^{1/n}$. The generic fibre of ${{\frak{X}}}$, denoted as before by ${{\frak{X}}}_{{{\operatorname{rig}}}}$, is defined to be the union of ${{\frak{X}}}^n_{{{\operatorname{rig}}}}$ via the above inclusions. The rigid spaces ${{\frak{X}}}^n_{{\operatorname{rig}}}$ form an admissible cover of ${{\frak{X}}}_{{\operatorname{rig}}}$. The construction yields a functor ${\bf
rig}$ whose target is the category of quasi-separated rigid $K$-spaces.
As an illustration, take ${{\frak{X}}}$ to be ${{\operatorname{Spf }}}(R[\![\xi_1,...,\xi_r]\!])$ with the ideal of definition $I=(\varpi,\xi_1,...,\xi_r)$. Then ${{\frak{X}}}_{{{\operatorname{rig}}}}$ is simply the open unit polydisc of dimension $r$, which is not quasi-compact, and ${{\frak{X}}}^n_{{{\operatorname{rig}}}} \subset {{\frak{X}}}_{{{\operatorname{rig}}}}$ is the affinoid subdomain over which $|\xi_i| \leq |\varpi|^{1/n}$, which is isomorphic to a closed unit polydisc, and hence is quasi-compact. Similarly, for ${{\frak{X}}}={{\operatorname{Spf }}}(R[\![x_1,x_2]\!]/(x_1x_2-a))$, where $a
\in R$, one sees that ${{\frak{X}}}_{{\operatorname{rig}}}$ is the open annulus over $K$ with radii $(|a|,1)$.
As in the admissible case, one can define a specialization map ${{\operatorname{sp }}}\colon {{\frak{X}}}_{{{\operatorname{rig}}}} \rightarrow {{\frak{X}}}$ by taking the direct limit of the maps ${{\frak{X}}}^n_{{{\operatorname{rig}}}} \overset{{{\operatorname{sp }}}}{\longrightarrow}
{{\frak{X}}}^n \rightarrow {{\frak{X}}}$. The following is Proposition 0.2.7. of [@Berthelot].
\[prop:Berthelot\] Let notation be as above. Let ${{\frak{Z}}}\subseteq {{\frak{X}}}_0$ be a closed subscheme. Let ${{\frak{X}}}^{\wedge {{\frak{Z}}}}$ denote the formal completion of ${{\frak{X}}}$ along ${{\frak{Z}}}$. Then ${{\operatorname{sp }}}^{-1}({{\frak{Z}}})$ is an admissible open subset of ${{\frak{X}}}_{{\operatorname{rig}}}$ and the canonical morphism ${{\frak{X}}}^{\wedge {{\frak{Z}}}}_{{\operatorname{rig}}}\rightarrow {{\frak{X}}}_{{\operatorname{rig}}}$ induces a functorial isomorphism ${{\frak{X}}}^{\wedge {{\frak{Z}}}}_{{\operatorname{rig}}}\cong
{{\operatorname{sp }}}^{-1}({{\frak{Z}}})$.
Algebraic geometric input {#subsection:algeom}
-------------------------
As in the Introduction, let ${{\mathcal{O}}}$ be the ring of integers of a finite extension $K$ of $\QQ_p$, $\varpi$ a uniformizer of ${{\mathcal{O}}}$ and $\kappa = {{\mathcal{O}}}/(\varpi)$ the residue field. Let $X, Y$ be relative curves over ${{\mathcal{O}}}$. We assume that $X, Y$ are regular schemes, $X \rightarrow {{\operatorname{Spec }}}({{\mathcal{O}}})$ is smooth and $\pi\colon Y
{{\; \longrightarrow \;}}X$ is a finite flat morphism of degree $e+1$. Moreover, we assume that (i) there exists a section $s\colon X\otimes \kappa
{{\; \longrightarrow \;}}Y\otimes \kappa$ to $\pi \otimes \kappa$, that (ii) the special fibre $Y\otimes \kappa$ is a reduced normal crossing divisor with two components, and that (iii) the set theoretic preimage $(\pi\otimes \kappa)^{-1}(\pi\otimes \kappa)(Q)$ is equal to $Q$ for any singular point $Q\in Y\otimes \kappa$.
The following lemma must be known to the experts; for lack of a reference we provide a proof.
\[lemma:classification of local rings\] Let $(A, {{\frak{m}}})$ be a regular two-dimensional complete local ring containing ${{\mathcal{O}}}$, such that ${{\mathcal{O}}}$ is integrally closed in $A$, ${{\frak{m}}}\cap {{\mathcal{O}}}=(\varpi)$, and $\kappa \subseteq
A/{{\frak{m}}}$ is an algebraic extension.
1. If $A\otimes \kappa$ is regular then $A \cong {{\mathcal{O}}}[\![x]\!]$.
2. If $A\otimes \kappa\cong \kappa[\![s, t]\!]/(st)$ then $A \cong {{\mathcal{O}}}[\![x, y]\!]/(xy - \varpi)$.
First note that $A/{{\frak{m}}}\supseteq \kappa$ and so the local homomorphism $W(A/{{\frak{m}}}) \rightarrow A$ has image containing $W(\kappa)$ viewed as a subring of ${{\mathcal{O}}}$. Since $A/{{\frak{m}}}$ is an algebraic extension of $\kappa$, $W(A/{{\frak{m}}})$ is integral over $W(\kappa)$. Since ${{\mathcal{O}}}$ is integrally closed in $A$ it follows that $W(A/{{\frak{m}}})$ is contained in ${{\mathcal{O}}}$. In particular, $A/{{\frak{m}}}=\kappa$.
If $A\otimes \kappa$ is regular it follows by Cohen’s Theorem that $A\otimes \kappa \cong \kappa[\![x]\!]$. This gives a morphism ${{\mathcal{O}}}[\![x]\!] \rightarrow A$ which is surjective by Nakayama’s lemma; since both rings are domains of the same dimension, we conclude that ${{\mathcal{O}}}[\![x]\!] \rightarrow A$ is an isomorphism (the kernel is a prime ideal of height $0$).
Assume then that $A\otimes \kappa\cong \kappa[\![s, t]\!]/(st)$. Let $x^\prime, y^\prime \in A$ be elements reducing to $s, t$, respectively. The homomorphism ${{\mathcal{O}}}[\![x, y]\!] \rightarrow A$, taking $x, y$ to $x^\prime, y^\prime$ respectively, is surjective by Nakayama’s lemma. Let ${{\frak{p}}}$ be the kernel; it is a prime ideal of height $1$. In fact ${{\frak{p}}}$ is a principal ideal, because ${{\mathcal{O}}}[\![x, y]\!]$ is a factorial ring and by a theorem of Krull every prime ideal of height $1$ is principal. We may therefore write ${{\frak{p}}}= (h(x, y))$, where $h(x, y) = xyv - \varpi
z$ for some $v, z \in {{\mathcal{O}}}[\![x, y]\!]$. It follows that $A\otimes \kappa \cong \kappa[\![x, y]\!]/(xy\bar{v})$, where $\bar{v}$ is the reduction of $v$ modulo $\varpi$. Since $\kappa[\![x,y]\!]/(xy\bar{v})\cong \kappa[\![s,t]\!]/(st)$ by the map taking $x\mapsto s$ and $y\mapsto t$, it follows that $\bar{v}$ is a unit. This implies that $v$ itself is a unit and so $A\cong {{\mathcal{O}}}[\![x, y]\!]/(xy - \varpi z)$.
We next claim that the ring $A$ is regular if and only if $z$ is a unit. Indeed, if $z$ is a unit then $A\cong {{\mathcal{O}}}[\![x,
yz^{-1}]\!]/(x\cdot yz^{-1}-\varpi)$, which is easily checked to be regular. Assume now that $A$ is regular. Then $(\varpi, x,
y)/I$ is a $2$-dimensional $\kappa = A/{{\frak{m}}}$ vector space, where $I = (\varpi, x, y)^2 + (xy - \varpi z)$ and ${{\frak{m}}}$, the maximal ideal of $A$, is the image of $(\varpi, x, y)$. Hence, for some $c_1, c_2, c_3 \in A$, not all in ${{\frak{m}}}$, we have $c_1
\varpi + c_2 x + c_3 y \in I$. Such a relation gives modulo $\varpi$ the relation $c_2x + c_3y \in (x, y)^2$. Since the cotangent space at the singular point is two dimensional with basis $\{x,y\}$, it follows that modulo $\varpi$ we have $c_2, c_3
\in (x,y)$. Thus, we must have $c_2, c_3 \in {{\frak{m}}}$. Therefore, $A$ is regular implies that $\varpi\in I$. Thus, $\varpi
\pmod{{{\frak{m}}}^2} \in I/{{\frak{m}}}^2 = (\varpi z)/{{\frak{m}}}^2$. It follows that $z$ is a unit modulo ${{\frak{m}}}^2$ and hence is a unit.
\[lem:localcoordinates\]Let $Q\in Y$ be a singular point and $ P= \pi(Q)$. There is a choice of local coordinates at $Q$ and $P$ giving ${{\mathcal{O}}}_Y^{\wedge Q}\cong {{\mathcal{O}}}[\![x, y]\!]/(xy - \varpi)$ and ${{\mathcal{O}}}_X^{\wedge P} \cong {{\mathcal{O}}}[\![t]\!]$, respectively, such that on the level of completed local rings at $Q$ and $P$ the morphism $\pi$ is given by $$t\mapsto x + (yu)^e +
f(y) + \varpi g,$$ where $f(y) \equiv 0
\pmod{(y^{e+1})}$ and $u$ is a unit congruent to $1$ modulo $\varpi$.
It follows from Lemma \[lemma:classification of local rings\] that the map $Y\rightarrow X$ can be written at a singular point $Q\in Y$ in the form of an ${{\mathcal{O}}}$-algebra local homomorphism $\pi^*\colon {{\mathcal{O}}}[\![t]\!] \rightarrow {{\mathcal{O}}}[\![x,
y]\!]/(xy - \varpi)$. Now, upon reduction modulo $\varpi$, we get a homomorphism of $\kappa$-algebras $\pi^*\otimes \kappa\colon
\kappa[\![t]\!] \rightarrow \kappa[\![x, y]\!]/(xy)$. By our assumptions on $\pi\otimes \kappa$, the compositions $\kappa[\![t]\!] \rightarrow \kappa[\![x, y]\!]/(xy)
\underset{y \mapsto 0}{\rightarrow} \kappa[\![x]\!]$ and $\kappa[\![t]\!] \rightarrow \kappa[\![x, y]\!]/(xy)
\underset{x \mapsto 0}{\rightarrow} \kappa[\![y]\!]$ are given, w.l.o.g., by $t \mapsto x$ and $t\mapsto y^e+ f_1(y)$, where $f_1(y) \equiv 0 \mod{(y^{e+1})}$ (the existence of the section implies that every ramification index is equal to $e$). Thus, the map $\pi^\ast \otimes \kappa$ is determined by the image of $t$ which has the form $x+y^e + f_1(y) + xyf_2(x, y)$.
Our goal now is to change coordinates on $A := {{\mathcal{O}}}[\![x,
y]\!]/(xy - \varpi)$ so as to simplify this map and still have the same presentation, namely, find $\hat x, \hat y\in A$ such that ${{\mathcal{O}}}[\![x, y]\!]/(xy - \varpi) = {{\mathcal{O}}}[\![\hat x, \hat
y]\!]/(\hat x\hat y - \varpi)$. First note that since $A$ is $\varpi$-adically complete the map of units $A^\times
\rightarrow (A\otimes \kappa)^\times$ is surjective. Let $u^\prime = (1 +
yf_2(x, y)) \in (A\otimes \kappa)^\times$ and $\hat u$ any lift of it to $A^\times$. Let $\hat x = x\hat u, \hat y = y \hat u^{-1}$. Then we have ${{\mathcal{O}}}[\![\hat x, \hat y]\!](\hat x \hat y - \varpi)
= {{\mathcal{O}}}[\![x, y]\!]/(xy-\varpi)$ and the map ${{\mathcal{O}}}[\![t]\!]
\rightarrow {{\mathcal{O}}}[\![\hat x, \hat y]\!]/(\hat x \hat y - \varpi)$ has the form $t \mapsto \hat x + (\hat y\hat u)^e + \hat f(\hat y)
+ \varpi \hat g$, where $\hat f$ is a lift of $f_1$ satisfying $\hat f(y) \equiv 0 \pmod{(y^{e+1})}$.
A measure of singularity {#section:measure}
------------------------
Let $\pi\colon Y \rightarrow X$ be a morphism of curves as in §\[subsection:algeom\]. We denote by ${{\frak{X}}}$, ${{\frak{Y}}}$ the formal schemes obtained from $X$, $Y$ by completion along their special fibres. Let $\beta_1,\dots,\beta_h$ be the singular points of $Y$. Let $\alpha_i=\pi(\beta_i)$ for $i=1,\dots,h$. Recall that by assumption the $\alpha_i$’s and $\beta_i$’s are defined over $\kappa$. Let $D_{\alpha_i}$ (respectively $D_{\beta_i}$) denote the inverse image of $\alpha_i$ (respectively $\beta_i$) under the specialization map ${{\operatorname{sp }}}\colon {{{{\frak{X}}}_\text{\rm rig}}}\rightarrow {{\frak{X}}}$ (respectively ${{\operatorname{sp }}}\colon {{{{\frak{Y}}}_\text{\rm rig}}}\rightarrow {{\frak{Y}}}$).
By Proposition \[prop:Berthelot\] $D_{\alpha_i}$ is the rigid space associated to ${{\operatorname{Spf }}}({{\mathcal{O}}}_X^{\wedge \alpha_i}) \cong
{{\operatorname{Spf }}}({{\mathcal{O}}}[\![t]\!])$, using Lemma \[lemma:classification of local rings\]. Therefore $D_{\alpha_i}$ is an open disc of radius $1$ with parameter $t$. This parameter is unique up to $t
\mapsto t^\prime=t\epsilon+\varpi z$, where $\epsilon \in
{{\mathcal{O}}}^\times$ and $z \in {{\mathcal{O}}}[\![t]\!]$. For a general closed point $P \in D_{\alpha_i}$ the value ${{\operatorname{val}}}(t(P))$ depends on $t$, however if ${{\operatorname{val}}}(t(P))<1$, then ${{\operatorname{val}}}(t(P))={{\operatorname{val}}}(t^\prime(P))$ for any $t^\prime$ as above. We abuse notation and define $$\nu_{{\frak{X}}}(P)={{\operatorname{val}}}(t(P)),$$ bearing in mind that this is well defined only if ${{\operatorname{val}}}(t(P)) <1$.
Similarly, $D_{\beta_i}$ is the rigid space associated to ${{\operatorname{Spf }}}({{\mathcal{O}}}_X^{\wedge \beta_i}) \cong
{{\operatorname{Spf }}}({{\mathcal{O}}}[\![x,y]\!]/(xy-\varpi))$. Therefore, $D_{\beta_i}$ is an open annulus of radii $(|\varpi|,1)$ with parameter $x$. For any closed point $Q$ in $D_{\beta_i}$, define $$\nu_{{\frak{Y}}}(Q)={{\operatorname{val}}}(x(Q)).$$This definition is independent of the choice of the parameters if chosen as in Lemma \[lem:localcoordinates\]. The reason is that any other such parameter $x^\prime$ is of the form $x^\prime=x\epsilon+\varpi z$, where $\epsilon \in {{\mathcal{O}}}^\times$ and $z \in
{{\mathcal{O}}}[\![x,y]\!]/(xy-\varpi)$, and ${{\operatorname{val}}}(x(Q))<1$.
Let ${{\mathscr{Z}}}$ denote the complement in ${{{{\frak{X}}}_\text{\rm rig}}}$ of ${{\operatorname{sp }}}^{-1}(\{\alpha_1,\dots,\alpha_h\})$. For a closed point $P$ in ${{\mathscr{Z}}}$ we define $\nu_{{\frak{X}}}(P)=0$. By Proposition \[prop:section over the ordinary locus\] below, the complement in ${{{{\frak{Y}}}_\text{\rm rig}}}$ of ${{\operatorname{sp }}}^{-1}(\{\beta_1,\dots,\beta_h\})$ has two connected components, ${{\mathscr{Z}}}^0={{\operatorname{sp }}}^{-1}\left( (Y\otimes
\kappa)^0-\{\beta_1,\dots,\beta_h\} \right)$, and ${{\mathscr{Z}}}^\infty={{\operatorname{sp }}}^{-1}\left( (Y\otimes
\kappa)^\infty-\{\beta_1,\dots,\beta_h\} \right)$. For points in ${{\mathscr{Z}}}^\infty$ we define $\nu_{{\frak{Y}}}$ to be $0$, and on ${{\mathscr{Z}}}^0$ we define $\nu_{{\frak{Y}}}$ to be $1$. We refer to $\nu_{{\frak{X}}}$ and $\nu_{{\frak{Y}}}$ as [*measures of singularity*]{}. For an interval $I$ of real numbers, we define ${{{{\frak{X}}}_\text{\rm rig}}}I$ to be the set of points of ${{{{\frak{X}}}_\text{\rm rig}}}$ where $\nu_{{\frak{X}}}$ belongs to $I$. For $U$ an admissible open subset of ${{{{\frak{X}}}_\text{\rm rig}}}$ we set $UI=U \cap {{{{\frak{X}}}_\text{\rm rig}}}I$. We use a similar notation for ${{{{\frak{Y}}}_\text{\rm rig}}}$. We call ${{\mathscr{Z}}}$ the [*ordinary locus*]{} of ${{{{\frak{X}}}_\text{\rm rig}}}$ and its complement ${{\operatorname{sp }}}^{-1}(\{\alpha_1,\dots,\alpha_h\})={{{{\frak{X}}}_\text{\rm rig}}}(0,\infty)$ the [*singular locus*]{} of ${{{{\frak{X}}}_\text{\rm rig}}}$. We have ${{\pi_\text{\rm rig}}}^{-1}({{{{\frak{X}}}_\text{\rm rig}}}(0,\infty))={{{{\frak{Y}}}_\text{\rm rig}}}(0,1)={{\operatorname{sp }}}^{-1}(\{\beta_1,\dots,\beta_h\})$ which we call the singular locus of ${{{{\frak{Y}}}_\text{\rm rig}}}$.
Main Theorem {#section:main theorem}
============
In this section we prove Theorem [**A**]{} of the Introduction, using the same notation. Our strategy is to construct sections separately on the ordinary locus and the singular locus and glue them by means of a general principle of rigid geometry. We start by constructing a section to ${{\pi_\text{\rm rig}}}$ over the ordinary locus of ${{{{\frak{X}}}_\text{\rm rig}}}$.
\[prop:section over the ordinary locus\] The map ${{\pi_\text{\rm rig}}}$ induces an isomorphism between ${{\mathscr{Z}}}^\infty$ and ${{\mathscr{Z}}}$. Therefore there is a unique section ${{\frak{s}}}_{{\operatorname{rig}}}^\infty\colon {{\mathscr{Z}}}\rightarrow {{{{\frak{Y}}}_\text{\rm rig}}}$ to ${{\pi_\text{\rm rig}}}$ whose image is ${{\mathscr{Z}}}^\infty$. Furthermore, both ${{\mathscr{Z}}}^\infty$ and ${{\mathscr{Z}}}^0$ are connected. If $e>1$, then any section to ${{\pi_\text{\rm rig}}}$ on ${{\mathscr{Z}}}$ coincides with ${{\frak{s}}}_{{\operatorname{rig}}}^\infty$.
We show the existence of the section on the level of the formal schemes. The curves $(Y \otimes \kappa)^\infty$, $(Y
\otimes \kappa)^0$ are connected reduced affine curves. Let $U$ be the open subset of ${{\frak{Y}}}$ equal to the underlying set of $(Y
\otimes \kappa)^\infty \cup (Y \otimes \kappa)^0$. Then $U$ is affine in the formal schemes sense, namely, we have an open immersion ${{\operatorname{Spf }}}(B) \rightarrow {{\frak{Y}}}$ whose set theoretic image is $U$. Under the specialization map ${{\operatorname{sp }}}\colon {{\frak{Y}}}_{{\operatorname{rig}}}\rightarrow {{\frak{Y}}}$ we have ${{\operatorname{sp }}}^{-1}(U) = {{\mathscr{Z}}}^0 \cup
{{\mathscr{Z}}}^\infty$ and, moreover, ${{\mathscr{Z}}}^0 \cup {{\mathscr{Z}}}^\infty = U_{{\operatorname{rig}}}$ (cf. the discussion in § \[subsection:recall\]). We conclude the following: We have a morphism ${{\operatorname{Spf }}}(B) \rightarrow {{\operatorname{Spf }}}(A)$, induced by a homomorphism of $\varpi$-adically complete ${{\mathcal{O}}}$-algebras $A \rightarrow B$, that yields the morphism ${{\mathscr{Z}}}^0 \cup {{\mathscr{Z}}}^\infty \rightarrow {{\mathscr{Z}}}$ and reduces to the morphism $Y\otimes\kappa\setminus\{ \beta_i\}_{i=1}^h
\longrightarrow X\otimes \kappa \setminus\{ \alpha_i\}_{i=1}^h$. It transpires that $B \otimes \kappa = (A\otimes \kappa) \oplus
B_1$. Using Hensel’s lemma to lift idempotents, we conclude that we have $B = A^+ \oplus B_1^+$, with $A^+ \otimes \kappa =
A\otimes \kappa, B_1^+ \otimes \kappa = B_1$. Using that $A
\rightarrow A^+$ is a finite flat homomorphism reducing to an isomorphism after $\otimes \kappa$, we conclude that $A = A^+$. This gives the existence of the section ${{\frak{s}}}^\infty\colon{{\operatorname{Spf }}}(A)
{{\; \longrightarrow \;}}{{\operatorname{Spf }}}(B)$, the analytification of which is the desired section ${{\frak{s}}}_{{\operatorname{rig}}}^\infty\colon {{\mathscr{Z}}}\rightarrow {{\mathscr{Z}}}^0\cup
{{\mathscr{Z}}}^\infty$ with image ${{\mathscr{Z}}}^\infty$. In particular, ${{\mathscr{Z}}}^\infty$, being isomorphic to ${{\mathscr{Z}}}$, which is a curve minus finitely many residue discs, is connected.
Furthermore, the morphism ${{\operatorname{Spf }}}(B_1^+) \rightarrow {{\operatorname{Spf }}}(A)$ is finite flat of degree $e$. To show ${{\mathscr{Z}}}^0$ is connected it is enough to show that ${{\operatorname{Spf }}}(B^+_1)$ is flat over ${{\operatorname{Spf }}}({{\mathcal{O}}})$, and has a reduced and connected special fibre (see Remark \[rmk:connected\]). But this is clear since ${{\operatorname{Spec }}}(B_1^+\otimes \kappa) = (Y\otimes \kappa)^0$.
For the final assertion, note that the image of any section to ${{\pi_\text{\rm rig}}}$ on ${{\mathscr{Z}}}$ must be a connected component of ${{\pi_\text{\rm rig}}}^{-1}({{\mathscr{Z}}})={{\mathscr{Z}}}^\infty \cup {{\mathscr{Z}}}^0$, and hence it must be either ${{\mathscr{Z}}}^\infty$ or ${{\mathscr{Z}}}^0$. The latter can’t happen since ${{\pi_\text{\rm rig}}}\colon {{\mathscr{Z}}}^0 \rightarrow {{\mathscr{Z}}}$ is $e$-to-$1$ and $e>1$.
\[rmk:connected\] Let ${{\frak{B}}}= {{\operatorname{Spf }}}(B)$ be an admissible formal scheme, with associated rigid space ${{\frak{B}}}_{{\operatorname{rig}}}$. It is possible that ${{\frak{B}}}_{{\operatorname{rig}}}$ is disconnected, yet the underlying topological space of ${{\frak{B}}}$ is connected. An example is provided when ${{\mathcal{O}}}$ is a ramified extension of $\ZZ_p$ and we let $B = {{\mathcal{O}}}\langle x,
y, T\rangle/(xy - p, (x+y)T - \varpi)$. The associated rigid space is a disjoint union of two annuli. The special fibre is three lines meeting at a single point. Note though that $B\otimes \kappa
= k[x, y, T]/(xy, (x+y)T)$ in which $xT$ is nilpotent.
On the other hand, assume ${{\frak{B}}}$ is an admissible formal scheme over ${{\mathcal{O}}}$ such that ${{\frak{B}}}_{{\operatorname{rig}}}$ is affinoid (in particular ${{\frak{B}}}_{{\operatorname{rig}}}={{\operatorname{Spm }}}(B \otimes_{{\mathcal{O}}}K)$ where $B=H^0({{\frak{B}}},{{\mathcal{O}}}_{{{\frak{B}}}})$). If ${{\frak{B}}}\otimes \kappa$ is *reduced*, then the connectedness of ${{\frak{B}}}$ implies the same for ${{\frak{B}}}_{{\operatorname{rig}}}$. Indeed, if not, then there is a non-trivial idempotent element $e\in B \otimes_{{\mathcal{O}}}K$. We show that $e \in
B$. Note that by flatness of ${{\frak{B}}}$ over ${{\mathcal{O}}}$ we know that $B{{\; \hookrightarrow \;}}B\otimes K$. If $e \not\in B$, we can write $e=
f/\varpi^n$, where $n>0$ is minimal, and $f\in B$. Then we have $f^2=\varpi^nf$. Reducing modulo $\varpi$, we get $\bar{f}\neq 0$ and $\bar{f}^2=0$ which contradicts our assumption on ${{\frak{B}}}\otimes \kappa$. Therefore $e\in B$. It then follows that the decomposition of the “generic fibre" $B\otimes
K$, namely of the rigid space, induces a decomposition of the formal scheme ${{\operatorname{Spf }}}(B)$.
Let ${{\mathcal{X}}}$ be a $K$-rigid analytic space, and ${{\mathcal{U}}}\subset
{{\mathcal{X}}}$ be an admissible affinoid subdomain. An affinoid subdomain ${{\mathcal{U}}}\subset {{\mathcal{V}}}\subset {{\mathcal{X}}}$ is called a [*strict neighborhood*]{} of ${{\mathcal{U}}}$ in ${{\mathcal{X}}}$ if the reduction of the inclusion $\iota\colon {{\mathcal{U}}}\rightarrow {{\mathcal{V}}}$ factors through an affine scheme which is finite over ${{\operatorname{Spec }}}(\kappa)$. See [@CGJ §3] for more details. Any strict neighborhood of ${{\mathscr{Z}}}$ in ${{{{\frak{X}}}_\text{\rm rig}}}$ contains a domain of the form ${{{{\frak{X}}}_\text{\rm rig}}}[0,a]$ for some positive $a \in \QQ$; cf. [@KisinLai Prop. 2.3.2]. The following is Lemma 6 of [@CGJ]. See also [@Berthelot].
Let $f\colon {\mathcal Y} \rightarrow {\mathcal X}$ be a finite flat morphism of rigid analytic curves. Let ${\mathcal U}$ be an affinoid subdomain of ${\mathcal X}$, and $s\colon {\mathcal U}
\rightarrow {\mathcal Y}$ a section to $f$. Then $s$ can be extended to a strict neighborhood of ${\mathcal U}$ in ${\mathcal
X}$.
\[cor:overconvergence\] The section ${{\frak{s}}}_{{\operatorname{rig}}}^\infty$ extends to a section ${{\frak{s}}}_{{\operatorname{rig}}}^\dagger$ over ${{{{\frak{X}}}_\text{\rm rig}}}[0,a]$ for some positive $a \in
\QQ$.
Next we discuss sections to ${{\pi_\text{\rm rig}}}$ over the singular locus, i.e. where $\nu_{{\frak{X}}}>0$.
\[prop:extending the section\]The map ${{\pi_\text{\rm rig}}}\colon {{{{\frak{Y}}}_\text{\rm rig}}}(0,1) \rightarrow
{{{{\frak{X}}}_\text{\rm rig}}}(0,\infty)$ admits a section ${{\frak{t}}}$ on ${{{{\frak{X}}}_\text{\rm rig}}}(0,e/(e+1))$ whose image is ${{{{\frak{Y}}}_\text{\rm rig}}}(0,e/(e+1))$. Such a section is unique. If $e>1$, then we have the following stronger uniqueness result: any section to ${{\pi_\text{\rm rig}}}$ on a connected admissible open subset of ${{{{\frak{X}}}_\text{\rm rig}}}(0,e/(e+1))$ which contains some circle $D_{\alpha_i}[a,a]$ is obtained by the restriction of ${{\frak{t}}}$.
We have ${{{{\frak{Y}}}_\text{\rm rig}}}(0,1)= \coprod_i D_{\beta_i}$ and ${{{{\frak{X}}}_\text{\rm rig}}}(0,\infty)=\coprod_i D_{\alpha_i}$. Since by our assumptions $\pi^{-1}(\alpha_i)=\{\beta_i\}$ as sets, we have ${{\pi_\text{\rm rig}}}^{-1}(D_{\alpha_i})=D_{\beta_i}$, and hence, for the first assertion, it suffices to show that for each $i$ the map ${{\pi_\text{\rm rig}}}\colon D_{\beta_i}
\rightarrow D_{\alpha_i}$ admits a section on $D_{\alpha_i}(0,e/(e+1))$ whose image is $D_{\beta_i}(0,e/(e+1))$. The map ${{\pi_\text{\rm rig}}}\colon D_{\beta_i}
\rightarrow D_{\alpha_i}$ is the analytification of the map $\pi\colon {{\operatorname{Spf }}}({{\mathcal{O}}}_Y^{\wedge \beta_i}) \rightarrow
{{\operatorname{Spf }}}({{\mathcal{O}}}_X^{\wedge \alpha_i })$ by Proposition \[prop:Berthelot\]. By Lemma \[lem:localcoordinates\], choosing local coordinates, this map is given by $${{\mathcal{O}}}[\![t]\!] \rightarrow {{\mathcal{O}}}[\![x, y]\!]/(xy - \varpi), \qquad
t \mapsto x + uy^e + f(y) + \varpi g,$$ where $f(y)
\equiv 0 \pmod{y^e}$, $u,g\in {{\mathcal{O}}}[\![x, y]\!]/(xy -\varpi)$, and $u$ is a unit. Let $\tilde{u}, \tilde{g}$ denote arbitrary liftings of $u,g$ to ${{\mathcal{O}}}[\![x,y]\!]$ and define $g_0(x)=\tilde{g}(x,\varpi/x)$, $u_0(x)=\tilde{u}(x,\varpi/x)$, and $f_0(x)=f(\varpi/x)$. Then the map ${{\pi_\text{\rm rig}}}\colon D_{\beta_i}
\rightarrow D_{\alpha_i}$ is the map characterized by $$t({{\pi_\text{\rm rig}}}Q)=x(Q)+ u_0(x(Q))(\varpi/x(Q))^e+f_0(x(Q))+\varpi g_0(x(Q)).$$
\[lem: val and pi rig\] Let $Q \in {{{{\frak{Y}}}_\text{\rm rig}}}$.
1. If $\nu_{{\frak{Y}}}(Q)<e/(e+1)$ then $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)=\nu_{{\frak{Y}}}(Q)$.
2. If $\nu_{{\frak{Y}}}(Q)>e/(e+1)$ then $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)=e(1-\nu_{{\frak{Y}}}(Q))<e/(e+1)$.
3. If $\nu_{{\frak{Y}}}(Q)=e/(e+1)$ then $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)\geq e/(e+1)$.
The statement is clear for $Q \in
{{\mathscr{Z}}}^\infty \cup {{\mathscr{Z}}}^0$. If $Q \in D_{\beta_i}$ satisfies $\nu_{{\frak{Y}}}(Q)={{\operatorname{val}}}(x(Q)) < e/(e+1)$, then $${{\operatorname{val}}}(x(Q))<\min\left\{ {{\operatorname{val}}}((\varpi/x(Q))^e ),\;
{{\operatorname{val}}}(f_0(x(Q))),\; {{\operatorname{val}}}(\varpi g_0(x(Q)))\right\}.$$This implies that ${{\operatorname{val}}}(t({{\pi_\text{\rm rig}}}Q))={{\operatorname{val}}}(x(Q))$. The other cases are similar.
From the lemma it follows that $${{\pi_\text{\rm rig}}}^{-1} D_{\alpha_i}(0,e/(e+1))=
D_{\beta_i}(0,e/(e+1)) \coprod D_{\beta_i}(e/(e+1),1).$$ Indeed the lemma proves something stronger: for any $a \in \QQ$ satisfying $0 < a < e/(e+1)$ we have
$$\label{eqn:radius}
{{\pi_\text{\rm rig}}}^{-1}(D_{\alpha_i}[a,a])= D_{\beta_i}[1-a/e,1-a/e] \cup
D_{\beta_i} [a,a].$$
This shows that the inverse image of $D_{\alpha_i}(0,e/(e+1))$ under ${{\pi_\text{\rm rig}}}$ has two connected components each of which maps onto $D_{\alpha_i}(0,e/(e+1))$ in a finite flat manner.
We next show that the finite flat morphism ${{\pi_\text{\rm rig}}}\colon
D_{\beta_i}(0,e/(e+1)) \rightarrow D_{\alpha_i}(0,e/(e+1))$ is of degree one and hence is an isomorphism. The inverse of this map provides the desired section ${{\frak{t}}}$. To calculate the degree we restrict the map to a circle $D_{\alpha_i}[a,a]$ with $0<a<e/(e+1)$. It is therefore enough to show that ${{\pi_\text{\rm rig}}}\colon
D_{\beta_i}[a,a] \rightarrow D_{\alpha_i}[a,a]$ has degree one. We show this by reduction modulo $\varpi$. Our argument is based on the following general principle.
Let $\phi\colon {{\operatorname{Spm }}}(B) \rightarrow {{\operatorname{Spm }}}(A)$ be a finite flat morphism of $K$-affinoids. Let $L$ be a finite field extension of $K$ and let $\phi_L\colon {{\operatorname{Spm }}}(B\otimes_K L) \rightarrow
{{\operatorname{Spm }}}(A\otimes_K L)$ be the induced morphism. Let $\theta$ be a uniformizer of $L$ and let $n$ be a positive integer; let $(B\otimes_K L)^\circ$ denote the ${{\mathcal{O}}}_L$-algebra of functions of supremum norm at most $1$. Define $\overline{B}=(B\otimes_K L)^\circ/(\theta^n)$, and similarly for $A$. Let $\overline{\phi}_L\colon
{{\operatorname{Spec }}}(\overline{B}) \rightarrow {{\operatorname{Spec }}}(\overline{A})$ be the induced map. Then, if $\overline{\phi}_L$ is an isomorphism so is $\phi$. The argument reduces to proving that $\phi_L^*\colon (A
\otimes_K L)^\circ \rightarrow (B \otimes_K L)^\circ$ is surjective, which, in turn, follows from Nakayama’s lemma.
To prove that the reduction of ${{\pi_\text{\rm rig}}}\colon D_{\beta_i}[a,a]
\rightarrow D_{\alpha_i}[a,a]$ is an isomorphism, we first re-scale. We pass to a finite extension $L$ of $K$ with uniformizer $\theta$ in which there exists an element $\lambda$ of valuation $a$. Setting $x=\lambda x_0$ and $t=\lambda t_0$ the map ${{\pi_\text{\rm rig}}}$ becomes a map between circles of radius one characterized by $$t_0({{\pi_\text{\rm rig}}}Q)= x_0(Q) + u_0(\lambda x_0(Q))(\varpi^e/\lambda^{e+1})
x_0(Q)^{-e}+\lambda^{-1}f_0(\lambda x_0(Q))+\lambda^{-1}\varpi
g_0(\lambda x_0(Q)).$$ Using $0<a<e(e+1)^{-1}$ and $f(y)\equiv 0 \pmod{y^{e+1}}$, one sees that this map reduces modulo $\theta$ to the identity map of $ {{\mathcal{O}}}_L/(\theta) [T,1/T]$.
For the second statement we argue as follows. Let $U \supseteq
D_{\alpha_i}[a,a]$ be a connected admissible open of ${{{{\frak{X}}}_\text{\rm rig}}}(0,e/(e+1))$ over which there is a section ${{\frak{t}}}^\prime$ to ${{\pi_\text{\rm rig}}}$. Then $U$, being connected, lies entirely within $D_{\alpha_i}(0,e/(e+1))$. By Lemma \[lem: val and pi rig\] the image of $U$ under ${{\frak{t}}}^\prime$ is either a subset of $D_{\beta_i}(0,e/(e+1))$, or a subset of $D_{\beta_i}(e/(e+1),1)$. In the former case, by the construction of ${{\frak{t}}}$, it is clear that ${{\frak{t}}}^\prime={{\frak{t}}}|_U$. In the latter case, ${{\frak{t}}}^\prime(D_{\alpha_i}[a,a])$ is a connected component of $D_{\beta_i}[1-a/e,1-a/e]$ by Equation (\[eqn:radius\]). However, since $D_{\beta_i}[1-a/e,1-a/e]$ is connected and ${{\pi_\text{\rm rig}}}\colon D_{\beta_i}[1-a/e,1-a/e] \rightarrow
D_{\alpha_i}[a,a]$ is $e$-to-$1$, we find that $e=1$, which contradicts our assumption.
\[cor:the section\] Assume $e>1$. The morphism ${{\pi_\text{\rm rig}}}\colon {{{{\frak{Y}}}_\text{\rm rig}}}\rightarrow {{{{\frak{X}}}_\text{\rm rig}}}$ admits a unique section $${{\frak{s}}}_{{\operatorname{rig}}}\colon {{\frak{X}}}_{{\operatorname{rig}}}[0,e/(e+1)) \rightarrow {{{{\frak{Y}}}_\text{\rm rig}}}$$ which extends ${{\frak{s}}}_{{{\operatorname{rig}}}}^\infty$.
By Corollary \[cor:overconvergence\], ${{\frak{s}}}_{{{\operatorname{rig}}}}^\infty$ extends to a section ${{\frak{s}}}_{{{\operatorname{rig}}}}^\dagger$ on ${{{{\frak{X}}}_\text{\rm rig}}}[0,a]$ for some positive rational number $a<e/(e+1)$. By the uniqueness assertion in Proposition \[prop:extending the section\] we know that the restriction of ${{\frak{s}}}_{{{\operatorname{rig}}}}^\dagger$ to ${{{{\frak{X}}}_\text{\rm rig}}}(0,a]$ is obtained as the restriction of ${{\frak{t}}}$. This implies that ${{\frak{s}}}_{{{\operatorname{rig}}}}^\infty$ and ${{\frak{t}}}$ glue together to form the desired unique section.
\[prop:Coleman\] Assume $e>1$. The section ${{\frak{s}}}_{{\operatorname{rig}}}$ constructed in Corollary \[cor:the section\] is maximal in the following sense: Let $U$ be a connected affinoid inside ${{{{\frak{X}}}_\text{\rm rig}}}(0,\infty)$ such that it intersects both ${{{{\frak{X}}}_\text{\rm rig}}}(0,e/(e+1))$ and ${{{{\frak{X}}}_\text{\rm rig}}}[e/(e+1),1)$ nontrivially . Then there is no section to ${{\pi_\text{\rm rig}}}$ on $U$.
Assume there is a section ${{\frak{s}}}$ to ${{\pi_\text{\rm rig}}}$ on such $U$. Since $U$ is connected it lies inside some $D_{\alpha_i}$, and intersects both $D_{\alpha_i}(0,e/(e+1))$ and $D_{\alpha_i}[e/(e+1),e/(e+1)]$ nontrivially. By [@BGR §9.7.2, Thm. 2] any connected affinoid of $D_{\alpha_i}$ is the complement of a union of finitely many disjoint open discs in a closed disc. A simple calculation using the non-archimedean property of the norm shows that a closed disc which intersects both $D_{\alpha_i}(0,e/(e+1))$ and $D_{\alpha_i}[e/(e+1),e/(e+1)]$ nontrivially, must contain all of $D_{\alpha_i}[e/(e+1),e/(e+1)]$. Therefore, $U$ contains the complement of a union of finitely many disjoint open discs $V_1, \dots, V_r$ (which we may assume to have radius $e/(e+1)$) in the circle $D_{\alpha_i}[e/(e+1),e/(e+1)]$.
We first re-scale as in the proof of Proposition \[prop:extending the section\]: let $\lambda \in L$ be such that ${{\operatorname{val}}}(\lambda)=e/(e+1)$. Setting $x=\lambda x_0$ and $t=\lambda t_0$ the map ${{\pi_\text{\rm rig}}}\colon D_{\beta_i}[e/(e+1),e/(e+1)] \rightarrow
D_{\alpha_i}[e/(e+1),\infty)$ becomes a map between a circle $C$ of radius one and the closed unit disc $D$ characterized by $$t_0({{\pi_\text{\rm rig}}}Q)= x_0(Q) + u_0(\lambda x_0(Q))(\varpi^e/\lambda^{e+1})
x_0(Q)^{-e}+\lambda^{-1}f_0(\lambda x_0(Q))+\lambda^{-1}\varpi
g_0(\lambda x_0(Q)).$$ The section ${{\frak{s}}}$ is defined on $W$ – the complement in $D$ of finitely many residue discs: the open unit disc together with $\lambda^{-1} V_1, \dots, \lambda^{-1}
V_r$. The reduction $\bar{{{\frak{s}}}}$ of ${{\frak{s}}}\colon W \rightarrow
C$, then, gives a map between $\AA^1_{{{\mathcal{O}}}_L/(\theta)}$ minus a finite number of points (with parameter $\bar{t}_0$), and $\AA^1_{{{\mathcal{O}}}_L/(\theta)}$ (with parameter $\bar{x}_0$) characterized by $$\bar{t}_0(Q)= \bar{x}_0({\bar{{\frak{s}}}} (Q)) + \overline{(\varpi^e/\lambda^{e+1})}
\bar{x}_0({\bar{{\frak{s}}}}(Q))^{-e}.$$Here $\bar{x}_0({\bar{{\frak{s}}}}(-))$ is a rational function in $\bar{t}_0$ and $\overline{\varpi^e/\lambda^{e+1}}$ is nonzero by our choice of $\lambda$. Degree considerations show that this is impossible.
We summarize the above results as a theorem (Theorem A of the Introduction).
Assume $e>1$. The morphism ${{\pi_\text{\rm rig}}}\colon {{{{\frak{Y}}}_\text{\rm rig}}}\rightarrow {{{{\frak{X}}}_\text{\rm rig}}}$ admits a section $${{\frak{s}}}_{{\operatorname{rig}}}\colon {{\frak{X}}}_{{\operatorname{rig}}}[0,e/(e+1)) \rightarrow {{{{\frak{Y}}}_\text{\rm rig}}}.$$ This section is maximal, namely, it can not be extended to any connected admissible open properly set containing ${{\frak{X}}}_{{\operatorname{rig}}}[0,e/(e+1))$.
The canonical subgroup of an elliptic curve can be thought of as a certain lifting the kernel of Frobenius from characteristic $p$ [@Katz Thm. 3.1]. We prove a similar result in our setting. The section ${{\frak{s}}}_{{\operatorname{rig}}}^\infty={{\frak{s}}}_{{\operatorname{rig}}}|_{{\mathscr{Z}}}$ was constructed on the level of formal schemes, and by its construction it reduces to $s$ mod $\varpi$.
Fix $D_{\alpha_i}$ and let $t$ be a coordinate on it, obtained from an isomorphism ${{\mathcal{O}}}_X^{\wedge \alpha_i} \cong
{{\mathcal{O}}}[\![t]\!]$ as in Lemma \[lem:localcoordinates\]. Also fix an isomorphism ${{\mathcal{O}}}_X^{\wedge \beta_i} \cong
{{\mathcal{O}}}[\![x,y]\!]/(xy-\varpi)$ as in loc. cit.; $x$ is a parameter on $D_{\beta_i}$ and $xy=\varpi$.
Let $\gamma_P\colon{{\operatorname{Spm }}}(L) \rightarrow D_{\alpha_i}$ correspond to a closed point $P$ which lies in $D_{\alpha_i}(0,e/(e+1))$. Thus $L$ is a finite extension of $K$. Let $\gamma_{{{\frak{s}}}_{{\operatorname{rig}}}(P)}={{\frak{s}}}_{{\operatorname{rig}}}\circ\gamma_P\colon {{\operatorname{Spm }}}(L)
\rightarrow D_{\beta_i}$ correspond to the image of $P$ under ${{\frak{s}}}_{{\operatorname{rig}}}$. Let $\tilde{\gamma}_P\colon{{\operatorname{Spf }}}({{\mathcal{O}}}_L) \rightarrow
{{\frak{X}}}$ denote the extension of $\gamma_P$ to the formal model, and similarly define $\tilde{\gamma}_{{{\frak{s}}}_{{\operatorname{rig}}}(P)}$. Let $\overline{\gamma}_P$, $\overline{\gamma}_{{{\frak{s}}}_{{\operatorname{rig}}}(P)}$ denote, respectively, the reductions of $\tilde{\gamma}_P$, $\tilde{\gamma}_{{{\frak{s}}}_{{\operatorname{rig}}}(P)}$ modulo the element $\varpi/t(P)$ of ${{\mathcal{O}_L}}$. Let $s^\prime$ denote the base change of $s\colon
X\otimes\kappa \rightarrow Y\otimes\kappa$ from $\kappa$ to ${{\mathcal{O}}}_L/(\varpi/t(P))$. For simplicity we denote the $\kappa$-algebra ${{\mathcal{O}}}_L/(\varpi/t(P))$ by $R$.
For closed points $P\in D_{\alpha_i}(0,e/(e+1))$ with $t(P)=r \in {{\mathcal{O}}}_L$ the section ${{\frak{s}}}_{{\operatorname{rig}}}$ reduces modulo $\varpi/r$ to $s^\prime$. More precisely, for any $P \in
D_{\alpha_i}(0,e/(e+1))$ we have $\overline{\gamma}_{{{\frak{s}}}_{{\operatorname{rig}}}(P)}=s^\prime \circ
\overline{\gamma}_P$.
Let us denote the image of an element $a \in {{\mathcal{O}}}_L$ in $R:={{\mathcal{O}}}_L/(\varpi/t(P))$ by $\overline{a}$. Since $P \in D_{\alpha_i}$, the map $\tilde{\gamma}_P\colon{{\operatorname{Spf }}}({{\mathcal{O}}}_L) \rightarrow {{\frak{X}}}$ factors through ${{\operatorname{Spf }}}({{\mathcal{O}}}_X^{\wedge\alpha_i})$. Similarly $\tilde{\gamma}_{{{\frak{s}}}_{{\operatorname{rig}}}(P)}$ factors through ${{\operatorname{Spf }}}({{\mathcal{O}}}_Y^{\wedge\beta_i})$. Therefore, it is enough to prove the statement after replacing ${{\frak{X}}}$ with ${{\operatorname{Spf }}}({{\mathcal{O}}}_X^{\wedge\alpha_i})$ and ${{\frak{Y}}}$ with ${{\operatorname{Spf }}}({{\mathcal{O}}}_Y^{\wedge\beta_i})$. Then $$\overline{\gamma}_P\colon{{\operatorname{Spec }}}(R) \rightarrow
{{\operatorname{Spec }}}({{\mathcal{O}}}_X^{\wedge\alpha_i} \otimes R)\cong{{\operatorname{Spec }}}(R[\![t]\!])$$ is given by $t \mapsto \overline{t(P)}$. Similarly, the map $$\overline{\gamma}_{{{\frak{s}}}_{{\operatorname{rig}}}(P)}\colon{{\operatorname{Spec }}}(R) \rightarrow
{{\operatorname{Spec }}}({{\mathcal{O}}}_Y^{\wedge\beta_i} \otimes
R)\cong{{\operatorname{Spec }}}(R[\![x,y]\!]/(xy))$$ is given by $x \mapsto
\overline{x({{\frak{s}}}_{{\operatorname{rig}}}(P))}, y \mapsto
\overline{y({{\frak{s}}}_{{\operatorname{rig}}}(P))}$. From the proof of Lemma \[lem:localcoordinates\], we see that the section $$s^\prime\colon{{\operatorname{Spec }}}(R[\![t]\!])\cong{{\operatorname{Spec }}}({{\mathcal{O}}}_X^{\wedge\alpha_i}
\otimes \kappa \otimes_\kappa R) \rightarrow
{{\operatorname{Spec }}}({{\mathcal{O}}}_Y^{\wedge\beta_i}\otimes \kappa \otimes_\kappa R)\cong
{{\operatorname{Spec }}}(R[\![x,y]\!]/(xy))$$ is given by $x\mapsto t, y\mapsto 0$. Hence, it is enough to show that $\overline{y({{\frak{s}}}_{{{\operatorname{rig}}}}(P))}=0$ and $\overline{x({{\frak{s}}}_{{{\operatorname{rig}}}}(P))}=\overline{t(P)}$. For the first equality notice that by Lemma \[lem: val and pi rig\] we have ${{\operatorname{val}}}(t(P))={{\operatorname{val}}}(x({{\frak{s}}}_{{\operatorname{rig}}}(P)))$ and hence $y({{\frak{s}}}_{{\operatorname{rig}}}(P))=\varpi/x({{\frak{s}}}_{{\operatorname{rig}}}(P))$ is divisible by $\varpi/t(P)$. Since ${{\operatorname{val}}}(t(P))={{\operatorname{val}}}(x({{\frak{s}}}_{{\operatorname{rig}}}(P)))$, to prove the second equality it is enough to show that $t(P)$ and $x({{\frak{s}}}_{{\operatorname{rig}}}(P))$ have the same reduction modulo $y({{\frak{s}}}_{{\operatorname{rig}}}(P))=\varpi/x({{\frak{s}}}_{{\operatorname{rig}}}(P))$. But that is clear since from the proof of Proposition \[prop:extending the section\] we have $$t(P)=x({{\frak{s}}}_{{\operatorname{rig}}}(P))+ u_0(x({{\frak{s}}}_{{\operatorname{rig}}}(P)))(y({{\frak{s}}}_{{\operatorname{rig}}}(P)))^e+f_0(x({{\frak{s}}}_{{\operatorname{rig}}}(P)))
+\varpi g_0(x({{\frak{s}}}_{{\operatorname{rig}}}(P))),$$ and $f_0(x({{\frak{s}}}_{{\operatorname{rig}}}(P)))\equiv
0 \pmod {y({{\frak{s}}}_{{\operatorname{rig}}}(P))^{e+1}}$.
Let $Q \in {{{{\frak{Y}}}_\text{\rm rig}}}$.
1. We say that $Q$ is [*canonical*]{} if $Q$ is in the image of ${{\frak{s}}}_{{\operatorname{rig}}}$. By the construction of ${{\frak{s}}}_{{\operatorname{rig}}}$, this is equivalent to having $\nu_{{\frak{Y}}}(Q) < e/(e+1)$. If $Q$ is canonical, then by Lemma \[lem: val and pi rig\] we have $\nu_{{\frak{Y}}}(Q)=\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)$.
2. We say that $Q$ is [*anti-canonical*]{} if $\nu_{{\frak{Y}}}(Q) >
e/(e+1)$. In this case by Lemma \[lem: val and pi rig\] we have $\nu_{{\frak{Y}}}(Q)=1-e^{-1}\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)$.
3. We say that $Q$ is [*too singular*]{} if $\nu_{{\frak{Y}}}(Q)=e/(e+1)$. This is equivalent to $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)
\geq e/(e+1)$.
Throwing in an involution {#section:involution}
=========================
In this section we prove the following theorem (Theorem B of the Introduction).
Let $w$ be an automorphism of ${{\frak{Y}}}$ that permutes the components of ${{\frak{Y}}}$. We denote by $w$ also the induced automorphism of ${{{{\frak{Y}}}_\text{\rm rig}}}$ and its effect of points by $Q \mapsto Q^w$. Then:
1. $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q) = 0 \;\Leftrightarrow\;
\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w) = 0$. In this case $Q$ is canonical if and only if $Q^w$ is anti-canonical.
2. If $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)
<(e+1)^{-1}$ and $Q$ canonical, then $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w) =
e\cdot \nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)$ and $Q^w$ is anti-canonical.
3. If $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)=(e+1)^{-1}$, and $Q$ is canonical, then $Q^w$ is too singular.
4. If $(e+1)^{-1} <
\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q) < e(e+1)^{-1}$, and $Q$ is canonical, then $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w) = 1 - \nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)$ and $Q^w$ is canonical.
5. If $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q) < e(e+1)^{-1}$, and $Q$ is anti-canonical, then $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w) =
e^{-1}\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)$, and $Q^w$ is canonical.
6. If $Q$ is too singular, then $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w)=(e+1)^{-1}$ and $Q^w$ is canonical.
We begin by proving the following lemma.
\[lem:valuations identity\] For any $Q \in {{{{\frak{Y}}}_\text{\rm rig}}}$ we have $$\nu_{{\frak{Y}}}(Q) +
\nu_{{\frak{Y}}}(Q^w)=1.$$
We first note that $w({{\mathscr{Z}}}^\infty)={{\mathscr{Z}}}^0$, and hence for $Q \in
{{\mathscr{Z}}}^\infty \cup {{\mathscr{Z}}}^0$ the result follows from the definition of $\nu_{{\frak{Y}}}$. Assume $Q \in D_{\beta_i}$ for some $1 \leq i \leq
h$. The involution $w$ induces an isomorphism between $D_{\beta_i}$ and $D_{\beta_j}$ where $\beta_j=\beta_i^w$. Let $x,y$ be coordinates on $D_{\beta_i}$ as in Lemma \[lem:localcoordinates\]. Then $\eta:=w^*x$ and $\xi:=w^*y$ are coordinates on $D_{\beta_j}$ such that $D_{\beta_j}$ is the analytification of ${{\operatorname{Spf }}}({{\mathcal{O}}}[\![\xi,\eta]\!]/(\xi\eta-\varpi))$. Because $w$ switches the two components of $Y \otimes \kappa$, $\xi$ is a local parameter on the component containing $(Y\otimes
\kappa)^\infty$ at the point $\beta_j$. Examination of the proof of Lemma \[lem:localcoordinates\] shows that there is a local parameter $\tau$ on $D_{\alpha_j}$, and local parameters $(\hat{\xi}, \hat{\eta})$ on $D_{\beta_j}$ such that $\hat{\xi}=\xi\hat{u}$, $\hat{\eta}=\eta\hat{u}^{-1}$, where $\hat{u}$ is a unit in ${{\mathcal{O}}}[\![\xi,\eta]\!]/(\xi\eta-\varpi)$, and such that $(\tau,\hat{\xi},\hat{\eta})$ are related as in the statement of Lemma \[lem:localcoordinates\].
By our definition, we can use $\hat\xi$ to calculate $\nu_{{\frak{Y}}}$ on $D_{\beta_j}$. Therefore $$\nu_{{\frak{Y}}}(Q^w)={{\operatorname{val}}}(\hat\xi(Q^w))={{\operatorname{val}}}(\xi(Q^w))={{\operatorname{val}}}(y(Q))=1-{{\operatorname{val}}}(x(Q))=1-\nu_{{\frak{Y}}}(Q).$$
We now prove the theorem.
\(1) is clear.
\(2) Since $Q$ is canonical, Lemma \[lem: val and pi rig\] implies that $\nu_{{\frak{Y}}}(Q)=\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)<(e+1)^{-1}$. Therefore by Lemma \[lem:valuations identity\] we have $\nu_{{\frak{Y}}}(Q^w) > e(e+1)^{-1}$, which means that $Q^w$ is anti-canonical. It now follows from Lemma \[lem: val and pi rig\] that $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w) = e(1 - \nu_{{\frak{Y}}}(Q^w)) = e
\nu_{{\frak{Y}}}(Q) = e \nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)$.
\(3) Since $Q$ is canonical, Lemma \[lem: val and pi rig\] implies that $\nu_{{\frak{Y}}}(Q)=\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)=(e+1)^{-1}$, and therefore $\nu_{{\frak{Y}}}(Q^w)=e(e+1)^{-1}$. This shows that $Q^w$ is too singular. It follows from Lemma \[lem: val and pi rig\] that $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w) \geq e(e+1)^{-1}$.
\(4) Since $Q$ is canonical, we have $\nu_{{\frak{Y}}}(Q)=\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)>(e+1)^{-1}$, and hence $\nu_{{\frak{Y}}}(Q^w)<e(e+1)^{-1}$. This shows that $Q^w$ is canonical. Therefore, $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w) = \nu_{{\frak{Y}}}(Q^w)=1 -
\nu_{{\frak{Y}}}(Q)=1-\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)$.
\(5) Since $Q$ is anti-canonical, Lemma \[lem: val and pi rig\] shows that $\nu_{{\frak{Y}}}(Q)=1-e^{-1}\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q) >
e(e+1)^{-1}$. Therefore, $\nu_{{\frak{Y}}}(Q^w)=1-\nu_{{\frak{Y}}}(Q)<(e+1)^{-1}$ and hence $Q^w$ is canonical. We have $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w) =
\nu_{{\frak{Y}}}(Q^w)=1-\nu_{{\frak{Y}}}(Q)=e^{-1}\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q)$.
\(6) Since $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q) \geq e(e+1)^{-1}$, by Lemma \[lem: val and pi rig\] we have $\nu_{{\frak{Y}}}(Q)=e(e+1)^{-1}$, and hence $\nu_{{\frak{Y}}}(Q^w)=(e+1)^{-1}$, This shows that $Q^w$ is canonical. Therefore, we have $\nu_{{\frak{X}}}({{\pi_\text{\rm rig}}}Q^w)=\nu_{{\frak{Y}}}(Q^w)=(e+1)^{-1}$.
Applications {#section:applications}
============
In this section we review some of the structure theory for Shimura curves and show that our results apply to these situations. Our main references are Drinfeld [@DrinfeldElliptic; @Drinfeld] and Carayol [@Carayol]. In particular, we reproduce the classical results on canonical subgroups [@Katz], as well as more recent developments [@Kassaei1; @Kassaei2].
Let $F$ be a totally real field of degree $d$ with ring of integers ${{\mathcal{O}}}_F$ and let $B/F$ be a quaternion algebra split at exactly one infinite prime of $F$. Let $R$ be a maximal order of $B$. Let ${{\frak{p}}}$ be a finite prime of $F$ at which $B$ splits, $F_{{\frak{p}}}$ the completion of $F$ at the prime ${{\frak{p}}}$, ${{\mathcal{O}}}_{F, {{\frak{p}}}}$ its ring of integers with a uniformizer $\varpi$, and identify $B\otimes_F F_{{\frak{p}}}$ with $M_2(F_{{\frak{p}}})$ so that $R\otimes_{{{\mathcal{O}}}_F} {{\mathcal{O}}}_{F, {{\frak{p}}}} =
M_2({{\mathcal{O}}}_{F, {{\frak{p}}}})$. With $B$ there is associated a projective system of Shimura curves, initially over the complex numbers but, by Shimura’s theory of canonical models, in fact over $F$. Let $G
= {\rm Res}_{F/\QQ}(B^\times)$. Let $X$ be the $G(\RR)$-conjugacy class of the homomorphism $\CC^\times \rightarrow G(\RR)$ sending $x+iy$ to $\left[\left(\begin{smallmatrix}x&y \\ - y &
x\end{smallmatrix}\right), 1, \dots, 1\right]\in {{\operatorname{GL}}}_2(\RR) \times
(\HH^\times)^{d-1}$. Let $K$ be an open compact subgroup of $G(\AA^f)$ of the form $K_{{\frak{p}}}\times K^{{\frak{p}}}$, where $K_{{\frak{p}}}$ is a subgroup of ${{\operatorname{GL}}}_2({{\mathcal{O}}}_{F, {{\frak{p}}}})$ and $K^{{\frak{p}}}$ is “away from ${{\frak{p}}}$". The Shimura curve associated with $K$ is $M_K(G,
X)(\CC) = G(\QQ)\backslash G(\AA^f)\times X /K$.
The case $F = \QQ$
------------------
In this case the Shimura curves $M_K(G, X)/\QQ$ afford a natural modular description. Consider the functor associating to a scheme $S$ the isomorphism classes of triples $(A, \iota, \alpha)/S$, where $A/S$ is an abelian scheme of relative dimension $2$, $\iota\colon R \rightarrow {{\operatorname{End}}}_S(A)$ is an injective ring homomorphism and $\alpha\colon
\underline{R/NR} \rightarrow A[N]$ is an isomorphism of $R$-group schemes; cf. [@Drinfeld §4], [@DiamondTaylor §4], [@BuzzardModels]. (Such objects are sometimes called “false elliptic curves" because of the similarity with the case of $B =
M_2(\QQ)$ and the usual modular curves.) This corresponds to the case where $K$ is $\Gamma(N)$ – the elements of $(R\otimes_\ZZ\widehat{\ZZ})^\times$ (viewed as a subgroup of $G(\AA^f)$) that reduce to the identity element under $(R\otimes_\ZZ\widehat{\ZZ})^\times \rightarrow (R\otimes_\ZZ
\ZZ/N\ZZ)^\times$. For a general $K$, $K$ contains $\Gamma(N)$ for some $N$ and we take $\alpha$ up to $K$-equivalence (étale locally). This makes sense in all characteristics once the level structure is understood in Drinfeld’s sense for which we refer to [@DrinfeldElliptic; @KatzMazur]. For $K$ small enough, there is therefore a scheme ${\bf M}_K$ over ${{\operatorname{Spec }}}(\ZZ)$ representing this functor such that ${\bf M}_K \otimes_\ZZ \QQ \cong M_K(G, X)$.
As a module over $R\otimes \ZZ_p = M_2(\ZZ_p)$, the $p$-divisible group $A[p^\infty]$ of $A/S$ is a direct sum $A[p^\infty]_1\oplus
A[p^\infty]_2$ of two isomorphic $p$-divisible groups over $S$, where the decomposition is determined by the orthogonal idempotents $\left(\begin{smallmatrix} 1 & 0
\\ 0 & 0 \end{smallmatrix}\right), \left(\begin{smallmatrix} 0 & 0
\\ 0 & 1 \end{smallmatrix}\right) \in M_2(\ZZ_p)$; furthermore, these idempotents are conjugate under $\left(\begin{smallmatrix} 0 & 1
\\ 1 & 0 \end{smallmatrix}\right)$, which induces the isomorphism $A[p^\infty]_1\cong A[p^\infty]_2$. Let $K^p$ be small enough and let $K_p$ be the standard Iwahori subgroup of ${{\operatorname{GL}}}_2(\ZZ_p)$. The open compact subgroup $K = K_p \times K^p$ corresponds to a choice of level structure away from $p$ (given by $K^p$) and a choice of a non-trivial ideal $H \subset
M_2(\ZZ/p\ZZ)$. Such $H$ corresponds, via the $K_p$-equivalence class of $\alpha$, to an $R$-invariant subgroup of $A[p]$ of degree $p^2$. The level structure at $p$ can therefore also be expressed as an isogeny $f\colon A_1 \rightarrow A_2$ of false elliptic curves whose kernel is of degree $p^2$ and is killed by $p$. The conditions on $f$ can also be formulated by requiring $f$ to have “false degree" $p$, i.e. that $f^t \circ f= [p]$ (see below for the exact meaning of this formula); cf. [@DiamondTaylor p. 453], [@Kassaei1 §§10-11].
Let $A/k$ be a false elliptic curve over an algebraically closed field $k$ of characteristic $p$. One can prove, by means of the idempotents we have chosen, that the functor of infinitesimal deformations of $A$ (resp., together with an Iwahori level structure $K_p \subset {{\operatorname{GL}}}_2(\ZZ_p)$) is equivalent to the functor of deformation of a $1$-dimensional $p$-divisible group of height $2$ over $k$ (resp., with a $\Gamma_0(p)$-level structure). Thus, this is exactly the situation arising for elliptic curves and is well understood; cf. [@BuzzardModels]. One concludes for such choice of $K$ that every geometric connected component of the special fibre ${\bf M}_K \otimes \FF_p$ of ${\bf M}_K$ consists of two smooth curves crossing transversely at the supersingular points and so is a normal crossing divisor. Moreover, the natural morphism ${\bf M}_K \otimes \FF_p {{\; \longrightarrow \;}}{\bf
M}_{{{\operatorname{GL}}}_2(\ZZ_p) \times K^p} \otimes \FF_p$ is finite flat of degree $p+1$ and admits the usual section taking a false elliptic curve $A$ with $K^p$-structure to $(A, {{\operatorname{Ker}}}({{\operatorname{Fr }}}_A))$ with the same $K^p$-structure. The other component is isomorphic to ${\bf
M}_{{{\operatorname{GL}}}_2(\ZZ_p) \times K^p} \otimes \FF_p$ as well. Indeed, the morphism ${\bf M}_K \otimes \FF_p {{\; \longrightarrow \;}}{\bf M}_{{{\operatorname{GL}}}_2(\ZZ_p)
\times K^p} \otimes \FF_p$ induces on it a map which is bijective on geometric points (the pre-image of a point $A$ is $(A,
{{\operatorname{Ker}}}({{\operatorname{Ver }}}_A)$). Hence the map is purely inseparable of degree $p$.
There is an involution $w$ on ${\bf M}_K$ that is best described by its action on objects: an Iwahori level structure $f\colon A_1
\rightarrow A_2$ of false elliptic curves is sent by duality to $f^t\colon A_2^t \rightarrow A_1^t$. We remark here that every false elliptic curve carries a principal polarization compatible with the $R$-action [@Drinfeld §4], hence we get $f^t\colon
A_2 \rightarrow A_1$, whose isomorphism class is well defined (independent of the choice of polarization). If the kernel of $f$ is connected (resp. étale) then the kernel of $f^t$ is étale (resp. connected). It follows that $w$ permutes the two irreducible components of every geometric connected component of ${\bf M}_K \otimes \FF_p$. Finally, there is a finite extension $\FF_q \supseteq \FF_p$ over which all the connected components of ${\bf M}_K\otimes \overline{\FF}_p$ and ${\bf
M}_{{{\operatorname{GL}}}_2(\ZZ_p) \times K^p}\otimes \overline{\FF}_p$ are defined and each connected component is a normal crossing divisor. Using argument as in Remark \[rmk:connected\], and the fact that ${\bf
M}_K\otimes W(\FF_q)$ is flat over $W(\FF_q)$ and has reduced special fibre, one find that the connected components of ${\bf
M}_K\otimes \FF_q$ (resp. ${\bf M}_{{{\operatorname{GL}}}_2(\ZZ_p) \times
K^p}\otimes \FF_q$) are in bijection with the connected components of the generic fibre. We conclude that each connected component $Y$ of ${\bf M}_K\otimes W(\FF_q)$ and its image $X\subseteq {\bf M}_{{{\operatorname{GL}}}_2(\ZZ_p) \times K^p}\otimes
W(\FF_q)$ satisfy the hypotheses of this paper. Moreover, a descent argument, using the uniqueness of the section on each connected component (see Proposition \[prop:extending the section\]), allows one to get a section over ${\bf M}_{{{\operatorname{GL}}}_2(\ZZ_p)
\times K^p}\otimes \QQ_p$ defined over $\QQ_p$. The application of our results gives a new proof for the existence and other properties of canonical subgroups of false elliptic curves, recovering Theorem 11.1 and Lemma 12.5 of [@Kassaei1].
The case $[F:\QQ] = d >1$
-------------------------
In contrast to the previous case, when $F\neq \QQ$ there is no natural modular description of the Shimura curves associated to $B$. Instead, by making an auxiliary choice of a CM field $L/F$, one can associate to the algebra $B\otimes_F L$ another algebraic group $G^\prime$ with the same derived group as that of $G$. The curves $M_{K^\prime}(G^\prime, X^\prime)/F$ associated to $G^\prime$ are PEL Shimura curves. These auxiliary curves play an important role in Carayol’s construction of an integral model for $M_K(G, X)/F$ over ${{\mathcal{O}}}=
{{\mathcal{O}}}_{F, {{\frak{p}}}}$, since they are closely related to the Shimura curves defined by $G$ [@Carayol §4]. Carayol proves that such a model ${\bf M}_K$ exists, and that there is a universal $p$-divisible ${{\mathcal{O}}}$-module ${{\mathscr{G}}}$ of (${{\mathcal{O}}}$-) height $2$ over the projective limit ${\bf M}_\infty$ of ${\bf
M}_K$ over $K$. This $p$-divisible group is constructed as a certain “piece" of the $p$-divisible group of the universal abelian variety with additional structure existing over (the projective limit of) the Shimura curves ${\bf M}_{K^\prime}(G^\prime,
X^\prime)/{{\mathcal{O}}}_F$. Note that the $p$-divisible group ${{\mathscr{G}}}$ does not carry an $R\otimes {{\mathcal{O}}}$-structure. In a moral sense, this structure was already used in reducing the height of the $p$-divisible ${{\mathcal{O}}}$-module to $2$ (this corresponds to choosing a particular piece of the $p$-divisible group of the universal abelian variety over ${\bf M}_{K^\prime}(G^\prime, X^\prime)/{{\mathcal{O}}}_F$ and is analogous to the process indicated above for $F = \QQ$). For details see [@Carayol], in particular §§3.3, 6.3. We discuss this further.
Assume first that $K={{\operatorname{GL}}}_2({{\mathcal{O}}}) \times K^{{\frak{p}}}$. Thus, no level structure is imposed at ${{\frak{p}}}$. Carayol constructs a $p$-divisible group ${{\mathscr{G}}}$ over ${\bf M}_\infty$, which is a $p$-divisible ${{\mathcal{O}}}$-module of height $2$. For any geometric point $x$ of ${\bf M}_K$, there is a way to define the fibre ${{\mathscr{G}}}_x$ by lifting $x$ to a geometric point of ${\bf M}_\infty$. Over a geometric characteristic $0$ point $x$ of ${\bf M}_K$ we have ${{\mathscr{G}}}_x \cong (F_{{\frak{p}}}/{{\mathcal{O}}})^2$. The prime-to-${{\frak{p}}}$ level structure plays a somewhat dormant role. For example, Carayol proves [@Carayol §6.6] a “Serre-Tate theorem" to the effect that the formal completion of the henselization of ${\bf M}_K$ at a geometric point $x$ of its special fibre pro-represents the functor of infinitesimal deformations for the $p$-divisible ${{\mathcal{O}}}$-module ${{\mathscr{G}}}_x$. There are two cases:
1. The *ordinary* case, where ${{\mathscr{G}}}_x$ is isomorphic to $F_{{\frak{p}}}/{{\mathcal{O}}}\oplus (F_{{\frak{p}}}/{{\mathcal{O}}})^t$, where $(-)^t$ denotes the dual $p$-divisible group;
2. The *supersingular* case where ${{\mathscr{G}}}_x$ is the “unique" formal ${{\mathcal{O}}}$-module of dimension $1$ and height $2$ [@DrinfeldElliptic Prop. 1.7].
The deformation theory was worked out by Drinfeld. One concludes that in either case the completed local ring is isomorphic to $\hat{{{\mathcal{O}}}}^{\rm nr}[\![t]\!]$ and hence that ${\bf M}_K$ is a regular surface with a smooth special fibre; cf. [@DrinfeldElliptic Prop. 4.2, 4.5],[@Carayol App. §3].
Carayol also considers the case of level structure $K_{{\frak{p}}}(n)
\times K^{{\frak{p}}}$, where $K_{{\frak{p}}}(n)$ is the subgroup consisting of matrices in ${{\operatorname{GL}}}_2({{\mathcal{O}}})$ congruent to $1$ modulo ${{\frak{p}}}^n$. There is a moduli interpretation of a sort to the ensuing morphism $\pi\colon{\bf M}_{K_{{\frak{p}}}(n) \times K^{{\frak{p}}}} \rightarrow
{\bf M}_{K_{{\frak{p}}}(0) \times K^{{\frak{p}}}}$; the group scheme ${{\mathscr{G}}}[{{\frak{p}}}^n]$ descends to ${\bf M}_{K_{{\frak{p}}}(n) \times
K^{{\frak{p}}}}$ and is equipped with a Drinfeld full ${{\frak{p}}}^n$-level structure, namely, a morphism of ${{\mathcal{O}}}$-group schemes $\alpha\colon \underline{({{\frak{p}}}^{-n}/{{\mathcal{O}}})}^2 \rightarrow
{{\mathscr{G}}}[{{\frak{p}}}^n]$, such that the closed subscheme $\sum_{P\in
({{\frak{p}}}^{-n}/{{\mathcal{O}}})^2}\alpha(P)$ is equal to ${{\mathscr{G}}}[{{\frak{p}}}^n]$. The scheme ${\bf M}_{K_{{\frak{p}}}(n) \times K^{{\frak{p}}}}$ is a torsor over ${{\operatorname{Aut}}}(\underline{({{\frak{p}}}^{-n}/{{\mathcal{O}}})}^2) \times {\bf M}_{K_{{\frak{p}}}(0)
\times K^{{\frak{p}}}}$ and the morphism $\pi$ is the natural one (in particular its fibres are principal homogenous spaces for ${{\operatorname{Aut}}}(\underline{({{\frak{p}}}^{-n}/{{\mathcal{O}}})}^2)$). Such level structures were introduced and studied by Drinfeld in [@DrinfeldElliptic p. 572], developed more in [@Carayol §7, Appendix], and studied extensively in [@KatzMazur]. Again Carayol proves a “Serre-Tate theorem" as to the nature of the completed local rings [@Carayol §7]. He also proves that the morphism $\pi$ extends the natural morphism $M_{K_{{\frak{p}}}(n)
\times K^{{\frak{p}}}}(G, X) \rightarrow M_{K_{{\frak{p}}}(0) \times
K^{{\frak{p}}}}(G, X)$ induced by the inclusion $K_{{\frak{p}}}(n) \times
K^{{\frak{p}}}{{\; \hookrightarrow \;}}K_{{\frak{p}}}(0) \times K^{{\frak{p}}}$.
As Carayol remarks [@Carayol §0.4], the construction and results extend to any choice of level subgroup at ${{\frak{p}}}$; in particular, for $K = K_{{\frak{p}}}\times K^{{\frak{p}}}$, where $K_{{\frak{p}}}$ is the Iwahori subgroup. The scheme ${\bf M}_{K_{{\frak{p}}}\times
K^{{\frak{p}}}}$ then carries a finite flat group scheme ${{\mathscr{H}}}$ (étale locally) with a Drinfeld level structure $\underline{{{\frak{p}}}^{-1}/{{\mathcal{O}}}}\rightarrow {{\mathscr{H}}}$ such that $\sum_{P\in {{\frak{p}}}^{-1}/{{\mathcal{O}}}}\alpha(P)$ is equal to ${{\mathscr{H}}}$ as a closed subscheme. The following conclusion follows from Carayol’s work: The completion of the henselization of ${\bf
M}_{K_{{\frak{p}}}\times K^{{\frak{p}}}}$ at a geometric characteristic $p$ point $x$ is the ring that pro-represents the functor of infinitesimal deformations of the divisible ${{\mathcal{O}}}$-module ${{\mathscr{G}}}_x$ together with an ${{\mathcal{O}}}$-subgroup scheme of order $q =
\vert {{\mathcal{O}}}/{{\frak{p}}}\vert$ killed by ${{\frak{p}}}$. This moduli problem can also be phrased in a balanced manner. It can be viewed as deforming a pair of divisible ${{\mathcal{O}}}$-modules of height $2$, say ${{\mathscr{G}}}_x, {{\mathscr{G}}}_x^\prime$, together with an ${{\mathcal{O}}}$-isogeny ${{\mathscr{G}}}_x\rightarrow {{\mathscr{G}}}_x^\prime$ of degree $q$ whose kernel is ${{\frak{p}}}$-torsion.
The situation is again very similar to elliptic curves with $\Gamma_0(p)$-level structure, and in particular the following holds. The scheme ${\bf M}_{K_{{\frak{p}}}\times K^{{\frak{p}}}}$ is a regular two dimensional scheme, flat over ${{\mathcal{O}}}_{F, {{\frak{p}}}}$, the morphism $\pi$ is finite flat of degree $q+1$ and the nature of $\pi$ at every point is completely understood. In particular, there are two pre-images to every ordinary point of ${\bf
M}_{{{\operatorname{GL}}}_2({{\mathcal{O}}})\times K^{{\frak{p}}}}$ and ${\bf M}_{K_{{\frak{p}}}\times
K^{{\frak{p}}}}$ is regular at each; there is a unique pre-image $y$ to any geometric supersingular point and the completed local ring of $y$ is isomorphic to $\hat{{{\mathcal{O}}}}^{\rm nr}[\![ s, t ]\!]/(st -
\varpi)$. For completeness we sketch an argument below. We remark that one can also argue using the results in [@Carayol] obtained for full ${{\frak{p}}}$-level structure. However, Carayol uses an explicit description of the formal ${{\mathcal{O}}}$-module to obtain his results. Since we do not anticipate such description to be available (or indeed useful) in higher-dimensional cases, using Carayol’s result will be contrary to our thesis. We therefore provide an argument that should extend to the more general situation we have in mind.
### A sample case
Firstly, we quickly recall the technique of local models in the particular situation of elliptic curves, which serves as a good sample case for our problem.
The deformation theory of elliptic curves (or abelian varieties) can be studied as follows. Given a characteristic $p$ closed point $x$ of a moduli space ${\bf M}$ of elliptic curves with level prime to $p$ and its universal object $f\colon {{\mathscr{E}}}\rightarrow {\bf M}$, choose an open affine neighborhood $U\ni x$ and a trivialization of $\HH^1_{\rm dR}({{\mathscr{E}}}/U)\cong {{\mathcal{O}}}_U^2$. The variation of Hodge structure $R^0f_\ast \Omega_{{{\mathscr{E}}}/U}
\rightarrow \HH^1_{\rm dR}({{\mathscr{E}}}/U)$ provides a morphism $U {{\; \longrightarrow \;}}{\bf Grass}$, where ${\bf Grass}$ is the Grassmann scheme of locally free, locally direct summands of rank $1$ of ${{\mathcal{O}}}_U^2$. One then shows, using the crystalline theory developed by Grothendieck, that this morphism is étale and so is an isomorphism on the level of completed local rings of $x$ and its image in ${\bf Grass}$; cf. [@deJongGamma0(p); @DP]. If one wants to work instead with the $p$-divisible groups, one may replace $\HH^1_{\rm dR}$ by a similar object provided by the theory of displays as developed by Zink, or by the theory of Cartier-Dieudonné modules, or any other theory studying deformations of $p$-divisible groups. For example, [@RapoportZink §3] choose the Lie algebra of the universal vectorial extension of the $p$-divisible group. By analyzing the Grassmann scheme, one therefore establishes that the completed local ring is $D=W(k(x))[\![t]\!]$.
Under this method, the formal scheme representing the infinitesimal deformation problem of an elliptic curve with a subgroup of order $p$ may be translated to a (formal) incidence variety. We think of the moduli problem as the one for a cyclic isogeny $h\colon E_1 \rightarrow E_2$ of degree $p$ between elliptic curves and we are interested in the completed local ring of the point on the moduli space that corresponds to such data over a finite field $k$ of characteristic $p$. One may choose the trivialization of the two $\HH^1_{\rm dR}({{\mathscr{E}}}/{{\operatorname{Spf }}}(D_i))$, $D_i$ ($\cong D$) the completed local ring at $E_i$, such that the isogeny is given by $\left(
\begin{smallmatrix} 1 & 0 \\ 0 & p
\end{smallmatrix}\right)$ [@DP §5.3 ff.] or [@deJongGamma0(p)]. We are then parameterizing a pair of locally free, locally direct summands $(L_1, L_2)$ of rank $1$ of $D^2$ such that $\left(
\begin{smallmatrix} 1 & 0 \\ 0 & p
\end{smallmatrix}\right)L_1 \subseteq L_2$. In the ordinary case we get an $L_1$ whose reduction modulo $p$ is not killed by $\left(
\begin{smallmatrix} 1 & 0 \\ 0 & 0
\end{smallmatrix}\right)$ and the deformation problem is represented by the completion of the local ring of a $k$-point $x$ of $\PP^1_{W(\FF_p)}$ and so is isomorphic to $W(k)[\![t]\!]$. In the supersingular case we get an $L_1$ whose reduction killed by $\left(
\begin{smallmatrix} 1 & 0 \\ 0 & 0
\end{smallmatrix}\right)$. Let $x$ be a $k$-rational point of $\PP^1_{W(k)}$ and let $\PP$ be the blow-up of $\PP^1_{W(k)}$ at $x$. Its special fibre has a unique singular point that we shall still denote by $x$. The deformation problem is pro-represented by the completion of the local ring of $x$ on $\PP$ and so is isomorphic to $W(k)[\![s,t]\!]/(st-p)$.
### The calculation of the completed local rings
Recall that the moduli problem is phrased in a balanced manner. Let $x^\prime$ be a closed point of ${\bf M}_{K_{{\frak{p}}}\times
K^{{\frak{p}}}}$ with finite residue field $k$, and let $x$ be a $\bar
k$-point supported on $x^\prime$, where $\bar k$ is an algebraic closure of $k$. The situation we have is of two divisible ${{\mathcal{O}}}$-modules ${{\mathscr{G}}}_x, {{\mathscr{G}}}_x^\prime$ of dimension $1$ and height $2$ over $\bar k$ and an ${{\mathcal{O}}}$-isogeny $h\colon {{\mathscr{G}}}_x
\rightarrow {{\mathscr{G}}}_x^\prime$ of degree $q = \vert {{\mathcal{O}}}/(\varpi)\vert$, whose kernel is killed by ${{\frak{p}}}$.
Let ${{\mathscr{G}}}$ be ${{\mathscr{G}}}_x$ or ${{\mathscr{G}}}_x^\prime$. The Lie algebra of the universal vectorial extension of ${{\mathscr{G}}}$, which serves as a substitute for the first de Rham cohomology, is a free $\bar{k}$-module of dimension $2$. As mentioned above, the functor of infinitesimal deformations of ${{\mathscr{G}}}$ is pro-representable by $R^u = \hat {{\mathcal{O}}}^{\rm nr}[\![t]\!]$, which carries a universal object ${{\mathscr{G}}}^u$. This can also be proven by the same technique of local models applied to the relative Lie algebra $\underline{{\rm
Lie}}({{\mathscr{G}}}^u)$ of ${{\mathscr{G}}}^u$ and the Lie algebra $\underline{{\rm
Lie}}(V{{\mathscr{G}}}^u)\cong (R^u)^2$ of its universal vectorial extension $V{{\mathscr{G}}}^u$, which identifies the completed local ring of $x$ with the completed local ring of the $\bar k$-point, still called $x$, on the formal Grassmann scheme $(\PP^1_{R^u})^{\wedge
x}$.
The analogue of Lemma 5.5 of [@DP] holds. Namely, one can choose isomorphisms $\underline{{\rm Lie}}(V{{\mathscr{G}}}_x^u)\cong
(R^u)^2$ and $\underline{{\rm Lie}}(V{{\mathscr{G}}}_x'^u)\cong (R^u)^2$ such that $h$ is given by the matrix $A=\left( \begin{smallmatrix}
1 & 0 \\ 0 & \varpi \end{smallmatrix}\right)$ (one should use that the $p$-divisible groups are in fact polarized, are “special ${{\mathcal{O}}}$-modules" in Drinfeld’s sense and that $h$ is compatible with the polarizations). Therefore, the completed local ring of $\bar x$ is isomorphic to the formal incidence variety in $(\PP^1_{R^u})^{\wedge x}\times (\PP^1_{R^u})^{\wedge x}$ given by $A$, i.e., by the closed subscheme over which we have $A\left(\underline{{\rm Lie}}({{\mathscr{G}}}_x^u)\right) \subseteq
\underline{{\rm Lie}}({{\mathscr{G}}}_x'^u)$.
In the ordinary case we find that the complete local ring is $\hat
{{\mathcal{O}}}^{\rm nr}[\![t]\!]$, and in the supersingular case we find that it is $\hat {{\mathcal{O}}}^{\rm nr}[\![s,t]\!]/(st-\varpi)$. Finally, one may conclude that the completed local ring of $x^\prime$ itself is ${{\mathcal{O}}}\otimes_{W(\kappa)} W(k)[\![t]\!]$ if $x$ is ordinary and is ${{\mathcal{O}}}\otimes_{W(\kappa)} W(k_1)[\![s,t]\!]/(st-\varpi)$ if $x$ is supersingular, where $[k_1: k]\leq 2$.
Given these results, it is straightforward to verify that the connected components of the generic fibre of a suitable unramified base-change of ${\bf M}_{K^{{\frak{p}}}\times K_{{\frak{p}}}} \rightarrow {\bf
M}_{{{\operatorname{GL}}}_2({{\mathcal{O}}}) \times K^{{\frak{p}}}}$ satisfy the assumptions of this paper, including the existence of an involution. In particular, one has a unique (partial) section on each pair of connected components of the generic fibres; a descent argument allows one to conclude that the section can already be defined before base-change.
One may, of course, carry the same analysis for the Shimura curves $M_{K^\prime}(G^\prime,X^\prime)$. If anything, the analysis is easier, since it is the one underlying Carayol’s results. Hence, the results of this paper apply to these cases as well.
As is clear from our discussion, whenever we are in a situation of curves $Y \rightarrow X$ such that $Y$ (or the fibres) parameterizes group schemes, e.g. in the case of usual modular curves where $Y$ has a $\Gamma_0(p)$-level structure, or for pairs ${\bf M}_{K_{{\frak{p}}}\times K^{{\frak{p}}}} \rightarrow {\bf
M}_{K_{{\frak{p}}}(0) \times K^{{\frak{p}}}}$ (or the analogous situations for the groups $G^\prime$), the construction of a section as in this paper provides one with a group scheme over the region where the section is defined. In particular our results reprove Theorems 3.1 and 3.10.7 of [@Katz], and Theorem 9.1 of [@Kassaei2] on canonical subgroups of abelian schemes parameterized by $M_{K^\prime}(G^\prime,X^\prime)$, and in addition provide an analogue of Theorem 3.10.7 of [@Katz] for such canonical subgroups.
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abstract: 'Trace distance is available to capture the dynamical information of the unital aspect of a quantum process. However, it cannot reflect the non-unital part. So, the non-divisibility originated from the non-unital aspect cannot be revealed by the corresponding measure based on the trace distance. We provide a measure of non-unital non-Markovianity of quantum processes, which is a supplement to Breuer-Laine-Piilo (BLP) non-Markovianity measure. A measure on the degree of the non-unitality is also provided.'
author:
- Jing Liu
- 'Xiao-Ming Lu'
- Xiaoguang Wang
title: 'Non-unital non-Markovianity of quantum dynamics'
---
Introduction
============
Understanding and characterizing general features of the dynamics of open quantum systems is of great importance to physics, chemistry, and biology [@breuer]. The non-Markovian character is one of the most central aspects of an open quantum process, and attracts increasing attentions [@Wolf2008; @Wolf2008-1; @Breuer2009; @Rivas2010; @Lu2010; @Chruscinski2010; @Zhang2012; @Hou2011; @Luo2012; @Vasile2011; @Laine2012; @Liu2011; @Bylicka2013; @Laine2010; @Rajagopal2010]. Markovian dynamics of quantum systems is described by a quantum dynamical semigroup [@breuer; @Lendi_book], and often taken as an approximation of realistic circumstances with some very strict assumptions. Meanwhile, exact master equations, which describe the non-Markovian dynamics, are complicated [@Zhang2012]. Based on the infinitesimal divisibility in terms of quantum dynamical semigroup, Wolf *et al.* provided a model-independent way to study the non-Markovian features [@Wolf2008-1; @Wolf2008]. Later, in the intuitive picture of the backward information flow leading to the increasing of distinguishability in intermediate dynamical maps, Breuer, Laine, and Piilo (BLP) proposed a measure on the degree of non-Markovian behavior based on the monotonicity of the trace distance under quantum channels [@Breuer2009], as shown in Fig. \[fig:sketch\]. The BLP non-Markovianity has been widely studied, and applied in various models [@Xu2010; @Chruscinski2011; @Rebentrost2011; @Wibmann2012; @Haikka2011; @Clos2012].
 Sketch of the information flow picture for non-Markovianity [@Breuer2009]. According to this scenario, the loss of distinguishability of the system’s states indicates the information flow from the system to the reservoir. If the dynamics is Markovian, the information flow is always outward, represented by the green thick arrow. Non-Markovian behaviors occurs when there is inward information flow, represented by the orange thin arrow, bringing some distinguishability back to the system.](fig_skecth.eps){width="6cm"}
Unlike for classical stochastic processes, the non-Markovian criteria for quantum processes is non-unique, and even controversial. First, the non-Markovian criteria from the infinitesimal divisibility and the backward information flow are not equivalent [@Haikka2011; @Chruscinski2011]. Second, several other non-Markovianity measures, based on different mechanism like the monotonicity of correlations under local quantum channels, have been introduced [@Rivas2010; @Luo2012]. Third, even in the framework of backward information flow, trace distance is not the unique monotone distance for the distinguishability between quantum states. Other monotone distances on the space of density operators can be found in Ref. [@Petz1996], and the statistical distance [@Wootters1981; @Braunstein1994] is another widely-used one. Different distance should not be expected to give the same non-Markovian criteria. The inconsistency among various non-Markovianity reflects different dynamical properties.
In this paper, we show that the BLP non-Markovianity cannot reveal the infinitesimal non-divisibility of quantum processes caused by the non-unital part of the dynamics. Besides non-Markovianity, “non-unitality” is another important dynamical property, which is the necessity for the increasing of the purity $\mathrm{Tr\rho^{2}}$ under quantum channels [@Lidar2006] and for the creating of quantum discord in two-qubit systems under local quantum channels [@Streltsov2011]. In the same spirit as BLP non-Markovianity, we define a measure on the non-unitality. As BLP non-Markovianity is the most widely used measure on non-Markovianity, we also provide a measure on the non-unital non-Markovianity, which can be conveniently used as a supplement to the BLP measure, when the quantum process is non-unital. We also give an example to demonstrate an extreme case, where the BLP non-Markovianity vanishes while the quantum process is not infinitesimal divisible.
This paper is organized as follows. In Sec. \[Review\], we give a brief review on the representation of density operators and quantum channels with Hermitian orthonormal operator basis, and various measures on non-Markovianity. In Sec. \[sec:Non-unital-NM\], we investigate the non-unitality and the non-unital non-Markovianity and give the corresponding quantitative measures respectively. In Sec. \[sec:EXAMPLE\], we apply the non-unital non-Markovianity measure on a family of quantum processes, which are constructed from the generalized amplitude damping channels. Section \[sec:Conclusion\] is the conclusion.
Review on quantum channels and non-Markovianity\[Review\]
=========================================================
Density operators and quantum channels represented by Hermitian operator basis.
-------------------------------------------------------------------------------
The states of a quantum system can be described by the density operator $\rho$, which is positive semidefinite and of trace one. Quantum channels, or quantum operations, are completely positive and trace-preserving (CPT) maps from density operators to density operators, and can be represented by Kraus operators, Choi-Jamiołkowski matrices, or transfer matrices [@Wolf_channels; @Kraus1983; @Choi1975; @Jamiolkowski1972].
In this work, we use the Hermitian operator basis to express operators and represent quantum channels. Let $\{\lambda_{\mu}\mid\mu=0,1,\cdots,d^{2}-1\}$ be a complete set of Hermitian and orthonormal operators on complex space $\mathbb{C}^{d}$, i.e., $\lambda_{\mu}$ satisfies $\lambda_{\mu}^{\dagger}=\lambda_{\mu}$ and $\langle\lambda_{\mu},\lambda_{\nu}\rangle:=\mathrm{Tr}(\lambda_{\mu}^{\dagger}\lambda_{\nu})=\delta_{\mu\nu}$. Any operator $O$ on $\mathbb{C}^{d}$ can be express by a column vector $r:=(r_{0},r_{1},\cdots,r_{d^{2}-1})^{\mathrm{T}}$ through $$O=\sum_{\mu=0}^{d^{2}-1}r_{\mu}(O)\lambda_{\mu}$$ with $r_{\mu}(O):=\langle\lambda_{\mu},O\rangle$. Every $r_{\mu}(O)$ is real if $O$ is Hermitian.
In the meantime, any quantum channel $\mathcal{E}\colon\rho\mapsto\mathcal{E}(\rho)$ can be represented by $T(\mathcal{E})\colon r(\rho)\mapsto r[\mathcal{E}(\rho)]$ via $$r[\mathcal{E}(\rho)]=T(\mathcal{E})r(\rho) \label{eq:1},$$ where $T(\mathcal{E})$ is a $d^2\times d^2$ real matrix with the elements $$T_{\mu\nu}(\mathcal{E}):=\left\langle \lambda_{\mu},\mathcal{E}(\lambda_{\nu})\right\rangle. \label{eq:T}$$ Furthermore, one can easily check that $$T(\mathcal{E}_{1}\circ\mathcal{E}_{2})
=T(\mathcal{E}_{1})T(\mathcal{E}_{2})$$ for the composition of quantum channels. Here $\mathcal{E}_{1}\circ\mathcal{E}_{2}$ denotes the composite maps $\mathcal{E}_{1}(\mathcal{E}_{2}(\rho))$.
Taking into the normalization of the quantum states, i.e., $\mathrm{Tr}(\rho)=1$, $r_{0}$ can be fixed as $r_{0}(\rho)=1/\sqrt{d}$ for any density operator $\rho$ by choosing $\lambda_{0}=\openone/\sqrt{d}$ with $\openone$ the identity operator. In such a case, $\lambda_{\mu}$ for $\mu=1,2,\cdots,d^{2}-1$ are traceless and generate the algebra $\mathfrak{su}(d)$. This real parametrization $r_{\mu}(\rho)$ for density operators is also called as coherent vector, or generalized Bloch vector [@Bloch1946; @Hioe1981; @Bengtsson2006]. In order to eliminate the degree of freedom for the fixed $r_{0}$, we use the decomposition $r=(r_{0},\mathbf{r})^{\mathrm{T}}$. Therefore, any density operator $\rho$ can be expressed as $$\rho=\frac{\openone}{d}+\mathbf{r}\cdot\bm{\lambda},\label{eq:rho}$$ with $\mathbf{r}$ the generalized Bloch vector and $\bm{\lambda}$ represents $(\lambda_{1},\lambda_{2},\cdots,\lambda_{d^{2}-1})^{\mathrm{T}}$. Under this frame, quantum channels can be represented by the affine map [@Lendi_book; @Nielsen] $$\mathbf{r}(\mathcal{E}(\rho))=M(\mathcal{E})\mathbf{r}(\rho)+\mathbf{c}
(\mathcal{E}) \label{eq:2},$$ where $M(\mathcal{E})$ is a real matrix with the dimension $d^2-1$ and the elements of the vector $\mathbf{c}(\mathcal{E})$ reads $$\left[\mathbf{c}(\mathcal{E})\right]_{\mu} = \left\langle\lambda_{\mu},\mathcal{E}(\openone)\right\rangle/d,$$ for $\mu=1,2,\cdots,d^{2}-1$. Comparing Eq. (\[eq:1\]) with Eq. (\[eq:2\]), one could find that $$T_{\mu\nu}(\mathcal{E})=[M(\mathcal{E})]_{\mu\nu},$$ for $\mu,\nu=1,2,\cdots,d^{2}-1$. Thus, $T(\mathcal{E})$ can be decomposed into the following sub-blocks: $$T(\mathcal{E})=\left[\begin{array}{c|c}
1 & 0_{1\times(d^{2}-1)}\\
\hline \sqrt{d}\mathbf{c} & M
\end{array}\right].$$
Reminding that a quantum channel $\mathcal{E}$ is said to be unital if and only if $\mathcal{E}(\openone/d)=\openone/d$ [@Nielsen], one could find that the necessary and sufficient condition for a unital map is that $\mathbf{c}(\mathcal{E})=0$, namely, $$\mathbf{c}(\mathcal{E})=0\Longleftrightarrow\mathcal{E}\text{ is unital}.$$ Thus, $\mathbf{c}(\mathcal{E})$ describes the non-unital property of the quantum channel $\mathcal{E}$. The necessary and sufficient condition above could be easily proved by realizing that the Bloch vector of $\openone/d$ is zero vector, i.e., $\mathbf{r}=0$. Based on the sub-block form of $T(\mathcal{E})$, $\mathbf{c}(\mathcal{E})=0$ is equivalent to that $T(\mathcal{E})$ is block diagonal, i.e., $T(\mathcal{E})=\mathrm{diag}(1,M(\mathcal{E}))$.
Whether a quantum channel $\mathcal{E}$ is completely positive (CP) can be reflected by the Choi-Jamiołkowski matrix [@Choi1975; @Jamiolkowski1972] $$C(\mathcal{E}):=\left(\mathcal{E}\otimes\openone\right)
(|\Omega\rangle\langle\Omega|), \label{eq:C}$$ where $|\Omega\rangle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}|j\rangle \otimes|j\rangle$ is the maximally entangled state. Here $\{|j\rangle\} $ is a basis in Hilbert space. $\mathcal{E}$ is CP if and only if the Choi-Jamiołkowski matrix is positive. With the Hermitian operator basis, $|\Omega\rangle\langle\Omega|$ is a $d^2\times d^2$ matrix and can be written in the form [@Yu2009] $$|\Omega\rangle\langle\Omega|=\frac{1}{d}\sum_{\nu=0}^{d^{2}-1}\lambda_{\nu}\otimes\lambda_{\nu}^{\mathrm{T}}.$$ Substituting this formula into Eq. (\[eq:C\]) and utilizing Eq. (\[eq:T\]), one could express the Choi-Jamiołkowski matrix as $$C(\mathcal{E})=\sum_{\mu,\nu=0}^{d^{2}-1}\frac{1}{d}T_{\mu\nu}
(\mathcal{E})\lambda_{\mu}\otimes\lambda_{\nu}^{\mathrm{T}}.$$ If $\mathcal{E}$ is unital, it can be reduced into $$C(\mathcal{E})=\frac{1}{d^2}\left(\openone_{d^2\times d^2}+d\sum_{\mu,\nu=1}^{d^2-1} M_{\mu\nu}\lambda_{\mu} \otimes \lambda^{\mathrm{T}}_{\nu}\right).$$
Non-divisibility and non-Markovianity
-------------------------------------
Without the presence of correlation between the open system and its environment in the initial states, the reduced dynamics for the open system from $t=0$ to any $t\geq0$ can be expressed as $$\mathcal{E}_{t,0}:\rho\mapsto\mathrm{Tr}_{E}\left[U(t)\left(\rho\otimes\rho_{E}\right)U(t)^{\dagger}\right],$$ which is a quantum channel. This indicates that $\mathcal{E}_{t,0}$ is CPT. The unitary operator $U(t)$ describes the time evolution of the closed entirety, and $\rho_{E}$ is the initial state of the environment. A quantum process $\mathcal{E}_{t}:=\mathcal{E}_{t,0}$ is said to be infinitesimal divisible, also called as time-inhomogeneous or time-dependent Markovian, if it satisfies the following composition law [@Wolf2008] $$\mathcal{E}_{t_{2},0}=\mathcal{E}_{t_{2},t_{1}}\circ\mathcal{E}_{t_{1},0}$$ for any $t_{2}\ge t_{1}\geq0$, where $\mathcal{E}_{t_{2},t_{1}}$ is also completely positive and trace preserving.
Various measures on the degree of the non-Markovian behavior of quantum processes have been proposed and investigated [@Breuer2009; @Rivas2010; @Vasile2011; @Luo2012; @Hou2011]. Almost all of the measures on the non-Markovianity can be classified into three kinds, base on the degree of the violation of the following properties owned by the infinitesimal divisible quantum process:
\(i) Monotonicity of distance $D$ under CPT maps. That is $D(\mathcal{E}(\rho_{1}),\mathcal{E}(\rho_{2}))\leq D(\rho_{1},\rho_{2})$ for any quantum channel $\mathcal{E}$, where $D(\rho_{1},\rho_{2})$ is an appropriate monotone distance under CPT maps on the space of density operators [@Petz1996], including trace distance, Bures distance, statistical distance, relative entropy, and fidelity (although fidelity itself is not a distance, it can be used to construct monotone distances) and so on. Some measures on non-Markovianity by increasing of the monotone distance during the mediate dynamical maps $\mathcal{E}_{t_{2},t_{1}}$ have been given and discussed in Refs. [@Breuer2009; @Vasile2011].
The typical measure of this type, which would be used later in this paper, was first proposed by Breuer, Laine, and Piilo in Ref. [@Breuer2009], based on the monotonicity of trace distance [@Nielsen; @Bengtsson2006] $$D_{\mathrm{tr}}(\rho_{1},\rho_{2}):=\frac{1}{2}\mathrm{Tr}\left|\rho_{1}-\rho_{2}\right|,$$ where $|O|:=\sqrt{O^{\dagger}O}$. Interpreting the increase of the trace distance during the time evolution as the information flows from the environment back to the system, the definition of the BLP non-Markovianity is defined by $$\mathcal{N}_{\mathrm{BLP}}(\mathcal{E}_{t}):=\max_{\rho_{1},\,\rho_{2}}\int_{\sigma>0}\mathrm{d}t\,\sigma\left(t,\rho_{1},\rho_{2}\right),$$ where $$\sigma\left(t,\rho_{1},\rho_{2}\right):=\frac{\mathrm{d}}{\mathrm{d}t}D_{\mathrm{tr}}\left(\rho_{1}(t),\rho_{2}(t)\right),$$ and $\rho_{i}(t)=\mathcal{E}_{t}(\rho_{i})$ for $i=1,2$ are two evolving states.
\(ii) Positivity of the Choi-Jamiołkowski matrix for CPT maps. The Choi-Jamiołkowski matrix $C(\mathcal{E})\geq0$ if and only if $\mathcal{E}$ is a quantum channel, namely, $\mathcal{E}$ is a CPT map. Some measures on non-Markovianity by the negativity of the Choi-Jamiołkowski matrix for mediate dynamical maps $\mathcal{E}_{t_{2},t_{1}}$have been given and discussed in Refs. [@Rivas2010; @Hou2011].
In this work we would use one of these measures, which was proposed by Rivas, Huelga and Plenio (RHP) in Ref. [@Rivas2010]. They utilize the negativity of the Choi-Jamiołkowski matrix $C$ for the mediate dynamical maps with the definition $$\mathcal{N}_{\mathrm{RHP}}(\mathcal{E}_{t}):=\int_{0}^{\infty}g(t)dt,$$ where $$g(t):=\underset{\epsilon\rightarrow0^{+}}{\mathrm{lim}}
\frac{\mathrm{Tr}|C(\mathcal{E}_{t+\epsilon,t})|-1}{\epsilon}.
\label{eq:gt}$$
\(iii) Monotonicity of correlations $E$ under local quantum channels. That is $E\left[\left(\mathcal{E}\otimes\openone\right)(\rho^{AB})\right]\leq E(\rho^{AB})$ for any local quantum channel $\mathcal{E}$, where $E$ is an appropriate measure for the correlations in the bipartite states $\rho^{AB}$, including entanglement entropy and the mutual information. The corresponding measures on non-Markovianity are given and discussed in Refs. [@Rivas2010; @Luo2012].
Non-unital non-Markovianity\[sec:Non-unital-NM\]
================================================
The non-Markovianity measure $\mathcal{N}_{\mathrm{BLP}}$ is available to capture the non-Markovian behavior of the unital aspect of the dynamics. But for the non-unital aspect, it is not capable. To show this, we use the Hermitian orthonormal operator basis to express states and quantum channels. Utilizing Eq. (\[eq:rho\]), the trace distance between two states $\rho_{1}$ and $\rho_{2}$ is given by $$\begin{aligned}
D_{\mathrm{tr}}\left(\rho_{1},\rho_{2}\right) & = & \frac{1}{2}\mathrm{Tr}\big|\left[\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})\right]\cdot\bm{\lambda}\big|.\end{aligned}$$ Therefore, for the two evolving states, we get $$D_{\mathrm{tr}}\left(\rho_{1}(t),\rho_{2}(t)\right)=\frac{1}{2}\mathrm{Tr}
\big|M(\mathcal{E}_{t})\left[\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})\right]
\cdot\bm{\lambda}\big|, \label{eq:dis}$$ where $\rho_{1}$, $\rho_{2}$ are initial states of the system.
From this equation one can see that the trace distance between any two evolved states is irrelevant to the non-unital part $\mathbf{c}(\mathcal{E}_{t})$ of the time evolution. Then if there are two quantum channels, whose affine maps are $\mathbf{r}\mapsto M\mathbf{r}+\mathbf{c}_{1}$ and $\mathbf{r}\mapsto M\mathbf{r}+\mathbf{c}_{2}$, respectively, the characteristic of trace distance between the evolving states from any two initial states cannot distinguish these two channels. More importantly, $\mathbf{c}(\mathcal{E}_{t})$ may cause the non-divisibility of the quantum process $\mathcal{E}_{t}$, and this cannot be revealed by $\mathcal{N}_{\mathrm{BLP}}$.
On the other hand, the non-unital part $\mathbf{c}(\mathcal{E}_{t})$ has its own physical meaning: $\mathbf{c}(\mathcal{E}_{t})\neq0$ is necessary for the increasing of the purity $\mathcal{P}(\rho)=\mathrm{Tr}(\rho^{2})$ [@Lidar2006]. In other words, $$\mathbf{c}(\mathcal{E}_{t})=0\ \Longrightarrow\ \mathcal{P}(\mathcal{E}_{t}(\rho))\leq\mathcal{P}(\rho),\ \forall\rho.$$ Besides the non-Markovian feature, the non-unitality is another kind of general feature of quantum processes. In analogy to the definition of BLP non-Markovianity, we defined the following measure on the degree of the non-unitality of a quantum process: $$N_{\mathrm{nu}}(\mathcal{E}_{t}) = \max_{\rho_{0}}
\int_{\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{P}(\mathcal{E}_{t}(\rho_{0}))>0}
\left| \frac{\mathrm{d} \mathcal{P}[\mathcal{E}_{t}(\rho_{0})] }{\mathrm{d}t} \right|
\mathrm{d}t,$$ where $\rho_{0}$ is the initial state. Obviously, $N_{\mathrm{nu}}(\mathcal{E}_{t})$ vanishes if $\mathbf{c}(\mathcal{E}_{t})=0$.
Since the non-unital aspect of the dynamics, which is not revealed by the trace distance, has its own speciality, we aim to measure the effect of non-unitality on non-Markovian behavior. However, a perfect separation of the non-unital aspect from the total non-Markovianity may be infeasible. Therefore we require a weak version $\mathcal{N}_\mathrm{nu}$ for measuring non-unital non-Markovianity to satisfy the following three conditions: (i) $\mathcal{N}_{\text{nu}}$ vanishes if $\mathcal{E}_{t}$ is infinitesimal divisible, (ii) $\mathcal{N}_{\text{nu}}$ vanishes if $\mathcal{E}_{t}$ is unital, (iii) $\mathcal{N}_{\mathrm{nu}}$ should be relevant to $\mathbf{c}(\mathcal{E}_{t})$. Based on these conditions, we introduce the following measure $$\mathcal{N}_{\mathrm{nu}}:=\underset{\varrho_{\tau}\in\mathcal{X}}{\max}\int\limits _{\sigma_{\text{nu}}>0}\sigma_{\text{nu}}(t,\,\varrho_{\tau})\text{d}t,\label{eq:definition_NUN}$$ where $\mathcal{X}:=\left\{ \varrho_{\tau}\mid0\leq\tau\leq\infty\right\} $ with $\varrho_{\tau}:=\mathcal{E}_{\tau}(\openone/d)$ is the set of the trajectory states which evolve from the maximally mixed state, and $$\sigma_{\text{nu}}(t,\varrho_{\tau}):=\frac{\text{d}}{\text{d}t}D\left[\mathcal{E}_{t}(\varrho_{0}),
\mathcal{E}_{t}(\varrho_{\tau})\right],$$ with $D(\rho_{1},\rho_{2})$ an appropriate distance which will be discussed below. The first condition is guaranteed if we require that $D$ is monotone under any CPT maps, i.e., $D[\mathcal{E}(\rho_{1}),\mathcal{E}(\rho_{2})]\leq D(\rho_{1},\rho_{2})$ for any quantum channel $\mathcal{E}$. For the unital time evolution, the set $\mathcal{X}=\{\openone/d\}$ only contains the maximally mixed state, so the above defined $\mathcal{N}_{\mathrm{nu}}$ vanishes, and the second condition is satisfied. The third condition excludes the trace distance.
In this paper, we use the Bures distance which is defined as $$D_{\mathrm{B}}(\rho_{1},\rho_{2})=\sqrt{2\left[1-F\left(\rho_{1},\rho_{2}\right)\right]},$$ where $$F(\rho_{1},\rho_{2})=\mathrm{Tr}|\sqrt{\rho_{1}}\sqrt{\rho_{2}}|
=\mathrm{Tr}\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}}}$$ is the Uhlmann fidelity [@Uhlmann1976; @Jozsa1994] between $\rho_{1}$ and $\rho_{2}$. Here $|O|=\sqrt{O^{\dagger}O}$. Bures distance is an appropriate distance for $\mathcal{N}_{\mathrm{nu}}$ because it obeys the monotonicity under CPT maps [@Petz1996] and is relevant to $\mathbf{c}(\mathcal{E}_{t})$. As here only the monotonicity of distance is relevant, for simplicity, we can also take the square of the Bures distance or just the opposite value of Unlmann fidelity as a simple version of monotone “distance” [@Vasile2011]. Quantum relative entropy [@Vedral2002] $S(\rho_{1}\Vert\rho_{2})=\text{Tr}\left[\rho_{1}(\ln\rho_{1}-\ln\rho_{2})\right]$, or its symmetric version $S_{\mathrm{sym}}(\rho_{1}\Vert\rho_{2}):=S(\rho_{1}\Vert\rho_{2})+S(\rho_{2}\Vert\rho_{1})$, is another qualified candidate for the distance. Noting that when the support of $\rho_{1}$ is not within the support of $\rho_{2}$, namely, $\mathrm{supp}(\rho_{1})\nsubseteq\mathrm{supp(\rho_{2})}$, $S(\rho_{1}\Vert\rho_{2})$ will be infinite, so in such cases, quantum relative entropy will bring singularity to the measure of non-Markovianity. Also, Hellinger distance [@Luo2004] is qualified. Although all of these distances are monotone under CPT maps, they may have different characteristics in the same dynamics, see Ref. [@Dajka2011].
The difference between non-unital non-Markovian measure defined by Eq. (\[eq:definition\_NUN\]) and the BLP-type measures, including those which use other alternative distances, is the restriction on the pairs of initial states. Comparing with the BLP-type measures relying on any pair of initial states, the non-unital non-Markovianity measure only relies on the pairs consisting of the maximally mixed state and its trajectory states. On one hand, this restriction makes the non-unital non-Markovianity measure vanish when the quantum processes are unital, no matter they are Markovian or non-Markovian; on the other hand, this restriction reflects that non-unital non-Markovianity measure reveals only a part of information concerning the non-Markovian behaviors.
EXAMPLE \[sec:EXAMPLE\]
=======================
To illustrate the non-unital non-Markovian behavior, we give an example in this section. We use the generalized amplitude damping channel (GADC) as a prototype to construct a quantum process. The GADC can be described by $\mathcal{E}(\rho)=\sum_{i}E_{i}\rho E_{i}^{\dagger}$ with the Kraus operators $\{E_{i}\}$ given by [@Fujiwara2004; @Nielsen] $$\begin{aligned}
E_{1} & = & \sqrt{p}\left(\begin{array}{cc}
1 & 0\\
0 & \sqrt{\eta}
\end{array}\right),\\
E_{2} & = & \sqrt{p}\left(\begin{array}{cc}
0 & \sqrt{1-\eta}\\
0 & 0
\end{array}\right),\\
E_{3} & = & \sqrt{1-p}\left(\begin{array}{cc}
\sqrt{\eta} & 0\\
0 & 1
\end{array}\right),\\
E_{4} & = & \sqrt{1-p}\left(\begin{array}{cc}
0 & 0\\
\sqrt{1-\eta} & 0
\end{array}\right),\end{aligned}$$ where $p$ and $\eta$ are real parameters. Note that for any $p\in[0,1]$ and any $\eta\in[0,1]$, the corresponding $\mathcal{E}$ is a quantum channel. For a two-level system, the Hermitian orthonormal operator basis can be chosen as $\bm{\lambda}=\bm{\sigma}/\sqrt{2}$, where $\bm{\sigma}=\{\sigma_{x},\sigma_{y},\sigma_{z}\}$ is the vector of Pauli matrices. With the decomposition in Eq. (\[eq:rho\]), the affine map for the Bloch vector is given by $\mathbf{r}(\mathcal{E}(\rho))\mapsto M(\mathcal{E})\mathbf{r}(\rho)+\mathbf{c}(\mathcal{E})$ [@Nielsen], where $$\begin{aligned}
M(\mathcal{E}) & = & \left(\begin{array}{ccc}
\sqrt{\eta} & 0 & 0\\
0 & \sqrt{\eta} & 0\\
0 & 0 & \eta
\end{array}\right),\\
\mathbf{c}(\mathcal{E}) & = & \left(0,0,\frac{(2p-1)(1-\eta)}{\sqrt{2}}\right)^{\mathrm{T}}.\end{aligned}$$ The GADC is unital if and only if $p=1/2$ or $\eta=1$. When $\eta=1$, $M(\mathcal{E})=\openone$, the map is identity.
A quantum process can be constructed by making the parameter $p$ and $\eta$ to be dependent on time $t$. For simplicity, we take $p_{t}=\cos^{2}\omega t$ and $\eta_{t}=e^{-t}$, where $\omega$ is a constant real number. This is a legitimate quantum process, because $\mathcal{E}_{t}$ is a quantum channel for every $t\geq0$, and $\mathcal{E}_{t=0}$ is the identity map.
First, let us consider the $\mathcal{N}_{\mathrm{BLP}}$ for this quantum process. For any two initial states $\rho_{1}$ and $\rho_{2}$, we have the trace distance $$\begin{aligned}
D_{\mathrm{tr}}\left[\mathcal{E}_{t}(\rho_{1}),\mathcal{E}_{t}(\rho_{2})\right] & =\frac{1}{2}\mathrm{Tr}\left|M(\mathcal{E}_{t})[\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})]\cdot\frac{\bm{\sigma}}{\sqrt{2}}\right|\nonumber \\
& =\frac{1}{\sqrt{2}}\left|M(\mathcal{E}_{t})[\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})]\right|,\end{aligned}$$ where $|\mathbf{r}|=\sqrt{\mathbf{r}\cdot\mathbf{r}}$ is the Euclidean length of the vector $\mathbf{r}$, and we used the equality $$(\mathbf{a}\cdot\bm{\sigma})(\mathbf{b}\cdot\bm{\sigma})=(\mathbf{a}\cdot\mathbf{b})\openone
+i\bm{\sigma}\cdot(\mathbf{a}\times\mathbf{b})$$ for Pauli matrices. Denoting $\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})$ by $(x,y,z)^{\mathrm{T}}$, we get $$D_{\mathrm{tr}}\left[\mathcal{E}_{t}(\rho_{1}),\mathcal{E}_{t}(\rho_{2})\right]=\frac{e^{-t/2}}{\sqrt{2}}\sqrt{x^{2}+y^{2}+e^{-t}z^{2}},\label{eq:GADC_tr_distance}$$ which implies $\frac{\mathrm{d}}{\mathrm{d}t}D_{\mathrm{tr}}\left[\mathcal{E}_{t}(\rho_{1}),\mathcal{E}_{t}(\rho_{2})\right]\leq0$ for every time point $t\geq0$ and for any real numbers $x$, $y$, and $z$. Thus, the BLP non-Markovianity vanishes, i.e., $\mathcal{N}_{\mathrm{BLP}}(\mathcal{E}_{t})\equiv0$, although $\mathcal{E}_{t}$ may be not infinitesimal divisible, which will become clear later.
![\[fig:DB\_Dtr\] (Color online) (a) Evolution of trace distance and Bures distance between two evolving states of a two-level system under the variant generalized amplitude damping channel, initially from the maximal mixed states $\varrho_{0}=\openone/2$ and its trajectory state $\varrho_{\tau}=\mathcal{E}_{\tau}(\varrho_{0})$, respectively. (b) The evolution of $g(t)$ defined by Eq. (\[eq:gt\]), whose integral with respect to time $t$ is RHP measure for non-Markovianity. In these figures, the parameters are taken as $\tau=10$ and $\omega=5$.](example){width="5cm"}
In order to investigate whether $\mathcal{E}_{t}$ is infinitesimal divisible or not, we shall apply $\mathcal{N}_{\mathrm{nu}}$ in the above model. The trajectory of the maximally mixed state under $\mathcal{E}_{t}$ reads $$\mathcal{E}_{t}(\varrho_{0})=\frac{1}{2}\openone+\mathbf{c}_{t}\cdot\frac{\bm{\sigma}}{\sqrt{2}}=\frac{1}{2}\left(\begin{array}{cc}
1+W_{t} & 0\\
0 & 1-W_{t}
\end{array}\right),$$ where $$W_{t}:=(2p_{t}-1)(1-\eta_{t})=\cos(2\omega t)(1-e^{-t}).$$ Taking these trajectory states as the initial states, we get the corresponding evolving states: $$\begin{aligned}
\mathcal{E}_{t}(\varrho_{\tau}) & = & \frac{1}{2}\openone+(M_{t}\mathbf{c}_{\tau}+\mathbf{c}_{t})\cdot\frac{\bm{\sigma}}{\sqrt{2}}\\
& = & \frac{1}{2}\left(\begin{array}{cc}
1+W_{t}+\eta_{t}W_{\tau} & 0\\
0 & 1-W_{t}-\eta_{t}W_{\tau}
\end{array}\right).\quad\end{aligned}$$ Then the fidelity reads $$F\left[\mathcal{E}_{t}(\varrho_{0}),\mathcal{E}_{t}(\varrho_{\tau})\right]=\frac{1}{2}(h_{+}+h_{-}),$$ where $$\begin{aligned}
h_{+} & := & \sqrt{\left(1+W_{t}\right)\left(1+W_{t}+\eta_{t}W_{\tau}\right)},\\
h_{-} & := & \sqrt{\left(1-W_{t}\right)\left(1-W_{t}-\eta_{t}W_{\tau}\right)}.\end{aligned}$$ To compare with the behavior of trace distance, we also get $D_{\mathrm{tr}}\left[\mathcal{E}_{t}(\varrho_{0}),\mathcal{E}_{t}
(\varrho_{\tau})\right]=\left|\eta_{t}W_{\tau}\right|/2$. With the expressions $\eta_{t}=e^{-t}$ and $p_{t}=\cos^{2}\omega t$, it is $$D_{\mathrm{tr}}\left[\mathcal{E}_{t}(\varrho_{0}),\mathcal{E}_{t}(\varrho_{\tau})\right]=\frac{e^{-t}}{2}\left|\cos2\omega\tau\right|(1-e^{-\tau}).\label{eq:GADC_nonunital}$$
In Fig. \[fig:DB\_Dtr\](a), we can see that while the trace distance between the evolving states $\mathcal{E}_{t}(\varrho_{0})$ and $\mathcal{E}_{t}(\varrho_{\tau})$ monotonously decreases with the time $t$, the Bures distance increases during some intermediate time intervals. From Eq. (\[eq:GADC\_nonunital\]), one can see although $D_{\mathrm{tr}}[\mathcal{E}_{t}(\varrho_{0}),\mathcal{E}_{t}(\varrho_{\tau})]$ depends on $W_{\tau}$, it does not depend on $W_{t}$. Actually, from Eq. (\[eq:GADC\_tr\_distance\]) one could find that for any two initial states, the trace distance between the evolving states is independent on $W_{t}$. In this sense, the BLP non-Markovianity treats a family of quantum processes, which only differ with $p_{t}$, as the same one. Meanwhile, $\mathcal{N}_{\mathrm{nu}}$ reveals the effects of $p_{t}$ on the infinitesimal non-divisibility and is capable of measuring it.
In order to compare with BHP measure, we also calculate the $g(t)$ defined by Eq. (\[eq:gt\]). We get $$g(t) = \frac{1}{2}\big[ |1-f(t) |+|f(t)|-1\big]$$ with $$f(t) := -\omega\sin(2\omega t)\left(1-e^{-t}\right)+\cos^{2}(\omega t).$$ The mediate dynamical maps $\mathcal{E}_{t+\epsilon,t}$ with infinitesimal $\epsilon$ are not completely positive when $g(t)>0$. From Fig. \[fig:DB\_Dtr\](b), we can see that the increasing of the Bures distance occurs in the regimes where $g(t)>0$, which coincides with the monotonicity of Bures distance under CPT maps.
Conclusion\[sec:Conclusion\]
============================
In conclusion, we have shown that the measure for non-Markovianity based on trace distance cannot reveal the infinitesimal non-divisibility caused by the non-unital part of the dynamics. In order to reflect effects of the non-unitality, we have constructed a measure on the non-unital non-Markovianity, and also defined a measure on the non-unitality, in the same spirit as BLP non-Markovianity measure.
Like non-Markovianity, the non-unitality is another interesting feature of the quantum dynamics. With the development of quantum technologies, we need novel theoretical approaches for open quantum systems. It is expected that some quantum information methods would help us to understand some generic features of quantum dynamics. We hope this work may draw attention to study more dynamical properties from the informational perspective.
This work was supported by NFRPC through Grant No. 2012CB921602, the NSFC through Grants No. 11025527 and No. 10935010 and National Research Foundation and Ministry of Education, Singapore (Grant No. WBS: R-710-000-008-271).
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|
---
author:
- Masayoshi Fujihala
- Hiroko Koorikawa
- Setsuo Mitsuda
- Katsuhiro Morita
- Takami Tohyama
- Keisuke Tomiyasu
- Akihiro Koda
- Hirotaka Okabe
- Shinichi Itoh
- Tetsuya Yokoo
- Soshi Ibuka
- Makoto Tadokoro
- Masaki Itoh
- Hajime Sagayama
- Reiji Kumai
- Youichi Murakami
title: 'Tomonaga-Luttinger spin liquid in the spin-1/2 inequilateral diamond-chain compound K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$'
---
Introduction {#introduction .unnumbered}
============
Identifying spin liquid phases in the ground state is one of hot topics in the field of low-dimensional quantum magnets. Three-dimensional orders induced by magnetic interactions between one-dimensional (1D) chains/two-dimensional layers, however, prevent the spin-liquid ground state. It is thus essential to search for quantum magnets that possess negligibly weak inter chain/layer interactions.
One of possible spin liquids in 1D is a Tomonaga-Luttinger (TL) liquid, where spin-spin correlation decays algebraically with distance. Azurite Cu$_3$(CO$_3$)$_2$(OH)$_2$ that contains spin-1/2 distorted diamond chains [@Kikuchi; @Rule_2005; @Rule_2008; @Rule_2011] is a possible candidate for the TL spin liquid, since the ground state of the distorted diamond-chain is expected to belong to an alternating dimer-monomer phase where neighboring monomers are connected via the dimer in between and an effective Heisenberg 1D chain controls low-energy excitations [@Okamoto_1999; @Okamoto_2003]. In fact, a recent theoretical approach based on density functional theory together with numerical many-body calculations has proposed a microscopic model for the azurite, which includes two energy scales coming from dimer singlet and from a 1D Heisenberg chain [@Jeschke]. Although the model predicts the TL spin liquid, three-dimensional magnetic interactions in the azurite cause the magnetic order at 1.85 K.
Recently a highly 1D inequilateral diamond-chain compound alumoklyuchevskite K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ has been reported by some of the present authors[@Fujihala]. There is no long-range magnetic order down to at least 0.5 K as evidenced by specific heat measurements. The magnetic susceptibility exhibits a double broad peak at around 200 K and 50 K. By analyzing the magnetic susceptibility, an effective model for K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ has been proposed [@Morita], where the dimer is formed by one of the four sides in the diamond and the remaining spins form a 1D Heisenberg chain, as shown in Fig. \[config\](d). The model is different from that for the azurite.
In this paper, we present detailed studies of K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ through single-crystal and powder X-ray diffraction (XRD), inelastic neutron scattering (INS), and muon spin rotation/relaxation ($\mu$SR) spectroscopy. These experimental results indicate that K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ is an appropriate model material for the investigation of the TL spin liquid state at low temperatures close to the ground state.
Results and Discussion {#results-and-discussion .unnumbered}
======================
Crystal structure refinement {#crystal-structure-refinement .unnumbered}
----------------------------
![(a) Synchrotron XRD intensity pattern (filled red circles) observed for K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ at room temperature, the result of Rietveld refinement using the computer program RIETAN-FP [@Fujio] (black solid line), and difference between the calculated and observed intensities (blue solid line). The green vertical bars indicate the position of Bragg reflection peaks. The inset shows the crystal structure of K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ featuring a large inter-chain spacing. (b) The diamond chain of K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$, which consists of Cu$^{2+}$ ions (grey spheres) along the $a$-axis with nearby oxygen (red spheres) and sulfate ions (yellow tetrahedral). (c) Effective spin model of K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ with the nearest-neighbor exchange couplings $J_i$ ($i=1$ to 5), and the next nearest-neighbor exchange couplings of $J_\mathrm{m}$, $J_\mathrm{d}$, and $J'_{\rm d}$. (d) Spin configuration of the ground state for K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$.[]{data-label="config"}](Fig1){width="\linewidth"}
The Cu-O-Cu angle significantly influences on the value of the exchange interactions, the variation of the angles can give strong bond-dependent exchange interactions [@Mizuno]. Therefore, the crystal structure refinement of K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ is necessary to determine the magnetic exchange interactions. The space group of the mineral alumoklyuchevskite K$_3$Cu$_3$Al$_{0.64}$Fe$_{0.36}$O$_2$(SO$_4$)$_4$, is $C$2, as reported by Krivovichev $et ~al.$[@Krivovichev]. However, the previous powder XRD data showed that it is not consistent with synthesized K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$.
The space group and structural parameters for the synthesized material are determined from single crystal XRD and synchrotron powder XRD to be $P$1 and $a=4.9338(5)$ Å, $b=11.923(5)$ Å, and $c=14.578(5)$ Å, with $\alpha=87.309(5)^{\circ}$, $\beta=80.837(3)^{\circ}$, and $\gamma=78.458(6)^{\circ}$, respectively. As shown by Rietveld refinement results, there is no impurity peaks \[Fig. \[config\](a)\]. The inset of Fig. \[config\](a) and Fig. \[config\](b) show that K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ contains magnetic Cu$^{2+}$ ions in an inequilateral diamond-chain arrangement along the $a$-axis direction. The nearest-neighbor magnetic couplings, $J_i$ ($i=1$ to 5), are the superexchange interactions through Cu-O-Cu bonds: $J_1$ ($J_4$) through Cu2-O-Cu3 bond with bond angle 101.59(17)$^{\circ}$ (105.38(18)$^{\circ}$), $J_2$ through Cu1-O-Cu3 with two bond angles 96.6(2)$^{\circ}$ and 96.7(2)$^{\circ}$, and $J_3$ ($J_5$) through Cu1-O-Cu2 with 104.51(18)$^{\circ}$ (127.8(2)$^{\circ}$), see Fig. \[config\](c). In addition, the exchange interactions through the Cu-O-S-O-Cu exchange paths are denoted by $J_{\rm m}$, $J_{\rm d}$, $J'_{\rm d}$ in Fig. \[config\](b). $J_5$ with the largest angle is expected to be the largest antiferromagnetic (AFM) interaction, while $J_2$ with the smallest angle is considered to be a ferromagnetic (FM) interaction [@Mizuno].
The values of the exchange interactions have been obtained by fitting the temperature dependence of the magnetic susceptibility with calculated one [@Morita]: $J_1=J_3=J_4=-30$ K, $J_2=-300$ K, $J_5=510$ K, and $J_{\rm m}=J_{\rm d}=J'_{\rm d}=75$ K. These values are completely different from those of azurite [@Jeschke]: the $J_2$ bond, where the singlet dimer is located in azurite, is FM, while the singlet dimer and a 1D chain is formed on the $J_5$ bond and $J_{\rm d}$ bond, respectively, as shown in Figs. \[config\](c) and (d). The alternating dimer-monomer model realized in azurite is, thus, not the case in K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$. The TL spin liquid in this compound is formed along the $J_\mathrm{d}$ bond. We will show below that our experimental results are in excellent agreement with this prediction.
Spinon continuum and spin-singlet dimer excitations in the inelastic neutron scattering spectra {#spinon-continuum-and-spin-singlet-dimer-excitations-in-the-inelastic-neutron-scattering-spectra .unnumbered}
-----------------------------------------------------------------------------------------------
![INS spectra for K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$. (a) Magnetic scattering contribution, extracted the 100 K data from the 4 K data corrected for the phonon population factor ($E_i = 45.95$ meV). (b) Single-crystal-like data obtained by applying the conversion method. The superimposed gray dashed lines indicate the lower and upper energy boundaries of the continuum given by $(\pi J_{\rm d}/2)|\sin(Qa)|$ and $\pi J_{\rm d}|\sin(Qa/2)|$ [@Lake], respectively, where $a$ is the lattice parameter in the chain direction. (c) Experimental raw data measured at 4 K with incident neutron energy of 205.8meV. (d) Simulated powder-averaged INS spectrum. []{data-label="ins"}](Fig2){width="\linewidth"}
The theoretical study predicted that a gapless low-energy spin excitation and gap excitation are observed by INS experiment, because the spin dimer together with a nearly isolated 1D Heisenberg spin chain characterizes magnetic properties of K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ [@Morita].
INS experiments are performed on the powder samples using the high resolution chopper spectrometer HRC at MLF of J-PARC. Figure \[ins\](a) shows the magnetic scattering contribution at 4 K, which is obtained by subtracting the phonon contribution from the observed spectrum at 100 K (see Supplementary Information Sec. I). The strong flat signal is seen at approximately 10 meV, indicating that there is the van Hove singularity of spinon continuum edges at this energy. Furthermore, the conversion method developed by Tomiyasu [*et al*]{}. [@Tomiyasu] is used to obtain single-crystal-like information on magnetic excitations along the chain direction from the powder INS spectrum \[Fig. \[ins\](b)\]. The spinon continuum edges rise up from the Brillouin zone centers in chain direction $Q=\pi/a=0.64$ Å$^{-1}$ and $3\pi/a=1.91$ Å$^{-1}$, which supports the ideal one-dimensionality of this compound.
Figure \[ins\](c) shows the data measured with $E_i= 205.8$ meV. A signal is observed at around $E=40$ meV and $Q=1.0$ Å$^{-1}$. The signal due to magnetic excitations is generally enhanced at low-$Q$ values, whereas phonon excitations are dominant at high-$Q$. Therefore, we consider that this signal comes from magnetic excitations.
The dynamical spin structure factor for the proposed model has been calculated by the dynamical density-matrix renormalization group [@Morita]. In order to compare the calculated spectrum with the powder INS spectra in Figs. \[ins\](a) and \[ins\](c), we simulate the powder-averaged INS spectrum including the magnetic form factor of Cu$^{2+}$ by using a conversion technique (see Eq. (2) of Ref. 13). Figure \[ins\](d) shows the simulated powder-averaged INS spectrum. The agreement with experimental data in Figs. \[ins\](a) and \[ins\](c) is fairly good. Therefore, we are confident that the proposed exchange interactions and the spin configuration shown in Fig. \[config\](c) are appropriate for this compound. In other words, the low-energy excitation is characterized by a TL spin liquid.
$\mu$SR evidence for a quantum spin liquid state in K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ {#musr-evidence-for-a-quantum-spin-liquid-state-in-k_3cu_3alo_2so_4_4 .unnumbered}
----------------------------------------------------------------------------------
![(a) ZF-$\mu$SR spectra (using a dilution refrigerator) at representative temperatures (see Supplementary Information Sec. II for the spectra obtained using $^4$He cryostat). The thick lines behind the data points are curves fitted by using the stretched exponential function (see text). The inset shows the ZF-$\mu$SR spectrum measured at 90 mK, which decreases continuously without oscillations up to 20 $\mu$s. (b) $\mu$SR spectra measured at 90 mK under ZF and representative external fields. The inset shows the field dependence of the muon spin relaxation rate $\lambda$. The blue solid line and gray dashed line are curves fitted by using the power law and the Redfield equation (see text). (c) Temperature dependence of $\lambda$. (d) Temperature dependence of the stretching exponent $\beta$. The gray solid lines in (c) and (d) are the guide to eyes.[]{data-label="msr"}](Fig3){width="\linewidth"}
Quantum spin fluctuations in K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ are investigated by using zero-field (ZF) and longitudinal-field (LF) $\mu$SR measurements for a powder sample in the temperature range from 90 mK to 300 K at MLF of J-PARC.
The ZF-$\mu$SR spectra are fitted by the stretched exponential function $a(t)=a_1\exp[-(\lambda t)^{\beta}] + a_{\rm BG}$, where $a_1$ is an intrinsic asymmetry $a_1 = 0.17$, $a_{\rm BG}$ is a constant background $a_{\rm BG} = 0.031$, $\lambda$ is the muon spin relaxation rate, and $\beta$ is the stretching exponent. The spectra at representative temperatures are presented in Fig. \[msr\](a). The combined effect of these multiple nuclear dipole fields leads to a phenomenologically described relaxation function of the stretched exponential. The field distribution and the stretching exponent at high temperatures are approximately given by $\Delta_{\rm nuclear}=\lambda/\gamma_{\mu}=0.6$ G and $\beta\approx 2$, respectively, which are typical for a nuclear dipole field. The ZF spectrum at the lowest temperature $90$ mK decreases continuously without oscillations up to 20 $\mu$s (see the inset of Fig. \[msr\](a)). If this ZF spectrum is due to static magnetism, the internal field should be approximately 1.1 G. However, the relaxation is clearly observed, even in the LF at 0.4 T, which is evidence for the fluctuation of Cu$^{2+}$ electron spins without static ordering down to 90 mK.
The plateau of the relaxation rate $\lambda$ below 1.5 K and the gradual decrease of $\beta$ are observed, as shown in Figs. \[msr\](c) and \[msr\](d), respectively. These characteristics have also been observed in other quantum spin-liquid candidates [@Gomilsek; @Mendels; @Adroja; @Kermarrec; @Quilliam]. The LF spectra at 0.005 T are also fitted by a stretched exponential function with $\beta=0.748$, because there is a complete decoupling of the nuclear component. Using the power law represented by $1/(a+ bH^\alpha)$ [@Kermarrec; @Quilliam] with an unconventional value of $\alpha=0.65$, where $a$ and $b$ are dependent on the fluctuation rate and fluctuating field, we obtain a good fitting to the data, as shown in the inset of Fig. \[msr\](b). Incidentally, the $1/(a+ bH^2)$ is a standard case that the $\lambda$ obeys the Redfield equation. The value obtained for $\alpha$ and the magnitude of the $\lambda$ plateau are close to those for Mg-herbertsmithite [@Kermarrec]. All of these $\mu$SR results strongly support the formation of a quantum spin liquid at very low temperature in K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$.
Conclusion {#conclusion .unnumbered}
==========
In summary, a spin-1/2 inequilateral diamond-chain compound K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ has been experimentally examined by single-crystal and powder X-ray diffraction, muon spin rotation/relaxation spectroscopy, and inelastic neutron scattering. By comparing the INS experimental data with calculations for a theoretically proposed model, we have confirmed that the compound is described by a composite structure consisting of singlet dimers and a one-dimensional Heisenberg chain, which is different from an alternating dimer-monomer model corresponding to azurite. Since the low-energy excitations are described by the one-dimensional Heisenberg model and there is no three-dimensional long-range order down to 90 mK, K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ is regarded as a typical compound that exhibits a TL spin liquid behavior at low temperatures close to the ground state. K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ would further contribute to experimental efforts in uncovering exotic properties of the TL spin liquid, such as spinon spin currents [@Hirobe] and ballistic thermal conduction [@Kawamata].
Methods {#methods .unnumbered}
=======
Single phase polycrystalline K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ was synthesized by solid-state reaction in which high-purity K$_2$SO$_4$, CuO, CuSO$_4$, and AlK(SO$_4$)$2$ powders were mixed in a molar ratio of 1:2:1:1, followed by heating at 600$^{\circ}$C for three days and slow cooling in air. Significant efforts were made to grow single crystals, whereby several tiny single crystals of K$_3$Cu$_3$AlO$_2$(SO$_4$)$_4$ were successfully grown by heating the preliminarily grown powder at temperatures as high as 600$^{\circ}$ in a sealed and evacuated quartz tube for one week with slow cooling. Single crystal XRD data were collected on a Bruker AXS Smart ApexII ULTRA CCD diffractometer using MoK${\alpha}$ ($\lambda=0.71073$ Å) radiation. Synchrotron powder XRD data were collected using an imaging plate diffractometer installed at BL-8B of the Photon Factory. An incident synchrotron X-ray energy of 18.0 keV (0.68892 Å) was selected. The INS experiments are performed on the HRC, installed at BL12 beamline at the Materials and Life Science Experimental Facility (MLF) of the Japan Proton Accelerator Research Complex (J-PARC). At the HRC, white neutrons are monochromatized by a Fermi chopper synchronized with the production timing of the pulsed neutrons. The energy transfer was determined from the time-of-flight of the scattered neutrons detected at position sensitive detectors. A 400 Hz Fermi chopper was used to obtain a high neutron flux. A GM-type closed cycle cryostat was used to achieve 100 K and 4 K. The energies of incident neutrons were $E_i$ = 205.8 meV and 45.95 meV (second frame), which yielded an energy resolutions of $E$ = 5 and 1 meV at the elastic position. ZF and LF $\mu$SR experiments are performed using the spin-polarized pulsed surface-muon ($\mu$$^+$) beam at the D1 beamline of the MLF of J-PARC. The spectra were collected in the temperature range from 90 mK to 300 K using a dilution refrigerator and $^4$He cryostat.
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Acknowledgements {#acknowledgements .unnumbered}
================
The authors are grateful to S. Takeyama, D. Nakamura, K. Okamoto and T. Goto for helpful discussions. The $\mu$SR and INS experiments were performed at the MLF of J-PARC under a user program (Proposal Nos. 2015A0314, 2016A0130 and 2015A0225). Synchrotron powder XRD measurements were performed with the approval of the Photon Factory Program Advisory Committee (Proposal Nos. 2015P001 and 2016G030). Theoretical study is in part supported by Creation of new functional devices and high-performance materials to support next-generation industries (GCDMSI) to be tackled by using post-K computer and by MEXT HPCI Strategic Programs for Innovative Research (SPIRE) (hp160222, hp170274). This study is partly supported by the Grant-in-Aid for Scientific Research (No. 26287079) and (No. 15H03692) from MEXT, Japan.
Author contributions statement {#author-contributions-statement .unnumbered}
==============================
M.F., K.H. and S.M. conceived the study. K.M. and T. T. provided theoretical insight and calculated powder-averaged dynamical spin structure factor. K.T. analyzed the INS data. A.K., H.O., M.F. and K.H. performed the $\mu$SR experiments. S.It, T.Y., S.I., M.F. and K.H. performed the neutron scattering experiments. M.T. and M.I. collected and analyzed the single-crystal XRD data. H.S., R.K. and Y.M. helped collect the Synchrotron powder XRD data. All the authors contributed to the writing of the manuscript.
Additional information {#additional-information .unnumbered}
======================
[**Supplementary Information**]{} accompanies this paper at http://www.nature.com/srep\
**Competing interests:** The authors declare no competing financial interests.
|
---
abstract: 'Doped Si is a promising candidate for quantum computing due to its scalability properties, long spin coherence times, and the astonishing progress on Si technology and miniaturization in the last few decades. This proposal for a quantum computer ultimately relies on the quantum control of electrons bound to donors near a Si/barrier (e.g. SiO$_2$) interface. We address here several important issues and define critical parameters that establish the conditions that allow the manipulation of donor electrons in Si by means of external electric and magnetic fields.'
author:
- María José Calderón
- André Saraiva
- Belita Koiller
- 'Sankar [Das Sarma]{}'
title: 'Quantum control and manipulation of donor electrons in Si-based quantum computing'
---
= 10000
Introduction
============
In the last few decades, the density of transistors in a chip has been consistently duplicated every two years, as given by Moore’s law. This demand is pushing the semiconductor devices fabrication (in particular, for Si) towards the atomic limit in which the number of dopants is so small that their exact position and distribution affects the performance of a transistor. [@shanida05] There is therefore a great interest in achieving an atomic control over the position of impurities in Si. [@ruessNANO04; @ruessSMALL07] This interest has been reinforced by the proposal of using shallow donors (typically P or As) as spin qubits for a Si-based quantum computer, [@kane98; @kane00] and related ones that followed, [@vrijen00] where qubit operations are performed by manipulating the donor electron with an external electric field provided by local gates. Spin is a natural candidate for a qubit, particularly in Si where very long spin coherence times have been measured (at least $\sim 1$ ms in natural Si). [@tyryshkin03; @sousa03; @witzel05; @tyryshkin06] Spin coherence time has to be orders of magnitude longer than the time required for operations, and enough to allow for quantum error correction.[@shorPRA95]
In the doped Si-based quantum computer, [@kane98; @kane00] schematically shown in Fig. \[fig:QCproposal\], single-qubit operations (rotations of the spin state on the Bloch sphere) are accomplished with an AC magnetic field in resonance with the level splitting of the hyperfine coupled nuclear spin-electron spin system. The hyperfine coupling is proportional to the probability density of the electron wave-function on the donor. Therefore, the on and off resonance condition is controlled by applying electric fields to manipulate the electron wave-function. Two-qubit operations could in principle be performed by letting electrons in neighboring donors interact via a transient exchange coupling during a specified period of time. [@loss98] Combinations of these exchange mediated operations and single spin rotations configures a controlled-NOT (CNOT) gate. [@nielsenchuang] With the one qubit rotations and the CNOT gate any operation on $n$ qubits may be performed. [@barenco95; @nielsenchuang]
We have considered the problem of a donor a distance $d$ from a Si/SiO$_2$ (001) interface and studied the manipulation of the donor electron by electric and magnetic fields. Our study applies for cases when $d$ is large enough so that, under an applied field, abrupt ionization (via tunneling) takes place. [@martins04] When no external fields are applied, the only attractive potential felt by the electron is the donor Coulomb potential and, at the low temperatures relevant here, the electron remains bound to the donor \[see Fig. \[fig:scheme-D-I\](a)\]. Under an electric field ($F$) applied perpendicular to the interface, a triangular potential well is formed at the interface, as shown in Fig. \[fig:scheme-D-I\](b). For fields larger than a certain minimum value, bound states exist at the triangular well. When $F$ reaches a characteristic strength $F_c$ such that the bound state at the interface is degenerate in energy with the bound state at the donor, tunelling between the two wells (the donor Coulomb potential, and the triangular well at the interface) is possible. In this way, we can reversibly manipulate the position of the electron between the donor and the interface by oscillating the electric field around $F_c$ \[namely, going back and forth from situation in Fig. \[fig:scheme-D-I\](b) to situation in Fig. \[fig:scheme-D-I\](c)\]. Note that the electron, when at the interface, may still remain bound to the donor along the xy-plane, and hence, the electron may be confined in all three space directions upon certain conditions that will be described below. We simplify the problem by assuming that the SiO$_2$ barrier is infinite.
The outline of this paper is the following. In Sec. \[sec:model\] we describe our model and discuss the peculiarities of the conduction band of Si, that has a multivalley structure. We also describe our basis set consisting on the lowest uncoupled donor and interface states. In particular, we point out the conditions that guarantee the confinement in all 3 directions of space of the electron at the interface. In Sec. \[sec:manipulation\] we discuss the conditions that allow the manipulation of the donor electron to be possible, in particular: [@calderonPRL06; @calderon07] (i) we calculate the value of the characteristic electric field at which shuttling of the electron between donor and interface may occur; and(ii) we estimate the tunneling time of this process. We also briefly discuss the effect of a magnetic field applied parallel to the electric field. [@calderon062] In Sec. \[sec:exchange\] we discuss how the 2-qubit operations could be performed with the electrons at the interface state [@calderon07] and show new results for the calculation of the exchange calculated with an improved Heitler-London method. [@saraiva07] We finish with a discussion and conclusions in Sec. \[sec:discuss\].
Model {#sec:model}
=====
The wavefunction of a donor electron in a semiconductor can be written as an expansion in terms of the Bloch waves close to the bottom of the conduction band. [@kohn] The conduction band of Si has six minima (valleys) located in the $\langle 100 \rangle$ directions at a distance $k_0=2\pi \,0.85/a_{Si}$ ($a_{Si}=5.4$ Å$\,\,$ the lattice parameter of Si) from the $\Gamma$ point. Therefore, combinations of Bloch waves from the 6 valleys have to be considered, and the donor ground state wave-function would be 6-fold degenerate. This degeneracy is lifted when the coupling between Bloch waves from different valleys due to the singular donor coulomb potential is taken into account (the so-called valley-orbit coupling). The resulting ground state is non-degenerate and it is a symmetric combination of the six valleys (symmetry A$_1$). [@kohn] Within the effective mass approximation, only the Bloch functions at the positions of the conduction band minima are involved, and the ground state of the electron at the donor is written [@kohn55] $$\Psi_D^{\rm gs}= {{1}\over{\sqrt{6}}} \sum_{\mu} F_D^{\mu} ({\bf r}) \phi_{\mu} ({\bf r},{\bf r}_D) \,,
\label{eq:psid}$$ where $F_D^\mu ({\bf r})$ are envelope functions ($\mu$ = +x,-x,+y,[-y,+z,-z]{}), and $\phi_{\mu} ({\bf r},{\bf r}_D)= \Psi_{\rm Bloch}^{{\bf k}_{\mu}} e^{-i {\bf k}_{\mu}\cdot {\bf r}_D} =u_{\mu} ({\bf r}) e^{i {\bf k}_{\mu}\cdot ({\bf r}-{\bf r}_D)}$. Here, ${\bf r}_D$ is a reference site, which represents a pinning site for the Bloch waves in the superposition state. In the case of a single Bloch state, ${\bf r}_D$ contributes with an irrelevant phase, but for superposition states it leads to interference effects. For an isolated impurity, ${\bf r}_D$ is naturally chosen to be the position of the donor ($r_D=0$).
For the description of the interface state it has to be taken into account that the (001) interface breaks the degeneracy of the 6 valleys, making the x, -x, y, and -y valleys much higher in energy than the z and -z valleys. [@kane00PRB] (A tensile strain has the same effect on the valley degeneracy). [@koiller02PRB] Therefore, the ground state at the interface only includes the Bloch states from the z and -z valleys. We write it $$\Psi_I^{\rm gs}={{1}\over{\sqrt{2}}} F_I ({\bf r}) \left[ \phi_{z} ({\bf r},z_I+d) \pm \phi_{-z}({\bf r},z_I+d)\right]$$ where the envelope function $F_I$ is obtained variationally assuming the form: $$F_I= {{\alpha^{\frac{5}{2}}}\over{2\sqrt{3}}} (z+d)^{2}\, e^{-{\alpha (z+d)}/{2}}\times{{\beta}\over{\sqrt{\pi}}}\, e^{{-\beta^2 \rho^2}/{2}}
\label{eq:FI}$$ with $\alpha$ and $\beta$ taken as variational parameters, which are the same as for the single valley approximation. [@calderonPRL06] The parameter $1/\alpha$ is related to the width of the wave-function at the interface in the z-direction while $1/\beta$ gives the width of the wavefunction along the xy plane. [@calderonPRL06; @calderon07] For the isolated interface, we assume the pinning site $z_I$ is exactly at the interface.
The value of $1/\beta$ depends on how far the donor is from the interface. When the electron is at the interface it still feels the attractive potential of both the donor and its image \[see Fig. \[fig:scheme-D-I\] (d)\]. Typical values of $1/\beta$ range from $6$ nm for $d=6$ nm to $22$ nm for $d=40$ nm. If we want to be able to shuttle the electron between the interface and the donor reversibly, we need the electron to be confined at the interface, and not to spread forming a two-dimensional gas. This gives us a limit for the maximum planar density of around $10^{10}$ cm$^{-2}$. [@calderonPRL06; @calderon07]
The fact that the 6-valley degeneracy is broken at the interface plus the fact that the smooth potential between the donor and the interface does not couple valleys in different directions, allows us, in a first approach to the problem, to consider only the z and -z valleys for the state at the donor. This simplification would be exact for strained Si. Therefore, it makes sense to use $$\Psi_D^{\rm gs}={{1}\over{\sqrt{2}}} F_D ({\bf r}) \left[ \phi_{z} ({\bf r},z_D) \pm \phi_{-z} ({\bf r},z_D)\right]$$ with $$F_D \propto (z+d)e^{-\sqrt{\rho^2 / a^2+z^2/b^2}}~,
\label{eq:FD}$$ the donor envelope function centered at ${\bf r}=0$. This is a hydrogen-like envelope with an anisotropic shape that arises due to the conduction band effective mass anisotropy in Si, multiplied by the factor $(z+d)$ that guarantees that the wavefunction fulfills the boundary condition for an infinite barrier at $z=-d$. We are interested here in the situation where the donor is relatively far from the interface, i.e, the ground state wavefunctions in each well do not overlap significantly, and in this range the variationally determined Bohr radii $a$ and $b$ are the same as for a donor in bulk ($a=2.365$ nm and $b=1.36$ nm). [@calderon07]
Note that, for each isolated well, the value of the splitting between the symmetric and antisymmetric combinations of the Bloch waves and which one is the ground state depend on the value of the pinning positions $z_D$ or $z_I$, as illustrated in Fig. \[fig:vo\].
Let us write now the full Hamiltonian for a donor (at $z=0$) in Si a distance $d$ from a (001) interface with SiO$_2$. The boundary problem between the two semi-infinite dielectrics is considered by including the image charges for the donor and for the electron (see Fig. \[fig:images\]). A uniform electric field, perpendicular to the interface, is also included. The resulting Hamiltonian is, [@calderonPRL06; @calderon07] in rescaled atomic units $a^*={{\hbar^2\epsilon_{Si}}/{m_\perp e^2}} =
3.157$ nm and $Ry^*={{m_\perp e^4}/{2\hbar^2\epsilon_{Si}^2}}=
19.98$ meV, $$H = T+\kappa e F z -{{2}\over{r}}+V_D^{\rm image}+V_e^{\rm image}
+H_{vo}~,
\label{eq:h-effunits}$$ where $T=- {{\partial^2}\over{\partial x^2}} - {{\partial^2}\over{\partial
y^2}}- \gamma {{\partial^2}\over{\partial
z^2}}$, $\gamma=m_\perp/ m_\|$, $m_\perp=0.191m$, $m_\|=0.916$, $\kappa=3.89 \times 10^{-7} \epsilon_{Si}^3\left({{m}/{m_\perp}}\right)^2$ cm/kV, and the electric field $F$ is given in kV/cm. The next terms are the donor image $$V_D^{\rm image}={{2Q}\over{\sqrt{\rho^2+(z+2d)^2}}} \, ,
\label{eq:vdimage}$$ and the electron image $$V_e^{\rm image}= -{{Q}\over{2(z+d)}} \, .
\label{eq:veimage}$$ $Q$ is a function of the dielectric constants $Q={{(\epsilon_{\rm SiO_{2}}-\epsilon_{\rm Si})}/{(\epsilon_{\rm
SiO_{2}}+\epsilon_{\rm Si})}}$, with $\epsilon_{\rm Si}=11.4$ and $\epsilon_{\rm SiO_{2}}=3.8$. With this $Q$ we get that the donor image potential is attractive while the electron image potential is repulsive.
The last term in the Hamiltonian Eq. (\[eq:h-effunits\]) describes valley-orbit effects, namely the coupling between different valleys due to the singular nature of both the donor $(D)$ and the interface $(I)$ potentials. These couplings are quantified by the parameters $V_{vo}^D$ and $V_{vo}^I$. $V_{vo}^D$ is known from the splitting of the 1S manyfold of the isolated donor spectrum $V_{vo}^D=-1.5$ meV. We use this value all throughout. $V_{vo}^I$ is not precisely known, although estimated to be in the order of $1$ meV, [@ando82] and probably dependent on the interface quality. [@takashina06; @saraiva09] We consider it as a parameter ranging from $0$ to $-10$ meV. The results described below for a finite value of the interface valley orbit coupling do not depend qualitatively on the particular value or sign of $V_{vo}^I$ for $|V_{vo}^I| \gtrsim 0.02$ meV.
In our two-valley formalism, assuming that $d$ is large enough so that no strong hybridization occurs between the donor and interface states, the problem may be restricted to the basis set of the lowest uncoupled donor and interface states, namely $\{F_D \Psi_{\rm Bloch}^{k_z} e^{-i k_0 z_D}$, $F_D \Psi_{\rm Bloch}^{-k_z} e^{i k_0 z_D}$, $F_I \Psi_{\rm Bloch}^{k_z} e^{-i k_0 z_I}$, $F_I \Psi_{\rm Bloch}^{-k_z} e^{i k_0 z_I}\}$, leading to the Hamiltonian matrix
$${\mathbf H} =\left( \begin{array}{cccc}
E_{D} & V_{vo}^D \cos(2 k_0 z_D) & H_{ID} e^{ i k_0 (z_D-z_I)} & 0 \\
V_{vo}^D \cos(2 k_0 z_D) &E_{D} & 0 & H_{ID} e^{ -i k_0 (z_D-z_I)}\\
H_{ID} e^{- i k_0 (z_D-z_I)} & 0 &E_I & V_{vo}^I \cos[2 k_0 (z_I+d)]\\
0 & H_{ID} e^{ i k_0 (z_D-z_I)} & V_{vo}^I \cos(2 k_0 (z_I+d)) &E_I
\end{array} \right)~,
\label{eq:2by2matrix}$$
where $E_D=\langle F_D|H|F_D \rangle$, $E_I=\langle F_I|H|F_I \rangle$, and $H_{ID}=\langle F_I|H|F_D \rangle$ are the same as the single-valley matrix elements. [@calderonPRL06]
Manipulation of a single donor electron close to a Si/SiO$_2$ interface {#sec:manipulation}
=======================================================================
We solve the Hamiltonian Eq. (\[eq:2by2matrix\]) for different $d$’s and a range of electric fields $F$. At each point, the pinning sites $z_I$ and $z_D$ are determined variationally such that the total ground state energy is minimized. For $F<<F_c$, the electron is at the donor and the pinning is on the donor, to be precise, $abs[\cos(2 k_0 z_D)]=1$ while $abs[\cos(2 k_0 (z_I+d))] < 1$. In the opposite limit, $F>>F_c$, the electron is confined at the interface and so is the pinning, namely, $abs[\cos(2 k_0 (z_I+d))] = 1$ while $abs[\cos(2 k_0 z_D)]<1$. [@calderon07]
The variational solution of the Hamiltonian Eq. (\[eq:2by2matrix\]) leads to four eigenvalues that, as a function of the electric field, show certain level anticrossings. As an illustration, two typical cases are shown in Fig. \[fig:antic\]. [@multivalley] When $V_{vo}^{I}=0$, the two states at the interface are degenerate and only two anticrossings, involving three levels each, occur. For the more general case of $V_{vo}^{I} \ne 0$, there are typically four anticrossings, involving two levels each. For particular values of $d$, corresponding to $\cos(k_0 d)=0$, the coupling between levels is so strong that one anticrossing can involve all four levels.
To determine the characteristic field $F_c$ at which the ’shuttling’ of the electron between donor and interface may occur and the times involved in the process, we need to look at the anticrossing between the lowest eigenvalues. The value of the electric field at which the anticrossing happens is $F_c$ and the gap between levels at anticrossing gives an estimate for the tunneling time $\tau\sim\hbar/E_{gap}$. In Fig. \[fig:Fc\] we show the value of $F_c$ versus $d$ for the single valley approximation, the 2-valley model with $V_{vo}^I=0$, and $V_{vo}^I=-1$ meV. The three curves show the same qualitative shape. The small shift between them arises due to the changes produced in the ground state energies when the valley orbit coupling is included (this change is illustrated, with respect to the single valley energy, in Fig. \[fig:vo\]). Therefore, the multivalley structure of the conduction band of Si does not significantly affect the value of $F_c$. This was, in fact, expected from previous tight-binding results that took into account the six valleys. [@martins04] Very recent experiments on a Si FinFET with a single donor [@lansbergen08] show the same kind of behavior for $F_c$ versus $d$ that we find here.
Now we look at the value of the gap between levels calculated at anticrossing: The results are shown in Fig. \[fig:gap\]. For the 2-valley model with degenerate states at the interface ($V_{vo}^I=0$) the result is qualitatively similar to the smooth decrease already observed in the single valley approximation. [@calderonPRL06] This is due to the fact that only one of the degenerate levels at the interface is coupled to the lowest level of the donor, leading to an effectively single valley result. However, a qualitative striking difference is observed when a finite valley orbit coupling is present at the interface: The gap versus $d$ shows oscillations that go as $abs[\cos(k_0 d)]$, namely, the oscillations are not commensurate with the lattice. The tunneling times we get within the single valley approximation range from subpicosecond to nanoseconds depending on $d$ (larger $d$ correspond to longer vales of $\tau$). The times corresponding to the multivalley result can be much larger for $d$’s such that $\cos(k_0 d)\sim 0$, but the statistical weight of such points (see the stars in Fig. \[fig:gap\]) is relatively small. Therefore, in an ion implanted sample, where the control on the donor depth positioning is limited, most donors would have tunneling times comparable to the single valley results, while others would take much longer.
We have also analyzed the effect of a magnetic field applied parallel to the electric field. [@CKDmag; @calderon07] The magnetic field shrinks the electron wavefunction in the direction parallel to the interface. The effect is much stronger for less confined wavefunctions. Consequently, the wavefunction at the interface is much more affected than at the donor. The increase in confinement is concomitant with an increase in energy, which is hence significant at the interface while very small at the donor for magnetic fields on the order of a few Tesla. This magnetic field induced shift in energy has an interesting consequence: Starting from an electric field just above $F_c$ (so the electron is at the interface), the application of a magnetic field can push the electron back to the donor. [@calderon07] The manipulation of electrons using simultaneously electric and magnetic fields can help us distinguish donor electron from other moving charges in real systems. [@kenton06]
Exchange between electrons at the interface {#sec:exchange}
===========================================
We have discussed above how the multivalley nature of the conduction band of Si produces oscillations in the tunneling time involved in the shuttling of an electron between donor and interface. Similar oscillations have been found in the exchange interaction between electrons in neighboring donors as a function of the distance $R$ between them. [@koiller02PRL]
Since the donor electron is much less confined at the interface than at the donor well, the transient exchange interaction between donors very distant from each other in the bulk, where they do not interact, could be performed by pulling (with an electric field) the two corresponding electrons towards the interface, where they would interact. Besides, if the two neighboring donors are at the same distance $d$ from the interface, we do not expect to get oscillations on the exchange versus interdonor distance $R$. The 2D interface potential formed by two donors separated a distance $R$ from one another and at the same depth $d$ from the interface has the double-well shape, depicted in Fig. \[fig:int-double-well\] along the axis ($x$) connecting the well minima, as given by the expression $$V(x)={{2(Q-1)}\over{\sqrt{(x-R/2)^2+d^2}}} +{{2(Q-1)}\over{\sqrt{(x+R/2)^2+d^2}}}.
\label{eq:int-double-well}$$
The distance $R$ between donors has to be large enough so that the single electron wavefunction is below the barrier between the wells, to guarantee that the electron shuttling between donor and interface is reversible. In Fig. \[fig:exchange\] we show the exchange calculated for donor pairs at three different distances from the interface as a function of the interdonor distance. The calculations were performed in 2D, using the Heitler-London approximation with an improved hybrid variational wave-function which is gaussian in the center, where the parabolic approximation for the potential is valid and has an exponential decay at long distances of the well minimum, where the potential saturates and deviates significantly from the parabolic behavior, [@saraiva07]
$$\phi_{MV}(\rho) = \left\{ \begin{array}{ll}
A_1 \exp{\left(-\frac{\beta^2 \rho^2}{2}\right)} &\mbox{ if $\rho<\mu$} \, ,\\
A_2 \exp{\left(-\frac{\eta \rho}{2}\right)} &\mbox{ if $\rho > \mu$}\, .
\end{array} \right.
\label{eq:MV}$$
This [*ansatz*]{} involves five parameters, three of which ($A_1$, $A_2$, and $\eta$) are obtained from the boundary conditions and normalization, and we are left with two variational parameters ($\beta$, and $\mu$). We note that $\eta$ is obtained from $\eta = 2 \beta ^2 \mu$. The adopted matched variational wavefunction gives an exchange coupling which fits well with the asymptotic values given by more rigorous calculations [@ponomarev992] and which is orders of magnitude larger than the one we previously calculated using a pure gaussian variational form for the electron wavefunction at the interface. [@calderon07] The exchange we are getting is of the order of the one estimated for gated quantum dots in GaAs. [@laird06]
Discussion and conclusion {#sec:discuss}
=========================
We have discussed here some of the relevant parameters that would allow elementary operations involved in the manipulation of donor electrons in Si close to a SiO$_2$ interface for quantum computing applications. For our calculations we have considered an ideal semi-infinite SiO$_2$ barrier. However, in real devices [@kenton06; @lansbergen08; @morello] metallic gates are located on top of a relatively thin (few nm wide) SiO$_2$ barrier. It seems then that a more realistic description of the devices should take into account the metallic gates. There exists a calculation similar to ours in the opposite limit of an infinitesimal SiO$_2$ barrier separating the Si layer from the metallic gate. [@slachmuylders08] In that case, $Q=1$ and the image charges have opposite sign to the real charges. In particular, the donor image potential will be repulsive instead of attractive and this can seriously deteriorate the confinement of the electrons along the xy-plane at the interface. For a realistic case of a SiO$_2$ barrier width of $\sim 2$ nm and an interface-donor distance $d=4-5$ nm, [@lansbergen08] we would be in between the two extreme cases of a semi-infinite SiO$_2$ barrier considered here and an infinitesimal one considered in Ref. . In particular, the width of the wavefunction at the interface $1/\beta$ would be larger than reported here. By comparing our $1/\beta$ with the results reported in Ref. \[see $\sqrt{\rho^2}/a_0$ in the inset of Fig. 4\] we can estimate at most a doubling of $1/\beta$ when metallic gates are included. This pushes the upper limit for the donor planar density towards smaller values, still keeping our qualitative conclusions valid.
Another relevant issue is related to the modification of the spin coherence time T$_2$ when donors are located close to a Si surface. Different experiments [@schenkel06; @huebl08] have revealed that T$_2$ gets smaller than in bulk when donors are located close to an interface: bulk values reach $T_2^{\rm bulk} \sim 60$ ms while close to a surface they range from $T_2^{\rm surf}\sim 0.1-1$ ms [@schenkel06] to $T_2^{\rm surf}\sim 1\,\mu s$ [@huebl08] for different samples. T$_2$ depends on the quality of the interface (it is longer for a hydrogen passivated surface than for a Si/SiO$_2$ interface [@schenkel06]) and the distance from the donor to the interface $d$. Some effects that may be responsible for the decrease of coherence times are the magnetic noise produced by impurities and other defects (like P$_b$ centers) at the interface, [@desousa07] and/or the recombination of donor electrons on the P$_b$ centers. [@huebl08] These observations imply that increasing the interface quality may be crucial for the practical implementation of a Si-based quantum computer.
In summary, we have described the basic conditions that would allow the manipulation of donor electrons in Si by external fields, taking into account the multivalley structure of the conduction band of Si. Experiments are getting close to the limit of isolated donors [@lansbergen08; @morello] where the operations discussed here are relevant. In fact, qualitative agreement for the behavior of $F_c$ versus $d$ has already been found in samples with isolated donors in a Si FinFET, [@lansbergen08] in a situation where the donor is closer to the interface than we considered in the present study. Both situations (small and large values of $d$ as compared to the effective electronic confinement lengths) are important, and should be carefully investigated in the context of applications of donor electrons as carriers of quantum information through its charge or spin degrees of freedom.
This work is supported by LPS and NSA. M.J.C. acknowledges support from Ramón y Cajal Program and MAT2006-03741 (MICINN, Spain). B.K. also acknowledges support from CNPq, FUJB, Millenium Institute on Nanotechnology - MCT, and FAPERJ.
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---
abstract: 'In this first article of a series on alternative cosmological models we present an extended version of a cosmological model in Weyl-Cartan spacetime. The new model can be viewed as a generalization of a model developed earlier jointly with Tresguerres. Within this model the non-Riemannian quantities, i.e. torsion $T^{\alpha}$ and nonmetricity $Q_{\alpha \beta}$, are proportional to the Weyl 1-form. The hypermomentum $\Delta_{\alpha \beta}$ depends on our ansatz for the nonmetricity and vice versa. We derive the explicit form of the field equations for different cases and provide solutions for a broad class of parameters. We demonstrate that it is possible to construct models in which the non-Riemannian quantities die out with time. We show how our model fits into the more general framework of metric-affine gravity (MAG).'
address: 'Institute for Theoretical Physics, University of Cologne, 50923 Köln, Germany'
author:
- Dirk Puetzfeld
title: 'A cosmological model in Weyl-Cartan spacetime:I. Field equations and solutions'
---
Introduction
============
At the moment cosmology is one of the fastest changing fields in physics. This fact might, on the one hand, be ascribed to the vast amount of new observational data (cf [@Perlmutter; @Schmidt; @Jaffe] e.g.), on the other hand there are still fundamental open questions within what is nowadays called the cosmological standard model [@KolbTurner; @Peacock]: Where does the inflaton field come from? Is there a something like the cosmological constant $\lambda$ which contributes to the dark energy etc.?
From a theoretical viewpoint one might divide efforts today within cosmology into two broad subclasses. Firstly, we have models which extend the standard model to a certain amount, inflation [@Guth1; @Linde1], e.g., can be viewed as an add-on for the classical Friedman-Lemaître-Robertson-Walker (FLRW) model. All of these models have in common that they do not affect the structure of spacetime itself, i.e., they are still bound to a four-dimensional Riemannian spacetime and, in addition, do not modify the underlying gravity theory, i.e., General Relativity (GR). Secondly, we have models which are no longer tied to a four-dimensional Riemannian spacetime and might also modify the underlying gravity theory. One example for the latter is the so-called metric-affine gauge theory of gravity (MAG), as proposed by Hehl in [@PhysRep]. Within this gauge theoretical formulation of a gravity theory there are new geometrical quantities, torsion $T^\alpha:=D \vartheta^\alpha$ and nonmetricity $Q_{\alpha \beta}:=-Dg_{\alpha\beta}$, which liberate spacetime from its Riemannian structure. In this paper we are going to work within this more general framework of a post-Riemannian gravity theory.
The main reason to consider more general structures within cosmology is the idea that new geometrical quantities might shed light on the problems of the cosmological standard model, e.g. provide an explanation for the rather artificial introduction of an additional scalar field, like the inflaton field. The new quantities couple to the spin, shear, and dilation currents of matter, which are supposed to come into play at high energy densities, i.e. at early stages of the universe [@HehlHeyde; @Bauerle]. Another open question, which might also be attacked, is the origin of the large amount of dark energy, predicted by recent supernova observations. A model which deals with a new kind of dark matter interaction within a post-Riemannian theory has been proposed by Tucker and Wang in [@TuckerDark].
Within this paper, we confine ourselves to a Weyl-Cartan spacetime. This type of spacetime can be viewed as a special case of the more general metric-affine framework, in which the tracefree part of the nonmetricity $Q_{\alpha \beta} \hspace{-0.8cm} \nearrow
\hspace{0.3cm}$ vanishes by definition. The reason to consider this kind of restriction is twofold. Firstly, computations are more feasible in a spacetime which is not endowed with the full MAG symmetries. Secondly, the Weyl-Cartan spacetime is, unlike the Riemann-Cartan or the Weyl spacetime, still able to carry both of the new field strengths nonmetricity and torsion. Let us note that the metric-affine framework incorporates all of the above mentioned types of spacetimes. The Einstein-Cartan theory, which is formulated in a Riemann-Cartan spacetime, represents a viable gravity theory with torsion. By switching off all non-Riemannian quantities in MAG we arrive at GR.
As becomes clear from the title, this paper stands at the beginning of series of articles on cosmological models in alternative gravity theories. In this first article we derive the field equations for an enhanced gauge Lagrangian and look for solutions of these equations. Thereby we extend earlier joint work with Tresguerres [@PuetzTres] and lay the foundation for the second article in this series, which will deal with the observational consequences of this new cosmological model.
The plan of this paper is as follows. In we derive the field equations and Noether identities for a new gauge Lagrangian on a formal level. After that we make use of computer algebra and provide the explicit form of the field equations and Noether identities for a rather general choice of the Weyl 1-form in . In we restrict our considerations to a special case, which leads to a more manageable set of field equations. In we look for exact solutions of these equations. We will draw our conclusion in and present some plots for one branch of our model. In \[MAG\_KAPITEL\] and \[WEYL\_CARTAN\_KAPITEL\] we provide a short introduction into MAG and the Weyl-Cartan spacetime. Additionally, we show in \[WEYL\_CARTAN\_KAPITEL\] how our model fits into the general framework of MAG as proposed in [@PhysRep]. In \[UNITS\_SECTION\], we provide an overview over the units used throughout the preceding sections. Note that we make extensive use of differential forms within this paper. A short compilation of symbols used within the paper can be found in \[OPERATIONS\_SECTION\], for a more rigorous treatment the reader should consult Appendix A of [@PhysRep].
Lagrangian, gauge, and matter currents \[TRESGUERRES\_KAPITEL\]
===============================================================
In [@PuetzTres] we considered the following gauge Lagrangian $$V_{\texttt{\tiny \rm old}}=\frac{\chi }{2\kappa }R_{\alpha }{}^{\beta }\wedge \,\eta
_{\beta }{}^{\alpha }+\sum_{I=1}^{6}a_{I}\,^{\left( I\right) }W_{\alpha
}{}^{\beta }\wedge \,^{\star }R_{\beta }{}^{\alpha }+b \, Z_{\alpha \beta
}\wedge \,^{\star }R^{\beta \alpha }. \label{lag_old}$$ Where $R_{\alpha \beta}=W_{\alpha \beta}+Z_{\alpha \beta} =$ antisymmetric + symmetric part of the curvature, and $\eta_{\alpha
\beta}:=\,^{\star}\left(\vartheta_\alpha \wedge \vartheta_\beta
\right)$. Numbers in parentheses in front of quantities correspond to the irreducible decompositions performed in [@McCrea].
Since we are interested in more general gauge Lagrangians (cf equation (\[general\_v\_mag\]) for a very general one proposed in MAG), we are going to extend (\[lag\_old\]). We will perform our calculations on the basis of a Weyl-Cartan spacetime, i.e. a spacetime in which the tracefree part of the nonmetricity $Q_{\alpha \beta}$ vanishes, a short introduction into Weyl-Cartan spacetime is given in \[WEYL\_CARTAN\_KAPITEL\]. The most obvious extension of (\[lag\_old\]) is given by $$V_{\texttt{\tiny \rm 1}}=\sum_{I=1}^{4}c_{I}\,\,^{\left( I\right) }Q_{\alpha \beta
}\,\wedge \,^{\star }Q^{\beta \alpha }. \label{quadratic_nonmet_term}$$ Since in a Weyl-Cartan space the nonmetricity is reduced to its trace part, i.e. $Q_{\alpha \beta }=g_{\alpha \beta }Q=\frac{1}{4}g_{\alpha \beta
}Q^{\gamma }{}_{\gamma }=\,^{\left( 4\right) }Q_{\alpha \beta }$, see equation (\[torsion\_in\_weyl\_cartan\_spacetime\]), equation (\[quadratic\_nonmet\_term\]) now reads $$V_{\texttt{\tiny \rm 1}}=c_{4}\,\,^{\left( 4\right) }Q_{\alpha \beta }\,\wedge \,^{\star
}\,^{\left( 4\right) }Q^{\beta \alpha } =: c\,Q_{\alpha \beta }\wedge \,^{\star }Q^{\beta \alpha }.
\label{short_new_lagrangian}$$ Our new Lagrangian now reads $$\begin{aligned}
V&=&V_{\texttt{\tiny \rm old}}+V_{\texttt{\tiny \rm 1}} \label{Langrangian_new_symbolic}\\
&=&\texttt{\rm Einstein-Hilbert} + \texttt{\rm quadratic rotational curvature} \nonumber \\
&&\texttt{\rm + quadratic strain curvature + quadratic nonmetricity.}
\label{lag_new}\end{aligned}$$ In contrast to [@Minkevich4] we included an explicit nonmetricity term in our Lagrangian. Note that we have the arbitrary constants $\chi ,$ $a_{I=1\dots
6},$ $b,$ $c, $ and the [*weak*]{} gravity coupling constant $\kappa $. The Lagrangian in (\[lag\_new\]) can be viewed as another step towards a better understanding of the full MAG Lagrangian as displayed in (\[general\_v\_mag\]). Since a treatment of the full Lagrangian is computationally not feasible at the moment it is necessary to successively study the impact of additional terms in the Lagrangian (a review of Lagrangians used in MAG and exact solutions of the corresponding field equations can be found in [@Exact2]). Together with the quadratic rotational curvature and quadratic strain curvature terms, which were already included in our previous work [@PuetzTres], we now have an additional post-Riemannian piece in form of a quadratic nonmetricity term which enhances the usual Einstein-Hilbert Lagrangian commonly used in general relativistic cosmological models. Note that our Lagrangian in (\[lag\_new\]) does not include a term with the usual cosmological constant. As we will show in section \[SPECIAL\_CASE\_SECTION\] our ansatz in (\[lag\_new\]) gives rise to an additional constant which, on the level of the field equations, will play the same role as the cosmological constant in the standard model. Hence we omit an explicit cosmological constant term at this stage. From (\[lag\_new\]) we can derive the gauge field excitations. They read $$\begin{aligned}
M^{\alpha \beta } &=&-4c\,^{\star }Q^{\beta \alpha }=-c\,^{\star
}\left( g^{\beta \alpha }Q^{\gamma }{}_{\gamma }\right) , \label{excit_1} \\
H_{\alpha } &=&0, \label{excit_3} \\
H^{\alpha }{}_{\beta } &=&-\frac{\chi }{2\kappa }\eta _{\beta }{}^{\alpha
}-2\sum_{I=1}^{6}a_{I}\,^{\star (I)}W_{\beta }{}^{\alpha }-\frac{b}{2}\delta
_{\beta }^{\alpha }\,^{\star }R^{\gamma }{}_{\gamma }. \label{excit_2}\end{aligned}$$ The canonical gauge energy-momentum is given by $$\begin{aligned}
E_{\alpha } &=&e_{\alpha }\rfloor V+\left( e_{\alpha }\rfloor R_{\beta
}{}^{\gamma }\right) \wedge H^{\beta }{}_{\gamma }+\left( e_{\alpha }\rfloor
T^{\beta }\right) \wedge H_{\beta }+\frac{1}{2}\left( e_{\alpha }\rfloor
Q_{\beta \gamma }\right) \wedge M^{\beta \gamma } \nonumber \\
&=&e_{\alpha }\rfloor V+\left( e_{\alpha }\rfloor R_{\beta }{}^{\gamma
}\right) \wedge H^{\beta }{}_{\gamma }+\frac{1}{2}\left( e_{\alpha }\rfloor
Q_{\beta \gamma }\right) \wedge M^{\beta \gamma }.\end{aligned}$$ In contrast to [@PuetzTres], we now have a non-vanishing gauge hypermomentum [^1] $$\begin{aligned}
E^{\alpha }{}_{\beta } &=&-\vartheta ^{\alpha }\wedge H_{\beta }-g_{\beta
\gamma }M^{\alpha \gamma } \nonumber \\
&\stackrel{N}{=}&4\, c\, g_{\beta \gamma }\,^{\star }Q^{\gamma \alpha }=\, c \,g_{\beta \gamma
}\,^{\star }\left( g^{\gamma \alpha }Q^{\nu }{}_{\nu }\right) .
\label{hypermomentum}\end{aligned}$$ The field equations now turn into $$\begin{aligned}
-E_{\alpha } &=&\Sigma _{\alpha }, \label{field1} \\
dH^{\alpha }{}_{\alpha }-E^{\alpha }{}_{\alpha } &\stackrel{N}{=}&\Delta , \label{field2_trace} \\
g_{\gamma \lbrack \alpha }DH^{\gamma }{}_{\beta ]}-E_{[\alpha \beta ]} &\stackrel{N}{=}&\tau _{\alpha \beta }. \label{field2_antisymmetric}\end{aligned}$$ Note that in eqs. (\[field2\_trace\]) and (\[field2\_antisymmetric\]) we decomposed the second field equation into its trace and antisymmetric part (cf \[WEYL\_CARTAN\_KAPITEL\]). Since we are interested in the behaviour induced by the new part in (\[lag\_new\]), we will confine ourselves to a [*non-massive medium without spin*]{}, i.e. $\tau _{\alpha \beta } \stackrel{A}{=}0$. Thus, eq. (\[field2\_antisymmetric\]) turns into $$g_{\gamma \lbrack \alpha }DH^{\gamma }{}_{\beta ]}-E_{[\alpha \beta ]}\stackrel{N}{=}0. \label{field2_antisymmtetric_vanishing_spin_current}$$ Because we have not specified a matter Lagrangian, we have to take into account the Noether identities (cf \[MAG\_KAPITEL\] and \[WEYL\_CARTAN\_KAPITEL\]), i.e. $$\begin{aligned}
D\Sigma _{\alpha } &=&\left( e_{\alpha }\rfloor T^{\beta }\right) \wedge
\Sigma _{\beta }-\frac{1}{2}\left( e_{\alpha }\rfloor Q\right) \sigma
^{\beta }{}_{\beta } +\left( e_{\alpha }\rfloor R_{[\beta \gamma ]}\right)
\wedge \tau ^{\beta \gamma } \nonumber \\ && +\frac{1}{4}\left( e_{\alpha }\rfloor R\right)
\wedge \Delta , \label{noether1} \\
\sigma _{\alpha \beta } &=&\frac{1}{4}g_{\alpha \beta }d\Delta +\vartheta
_{(\alpha }\wedge \Sigma _{\beta )}, \label{noether2_1} \\
0 &=&\vartheta _{\lbrack \alpha }\wedge \Sigma _{\beta ]}.
\label{noether2_2}\end{aligned}$$ We can rewrite eq. (\[noether2\_1\]) by using (\[field2\_trace\]) $$\sigma _{\alpha \beta }\stackrel{N}{=}-\frac{1}{4}g_{\alpha \beta }\,dE^{\gamma }{}_{\gamma }+\vartheta _{(\alpha
}\wedge \Sigma _{\beta )}. \label{metric_stress_energy_relation}$$ Note that (\[noether2\_1\])-(\[noether2\_2\]) represent the decomposed second Noether identity in case of a vanishing spin current. With (\[noether2\_2\]), eq. (\[noether1\]) turns into $$\begin{aligned}
D\Sigma _{\alpha }&\stackrel{N}{=}&\left( e_{\alpha }\rfloor T^{\beta }\right) \wedge \Sigma _{\beta }-\frac{1}{2}\left( e_{\alpha }\rfloor Q\right) \vartheta ^{\beta }\wedge \Sigma
_{\beta }+\frac{1}{8}\left( e_{\alpha }\rfloor Q\right) \,g^{\beta
}{}_{\beta }\,dE^{\gamma }{}_{\gamma }\nonumber \\
&&+\frac{1}{4}\left( e_{\alpha }\rfloor
R\right) \wedge \Delta . \label{noether1_simp1}\end{aligned}$$ Thus, we have to solve (\[field1\]), (\[field2\_trace\]), (\[field2\_antisymmtetric\_vanishing\_spin\_current\]), and (\[noether2\_2\])–(\[noether1\_simp1\]) in order to obtain a solution for our model proposed in (\[lag\_new\]). Now let us investigate the matter sources of our model. For a vanishing spin current, the hypermomentum $\Delta _{\alpha \beta }$ becomes proportional to its trace part, i.e. the dilation current cf $$\Delta _{\alpha \beta }=\frac{1}{4}g_{\alpha \beta }\Delta ^{\gamma
}{}_{\gamma }. \label{general_hypermomentum_for_vanishing_spin_current}$$ The trace part of the second field equation (\[field2\_trace\]) yields $$\Delta _{\alpha \beta }=\frac{1}{4}g_{\alpha \beta }\left( dH^{\gamma
}{}_{\gamma }-E^{\gamma }{}_{\gamma }\right) . \label{dilation_current}$$ In contrast to [@PuetzTres], the dilation current is no longer a conserved quantity since $d\Delta_{\alpha \beta }\not=0$. From (\[excit\_2\]), and (\[hypermomentum\]) we can infer that $$\Delta _{\alpha \beta }=-\frac{1}{4}g_{\alpha \beta }\left( 2\,b\,d\,^{\star
}R^{\gamma }{}_{\gamma }+c\,g_{\gamma \mu }\,^{\star }\left( g^{\mu \gamma
}Q^{\nu }{}_{\nu }\right) \right) . \label{dilation_current_explicit}$$ Because of , i.e. $R^{\gamma }{}_{\gamma }\sim dQ$, this equation turns into $$\Delta _{\alpha \beta }\stackrel{N}{=}-\frac{1}{4}g_{\alpha \beta }\left( \,b\,d\,^{\star }dQ^{\gamma }{}_{\gamma
}+c\,g_{\gamma \mu }\,^{\star }\left( g^{\mu \gamma }Q^{\nu }{}_{\nu
}\right) \right) . \label{hypermomentum_with_inserted_strain_curvature}$$ Thus, for our ansatz the hypermomentum $\Delta _{\alpha \beta }$ depends on the nonmetricity and vice versa. Note that the second term in (\[hypermomentum\_with\_inserted\_strain\_curvature\]) depends on the coupling constant introduced in eq. (\[excit\_1\]). Now let us specify the remaining quantities in our model. forces the components of energy-momentum 3-form to be symmetric, thus we choose[^2] $$\Sigma _{\alpha }\stackrel{A}{=}\Sigma _{\alpha \beta }\,\eta ^{\beta }\quad \quad \texttt{\rm , with}\quad \Sigma
_{\alpha \beta }=\texttt{\rm diag}\left( \mu (t),p_{r}(t),p_{t}(t),p_{t}(t)\right)
\texttt{\rm .} \label{energy_momentum_ansatz}$$ Subsequently, we can calculate the metric stress-energy $\sigma _{\alpha
\beta }$ from eq. (\[metric\_stress\_energy\_relation\]) $$\sigma _{\alpha \beta }=-\frac{1}{4}g_{\alpha \beta }\,\,d\left(
c\,\,g_{\gamma \mu }\,^{\star }\left( g^{\mu \gamma }Q^{\nu }{}_{\nu
}\right) \right) +\vartheta _{(\alpha }\wedge \left( \Sigma _{\beta )\gamma
}\,\eta ^{\gamma }\right) . \label{metric_stress_energy_explicit}$$ Again we obtained a quantity which depends on the Weyl 1-form, i.e. the trace of the nonmetricity. Since we want to compare our model with the cosmological standard model, we take the Robertson-Walker line element as starting point of our considerations $$\vartheta ^{\hat{0}}=dt,\quad \vartheta ^{\hat{1}}=\frac{S(t)}{\sqrt{1-kr^{2}%
}}dr,\quad \vartheta ^{\hat{2}}=S(t)\,r\,d\theta ,\quad \vartheta ^{\hat{3}%
}=S(t)\,r\sin \theta \, d\phi , \label{robertson_walker_coframe}$$ with $$ds^{2}\stackrel{A}{=}\vartheta ^{\hat{0}}\otimes \vartheta ^{\hat{0}}-\vartheta ^{\hat{1}}\otimes
\vartheta ^{\hat{1}}-\vartheta ^{\hat{2}}\otimes \vartheta ^{\hat{2}%
}-\vartheta ^{\hat{3}}\otimes \vartheta ^{\hat{3}}.
\label{robertson_walker_line_element}$$ As usual, $S(t)$ denotes the cosmic scale factor and $k=-1,0,$ or $1$ determines whether the three-dimensional spatial sections of spacetime are of constant negative, vanishing, or positive Riemannian curvature. Following the model proposed in [@PuetzTres], we will choose the torsion to be proportional to its vector piece $T^{\alpha }\sim
\,^{(2)}T^{\alpha }$ and relate it to the Weyl 1-form as follows $$T^{\alpha }\stackrel{A}{=}\frac{1}{2}Q\wedge \vartheta ^{\alpha }. \label{ansatz_torsion_1}$$ The only thing missing for setting up the field equations is a proper ansatz for the Weyl 1-form $Q$. In [@PuetzTres] we were able to derive $Q$ from an ansatz for the potential of the hypermomentum $\Delta$, the so called polarization 2-form $P$. Here we will adopt a slightly different point of view. Since we are interested in the impact of different choices of the non-Riemannian quantity $Q$ on cosmology, we will directly prescribe it in the following. Besides of the fact that we gain direct control of the post-Riemannian features of our model, we circumvent the question which type of matter might generate the corresponding hypermomentum. This question and the investigation of models with a more sophisticated matter model, like the hyperfluid of Obukhov [@Obukhov2], will be postponed to later articles. Let us note that our ansatz in equation (\[energy\_momentum\_ansatz\]) is in general not compatible with the energy-momentum obtained in ([@Obukhov2], eq. (3.28)). Both quantities are only equal in special cases like the one we will investigate in section \[SPECIAL\_CASE\_SECTION\]. Since we do not prescribe a matter Lagrangian and use the Noether identities as constraints on the matter variables, our approach could be termed [*phenomenological*]{} as suggested in the first part of [@Obukhov2].
Field equations and Noether identities\[FIELD\_NOETHER\_ARB\_SECTION\]
======================================================================
In this section we will derive the field equations and Noether identities resulting from specific choices of the 1-form $Q$ which controls nearly every feature of our model. We start with a rather general form of $Q$, namely $$Q=\frac{\xi (t,r)}{S(t)}\,\vartheta ^{\hat{0}},
\label{specific_ansatz_for_nonemtricity_trace_no_1}$$ where $\xi (t,r)$ denotes an arbitrary function[^3] of the radial and the time coordinate, and $S(t)$ represents the cosmic scale factor of (\[robertson\_walker\_coframe\]). With the help of computer algebra we find that the field equations (\[field1\]), (\[field2\_trace\]), and (\[field2\_antisymmtetric\_vanishing\_spin\_current\]) yield a set of four equations. In order to compare these new field equations with the ones derived in [@PuetzTres] (cf eqs. (40)-(43) therein) we write them as follows: $$\begin{aligned}
&&\chi \left( \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right)
-\left( a_{4}+a_{6}\right) \kappa \left( \left( \frac{\ddot{S}}{S}\right)
^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right]
^{2}\right) \nonumber \\
&& \quad \quad \quad = \frac{\kappa }{3}\left( \mu -4c\left( \frac{\xi }{S}\right) ^{2}+b\left(
1-kr^{2}\right) \frac{\xi _{,r}^{2}}{S^{4}}\right) , \label{nexp_field_1} \\
&&\chi \left( 2\frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+\frac{%
k}{S^{2}}\right) +\left( a_{4}+a_{6}\right) \kappa \left( \left( \frac{\ddot{%
S}}{S}\right) ^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}%
}\right] ^{2}\right) \nonumber \\
&&\quad \quad \quad = -\kappa \left( p_{r}-4c\left( \frac{\xi }{S}\right) ^{2}-b\left(
1-kr^{2}\right) \frac{\xi _{,r}^{2}}{S^{4}}\right) , \label{nexp_field_2} \\
&&\chi \left( 2\frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+\frac{%
k}{S^{2}}\right) +\left( a_{4}+a_{6}\right) \kappa \left( \left( \frac{\ddot{%
S}}{S}\right) ^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}%
}\right] ^{2}\right) \nonumber \\
&&\quad \quad \quad = -\kappa \left( p_{t}-4c\left( \frac{\xi }{S}\right) ^{2}+b\left(
1-kr^{2}\right) \frac{\xi _{,r}^{2}}{S^{4}}\right) , \label{nexp_field_3} \\
&&\frac{d}{dt}\left( \frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+%
\frac{k}{S^{2}}\right) =0. \label{nexp_field_4}\end{aligned}$$ Since we have not specified a matter Lagrangian, we have to be careful with the Noether identities (\[noether1\])-(\[noether2\_2\]). Our ansatz (\[specific\_ansatz\_for\_nonemtricity\_trace\_no\_1\]) yields two equations $$\begin{aligned}
\hspace{-1cm} \dot{S}S^{3}\left( 3\mu +p_{r}+2p_{t}\right) +\dot{\mu}S^{4}-16\dot{S}cS\xi
^{2}+2\xi _{,rt}\xi _{,r}\, b\left( 1-kr^{2}\right) -8\xi _{,t} \, cS^{2}\xi &=&0,
\label{exp_noether_1} \\
\xi _{,rr}\xi _{,r}\, br\left( 1-kr^{2}\right) +\xi _{,r}^{2}\,b\left(
2-3kr^{2}\right) +4\xi _{,r}\, crS^{2}\xi +S^{4}\left( p_{t}-p_{r}\right) &=&0.
\label{exp_noether_2}\end{aligned}$$ Note that in eq. (\[nexp\_field\_4\]) we assumed that $a_{4}\not=-a_{6}$. As one realizes immediately, eqs. (\[nexp\_field\_1\])-(\[nexp\_field\_4\]) are very similar to the ones we obtained in ([@PuetzTres], (40)-(43)). There is only a change on the rhs, i.e. the matter side, of the above equations in form of additional terms contributing to the pressure and energy density. As we can see from eqs. (\[nexp\_field\_1\])-(\[nexp\_field\_3\]), the terms proportional to $\xi _{,r}$ vanish if we make the same assumptions as in [@PuetzTres], i.e. $\xi (t,r)\rightarrow \xi (t)$. Apart from this feature, there is another, more subtle change in (\[nexp\_field\_1\])-(\[nexp\_field\_4\]), i.e. a term of the order $\xi ^{2}$ controlled by the new coupling constant $c$ (cf eq. (\[short\_new\_lagrangian\]))[^4]. The Noether identities (\[exp\_noether\_1\]) and (\[exp\_noether\_2\]) can be transformed to $$\begin{aligned}
\hspace{-1cm}\frac{\partial }{\partial t}\left( \mu S^{4}+\xi _{,r}^{2}\, b\left(
1-kr^{2}\right) -8c\left( S\xi \right) ^{2}\right) +4cS^{2}\frac{\partial
\xi ^{2}}{\partial t}=\frac{1}{4}\frac{dS^{4}}{dt}\left( \mu
-p_{r}-2p_{t}\right) , \label{nexp_noether_1} \\
p_{r}-p_{t}=\frac{2cr}{S^{2}}\frac{\partial \xi ^{2}}{\partial r}+\frac{b}{%
S^{4}}\left( \frac{r}{2}\left( 1-kr^{2}\right) \frac{\partial \xi _{,r}^{2}}{%
\partial r}+\left( 2-3kr^{2}\right) \xi _{,r}^{2}\right) .
\label{nexp_noether_2}\end{aligned}$$ Comparison of (\[nexp\_noether\_2\]) with ([@PuetzTres], eq. (37)) yields a more sophisticated relation between the radial and tangential stresses.
Let us now extract some more information from the field equations. Addition of (\[nexp\_field\_1\]) and (\[nexp\_field\_3\]) yields $$\hspace{-1cm}2\chi \left( \frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k%
}{S^{2}}\right) =\frac{\kappa }{3}\left( \mu -3p_{t}+8c\left( \frac{\xi }{S}%
\right) ^{2}-2b\left( 1-kr^{2}\right) \frac{\xi _{,r}^{2}}{S^{4}}\right).
\label{nfield1_+_nfield3}$$ Subtracting (\[nexp\_field\_3\]) from (\[nexp\_field\_1\]) yields $$\begin{aligned}
&&2\chi \frac{\ddot{S}}{S}+2\kappa \left( a_{4}+a_{6}\right) \left( \left(
\frac{\ddot{S}}{S}\right) ^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+%
\frac{k}{S^{2}}\right] ^{2}\right) \nonumber \\
&&\quad\quad\quad =-\frac{\kappa }{3}\left( \mu +3p_{t}-16c\left( \frac{\xi }{S}\right)
^{2}+4b\left( 1-kr^{2}\right) \frac{\xi _{,r}^{2}}{S^{4}}\right) .
\label{nfield1_-_nfield3}\end{aligned}$$ Let us now combine (\[nexp\_field\_4\]) and (\[nfield1\_+\_nfield3\]) $$\begin{aligned}
0 &&\stackrel{\texttt{\rm(\ref{nexp_field_4})}}{=}2\chi \frac{d}{dt}\left( \frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right)
^{2}+\frac{k}{S^{2}}\right) \nonumber \\
&&\stackrel{\texttt{\rm(\ref{nfield1_+_nfield3})}}{=}\frac{\kappa }{3}\frac{d}{dt}\left( \mu -3p_{t}+8c\left( \frac{\xi }{S}%
\right) ^{2}-2b\left( 1-kr^{2}\right) \frac{\xi _{,r}^{2}}{S^{4}}\right) .
\label{combined_field4_and_addition}\end{aligned}$$ The trace of the energy-momentum reads $$\begin{aligned}
\hspace{-0.5cm}\Sigma ^{\gamma }{}_{\gamma } =&\mu -p_{r}-2p_{t} \nonumber \\
\stackrel{\texttt{\rm(\ref{nexp_noether_2})}}{=}& \mu -3p_{t}-\frac{2cr}{S^{2}}\frac{\partial \xi ^{2}}{\partial r}+\frac{b}{S^{4}}\left( \frac{r}{2}\left( 1-kr^{2}\right) \frac{\partial \xi _{,r}^{2}}{\partial r}+\left( 2-3kr^{2}\right) \xi _{,r}^{2}\right).
\label{energy_momentum_trace_field_inserted}\end{aligned}$$ Since we encountered a system of coupled PDEs, we will confine us to a special case in the following in which the field equations turn into a set of coupled ODEs. At this point we would like to note that the above situation is reminiscent to the extensions of the classical FLRW model to anisotropic and inhomogeneous metrical structures. For completeness we list the surviving curvature pieces for the ansatz in equation $$\begin{aligned}
^{(4)}W^{\alpha \beta } &=&\frac{\ddot{S}S-\dot{S}^{2}-k}{2S^{2}} \, \vartheta
^{\alpha }\wedge \vartheta ^{\beta },
\label{EXTWEYL_ricsymf_for_general_ansatz} \\
^{(6)}W^{\alpha \beta } &=&\frac{\ddot{S}S+\dot{S}^{2}+k}{2S^{2}}\, \vartheta
^{\alpha }\wedge \vartheta ^{\beta },
\label{EXTWEYL_scalar_for_general_ansatz} \\
^{(4)}Z_{\hat{0}\hat{0}} &=&-\,^{(4)}Z_{\hat{1}\hat{1}}=\,-^{(4)}Z_{\hat{2}%
\hat{2}}=-\,^{(4)}Z_{\hat{3}\hat{3}}=-\frac{\xi _{,r}\sqrt{-kr^{2}+1}}{2S^{2}%
}\, \vartheta ^{\hat{0}}\wedge \vartheta ^{\hat{1}}.
\label{EXTWEYL_dilcurv_for_general_ansatz}\end{aligned}$$
Special case $\xi (t,r)\rightarrow \zeta (t)$ \[SPECIAL\_CASE\_SECTION\]
========================================================================
In this section we will investigate the interesting special case in which $Q$, cf eq. (\[specific\_ansatz\_for\_nonemtricity\_trace\_no\_1\]), is given by a closed 1-form, i.e. $$Q=\frac{\zeta (t)}{S(t)}\vartheta ^{\hat{0}}.
\label{specific_ansatz_for_nonemtricity_trace_no_2}$$ The field equations are now given by $$\begin{aligned}
&&\chi \left( \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right)
-\left( a_{4}+a_{6}\right) \kappa \left( \left( \frac{\ddot{S}}{S}\right)
^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right]
^{2}\right) \nonumber \\
&&\quad \quad \quad = \frac{\kappa }{3}\left( \mu -4c\left( \frac{\zeta }{S}\right) ^{2}\right)
, \label{nfield_zeta_t_1} \\
&&\chi \left( 2\frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+\frac{%
k}{S^{2}}\right) +\left( a_{4}+a_{6}\right) \kappa \left( \left( \frac{\ddot{%
S}}{S}\right) ^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}%
}\right] ^{2}\right) \nonumber \\
&&\quad \quad \quad = -\kappa \left( p_{r}-4c\left( \frac{\zeta }{S}\right) ^{2}\right) ,
\label{nfield_zeta_t_2} \\
&&\chi \left( 2\frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+\frac{%
k}{S^{2}}\right) +\left( a_{4}+a_{6}\right) \kappa \left( \left( \frac{\ddot{%
S}}{S}\right) ^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}%
}\right]^{2}\right) \nonumber \\
&&\quad \quad \quad = -\kappa \left( p_{t}-4c\left( \frac{\zeta }{S}\right) ^{2}\right) ,
\label{nfield_zeta_t_3} \\
&&\frac{d}{dt}\left( \frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+%
\frac{k}{S^{2}}\right) =0. \label{nfield_zeta_t_4}\end{aligned}$$ Thus, the function $\zeta $ contributes to the energy density and pressure in a similar way as the function $\xi $ in [@PuetzTres]. We note that there is no additional contribution from the strain curvature in eqs. (\[nfield\_zeta\_t\_1\])–(\[nexp\_field\_4\]), i.e. no term controlled by the coupling constant $b$ of our Lagrangian (cf eq. (\[lag\_old\])). This behaviour is explained by the fact that the strain curvature vanishes identically for closed 1-forms, like $Q$ from eq. (\[specific\_ansatz\_for\_nonemtricity\_trace\_no\_2\]), in a Weyl-Cartan spacetime. The Noether identities now read: $$\begin{aligned}
&&\frac{d}{dt}\left( \mu S^{4}-8c\left( S\zeta \right) ^{2}\right) +4cS^{2}%
\frac{d\zeta ^{2}}{dt}=\frac{1}{4}\frac{dS^{4}}{dt}\left( \mu
-p_{r}-2p_{t}\right) , \label{nnoether_zeta_t_1} \\
&&p_{r}-p_{t}=0. \label{nnoether_zeta_t_2}\end{aligned}$$ In contrast to (\[nexp\_noether\_2\]), eq. (\[nnoether\_zeta\_t\_2\]) forces the radial stress to be equal to the tangential stress. Addition of (\[nfield\_zeta\_t\_1\]) and (\[nfield\_zeta\_t\_3\]), i.e. eq. (\[nfield1\_+\_nfield3\]), yields $$2\chi \left( \frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k%
}{S^{2}}\right) =\frac{\kappa }{3}\left( \mu -3p_{r}+8c\left( \frac{\zeta }{S%
}\right) ^{2}\right) . \label{nfield_zeta1_+_nfield_zeta_3}$$ Subtracting (\[nfield\_zeta\_t\_3\]) from (\[nfield\_zeta\_t\_1\]) (cf eq. \[nfield1\_-\_nfield3\]) yields $$\begin{aligned}
&&2\chi \frac{\ddot{S}}{S}+2\kappa \left( a_{4}+a_{6}\right) \left( \left(
\frac{\ddot{S}}{S}\right) ^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+%
\frac{k}{S^{2}}\right] ^{2}\right) \nonumber \\
&&\quad \quad \quad = -\frac{\kappa }{3}\left( \mu +3p_{r}-16c\left( \frac{\zeta }{S}\right)
^{2}\right) . \label{nfield_zeta_1_-_nfield_zeta_3}\end{aligned}$$ Combination of (\[nfield\_zeta1\_+\_nfield\_zeta\_3\]) with the field equations (cf eq. (\[combined\_field4\_and\_addition\])) leads to $$\begin{aligned}
0 &\stackrel{\texttt{\rm(\ref{nfield_zeta_t_4})}}{=}&\frac{\kappa }{3}\frac{d}{dt}\left( \mu -3p_{r}+8c\left( \frac{\zeta }{S}%
\right) ^{2}\right) \nonumber \\
&=&\frac{\kappa }{3}\frac{d}{dt}\left( \Sigma ^{\gamma }{}_{\gamma
}+8c\left( \frac{\zeta }{S}\right) ^{2}\right) \Rightarrow \Sigma ^{\gamma
}{}_{\gamma }+8c\left( \frac{\zeta }{S}\right) ^{2}=\texttt{\rm const}=:\Xi
\label{conserverd_energy_momentum_trace_plus_additional_stress}\end{aligned}$$ Thus, we obtained a conserved quantity similar to the one in ([@PuetzTres], eq. (47)). The first Noether identity (\[nnoether\_zeta\_t\_1\]) takes the form $$\begin{aligned}
&&\frac{d}{dt}\left[ S^{4}\left( \mu -8c\left( \frac{\zeta }{S}\right)
^{2}\right) \right] +4cS^{2}\frac{d\zeta ^{2}}{dt} =\frac{1}{4}\frac{dS^{4}%
}{dt}\Sigma ^{\gamma }{}_{\gamma } \nonumber \\
&&\quad \quad \quad \stackrel{\texttt{\rm(\ref{conserverd_energy_momentum_trace_plus_additional_stress})}}{\Leftrightarrow} 4\frac{\dot{S}}{S}\left( \Xi -\mu -\frac{3}{4}\Sigma
^{\gamma }{}_{\gamma }\right) -\dot{\mu}\ =8c\left( \frac{\zeta }{S}%
\right) ^{2}\left( 2\frac{\dot{S}}{S}-\frac{\dot{\zeta}}{\zeta }\right) .
\label{noether2_for_zeta_simplified}\end{aligned}$$ Before we proceed with the search for explicit solutions, we will collect the remaining field equations $$\begin{aligned}
&&\chi \left( \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right)
-\left( a_{4}+a_{6}\right) \kappa \left( \left( \frac{\ddot{S}}{S}\right)
^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right]
^{2}\right) \nonumber \\
&&\quad \quad \quad =\frac{\kappa }{3}\left( \mu -4c\left( \frac{\zeta }{S}\right) ^{2}\right)
, \label{field_final_zeta_XI_<>_0_1} \\
&&\chi \left( \Lambda +\frac{\ddot{S}}{S}\right) +\left( a_{4}+a_{6}\right)
\kappa\left( \left( \frac{\ddot{S}}{S}\right) ^{2}-\left[ \left( \frac{%
\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right] ^{2}\right) \nonumber \\
&&\quad \quad \quad = -\kappa \left( p_{r}-4c\left( \frac{\zeta }{S}\right) ^{2}\right) ,
\label{field_final_zeta_XI_<>_0_2} \\
&&\frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}
=\Lambda , \label{field_final_zeta_XI_<>_0_3} \\
&&4\frac{\dot{S}}{S}\left( \Xi -\mu -\frac{3}{4}\Sigma ^{\gamma }{}_{\gamma
}\right) -\dot{\mu}\ =8c\left( \frac{\zeta }{S}\right) ^{2}\left( 2\frac{%
\dot{S}}{S}-\frac{\dot{\zeta}}{\zeta }\right) .
\label{field_final_zeta_XI_<>_0_4}\end{aligned}$$ Note that the new constant $\Lambda $ in (\[field\_final\_zeta\_XI\_<>\_0\_2\]) is defined via (\[field\_final\_zeta\_XI\_<>\_0\_3\]). Comparison of (\[field\_final\_zeta\_XI\_<>\_0\_3\]) with the Friedman equation in standard cosmology reveals that $\Lambda$ plays the same role as the usual cosmological constant. Since we did not include this additional constant in our Lagrangian right from the beginning $\Lambda$ might be termed [*induced*]{} cosmological constant. Now let us exploit the fact that we are allowed to set the constant $\Xi
\stackrel{A}{=}0$, which leads to an additional constraint, i.e. $$\mu =3p_{r}-8c\left( \frac{\zeta }{S}\right) ^{2}.
\label{xi_=_0_energy_and_stresses_relation}$$ Subsequently eq. (\[field\_final\_zeta\_XI\_<>\_0\_1\]) turns into $$\begin{aligned}
\chi \left( \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right)
-\left( a_{4}+a_{6}\right) &\kappa &\left( \left( \frac{\ddot{S}}{S}\right)
^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right]
^{2}\right) \nonumber \\
\quad \quad \quad = \kappa \left( p_{r}-4c\left( \frac{\zeta }{S}\right) ^{2}\right) ,
\label{field_final_zeta_XI_=_0_1}\end{aligned}$$ and the second Noether identity (\[field\_final\_zeta\_XI\_<>\_0\_4\]) now reads $$4\frac{\dot{S}}{S}\mu +\dot{\mu}\ =8c\left( \frac{\zeta }{S}\right)
^{2}\left( \frac{\dot{S}}{S}+\frac{\dot{\zeta}}{\zeta }\right) .
\label{field_final_zeta_XI_=_0_4}$$ Note that we collected all assumptions made up to this point in .
[@lll]{} Ansatz/Assumption&Resulting quantity/equation&Equation\
$\tau_{\alpha \beta}=0$&Affects the form of the second field equation&\
$T^{\alpha}=\frac{1}{2} Q \wedge \vartheta^{\alpha}$&Affects the form of the connection&\
$Q=\frac{\xi(t,r)}{S(t)}\vartheta^{\hat 0}$&Controls non-Riemannian features/&,,,\
&Affects the form of the field equations&-\
$Q=\frac{\zeta(t)}{S(t)}\vartheta^{\hat 0}$&Controls non-Riemannian features/&,,,\
&Simplifies field equations&-\
$a_4 \not = -a_6$&Affects the form of the second field equation&\
$\Xi=\Sigma^{\alpha}{}_{\alpha}=0$&Relation between $\mu$ and $p_r$&\
$\Lambda$&Affects the form of the field equations&–\
Solutions\[special\_case\_xi\_=\_zeta\_solutions\_section\]
===========================================================
$\Lambda \neq 0$ solutions\[special\_case\_xi\_=\_zeta\_solutions\_lambda\_<>\_0\_section\]
-------------------------------------------------------------------------------------------------
We are now going to solve eq. (\[field\_final\_zeta\_XI\_<>\_0\_3\]) for nonvanishing $\Lambda $. Note that eq. (\[field\_final\_zeta\_XI\_<>\_0\_3\]) does not depend on the relation between the energy density and pressure and therefore can be solved independently. This ODE, after a substitution, turns into a Bernoulli ODE, which in turn can be transformed into a linear equation. After this procedure we obtain two branches for the scale factor. They read as follows: $$S=\pm \frac{1}{\sqrt{2\Lambda }}\sqrt{e^{-\sqrt{2\Lambda }\, t}\left( 2 k
e^{\sqrt{2\Lambda }\, t}-\sqrt{2\Lambda }\varkappa_{1}e^{2\sqrt{2\Lambda
}\, t}+\sqrt{2\Lambda }\varkappa _{2}\right) }, \label{scale_factor_lambda<>0_1}$$ where $\varkappa_{1}$, and $\varkappa_{2}$ are constants. This solution for the scale factor is valid for all three possible choices of $k$. Let us now proceed by fixing the equation of state.
We will start with the most simple ansatz, i.e. with the introduction of an additional constant $w$ into the equation of state, which parameterizes the ratio of the energy density and the pressure in our model, $$w\,\,\mu (t)=p_{r}(t). \label{parameter_equation_of_state_1_ansatz}$$ Now let us derive the impact of (\[parameter\_equation\_of\_state\_1\_ansatz\]) on our set of field equations given by (\[field\_final\_zeta\_XI\_<>\_0\_1\])–(\[field\_final\_zeta\_XI\_=\_0\_4\]). Equation (\[xi\_=\_0\_energy\_and\_stresses\_relation\]) yields $$\mu =-\frac{8c}{1-3w}\left( \frac{\zeta }{S}\right) ^{2}.
\label{energy_dust_lambda_neq_0}$$ The field equations are now given by $$\begin{aligned}
&&\chi \left( \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right)
-\left( a_{4}+a_{6}\right) \kappa \left( \left( \frac{\ddot{S}}{S}\right)
^{2}-\left[ \left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right]
^{2}\right) \nonumber \\
&&\quad \quad \quad \quad =-\frac{4c\kappa }{3}\left( \frac{\zeta }{S}\right) ^{2}\left( \frac{1+3w}{%
1-3w}\right) , \label{parametrized_field_lambda_ansatz_1_0} \\
&&\chi \left( \Lambda +\frac{\ddot{S}}{S}\right) +\left( a_{4}+a_{6}\right)
\kappa\left( \left( \frac{\ddot{S}}{S}\right) ^{2}-\left[ \left( \frac{%
\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}\right] ^{2}\right) \nonumber \\
&&\quad \quad \quad \quad = 4c\kappa \left( \frac{\zeta }{S}\right) ^{2}\left( \frac{1-w}{1-3w}%
\right) , \label{parametrized_field_lambda_ansatz_1_1} \\
&&\frac{\ddot{S}}{S}+\left( \frac{\dot{S}}{S}\right) ^{2}+\frac{k}{S^{2}}
=\Lambda , \label{parametrized_field_lambda_ansatz_1_2} \\
&&\frac{24c\left( 1-w\right) }{3w-1}\left( \frac{\zeta }{S}\right) ^{2}\left(
\frac{\dot{S}}{S}+\frac{\dot{\zeta}}{\zeta }\right) =0.
\label{parametrized_field_lambda_ansatz_1_4}\end{aligned}$$ Equation (\[parametrized\_field\_lambda\_ansatz\_1\_4\]) has two non-trivial solution, namely $$\zeta =\frac{\iota }{S},\texttt{\rm\thinspace \thinspace with }\iota
=\texttt{\rm const,}\qquad \texttt{\rm and \ } \quad w=1. \label{parametrized_first_condition}$$ Solving the remaining field equations with respect to the first solution in eq. (\[parametrized\_first\_condition\]), we obtain constraints among the coupling constants which are summarized in (note that every set of parameters on the rhs corresponds to a solution of the field equations).
[@lll]{} $\zeta$&$S$&Additional constraints\
$\zeta=\frac{\iota}{S}$&$S$ from eq.(\[scale\_factor\_lambda<>0\_1\])&$\{a_{4}=a_{6}=c=\chi =0\},$\
&& $\{a_{4}=a_{6}=\iota =0,w=\frac{1}{3}\},$\
&&$\{a_{4}=a_{6}=\chi =0,w=1\},$\
&&$\{a_{4}=a_{6}=c=0,$ $w=\frac{1}{3}\}$,\
&& $\{a_{4}=-a_{6},\Lambda =0, w= \texttt{\rm const}\}$\
These solutions are not very satisfactory since they either lead to vanishing post-Riemannian quantities or, in case of $\chi =c=0$, to a restriction on the Lagrangian level.
Let us switch to another ansatz for the equation of state, namely $$w(t)\mu (t)=p_{r}(t). \label{parameter_equation_of_state_2_ansatz}$$ Thus, we introduced an additional function into the equation of state which controls the relation between the energy density and stresses in a *dynamical* way. The field equation which changes with respect to the set (\[parametrized\_field\_lambda\_ansatz\_1\_0\])–(\[parametrized\_field\_lambda\_ansatz\_1\_4\]), besides of the fact that $w$ is no longer a constant, is the Noether identity in eq. (\[parametrized\_field\_lambda\_ansatz\_1\_4\]), which now reads $$\begin{aligned}
\hspace{-1cm}&&-\frac{24c}{\left( 3w-1\right) ^{2}}\left( \frac{\zeta }{S%
}\right) ^{2}\left( \frac{\dot{S}}{S}\left( 1+3w^{2}-4w\right) +\frac{\dot{%
\zeta}}{\zeta }\left( 1+3w^{2}-4w\right) +\dot{w}\right) =0.
\label{parametrized_field_lambda_ansatz_2_4}\end{aligned}$$ In case of an arbitrary choice of $\zeta$, this equation is solved by $$w=\frac{S^{2}\zeta ^{2}\varkappa _{3}-1}{S^{2}\zeta ^{2}\varkappa _{3}-3}.
\label{solution_for_w_lambda<>0_case_ansatz_2}$$ Reinsertion of this solution for $w$ into the remaining field equations yields additional parameter constraints which are summarized in . As one realizes immediately, none of the solutions collected in is of use for us, since they all lead to unrealistic or forbidden restrictions among the coupling constants in our model. Therefore, in the following section, we will switch to the case in which the induced cosmological constant $\Lambda$ vanishes.
[@llll]{} $\zeta $ & $S$ & Additional constraints &\
$\zeta $ arbitrary, $w=\frac{\zeta ^{2}S^{2}\varkappa _{3}-1}{\zeta ^{2}S^{2}\varkappa _{3}-3}$ & $S$ from eq. (\[scale\_factor\_lambda<>0\_1\]) & $%
\{a_{4}=-a_{6},\Lambda =0\},$ &\
& & $\{a_{4}=-a_{6},\chi =0,w=1\},$ &\
& & $\{a_{4}=-a_{6},\iota =\chi =0\}$ &\
$\Lambda =0$ solutions\[special\_case\_xi\_=\_zeta\_solutions\_lambda\_=\_0\_section\]
--------------------------------------------------------------------------------------
Solving equation (\[field\_final\_zeta\_XI\_<>\_0\_3\]) for vanishing $\Lambda$, yields a solution for the scale factor which depends on the value of the constant $k$: $$\begin{aligned}
k &\not=&0:\quad S=\pm \sqrt{\frac{1}{k}\left( \varkappa _{1}-k^{2}\left(
\varkappa _{2}+t\right) ^{2}\right) }\, ,\quad \texttt{\rm with \quad }\varkappa
_{1},\varkappa _{2}=\texttt{\rm const,}
\label{special_vacuum_solution_scale_factor_k<>0} \\
k &=&0:\quad S=\varkappa _{1}\texttt{\rm or }S=\pm \,\sqrt{2\varkappa _{1}\left(
t+\varkappa _{2}\right) }\, ,\quad \texttt{\rm with\quad }\varkappa _{1},\varkappa
_{2}=\texttt{\rm const}.
\label{special_vacuum_solution_scale_factor_k=0}\end{aligned}$$ Motivated by the results in the previous section for the $\Lambda \not=0$ case, we will directly start with the more general equation of state as given in (\[parameter\_equation\_of\_state\_2\_ansatz\]). The field equations are now given by eqs. (\[parametrized\_field\_lambda\_ansatz\_1\_0\])-( \[parametrized\_field\_lambda\_ansatz\_1\_4\]) but with $\Lambda =0$. The parameter constraints for this solution are summarized in the second part of .
Additionally, we investigated the case in which made use of the old solution for $\zeta $, i.e. as given in (\[parametrized\_first\_condition\]). With this ansatz for $\zeta $ the second Noether identity, as given in eq. (\[parametrized\_field\_lambda\_ansatz\_2\_4\]), turns into: $$\dot{w}=0. \label{second_noether_for_old_zeta_=_iota_div_S_ansatz}$$ Thus, $w$ has to be a constant which, subsequently, can be determined from the remaining field equations after choosing the branch for $S$ from eqs. (\[special\_vacuum\_solution\_scale\_factor\_k<>0\])–(\[special\_vacuum\_solution\_scale\_factor\_k=0\]). The additional constraints for this parameter choice are listed in the first part of . Most interestingly it turns out that the parameter $w$, which controls the equation of state, is restricted by the choice of a certain set of constants in our theory (cf rhs in ).
[@llll]{} $\zeta $ & $S$ & Additional constraints &\
$\zeta =\frac{\iota }{S},$ $w=$ const & $k\not=0,$ $S$ from eq. (\[special\_vacuum\_solution\_scale\_factor\_k<>0\]) & $w=\frac{4c\iota ^{2}\kappa
+\varkappa _{1}\chi }{4c\iota ^{2}\kappa +3\varkappa _{1}\chi }$ &\
& $k=0,$ $S=$ const cf eq. (\[special\_vacuum\_solution\_scale\_factor\_k=0\]) & $w=1$ &\
& $k=0,$ $S$ from eq. (\[special\_vacuum\_solution\_scale\_factor\_k=0\]) & $w=%
\frac{4c\iota ^{2}\kappa +\varkappa _{1}^{2}\chi }{4c\iota ^{2}\kappa
+3\varkappa _{1}^{2}\chi }$ &\
$w=\frac{\zeta ^{2}S^{2}\varkappa _{3}-1}{\zeta ^{2}S^{2}\varkappa _{3}-3}$, $\zeta $ arbitrary & $k\not=0,$ $S$ from eq. (\[special\_vacuum\_solution\_scale\_factor\_k<>0\]) & $\varkappa _{3}=-4\frac{%
c\kappa }{\varkappa _{1}\chi }$ &\
& $k=0,$ $S=$ const cf eq. (\[special\_vacuum\_solution\_scale\_factor\_k=0\]) & $c=0$ &\
& $k=0,$ $S$ from eq. (\[special\_vacuum\_solution\_scale\_factor\_k=0\]) & $%
\varkappa _{3}=-4\frac{c\kappa }{\varkappa _{1}^{2}\chi }$ &\
(80,50) (-60,4) (-70,30) (-77,26) (-25,4)
(80,50) (-60,4) (-72,33) (-77,26) (-25,4)
(80,50) (-25,4) (-34,12) (-77,26) (-60,4)
(80,50) (-16,26) (-20,38) (-58,8) (-63,40)
(80,50) (-34,29) (-64,7) (-30,3) (-79,30)
(80,50) (-30,16) (-59,6) (-20,6) (-77,25)
(80,50) (-34,14) (-60,6) (-22,6) (-79,27)
(80,50) (-35,31) (-19,41) (-62,12) (-30,10)
Conclusion \[CONCLUSION\_SECTION\]
==================================
In the the last section we have shown that it is possible to find exact solutions of the field equations within our model. We were able to generate a rather broad class of solutions which allows for a flexible equation of state. We collected the resulting constraints of the parameters in our model in tables \[tabelle\_1\]–\[tabelle\_3\]. There seem to be [*no*]{} reasonable solutions in case of a non-vanishing induced cosmological constant $\Lambda$, unless one wants to introduce strong restrictions on the Lagrangian level. Thus, we are going to focus on the solutions with [*vanishing*]{} $\Lambda$ in the following.
In we plotted the scale factor for all three possible values of $k$ and for different values of the parameter $\varkappa_1$. As becomes clear from the plot at bottom right, we have three qualitatively different behaviours depending on the value of $k$. As in the Friedman case the collapsing scenario corresponds to a universe with positive spatial curvature. In we plotted the function $Q$ for the ansatz mentioned in equation . As stated before $Q$, the Weyl 1-form controls the non-Riemannian features of our model. From the plots it becomes clear that it is possible to construct models in which $Q$ vanishes at later times. Thus, the non-Riemmanian quantities [*die out*]{} with time. This is a rather desirable behaviour, since the spacetime we are living in nowadays seems to be a Riemannian one. At least all experiments carried out so far point into this direction [@ProceedingsHonnef]. Nevertheless our model is flexible enough to cope with both situations, i.e. if there is evidence for non-Riemannian structures at the present time, we are able to implement this fact by modifying our ansatz in and , respectively.
In comparison with the usual FLRW model of cosmology we still have three distinct cases for the evolution of the scale factor, which correspond to the three different choices for $k$ in the ansatz for the metric in equation (\[robertson\_walker\_coframe\]). Since one of our field equations (\[field\_final\_zeta\_XI\_<>\_0\_3\]) is very similar to the Friedman equation in standard cosmology we obtain a similar root type behaviour for the scale factor as displayed in figure \[fig\_scale\_factors\_Lambda\_0\]. As shown in (\[scale\_factor\_lambda<>0\_1\]) an induced cosmological constant leads to inflationary like solutions. In contrast to our old model [@PuetzTres] we were not able to find meaningful parameter constraints for this branch of the model (cf. tables \[tabelle\_1\] and \[tabelle\_2\]). This drawback might be relaxed in the future if we switch to another ansatz for the Weyl 1-form $Q$. Most interestingly the non-Riemannian quantities lead to a contribution to the total energy density of the universe as shown in (\[energy\_dust\_lambda\_neq\_0\]). Thus, the energy density $\mu$ is no longer a quantity which is determined by the evolution of the scale factor only, like in the FLRW scenario. As we will show in the next article of this series this contribution might be used to define a new energy density parameter which adds to the total energy budget of the universe. Thereby leading to an interesting new source for a possible dark energy component. Since the field equations differ from the Friedman equations one can expect several observational changes with respect to the standard FLRW model. Note that an ansatz with a position dependent Weyl 1-form at very early stages of the universe might contribute to the observed inhomogeneities in the cosmic microwave background. Although speculative at this time, small inhomogeneities in the new geometric quantities might also have served as seeds for structure formation at early stages, thereby yielding an interesting supplement to the quantities within the standard paradigm.
Let us summarize that the solutions found above contribute to the collection of known exact solutions in MAG, see [@Exact2]. Additionally, we managed to extend the model proposed in [@PuetzTres]. We provided the foundation for upcoming articles which will deal with the observational consequences of this new model. The most pressing task will be to look for realistic parameter choices in order to determine whether the model is in agreement with recent observational data. Within the next article of this series [@ExtweylObs], we will use the supernova data of Perlmutter [@Perlmutter] and Schmidt [@Schmidt] in order to constrain the free parameters in our model.
MAG in general\[MAG\_KAPITEL\]
==============================
In MAG we have the metric $g_{\alpha \beta }$, the coframe $\vartheta
^{\alpha }$, and the connection 1-form $\Gamma _{\alpha}{}^{\beta}$ (with values in the Lie algebra of the four-dimensional linear group $GL(4,R)$) as new independent field variables. Here $\alpha ,\beta ,\ldots =0,1,2,3$ denote (anholonomic) frame indices. Spacetime is described by a metric-affine geometry with the gravitational field strengths nonmetricity $%
Q_{\alpha \beta }:=-Dg_{\alpha \beta }$, torsion $T^{\alpha }:=D\vartheta
^{\alpha }$, and curvature $R_{\alpha }{}^{\beta }:=d\Gamma _{\alpha
}{}^{\beta }-\Gamma _{\alpha }\,^{\gamma }\wedge \Gamma _{\gamma }{}^{\beta
} $. A Lagrangian formalism for a matter field $\Psi $ minimally coupled to the gravitational potentials $g_{\alpha \beta }$, $\vartheta ^{\alpha }$, $%
\Gamma _{\alpha }{}^{\beta }$ has been set up in [@PhysRep]. The dynamics of an ordinary MAG theory is specified by a total Lagrangian $$L=V_{{\rm MAG}}(g_{\alpha \beta },\vartheta ^{\alpha },Q_{\alpha \beta
},T^{\alpha },R_{\alpha }{}^{\beta })+L_{{\rm mat}}(g_{\alpha \beta
},\vartheta ^{\alpha },\Psi ,D\Psi ).$$ The variation of the action with respect to the independent gauge potentials leads to the field equations: $$\begin{aligned}
\frac{\delta L_{{\rm mat}}}{\delta \Psi } &=&0, \label{matter} \\
DM^{\alpha \beta }-m^{\alpha \beta } &=&\sigma ^{\alpha \beta },
\label{zeroth} \\
DH_{\alpha }-E_{\alpha } &=&\Sigma _{\alpha ,} \label{first} \\
DH^{\alpha }{}_{\beta }-E^{\alpha }{}_{\beta } &=&\Delta ^{\alpha }{}_{\beta
}. \label{second}\end{aligned}$$ Equations (\[zeroth\]) and (\[first\]) are the generalized Einstein equations with the symmetric energy-momentum 4-form $\sigma^{\alpha \beta}$ and the canonical energy-momentum 3-form $\Sigma _{\alpha }$ as sources. Equation (\[second\]) is an additional field equation which takes into account other aspects of matter, such as spin, shear and dilation currents, represented by the hypermomentum $\Delta ^{\alpha }{}_{\beta }$. We made use of the definitions of the gauge field excitations, $$H_{\alpha }:=-\frac{\partial V_{{\rm MAG}}}{\partial T^{\alpha }},\quad
H^{\alpha }{}_{\beta }:=-\frac{\partial V_{{\rm MAG}}}{\partial R_{\alpha
}{}^{\beta }},\quad M^{\alpha \beta }:=-2\frac{\partial V_{{\rm MAG}}}{%
\partial Q_{\alpha \beta }}, \label{exications}$$ of the canonical energy-momentum, the metric stress-energy, and the hypermomentum current of the gauge fields, $$E_{\alpha }:=\frac{\partial V_{{\rm MAG}}}{\partial \vartheta ^{\alpha }}%
,\quad m^{\alpha \beta }:=2\frac{\partial V_{{\rm MAG}}}{\partial g_{\alpha
\beta }},\quad E^{\alpha }{}_{\beta }=-\vartheta ^{\alpha }\wedge H_{\beta
}-g_{\beta \gamma }M^{\alpha \gamma }, \label{gauge_currents}$$ and of the canonical energy-momentum, the metric stress-energy, and the hypermomentum currents of the matter fields, $$\Sigma _{\alpha }:=\frac{\delta L_{{\rm mat}}}{\delta \vartheta ^{\alpha }}%
,\quad \sigma ^{\alpha \beta }:=2\frac{\delta L_{{\rm mat}}}{\delta
g_{\alpha \beta }},\quad \Delta ^{\alpha }{}_{\beta }:=\frac{\delta L_{{\rm %
mat}}}{\delta \Gamma _{\alpha }{}^{\beta }}. \label{matter_currents}$$ Provided the matter equations (\[matter\]) are fulfilled, the following Noether identities hold: $$\begin{aligned}
D\Sigma _{\alpha } &=&\left( e_{\alpha }\rfloor T^{\beta }\right) \wedge
\Sigma _{\beta }-\frac{1}{2}\left( e_{\alpha }\rfloor Q_{\beta \gamma
}\right) \sigma ^{\beta \gamma }+\left( e_{\alpha }\rfloor R_{\beta
}{}^{\gamma }\right) \wedge \Delta ^{\beta }{}_{\gamma },
\label{noether_ident_1} \\
D\Delta ^{\alpha }{}_{\beta } &=&g_{\beta \gamma }\sigma ^{\alpha \gamma
}-\vartheta ^{\alpha }\wedge \Sigma _{\beta }. \label{noether_ident_2}\end{aligned}$$ They show that the field equation (\[zeroth\]) is redundant, thus we only need to take into account (\[first\]) and (\[second\]).
As suggested in [@Exact2], the most general parity conserving quadratic Lagrangian expressed in terms of the irreducible pieces of the nonmetricity $%
Q_{\alpha \beta }$, torsion $T^{\alpha }$, and curvature $R_{\alpha \beta }$ reads $$\begin{aligned}
V_{{\rm MAG}}=&\frac{1}{2\kappa }\biggl[ &-a_{0}R^{\alpha \beta }\wedge \eta
_{\alpha \beta }-2\lambda \eta +T^{\alpha }\wedge \,^{\star }\left(
\sum_{I=1}^{3}a_{I}\,^{(I)}T_{\alpha }\right) \nonumber \\
&&+Q_{\alpha \beta }\wedge \,^{\star }\left(
\sum_{I=1}^{4}b_{I}\,^{(I)}Q^{\alpha \beta }\right)\nonumber \\ &&+b_{5}\left(
^{(3)}Q_{\alpha \gamma }\wedge \vartheta ^{\alpha }\right) \wedge \,^{\star
}\left( \,^{(4)}Q^{\beta \gamma }\wedge \vartheta _{\beta }\,\right) \nonumber
\\
&&+2\left( \sum_{I=2}^{4}c_{I}\,^{(I)}Q_{\alpha \beta }\right) \wedge
\vartheta ^{\alpha }\wedge \,^{\star }T^{\beta }\biggr] \nonumber \\
-\frac{1}{2\rho }\, R^{\alpha \beta }\wedge \,^{\star
}&&\hspace{-0.5cm}\biggl[\sum_{I=1}^{6}w_{I}\,^{(I)}W_{\alpha \beta
}+\sum_{I=1}^{5}z_{I}\,^{(I)}Z_{\alpha \beta }+w_{7}\vartheta _{\alpha
}\wedge \left( e_{\gamma }\rfloor \,^{(5)}W^{\gamma }{}_{\beta }\right)
\nonumber \\
&&\hspace{-0.5cm}+z_{6}\vartheta _{\gamma }\wedge \left( e_{\alpha }\rfloor
\,^{(2)}Z^{\gamma }{}_{\beta }\right) +\sum_{I=7}^{9}z_{I}\vartheta _{\alpha
}\wedge \left( e_{\gamma }\rfloor \,^{(I-4)}Z^{\gamma }{}_{\beta }\right) \biggr].
\label{general_v_mag}\end{aligned}$$ The constants entering (\[general\_v\_mag\]) are the cosmological constant $%
\lambda $, the weak and strong coupling constant $\kappa $ and $\rho $[^5], and the 28 dimensionless parameters $$a_{0},\dots ,a_{3},b_{1},\dots ,b_{5},c_{2},\dots ,c_{4},w_{1},\dots
,w_{7},z_{1},\dots ,z_{9}. \label{general_coupling}$$ This Lagrangian and the presently known exact solutions in MAG have been reviewed in [@Exact2]. We note that this Lagrangian incorporates the one used in section \[TRESGUERRES\_KAPITEL\] eq. (\[Langrangian\_new\_symbolic\]), as can be seen easily by making the following choice for the constants in (\[general\_v\_mag\]): $$\lambda ,a_{1},\dots ,a_{3},b_{1},\dots ,b_{3},b_{5},c_{2},\dots,c_{4},w_{7},z_{1},\dots ,z_{3},z_{5},\dots ,z_{9}=0.\quad$$ In order to obtain exactly the form of (\[Langrangian\_new\_symbolic\]), one has to perform the additional substitutions: $$a_{0}\rightarrow -\chi ,\, w_{1},\dots ,w_{6}\rightarrow -2\rho
a_{1},\dots ,-2\rho a_{6},\, b_{4}\rightarrow c,\, z_{4}\rightarrow -2\rho b.$$ In table \[tabelle\_9\] we collected some of symbols defined within this appendix.
[@lllll]{} Potentials&Field strengths&Excitations&Gauge currents&\
&[$Q_{\alpha \beta }:=-Dg_{\alpha \beta}$]{}& $M^{\alpha
\beta}:=-2\frac{\partial V}{\partial Q_{\alpha \beta}}$ &$m^{\alpha \beta}:=2\frac{\partial V}{\partial g_{\alpha \beta}}$ &\
[$\vartheta^\alpha$]{}&[$T^{\alpha }\hspace{0.2cm}:=D\vartheta^\alpha$]{}& $H_\alpha\hspace{0.22cm}:=-\frac{\partial V}{\partial T^{\alpha}}$ & $E_\alpha\hspace{0.21cm}:=\frac{\partial V}{\partial \vartheta^{\alpha}}$ &\
[$\Gamma_{\alpha}{}^{\beta}$]{}&[$R_{\alpha}{}^{\beta
}:=$”$D$”$\Gamma_{\alpha}{}^{\beta}$]{}& $H^{\alpha}{}_{\beta}:=-\frac{\partial V}{\partial R_{\alpha}{}^{\beta}}$ & $E^{\alpha}{}_{\beta}:=\frac{\partial V}{\partial \Gamma_{\alpha}{}^{\beta}}$ &\
Weyl-Cartan spacetime\[WEYL\_CARTAN\_KAPITEL\]
==============================================
The Weyl-Cartan spacetime ($Y_{n}$) is a special case of the general metric-affine geometry in which the tracefree part $Q_{\alpha \beta
}\!\!\!\!\!\!\!\!\!\!\!\!\!\nearrow \,\,\,\,\,\,\,$ of the nonmetricity $Q_{\alpha \beta }\,$ vanishes. Thus, the whole nonmetricity is proportional to its trace part, i.e. the Weyl 1-form $Q:=%
\frac{1}{4}Q^{\alpha }{}_{\alpha }$, $$Q_{\alpha \beta }=g_{\alpha \beta }\,Q=\frac{1}{4}g_{\alpha \beta
}\,Q^{\gamma }{}_{\gamma }. \label{torsion_in_weyl_cartan_spacetime}$$ Therefore the general MAG connection reduces to $$\begin{aligned}
\Gamma _{\alpha \beta } &=&\frac{1}{2}dg_{\alpha \beta }+\left( e_{[\alpha
}\rfloor dg_{\beta ]\gamma }\right) \vartheta ^{\gamma }+e_{[\alpha }\rfloor
C_{\beta ]}-\frac{1}{2}\left( e_{\alpha }\rfloor e_{\beta }\rfloor C_{\gamma
}\right) \vartheta ^{\gamma } \nonumber \\
&&-e_{[\alpha }\rfloor T_{\beta ]}+\frac{1}{2}\left( e_{\alpha }\rfloor
e_{\beta }\rfloor T_{\gamma }\right) \vartheta ^{\gamma }+\frac{1}{2}%
g_{\alpha \beta }\,Q+\left( e_{[\alpha }\rfloor Q\right) \vartheta _{\beta ]}
\\
&=&\Gamma _{\alpha \beta }^{\{\,\}}-e_{[\alpha }\rfloor T_{\beta ]}+\frac{1}{%
2}\left( e_{\alpha }\rfloor e_{\beta }\rfloor T_{\gamma }\right) \vartheta
^{\gamma }+\frac{1}{2}g_{\alpha \beta }\,Q+\left( e_{[\alpha }\rfloor
Q\right) \vartheta _{\beta ]}. \label{Weyl_Cartan_Konnektion}\end{aligned}$$ Thus, it does not include any more a symmetric tracefree part. Let us now recall the definition of the material hypermomentum $\Delta _{\alpha \beta }$ given in (\[matter\_currents\]). Due to the absence of a symmetric tracefree piece in (\[Weyl\_Cartan\_Konnektion\]), $\Delta _{\alpha \beta }$ decomposes as follows $$\begin{aligned}
\Delta _{\alpha \beta } &=&\texttt{\rm antisymmetric piece + trace piece} \nonumber
\\
&=&\tau _{\alpha \beta }+\frac{1}{4}g_{\alpha \beta }\,\Delta =\tau _{\alpha
\beta }+\frac{1}{4}g_{\alpha \beta }\,\Delta ^{\gamma }{}_{\gamma } \nonumber
\\
&=&\texttt{\rm spin current + dilation current.} \label{Weyl_Cartan_hypermomentum}\end{aligned}$$ According to (\[Weyl\_Cartan\_hypermomentum\]) the second MAG field equation (\[second\]) decomposes into $$\begin{aligned}
dH^{\alpha }{}_{\alpha }-E^{\alpha }{}_{\alpha } &=&\Delta , \\
g_{\gamma \lbrack \alpha }DH^{\gamma }\,_{\beta ]}-E_{[\alpha \beta ]}
&=&\tau _{\alpha \beta },\end{aligned}$$ while the first field equation is still given by (\[first\]). Additionally, we can decompose the second Noether identity (\[noether\_ident\_2\]) into $$\begin{aligned}
\frac{1}{4}g_{\alpha \beta }\,d\Delta +\vartheta _{(\alpha }\wedge \Sigma
_{\beta )} &=&\sigma _{\alpha \beta }, \label{weyl_cartan_second_noether_1}
\\
D\tau _{\alpha \beta }+Q\wedge \tau _{\alpha \beta }+\vartheta _{\lbrack
\alpha }\wedge \Sigma _{\beta ]} &=&0. \label{weyl_cartan_second_noether_2}\end{aligned}$$ Thus, the first Noether identity (\[noether\_ident\_1\]) with inserted Weyl 1-form and hypermomentum reads $$D\Sigma _{\alpha }=\left( e_{\alpha }\rfloor T^{\beta }\right) \wedge \Sigma
_{\beta }-\frac{1}{2}\left( e_{\alpha }\rfloor Q\right) \sigma ^{\beta
}{}_{\beta }+\left( e_{\alpha }\rfloor R_{[\beta \gamma ]}{}\right) \wedge
\tau ^{\beta \gamma }+\frac{1}{4}\left( e_{\alpha }\rfloor R\right) \wedge
\Delta . \label{connection_weyl_cartan_final}$$ Finally, we note that in a $Y_{n}$ spacetime the symmetric part of the curvature $R_{(\alpha \beta )}=Z_{\alpha \beta }\,$, i.e. the strain curvature, reduces to the trace part $$Z_{\alpha \beta }=\frac{1}{4}g_{\alpha \beta }R=\frac{1}{4}g_{\alpha \beta
}R^{\gamma }{}_{\gamma }=\frac{1}{2}g_{\alpha \beta }\,dQ. \label{strain_curvature_in_weyl_cartan_general}$$
Differential geometric formalism\[OPERATIONS\_SECTION\]
=======================================================
We assume a connected $n$-dimensional differential manifold $Y_n$ as underlying structure throughout the paper. A vector basis of its tangent space $T_p Y_n$ is denoted by $e_\alpha$, which is dual (i.e. $e_\alpha \rfloor
\vartheta^\beta=\delta_\alpha^\beta$) to the basis $\vartheta^{\alpha}$ of the cotangent space $T_p^{\ast} Y_n$. A $p$-form $\Xi$ can be expanded with respect to this basis as follows $$\Xi=\frac{1}{p!}\, \Xi_{\beta_1 \dots \beta_p} \, \vartheta^{\beta_1} \wedge
\dots \wedge \vartheta^{\beta_p}.$$ Table \[tabelle\_8\] provides a rough overview of the operators used throughout the paper. For a more comprehensive treatment the reader should consult [@Nakahara] or section 3, and Appendix A of [@PhysRep].
[@lllll]{} Operation&Symbol&Input&&Output\
Exterior multiplication&$\wedge$&$p$-form $\wedge$ $q$-form&$\rightarrow$&$ (p+q)$-form\
Interior multiplication&$\rfloor$&vector $\rfloor$ $p$-form&$\rightarrow$&$ (p-1)$-form\
Exterior derivative&$d$&$d$ $p$-form&$\rightarrow$&$ (p+1)$-form\
Hodge star in a $n$-dimen. space&$^{\star}$&$^{\star} p$-form&$\rightarrow$&$ (n-p)$-form\
Units\[UNITS\_SECTION\]
=======================
In this work we made use of *natural units*, i.e. $\hbar =c=1$ (cf table \[tabelle\_6\]).
[@llll]{} & \[mass\] & \[time\] & \[length\]\
length$^{-1}$ & length$^{-1}$ & length & length\
Additionally, we have to be careful with the coupling constants and the coordinates within the coframe. In order to keep things as clear as possible, we provide a list of the quantities emerging throughout all sections in table \[tabelle\_7\].
[@lll]{} Quantities & *1* & *Length*\
Gauge potentials & $[g_{\alpha \beta }],[\Gamma _{\alpha }{}^{\beta }]$ & $%
[\vartheta ^{\alpha }]$\
Gauge field strengths & $[Q_{\alpha \beta }],[R_{\alpha \beta }]$ & $[T^{\alpha }]$\
Gauge field excitations & $[M^{\alpha \beta }],[H^{\alpha }{}_{\beta }]$ & $%
[H_{\alpha }]^{-1}$\
Gauge field currents & $[E^{\alpha }{}_{\beta }],[m^{\alpha \beta }]$ & $%
[E_{\alpha }]^{-1}$\
Matter currents & $[\Delta _{\alpha \beta }],[\sigma ^{\alpha \beta }],[\tau
_{\alpha \beta }]$ & $[\Sigma _{\alpha }]^{-1}$\
Coordinates & $[\theta ],[\phi ],[r]$ & $[t]$\
Functions & $[\xi (t,r)],[\zeta (t)]$ & $[S(t)],[\mu (t)]^{-\frac{1}{4}%
},[p_{r}(t)]^{-\frac{1}{4}}$\
& & $[p_{t}(t)]^{-%
\frac{1}{4}}$\
Miscellany & & $[\Sigma _{\alpha \beta }]^{-\frac{1}{4}}$\
Constants & $[\chi ],[b],[k],[a_{I}]$ & $[\kappa ]^{\frac{1}{2}},[\Lambda
]^{-\frac{1}{2}},[c_{I}]^{-\frac{1}{2}},[c]^{-\frac{1}{2}}$\
& & $[\Xi ]^{-\frac{1}{4}}$\
eq. (\[special\_vacuum\_solution\_scale\_factor\_k<>0\]) & & $[\varkappa_{1}]^{\frac{1}{2}},[\varkappa_{2}]$\
eq. (\[special\_vacuum\_solution\_scale\_factor\_k=0\]) & & $[\varkappa
_{1}],[\varkappa _{2}]$\
eq. (\[scale\_factor\_lambda<>0\_1\]) & & $[\varkappa _{1}],[\varkappa _{2}]$\
eq. (\[parameter\_equation\_of\_state\_1\_ansatz\]) & $[w]$ &\
eq. (\[parametrized\_first\_condition\]) & & $[\iota ]$\
eq. (\[solution\_for\_w\_lambda<>0\_case\_ansatz\_2\]) & & $[\varkappa _{3}]^{-%
\frac{1}{2}}$\
Note that $[d]=1$ and $[\,^{\star }]=$ length$^{n-2p}$, where $n=$ dimension of the spacetime, $p=$ degree of the differential form on which $^{\star }$ acts.
The author is grateful to F.W. Hehl and to the members of the gravity group at the University of Cologne for their support. Additional thanks go to an anonymous referee for useful suggestions concerning the comprehensibility of this work.
[9]{}
E.W. Kolb, Michael S. Turner: *The early universe.* Addison-Wesley (1990)
J.A. Peacock: *Cosmological physics.* Cambridge University Press (2000)
C. Lämmerzahl, C.W.F. Everitt, F.W. Hehl (eds.): *Gyros, clocks, interferometers $\dots$: Testing relativistic gravity in space* Springer, Lecture notes in physics: Vol. 562 (2001)
M. Nakahara: *Geometry, Topology and Physics.* Adam Hilger, Bristol (1990)
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D. Puetzfeld, R. Tresguerres: *A cosmological model in Weyl-Cartan spacetime.* Class. Quantum Grav. **18** (2001) 6677-693 Los Alamos e-Print Archive `gr-qc/0101050`
D. Puetzfeld: *A cosmological model in Weyl-Cartan spacetime: II. Magnitude-redshift relation* in preparation
J.D. McCrea: *Irreducible decompositions of the non-metricity, torsion, curvature and Bianchi identities in metric-affine spacetimes.* Class. Quantum Grav. 9 (1992) 553-568
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F.W. Hehl, A. Macías: *Metric-affine gauge theory of gravity: II. Exact solutions.* Int. J. Mod. Phys. D, Vol.**8**, No. 4 (1999) 399-416
R.W. Tucker, C. Wang: *Dark matter gravitational interactions* Class. Quantum Grav. 15 (1998) 933-954
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A.V. Minkevich, A.S. Garkun: *Isotropic cosmology in metric-affine gauge theory of gravity.* Los Alamos e-print Archive `gr-qc/9805007`
S. Perlmutter et al.: *Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae.* Astrophys. J. **517** (1999) 565
B.P. Schmidt et al.: *The high-z supernova search: Measuring cosmic deceleration and global curvature of the universe using type Ia supernovae.* Astrophys. J. **507** (1998) 46-63
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[^1]: Additional assumptions are marked with an ”$A$”. Note that we mark new relations, new with respect to [@PuetzTres], with an ”$N$”.
[^2]: Here we made use of $\eta ^{\alpha }:=\,^{\star }\vartheta ^{\alpha }$.
[^3]: Note that this function is *not* identical with the one used in ([@PuetzTres], eq. (24)). It has a slightly different meaning since we use it here directly in our ansatz for the nonmetricity.
[^4]: Formerly the term of the order $\xi ^{2}$ was controlled by the coupling constant $b$ (cf eq. (\[lag\_old\])), and ([@PuetzTres], (41)-(43)).
[^5]: $[\lambda ]=$length$^{-2}$, $[\kappa ]=$length$^{2}$, $[\rho]=[\hbar ]=[c]=1.$
|
---
abstract: '*Objective:* The detection of epileptic seizures from scalp electroencephalogram (EEG) signals can facilitate early diagnosis and treatment. Previous studies suggested that the Gaussianity of EEG distributions changes depending on the presence or absence of seizures; however, no general EEG signal models can explain such changes in distributions within a unified scheme. *Methods:* This paper describes the formulation of a stochastic EEG model based on a multivariate scale mixture distribution that can represent changes in non-Gaussianity caused by stochastic fluctuations in EEG. In addition, we propose an EEG analysis method by combining the model with a filter bank and introduce a feature representing the non-Gaussianity latent in each EEG frequency band. *Results:* We applied the proposed method to multichannel EEG data from twenty patients with focal epilepsy. The results showed a significant increase in the proposed feature during epileptic seizures, particularly in the high-frequency band. The feature calculated in the high-frequency band allowed highly accurate classification of seizure and non-seizure segments \[area under the receiver operating characteristic curve (AUC) = 0.881\] using only a simple threshold. *Conclusion:* This paper proposed a multivariate scale mixture distribution-based stochastic EEG model capable of representing non-Gaussianity associated with epileptic seizures. Experiments using simulated and real EEG data demonstrated the validity of the model and its applicability to epileptic seizure detection. *Significance:* The stochastic fluctuations of EEG quantified by the proposed model can help detect epileptic seizures with high accuracy.'
author:
- 'Akira Furui${}^{*}$, Ryota Onishi, Akihito Takeuchi, Tomoyuki Akiyama${}^{*}$, and Toshio Tsuji${}^{*}$, [^1] [^2] [^3] [^4]'
bibliography:
- 'ref.bib'
title: 'Non-Gaussianity Detection of EEG Signals Based on a Multivariate Scale Mixture Model for Diagnosis of Epileptic Seizures'
---
Electroencephalogram (EEG), stochastic model, stochastic fluctuation, multivariate scale mixture model, epileptic seizure, non-Gaussianity.
Introduction
============
is a chronic neurological disorder characterized by recurrent and unprovoked epileptic seizures caused by abnormal electric activities of brain neurons. It is reported that there are approximately 50 million epilepsy patients worldwide [@WHO]. Electroencephalogram (EEG) can record neuronal electric activities and is routinely utilized for the diagnosis of epilepsy. Especially prolonged EEG recording is a powerful tool to confirm an epileptic seizure when a patient manifests signs and/or symptoms that are difficult to judge from clinical observation or when there are no visible symptoms. It is important to recognize seizures early because prolonged seizures can lead to increased brain damage. However, the correct interpretation of EEG requires extensive training, and EEG specialists may not be readily available in emergency settings. Tools that aid non-expert clinicians in interpreting EEG can eventually improve neurological outcomes through early recognition and treatment of seizures.
Numerous studies have attempted to detect epileptic seizures quantitatively using EEGs [@Acir2005; @Subasi2005; @Nishad2020-jx; @Alotaiby2015-bg; @kellaway1979precise; @Deburchgraeve2008; @Greene2008; @Sharma2020-ko; @Sharma2020-rg; @mooij2020skew; @Srinivasan2007-yw; @Ocak2009-rh; @Arunkumar2018-cq]. In these studies, EEGs are generally processed by two steps: frequency decomposition and feature extraction. First, frequency decomposition divides EEGs into sub-bands to obtain frequency components that strongly reflects the effects of epileptic seizures. Filter bank techniques based on wavelet transform [@Subasi2005; @Nishad2020-jx] and band-pass filters [@Alotaiby2015-bg] have been widely used in this step. In the feature extraction step, features that characterize the difference between seizures and non-seizures are then calculated from the EEG sub-bands. Feature extraction plays an important role in detecting epileptic seizures, as the extracted features are generally used as input for machine learning-based classification schemes. In many cases, simple amplitude information is used to characterize EEGs during epileptic seizures [@kellaway1979precise; @Deburchgraeve2008; @Greene2008]. However, detecting epileptic seizures by evaluating only the magnitude of the amplitude in specified sub-bands is sometimes difficult because EEGs recorded during epileptic seizures will differ significantly with the patient’s age and epilepsy type. Therefore, in recent years, the relationship between epileptic seizures and amplitude-independent features, such as higher-order statistics [@Sharma2020-ko; @Sharma2020-rg; @mooij2020skew] and entropy [@Srinivasan2007-yw; @Ocak2009-rh; @Arunkumar2018-cq], has been studied by many researchers. Meanwhile, the probability distribution shapes of EEGs have also been studied as a feature independent of amplitude [@Saunders1963; @Campbell1967; @Weiss1973; @Charles1999; @Nurujjaman2009; @Gonen2012]. Previous studies typically assumed that EEG signals followed a steady Gaussian distribution [@Saunders1963; @Gonen2012]; however, after Campbell *et al.* statistically indicated that EEG signals had non-Gaussian properties [@Campbell1967], various studies examined the probability distribution shapes of EEG signals, focusing on their Gaussianity [@Weiss1973; @Charles1999; @Nurujjaman2009; @Gonen2012]. Although few previous studies investigated the relationship between epileptic seizures and the Gaussianity of EEG signals, they reported that EEG signals during epileptic seizures follow a non-Gaussian distribution [@Charles1999; @Nurujjaman2009]. However, these studies are only experimental reports, and currently, there is no theoretical framework for quantitatively addressing the non-Gaussianity of EEG signals. Building such a framework could lead to the development of a novel feature for detecting epileptic seizures.
In this study, we assume that non-Gaussianity in EEG signals is caused by stochastic fluctuations in amplitude, and propose a multivariate scale mixture model that can estimate these fluctuations from recorded EEG signals. In the proposed model, a scalp EEG signal recorded at a certain time using multichannel electrodes follows a multivariate Gaussian distribution, and its variance-covariance matrix can be considered as a random variable following an inverse Wishart distribution. Thereby, it allows the evaluation of the stochastic fluctuation of the variance-covariance matrix corresponding to the EEG amplitude. In addition, by combining this model with a filter bank, we develop an analysis method that can calculate a feature representing time-series stochastic fluctuations in each EEG frequency band. In the simulation experiment, parameter estimation based on the proposed method is conducted on artificial data, and the error between true values and estimated values observed when conditions are changed is discussed. In the analysis experiment, the proposed method analyzes EEG signals recorded during focal seizures. We also verify the goodness-of-fit of the proposed model and investigate the correspondence between (1) estimated stochastic fluctuations of EEG signals and (2) diagnoses by an epileptologist.
The reminder of this paper is organized as follows: Section II outlines the structure of the multivariate scale mixture model, the parameter estimation method, and the EEG analysis method considering frequency characteristics. Section III details the simulation experiments and EEG analysis experiments. Section IV presents the results of these experiments, and Section V provides related discussions. Finally, Section VI offers the conclusions of this study.
Methods
=======
Multivariate Scale Mixture Model of Scalp EEG Signals
-----------------------------------------------------
Fig. \[fig:model\] shows the stochastic relationship between the recorded EEG signal $\mathbf{x} \in \mathbb{R}^{D}$ ($D$ is the number of electrodes) and its variance-covariance matrix $\mathbf{\Sigma} \in \mathbb{R}^{D \times D}$ as a graphical model. In the proposed model, $\mathbf{x}$ is handled as a random variable that follows a multivariate Gaussian distribution with mean zero and a variance-covariance matrix $\mathbf{\Sigma}$. The variance-covariance matrix $\mathbf{\Sigma}$ is also a random variable for which the distribution is determined by the degrees of freedom parameter $\nu \in \mathbb{R}^+$ and the scale matrix parameter $\mathbf{\Psi} \in \mathbb{R}^{D \times D}$. Therefore, the variance-covariance matrix $\mathbf{\Sigma}$ is interpreted as a latent variable in the model.
![Graphical representation of the proposed multivariate scale mixture model, which describes the stochastic relationship between EEG signals and its variance-covariance matrix. The white nodes are random variables, and the black nodes are parameters to be estimated. In the model, an EEG signal $\mathbf{x} \in \mathbb{R}^{D}$ is handled as a random variable that follows a multivariate Gaussian distribution with mean vector $\mathbf{0}$ and a variance-covariance matrix $\mathbf{\Sigma} \in \mathbb{R}^{D \times D}$ which is also a random variable following the inverse Wishart distribution determined by the degrees of freedom $\nu \in \mathbb{R}^+$ and the scale matrix $\mathbf{\Psi} \in \mathbb{R}^{D \times D}$. The parameter $\nu$ represents stochastic fluctuations of the EEG signals. The variance-covariance matrix distribution parameters are estimated via marginal likelihood maximization from the recorded EEG signals.[]{data-label="fig:model"}](figure/furui1.eps){width="1.0\hsize"}
First, the conditional distribution of EEG signal $\mathbf{x}$ given $\mathbf{\Sigma}$ is expressed via the following multivariate Gaussian distribution with a mean vector of zero: $$\begin{aligned}
p(\mathbf{x}|{\mathbf{\Sigma}}) &= {\mathcal N}(\mathbf{x}|\mathbf{0}, {\mathbf{\Sigma}}) \nonumber\\
&= \frac{1}{(2\pi)^{\frac{D}{2}} |\mathbf{\Sigma}|^{\frac{1}{2}}} \mathrm{exp} \left[-\frac{1}{2}\mathbf{x}^\mathrm{T} {\mathbf{\Sigma}}^{-1} \mathbf{x}\right]. \label{eq:gauss_x} $$ The the variance-covariance matrix is assumed to obey an inverse Wishart distribution ${\mathcal {IW}}(\mathbf{\Sigma}; \nu, \mathbf{\Psi})$, which is known as a conjugate prior for the variance-covariance matrix of a multivariate Gaussian distribution [@t2006]: $$\begin{aligned}
p(\mathbf{\Sigma}) &= {\mathcal {IW}}(\mathbf{\Sigma};\nu,\mathbf{\Psi}) \nonumber\\
&= \frac{|\mathbf{\Psi}|^{\frac{\nu}{2}}}{2^{\frac{\nu D}{2}} \Gamma_D \left(\frac{\nu}{2}\right)} |\mathbf{\Sigma}|^{-\frac{\nu+D+1}{2}} \mathrm{exp} \left[-\frac{\mathrm{tr}(\mathbf{\Psi} \mathbf{\Sigma}^{-1})}{2}\right],\label{eq:p_sigma} $$ where $\nu$ and $\mathbf{\Psi}$ determine the inverse Wishart distribution and are referred to as the degrees of freedom and the scale matrix, respectively. Considering the marginal distribution of $\mathbf{x}$, $\mathbf{\Sigma}$ can be integrated out as follows: $$\begin{aligned}
p(\mathbf{x}) &= \int p(\mathbf{\Sigma})p(\mathbf{x}|\mathbf{\Sigma}) \mathrm{d}{\mathbf{\Sigma}} \nonumber \\
&= \int {\mathcal {IW}}(\mathbf{\Sigma}; \nu, {\mathbf{\Psi}}) {\mathcal N}(\mathbf{x}|\mathbf{0}, \mathbf{\Sigma}) \mathrm{d}{\mathbf{\Sigma}} \label{eq:marginal_x} \\ &= \int \frac{|{\mathbf{\Psi}}|^{\frac{\nu}{2}}}{2^{\frac{\nu D}{2}} \Gamma_D \left(\frac{\nu}{2}\right)} |\mathbf{\Sigma}|^{-\frac{\nu+D+1}{2}}\mathrm{exp} \left[-\frac{\mathrm{tr}(\mathbf{\Psi} \mathbf{\Sigma}^{-1})}{2}\right] \nonumber \\
&\quad \times\frac{1}{(2\pi)^{\frac{D}{2}} |\mathbf{\Sigma}|^{\frac{1}{2}}} \mathrm{exp} \left[-\frac{1}{2}\mathbf{x}^\mathrm{T} {\mathbf{\Sigma}}^{-1} \mathbf{x}\right] \mathrm{d}{\mathbf{\Sigma}} \nonumber \\
&= \frac{|{\mathbf{\Psi}}|^{\frac{\nu}{2}}}{2^{\frac{\nu D}{2}} (2\pi)^\frac{D}{2} \Gamma_D \left(\frac{\nu}{2}\right)} \int |\mathbf{\Sigma}|^{-\frac{\nu+D+2}{2}}\nonumber\\
&\quad \times \mathrm{exp} \left[-\frac{1}{2}\mathrm{tr}\{(\mathbf{\Psi} + \mathbf{x} \mathbf{x}^\mathrm{T})\mathbf{\Sigma}^{-1}\}\right] \mathrm{d}{\mathbf{\Sigma}} \nonumber \\
\label{eq:p_x}
&= \frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\frac{\nu-D+1}{2})} \frac{|\frac{1}{\nu-D+1} {\mathbf{\Psi}}|^{-\frac{1}{2}}}{\left[\pi(\nu-D+1) \right]^{\frac{D}{2}}} (1+\Delta)^{-\frac{\nu+1}{2}} ,$$ where $\Delta$ is the square of the Mahalanobis distance, as follows: $$\begin{aligned}
\Delta &= \mathbf{x}^\mathrm{T}{\mathbf{\Psi}}^{-1}\mathbf{x}.\end{aligned}$$ From (\[eq:gauss\_x\]) and (\[eq:marginal\_x\]), $p(\mathbf{x})$ is obtained by summing an infinite number of multivariate Gaussian distributions having different variance-covariance matrices, meaning it can be interpreted as the scale mixture of multivariate Gaussian distributions. As stated above, the marginal distribution of multidimensional EEG signals can be modeled based on a multivariate scale mixture distribution.
Parameter Estimation Based on Marginal Maximum Likelihood
---------------------------------------------------------
Let us consider the estimation of $\nu$ and $\mathbf{\Psi}$, given $N$ samples of EEG signals $\mathbf{X} = \{\mathbf{x}_n \in \mathbb{R}^{D}; n=1,2,\cdots,N \}$. The model parameters can be estimated by maximizing the marginal likelihood $p(\mathbf{X}) = \prod_{n=1}^{N} p(\mathbf{x}_n)$. However, obtaining the maximum likelihood solution of a marginal likelihood is generally complex, and thus it is difficult to optimize the solution analytically [@t2006]. Therefore, we conduct this optimization for $\nu$ and $\mathbf{\Psi}$ based on the expectation-maximization (EM) algorithm [@Models1998]. The EM algorithm is iterated via application of an expectation step (E-step) and a maximization step (M-step).
To simplify (\[eq:p\_x\]), we define the new parameters as: $$\begin{aligned}
\label{eq:eq6}
\nu&= \nu' + D - 1, \\
\label{eq:eq7}
\mathbf{\Psi}&= \nu' \mathbf{\Psi}'.\end{aligned}$$ Accordingly, the marginal distribution can be expressed as $$\begin{aligned}
\label{eq:eq8}
p(\mathbf{x}_n) &=\int \mathcal{IW}(\mathbf{\Sigma}_n; \nu'+D-1, \nu' \mathbf{\Psi}') \mathcal{N}(\mathbf{x}_n|\mathbf{0}, \mathbf{\Sigma}_n) \mathrm{d} {\mathbf{\Sigma}_n} \\
\label{eq:eq9}
&=\frac{\Gamma(\frac{\nu'+D}{2})}{\Gamma(\frac{\nu'}{2})} \frac{|{\mathbf{\Psi}'}|^{-\frac{1}{2}}}{\left(\pi \nu' \right)^{\frac{D}{2}}} \left(1+\frac{\Delta '}{\nu '} \right)^{-\frac{\nu'+D}{2}},\end{aligned}$$ where $$\begin{aligned}
\Delta ' = \mathbf{x}_n^\mathrm{T} ({\mathbf{\Psi} '})^{-1} \mathbf{x}_n.\end{aligned}$$ Equation (\[eq:eq9\]) is equivalent to multivariate Student-$t$ distribution $\mathrm{St}(\mathbf{x}_n|\nu', \mathbf{\Psi}')$ [@t2006]. Then, we redefine the latent variable and replace (\[eq:eq8\]) with the following equivalent expression, thereby allowing an efficient calculation (refer to Appendix).
$$\label{eq:eq11}
p(\mathbf{x}_n) = \int \mathrm{IG}(\tau_n;\nu'/2,\nu'/2) \mathcal{N}(\mathbf{x}_n|\mathbf{0}, \tau_n \mathbf{\Psi}') \mathrm{d}{\tau_n},$$
where $\tau_n$ is a new latent variable following an inverse Gamma distribution $\mathrm{IG(\cdot)}$. The model parameters $\nu'$ and $\mathbf{\Psi}'$ are estimated by maximizing the marginal likelihood as outlined below.
1. Initialize each parameter by selecting arbitrary starting values.
2. *E-step*. Calculate the expectation of the complete-data log-likelihood, denoted as $Q(\nu',\mathbf{\Psi}')$. $$\begin{aligned}
Q(\nu',&\mathbf{\Psi}') \notag \\
&=\mathbb{E} \left[\ln{\prod_{n=1}^{N}\mathrm{IG}(\tau_n;\nu'/2,\nu'/2) \mathcal{N}(\mathbf{x}_n|\mathbf{0}, \tau_n \mathbf{\Psi}')}\right] \notag \\
&=\sum_{n=1}^{N} \biggl[-\frac{D}{2}\ln{(2\pi)}-\frac{D}{2}\mathbb{E}\left[\ln{\tau_n}\right]-\frac{1}{2}\ln{|\mathbf{\Psi}'|} \notag \\
&\quad-\frac{1}{2}\mathbb{E}\left[\tau^{-1}_n\right]\Delta'+\frac{\nu'}{2} \ln{\frac{\nu'}{2}} - \ln{\Gamma{\left(\frac{\nu'}{2}\right)}} \notag \\
&\quad- \left(\frac{\nu'}{2}+1\right)\mathbb{E}\left[\ln{\tau_n}\right]-\frac{\nu'}{2}\mathbb{E}\left[\tau^{-1}_n \right] \biggr],\end{aligned}$$ where $\mathbb{E}\left[\tau^{-1}_n\right]$ and $\mathbb{E}\left[\ln{\tau_n}\right]$ are derived by calculating the posterior distribution $p(\tau_n|\mathbf{x}_n)$ of the latent variable $\tau_n$ as follows: $$\mathbb{E}\left[\tau^{-1}_n\right]=\frac{\nu'+D}{\nu'+\Delta'},$$ $$\mathbb{E}\left[\ln{\tau_n}\right]=-\ln{\mathbb{E}\left[\tau^{-1}_n\right]}+\ln{\left(\frac{\nu'+D}{2}\right)}-{\psi}\left(\frac{\nu'+D}{2}\right),$$ where $\psi(\cdot)$ is a digamma function.
3. *M-step*. Update the parameters by maximizing $Q(\nu',\mathbf{\Psi}')$. By setting the derivative of $Q(\nu',\mathbf{\Psi}')$ with $\mathbf{\Psi}'$ equal to zero, the new scale matrix is obtained as
$${^\mathrm{new} \mathbf{\Psi}}' = \frac{1}{N} \sum_{n=1}^{N} \mathbb{E}\left[\tau^{-1}_n\right] \mathbf{x}_n \mathbf{x}_n^\mathrm{T}.$$
Because there is no closed-form expression for the degrees of freedom parameter $\nu'$, we estimate $\nu'$ by iteratively maximizing $Q(\nu',\mathbf{\Psi}')$ using the bisection method.
$${^\mathrm{new} \nu}' = {\mathop{\rm arg\ max}\limits}_{\nu'} Q(\nu', {^\mathrm{new} \mathbf{\Psi}}').$$
4. Evaluate the log-likelihood $\ln{p(\mathbf{X})}$ and repeat steps (ii)–(iv) until the calculation converges. Finally, estimated parameters $\nu'$ and $\mathbf{\Psi}'$ are transformed to variance-covariance matrix distribution parameters $\nu$ and $\mathbf{\Psi}$.
Using these procedures, the parameters of the proposed model can be estimated from the recorded EEG signals. Here, the parameter $\nu$ corresponds to the degrees of freedom of the multivariate Student-$t$ distribution parameter (refer to (\[eq:eq6\])); hence, $\nu$ controls the Gaussianity of the distribution. In the framework of the scale mixture model, the Gaussianity of EEG is considered to change owing to the stochastic fluctuation of the scale parameter (i.e., the variance-covariance matrix). Therefore, the fluctuation of the variance-covariance matrix can be evaluated by estimating $\nu$ from recorded EEG signals.
Proposed EEG Analysis Methods
-----------------------------
{width="0.75\hsize"}
Fig. \[fig:system\] shows the overall outline of the proposed analysis method. In the proposed method, observed EEG signals are decomposed into multiple frequency bands by using a filter bank consisting of parallel band-pass filters. Additionally, we estimate non-Gaussianity for signals in each frequency band based on the proposed multivariate scale mixture model.
First, the EEG signal at time $t$ recorded from the $D$ pair of electrodes is defined as $\mathbf{x}_t \in \mathbb{R}^D$. Then, $\mathbf{x}_t$ is divided into $M$ frequency bands ($b_1, \cdots, b_M$) by applying a filter bank consisting of a third-order Butterworth band-pass filter; the obtained signal is defined as $x_t^{(b_m)}$ ($m = 1,\cdots, M$). Second, parameter estimation for the variance-covariance matrix distribution based on the proposed model is performed on $\mathbf{x}^{(b_m)}_t$ for each frequency band, and $\nu$ characterizing the stochastic fluctuation is obtained. Because the characteristics of EEG signals change significantly in a time series, a sliding window of length $W$ (s) is applied to $\mathbf{x}^{(b_m)}_t$ in each band, and $\nu$ is estimated from the sample in the window following the procedure described in Section II-*B*. The time window is slid by $S$ (s) continuously, resulting in the estimation of $ \nu$ in a time series.
Here, $\nu$ is a parameter that controls the Gaussianity of EEG signals. The EEG distribution becomes closer to a Gaussian form as the value of $\nu$ approaches $\infty$; that is, the stochastic fluctuations decrease as $\nu$ increases. In this paper, we calculate $1/\nu$, which is the reciprocal of $\nu$, to provide intuitive meaning and to serve as a feature that characterizes the non-Gaussianity. The feature $1/\nu$ indicates that the larger its value, the larger the stochastic fluctuation of the EEG signal.
From the above, the stochastic fluctuations latent in each frequency band of EEG signals can be estimated continuously. It is also possible to obtain the spatial distribution of stochastic fluctuations by dividing the EEG electrode arrangement into multiple regions in advance and performing the above analysis for each region.
Experiments
===========
![International 10–20 electrode montage[]{data-label="fig:electrodes"}](figure/furui3.eps){width="0.5\hsize"}
Simulation
----------
To verify the accuracy of parameter estimation based on the proposed model, we performed a simulation experiment to evaluate the error rate between the true and estimated values of the parameters. Since the marginal distribution of the proposed model is equivalent to the multivariate Student-$t$ distribution, a random number sequence $\{\mathbf{x}_t \in \mathbb{R}^D; t = 1, \cdots, T\}$ following a Student-$t$ distribution $\mathrm{St}(\mathbf{x}_t|\nu'_0, \mathbf{\Psi}'_0)$ was generated. Here, the $\{\mathbf{x}_t \}$ values were regarded as a time series of an EEG signal recorded at a sampling frequency of $f_s$. The accuracy of the distribution estimation was verified by comparing the true values $\nu_0$ and $\mathbf{\Psi}_0$ with estimated values $\nu$ and $\mathbf{\Psi}$, after converting $\nu'_0$ and $\mathbf{\Psi}'_0$ to inverse Wishart distribution parameters based on (\[eq:eq6\]) and (\[eq:eq7\]). As an index of estimation accuracy for each parameter, the absolute percentage error was defined as ${|\nu_0 -\nu|}/{|\nu_0|}\times100$ and ${\|\mathbf{\Psi}_0-\mathbf{\Psi}\|_F}/{\|\mathbf{\Psi}_0\|_F}\times100$. $\|\mathbf{\Psi}\|_F$ is the Frobenius norm [@GeneHowardGolub2013] of $\mathbf{\Psi}$ and is obtained as $$\begin{aligned}
\|\mathbf{\Psi}\|_F=\sqrt{\sum_{i=1}^D \sum_{j=1}^D |{\psi_{ij}}|^2},\end{aligned}$$ where ${\psi_{ij}}$ is the element of $\mathbf{\Psi}$. In the estimation of each parameter, the first $W$ values of the signals $\{\mathbf{x}_t \}$ were used. The window length $W$ took values of 1, 2, 5, 10, 15, 20, 30, 50, and 100 s, and the number of dimensions $D$ took values of 1, 2, 4, 8, 16, and 19. The average absolute percentage errors were calculated by changing the true values of multivariate $t$-distribution parameters 400 times ($\nu'_0 = 0.5, 1.0, 1.5, \cdots, 10.0, {\psi'_0}_{ii} = 1.0, 2.0, 3.0, \cdots, 20.0$). Note that $\mathbf{\Psi}'_0$ was changed only in terms of its diagonal components ${\psi'_0}_{ii}$, and the off-diagonal components were fixed at 0.5. To evaluate the calculation cost of the proposed analysis method, the computation time required for each parameter estimation was measured simultaneously. The $T$ and $f_s$ values in artificial data generation were set as 100 s and 500 Hz. The computer used in the experiment was an Intel Core (TM) i7–6900K (3.2 GHz), 64.0 GB RAM.
EEG analysis
------------
Experimental analyses were conducted to evaluate the validity of the proposed model and the effectiveness of the proposed feature $1/\nu$ for epileptic seizure detection. Twenty epileptic patients with focal epilepsy participated in the experiments. Table \[table:conditions\] summarizes patient information, analysis times, and duration of the epileptic seizures. The EEG signals were recorded with a digital sampling frequency of 500 Hz using an electroencephalograph system (Neurofax EEG-1218, Nihon Kohden, Tokyo, Japan) while the patients were in the supine position. The 19-channel surface electrodes ($D=19$) were placed on the scalp according to the international 10–20 electrode system, with reference electrodes on both earlobes: A1 and A2 (see Fig. \[fig:electrodes\]). The experiments were approved by the Okayama University Ethics Committee (approval No: 1706-019). The onset and offset of a focal seizure in each EEG recording were marked by a board-certified epileptologist (T.A.).
--- -------- -------- ------------ --------------
Age Total data Seizure
(year) length (s) duration (s)
A Male 2 300 71
B Male 23 300 54
C Female 4 300 31
D Male 4 380 93
E Male 0.5 320 39
F Male 41 300 32
G Female 3 240 53
H Male 19 390 36
I Male 0.8 390 98
J Male 20 360 16
K Male 36 300 23
L Male 9 300 43
M Male 13 300 43
N Male 15 300 48
O Male 8 300 17
P Female 19 300 69
Q Male 27 420 62
R Male 38 300 65
S Male 17 300 17
T Male 19 300 65
--- -------- -------- ------------ --------------
: Patient conditions[]{data-label="table:conditions"}
First, the proposed model was fitted to the recorded EEG signals for all participants, and the feature $1/\nu$ was calculated. We set the frequency bands in the proposed method to $b_m \in\{\delta, \theta, \alpha,\beta, \gamma\}$, and decomposed EEG signals into $\delta$ (1–3 Hz), $\theta$ (4–7 Hz), $\alpha$ (8–12 Hz), $\beta$ (13–24 Hz), and $\gamma$ (25–100 Hz). These frequency bands are generally used to extract features of EEG signals [@ep1994]. Here, the fitting of the model and the calculation of $1/\nu$ were performed continuously using a sliding window with length $W$ = 15 s and sliding width $S$ = 1 s.
We then performed a goodness-of-fit test to validate the proposed model for EEG signals in each frequency band. As an evaluation index for the goodness-of-fit, we used the Bayesian information criterion (BIC) [@Schwarz1978], which balances the fitness and complexity of the model. $$\begin{aligned}
\mathrm{BIC} = -2 \mathrm{ln}L(\hat{\theta}) + k \mathrm{ln}(N_W).\end{aligned}$$ Here, $\mathrm{ln}L(\hat{\theta})$ is the log-likelihood of the model, $k$ is the number of estimated parameters in the model, and $N_W$ is the sample size in the sliding window. BIC was calculated for the fitting results for each sliding window. For comparison, BIC was also calculated for a multivariate Gaussian and Cauchy distribution models, having heavier tail than the Gaussian distribution.
Next, to verify the effectiveness of the evaluation of EEG non-Gaussianity based on the proposed method, we investigated whether epileptic seizures and non-seizures were classified accurately by using the calculated feature $1/\nu$ of each frequency band ($\delta$–$\gamma$ bands). The calculated $1/\nu$ results for each patient were divided into those obtained from seizure and non-seizure segments. Here, a non-seizure segment is defined as the period from the start of the recording to the onset of seizures. Non-seizure segments that include extraneous noise owing to unexpected body movements or electrode shifts in the original EEG signals were excluded from the analysis. In addition, because the sample size for calculating $1/\nu$ differed significantly between the seizure and non-seizure segments, the calculated $1/\nu$ in the non-seizure segment was randomly sampled based on the sample size for calculating $1/\nu$ in the seizure segment, to obtain uniform sample size for each patient.
As an evaluation index of classification performance, we calculated the area under the receiver operating characteristic curve (AUC) based on the receiver operating characteristic (ROC) analysis. The AUC is an evaluation scale calculated from the ROC curve plotting the relationship between the false positive and true positive rates. The closer the AUC value is to 1, the higher the classification performance.
For comparison, the AUC was calculated in the same manner using the three time-domain features conventionally used for EEG-based epileptic seizure detection: root mean square (RMS), absolute value of third-order cumulant ($|$ToC$|$), and approximate entropy (ApEn).
- RMS is a common feature that characterizes the amplitude information of EEGs [@Hamedi2014] and can be calculated as follows: $$ \mathrm{RMS} = \sqrt{\frac{1}{N_W} \sum_{i=1}^{N_W} (x_i)^2},$$ where $x_i$ is the EEG signal at an arbitrary electrode.
- ToC is a higher-order statistic and is used to investigate nonlinear variations in the signal. For zero-mean signals, the ToC is the same as the third-order moments (i.e., skewness), which is a measure of the asymmetry of the probability distribution [@Chua2010-qq]. The effectiveness of ToC in epileptic seizure detection has been reported by earlier works [@Sharma2020-ko; @Sharma2020-rg]. In this experiment, the absolute value of ToC, $|$ToC$|$, was calculated.
- ApEn proposed by Pincus [@Pincus1991] measures the regularity and unpredictability of fluctuations over a time-series signal. Smaller values of ApEn indicate strong regularity in a data sequence. The ApEn is also a commonly used feature for epileptic seizure detection, and ApEn of EEG is shown to decrease with epileptic seizures [@Srinivasan2007-yw; @Ocak2009-rh; @Arunkumar2018-cq]. The embedded dimension $m$ and vector comparison distance $r$ in ApEn were set to recommended values [@Pincus1991; @Richman2000]; $m = 2$ and $r = 0.2sd$, where $sd$ is the standard deviation of the data.
These conventional features were calculated continuously using a sliding window with the same settings as those used in calculating $1/\nu$. The frequency decomposition before the calculation of each feature was also conducted using the same filter bank ($\delta$–$\gamma$) as the proposed method. In addition, although the proposed feature is defined for an arbitrary set of electrodes, the conventional features are calculated for each electrode. Therefore, the conventional features were calculated for the Cz channel (see Fig. \[fig:electrodes\]) obtained from the top of the head. Other experimental conditions were the same as the simulation experiment.
![Examples of artificially generated EEG signals ($D$ = 4) with $\nu_0$ set to (a) $\nu_0 = 5.0$, (b) $\nu_0 = 8.0$, and (c) $\nu_0 = 13.0$; in each case, ${{\psi_0}_{ii}}$ was set to ${{\psi_0}_{ii}} = 10.0$, ${{\psi_0}_{ii}} = 20.0$, and ${{\psi_0}_{ii}} = 40.0$.[]{data-label="fig:sim_EEG"}](figure/furui4.eps){width="1.0\hsize"}
![Average absolute percentage errors and computation time for each combination of the number of input dimensions ($D$) and sliding window length ($W$) in the estimation of variance-covariance distribution parameters. (a) Average absolute percentage error for degrees of freedom $\nu$. (b) Average absolute percentage error for scale matrix $\mathbf{\Psi}$. (c) Average computation time.[]{data-label="fig:ES_param"}](figure/furui5.eps){width="0.75\hsize"}
Results
=======
In the simulation experiments, we generated time-series simulation data and verified the estimation accuracy of the distribution parameter. Fig. \[fig:sim\_EEG\] shows examples of time-series waveforms of $\mathbf{x}_t$, where the parameters of the inverse Wishart distribution used in artificial data generation were set as $\nu_0$ = 5.0, $\nu_0$ = 8.0, and $\nu_0$ = 13.0, and ${\psi_0}$ was changed to ${\psi_0}_{ii}$ = 10.0, ${\psi_0}_{ii}$ = 20.0, and ${\psi_0}_{ii}$ = 40.0 for each $\nu_0$. In the examples, the number of dimensions for artificial EEG signals is set to $D = 4$, and the signals for each dimension are shown. The vertical and horizontal axes indicate the signal values and time, respectively. Fig. \[fig:ES\_param\](a) and (b) show the average absolute percentage errors in the estimation of $\nu$ and $\mathbf{\Psi}$ resulting from changing the window length $W$ and the number of dimensions $D$. The average computation time for each estimation is shown in Fig. \[fig:ES\_param\](c).
In the EEG analysis experiments, we performed a goodness-of-fit test for the EEG signals and evaluated non-Gaussianity using the proposed feature. Table \[table:BIC\] lists the percentage of times that the BIC of each model became a minimum in the EEG signals of the $\delta$–$\gamma$ bands.
[llll]{} &\
(r)[2-4]{} [Frequency band]{} & [Proposed]{} & Gaussian & Cauchy\
$\gamma$ (25–100 Hz) & 99.28% & 0.21% \*\*\* &0.51% \*\*\*\
$\beta$ (13–24 Hz) & 99.05% & 0.59% \*\*\* &0.36% \*\*\*\
$\alpha$ (8–12 Hz) & 96.79% & 3.09% \*\*\* &0.12% \*\*\*\
$\theta$ (4–7 Hz) & 95.47% & 4.42% \*\*\* &0.11% \*\*\*\
$\delta$ (1–3 Hz) & 81.80% & 17.92% \*\*\* &0.28% \*\*\*\
\*\*\*: significant difference with the proposed model as indicated by a McNemar test with a Holm adjustment (*p* $<$ 0.001)
The table also lists the McNemar test results (significance level: 0.1%) adjusted by the Holm method using the proposed model as a control group. In all frequency bands, significant differences were observed between the proposed and other models ($p < 0.001$). In Fig. \[fig:Colormap\], color maps show the $1/\nu$ calculations corresponding to the raw EEG waveforms of patients A and B.
{width="0.9\hsize"}
In the color maps, $1/\nu$ was normalized to ensure that the maximum and minimum values of all frequency bands would be 1, and 0, respectively. The shaded areas in the waveforms and the areas surrounded by white dotted lines indicate epileptic seizure occurrences diagnosed by an epileptologist.
![Experimental $1/\nu$ distributions estimated from EEG signals. The *p*-value from the paired $t$-test and effect size *g* are also shown. (a) $\gamma$ band. (b) $\beta$ band. (c) $\alpha$ band. (d) $\theta$ band. (e) $\delta$ band.[]{data-label="fig:dens"}](figure/furui7.eps "fig:"){width="1.00\hsize"}
Fig. \[fig:dens\] shows the $1/\nu$ distribution in each frequency band for all patients, as calculated by kernel density estimation [@Parzen1962] for each seizure and non-seizure segment. The figure also shows the results of the paired $t$-test (significance level: 0.1%) and the effect size $g$ [@Hedges1981] between the $1/\nu$ distributions from seizure and non-seizure segments. The effect size $g$ is a statistical index indicating the magnitude of the difference between the mean values of the two distributions. In general, $0.2 \leq g < 0.5$ is interpreted as a small effect size, $0.5 \leq g < 0.8$ indicates medium effect size, and $0.8 \leq g$ indicates large effect size [@Cohen2013].
[lllll]{} &\
(r)[2-5]{} [Frequency band]{} & [$1/\nu$]{} & RMS &$|$ToC$|$ & ApEn\
$\gamma$ (25–100 Hz) & **0.881** & 0.878 &0.853 &0.552\
$\beta$ (13–24 Hz) & 0.727 & 0.822 &0.786 &0.548\
$\alpha$ (8–12 Hz) & 0.615 & 0.817 &0.773 &0.555\
$\theta$ (4–7 Hz) & 0.647 & 0.746 &0.717 &0.557\
$\delta$ (1–3 Hz) & 0.709 & 0.721 &0.757 &0.560\
Table \[table:AUC\] summarizes the AUCs of each feature based on the results of the ROC analysis for different frequency bands. The highest AUC for each feature was $1/\nu$ in the $\gamma$ band (0.881), RMS in the $\gamma$ band (0.878), $|$ToC$|$ in the $\gamma$ band (0.853), and ApEn in the $\delta$ band (0.560). The pairwise comparison for all AUCs in the table was performed using DeLong test with the Holm adjustment (significance level: 0.1%). The ROC curves obtained via seizure and non-seizure classification and the statistical test results for the AUCs are partially presented in Fig. \[fig:roc\] and \[fig:roc\_best\]. Fig. \[fig:roc\] shows the results for each frequency band of $1/\nu$. The AUC of $1/\nu$ in the $\gamma$ band was significantly higher than that of $1/\nu$ in the other bands. Fig. \[fig:roc\_best\] shows the results of each feature for the frequency band having the best AUC. The AUC of $1/\nu$ in the $\gamma$ band was significantly higher than that of $|$ToC$|$ and ApEn in the $\gamma$ and $\delta$ bands, respectively.
![Results of ROC analysis based on $1/\nu$ for each frequency band. (a) ROC curves for seizure and non-seizure segments. (b) Corresponding AUCs. Statistical test results using DeLong test for two correlated ROC curves with the Holm adjustment are also shown ($p<0.001$).[]{data-label="fig:roc"}](figure/furui8.eps){width="0.75\hsize"}
The confusion matrix for each feature obtained from the ROC analysis is shown in Fig. \[fig:conf\_mat\]. In the figure, each feature is for the band showing the best AUC. The rows and columns correspond to the epileptologist assessments (actual label) and the classification results based on ROC analysis (predicted label), respectively. The first column represents the true positives and false positives, and the second column represents the false negatives and true negatives.
![Results of ROC analysis based on each feature for the frequency band having the best AUC. (a) ROC curves for seizure and non-seizure segments. (b) Corresponding AUCs. Statistical test results using DeLong test for two correlated ROC curves with the Holm adjustment are also shown ($p<0.001$).[]{data-label="fig:roc_best"}](figure/furui9.eps){width="0.75\hsize"}
![Normalized confusion matrix given by ROC analysis. Each feature represents the band showing the best AUC. (a) $1/\nu$ in the $\gamma$ band. (b) RMS in the $\gamma$ band. (c) $|$ToC$|$ in the $\gamma$ band. (d) ApEn in the $\delta$ band. The diagonal number represents the normalized number of cases where the predicted label is equal to the actual label.[]{data-label="fig:conf_mat"}](figure/furui10.eps){width="0.88\hsize"}
Discussion
==========
In the simulation experiment, as the value of $\nu_0$ changes from $\nu_0 = 5.0$ to $\nu_0 = 13.0$, the outlier-like variance of the waveform does not appear frequently, and the waveform stabilizes (Fig. \[fig:sim\_EEG\]). This indicates that the distribution approaches a Gaussian distribution as $\nu$ increases because this parameter determines the Gaussianity of the distribution. It can also be seen that the amplitude of the artificial data increases as the value of ${\psi_0}_{ii}$ changes from ${\psi_0}_{ii}=10.0$ to ${\psi_0}_{ii}=40.0$. This is because $\psi_{ii}$, the diagonal component of $\mathbf{\Psi}$, is a parameter that characterizes the scale of the variance of EEG signals in each dimension.
The average absolute percentage errors in the estimation of $\nu$ and $\mathbf{\Psi}$ are approximately 2% at $W$ = 100 s, which indicates that the estimation is accurate (Fig. \[fig:ES\_param\]). However, the error rate increases as the window length $W$ decreases; the error rate is approximately 25% in $\nu$ and approximately 35% in $\mathbf{\Psi}$ at $W$ = 1 s. These results indicate that the estimation accuracy depends on the window length used for parameter estimation. When the number of dimensions was increased, the average absolute percentage error of $\nu$ decreased, and that of $\mathbf{\Psi}$ decreased initially and then increased again. Here, $\nu$ is a one-dimensional parameter defined for all input dimensions; therefore, a substantial increase in the number of dimensions is considered to increase the sample size used in the estimation of $\nu$, thereby decreasing the average absolute percentage error in the estimation. The scale matrix $\mathbf{\Psi}$ is affected by the estimation result of $\nu'$ because $\mathbf{\Psi} = \nu' \mathbf{\Psi}'$; therefore, the estimation error of $\mathbf{\Psi}$ is considered to decrease synergistically when the number of dimensions increases. Indeed, the estimation error decreased as the number of dimensions ranged from $D = 1$ to $D = 8$, whereas it increased as the number of dimensions ranged from $D = 8$ to $D = 16$. This is probably because the calculation of the average absolute percentage error was performed using the Frobenius norm. Even if the estimation accuracy for each element in the matrix does not change, the overall error increases as the number of elements increases with the number of dimensions.
From the above, it is found that as the window length increases, the estimation of $\nu$ and $\mathbf{\Psi}$ is performed more accurately. As the number of input dimensions increases, the estimation of these parameters is also performed more accurately; however, the estimation accuracy of $\mathbf{\Psi}$ decreases for higher-dimensional inputs. Additionally, assuming the use of an international 10–20 electrode montage (i.e., the input signal has 19 dimensions), the model parameters can be estimated with an error rate of 10% or less when the window length exceeds 5 s. In particular, setting the window length to 15 s can reduce the error rate to 5%.
In Fig. \[fig:ES\_param\](c), the computation time tends to increase with the increasing window length $W$ and number of dimensions $D$, and the maximum average computation time was approximately 2 s for $W=100$ s and $D=19$. However, even if all the channels in an international 10–20 electrode montage are used, it can be estimated in less than 0.5 s when $W$ is set to less than 20 s. Therefore, the estimation results could be obtained almost in real-time if the sliding width $S$ of the window is set appropriately. If the EEG is divided into multiple sub-bands using a filter bank, this calculation must be iterated for the number of sub-bands. However, since the parameter estimation in each band is completely independent, it can be estimated without loss of real-time performance by parallel computation using multiple threads. The EEG analysis experiment showed that the proposed model is selected the greatest number of times in all frequency bands based on BIC (Table \[table:BIC\]). However, the ratio of the minimum BIC of the multivariate Gaussian distribution model was 17.92% in the $\delta$ band, which was relatively higher than the others. One possible explanation is that the distribution shape for each window length differs significantly in the low-frequency bands. In addition, the number of parameters in the proposed model ($k = 191$) is slightly higher than that in the multivariate Gaussian distribution model ($k = 190$). Accordingly, when the signal fits both the proposed and Gaussian models almost equally, the simpler Gaussian model shows better goodness-of-fit owing to the nature of BIC. However, in the other frequency bands, the percentage at which the proposed model was selected based on the minimum BIC exceeds 95%. This is because the proposed model can change parameter $\nu$ to adapt to the shape of EEG distributions (i.e., Gaussianity) that change momentarily according to the state of brain activity. By contrast, the Gaussian and Cauchy models have only a certain Gaussianity, and these two models are included in the proposed model as special cases (Gaussian: $\nu \rightarrow \infty$, Cauchy: $\nu = D$). From the above, we can conclude that the proposed model is more suitable for EEG signals than the other models. In addition, the ratio of the minimum BIC of the proposed model increases as the frequency band becomes higher. This is because the EEG distribution changes depending on frequency characteristics, and those of high-frequency bands (at which signals change rapidly) are better fitted to the proposed model.
In Fig. \[fig:Colormap\], the $1/\nu$ in the high-frequency band is particularly large in the epileptic seizure segments. This can be confirmed from the results of all patients; therefore, the proposed method can quantitatively evaluate the change in non-Gaussianity during epileptic seizures as stochastic fluctuations. However, at approximately 220 s in Fig. \[fig:Colormap\](a), a partial increase in $1/\nu$ was observed in intervals other than epileptic seizure segments. This may have occurred because of artifacts in high-frequency bands, e.g., electromyograms caused by body movements. The proposed feature, which characterizes stochastic fluctuations, may increase to some extent owing to non-stationary noise components superimposed onto the EEG.
In Fig. \[fig:dens\], the $1/\nu$ distributions of all patients in each frequency band show that the $1/\nu$ of non-epileptic seizure segments is distributed in a region smaller than 0.05 regardless of the frequency band. By contrast, the $1/\nu$ of epileptic seizure segments is distributed more widely than that of non-epileptic seizure segments, and this tendency is most noticeable in the high-frequency $\gamma$ band. This can also be confirmed from the effect size $g$, which is the standardized mean difference between the two groups. It is considered that $1/\nu$ in the high-frequency band best reflects the characteristics of epileptic seizures because a large effect size was obtained in the $\gamma$ band. This indicates that EEG signals in the high-frequency band exhibited a strong non-Gaussianity owing to epileptic seizures. Previous studies reported that the activity in the $\gamma$ band, which is the high-frequency band of EEG signals, becomes intense during epileptic seizures [@Kobayashi2004; @Kobayashi2009; @Benedek2016]. The detailed mechanism of such $\gamma$ activity is still unknown; however, activities of inhibitory interneurons and electrical coupling through gap junctions were previously suggested [@Park2012]. These activities cause intermittent amplitude changes specific to epileptic seizures; consequently, the non-Gaussianity in the high-frequency band may be emphasized and $1/\nu$ increases.
For all features except ApEn, the maximum AUC was observed in the $\gamma$ band (Table. \[table:AUC\]). In particular, the proposed $1/\nu$ in the $\gamma$ band showed the highest AUC compared to the other features and frequency bands, indicating that it has the highest classification ability for seizures and non-seizures. This is also supported by the ROC curve and corresponding statistical test results shown in Fig. \[fig:roc\] and \[fig:roc\_best\]. Although no significant difference was found between $1/\nu$ and RMS, their differences are highlighted from a different perspective in Fig. \[fig:conf\_mat\]. From the results of the confusion matrices, the conventional features, including RMS, tended to have high specificity (accuracy of detecting non-seizures) but relatively low sensitivity (accuracy of detecting seizures). By contrast, the sensitivity of $1/\nu$ was higher than that of the conventional features, indicating that the proposed feature is superior in terms of accuracy in detecting epileptic seizures. Furthermore, the diagonal elements of the confusion matrix of $1/\nu$ were equal, which means that the proposed feature in the $\gamma$ band can classify seizures and non-seizures in a balanced manner. The confusion matrices having different characteristics in the proposed and conventional features, although with relatively close overall accuracy, suggests that these features reflect different aspects of EEG activity. Thus, combining these features could result in higher classification performances.
These results revealed that the proposed feature defined by the stochastic EEG model has a relatively better seizure classification performance than the amplitude-based and amplitude-independent features that have been conventionally validated for epileptic seizure detection.
Conclusion
==========
In this paper, EEG signals are modeled using a multivariate scale mixture model, allowing the representation of stochastic fluctuations in the variance-covariance matrix of EEGs. We also proposed an EEG analysis method by combining the scale mixture model and a filter bank and introduced a feature $1/\nu$ characterizing the non-Gaussianity. In this method, EEG signals are decomposed into several frequency bands, and time-series features for each frequency band were calculated based on the sliding window.
The simulation experiment evaluated the estimation accuracy of each parameter, which varied depending on the sample size and the number of dimensions. We demonstrated the proposed model to be most suitable in studying EEG signals that included epileptic seizures. In addition, high accuracy (AUC = 0.881) in classifying seizure and non-seizure segments was obtained by focusing on the proposed feature $1/\nu$ in the $\gamma$ band.
This study focuses on the feature extraction part of the epileptic seizure detection problem. Therefore, in addition to using a simple filter bank in the $\delta$–$\gamma$ bands for the frequency decomposition part, the classification of seizures/non-seizures was also performed by a simple threshold-based ROC analysis. Applying the proposed method to automated detection systems for epileptic seizures requires searching for more effective EEG sub-bands and introducing machine learning-based classification techniques. Furthermore, the proposed feature $1/\nu$ may be affected by artifacts such as electromyograms caused by body movement or stiffness owing to seizures. Hence, an algorithm that detects and removes these artifacts will be introduced in the future. We also plan to verify the effectiveness of the proposed method for epilepsy other than the focal seizures targeted in this study. This appendix shows the equivalent expressions of (\[eq:eq9\]) and (\[eq:eq11\]). Equation (\[eq:eq11\]) can be calculated as $$\begin{aligned}
p(&\mathbf{x}_n) \notag \\
&= \int \mathrm{IG}(\tau_n;\nu'/2,\nu'/2) \mathcal{N}(\mathbf{x}_n|\mathbf{0}, \tau_n \mathbf{\Psi}') \mathrm{d}{\tau_n} \nonumber\\
&= \int \frac{\left(\frac{\nu'}{2}\right)^{\frac{\nu'}{2}}}{\Gamma \left(\frac{\nu'}{2}\right)} (\tau_n)^{-\frac{\nu'}{2}-1} \mathrm{exp} \left[-\frac{1}{\tau_n} \left(\frac{\nu'}{2} \right) \right] \nonumber\\
&\quad\times\frac{1}{(2\pi)^{\frac{D}{2}}(\tau_n)^{\frac{D}{2}}|\mathbf{\Psi'}|^{\frac{1}{2}}} \mathrm{exp} \left[-\frac{1}{2}\mathbf{x}_n^\mathrm{T} (\tau_n\mathbf{\Psi'})^{-1} \mathbf{x}_n\right] \mathrm{d}{\tau_n} \nonumber\\
&=\frac{1}{(2\pi)^{\frac{D}{2}}}\frac{\left(\frac{\nu'}{2}\right)^{\frac{\nu'}{2}}}{\Gamma \left(\frac{\nu'}{2}\right)}\frac{1}{|\mathbf{\Psi'}|^{\frac{1}{2}}} \nonumber\\
&\quad\times \int (\tau_n)^{-\frac{\nu'+D}{2}-1} \mathrm{exp} \left[-\frac{1}{\tau_n} \left(\frac{\nu' + \Delta'}{2}\right) \right] \mathrm{d}{\tau_n} \nonumber \\
&=\frac{1}{(2\pi)^{\frac{D}{2}}}\frac{\left(\frac{\nu'}{2}\right)^{\frac{\nu'}{2}}}{\Gamma \left(\frac{\nu'}{2}\right)}\frac{1}{|\mathbf{\Psi'}|^{\frac{1}{2}}} \frac{\Gamma \left(\frac{\nu'+D}{2}\right)}{\left(\frac{\nu'+\Delta'}{2}\right)^{\frac{\nu'+D}{2}}}\nonumber\\
&\quad\times \int \frac{\left(\frac{\nu'+\Delta'}{2}\right)^{\frac{\nu'+D}{2}}}{\Gamma \left(\frac{\nu'+D}{2}\right)} (\tau_n)^{-\frac{\nu'+D}{2}-1} \nonumber\\
&\quad\times \mathrm{exp} \left[-\frac{1}{\tau_n} \left(\frac{\nu' + \Delta'}{2}\right) \right] \mathrm{d}{\tau_n} \nonumber\\
&=\frac{1}{(2\pi)^{\frac{D}{2}}}\frac{\left(\frac{\nu'}{2}\right)^{\frac{\nu'}{2}}}{\Gamma \left(\frac{\nu'}{2}\right)}\frac{1}{|\mathbf{\Psi'}|^{\frac{1}{2}}} \frac{\Gamma \left(\frac{\nu'+D}{2}\right)}{\left(\frac{\nu'+\Delta'}{2}\right)^{\frac{\nu'+D}{2}}}\nonumber\\
&\quad\times \int \mathrm{IG} \left(\tau_n;\frac{\nu'+D}{2},\frac{\nu'+\Delta'}{2} \right) \mathrm{d}{\tau_n}.
\label{eq:20}\end{aligned}$$ Here, the integral of the probability density function over the entire space is equal to 1. Hence, (\[eq:20\]) can be expressed in the same form as (\[eq:eq9\]). $$\begin{aligned}
p(\mathbf{x}_n) &= \frac{\Gamma(\frac{\nu'+D}{2})}{\Gamma(\frac{\nu'}{2})} \frac{|{\bm \Psi'}|^{-\frac{1}{2}}}{\left(\pi \nu' \right)^{\frac{D}{2}}} \left(1+\frac{\Delta '}{\nu '} \right)^{-\frac{\nu'+D}{2}}.\end{aligned}$$
[^1]: ${}^{*}$A. Furui, R. Onishi, and ${}^{*}$T. Tsuji are with the Graduate School of Engineering, Hiroshima University, Higashi-hiroshima, 739-8527 Japan (e-mail: [email protected]; [email protected]).
[^2]: A. Takeuchi is with the Department of Neonatology, Okayama Medical Center, National Hospital Organization.
[^3]: ${}^{*}$T. Akiyama is with the Department of Child Neurology, Okayama University Hospital (e-mail: [email protected]).
[^4]: Copyright (c) 2020 IEEE. Personal use of this material is per-mitted. However, permission to use this material for any other pur-poses must be obtained from the IEEE by sending an email to [email protected]. This paper is an accepted version for publication in IEEE Transactions on Biomedical Engineering.
|
---
author:
- 'Tetsuya Mitsudo[^1] and Shinji Takesue$^1$'
title: Numerical Estimation of the Current Large Deviation Function in the Asymmetric Simple Exclusion Process with Open Boundary Conditions
---
Introduction
============
Current, such as electric current, heat flow, or current of matter, plays important roles in nonequilibrium systems. The presence of macroscopic current is a signature of nonequilibrium states. Near equilibrium, the current is proportional to the conjugate external field and the linear coefficient is given by the autocorrelation function of the current in equilibrium states[@Kubo]. Beyond the linear regime, although we do not know much about general properties of the current in strongly interacting systems, current fluctuations are considered as key quantities to understand the dynamical behavior of nonequilibrium systems. In this respect, the large deviation property of the current is being studied intensively[@Spo; @Der07; @Tou10] in various systems. The large deviation function characterizes the probability of the time-integrated current in the long-time limit and its Legendre transform gives the cumulant generating function. Thus, it contains information about not only the mean and the variance but also higher-order cumulants, which are related to the nonlinear transport coefficients in the full counting statistics[@Sai08]. In the context of statistical physics, the time-integrated current called the Helfand moment is related to the Green-Kubo formula[@Hel60]. The shear viscosity and the thermal conductivity calculated with the Helfand moment in the recent simulations are in excellent agreement with those calculated with the Green-Kubo formula[@Vis07-1; @Vis07-2]. It is also suggested that the current large deviation is relevant to understanding the dynamical behavior of glasses[@Gar07].
The simple exclusion process (SEP) is probably the simplest and best studied model of nonequilibrium systems[@Schu01; @Schad10]. The SEP is defined on the one-dimensional lattice, where particles are allowed to move to one of the nearest neighbor sites when the site to move to is empty. If the leftward and rightward hopping rates are the same, the model is called the symmetric simple exclusion process (SSEP). If the hopping rates are different, the model is called the asymmetric simple exclusion process (ASEP). Two extreme cases of the ASEP have special names. When the difference is as small as the inverse of the system size, the model is referred to as the weakly ASEP (WASEP). The totally ASEP (TASEP) means that one of the hopping rates becomes zero. The ASEP is applied to various phenomena such as traffic flow[@Cho00], molecular motor transportation[@Nis05], the process of copying messenger RNA[@Mac68], and sequence alignment[@Bun02]. The ASEP in the infinite system with a step initial condition can be mapped to a model of surface growth, and recent experiments show that the height fluctuations of the growing surface in electroconvection, which correspond to the current fluctuations in the ASEP, obey the Tracy-Widom distribution function[@Tak10] that appear in the exact solution of the one-dimensional Karar-Parisi-Zhang equation[@Sas10-2].
For the ASEP with open boundary conditions, the stationary state is exactly calculated with the matrix product method[@Der93; @Sas98; @Uch00] or with the recursion relation[@Schu93]. In particular, it is clarified that the system has three phases, the low-density phase, the high-density phase, and the maximum-current phase, depending on the particle’s entering and exiting rates at the boundaries. The transition between the maximum-current phase and the other phases is second order and that between the low-density and high-density phases is first order. Thus, in the latter case, the two phases can coexist on the phase transition line, where a kink appears between the low-density region and the high-density region. The random walk motion of the kink contributes to the power-law behavior of the power spectrum[@Tak03; @Tak08].
The current large deviation functions for the SSEP with the open boundary condition were studied by Derrida et al.[@Der04; @Der07], who obtained the scaling form for the generating function. The TASEP[@Der98-1; @Der98-2] and the WASEP[@Pro09] on a ring, and the SSEP in the infinite system with the step initial condition[@Der09] were studied by applying the Bethe ansatz. The direct calculation of the probability of the number of particles passing through a certain site was performed on the TASEP and the WASEP in the infinite system with some special initial conditions,[@Sas08; @Sas10] and it turned out that different initial conditions lead to different types of scaling behavior. The current moments of the ASEP in the infinitely large system have recently been studied by Imamura and Sasamoto[@Ima11], who showed that the calculation of the $n$th moment is reduced to the problem of the ASEP with a system with particles less than or equal to $n$. The macroscopic fluctuation theory has been developed as a theory for the large deviation function[@Ber01; @Ber06; @Bod06; @Bod07]. It is based on the postulate that the large deviation function is characterized only by the mean and variance of the current. The macroscopic fluctuation theory is applied to the SSEP[@Bod04] and the WASEP[@Bod06]. In the ASEP, the joint probability of the current and the density is calculated for some restricted boundary conditions by using the matrix product method[@Dep05]. For the current through the boundary, the generating function in the ASEP with open boundary conditions where both input and output are allowed at each boundary is calculated by using the Bethe ansatz[@Gie11] for the low- and high- density phases, and the symmetry relation for the current is also discussed[@Gie06].
In addition to the exact calculations, many numerical studies have been carried out. A population Monte Carlo method[@Iba01] was devised to calculate the large deviation functions first for discrete time dynamics[@Gia06], and it was extended to continuous time[@Lec07]. By using these methods, the large deviation functions in the zero range process[@Rak08] and in the Kipnis-Marchioro-Pressuti (KMP) model[@Hur10], which is a simple model of heat conduction, were computed. The density matrix renormalization group (DMRG) method was also applied to obtain the current large deviation for the TASEP with open boundary conditions[@Gor09].
Despite these extensive studies and the importance of the applications, the current large deviation function in the ASEP with open boundary conditions has not yet been fully understood. Furthermore, the definition of the current varies in the studies. Some studies consider the current through a bond and other studies consider the total current, which is the sum of the former over all bonds. Although the averages of these two quantities are trivially equivalent, their fluctuations may behave differently.
In the present study, we numerically calculate the current large deviation function in the SEP with open boundary conditions and find several properties that have not been reported in the exactly solvable cases. The large deviation function shows a cusp near zero current when the asymmetry is large. In both the SSEP and the ASEP, the even part of the generating function depends linearly on the system size, while the odd part does not. Our finding indicating that the even-order cumulants are proportional to system size $L$ differs from that obtained in the preceding study,[@Bod04] where the $n$th-order cumulant is proportional to $L^{n-1}$. We compare the generating functions for the two definitions of the current and find that they coincide with each other in the direct evaluation using the largest eigenvalue method, while the population Monte Carlo method fails when the current through a boundary is employed. Thus, the use of the total current has an advantage in the Monte Carlo simulation. We estimate numerical errors in the Monte Carlo simulation using the symmetry relation and find that the deviation is usually small. However, the Monte Carlo simulation sometimes shows an unexpectedly slow convergence and produces no reliable results. In those cases, the difference between the largest and second largest eigenvalues of the modified Master equation decays faster than the exponential of the system size.
This paper is organized as follows. In $\S$2, we give a brief review for the SEP and the current large deviation function, and explain two numerical methods that we use in the present simulations involving the use of the largest eigenvalue of the modified Master equation and the population Monte Carlo method. We also give a short summary of the studies of the current large deviation for the SSEP and the ASEP under open boundary conditions. In the next section, we show our numerical results. In particular, we focus on the difference induced by different definitions of current. The convergence problem in the population Monte Carlo method is also discussed. The last section is devoted to discussion and conclusions.
Model and Algorithm
===================
ASEP and Master equation
------------------------
The SEP is a continuous-time Markov process defined on a one-dimensional chain. We denote $\tau_j=1$ if site $j$ is occupied by a particle and $\tau_j=0$ if the site $j$ is empty. The configuration of the system of size $L$ is described by the set $C=\{\tau_1,\tau_2,\dots,\tau_L\}$. A particle can hop to the left or right nearest-neighbor site if the destination site is empty. Namely, the particles are subject to the exclusion interaction. The hopping rate to the right is denoted by $p_r$ and that to the left by $p_l$. If site 1 is empty, a particle enters with rate $\a$, and if the site is occupied, the particle exits with rate $\g$. Similarly, if site $L$ is empty, a particle enters with rate $\d$, and if the site is occupied, the particle exits with rate $\b$.
Let us denote the probability for the system to be in configuration $C$ at time $t$ by $P(C,t)$. By using the transition rate from the configuration $C'=\{\t'_1,\dots,\t'_L\}$ to $C=\{\t_1,\dots,\t_L\}$, $W_{CC'}$, the time evolution of $P(C,t)$ is described by the Master equation $$\frac{{\rm d}}{{\rm d} t}P(C,t)
=\sum_{C'(\ne C)}\left[W_{CC'}P(C',t)-W_{C'C}P(C,t)\right].
\label{master}$$ In the ASEP, the transition rate is written as $$\begin{aligned}
W_{CC'} &=& w_1 \d_{\t_2\t'_2}\cdots\d_{\t_L\t'_L} \notag \\
& & +\d_{\t_1\t'_1}\cdots\d_{\t_{j-1}\t'_{j-1}}\sum_{j=1}^{L-1}w_{j,j+1}\d_{\t_{j+2}\t'_{j+2}}\cdots\d_{\t_L\t'_L}
\notag \\
& & +\d_{\t_1\t'_1}\cdots\d_{\t_{L-1}\t'_{L-1}}w_L,\end{aligned}$$ where $\d_{\t_i\t'_i}=(1-\t_i)(1-\t'_i)+\t_i\t'_i$, $$\begin{aligned}
w_1=\a(1-\t'_1)\t_1+\g\t'_1(1-\t_1),\\
w_L=\d(1-\t'_L)\t_L+\b\t'_L(1-\t_L),\end{aligned}$$ and $$\begin{aligned}
\lefteqn{w_{j,j+1}= p_r\t'_j(1-\t'_{j+1})(1-\t_j)\t_{j+1}}\notag\\
& & +p_l(1-\t'_j)\t'_{j+1}\t_j(1-\t_{j+1}).\end{aligned}$$ If we denote $r(C)=\sum_{C'(\neq C)}W_{C'C}$, eq. (\[master\]) is rewritten as $$\frac{{\rm d}}{{\rm d} t}P(C,t)
=\sum_{C'(\ne C)}W_{CC'}P(C',t)-r(C)P(C,t).
\label{master2}$$ Thus, $r(C)$ means the rate of transition from $C$ to any other configurations.
The time evolution of the system is decomposed into the set of one-particle movements that we call a path. Consider that the system starts with initial configuration $C_0$ at time $t_0$. The configuration changes from $C_{i-1}$ to $C_{i}$ when a particle changes its position or a particle enters or exits from the system at time $t_i$ ($i=1,\dots,K$), reaches $C_K$ at $t_{K}$, and remains in the same state until time $t$. Thus, the state at time $t$ is $C=C_K$. This path is represented as $(C,t;C_K,t_K;\dots;C_1,t_1|C_0,t_0)$. Now, we denote the probability that the transition from $C_{i-1}$ to $C_{i}$ occurs between time $t_i$ and $t_i+{\rm d}t_{i}$ for $i=1,2,\dots,K$ by $$P(C,t;C_K,t_K;\dots;C_1,t_1|C_0,t_0){\rm d}t_K\dots {\rm d}t_{2}{\rm d}t_{1}.$$ Then the path probability density $P(C,t;C_{K},t_{K};\dots;C_1,t_1|C_0,t_0)$ for $K\ge 1$ is written as $$\begin{aligned}
\lefteqn{ P(C,t;C_K,t_K;\dots;C_1,t_1|C_0,t_0)}\notag\\
&=&\delta_{CC_K}e^{-r(C_K)(t-t_K)}W_{C_KC_{K-1}}\notag \\
& & \cdots
e^{-r(C_1)(t_2-t_1)}W_{C_1C_0}e^{-r(C_0)(t_1-t_0)} \notag\\
&=&\delta_{CC_K}\exp\left[-\sum_{i=0}^{K}r(C_i)(t_{i+1}-t_{i})\right]
\prod_{j=0}^{K-1}W_{C_{j+1}C_j}, \end{aligned}$$ where $t_{K+1}$ should be understood as $t$. Because $K$ represents the number of one-particle movements, if $K=0$, $P(C,t;C_K,t_K;\dots;C_1,t_1|C_0,t_0)$ should be interpreted as representing the probability that no transitions occur at the time interval $[t_0,t]$, which equals $\delta_{CC_0}e^{-r(C_0)(t-t_0)}$. The relationship between the conditional probability $P(C,t|C_0,t_0)$ and the path probability density is given by $$\begin{aligned}
\lefteqn{ P(C,t|C_0,t_0)}\notag\\
&=&\sum_{K=0}^{\infty}\sum_{C_1,C_2,\dots,C_K}\int_{t_0}^{t}{\rm d}t_{K}
\int_{t_0}^{t_{K}}{\rm d}t_{K-1}\cdots\int_{t_0}^{t_2}{\rm d}t_1\notag\\
& & P(C,t;C_K,t_K;\dots;C_1,t_1|C_0,t_0).\end{aligned}$$ If we differentiate the above equation with respect to $t$, we reobtain the Master equation.
Large deviation function and generating function
------------------------------------------------
Let us denote the probability that the mean current takes the value $q$ as $$q=\frac{1}{t}\sum_{i=0}^{K-1} Q(C_{i+1},C_{i})$$ by $P(q)$. The microscopic current $Q(C_{i+1},C_i)$ takes the value $1(-1)$ when the particle moves forward (backward) during the configuration change from $C_i$ to $C_{i+1}$. A precise definition of the microscopic current is given in the next subsection. When $t$ is large, this probability asymptotically behaves as $$P(q)\simeq e^{-tf(q)},$$ where $f(q)$ is called the large deviation function. As a result of the law of large number, $f(q)$ must have a minimum at a certain value $q^*$ and satisfy $f(q^*)=0$. The Legendre transform of $f(q)$ is called the generating function, which we write $$\mu(\lambda)=\max_{q}\left[\lambda q-f(q)\right].$$ The generating function satisfies $\mu(0)=0$ that corresponds to $f(q^*)=0$. Similarly, the large deviation function is given by the Legendre transform of the generating function as $$f(q)=\max_{\l}\left[q\l-\m(\l)\right].$$ The generating function is also rewritten as $$\label{braket}
e^{\m(\l)t}=\bra e^{\l q t} \ket$$ for large $t$. This means that the generating function is expanded as $$\label{expand}
\m(\l)=\sum_{n=1}^{\infty}\frac{1}{n!}\m^{(n)}(0)\l^n=
\sum_{n=1}^{\infty}\frac{1}{n!}\bra q^n\ket_c t^{n-1}\l^n$$ with respect to the cumulants $\bra q^n\ket_c$, or the cumulants are generated as $$\bra q^n \ket_c=\frac{1}{t^{n-1}} \left. \frac{\partial^n \m(\l)}{\partial
\l^n}\right|_{\l=0},$$ which is the origin of the name of the function.
The expectation in the rhs of eq. (\[braket\]) is taken with respect to the path probability. Thus, we obtain $$\begin{aligned}
\lefteqn{ \bra e^{\l q t} \ket
= \sum_{K=0}^{\infty} \sum_{C_{K},\cdots,C_1,C}
\int_{t_0}^{t}{\rm d}t_{K}\int_{t_0}^{t_{K}}{\rm d}
t_{K-1}\cdots\int_{t_0}^{t_2} {\rm d}t_{1}}\notag \\
& &\d_{C_{K}C}
e^{\l\sum_{i=0}^{K-1} Q(C_{i+1},C_{i})} e^{-r(C_K)(t-t_K)}
\nonumber \\
& &
\prod_{i=0}^{K-1}W_{C_{i+1}C_{i}}e^{-r(C_i)(t_{i+1}-t_{i})}.\end{aligned}$$ If we introduce $W^{\l}_{CC'}=W_{CC'}e^{\l Q(C,C')}$ and $$\begin{aligned}
\lefteqn{P^{\l}(C,t|C_0,t_0)}\nonumber\\
&=&\sum_{K=0}^{\infty} \sum_{C_{K},\cdots,C_1}
\int_{t_0}^{t}{\rm d}t_{K}\int_{t_0}^{t_{K}}{\rm d}
t_{K-1}\cdots\int_{t_0}^{t_2} {\rm d}t_{1} \nonumber \\
& &\d_{C_{K}C}
e^{-\sum_{i=1}^{K}r(C_i)(t_{i+1}-t_{i})}
\prod_{j=0}^{K-1}W^{\l}_{C_{j+1}C_{j}},\end{aligned}$$ the generating function is obtained as $$\m(\l)=\lim_{t\to\infty}\frac{1}{t}\log\left[\sum_{C}P^{\l}(C,t|C_0,t_0)\right],
\label{mula}$$ or we may assume some initial distribution $P_{0}(C_0)$ and substitute $$P^{\l}(C,t)=\sum_{C_0}P^{\l}(C,t|C_0,t_0)P_{0}(C_0)$$ in place of $P^{\l}(C,t|C_0,t_0)$ in the rhs of eq. (\[mula\]). The time evolution of $P^{\l}(C,t)$ obeys the modified master equation $$\begin{aligned}
\frac{{\rm d}}{{\rm d} t}P^{\l}(C,t)=
\sum_{C'\ne C}W^{\l}_{CC'}P^{\l}(C',t)-r(C)P^{\l}(C,t).
\label{masterL}\end{aligned}$$ We note here that because $\sum_{C'(\ne C)}W^{\l}_{CC'}\ne r(C)$, this equation cannot be interpreted as an evolution of probability. Namely, $\sum_{C}P^{\l}(C,t)$ varies in time. In the vector form, eq. (\[masterL\]) is written as $$\frac{\rm d}{{\rm d}t}\mathsf{P}^{\l}(t)=\mathsf{W}^{\l}\mathsf{P}(t),$$ where $\mathsf{P}^{\l}(t)$ is the vector whose $C$th element is $P^{\l}(C,t)$, and the $\l$-modified transition matrix $\mathsf{W}^{\l}$ is defined as $$\begin{aligned}
\left(\mathsf{W}^{\l}\right)_{CC'} &=& \left\{
\begin{array}{l} W^{\l}_{CC'} \; \mbox{for} \;C\neq C' \\
-r(C) \; \mbox{for} \; C=C' \end{array}\right. .\end{aligned}$$ Because eq. (\[masterL\]) is linear, the solution is spectrally decomposed into $$P^{\l}(C,t)=\sum_{n}e^{\z_n t}\psi_n(C)\sum_{C'}\psi'_n(C')P_0(C'),$$ where $\z_n$ is the eigenvalue and $\psi'_n(C)$ and $\psi_n(C)$ are the corresponding left and right eigenvectors of $\mathsf{W}^{\l}$. In the long time limit, only the term with the largest eigenvalue $\z_{\mathrm{max}}$ is dominant, and accordingly the generating function becomes equal to $\z_{\mathrm{max}}$. Thus, we can estimate the generating function by numerically calculating the largest eigenvalue of the $\l$-modified transition matrix if exact calculations are not possible. This method is powerful but restricted to small systems because of memory limitations.
Definition of the microscopic current
-------------------------------------
There are two different definitions of current. One is the current through the boundary of the system. From the equation of continuity, this current determines changes of the amount of the conserved quantity in the system. The other is the total current given as the spatial integration of the current density. It is natural to use the total current in the linear response theory when we consider the response to a static and uniform external field or thermal force.
Both definitions appear in the literature of SEP. In the studies of current large deviations of the SEP with periodic boundaries, the total current is employed by Derrida and Lebowitz[@Der98-1], Derrida and Appert,[@Der98-2] and Prolhac and Mallick[@Pro09]. On the contrary, the current through the left boundary is used by Derrida et al. [@Der04; @Der07] or de Gier and Essler[@Gie06; @Gie11], who study the systems with open boundaries.
We let $Q_A$ denote the total current defined as $$\begin{aligned}
\label{currenta}
Q_A(C_{i+1},C_i) &=& \sum_{j=1}^{L-1} \left[ -\t^i_j(1-\t^i_{j+1})
(1-\t^{i+1}_j)\t^{i+1}_{j+1}\right.\notag \\
& & \mbox{}\left. +(1-\t^i_j)\t^i_{j+1}\t^{i+1}_j(1-\t^{i+1}_{j+1})
\right] \notag \\
& & \mbox{}-(1-\t^i_1)\t^{i+1}_1+\t^i_1(1-\t^{i+1}_1)\notag \\
& & +(1-\t^i_L)\t^{i+1}_L-\t^i_L(1-\t^{i+1}_L),\end{aligned}$$ and $Q_B$ denote the current through the left boundary as $$\begin{aligned}
\label{currentb}
Q_B(C_{i+1},C_i) &=&
-(1-\t^i_1)\t^{i+1}_1+\t^i_1(1-\t^{i+1}_1).\end{aligned}$$ For each current, we define the mean current $q_X$ ($X=A$ or $B$) as $$q_X=\frac{1}{t}\sum_{i=0}^{K-1}Q_X(C_{i+1},C_i),$$ and the generating function $\m_X(\l)$ as $$e^{\m_X(\l)}=\bra e^{\l q_X t}\ket.$$ We note that the positive direction of the currents (\[currenta\]) and (\[currentb\]) is defined as leftward in this paper. We use the character $q$ for a current if we do not need to specify $q_A$ or $q_B$, or for a current in general. Since the system goes to a stationary state in the long-time limit, we obtain $$\label{qAB}
\bra q_A\ket = (L+1) \bra q_B\ket.$$ Thus, the two definitions are equivalent with respect to expectation values except the trivial factor.
Our question here is whether the large deviation properties of the two currents appear the same or not. Theoretically, they are considered equivalent as follows[@referee]. If a path is fixed, most particles that enter the system through a boundary exit from either boundary and the contribution from such particles to $q_A$ is exactly $L+1$ times the contribution to $q_B$. The number of such particles is proportional to $t$. On the contrary, the particles initially present in the system or those remaining in the system at time $t$ contribute to $q_A$ and $q_B$ differently. However, the number of those particles should be $t$-independent and the contribution from those particles is negligible for a large time. Thus, $q_A$ and $q_B$ are supposed to be essentially the same. Despite such consideration, how they appear in the numerical experiment is a different problem. Actually, as will be clarified later, the use of the total current has an advantage in the Monte Carlo simulations.
Monte Carlo simulations
-----------------------
If the system size is so large that the direct evaluation of the largest eigenvalue is not possible, we can employ a Monte Carlo method. However, because $P_{\lambda}(C,t)$ is not a probability distribution function, conventional Monte Carlo methods cannot be used. In fact, an algorithm based on the diffusion Monte Carlo method is devised by Giardinà et al.[@Gia06] and extended to a continuous-time case by Lecomte and Tailleur[@Lec07]. In those algorithms, besides the conventional Monte Carlo dynamics, we introduce clones of configuration $C$. The clones are duplicated or pruned with the rate $y(C)=e^{(r_{\l}(C)-r(C))\D t}$, where $\D t$ is the Poisson-distributed waiting time and its distribution function is given as $$r_{\l}(C)e^{-r_{\l}(C)\D t},$$ where $$r_{\l}(C)=\sum_{C'\neq C}W^{\l}_{CC'}.$$ Thus, the mean waiting time is $1/r_{\l}(C)$. Since the number of clones must be limited in the computer, we actually need an additional algorithm to keep the number of clones constant. If the clone is to be pruned, we choose a random clone and copy over the pruned one, and if the clone is to be duplicated, we copy the clone on the randomly chosen clone.
The precise algorithm is as follows.
1. Set initial conditions for $N$ clones.
2. Choose which clone to evolve. Here, we name it $c^{\alpha}$. First, the clones are chosen orderly. After all the clones evolve in the initial step, the clone with the earliest time is chosen.
3. Calculate the transition probability $W^{\l}_{CC'}/r_{\l}(C)$, and let a transition take place. The time interval $\D t$ is chosen from the Poisson law.
4. Calculate $y(C)$ and set $y=[y(C)+\xi]$, where $\xi$ is a uniform random number in $0\leq\xi<1$ and $[x]$ is the integer part of $x$. If $y=1$, the clone $c^{\a}$ is preserved; if $y=0$, the clone $c^{\a}$ is erased and overwritten by another randomly chosen clone; and if $y>1$, $y-1$ clones are chosen randomly and overwritten with the clone $c^{\a}$.
Thus, the total number of clones is kept constant. We denote $y$ at the $i$th time step by $y_i$ and define $X_i=(N+y_i-1)/N$. Thus, the generating function is given by the following formula in the long-time limit:[@Lec07] $$\label{direct}
\m(\l)=\frac{1}{t}\ln(X_1\cdots X_K),$$ where $K$ is the number of state changes.
Lecomte and Tailleur[@Lec07] introduced another method called thermodynamic integration. Because the generating function is written as $$\m(\l)=\frac{1}{t}\ln\bra e^{q\l t}\ket,$$ its derivative is $$\m'(\l)=\frac{\bra qe^{q\l t}\ket}{\bra e^{q\l t}\ket}=\bar{q}^{\l},$$ which can be interpreted as the average value of $q$ with respect to the population of clones. Thus, obtaining $\bar{q}^{\l}$ numerically and integrating it from $0$ to $\l$, we obtain $$\m(\l)=\int_0^{\l}{\rm d}\l' \bar{q}^{\l'}.$$ This thermodynamic integration gives a smoother result than the direct evaluation (\[direct\]). In this study, we use both methods and adopt a better result.
Known results and the symmetry relations
----------------------------------------
There are some known results about the current large deviation for the SEP with open boundaries.
The cumulants of the current in the SSEP were calculated [@Bod04] under the conjecture of the additivity principle as $$\begin{aligned}
\label{bod04}
\bra q_b\ket &=& \frac{I_1}{L} \\
\bra q_b^2\ket_c &=& \frac{tI_2}{LI_1} \\
\bra q_b^3\ket_c &=& \frac{t^2}{L}\frac{3(I_3I_1-I_2^2)}{I_1^3} \\
\bra
q_b^4\ket_c &=& \frac{t^3}{L}\frac{3(5I_4I_1^2-14I_1I_2I_3+9I_2^3)}{I_1^5} \\
I_n &=& \int_{\r_b}^{\r_a}{\rm d}\r D(\r)\s(\r)^{n-1}.\end{aligned}$$ Here, the macroscopic parameters are $D(\r)=1$ and $\s(\r)=2\r(1-\r)$, and the boundary parameters are $\r_a=\frac{\a}{\a+\g}$ and $\r_b=\frac{\d}{\b+\d}$ . Note that all the cumulants are proportional to $1/L$.
Next, a fluctuation-theorem-like symmetry was found for the current large deviation function for the SSEP [@Der04; @Der07] as $$\label{symmetryfq}
f(q_B)-f(-q_B)=A_{B}q_B,$$ where $A_{B}$ is the constant written as $$\label{abssep}
A_{B}=\ln{\frac{\a\b}{\g\d}}.$$ De Gier and Essler[@Gie06] generalized this to the ASEP, where the same equation (\[symmetryfq\]) holds with the coefficient $A_{B}$ given as $$\label{abasep}
A_{B}= \ln{\frac{\a\b}{\g\d}}+(L-1)\ln{\frac{p_l}{p_r}}.$$ The coefficient $A_{B}$ was derived from the detailed balance condition given by Enaud and Derrida[@Ena04], which is of the form $$\label{dbalance}
\frac{\a\b}{\g\d}\left(\frac{p_l}{p_r}\right)^{L-1}=1.$$ In the same manner, we can further extend the symmetry relation for the mean total current $q_A$ as $$f(q_A)-f(-q_A)=A_{A}q_A,$$ where $$\label{aaasep}
A_{A}=\frac{1}{L+1}A_{B}.$$
We briefly review other studies related to our study. The WASEP has been well studied. Bodineau and Derrida [@Bod06] assumed the macroscopic fluctuation theory and extended the results on the WASEP to the strong asymmetry limit, where they estimated the boundary effect on the local Gaussian fluctuation. Depken and Stinchcombe[@Dep05] calculated the joint probability of the density and the current in the TASEP when $\g=\d=0$. Prolhac and Mallick[@Pro09] calculated the cumulants of the current in the WASEP with periodic boundary conditions using the Bethe ansatz.
Now, we see that the large deviation for the total current in the SEP with open boundary conditions has not been well studied.
Simulation Results
==================
Open boundary SSEP
------------------
![Generating function $\m_A(\l)$ calculated by the largest eigenvalue method when $L=10$ and those calculated by the Monte Carlo method when $L=50$ and $100$. The line is interpolated to bring clearness to the graph.[]{data-label="fig1"}](fig1.eps)
First, we show the results on the open boundary SSEP where the hopping rates are set as $p_l=p_r=1.0$. Figure \[fig1\] shows some examples of the generating function of $q_A$ calculated by the two methods; the largest eigenvalue method ($L=10$) and the Monte Carlo method ($L=50$ and $100$). The boundary parameters are chosen as $(\a,\b,\g,\d)=(0.1,0.2,0.9,0.8)$ just for illustration. The generating functions depict parabola-like curves, whose width decreases as the system size increases.
We fit $\m_A(\l)$ to the power series up to the sixth order using the least-squares method and examine the system size dependence of the coefficients, which are the cumulants of the current. The fitting is performed in the range $-2\le\l\le 2$. The first-order cumulant is fixed to the steady current. As seen from Fig. \[fig2\], the fitting is successfully performed in the entire range with $L=50$, whereas the fitting up to the second order significantly deviates from the simulation results in the range $|\lambda|\gtrsim 0.5$.
The system size dependence of some low-order cumulants is shown in Fig. \[fig3\], where the data of $7\le L\le 11$ are obtained from the largest eigenvalues and those of the larger sizes from the Monte Carlo simulations. The boundary parameters are $(\a,\b,\g,\d)=(0.1,0.1,0.1,0.1)$, $(0.1,0.1,0.9,0.9)$, $(0.1,0.3,0.8,0.5)$, and $(0.3,0.3,0.7,0.7)$. In every case, the cumulants of even order are proportional to the system size but those of odd order are not. Thus, the large deviation function cannot be estimated by the mean and the variance only; its characterization needs higher-order statistical quantities. However, we must note that the present fitting is not very reliable, because the result depends on the range of data and on the function to fit with. To obtain more accurate results for the higher-order cumulants, we need a wider range of data. In particular, the cumulants of odd order have relatively small magnitudes and fitting errors are significantly large. Thus, we could not determine how these cumulants really change with the system size.
![Comparison between results obtained by the Monte Carlo simulation with $L=50$ and fitting function of the 6th-order. The broken line shows the quadratic part of the 6th-order fitting function.[]{data-label="fig2"}](fig2.eps)
![2nd- through 6th-order cumulants predicted from the fitting of the generating functions given by the simulations. The data of $7\leq L\leq11$ are given by the largest eigenvalue and the other data are given by the Monte Carlo methods. The data are obtained using the parameters $(\a,\b,\g,\d)=(0.1,0.1,0.1,0.1)$, $(0.1,0.1,0.9,0.9)$, $(0.1,0.3,0.8,0.5)$, and $(0.3,0.3,0.7,0.7)$. []{data-label="fig3"}](fig3a.eps "fig:") ![2nd- through 6th-order cumulants predicted from the fitting of the generating functions given by the simulations. The data of $7\leq L\leq11$ are given by the largest eigenvalue and the other data are given by the Monte Carlo methods. The data are obtained using the parameters $(\a,\b,\g,\d)=(0.1,0.1,0.1,0.1)$, $(0.1,0.1,0.9,0.9)$, $(0.1,0.3,0.8,0.5)$, and $(0.3,0.3,0.7,0.7)$. []{data-label="fig3"}](fig3b.eps "fig:") ![2nd- through 6th-order cumulants predicted from the fitting of the generating functions given by the simulations. The data of $7\leq L\leq11$ are given by the largest eigenvalue and the other data are given by the Monte Carlo methods. The data are obtained using the parameters $(\a,\b,\g,\d)=(0.1,0.1,0.1,0.1)$, $(0.1,0.1,0.9,0.9)$, $(0.1,0.3,0.8,0.5)$, and $(0.3,0.3,0.7,0.7)$. []{data-label="fig3"}](fig3c.eps "fig:") ![2nd- through 6th-order cumulants predicted from the fitting of the generating functions given by the simulations. The data of $7\leq L\leq11$ are given by the largest eigenvalue and the other data are given by the Monte Carlo methods. The data are obtained using the parameters $(\a,\b,\g,\d)=(0.1,0.1,0.1,0.1)$, $(0.1,0.1,0.9,0.9)$, $(0.1,0.3,0.8,0.5)$, and $(0.3,0.3,0.7,0.7)$. []{data-label="fig3"}](fig3d.eps "fig:") ![2nd- through 6th-order cumulants predicted from the fitting of the generating functions given by the simulations. The data of $7\leq L\leq11$ are given by the largest eigenvalue and the other data are given by the Monte Carlo methods. The data are obtained using the parameters $(\a,\b,\g,\d)=(0.1,0.1,0.1,0.1)$, $(0.1,0.1,0.9,0.9)$, $(0.1,0.3,0.8,0.5)$, and $(0.3,0.3,0.7,0.7)$. []{data-label="fig3"}](fig3e.eps "fig:")
From our results, we conjecture that the generating function split into three parts as $$\label{threep}
\m_A(\l)=\bra q_A \ket \l+\m_{A,\mathrm{odd}}(\l)+\m_{A,\mathrm{even}}(\l).$$ The first term in the rhs represents the mean current, which does not depend on the system size for the SSEP and is proportional to the system size for the ASEP. The second term is the even part, which is proportional to the system size $L$. We are not certain how the odd part behaves.
![The function $(\m_A(\l)+\m_A(-\l))/(L+1)$ is plotted with the system sizes of $L=8(\times)$, $10(+)$, $20(\ast)$, $30(\Box)$, $40(\bigcirc)$, $50(\triangle)$, and $60(\triangledown)$.[]{data-label="fig4"}](fig4.eps)
We show the data of $(\m_A(\l)+\m_A(-\l))/(L+1)=\frac{2}{L+1}\m_{A,\mathrm{even}}(\l)$ when $L=8$, $10$, $20$, $30$, $40$, $50$, and $70$ in Fig. \[fig4\], which shows data collapse. This supports the conjecture that the even part of the generating function is proportional to the system size.
ASEP with the open boundary conditions
--------------------------------------
Now, we present the results on the open boundary ASEP, where the transition rates satisfy $p_l<p_r=1$, and observe whether there are any differences among phases. Because the high-density phase is equivalent to the low-density phase due to the particle-hole symmetry, we examine the remaining two phases (the low-density phase and the maximum-current phase) and the coexisting state of the low- and high-density phases. Figure \[fig5\] illustrates the generating function $\m_A(\l)$ in each phase of the ASEP with system size $L=10$ and $p_l=0.5$. We note that no corresponding analytical or numerical results are indicated in the literature. It is remarkable that any qualitative differences are not observed among the three states.
![Generating function $\m_A(\l)$ in the ASEP when $L=10$ and $p_l=0.5$. Crosses, pluses, and stars correspond to the coexisting phase ($(\a,\b,\g,\d)=(0.1,0.1,0.1,0.1)$), the maximum-current phase ($(0.9,0.9,0.1,0.1)$), and the low-density phase ($(0.1,0.9,0.9,0.1)$), respectively.[]{data-label="fig5"}](fig5.eps)
As in the SSEP, we have attempted to fit the data with the 6th-order polynomial, and the result for $L=50$, $p_l=0.5$, and $(\a,\b,\g,\d)=(0.9,0.9,0.1,0.1)$ is shown in Fig. \[fig6\]. The fitting seems to work well if a certain neighborhood of the minimum is excluded. We confirmed this property also in other phases. This flattening of the minimum has been reported also in a system with periodic boundary condition[@Pro09]. Clearly, the generating function cannot be assumed to be quadratic.
![Fitting of the generating function $\m_A(\l)$ with 6th-order function in the maximum-current phase. We may fit $\m_A(\l)$ well in the region outside the minimum, and this can be seen in other phases.[]{data-label="fig6"}](fig6.eps)
Similarly to the case of the SSEP, we may divide the generating function for the ASEP into three parts as in eq. (\[threep\]). We illustrate $(\m_A(\l)+\m_A(-\l))/(L+1)$ in Fig. \[fig7\] for various system sizes $L=8$, $10$, $20$, $30$, and $40$.
![The function $(\m_A(\l)+\m_A(-\l))/(L+1)$ is plotted with the system sizes of $L=8(\times)$, $10(+)$, $20(\ast)$, $30(\Box)$, and $40(\bigcirc)$, in the low-density phase (top), maximum-current phase (middle) and coexisting phase (bottom).[]{data-label="fig7"}](fig7a.eps "fig:") ![The function $(\m_A(\l)+\m_A(-\l))/(L+1)$ is plotted with the system sizes of $L=8(\times)$, $10(+)$, $20(\ast)$, $30(\Box)$, and $40(\bigcirc)$, in the low-density phase (top), maximum-current phase (middle) and coexisting phase (bottom).[]{data-label="fig7"}](fig7b.eps "fig:") ![The function $(\m_A(\l)+\m_A(-\l))/(L+1)$ is plotted with the system sizes of $L=8(\times)$, $10(+)$, $20(\ast)$, $30(\Box)$, and $40(\bigcirc)$, in the low-density phase (top), maximum-current phase (middle) and coexisting phase (bottom).[]{data-label="fig7"}](fig7c.eps "fig:")
In relatively small systems, the maximum current phase shows very good data collapse, while the degree of data collapse is not so good in other phases. In every case, however, the data seem to converge in the large-system limit. Thus, as in the SSEP, the even part of the generating function is proportional to the system size when it is sufficiently large.
The large deviation function is obtained from the generating function via the Legendre transformation. Figure \[fig8\] shows the plot for $L=10$, $(\a,\b,\g,\d)=(0.1,0.9,0.9,0.1)$, and $p_l=0.1,0.2,0.3,0.5$, all of which correspond to the low density phase, in the range $-4\leq q\leq4$. A blowup near the minimum of $f(q_A)$ is shown on left of Fig. \[fig8\]. We note that a cusp appears near $q=0$ and its sharpness increases as $p_l$ decreases. This is confirmed by the plot of the derivative of the large deviation function shown on right of Fig. \[fig8\]. The cusp means that the probability of positive current rapidly decreases with $L$. Such a cusp is seen also in other phases.
![Large deviation function given by the Legendre transform in the range $-5\leq q_A\leq5$. The parameters are $L=10$, $(\a,\b,\g,\d)=(0.1,0.9,0.9,0.1)$ (low-density phase), and $p_l=0.1$, $0.2$, $0.3$, and $0.5$. The derivative of the left figure is shown in the right figure.[]{data-label="fig8"}](fig8a.eps "fig:") ![Large deviation function given by the Legendre transform in the range $-5\leq q_A\leq5$. The parameters are $L=10$, $(\a,\b,\g,\d)=(0.1,0.9,0.9,0.1)$ (low-density phase), and $p_l=0.1$, $0.2$, $0.3$, and $0.5$. The derivative of the left figure is shown in the right figure.[]{data-label="fig8"}](fig8b.eps "fig:")
In the limit $p_l\rightarrow 0$, the particles do not move leftward except at the boundaries. Thus, the large deviation function is expected to diverge for $q_A>0$ in the limit. This is why the cusp is generated.
Comparing the two currents
--------------------------
Thus far, we have presented the simulation results for the total currents. In this subsection, we compare the characteristic functions of $q_A$ and $q_B$. First, we introduce the dimensionless current $q^N_X$ defined by $$q^N_X=\frac{q_X}{\bra q_X\ket},$$ where $\bra q_X\ket$ is the expectation, and then we write the generating function $\m^N_X(\l)$ of the current $q_X^N$. We compare $\m^N_A(\l)$ and $\m^N_B(\l)$ in the simulations.
The dimensionless current defined here is effective when $\bra q_X\ket$ is known. Fortunately, the steady current is exactly calculated for both the SSEP[@Der01] and the ASEP[@Uch00] as $$\label{qbssep}
\bra q_B\ket_{SSEP}=-\frac{\frac{\d}{\b+\d}
-\frac{\a}{\g+\a}}{L+\frac{1}{\a+\g}+\frac{1}{\b+\d}-1},$$ for the SSEP, and $$\bra q_B\ket_{ASEP}=\left(
\begin{array}{cc}
(p_l-1)\r_a(1-\r_a), & \mbox{low-density and coexisting phases} \\
(p_l-1)\r_b(1-\r_b), & \mbox{high-density phase} \\
\frac{p_l-1}{4} & \mbox{maximum-current phase}
\end{array}\right.$$ for the ASEP with $p_l<p_r=1$, where $\r_a$ and $\r_b$ are given as $$\r_a=\frac{1}{1+a} \qquad a=\frac{1-p_l-\a+\g+\sqrt{(1-p_l-\a+\g)^2+4\a\g}}{2\a}$$ and $$\r_b=\frac{b}{1+b} \qquad b=\frac{1-p_l-\d+\b+\sqrt{(1-p_l-\d+\b)^2+4\b\d}}{2\b}.$$ Henceforth, we omit the suffix ‘SSEP’ or ‘ASEP’ from the steady current $\bra q_X\ket$, for it can be easily understood from the context.
The simulation results from the method of largest eigenvalues are shown in Fig. \[fig9\].
![Plot of $\m^N_A(\l)$($\times$) and $\m^N_B(\l)$($+$) for three sets of parameters. We observe a clear coincidence of $\m^N_A(\l)$ and $\m^N_B(\l)$.[]{data-label="fig9"}](fig9a.eps "fig:") ![Plot of $\m^N_A(\l)$($\times$) and $\m^N_B(\l)$($+$) for three sets of parameters. We observe a clear coincidence of $\m^N_A(\l)$ and $\m^N_B(\l)$.[]{data-label="fig9"}](fig9b.eps "fig:") ![Plot of $\m^N_A(\l)$($\times$) and $\m^N_B(\l)$($+$) for three sets of parameters. We observe a clear coincidence of $\m^N_A(\l)$ and $\m^N_B(\l)$.[]{data-label="fig9"}](fig9c.eps "fig:")
It is clear that $\m^N_A(\l)$ and $\m^N_B(\l)$ coincide with each other.
However, the two large deviation functions show differences in the Monte Carlo simulation, as shown in Fig. \[fig10\]. They agree with each other only when $|\l|$ is very small and differ greatly outside the region.
![Plot of $\m^N_A(\l)$ and $\m^N_B(\l)$ for the SSEP and the ASEP. The inset shows the extended plot of the coinciding area.[]{data-label="fig10"}](fig10a.eps "fig:") ![Plot of $\m^N_A(\l)$ and $\m^N_B(\l)$ for the SSEP and the ASEP. The inset shows the extended plot of the coinciding area.[]{data-label="fig10"}](fig10b.eps "fig:")
This difference means that $\m^N_B(\l)$ is not appropriately calculated in the Monte Carlo simulation. One reason for this is the lack of statistics. Because $q_B$ counts only the flux at the left boundary, the number of such events is much smaller than that in the case of $q_A$. Another possibility is that the number of clones is not sufficiently large. However, no significant change has been seen in a simulation where we doubled the number of clones and the time duration, as seen in Fig. \[fig11\].
![Plot of $\m^N_B(\l)$ for different time step $K$ and clone number $N$ values.[]{data-label="fig11"}](fig11.eps)
Thus, the difference between the two currents can affect results of the Monte Carlo simulations, though they represent essentially the same physical quantity. The use of the total current $q_A$ has an advantage in this respect.
Symmetry relations
------------------
The symmetry relation $$\label{symrel}
f(q_A)-f(-q_A)=Aq_A$$ should hold in the SEP. First, we show the results for small system sizes obtained from the direct evaluation of the largest eigenvalue. We plot $f(q_A)-f(-q_A)$ and $A_Aq_A$ with $A_A$ given by eqs. (\[abasep\]) and (\[aaasep\]) in Fig. \[fig12\]. The parameter sets used are $(\a,\b,\g,\d)=(0.1,0.9,0.8,0.2)$ for the low-density phase that we denote by $\times$, $(0.1,0.1,0.1,0.1)$ for the coexisting phase that we denote by $+$, and $(0.9,0.9,0.1,0.1)$ for the maximum-current phase that we denote by $\ast$, and the other parameters $L=10$, $p_l=0.5$, and $p_r=1.0$ used are the same.
![Plot of $f(q_A)-f(-q_A)$ from the direct evaluation results represented by $\times,+,$ and $\ast$, and that of $A_Aq_A$ from the analytical result obtained using eq. (\[aaasep\]), which is shown by the solid line.[]{data-label="fig12"}](fig12.eps)
The solid line shows $A_Aq_A$. We see that the symmetric relation is well satisfied in this case.
Next, we consider the results of the Monte Carlo method. In this case, we used the thermodynamic integral to compute the generating function and obtained large deviation functions via the Legendre transform. In Fig. \[fig13\] we plot $f(q_A)-f(-q_A)$ for the SSEP and the ASEP. The parameter sets are given in the inset of the figure. A negligible deviation is observed for the SSEP, while a small deviation is apparent in the ASEP.
![Comparison of $f(q_A)-f(-q_A)$ values for the SSEP (top) and ASEP (the others). For the SSEP, the system sizes of $L=30$ and $50$ are plotted. For the ASEP, the plot of $L=30$ is given in the second figure and the plot of $L=50$ is given in the third figure. The parameters are given in the inset. The solid and dotted lines give the analytical results.[]{data-label="fig13"}](fig13a.eps "fig:") ![Comparison of $f(q_A)-f(-q_A)$ values for the SSEP (top) and ASEP (the others). For the SSEP, the system sizes of $L=30$ and $50$ are plotted. For the ASEP, the plot of $L=30$ is given in the second figure and the plot of $L=50$ is given in the third figure. The parameters are given in the inset. The solid and dotted lines give the analytical results.[]{data-label="fig13"}](fig13b.eps "fig:") ![Comparison of $f(q_A)-f(-q_A)$ values for the SSEP (top) and ASEP (the others). For the SSEP, the system sizes of $L=30$ and $50$ are plotted. For the ASEP, the plot of $L=30$ is given in the second figure and the plot of $L=50$ is given in the third figure. The parameters are given in the inset. The solid and dotted lines give the analytical results.[]{data-label="fig13"}](fig13c.eps "fig:")
The deviation increases with the system size. The error may stem from the insufficient number of clones or the resolution of the data, which affects the precision of the Legendre transform. For large system sizes, the number of necessary clones becomes so large that we could not achieve a higher accuracy. Thus, deviations in Fig. \[fig13\] should be interpreted as representing the precision and limitation of the present Monte Carlo simulations.
Convergence problem in the Monte Carlo method
---------------------------------------------
The Monte Carlo method sometimes presents non-negligible errors. It happens when the hopping asymmetry and the system size are large. For example, Fig. \[fig14\] illustrates the instability observed in the numerically obtained generating function. Here, we have chosen the parameters $L=50$, $(\a,\b,\g,\d)=(0.9,0.9,0.1,0.1)$ (maximum-current phase), and $p_l=0.2$. This figure shows characteristic fluctuations in the region $0.2<\l<1.5$. The true generating function should be a smooth or at least convex function. However, the obtained data does not satisfy the expected convexity.
![$\m(\l)$ when $L=50$, $(\a,\b,\g,\d)=(0.9,0.9,0.1,0.1)$ (maximum-current phase), and $p_l=0.2$.[]{data-label="fig14"}](fig14.eps)
To find the cause of the fluctuations, we investigated the initial condition dependence of our Monte Carlo simulation. In Fig. \[fig15\], we show how $t^{-1}\ln(X_1\cdots X_K)$ evolves with time in the cases of $\l=-1$ and $\l=1$ each with three different initial configurations, namely, the empty initial condition (all $\t_i=0$), the random initial condition ($\t_i$ is randomly chosen with probability 1/2), and the half-filled initial condition ($\t_i=1$ for $i<L/2$ and $\t_i=0$ for the other value of $i$). The other parameters used are the same as those indicated in Fig. \[fig14\]. Note that $\l=-1$ corresponds to the stable region and $\l=1$ the unstable region. The top of Fig. \[fig15\] shows the case $\l=-1$, where the three samples exhibit convergence to a common value at $t=10^4$. The bottom shows the case $\l=1$, where the convergence is very slow and the initial-condition dependence remains even at $t=10^5$. This slow convergence is considered to be related to the instability shown in Fig. \[fig14\].
![Time evolutions of $\m(-1)$ (top) and $\m(1)$ (bottom) when $L=50$ and $p_l=0.2$ starting from three initial conditions: empty, random, and half-filled.[]{data-label="fig15"}](fig15a.eps "fig:") ![Time evolutions of $\m(-1)$ (top) and $\m(1)$ (bottom) when $L=50$ and $p_l=0.2$ starting from three initial conditions: empty, random, and half-filled.[]{data-label="fig15"}](fig15b.eps "fig:")
The convergence property should be reflected in the spectrum of eigenvalues of the $\l$-modified transition matrix $\mathsf{W}^\l$. Thus, we calculated the second largest eigenvalue $\z_2$. In Fig. \[fig16\], we show the system size dependence of the gap $\z_{\mathrm{max}}-\z_2$ for several $\l$ values in the systems of sizes $L=6,\cdots,13$, where $\z_{\mathrm{max}}$ is the largest eigenvalue. The top figure shows the cases $\l=-0.1$, $0.0$, $0.4$, and $1.6$ on the log-log scale and the bottom figure shows the cases $\l=0.0$, $0.1$, $0.16$, $0.2$, and $0.24$ on the semi-log scale.
![$\z_{\mathrm{max}}-\z_2$ versus the system size $L=6,\cdots,13$. The top figure shows the cases $\l=-0.1$, $0.0$, $0.4$, and $1.6$ in the log-log scale and the bottom figure shows the cases $\l=0.0$, $0.1$, $0.16$, $0.2$, and $0.24$ in the semi-log scale. The dotted line in the bottom figure is a fitted exponential function.[]{data-label="fig16"}](fig16a.eps "fig:") ![$\z_{\mathrm{max}}-\z_2$ versus the system size $L=6,\cdots,13$. The top figure shows the cases $\l=-0.1$, $0.0$, $0.4$, and $1.6$ in the log-log scale and the bottom figure shows the cases $\l=0.0$, $0.1$, $0.16$, $0.2$, and $0.24$ in the semi-log scale. The dotted line in the bottom figure is a fitted exponential function.[]{data-label="fig16"}](fig16b.eps "fig:")
The figure shows that the magnitude of the gap is proportional to a certain negative power of $L$ in the stable region, whereas the gap decreases faster than the exponential decrease with $L$ in the unstable region. In the bottom of Fig. \[fig16\], we see that the system size dependence of $\z_{\mathrm{max}}-\z_2$ changes from the power to exponential and finally to stretched exponential-like. Thus, one reason for the instability of the Monte Carlo calculation found here is considered to be that $\z_2$ becomes close to $\z_{\mathrm{max}}$ as $L$ increases. This behavior suggests that a certain type of phase transition may occur as reported recently[@Rak08; @Tch10].
Discussion and conclusions
==========================
Our simulation results show $\m_A^N(\l)=\m_B^N(\l)$, which implies $$\bra q_A^n \ket_c\sim L^n \bra q_B^n\ket_c.$$ From the calculation given by Bodineau and Derrida for the SSEP[@Der04], we may assume $\bra q_B^n\ket_c\sim\frac{1}{L}$, and thus $\bra q_A^n\ket_c\sim L^{n-1}$. This relation does not coincide with our result $\bra q_A^{2n} \ket_c\sim L$. These conflicting results suggest the necessity of further study. The calculation by Bodineau and Derrida assumes the additivity principle, the scaling relation $f(q_B)\sim g(q_BL)/L$, and the large deviation locally defined to be quadratic. Our results do not support the scaling relation and the Gaussian property of the local large deviation function. Both assumptions may need revision. However, our results have some drawbacks. The estimation by Bodineau and Derrida is given in the entire parameter region, while our simulation is limited to several sets of parameters. Furthermore, the present Monte Carlo method has errors in several cases. The improvement of the Monte Carlo method to obtain the generating function is necessary. We must note that the DMRG method is another method for obtaining a large deviation[@Gor09], and not only the continuous time, which is well studied, but also the discrete time case based on the DMRG method for the steady state[@Hie98] can be established.
The relation $q_A\sim Lq_B$ can be derived from the plausible argument considering the time-dependent and time-independent contributions to the current. The details are presented in $\S$ 2. Actually, the two generating functions for the two currents coincide with each other in relatively small systems, as shown in Fig. 9. Although they are considered to be the same also in large systems, we could not confirm it because the Monte Carlo calculation is not suitable for determining $q_B$. Thus, in a practical sense, they have differences.
We observed the cusp around $q=0$ in the current large deviation function in the ASEP. A similar cusp in the large deviation function for entropy production was reported by Mehl et al.[@Meh08] in a system of an overdamped Brownian particle under the periodic potential and uniform external field. They attributed the generation of the cusp to the sublinear response of the particle to the external field, which was derived from the potential within the site. In our study for the ASEP, the sublinear response may be caused by the interactions of particles, where the cause is different from that in the case of a one-particle system. The physical basis of the cusp we consider is that the probability of observing the current toward a lower hopping rate becomes $0$, which means that the current becomes zero in that direction. This explains why the cusp is observed, though this does not explain how the cusp is generated.
In conclusion, in this study we have calculated numerically the current large deviation function in the SSEP and ASEP with open boundaries. We have found that the generating function and the large deviation function deviate from the quadratic form. The system size dependence of the even order cumulants is found to be proportional to $L$ in both the SSEP and the ASEP. The generating functions defined by different definitions of the current appear to be the same for small systems. For the large system size, the Monte Carlo simulation gives unsatisfying results for the current through a boundary, from which we conjecture that the use of the total current is suitable for the Monte Carlo calculation of the large deviation function. The symmetry relation for the total current is calculated both numerically and analytically, and the obtained results are in good agreement. For the Monte Carlo calculations, a small deviation is observed in the symmetry relations and the deviation depends on the boundary parameters. We found that the convergence of the Monte Carlo simulation for calculating the large deviation function in the ASEP depends on the system size and the hopping parameter of a particle. The convergence is not good when the system size is large and the asymmetry of the hopping is large. In the region where the convergence is not good, the difference between the largest and second largest eigenvalues of the modified transition matrix depends on the system size as the stretched exponential, which is faster than the exponential decay.
Acknowledgments {#acknowledgments .unnumbered}
===============
One of the authors (T.M) wishes to express gratitude to Hisao Hayakawa for all the support and helpful discussions. We would like to acknowledge Yasuhiro Hieida for giving us many helpful comments on the paper. The numerical calculations were carried out at YITP in Kyoto University. This work was supported by the Grant-in-Aid for the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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[^1]: E-mail address:[email protected]
|
---
abstract: |
In this paper, we develop a method for unsupervised clustering of two-way (matrix) data by combining two recent innovations from different fields: the Sparse Subspace Clustering (SSC) algorithm [@Elhamifar:0zr], which groups points coming from a union of subspaces into their respective subspaces, and the t-product [@Kilmer:2011vn], which was introduced to provide a matrix-like multiplication for third order tensors. Our algorithm is analogous to SSC in that an “affinity” between different data points is built using a sparse self-representation of the data. Unlike SSC, we employ the t-product in the self-representation. This allows us more flexibility in modeling; in fact, SSC is a special case of our method.
When using the t-product, three-way arrays are treated as matrices whose elements (scalars) are n-tuples or tubes. Convolutions take the place of scalar multiplication. This framework allows us to embed the 2-D data into a vector-space-like structure called a free module over a commutative ring. These free modules retain many properties of complex inner-product spaces, and we leverage that to provide theoretical guarantees on our algorithm. We show that compared to vector-space counterparts, SSmC achieves higher accuracy and better able to cluster data with less preprocessing in some image clustering problems. In particular we show the performance of the proposed method on Weizmann face database, the Extended Yale B Face database and the MNIST handwritten digits database.
author:
- Eric Kernfeld
- Shuchin Aeron
- Misha Kilmer
bibliography:
- 'bibtensor.bib'
- 'SAbib.bib'
- 'elmbib.bib'
- 'ZZbib.bib'
- 'SSmCbibliography.bib'
date: 'Received: date / Accepted: date'
title: 'Clustering multi-way data: a novel algebraic approach [^1] '
---
[^1]: This research is funded in part by NSF:1319653. The first author acknowledges support for this work by the Tufts Summer Scholars’s Program 2013.
|
---
abstract: |
It is believed that hippocampus functions as a memory allocator in brain, the mechanism of which remains unrevealed. In Valiant’s neuroidal model [@hippo], the hippocampus was described as a randomly connected graph, the computation on which maps input to a set of activated neuroids with stable size. Valiant proposed three requirements for the hippocampal circuit to become a stable memory allocator (SMA): stability, continuity and orthogonality. The functionality of SMA in hippocampus is essential in further computation within cortex, according to Valiant’s model.
In this paper, we put these requirements for memorization functions into rigorous mathematical formulation and introduce the concept of capacity, based on the probability of erroneous allocation. We prove fundamental limits for the capacity and error probability of SMA, in both data-independent and data-dependent settings. We also establish an example of stable memory allocator that can be implemented via neuroidal circuits. Both theoretical bounds and simulation results show that the neural SMA functions well.
author:
- 'Wenlong Mou[^1]\'
- 'Zhi Wang[^2]\'
- 'Liwei Wang[^3]'
bibliography:
- 'sma.bib'
title: |
Stable Memory Allocation in the Hippocampus:\
Fundamental Limits and Neural Realization
---
Introduction
============
It is well known that the computational speed of human brain is several orders of magnitude slower than that of current computer, while almost everything human can do easily are still beyond the capability of the most powerful computers. Thus, the investigation into computational mechanisms of human brains leads a promising direction for theoretical aspect of computation, machine learning and neuroscience. In particular, the formation of memory, during which the neural circuit creates a representation for an object, was not fully understood, though it has been observed that the hippocampus plays a significant role in this process.
A research line developed by Valiant [@neurobook; @valiant2000neuroidal; @valiant2006quantitative; @valiant2014must] focused on the connectivity structure of neuroids, i.e., the computational abstraction of neurons, on which local computation can be performed. This formal and conservative model assumes the brain to be a randomly connected graph of neuroids. Each of them can be either active or inactive, depending on the states of neighbors that have a directed edge to it. Biological constraints are also taken into consideration, such as weak synaptic connections and sparsity of graphs. With biological evidences, this model captures the capability of realistic neurons. Specific supervised and unsupervised learning tasks such as inductive learning, rational expressions, and reasoning are viable in this model, using the vicinal algorithms [@memo; @valiant2006quantitative; @DBLP:journals/neco/FeldmanV09]. In Valiant’s model, those complicated processes are based on basic operations of $JOIN$ and $LINK$. Recently, a new operation called $PJOIN$, namely, predictive $JOIN$, was proposed for unsupervised learning of patterns [@papadimitriou2015cortical; @DBLP:conf/podc/PapadimitriouV15]. All those operations, as well as experimental evidences from neuroscience, requires the representation for an item to be allocated with a stable amount of neurons, which is done by hippocampus, before the computation in cortex.
Valiant [@hippo] first discussed the role and computational mechanisms of hippocampus in this model. The paper stated that the function of hippocampus is to “identify the set of neurons in cortex at which a new compound concept or chunk will be represented, and to enable that set of neurons to take on that role”. The stability of the amount of neurons allocated plays a key role in Valiant’s neuroidal models in order to control each new chunk within a limited range so as to avoid the overall system becoming unstable. To guarantee this, three requirements for stable memory allocators (SMA) was proposed in their paper, namely, stability, continuity and orthogonality. Valiant pointed out the direction by discussing biological constraints and proposing an example in his paper, yet rigorous mathematical treatment is still in need. Our systematic analysis of stable memory allocators is therefore important in the neuroidal model.
Memorization process creates neural representation for arbitrary set of input item, either randomly or through learning from samples. Performance and characteristics of such mapping functions have been vastly studied in machine learning, theoretical computer science and computational neuroscience. For example, in [@signcons], a sign-consistent sparse JL transform is constructed to explain the formation of memory, under a different model of neurons. [@DBLP:conf/focs/ArriagaV99] proposed a neuron-friendly random projection for robust concept learning via dimension reduction. Hopfield networks [@hopfield] dynamically adjust network weights to create memory for new items. Locality sensitive hashing [@lsh] and binary embeddings [@DBLP:conf/icml/YiCP15] randomly maps input vector into a discrete space while preserving locality. Our work relates to them in terms of compression of information and preservation of locality.
In this paper, we first manage to put Valiant’s idea into a rigorous mathematical formulation. We give basic theoretical results and a simple construction in Section 2. By introducing concepts of error probability and capacity into our formulation, we are able to better characterize the fundamental possibilities and impossibilities of memory allocation. Their theoretical bounds are discussed under both data-dependent and data-independent settings; see Section 3. While the network designed by Valiant works in special cases [@hippo], it could not be generalized to the scale of real hippocampus. That’s why we propose, in Section 4, a feed-forward neural network model to realize stable memory allocation in hippocampus, and at the same time consistent with the biological constraints. In contrast to subtractive models [@hippo], the inhibition patterns of neurons are divisive in our design. We show that this stable memory allocator can better achieves the requirements with reasonable capacity.
Formulation of SMA
==================
In this section, we will first refine the requirements for the stable memory allocation task [@hippo], and put it into mathematical formulations.
Basic requirements and motivation
---------------------------------
As proposed in [@hippo], a valid stable memory allocator (SMA) for the cortex should subject to three basic requirements: stability, continuity and orthogonality. In the following we present high-level ideas in the formulation of SMA, and discuss basic requirements for a stable memory allocator to function well in Valiant’s model.
- Stability: The mapping is stable in that for inputs within a wide range of activity levels, the output will have activity level differing little. This condition is fundamental in vicinal algorithms in the cortex, and also reasonable in neural representation of abstract concepts.
- Continuity: A noise in the input pattern should not cause huge variations in the output. Continuity can provide neural systems with robustness (error-correcting) and help to realize pattern recognition functions (nearest-neighbor search).
- Orthogonality: With this property, distinct items will not be confused. Pairs of inputs that differ by much will be mapped to outputs that also differ sufficiently, so that they can be treated by cortex as distinct. We weaken the orthogonality requirement in [@hippo] and broaden the class of stable memory allocators, yet the biological functionality can still be guaranteed.
The randomness of mapping function plays an essential role in a stable memory allocator, since any data-independent deterministic allocation function will fail upon adversarial inputs: Suppose we have a fixed SMA which maps vectors in $\{0,1\}^n$ to a stable number of activated neurons. By pigeon-hole principle there must be $2^{\Omega(n)}$ vectors mapped to the same set of neurons. Among those colliding inputs, there must be a pair of vectors with Hamming distance $\Omega(n)$, which leads to the failure of orthogonality. On the contrary, a randomized SMA, like random projections, can fulfill these requirements for arbitrary input set of bounded size, with high probability. This observation is important and triggers the concept of error probability in our framework. We can thereby define the concept of capacity as the maximal number of objects that a stable memory allocator can handle with low error probability. In other words, the capacity of stable memory allocator stands for the number of items one can memorize without being confused.
Mathematical formulation
------------------------
In our framework, SMA is formulated as a series of distributions with respect to size $n$ over function space. Although the growth of synapses is quite random, there are some underlying rules that govern the formation of random graph so that almost everyone can allocate memory for new incoming items without any difficulty, no matter the difference of inner connections for each individual. We therefore demand the three properties here to hold uniformly with high probability, which guarantees more robustness than the model based on expected value [@hippo]. It is also worth noticing that we do not make any assumptions on the structure or distribution of input vectors.
A series of distributions $\{\mathcal{H}_n\}$ over function space $\left\{0,1\right\}^n\rightarrow\left\{0,1\right\}^n$ is called a stable memory allocator (SMA) with parameter $\langle \delta,\mu,\kappa,A_n,B_n,r_n\rangle$ and capacity $K_n$, if $\forall \varepsilon>0$, $\exists N\in \mathbb{N}$ s.t. $\forall n>N$ $\forall S_n\subset\left\{0,1\right\}^n \setminus \left\{\bm{x} | \vert \bm{x} \vert \leq \kappa n \right\}, \vert S_n\vert \leq K_n$, the following three properties hold $$\begin{split}
& \mathop{Pr}_{h\sim\mathcal{H}_n}{\left\{\forall \bm{x}\in S_n, \vert h(\bm{x})\vert\ \in \left((1-\delta)r_n,(1+\delta)r_n\right)\right\}}>1-\varepsilon\\
&\mathop{Pr}_{h\sim\mathcal{H}_n} \left\{\forall \bm{x}\in S_n, \bm{y} \in \{0,1\}^n, d_H(h(\bm{x}),h(\bm{y}))\leq \mu d_H(\bm{\bm{x},\bm{y}})\right\}>1-\varepsilon\\
&\mathop{Pr}_{h\sim\mathcal{H}_n}{\left\{\forall \bm{x},\bm{y}\in S_n,d_H(\bm{x},\bm{y})>A_n, d_H(h(\bm{x}),h(\bm{y}))>B_n\right\}}>1-\varepsilon
\end{split}$$
It is natural to assume $\forall \bm{x},\bm{y}\in S_n,d_H(\bm{x},\bm{y})>A_n$ according to our interpretation of orthogonality. There are some requirements for parameters of SMA: as the output vector is stably sparse, the number of activated neurons in the output $r_n$ should satisfy $r_n/n \ll 1$. Also, $A_n$ and $B_n$ should increase with respect to size $n$ and one typical assumption is that they grow linearly: $A_n= an, B_n= br_n$. Notice that $a$ is also a small number compared to 1, otherwise only few items are considered “truly different” and orthogonality would fail.
Not all SMAs have good performance. When judging a SMA, we should first consider those parameters that define it: smaller Lipschitz factor $\mu$ in the continuity requirement means better preservation of input information and stronger error-correcting functions. SMA with larger $B_n$ and smaller $A_n$ can distinguish items better. The parameter $\kappa$ represents the lowest input density required. The following discussion, without pointing out explicitly, regards elements in $S_n$ as not too sparse (with density larger than $\kappa$) for all the $S_n$ mentioned. A small $\delta$ stands for good performance in stability. The extreme case is that $\delta=0$, which is a strong version that realize stability precisely:
Strong stability
: $$\mathop{Pr}_{h\sim\mathcal{H}_n} \left\{\forall \bm{x}\in S_n, h(\bm{x})\in T_n^{(r_n)} \right\} >1-\varepsilon$$
where $T_n^{(r_n)} \triangleq \left\{\bm{x}\in\left\{0,1\right\}^n\big| \vert\bm{x} \vert=r_n \right\}$ and we call a specific allocator function $h(\bm{x})$ strong stable with respect to input set $S_n$ if $ \forall \bm{x} \in S_n, h(\bm{x})\in T_n^{(r_n)}$.
An SMA that satisfies the strong stability defined above is called a strong SMA.
Relating error probability to capacity
--------------------------------------
The key issue involved within capacity is error probability. The more items are there to be allocated, the more conflicts are likely to be happen in the outputs. Thus, one target of designing SMA is to maximize its capacity, i.e. to avoid the possible conflicts.
Theorem \[lower bound\] reveals the relationship between pairwise error probability and capacity, in order to fulfill the requirements uniformly with high probability. In the design and analysis of SMA, this theorem serves as a useful tool to guarantee capacity.
\[lower bound\] If a series of distributions $\{\mathcal{H}_n\}$ over function space $\left\{0,1\right\}^n\rightarrow \left\{0,1\right\}^n$ satisfies the following properties:
- For $\forall \bm{x}\in \left\{0,1\right\}^n$ fixed ,with probability at least $1-\varepsilon_n$: $$\vert h(\bm{x})\vert\ \in \left((1-\delta)r_n,(1+\delta)r_n\right)$$
- For $\forall \bm{x}\in\left\{0,1\right\}^n$ fixed, with probability at least $1-\varepsilon_n$: $$\forall \bm{y} \in \left\{0,1\right\}^n, d_H(h(\bm{x}),h(\bm{y}))\leq \mu d_H(\mathbf{\bm{x},\bm{y}})$$
- For $\forall \bm{x},\bm{y}\in \left\{0,1\right\}^n$ fixed, $d_H(\bm{x},\bm{y})>A_n$, we have: $$\mathop{Pr}_{h\sim\mathcal{H}_n}{\left\{ d_H\left(h(\bm{x}),h(\bm{y})\right)>B_n\right\}}>1-\varepsilon_n$$
where $\lim\limits_{n\rightarrow 0} \varepsilon_n=0$. Then $\{\mathcal{H}_n\}$ is a strong SMA with parameters $\langle \mu,A_n,B_n,r_n \rangle$ and capacity at least $\frac{o(1)}{\sqrt{\varepsilon_n}}$, as $n$ goes to infinity.
The proof of this theorem estimates the overall error probability for a given set of items via union bound, the technical detail of which is deferred to the Appendix.
(Sketch)
A simple example for SMA
------------------------
The following simple construction serves as a proof for the existence of SMA. The basic idea is that we first randomly select $2r_n$ bits from the input item, then flip each of these bits with probability $\frac{1}{2}$.
Suppose that $\left\{i_1,i_2,\cdots,i_{2r_n}\right\}$ is uniformly distributed on all the $2r_n$-sized subsets of $\left\{1,2,\cdots,n\right\}$, and that $b_1,b_2,\cdots,b_{2r_n} \overset{i.i.d.}{\sim} \mathcal{B}\left(1,\frac{1}{2}\right)$. We define $h(\bm{x})$ along with its distribution $\mathcal{H}_n$ as follows, with operation $\oplus$ denoting the modulo 2 addition. $$h(\bm{x})_k= \left\{
\begin{array}{cc}
b_j \oplus \bm{x}_k & k=i_j\\
0 & otherwise
\end{array}
\right.$$
We argue that $\forall a,p,\delta\in (0,1)$, the construction above is an SMA with parameter $\langle \delta,0, 1 ,A_n=a\cdot n, (1-\delta) p A_n ,r_n=p\cdot n\rangle$ and capacity at least $o(1) e^{2pa^2\delta^2n}$, according to Theorem \[lower bound\].
Apparently, the flips would not change Hamming distance and naturally meet the requirement of continuity. As $\forall \bm{x} \in \{0,1\}^n$,$\vert h(\bm{x})\vert \sim \mathcal{B}\left(2r_n,\frac{1}{2}\right)$, we know stability by using Chernoff bound. For orthogonality, let $d=d_H(\bm{x},\bm{y})$. According to Chv[á]{}tal’s estimation on tail probability of hypergeometric distribution [@chvatal1979tail], we have $\mathop{Pr}_{h\sim\mathcal{H}_n}\left\{ d_H(h(\bm{x}),h(\bm{y}))\leq (1-\delta)\frac{2r_n}{n}d \right\} \leq e^{-4r_n (\frac{d}{n})^2 \delta^2}$
However, this example is still far from actual memorization mechanisms, since it could not be fit into a biologically plausible neural realization.
Theoretical analysis
====================
In this section, we will consider the theoretical bounds for capacity and error probability of SMA, which are two important characteristics to judge its performance other than those parameters that define it. Lower bound of error probability and upper bound of capacity are first proposed. We also consider a different scheme where the memory allocator can be learned from input, and establish upper and lower bounds for the capacity.
Lower bound of pairwise error probability
-----------------------------------------
Lower bounds for pairwise error probability is important, since it indicates what capacity bound we can get via Theorem \[lower bound\]. In this subsection, we consider a slightly modified scheme, where the orthogonality condition is strengthened in Lipschitz form, and combined with continuity condition to constitute a bi-Lipschitz condition on the input set, as in Valiant’s paper [@hippo].
\[error probability lower bound\] If $\{\mathcal{H}_n\}$ is a series of distribution over functions that satisfies the following requirements
- $\forall h \in$ supp$(\mathcal{H}_n)$, $\frac{|h(\bm{x})|}{r_n} \in (1-\delta,1+\delta)$
- $\forall \bm{x}, \bm{y} \in \{0,1\}^{n}$fixed, with probability at least $1-\varepsilon_n$: $$\lambda d_H(\bm{x},\bm{y})\leq d_H(h(\bm{x}),h(\bm{y}))\leq \mu d_H(\bm{x},\bm{y})$$
then we have the lower bound for error $$\varepsilon_n = \Omega \left(\left[2 r_n \delta \binom{n}{r_n(1+\delta)}\right]^{-(\frac{\mu-\lambda}{\mu+\lambda})^2 \frac{1}{log n}} \right)$$
Basically, the theorem states information-theoretic limitations on a random mapping which can compress $\{0,1\}^n$ to a much smaller one while preserving distance. The proof is based on reduction to one-way public-coin randomized communication complexity, which is deferred to Appendix.
(Sketch)
The proof is based on reduction to one-way public-coin randomized communication complexity for approximating gap Hamming distance, the lower bound of which is proved by Jayram and Woodruff [@jlopt]. They have showed we need at least $\Omega(\frac{1}{\epsilon^2}\log n\log \frac{1}{\delta})$ communication bits to approximate Hamming distance of two $n$-bit vectors with $(1\pm \epsilon)$ multiplicative error and $\delta$ error probability.
Our reduction uses SMA to encode the vector: for a given SMA described in the theorem, Alice and Bob use public coin to sample a function from $\mathcal{H}$, and Alice sends the allocated memory $h(\bm{x})$ to Bob. According to the stability requirement, the SMA has at most $\sum_{j=r_n(1-\delta)}^{r_n(1+\delta)} \binom{n}{j} \leq 2r_n\delta \binom{n}{r_n(1+\delta)}$ different possible outputs, the logarithm of which is the number of communication bits needed.
This protocol results in approximation error $1\pm \frac{\mu-\lambda}{\mu+\lambda}$. By plugging into the communication complexity lower bound, we get the bound on $\epsilon_n$.
Upper bound of capacity
-----------------------
The following theorem gives an upper bound for the capacity when strong stability holds. This results mainly captures the limitations against achieving orthogonality posed by stability requirement.
\[upperbound for capacity\] For an SMA with parameters $\langle 0,0,\mu,A_n,B_n,r_n\rangle$, if $ max \big\{|S_n|\big|\forall \bm{x},\bm{y} \in S_n, d_H(\bm{x},\bm{y})>A_n\big\} \geq K_n$, then the capacity is upper bounded with: $$\log K_n \leq \left(H(\alpha)-\alpha H(\beta)-(1-\alpha)H\left(\frac{\alpha\beta}{1-\alpha}\right)+o(1)\right)n$$ where $\alpha=\frac{r_n}{n}$, $\beta=\frac{B_n}{4r_n}$ and $H(\cdot)$ indicates entropy.
This theorem also discussed information-theoretic limitations on stable memory allocator, from a different perspective: the output space can only contain a limited number of “truly different” objects. The proof uses “sphere packing” arguments, which is deferred to the Appendix.
(Sketch)
We obtain a subset such that $\forall \bm{x},\bm{y}\in S_n, d_H(\bm{x},\bm{y})\geq A_n$ and $|S_n|=K_n$. According to the definition, given this input set, there is a function $h\in supp(\mathcal{H})$, such that $\forall \bm{x} \in S_n, |h(\bm{x})|=r_n $ and $\forall \bm{x}, \bm{y} \in S_n, d_H(\bm{x},\bm{y})>B_n$.
Now we have converted the SMA to deterministic encoding, whose upper bound can be obtained by geometric argument in Hamming space: we have to allocate $K_n$ objects in $T_n^{(r_n)}=\{\bm{x}\in\{0,1\}^n:|\bm{x}|=r_n\}$, with pairwise distance at least $B_n$. Let $U(\bm{x},r)$ denote $\left\{\bm{x'}\in T_n^{(r_n)}\vert d_H(\bm{x},\bm{x}')\leq r\right\}$, we have $\bigcup_{\bm{x}\in S_n} U(h(\bm{x}))\subset T_n^{(r_n)}$, and $$\forall \bm{x},\bm{y} \in S_n, U\left(h(\bm{x}),\frac{B}{2}\right)\cap U\left(h(\bm{y}),\frac{B}{2}\right)=\varnothing$$ As in standard “sphere packing” problems, we can bound the number of items with volume $K_n\leq \frac{|T_n^{(r_n)}|}{|U(x,B_n/2)|}$. Direct computations on the sizes of sets imply the result.
Theretical results for data-dependent allocators
------------------------------------------------
Till now, our discussions are data-independent without exploring whether the memory allocator can adjust according to data. We focus in this part on the theoretical possibilities for a data-dependent memory allocator which can learn from items. Though it is impossible for hippocampus to be aware of all the items to encounter beforehand, our theoretical analysis in data-dependent setting reveals the possibilities and limits for online learning in memory allocation, where the memory allocator can adjust its mapping function after seeing each instance. In the following we propose both upper and lower bounds for the capacity in the data-dependent setting.
The definition of SMA indicates that, for each fixed input set with limited size, there is at least one allocator $h$ that satisfies the three conditions regardless of the distribution $\mathcal{H}_n$ on the function space. Therefore, $\forall S_n, \exists$ $h$ dependent on $S_n$ such that $\forall \bm{x},\bm{y}\in S_n, d_H(\bm{x},\bm{y})>A_n \Rightarrow d_H(h(\bm{x}),h(\bm{y})\geq B_n$. An equivalent description of Theorem \[upperbound for capacity\] from a data-dependent perspective is as follows:
If an allocator function is strong stable, then the number of “truly different” items it can discriminate would be upper bounded with: $$\exp\left\{ \left(H(\alpha)-\alpha H(\beta)-(1-\alpha)H\left(\frac{\alpha\beta}{1-\alpha}\right)+o(1)\right)n \right\}$$ Conversely, we consider that for a given set of “truly different” items $S_n$, whether an SMA that guarantees strong stability, orthogonality and uniform Lipschitz continuity exists. The following theorem guarantees the existence of data-dependent memory allocator with high capacity. We defer the technical proof to Appendix.\
\[existence of hashing function\] Given input set $S_n \subset \left\{0,1\right\}^n$ that satisfies $\forall \bm{x},\bm{y}\in S_n, d_H(\bm{x},\bm{y})> A_n$. If $\log{\vert S_n \vert }\leq \frac{1}{2} \left(H(\alpha)-\alpha H(\beta)-(1-\alpha)H \left(\frac{\alpha\beta}{1-\alpha}\right)\right)n$, then there exists a function $h_n: \left\{0,1\right\}^n\rightarrow \{0,1\}^n $ such that $$\begin{split}
\forall \bm{x} \in S_n, h(\bm{x}) \in T_n^{(r_n)}; \forall \bm{x},\bm{y}\in S_n, d_H(h_n(\bm{x}),h_n(\bm{y}))> B_n;\\
\forall \bm{x} \in S_n, \forall \bm{y} \in \{0,1\}^n, d_H(h_n(\bm{x}),h_n(\bm{y})) \leq \frac{8 r_n}{A_n} d_H(\bm{x},\bm{y})
\end{split}$$ where $r_n=|S_n|$, $\alpha=\frac{r_n}{n}$, $\beta =\frac{B_n}{2r_n}$ and $H(\cdot)$ is the entropy.
(Sketch)\
We will first find a mapping $\hat{h}_n$ on $S_n$ that meets the first two requirements using probability method of proofs, and then extend it to $h_n$ on $\left\{0,1\right\}^n$ that satisfies all the three requirements by Kirszbraun theorem [@kirszbraun].
Let $\mathcal{U}$ denote the uniform distribution on all the mappings from $S_n$ to $ T_n^{(r_n)}$. By using union bound we have $\mathop{Pr}_{\tilde{h}_n\sim \mathcal{U}}\left\{\exists \bm{x}\neq \bm{y}\in S_n, d_H(\tilde{h}_n(\bm{x}),\tilde{h}_n(\bm{y})) \leq B_n\right\} \leq \frac{|S_n|^2}{2} \mathop{Pr}_{\tilde{h}_n\sim \mathcal{U}}\left\{d_H(\tilde{h}_n(\bm{x}),\tilde{h}_n(\bm{y})) \leq B_n \right\}$. By further calculating the probability with finite sums and plugging in the condition for $\log |S_n|$, we have this probability strictly less than $1$. Thus, there exists at least one mapping $\hat{h}_n$ satisfying the orthogonality requirement.
Then, we deem Hamming space as an Euclidean space with $d(\bm{x},\bm{y})= \sqrt{d_H(\bm{x},\bm{y})}$. Let $\omega(\bm{x})$ denote the translation $\bm{x}-\frac{r_n}{n} \bm{1}_n$, then $\omega\circ \hat{h}_n$ is a map from the subset $S_n$ of Hilbert space $(R^n,d)$ to Hilbert space $(W_{n-1},d)$ with Lipschitz constant $\sqrt{\frac{2r_n}{A_n}}$, where $W^{n-1}=\{\bm{x} \in R^{n-1}| \sum_{i=1}^{n} x_i =0\}$. According to Kirszbraun theorem, $\omega\circ \hat{h}_n$ can be extended to $h_0: R^n \rightarrow W^{n-1}$ with Lipschitz constant remained. The function we want is defined as $$h_n(\bm{x})_i = \left\{ \begin{array}{ll}
0 & |h_0(\bm{x})_i+\frac{r_n}{n}| \leq |h_0(\bm{x})_i-\frac{n-r_n}{n}|\\
1 & |h_0(\bm{x})_i+\frac{r_n}{n}| > |h_0(\bm{x})_i-\frac{n-r_n}{n}|
\end{array} \right.$$ and its Lipschitz constant in the Hamming space is $\frac{8 r_n}{A_n}$.
Notice that the number of input items in Theorem \[existence of hashing function\] is approximately the square root of the upper bound indicated by Theorem \[upperbound for capacity\]. The lower bound is strong in this sense.
A feedforward neural network for SMA
====================================
In this section, we propose a feedforward neural network model that realizes SMA functionality. Compared with Valiant’s previous neural circuits [@hippo], not only does our construction achieve better performance of SMA, with capacity guarantees, but also it is biologically more reasonable, in terms of number of synapse connected to a neuron and their strengths.
Biologically reasonable assumptions
-----------------------------------
We adopt the randomly-connected feed-forward network architecture and unmodifiable synapse in Valiant’s paper [@hippo]. Actually, as Valiant has suggested in his paper [@hippo], there are also experimental evidences supporting this architecture. The information flow within hippocampus appears to be uni-directional, with much less reciprocal connections [@andersen2006hippocampus]; it has also been reported that both modifiable and unmodifiable synapse exists in the hippocampus [@debanne1999heterogeneity]. As in Valiant’s paper, our theory focuses on explaining the mechanism of memory allocation component, with conservative assumptions on neurons, though the overall infrastructure of hippocampus can be much more complicated.
A major difference we made in the design of neural circuit is that, the inhibition patterns of neurons are divisive, instead of subtractive. Whether the way of inhibition is subtractive or divisive is an important open question in neuroscience [@doiron2001subtractive; @mejias2014subtractive]. In divisive inhibition model, each neuron is activated if and only if the ratio of its total positive synaptic inputs $\sum_{+} W_i$ to the sum of all the synaptic weights that come from input neurons $(\sum_{+} W_i) +(\sum_{-} W_i)$ is larger than the threshold value $C$, i.e. $\frac{\sum_{+} W_i}{\sum_{+} W_i +\sum_{-} W_i}>C$. For the purpose of memory allocation, subtractive models [@hippo] have to narrow down the range of density in each layer and use contraction mapping theorem for stability. Divisive inhibition, on the contrary, enables us to control the expected level of activity in second layer directly, for a wide range of input density.
Description of the network
--------------------------
Our neural realization is based on a unidirectional three-layer network architecture. For simplicity we assume each layer has $n$ neurons. Firing neurons in the first layer represents the input vector $\bm{x}\in\{0,1\}^n$, and the output $h(\bm{x})$ is taken from the third layer. For $k\in\{1,2\}$, the edges from $k$-th layer to $(k+1)$-th layer is randomly and independently connected: each edge exists with probability $2p$, and the chances for it to become positive and inhibitive are both $p$. The threshold value in divisive inhibition is denoted as $C_1$ and $C_2$, for second and third layer neurons respectively.
Proof of SMA functionality
--------------------------
In this subsection, we give proofs on the validity of our construction as a stable memory allocator, with reasonable parameters and capacity.
We first give the following technical lemmas needed in our analysis. Lemma \[middle layer stablity\], \[middle one-bit change probablity\], \[CLT second-layer change probability\] and \[CLT third-layer change probability\] described the consequences of passing through one layer in our network model. We defer the proofs for the lemmas and theorem to Appendix.
\[middle layer stablity\] For $\forall \bm{x}\in\left\{0,1\right\}^n$ fixed, let $\beta\in \mathbb{R}^n$ with all entries $i.i.d.$, $Pr\{\beta_i=1\}=Pr\{\beta_i=-1\}=p$, $Pr\{\beta_i=0\}=1-2p$. Then we have: $$0\leq \frac{1}{2}-\mathop{Pr}\{\beta^T\bm{x}>0\}\leq \frac{1}{2\sqrt{\pi p |\bm{x}|}}+\frac{1}{2}e^{-2(2\ln 2-1)p|\bm{x}|}$$
The following lemma estimates how one-bit difference in the input influence the output.
\[middle one-bit change probablity\] For $\forall \bm{x},\bm{y}\in\left\{0,1\right\}^n$ fixed, with $d_H(\bm{x},\bm{y})=1$, let $\beta\in \mathbb{R}^n$ with all entries $i.i.d.$, $Pr\{\beta_i=1\}=Pr\{\beta_i=-1\}=p$, $Pr\{\beta_i=0\}=1-2p$. For $\forall \bm{v}\in\{0,1\}^n$, let $g(\bm{v})=I[\beta^T\bm{v}]>0$. Then we have: $$Pr\left\{g(\bm{x})\neq g(\bm{y})\right\}\leq 2\sqrt{\frac{p}{\pi |\bm{x}|}}+2pe^{-2(2\ln 2-1)p|\bm{x}|}$$
(Sketch of Lemma \[middle layer stablity\] and Lemma \[middle one-bit change probablity\])\
We need to estimate $Pr\{\beta^T\bm{x}>0\}$ in Lemma \[middle layer stablity\], and $Pr\{g(\bm{x})\neq g(\bm{y})\}$ in Lemma \[middle one-bit change probablity\]. Both problems can be reduced to estimating $Pr\{\beta^T\bm{x}=0\}$ and $Pr\{\beta^T\bm{x}=1\}$, which can be resolved using Stirling formula and Chernoff bound.
For a more general case, we cannot write the probability of changed bits in closed form, while its asymptotic behavior can be estimated using central limit theorem, as stated in the following two lemmas.
\[CLT second-layer change probability\] For $\forall \bm{x},\bm{y}\in\left\{0,1\right\}^n$ fixed, with $d_H(\bm{x},\bm{y})=L$, let $\beta\in \mathbb{R}^n$ with all entries $i.i.d.$, $Pr\{\beta_i=1\}=Pr\{\beta_i=-1\}=p$, $Pr\{\beta_i=0\}=1-2p$. For $\forall \bm{v}\in\{0,1\}^n$, let $g(\bm{v})=I[\beta^T\bm{v}]>0$. Then we have: $$Pr\left\{g(\bm{x})\neq g(\bm{y})\right\}\approx\frac{1}{\pi}\cos^{-1}\frac{|\bm{x}\cap \bm{y}|}{\sqrt{|\bm{x}|\cdot |\bm{y}|}}$$
\[CLT third-layer change probability\] For $n$ independent and identically distributed pairs $\{(x_i,y_i)\}_{i=1}^{n}$ with $Pr\{x_i=1\}=Pr\{y_i=1\}\approx \frac{1}{2}$ and $Pr\{x_i\neq y_i\}=\eta$, we concatenate them and get two random vectors $\bm{x},\bm{y}\in \{0,1\}^n$. Let $\beta\in \mathbb{R}^n$ with all entries $i.i.d.$, $Pr\{\beta_i=1-c\}=Pr\{\beta_i=-c\}=p$, $Pr\{\beta_i=0\}=1-2p$. For $\forall \bm{v}\in\{0,1\}^n$, let $g(\bm{v})=I[\beta^T\bm{v}]>0$. We have $Pr\{g(\bm{x})\neq g(\bm{y})\}$ is approximately equal to: $$\frac{2}{\sqrt{1-\frac{1}{2}\eta}}\int_{\mathbb{R}}{\varphi(\frac{t}{\sqrt{1-\frac{1}{2}\eta}})\Phi(\frac{-1}{\sqrt{\eta}}\big|\sqrt{2}t-\frac{(2c-1)p\sqrt{n}}{\sqrt{v(c,p)}}\big|)}dt$$ ($\varphi(\cdot)$ and $\Phi(\cdot)$ denotes the density and cumulative density function for $\mathcal{N}(0,1)$, respectively)\
($v(c,p)=(2c^2-2c+1)p-(1-2c)^2p^2$ is variance of $\beta_i$)
(Sketch of Lemma \[CLT second-layer change probability\] and Lemma \[CLT third-layer change probability\])\
Let $\bm{w}=\bm{x}\cap \bm{y}$, we have $\beta^T\bm{w},\beta^T(\bm{x}-\bm{w}),\beta^T(\bm{y}-\bm{w})$ independent. Central limit theorem allows us to approximate joint distribution of $\langle \beta^T\bm{x},\beta^T\bm{y}\rangle$ a bivariate Gaussian. $Pr\{g(\bm{x})\neq g(\bm{y})\}$ can be then approximated by the quadrant integral of Gaussian pdf, as appearing in lemmas.
By combining them together, we conclude the functionality of neural stable memory allocator:
\[neural SMA\] Given $r_n,A_n=a\cdot n,s_0$, $\forall \gamma\in(0,1)$, we construct the neural network with parameters: $p=\frac{1}{n^{\gamma}}$, and $C_1=\frac{1}{2}$, $C_2=\frac{1}{2}-
\frac{s}{2\sqrt{2-s^2}}$, where $s=\Phi^{-1}\left(\frac{r_n}{n}\right)\sqrt{\frac{2}{np}}$. Let $\mathcal{H}$ denote the mapping function defined by this neural network. There exists a constant $b$ depending solely on $a$, and $\mu_n=Z_0r_ns_0^{-\frac{1}{4}}n^{-\frac{1+\gamma}{4}}(\log n)^2$ with some absolute constant $Z_0$, such that $\forall \bm{x},\bm{y},\bm{z}\in \{0,1\}^n$with $|\bm{x}|,|\bm{y}|,|\bm{z}|\geq s_0n$, and $d_H(\bm{x},\bm{z})>A_n)$, the following hold if $n$ is large enough: $$\begin{split}
&\mathop{Pr}_{h\sim \mathcal{H}}\left\{\frac{|h(\bm{x})-r_n|}{n}>\epsilon\right\}\leq \exp\left\{-2n\left(\epsilon-\frac{\log(n/r_n)}{\sqrt{s_0n^{1-\gamma}}}\right)^2\right\}\\
&\mathop{Pr}_{h\sim \mathcal{H}}\left\{\frac{d_H(h(\bm{x}),h(\bm{y}))}{\sqrt{d_H(\bm{x},\bm{y})}}\geq t\mu_n \right\}\leq \exp\left\{-(t\ln t-(t-1))\mu_n\sqrt{d_H(\bm{x},\bm{y})}\right\}.\quad \forall t\geq 1\\
&\mathop{Pr}_{h\sim\mathcal{H}}\left\{d_H(h(\bm{x}),h(\bm{z}))\leq (b-\epsilon)r_n\right\} \leq e^{-2n\epsilon^2}
\end{split}$$
(Sketch)\
**Stability**: Lemma \[middle layer stablity\] estimates the range of firing probability of middle-layer neurons (denoted as $s_1$). Using central limit theorem we can then calculate the firing probability for a third-layer neuron as $\Phi(\sqrt{n}\frac{(1-2c)ps_1}{\sqrt{v(c,ps_1)}})$. With the $C_2$ value stated in this theorem, we prove this probability is restricted within interval $\frac{r_n}{n}\pm\frac{log(n/r_n)}{\sqrt{s_0n^{1-\gamma}}}$. By applying Chernoff bound to the $i.i.d.$ output neurons we get the result.\
**Continuity**: Fix $d_H(\bm{x},\bm{y})=L$, we construct a sequence $\bm{x}=\bm{v}_0,\bm{v}_1,\bm{v}_2,\cdots,\bm{v}_L=\bm{y}$ with $d_H(\bm{v}_{i-1},\bm{v}_i)=1$. By union bound we have $Pr\{g(\bm{x})\neq g(\bm{y})\}\leq \sum_{i=1}^{L}Pr\{g(\bm{v}_i)\neq g(\bm{v}_{i-1})\}$, each term of which is bounded by Lemma\[middle one-bit change probablity\]. Lemma \[middle layer stablity\] tells us that both $\bm{x}$ and $\bm{y}$ activate approximately $\frac{n}{2}$ neurons in the middle layer. By applying Lemma \[CLT third-layer change probability\] with $\eta=O(L\sqrt{\frac{p}{s_0n}})$, we derive the following (approximate) upper bound for probability of changed bits in third layer: $$O(1)\cdot\int_{\mathbb{R}}{\varphi(t)\Phi\left(-\frac{(s_0n^{1+\gamma})^{\frac{1}{4}}}{\sqrt{L}}|t-\Phi^{-1}(\frac{r_n}{n})|\right)dt}$$ By dividing the integral into three parts at splitting points $\Phi^{-1}(\frac{r_n}{n})\pm\frac{2\log n\sqrt{L}}{(s_0n^{1+\gamma})^{\frac{1}{4}}}$, we get an approximate upper bound $\frac{3\log n\sqrt{L}}{(s_0n^{1+\gamma})^{\frac{1}{4}}}\varphi(\Phi^{-1}(\frac{r_n}{n}))\leq \frac{\mu_n\sqrt{L}}{n}$. We then apply the Chernoff bound in relative entropy form and conclude the inequality.\
**Orthogonality**: For $\bm{x},\bm{z}\in \{0,1\}^n$ such that $d_H(\bm{x},\bm{z})\geq an$, we have $\frac{|\bm{x}\cap \bm{z}|}{\sqrt{|\bm{x}|\cdot |\bm{z}|}}\leq \sqrt{1-a}$. According to Lemma \[CLT second-layer change probability\], the probability of making differences at second layer is at least $b_1=\frac{1}{\pi}\cos^{-1}\sqrt{1-a}$. By Lemma \[CLT third-layer change probability\] with $\eta=b_1$, we get $$\int_{\mathbb{R}}\varphi(\frac{t}{\sqrt{1-\frac{1}{2}b_1}})\Phi(\frac{|t-\Phi^{-1}(r_n/n)|}{\sqrt{b_1}})dt$$ The integral is lower bounded by limiting its range within $\frac{r_n}{n}\pm b_1$. By simple calculation we see that it is at least $b\frac{r_n}{n}$ for some $b$ depending solely on $b_1$, no matter how $r_n$ is related to $n$,. We then apply Chernoff bound to the $i.i.d.$ output units and get this result.
Assume $r_n$ grows linearly with $n\rightarrow +\infty$, while $\epsilon, b$ remains absolutely constant, and $t=\Theta(n^{\delta})$ with small absolute constant $\delta$, the error probabilities in Theorem \[neural SMA\] diminish quasi-exponentially as $n$ increases. Combined with Theorem \[lower bound\], we conclude that the neural stable memory allocator achieves a capacity which is exponential in $poly(n)$. The continuity requirement in this neural SMA deviates from standard definition: on the one hand, though the Lipschitz factor $t\mu_n$ grows unboundedly as $n\rightarrow+\infty$, it is much smaller than $r_n$ so that error-correcting functions can still be performed; on the other hand, the theorem provides more guarantees than Lipschitz constant, since the number of different output bits grows linearly with $\sqrt{d_H(\bm{x},\bm{y})}$ instead of $d_H(\bm{x},\bm{y})$, making it more robust for input errors of more than constant bits.
Simulation results
------------------
In this subsection, we verify the theoretical guarantees via simulating the neural network. Numerical results show that, the neural stable memory allocator is consistent with theoretical predictions, and thus functions well as an SMA.
We construct three-layer network as described in this section. Guided by Theorem \[neural SMA\], parameters are set as follows: $$n=10^5,p=2.5\times 10^{-3},C_1=0.5,C_2=0.57$$ To illustrate the stability of allocated neurons, we observe the proportion of activated neurons in second and third layer for input density ranging from 0.05 to 0.5, as plotted in Figure 1.a. The density of firing neurons is restricted in $(0.46,0.49)$ in the second layer, and further narrowed down in range $(0.014,0.017)$ in the third layer. By adjusting $C_2$ we can also control the density in output layer arbitrarily.
Figure 1.b shows the relationship between expansion rate $\frac{d_H(h(\bm{x}),h(\bm{y}))}{d_H(\bm{x},\bm{y})}$ and the proportion of activated input neurons, which demonstrates continuity and orthogonality properties of our memory allocator. The values shown in the figure is by taking average from $10$ randomly chosen inputs. Different curves stand for the behaviors of expansion rate for $d_H(\bm{x},\bm{y})=1,3,10,30,100$, respectively. As the figure shows, difference in inputs for a few bits will not cause the large Hamming distance between outputs, which verifies continuity. On the other hand, the expansion rate is lower bounded with some positive constant, which illustrates orthogonality. It is also worth noticing that expansion rate decreases as $d_H(\bm{x},\bm{y})$ increases, which is consistence with continuity result in Theorem \[neural SMA\]. Therefore, our experimental results verify the validity of Theorem \[neural SMA\], and demonstrate the performance of our construction as a stable memory allocator.
Conclusion and open questions
=============================
In this paper we give definitions, fundamental limits and a neural realization for stable memory allocation in the hippocampus, under Valiant’s neuroidal framework. We first push ahead rigorous formulations from the three properties of SMA that Valiant has proposed in his paper [@hippo]. Some important characteristics of SMAs in our definition such as capacity and error probability are discussed with their upper and lower bounds presented. We also explore the theoretical possibilities of memory allocators which can learn from input items.
We explicitly construct a stable memory allocator based on a randomly-connected feed-forward neural network with biologically sound parameters. We use divisive inhibition of neurons in our model, which allows us to obtain arbitrary level of stability and preserves continuity better. Rigorous proofs are given for the performance of this SMA. We also verify our theory through computer simulations.
There are a lot of promising future works we can do. By making full use of the three conditions in definition, we can possibly give tighter bounds for theoretical limits of SMAs. On the other hand, though we have constructed a neural network model that realizes SMA, it is far from optimal. The question whether we can construct a neural stable memory allocator with constant Lipschitz factor for continuity remains open. Besides, it is worthwhile to investigate more complicated computational functions of the hippocampus, such as the interaction between hippocampus and cortex, and the retrieval of allocated memory.
Appendix {#appendix .unnumbered}
========
Proof of Theorem \[lower bound\] {#proof-of-theoremlower-bound .unnumbered}
--------------------------------
We bound the overall error probability using pairwise error probabilities via union bound. For orthogonality, we have: $$\begin{aligned}
&\mathop{Pr}_{h\sim\mathcal{H}_n}{\left\{\forall \bm{x},\bm{y}\in S_n,d_H(\bm{x},\bm{y})>A_n, d_H(h(\bm{x}),h(\bm{y}))>B_n\right\}}\\
=&1-\mathop{Pr}_{h\sim\mathcal{H}_n}{\left\{\exists \bm{x},\bm{y}\in S_n,d_H(\bm{x},\bm{y})>A_n, d_H(h(\bm{x}),h(\bm{y}))<B_n\right\}}\\
=&1-\mathop{Pr}_{h\sim\mathcal{H}_n}{\left\{\bigcup_{\bm{x},\bm{y}\in S_n,d_H(\bm{x},\bm{y})>A_n} \{d_H(h(\bm{x}),h(\bm{y}))<B_n\}\right\}}\\
\geq &1-\sum_{\bm{x},\bm{y}\in S_n, d_H(\bm{x},\bm{y})>A_n}\mathop{Pr}_{h\sim\mathcal{H}_n}{\left\{d_H(h(\bm{x}),h(\bm{y}))<B_n\right\}}\\
\geq &1-\vert S_n\vert^2 \varepsilon_n
\end{aligned}$$ Similarly, for continuity and stability we have: $$\mathop{Pr}_{h\sim \mathcal{H}_n}\{ \forall \bm{x}\in S_n,\bm{y}\in \{0,1\}^n, d_H(\bm{x},\bm{y})\leq \mu d_H(\bm{x},\bm{y}) \}\geq 1-|S_n| \varepsilon_n$$ $$\mathop{Pr}_{h\sim \mathcal{H}_n}\{ \forall \bm{x}\in S_n, |h(\bm{x})|\in (r_n(1-\delta),r_n(1+\delta)) \}\geq 1-|S_n| \varepsilon_n$$ For any sequence $\xi_n\rightarrow 0$, let $|S_n|=K_n=\xi_n\frac{1}{\sqrt{\varepsilon_n}}$. The probability of violating any of the requirements is $O(\xi_n)$, which goes to zero. Therefore, the capacity of this SMA is at least $\frac{o(1)}{\sqrt{\epsilon_n}}$.
Proof of Theorem \[error probability lower bound\] {#proof-of-theoremerror-probability-lower-bound .unnumbered}
--------------------------------------------------
The proof is based on the lower bound of one-way randomized approximate communication complexity of Hamming distance by [@jlopt]. They showed that: If Alice has vector $\bm{x}$ of length $n$, and Bob has vector $\bm{y}$ of length $n$, to estimate $d_H(\bm{x},\bm{y})$ within a factor of $1\pm \delta$ with probability at least $1-\varepsilon$, Alice has to send at least $\Omega \left( \frac{1}{\delta^2}\log\frac{1}{\varepsilon} log n\right)$ bits to Bob.
We can use SMA to construct an one-way communication protocol: For a given SMA described in the theorem, a common hashing function $h\sim \mathcal{H}_n$ for Alice and Bob is sampled first. Upon receiving the vector $\bm{x}$, Alice computes $h(\bm{x})$ and sends it to Bob. Bob has vector $\bm{y}$ and gets $h(\bm{x})$, so he can compute $d_H(h(\bm{x}),h(\bm{y}))$. Because there are $\sum_{j=r_n(1-\delta)}^{r_n(1+\delta)} \binom{n}{j} \leq 2r_n\delta \binom{n}{r_n(1+\delta)}$ different possible values for $h(\bm{x})$, the number of bits needed to encode these values is at least: $$\label{encode bit}
K=\log\left(2r_n \delta \binom{n}{r_n(1+\delta)}\right)$$ We can write the Lipschitz condition in normalized form: $$\mathop{Pr}_{h\sim\mathcal{H}_n} \left((1-\frac{\mu-\lambda}{\mu+\lambda}) d_H(\bm{x},\bm{y}) \leq \frac{2}{\mu+\lambda}d_H(h(\bm{x}),h(\bm{y})) \leq (1+\frac{\mu-\lambda}{\mu+\lambda}) d_H(\bm{x},\bm{y}) \right) > 1-\varepsilon_n$$ By plugging into the result of [@jlopt], we have $$\label{one way communication complexity}
K=\Omega \left( (\frac{\mu+\lambda}{\mu-\lambda})^2 log \frac{1}{\varepsilon_n} log n \right)$$ By combining equation (\[encode bit\])(\[one way communication complexity\]), we have $$\varepsilon_n = \Omega \left(\big[2 r_n \delta \binom{n}{r_n(1+\delta)}\big]^{-(\frac{\mu-\lambda}{\mu+\lambda})^2 \frac{1}{log n}} \right)$$
Proof of Theorem \[upperbound for capacity\] {#proof-of-theoremupperbound-for-capacity .unnumbered}
---------------------------------------------
For a strong SMA with $A_n$ not too large, $ max \big\{|S_n|\big|\forall \bm{x},\bm{y} \in S_n, d_H(\bm{x},\bm{y})>A_n\big\} \geq K_n$, we can select set of input items $S_n \subseteq \{0,1\}^n$ that satisfies $|S_n|=K_n$ and $\forall \bm{x},\bm{y} \in S_n, d_H(\bm{x},\bm{y})>A_n$. According to the definition of strong SMA, there exists a hashing function $h$ such that $\forall \bm{x} \in S_n, h(\bm{x}) \in T_n^{(r_n)} $ and $\forall \bm{x}, \bm{y} \in S_n, d_H(\bm{x},\bm{y})>B_n$.
Let $U(\bm{x},r)$ denote $\left\{\bm{x'}\in T_n^{(r_n)}\vert d_H(\bm{x},\bm{x}')\leq r\right\}$, we have $$\forall \bm{x},\bm{y} \in S_n, U\left(h(\bm{x}),\frac{B}{2}\right)\cap U\left(h(\bm{y}),\frac{B}{2}\right)=\varnothing$$ Thus, we have $$\sum_{\bm{x}\in S_n}{\big| U\left(h(\bm{x}),\frac{B_n}{2}\right) \big|}\leq\vert T_n^{(r_n)}\vert= \binom{n}{r_n}$$ To estimate the number of elements in set $\big| U\left(h(\bm{x}),\frac{B_n}{2}\right) \big|$, notice that to guarantee $h(\bm{x}) \in T_n^{(r_n)}$, we can change $k \leq B_n/4$ bits at most from 0 to 1 and from 1 to 0 at the same time. Therefore,$$\left| U\left(h(\bm{x}),\frac{B_n}{2}\right) \right| = \sum_{k=1}^{\frac{B_n}{4}} \binom{r_n}{k}\cdot \binom{n-r_n}{k}$$ Also, $$B_n \leq d_H(h(\bm{x}),h(\bm{y})) \leq 2r_n \Longrightarrow k \leq \frac{B_n}{4} \leq \frac{r_n}{2}$$ The sparsity of output suggests that $r_n \ll n$, hence $$\label{monotonous}
\frac{\binom{r_n}{k+1}\cdot \binom{n-r_n}{k+1}}{\binom{r_n}{k}\cdot \binom{n-r_n}{k}}=\frac{1}{(k+1)^2} (r_n-k)(n-r_n-k) \geq \frac{1}{(\frac{r_n}{2}+1)^2} \cdot \frac{r_n}{2}(n-r_n-\frac{r_n}{2}) \gg 1$$ and $$\binom{r_n}{\frac{B_n}{4}}\cdot \binom{n-r_n}{\frac{B_n}{4}} \leq \big| U\left(h(\bm{x}),\frac{B_n}{2}\right) \big|$$ is a good estimation for $\big| U\left(h(\bm{x}),\frac{B_n}{2}\right) \big|$ ,and we have $$\label{bound for S_n}
|S_n| \leq \frac{\binom{n}{r_n}}{\binom{r_n}{\frac{B_n}{4}}\cdot \binom{n-r_n}{\frac{B_n}{4}}}$$ As $n,r_n$ and $B_n$ are large, we can use Stirling’s formula $\log n!=n\log n-n+O(\log n)$ to approximate $\binom{n}{r_n}$: $$log \binom{n}{r_n}= (-\frac{r_n}{n}log\frac{r_n}{n}-(1-\frac{r_n}{n})log(1-\frac{r_n}{n})+o(1))n= (H(\alpha)+o(1))n$$Similar results can be derived for $\binom{r_n}{\frac{B_n}{4}}$ and $\binom{n-r_n}{\frac{B_n}{4}}$. By using inequality (\[bound for S\_n\]), we get $$\log K_n \leq \left(H(\alpha)-\alpha H(\beta)-(1-\alpha)H\left(\frac{\alpha\beta}{1-\alpha}\right)+o(1)\right)n$$ where $\alpha=\frac{r_n}{n}$, $\beta=\frac{B_n}{4r_n}$ and $H(p)=-p\log p-(1-p)\log{(1-p)}$.
Proof of Theorem \[existence of hashing function\] {#proof-of-theoremexistence-of-hashing-function .unnumbered}
--------------------------------------------------
Let $\mathcal{U}$ denotes the uniform distribution on all the function that maps from $S_n$ to $ T_n^{(r_n)}$. $\tilde{h}_n\sim \mathcal{U}$ indicates that given $\bm{x},\bm{y}$, $\tilde{h}_n(\bm{x})$ and $\tilde{h}_n(\bm{y})$ are independently and uniformly distributed on $T_n^{(r_n)}$. We have $$\begin{aligned}
&\mathop{Pr}_{\tilde{h}_n\sim \mathcal{U}}\left\{\exists \bm{x}\neq \bm{y}\in S_n, d_H(\tilde{h}_n(\bm{x}),\tilde{h}_n(\bm{y})) \leq B_n\right\}\\
=& \mathop{Pr}_{\tilde{h}_n\sim \mathcal{U}}\left\{\bigcup_{\bm{x}\neq \bm{y}
\in S} \{d_H(\tilde{h}_n(\bm{x}),\tilde{h}_n(\bm{y})) \leq B_n\}\right\}\\
\leq& \sum_{\bm{x}\neq \bm{y}\in S_n}{\mathop{Pr}_{\tilde{h}_n\sim \mathcal{U}}\left\{d_H(\tilde{h}_n(\bm{x}),\tilde{h}_n(\bm{y})) \leq B_n \right\}}\\
<& \frac{|S_n|^2}{2} \mathop{Pr}_{\tilde{h}_n\sim \mathcal{U}}\left\{d_H(\tilde{h}_n(\bm{x}),\tilde{h}_n(\bm{y})) \leq B_n \right\}\\
< & \frac{|S_n|^2}{2} \frac{\sum_{k=0}^{\frac{B_n}{2}} \binom{r_n}{k}\cdot \binom{n-r_n}{k}} {\binom{n}{r_n}}
\end{aligned}$$ From equation (\[monotonous\]) in the proof of Theorem \[upperbound for capacity\], we know that $$\sum_{k=0}^{\frac{B_n}{2}} \frac{\binom{r_n}{k}\cdot \binom{n-r_n}{k}}{ \binom{n}{r_n}} \leq 2 \frac{\binom{r_n}{\frac{B_n}{2}} \cdot \binom{n-r_n}{\frac{B_n}{2}}} {\binom{n}{r_n}}$$ By using the condition in the statement $$\log{\vert S_n \vert }\leq \frac{1}{2} \left(H(\alpha)-\alpha H(\beta)-(1-\alpha)H \left(\frac{\alpha\beta}{1-\alpha}\right)+o(1)\right)n,$$ we derive $$\begin{split}
&\mathop{Pr}_{\tilde{h}_n\sim \mathcal{U}}\left\{\exists \bm{x}\neq \bm{y}\in S_n, d_H(\tilde{h}_n(\bm{x}),\tilde{h}_n(\bm{y})) \leq B_n\right\} <1\\
\Rightarrow& \exists \hat{h}_n: S_n \rightarrow T_n^{(r_n)}, \forall \bm{x},\bm{y}\in S_n, d_H(\hat{h}_n(\bm{x}),\hat{h}_n(\bm{y}))> B_n
\end{split}$$ Notice that $\hat{h}_n$ has Lipschitz constant $\frac{2r_n}{A_n}$ on $S_n$. To extend $\hat{h}_n$ to $h_n$ defined on $\{0,1\}^n$, we use Kirszbraun theorem [@kirszbraun].
Consider the Euclidean space $R^n$ and its subspace $W^{n-1}=\{\bm{x} \in R^{n-1}| \sum_{i=1}^{n} x_i =0\}$. The relationship between Euclidean distance and Hamming distance is that $d(\bm{x},\bm{y})= \sqrt{d_H(\bm{x},\bm{y})}, \forall \bm{x},\bm{y} \in S_n$. Let $\omega$ denote the translation $\omega(\bm{x})=\bm{x}-\frac{r_n}{n} \bm{1}_n$, then we have $\omega\circ \hat{h}_n: S_n \rightarrow W_{n-1}$ is a Lipschitz-continuous map from the subset $S_n$ of Hilbert space $(R^n,d)$ to Hilbert space $(W_{n-1},d)$ with Lipschitz constant $\sqrt{\frac{2r_n}{A_n}}$. According to Kirszbraun theorem, there exists a Lipschitz-continuous map $h_0$ that extends $\omega\circ \hat{h}_n$ and has the same Lipschitz constant: $h_0: R^n \rightarrow W^{n-1}$ $$\forall \bm{x},\bm{y} \in R^n, d(h_0(\bm{x}),h_0(\bm{y})) \leq \sqrt{\frac{2r_n}{A_n}} d(\bm{x},\bm{y})$$ We define $h_n$ on $\{0,1\}^n$ as follows: $$\forall \bm{x} \in \{0,1\}^n, h_n(\bm{x})_i = \left\{ \begin{array}{ll}
0 & |h_0(\bm{x})_i+\frac{r_n}{n}| \leq |h_0(\bm{x})_i-\frac{n-r_n}{n}|\\
1 & |h_0(\bm{x})_i+\frac{r_n}{n}| > |h_0(\bm{x})_i-\frac{n-r_n}{n}|
\end{array} \right. 1 \leq i \leq n$$ Given an arbitrary $\bm{y} \in S_n$, we have $\forall \bm{x} \in \{0,1\}^n$, $$|h_0(\bm{x})_i-h_0(\bm{y})_i| < \frac{1}{2} \Rightarrow h_n(\bm{x})_i=h_n(\bm{y})_i$$ Let $k$ denote the number of bits such that $|h_0(\bm{x})_i-h_0(\bm{y})_i| \geq \frac{1}{2}$, we have $$\sqrt{\frac{k}{4}} \leq d(h_0(\bm{x}),h_0(\bm{y})) \leq \sqrt{\frac{2r_n}{A_n}} d(\bm{x},\bm{y}) \Rightarrow k \leq \frac{8r_n}{A_n} d_H(\bm{x},\bm{y})$$ Therefore $$d_H(h_n(\bm{x}),h_n(\bm{y})) =\sum_{i=1}^n |h_n(\bm{x})_i-h_n(\bm{y})_i| \leq k \leq \frac{8r_n}{A_n} d_H(\bm{x},\bm{y})$$
Proof of Theorem \[neural SMA\] {#proof-of-theoremneural-sma .unnumbered}
-------------------------------
Theorem \[neural SMA\] guarantees stable memory allocation properties of a three-layer neural circuit with divisive inhibition. Key issue in the proof is to estimate high probability upper and lower bounds for output-layer proportion of activation, as well as expansion rates. From a high level point of view, the second layer of this network makes it possible for a wide range of input density to have approximately the same output density, while in the third layer, the parameter for divisive inhibition enables us to control the output density at an arbitrary level. The continuity and orthogonality condition is satisfied since random projection preserves locality.\
Lemma \[middle layer stablity\] describes the probability of activation in the second layer:
By symmetry, the probability of $\beta^T\bm{x}>0$ and $\beta^T\bm{x}<0$ is equal. So we have $\mathop{Pr}\{\beta^T\bm{x}>0\}=\frac{1}{2}\left(1-\mathop{Pr}\{\beta^T\bm{x}>0\}\right).$ We can then write this probability as: $$Pr\{\beta^T\bm{x}=0\}=\sum_{i=1}^{\lfloor|x|/2\rfloor}\binom{|x|}{2i}\binom{2i}{i}p^{2i}(1-2p)^{|x|-2i}$$ By applying Chernoff bound in relative entropy form, we have: $$\sum_{i=2p|x|+1}^{\lfloor|x|/2\rfloor}\binom{|x|}{2i} 2^{2i}p^{2i}(1-2p)^{|x|-2i}\leq e^{-2(2\ln 2-1)p|x|}$$ For the first $2p|x|$ terms, we use Stirling approximation $2^{-k}\binom{2k}{k}=\frac{1}{\sqrt{\pi k}}+O(\frac{1}{k})$ for each term. Since first $p|x|$ terms are smaller than terms with $k\in [p|x|+1,2p|x|]$, we have $$\sum_{i=1}^{2p|x|}\binom{|x|}{2i} \binom{2i}{i}p^{2i}(1-2p)^{|x|-2i}\leq \frac{2}{\sqrt{\pi p|x|}}\sum_{i=p|x|+1}^{2p|x|}\binom{|x|}{2i} (2p)^{2i}(1-2p)^{|x|-2i}\leq \frac{1}{\sqrt{\pi p|x|}}$$ which concludes the proof of this lemma.
Let $q$ be the index where $x_q\neq y_q$, we assume $x_q=0,y_q=1$ without loss of generality. On such circumstance, the only chance to make $g(x)\neq g(y)$ is $\langle \beta_q=1,\beta^Tx=0\rangle$ or $\langle\beta_q=-1,\beta^Tx=1\rangle$. Using the estimate in Lemma \[middle layer stablity\] we can obtain the probability of such events, which leads directly to the result.
In the proof we use Gaussian approximation for binomial variables, the error term can be estimated via Berry-Esseen bound.\
Let the common bits of $x$ and $y$ be vector $w=x\cap y$, we have $x=w+\tilde{x},y=w+\tilde{y}$. We obtain independent random variables $\beta^Tw,\beta^T\tilde{x},\beta^T\tilde{y}$. By applying bivariate central limit theorem to them, we have $$\left[
\begin{matrix}
\beta^Tx\\
\beta^Ty
\end{matrix}
\right]\sim
\mathcal{N}\left(
p(1-2c)
\left[
\begin{matrix}
|x|\\
|y|
\end{matrix}
\right],
v(c,p)
\left[
\begin{matrix}
|x|&|w|\\
|w|&|y|
\end{matrix}
\right]
\right)$$ Then the probability can be written as quadrant integral of bivariate Gaussian distribution, as appears in the lemmas. $$Pr\left\{g(x)\neq g(y)\right\}\approx \int\int_{u\cdot v<0} \mathcal{N}\left(
p(1-2c)
\left[
\begin{matrix}
|x|\\
|y|
\end{matrix}
\right],
v(c,p)
\left[
\begin{matrix}
|x|&|w|\\
|w|&|y|
\end{matrix}
\right]
\right)dudv$$ The integral in Lemma \[CLT second-layer change probability\] centers at origin and admits a closed-form solution. While in Lemma \[CLT third-layer change probability\] we write it as univariate integral with CDF and PDF of Gaussian. Later we will estimate its value given specific parameters.
According to Berry-Esseen inequality, the error introduced in cumulative density functions by Gaussian approximator is bounded with $O(\frac{E[|\beta_i|^3]}{var(\beta_i)^{1.5}\sqrt{n}})=O(\frac{1}{\sqrt{pn}})$.
Now we proceed to prove Theorem \[neural SMA\].
We give proofs for three requirements one by one:
**Stability**: Let $s_1$ denote the probability for middle-layer neurons to fire. According to Lemma \[middle layer stablity\], $s_1$ is restricted in a narrow range: $0\leq \frac{1}{2}-s_1\leq \frac{1}{2\sqrt{\pi p |x|}}+\frac{1}{2}e^{-2(2\ln 2-1)p|x|}$.\
Now that each unit in third layer depends upon the sum of several $i.i.d.$ random variables. By applying central limit theorem to the sum, we can estimate the probability for a output-layer neuron to fire is approximately $$Pr\{h(x)_i=1\}\approx\Phi\left(\sqrt{n}\frac{(1-2c)ps_1}{\sqrt{v(c,ps_1)}}\right)$$ By plugging in the parameter $C_2=\frac{1}{2}-
\frac{s}{2\sqrt{2-s^2}}$ and $s=\Phi^{-1}\left(\frac{r_n}{n}\right)\sqrt{\frac{2}{np}}$, we can show that $$\leq\frac{r_n}{n}-Pr\{h(x)_i=1\}\leq \frac{log(n/r_n)}{\sqrt{s_0n^{1-\gamma}}}$$ Since all output units are $i.i.d.$ with respect to randomness of last layer, we can therefore apply Chernoff’s bound to conclude the stability argument.
**Continuity**: The technique used in our proof is based on bounding the output difference between two vector who has exact one different bit. Fix $d_H(x,y)=L$, we construct a sequence $x=v_0,v_1,v_2,\cdots,v_L=y$ with $d_H(v_{i-1},v_i)=1$.\
For $v_{i-1}$ and $v_{i}$, consider the probability for each second-layer neuron to be different. According to Lemma \[middle one-bit change probablity\], we can estimate $Pr\{g(v_i)\neq g(v_{i-1})\}\approx \frac{1}{\pi}\cos^{-1}\frac{|v_{i-1}\cap v_{i}|}{\sqrt{|v_{i}|\cdot |v_{i-1}|}}$. By union bound we have the following probabilistic triangle inequality: $$Pr\{g(x)\neq g(y)\}\leq Pr\{\bigcup_{i=1}^{L}g(v_i)\neq g(v_{i-1})\}\leq \sum_{i=1}^{L}Pr\{g(v_i)\neq g(v_i-1)\}$$ Lemma \[middle layer stablity\] tells us that both $x$ and $y$ activate approximately $\frac{n}{2}$ neurons in the middle layer, which satisfies the requirement of Lemma \[CLT third-layer change probability\]. By applying Lemma \[CLT third-layer change probability\] with $\eta=O(L\sqrt{\frac{p}{s_0n}})$, we derive the following (approximate) upper bound for probability of changed bits in third layer: $$O(1)\cdot\int_{\mathbb{R}}{\varphi(t)\Phi\left(-\frac{(s_0n^{1+\gamma})^{\frac{1}{4}}}{\sqrt{L}}|t-\Phi^{-1}(\frac{r_n}{n})|\right)dt}$$ We estimate the integral by dividing it into three parts:\
Let $t_{1},t_{2}=\Phi^{-1}(\frac{r_n}{n})\pm\frac{2\log n\sqrt{L}}{(s_0n^{1+\gamma})^{\frac{1}{4}}}$ respectively. We have $$\int_{(t_1,t_2)}{\varphi(t)\Phi\left(-\frac{(s_0n^{1+\gamma})^{\frac{1}{4}}}{\sqrt{L}}|t-\Phi^{-1}(\frac{r_n}{n})|\right)dt}\leq \int_{(t_1,t_2)}\phi(t)dt\leq \frac{1}{\sqrt{2\pi}}(t_2-t_1)$$ For the integral value outside this interval, we bound it with Gaussian tail: $$\int_{t>t_2}{\varphi(t)\Phi\left(-\frac{(s_0n^{1+\gamma})^{\frac{1}{4}}}{\sqrt{L}}|t-\Phi^{-1}(\frac{r_n}{n})|\right)dt}\leq \int_{t>t_2}{\varphi(t)\Phi\left(-2\log n\right)dt}=O\left( \frac{1}{n^2}\right)$$ (For $t<t_1$, we have very similar bounds)\
Summarizing the calculation above, we can get an approximate upper bound $$Pr\{h(x)_i\neq h(y)_i\}=O\left(\frac{\log n\sqrt{L}}{(s_0n^{1+\gamma})^{\frac{1}{4}}}\varphi(\Phi^{-1}(\frac{r_n}{n}))\right)$$ So the probability for each output neuron to differ is upper bounded with $\frac{\mu_n\sqrt{L}}{n}$. We then apply the Chernoff bound in relative entropy form and conclude the probabilistic upper bound.\
**Orthogonality**: For $x,y\in \{0,1\}^n$ such that $d_H(x,y)\geq an$, we have $$\frac{|x\cap y|}{\sqrt{|x|\cdot |y|}}\leq \sqrt{\frac{|x\cap y|}{|x\cup y|}}\leq \sqrt{1-a}$$ According to Lemma \[CLT second-layer change probability\], the probability of making differences at second layer is at least $$Pr\{g(x)\neq g(y)\}\geq b_1\triangleq\frac{1}{\pi}\cos^{-1}\sqrt{1-a}$$ Then we apply Lemma \[CLT third-layer change probability\] with $\eta=b_1$: $$Pr\{h(x)_i\neq h(y)_i\}\geq\int_{\mathbb{R}}\varphi(\frac{t}{\sqrt{1-\frac{1}{2}b_1}})\Phi(\frac{|t-\Phi^{-1}(r_n/n)|}{\sqrt{b_1}})dt$$ Assuming $b_1$ constant, we can obtain the following lower bound for this integral: $$\begin{split}
Pr\{h(x)_i\neq h(y)_i\}\geq &\int_{\mathbb{R}}\varphi(\frac{t}{\sqrt{1-\frac{1}{2}b_1}})\Phi(\frac{|t-\Phi^{-1}(r_n/n)|}{\sqrt{b_1}})dt\\
\geq&\int_{|t-\Phi^{-1}(\frac{r_n}{n})|\leq \sqrt{b_1}}\varphi(\frac{t}{\sqrt{1-\frac{1}{2}b_1}})\Phi(\frac{|t-\Phi^{-1}(r_n/n)|}{\sqrt{b_1}})dt\\
\geq&\int_{|t-\Phi^{-1}(\frac{r_n}{n})|\leq \sqrt{b_1}}\varphi(\frac{t}{\sqrt{1-\frac{1}{2}b_1}})\Phi(-1)dt\\
\geq&\Phi(-1)\Phi\left(\frac{\Phi^{-1}(r_n/n)+\sqrt{b_1}}{\sqrt{1-\frac{1}{2}b_1}}\right)-\Phi(-1)\Phi\left(\frac{\Phi^{-1}(r_n/n)-\sqrt{b_1}}{\sqrt{1-\frac{1}{2}b_1}}\right)\\
\geq&\frac{br_n}{n}
\end{split}$$ The value of $b$ depends upon neither $r_n$ nor $n$, but only $b_1$, which is assumed to be a constant. So we have proved a lower bound for the probability of third-layer nodes to be different. Since those nodes are $i.i.d.$ with respect to the randomness in the last layer. By applying Chernoff bound we can get this result.
[^1]: Key Laboratory of Machine Perception, School of EECS, Peking University. Email: `[email protected]`
[^2]: Key Laboratory of Machine Perception, School of Mathematics, Peking University. Email: `[email protected]`
[^3]: Key Laboratory of Machine Perception, School of EECS, Peking University. Email: `[email protected]`
|
---
abstract: 'We propose that a significant fraction of the ultracompact regions found in massive star-forming clouds are the result of the interaction of the wind and ionizing radiation from a young massive star with the clumpy molecular cloud gas in its neighbourhood. Distributed mass loading in the flow allows the compact nebulae to be long-lived. In this paper, we discuss a particularly simple case, in which the flow in the region is everywhere supersonic. The line profiles predicted for this model are highly characteristic, for the case of uniform mass loading. We discuss briefly other observational diagnostics of these models.'
author:
- |
J.E. Dyson, R.J.R. Williams and M.P. Redman\
Department of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL
date: 'Received \*\*INSERT\*\*; in original form \*\*INSERT\*\*'
title: |
Clumpy ultracompact H[**II**]{} regions I:\
Fully supersonic wind-blown models
---
\#1\#2[[\#1]{}$\,\scriptstyle\rm\uppercase{#2}$]{} \#1[[(\[\#1\])]{}]{} \#1[[[\#1]{}]{}]{} \#1\#2[\#20=]{} \#1\#2\#3[\#2\#2\#3]{} \#1[\_[\#1]{}]{} ‘|=|=16=2.7pt17=2.7pt 16=3.0pt17=3.0pt \#1[$\times10^{#1}$]{}
\[firstpage\]
hydrodynamics – stars: mass-loss – ISM: structure – regions – radio lines: ISM
Introduction
============
The formation of massive stars is an important, but vexed, question, largely because such stars disrupt their natal environment by their powerful winds and ultraviolet radiation fields. The ultracompact regions ([[UCH[ii]{}R]{}]{}), which are found deep within molecular clouds, may provide significant information on the properties and environment of the very youngest massive stars. [[UCH[ii]{}R]{}]{} are highly obscured at optical wavelengths and most information on them derives from infrared and high frequency radio data. A comprehensive overview of their properties is given by Churchwell (1990). They have high emission measures ($\langle n|e^2 \rangle L \ga 10^7\cm^{-6}\parsec$) and small scale sizes ($L\la 10^{17}\cm$) and thus high r.m.s. electron densities ($\langle n|e^2\rangle^{1/2} \ga 10^4\cm^{-3}$). The ionizing photon supply rates necessary to maintain the observed ionization are in the approximate range $10^{44}$–$10^{49}\secnd^{-1}$, corresponding roughly to ZAMS spectral types B2–O5 for single star ionization. There is, however, considerable evidence from comparison of IR and radio data, that groups of stars may be embedded within a given [[UCH[ii]{}R]{}]{} and also that the dust content of the region may absorb nearly all the UV photons generated by the stars (Kurtz, Churchwell & Wood 1994). Consequently, the assignment of a unique spectral type to the exciting source is an extremely uncertain procedure.
[[UCH[ii]{}R]{}]{} have a wide range of observed morphologies. Churchwell (1990) defines 5 main morphological types: cometary ($\sim 20$ per cent), core-halo ($\sim 16$ per cent), shell ($\sim 4$ per cent), irregular or multiply peaked ($\sim 17$ per cent) and spherical and unresolved ($\sim 43$ per cent). Most theoretical attention has been given to cometary regions ( Van Buren & Mac Low 1992). They can be modelled fairly convincingly as the steady-state partially ionized structures behind bow shocks driven into molecular cloud material by the winds of moving stars.
Although Mac Low (1991) suggest that morphologies other than cometary (in particular core-halo) can be explained as cometary structures viewed along the axis of symmetry, it seems likely that other models need consideration. For example, Kurtz, Churchwell & Wood (1994) give evidence for a size-density relationship for spherical or unresolved regions. This is neither present nor would be expected for cometary [[UCH[ii]{}R]{}]{}. Dyson (1994) suggested that a natural explanation for at least some [[UCH[ii]{}R]{}]{} arises from the interaction between an early type star and a very clumpy, as opposed to relatively homogeneous, molecular cloud. Molecular clouds are well established to be clumpy on length scales down to the limits of observational resolution. Clumps can act as localised reservoirs of gas which can be injected into the surroundings by photoionization and/or hydrodynamic ablation. The continuous injection of material into a region can lead to a quasi steady-state configuration which is bounded by a recombination front (RF) as opposed to the ionization front (IF) which bounds standard regions. This particular configuration avoids the expansion problems associated with a ‘classical’ region model for [[UCH[ii]{}R]{}]{}, which would expand on a timescale far shorter than the characteristic $10^{5-6}\yr$ lifetime of [[UCH[ii]{}R]{}]{} estimated from the statistics of [[UCH[ii]{}R]{}]{} association with early type stars (Churchwell 1990). regions bounded by recombination fronts can be held in equilibrium by gas pressure or recoil pressure depending on whether the flow exits sub- or super-sonically.
We discuss here a particularly simple model where a star with both a UV radiation field and a powerful hypersonic wind interacts with clumpy material and show that it may explain the structure of shell-like [[UCH[ii]{}R]{}]{}. Further papers in this series will investigate in detail the full range of models outlined by Dyson (1994). A future key test of all these models will be the observational determination of the kinematics of ionized gas, molecular gas and neutral hydrogen associated with such [[UCH[ii]{}R]{}]{}.
Non-steady models with photoionization-induced mass injection have also been suggested by Lizano (1995). One major difference between their models and those given here and by Dyson (1994) is that Lizano assume a very specific model for the mass injection process which leads to a close coupling between the spatial and temporal behaviour of the mass injection and the radiation field. We treat mass injection as a free parameter and assume here that it is constant in space and time. Dyson (1994) has discussed some of the factors involved in the mass injection process and emphasised that both hydrodynamics and photoinjection may play roles. It is clear that both approaches need investigation.
[[UCH[ii]{}R]{}]{} as wind driven flows in clumpy molecular clouds
==================================================================
In this initial paper, we assume that the hypersonic wind from a single star blows into a clumpy molecular cloud and that mass injection into the flow occurs at a constant volume rate $\dot{q}$ near to the star. Mass injected into a fast wind will slow the wind down. The frictional energy dissipated in this process may be radiated away by, for example, enhanced radiative losses in boundary layers (Hartquist & Dyson 1993). We assume that this occurs here and that the flow is photoionized by the stellar radiation field and remains isothermal at the usual photoionization equilibrium temperature $T\simeq 10^4\Kelv$. A more detailed discussion of the isothermality of mass loaded winds is given elsewhere (Williams, Hartquist & Dyson 1995).
We assume that mass loading occurs only in the ionized region and that the flow always remains supersonic. We neglect the possibility of global shocks in the flow produced by mass injection. This will be a valid assumption provided that the Mach number in the ionized region is not predicted to fall below about 2 (Williams 1995). The recombination front which then occurs has supersonic inflow and outflow ( Newman & Axford 1968). A discussion of the consequences of global shocks and subsequent transonic flow for RF structures is given elsewhere (Williams & Dyson, in preparation).
For steady supersonic flow with uniform mass loading, the mass and momentum conservation equations are respectively $$\rho u = {1\over3}\dot{q}r\label{e:masscons}$$ and $$4\pi r^2\rho u^2 = \dot{M}_\star v_\star \equiv {{\dot{\mu}_\star}},\label{e:momm}$$ where $u$ and $\rho$ are respectively the flow velocity and density, $r$ is the radial coordinate and $\dot{M}_\star$ and $v_\star$ the stellar wind mass loss rate and velocity. We have assumed that the mass flux in the flow is dominated by gas ablated from the clumps. Then from equations and , $$u = {3{{\dot{\mu}_\star}}\over4\pi\dot{q}r^3};\quad
\rho = {4\pi\dot{q}^2r^4\over9{{\dot{\mu}_\star}}}.\label{e:struct}$$ Clearly, radiation such as bremsstrahlung and recombination lines, which have emissivities proportional to $\rho^2$, will be dominated by the outer regions of the flow.
The RF occurs at a radius, $R|R$, where the photon output rate balances the recombination rate. We assume an effective stellar output of $S_\star\,\mbox{photons}\secnd^{-1}$ in the Lyman continuum (which may be only a small fraction of that actually generated by the star because of absorption by dust). We assume (and justify later) that if mass injection is a result of photoionization, it needs only a negligibly small fraction of the available UV. If the observed nebula is ionization bounded, then it must be powered by a Lyman continuum flux of $$S_\star =\int_0^{R|R}4\pi {r'}^2 n^2 \beta_2 \id{r'},\label{e:front}$$ where $n$ ($\equiv \rho/\bar m$) follows from equation , $\bar m$ is the mean mass per nucleon and is the hydrogen recombination coefficient. Thus from equation $$\dot{q} = \left(891\over64\pi^3\right)^{1/4}
{{\dot{\mu}_\star}}^{1/2} \bar{m}^{1/2}S_\star^{1/4}\beta_2^{-1/4}R|R^{-11/4}.
\label{e:massrate}$$ We treat $\dot{q}$ as a free parameter since it is presumably determined by factors such as the density of mass loading centres and the mode of ablation (Section 1).
Equation implies that we can neglect any appreciable contributions to the radio emission from the mass loading centres themselves ( from dense ionized clump surfaces), and so the measured r.m.s. electron density is that of the flow and is given by $$\langle n|e^2\rangle^{1/2} =
\left[{3\over R|R^3}\int_0^{R|R}
\left(16\pi^2\dot{q}^4\over81{{\dot{\mu}_\star}}^2\bar{m}^2\right)
r^{10}\id{r}\right]^{1/2}.$$ From equation therefore $$\langle n|e^2\rangle^{1/2} =
\left(3\over 4\pi\right)^{1/2} S_\star^{1/2} R|R^{-3/2}\beta_2^{-1/2}.
\label{e:modelne}$$ The Mach number just before the RF is $$M|R \equiv {u|R\over c|i} = {3{{\dot{\mu}_\star}}\over4\pi\dot{q}R|R^3c|i}
\label{e:modelmach}$$ Inserting characteristic values $S_{48} \equiv S_\star/10^{48}\secnd^{-1}$; $\dot\mu_{28} \equiv {{\dot{\mu}_\star}}/10^{28}\gram\cm\secnd^{-2}$; $R_{17} \equiv R|R/10^{17}\cm$; $c|i =10\kms$ and $\bar{m} = 2\ee{-24}\gram$, equations and become $$\begin{aligned}
\langle n|e^2\rangle^{1/2} &= &
3.5\ee{4} S_{48}^{1/2} R_{17}^{-3/2}\cm^{-3}\label{e:densmod} \\
M|R & = &
0.78 S_{48}^{-1/4}R_{17}^{-1/4}\dot\mu_{28}^{1/2}.\label{e:machmod}\end{aligned}$$ The mass injection rate required is $$\dot{q} =
2.4\ee{-30}\dot\mu_{28}R_{17}^{-3}M|R^{-1} \gram\cm^{-3}\secnd^{-1}.$$
The Mach number at the recombination front from equation is smaller than required, for the values of $S_{48}$ and $R_{17}$ derived from observation, unless the momentum input (for the observed $S_{48}$) is considerably larger than that found for field OB stars. Dust significantly reduces the fraction of $S_\star$ which ionizes the nebula (and thus increases the effective $\mu_{28}^2/S_{48}$); however, in the sample of Kurtz , the fraction of the stellar luminosity absorbed by dust is less for the earlier type stars which would have stronger winds. The presence of multiple stars within a single [[UCH[ii]{}R]{}]{} may also have an effect.
It is certain that the flow will have significant density inhomogeneities. These can decrease the radius of the recombination front for given mass fluxes and ionizing luminosity. We assume that the inhomogeneities have a large density contrast, and that the dense gas dominates both the emission and the mass flux compared to the more tenuous surrounding gas. In this case, the Mach number of the recombination front \[equation \] scales as $M|R\propto
f^{-1/4}$, where $f$ is the volume filling factor of the dense regions; the mean density of the flow at any radius is $f^{1/2}$ times smaller than that derived (for observed $S_\star$ and $R|R$) by assuming a smooth flow. High resolution images of [[UCH[ii]{}R]{}]{} do indeed show a strongly clumped structure ( Kurtz 1994). In the present paper we assume that the dense gas has a large ($\simeq 1$) covering factor as seen from the central star, and that the dense gas is well coupled to the global flow. Exploration of these assumptions demands a treatment of intermediate scale structure in the flow (Dyson, Hartquist & Williams, in preparation). We have also assumed here that the filling factor is independent of radius.
To see whether the required mass injection is plausible and also compatible with the assumptions in the model we treat the very special case where the mass injection comes from the surfaces of pressure confined self-gravitating isothermal clumps (Dyson 1968; Kahn 1969). Kahn (1969) showed that the mass loss rate from such a photoionized globule had the remarkable property of being dependent only on the sound speed $c|n$ in the neutral gas of the globule. We adopt his mass loss rate from a globule $\dot{m}|g \simeq 10^{20}
c_3^4\gram\secnd^{-1}$, where the sound speed in the cloud is $c|n =
0.3c_3\kms$ to allow for its being cold and molecular. The number of clumps $N|c$ required within the [[UCH[ii]{}R]{}]{} is thus $N|c \simeq
(4\pi/3)R|R^3\dot{q}/\dot{m}|g \simeq 130
R_{17}^{1/4}f^{1/4}S_{48}^{1/4}\dot\mu_{28}^{1/2}c_3^{-4}$ which appears plausible. The fraction, $\phi$, of photons used up in the photoinjection process itself is $\phi = (4\pi
R|R^3\dot{q}/3\bar{m}S_\star) \simeq 6\ee{-3}
R_{17}^{1/4}f^{1/4}S_{48}^{-3/4}\dot\mu_{28}^{1/2}$, which is consistent with the assumptions made.
Mass loss driven by ionization from gravitationally confined clumps is just one of a wide range of possibilities. Hydrodynamical ablation may play an important role in releasing gas from the clumps, for instance, as may magnetic fields in maintaining inhomogeneities within a global flow. There must be several tens of mass loading clumps within the [[UCH[ii]{}R]{}]{}, if our continuum approximation is to be reasonably valid. There is strong observational evidence for such clumping in the emitting gas, ten or more individual emission peaks being visible in the better-resolved images in the survey by Kurtz, Churchwell & Wood (1994).
This characteristic number of clumps implies a space density of $\sim
10^6 \parsec^{-3}$. The clumps may be related to the partially ionized globules (PIGS) and protoplanetary disks around low mass stars (proplyds, O’Dell & Wen 1994) observed distributed in Orion nebula. The peak number density of stars in the low mass stellar cluster around the Trapezium OB stars is at least $5\ee4 \parsec^{-3}$ (McCaughrean & Stauffer 1995). While the observed velocity structure of these sources (Massey & Meaburn 1995) may well be similar to that of the mass-loading clumps in [[UCH[ii]{}R]{}]{}, the Orion nebula is a far older region than those discussed here. It is likely that in the near vicinity of a very young massive star, there will also be many, more transient, local density enhancements which have not quite passed the threshold for star formation. In $10^5\yr$, between 1 and $10\Msun$ of gas will pass through a near-sonic recombination front, suggesting a mean density of clump gas in the region ($3\ee5-3\ee6\cm^{-3}$), comparable to the mean mass density of the proplyd stars in Orion.
It is also interesting to compare the Keplerian velocity at the edge of an [[UCH[ii]{}R]{}]{} (around $1\kms$) and the relative velocity necessary to transit the [[UCH[ii]{}R]{}]{} in its lifetime (around $0.3\kms$) to the mean velocity dispersion of the Trapezium stars (around $3\kms$). A significant flux of new clumps will enter the [[UCH[ii]{}R]{}]{} during its lifetime, either through random motions or systematic infall; the density of clumps within the region may also be significantly enhanced by gravitational focussing effects.
A simple model [[UCH[ii]{}R]{}]{}
=================================
In order to satisfy the requirement that no global shocks occur in the [[UCH[ii]{}R]{}]{}, we require $M|R\ga 2$ (Section 2), $(fS_{48}R_{17}/\dot\mu_{28}^2)\la 0.03$. We therefore choose as an illustrative set of parameters $f=0.02$, $S_{48} = 1$, $R_{17} = 1$, $\dot\mu_{28} = 1$. We then find the observed r.m.s. density, $\langle n|e^2\rangle^{1/2} \simeq 3.5\ee4\cm^{-3}$ (equivalent to an r.m.s. density in the smoothed flow of $5\ee3\cm^{-3}$), $\dot{q}\simeq 1.2\ee{-30}\gram\cm^{-3}\secnd^{-1}$, $\phi=2.3\ee{-3}$ and $M|R=2.2$.
------- -------
=by 2 =by 2
------- -------
In Figure \[f:emplot\], we show the emission measure, ${\rm EM} =
\int n^2 \id{l}$, as a function of offset $z$ from the star (where $z$ is given in units of $R|R$). The total emission is normalized so that $$\int_0^1 {\rm EM}(z) 2\pi z \id{z} = 1.$$ The rapid rise of density close to the recombination front, equation , leads to an edge brightened structure in both line flux and in the far stronger free-free continuum. The peak intensity is more than twice that at the centre of the of the region, for $M|R=2.2$.
In Figure \[f:lineprof\], we show line profiles for an optically thin recombination line ($I\propto n^2$) at $z=0.1$ and $0.7$. For flows with Mach numbers $m\ga2$ at the recombination front, the predicted line profiles are symmetric but noticeably double-peaked, particularly in the weaker emission lines close to the centre of the nebula. For smaller Mach numbers at the RF, the line profiles will be singly peaked, although noticeably broadened and variable across an individual [[UCH[ii]{}R]{}]{}. Continuum optical depths may lead to a systematic blueshift of the emission in lower frequency recombination lines. Clearly, observations of radio recombination line profiles is a key test of the model presented here.
In Figure \[f:simplemod\], we show the structure of the [[UCH[ii]{}R]{}]{}calculated for the simple model parameters given above, assuming that the gas is everywhere isothermal, at $T = 8000\Kelv$ (cf. Arthur, Dyson & Hartquist 1994). This is a fairly reasonable assumption within the ionized nebula. Outside the recombination front, the gas would be expected to cool adiabatically; however, since the gas is supersonic here, the cooling will have little dynamical effect. Beyond the ionized region, the Mach number plotted should be interpreted as the flow velocity scaled to the isothermal sound speed. Note that the neutral hydrogen emerging from the RF has a velocity of $45\kms$. Clearly, the observation of neutral material with high radial velocities close to the projected centre of an [[UCH[ii]{}R]{}]{} would again constitute a crucial test of such models.
Discussion
==========
In this paper, we have discussed one of the mechanisms for the formation of ultracompact regions introduced by Dyson (1994). Winds around young massive stars mass-loaded by the remaining shreds of the molecular cloud from which the star initially formed can naturally explain many of the properties of spherical [[UCH[ii]{}R]{}]{}.
For parameters characteristic of spherical [[UCH[ii]{}R]{}]{}, the mass-loaded stellar wind will remain supersonic at all radii, so long as the radiating gas has a filling factor $f\la 0.02$. This filling factor is close to the preferred range $0.03-0.10$ quoted by Afflerbach (1994) from a non-LTE analysis of recombination line ratios for the (cometary) [[UCH[ii]{}R]{}]{} G29.96$-$0.02, although limb brightening on a finer scale than the observational resolution would bias their results towards small values.
We have shown predicted line profiles and distribution of emission measure across and [[UCH[ii]{}R]{}]{}. The double-peaks of line profiles, particularly close to the centre of the nebula, are a strong observational diagnostic, so long as sufficient spatial and spectral resolution can be obtained. Observations of sufficient quality are now beginning to become available for analysis.
In future papers, we will extend the work presented here to treat centre-bright nebulae (most naturally understood as flows which are subsonic for a substantial range of radii). We will also include more details of the radiation transfer process (such as continuum optical depths and stimulated emission), of the wind-clump interfaces and of the distribution and dynamical effects of dust. Preliminary models incorporating more realistic heating and cooling rates predict temperatures appreciably greater than $10^4\Kelv$ in the inner regions. This will result in edge brightening even more pronounced than that shown above (Williams, Dyson & Redman, in preparation).
Radio recombination lines from helium and carbon, while fainter than those of hydrogen will give better velocity resolution; the differences in ionization potential between these atoms means that comparison of their line profiles will yield important additional information on the radial structure of the nebula.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish thank the anonymous referee, in particular for suggesting that we extend the discussion of the nature of clumps, and Robert Massey for discussions on the Trapezium cluster. This work was supported by PPARC both through the Rolling Grant to the Astronomy Group at Manchester (RJRW) and through a Graduate Studentship (MPR).
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\[lastpage\]
|
---
abstract: 'For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\F_q$ that do not share a common component and intersect in exactly $k\ \F_q$-rational points. We set this up as a problem about a weight enumerator of a certain projective Reed-Muller code. The main inputs to the proof include counting pairs of cubic curves that do share a common component, counting configurations of points that fail to impose independent conditions on cubics, and a variation of the MacWilliams theorem from coding theory.'
address:
- |
Mathematics Department\
University of California Irvine\
Irvine, CA 92697
- |
Mathematics Department\
University of California Irvine\
Irvine, CA 92697
author:
- Nathan Kaplan
- Vlad Matei
title: Counting Plane Cubic Curves over Finite Fields with a Prescribed Number of Rational Intersection Points
---
Introduction {#sec:intro}
============
The goal of this paper is to answer the following question:
\[Q1\] Bézout’s theorem implies that a pair of plane cubic curves that do not share a common component intersect in at most $9$ points. Let $k \in [0,9]$ be an integer. **How many pairs of plane cubic curves defined over $\F_q$ do not share a common component and intersect in exactly $k\ \F_q$-rational points?**
**Note**: In this paper we use the phrase *common component* to mean common component defined over $\overline{\F_q}$.
Let $\F_q[x_0,\ldots, x_n]_d$ denote the $\binom{n+d}{d}$-dimensional vector space of homogeneous degree $d$ polynomials in $\F_q[x_0,\ldots, x_n]$. It does not make sense to evaluate a polynomial $f \in \F_q[x_0,\ldots, x_n]_d$ at a point of $\Proj^n(\F_q)$, since for any $\alpha \in \F_q^*$, $$f(\alpha x_0, \alpha x_1,\ldots, \alpha x_n) = \alpha^d f(x_0,x_1,\ldots, x_n).$$ However, it does make sense to ask whether $f$ is zero at a point of $\Proj^n(\F_q)$ or not. Therefore, we can rephrase Question \[Q1\] as follows.
\[Q2\]
1. For each $k \in [0,9]$, how many of the $q^{20}$ ordered pairs $(f,g)$ with $f,g \in \F_q[x,y,z]_3$ have exactly $k$ common zeroes in $\Proj^2(\F_q)$?
2. More precisely, how many such pairs $(f,g)$ do not have a common irreducible factor over $\overline{\F_q}$ and have exactly $k$ common zeroes in $\Proj^2(\F_q)$?
This is the main question that we answer in this paper.
\[ThmIntCubics\] Let $\F_q$ be a finite field of size $q > 2$. For each $k \in [0,9]$, let $c_k$ denote the number of pairs $(f,g)$ with $f,g \in \F_q[x,y,z]_3$, both nonzero, that do not have a common irreducible factor over $\overline{\F_q}$ and have exactly $k$ common zeros in $\Proj^2(\F_q)$. We have $$\begin{aligned}
c_0 & = &
\frac{16687}{45360} \cdot (q + 1)^{2} \cdot (q - 1)^{3}
\cdot q^{5} \cdot (q^{2} + q + 1) \cdot \\
& & \left(q^{8} - q^{7} +
\frac{15988}{16687} q^{6} + \frac{882}{16687} q^{5} - \frac{126}{451}
q^{4} + \frac{3192}{16687} q^{3} + \frac{4397}{16687} q^{2} -
\frac{2507}{16687} q - \frac{2170}{16687}\right) \\
c_1 & = & \frac{2119}{5760} \cdot (q + 1) \cdot (q - 1)^{2} \cdot
q^{3} \cdot (q^{2} + q + 1) \cdot \\
& & \bigg(q^{12} - \frac{1}{14833} q^{11} +
\frac{2390}{2119} q^{10} - \frac{10240}{14833} q^{9} +
\frac{2459}{2119} q^{8} + \frac{99}{2119} q^{7} + \frac{3440}{2119}
q^{6} \\
& & - \frac{8630}{14833} q^{5} - \frac{4748}{2119} q^{4} +
\frac{76978}{14833} q^{3} + \frac{100}{2119} q^{2} - \frac{14160}{2119}
q + \frac{5760}{2119}\bigg) \\
c_2 & = & \frac{103}{560} \cdot (q - 1)^{2} \cdot (q + 1)^{2} \cdot
q^{4} \cdot (q^{2} + q + 1) \cdot \bigg(q^{10} + \frac{1}{927} q^{9} +
\frac{1634}{927} q^{8} + \frac{742}{927} q^{7} \\
& & + \frac{1589}{927} q^{6}
+ \frac{1729}{927} q^{5} + \frac{4106}{927} q^{4} - \frac{2818}{103}
q^{3} + \frac{21608}{309} q^{2} - \frac{22610}{309} q +
\frac{2520}{103}\bigg) \\
c_3 & = & \frac{53}{864} \cdot (q + 1)^{2} \cdot (q - 1)^{3} \cdot
q^{4} \cdot (q^{2} + q + 1) \cdot \bigg(q^{9} + \frac{527}{265} q^{8} +
\frac{221}{53} q^{7} + \frac{1533}{265} q^{6}\\
& & + \frac{738}{53} q^{5} +
\frac{5958}{265} q^{4} - \frac{3956}{53} q^{3} + \frac{67402}{265} q^{2}
- \frac{11348}{53} q + \frac{2376}{53}\bigg) \\
c_4 & = & \frac{11}{720} \cdot (q + 1)^{2} \cdot (q - 1)^{3} \cdot
q^{4} \cdot (q^{2} + q + 1) \cdot \bigg(q^{9} + \frac{34}{11} q^{8} +
\frac{48}{11} q^{7} + \frac{182}{11} q^{6} \\
& & + \frac{109}{11} q^{5} -
\frac{1564}{11} q^{4} + 712 q^{3} - \frac{13292}{11} q^{2} +
\frac{7120}{11} q - \frac{600}{11}\bigg) \\
c_5 & = & \frac{1}{320} \cdot (q + 1)^{2} \cdot (q - 1)^{4} \cdot
q^{4} \cdot (q^{2} + q + 1) \cdot \bigg(q^{8} + \frac{40}{9} q^{7} +
\frac{151}{9} q^{6} + \frac{50}{9} q^{5} \\
& & - \frac{874}{9} q^{4} +
\frac{2890}{3} q^{3} - \frac{7022}{3} q^{2} + \frac{4940}{3} q - 40\bigg) \\
c_6 & = & \frac{1}{2160} \cdot (q + 1)^{2} \cdot (q - 1)^{4} \cdot
q^{5} \cdot (q^{2} + q + 1) \cdot \\
& & \bigg(q^{7} + 9 q^{6} - 5 q^{5} - 17 q^{4} + 910 q^{3} - 4316 q^{2} + 7416 q - 4670\bigg) \\
c_7 & = & \frac{1}{10080} \cdot (q - 2) \cdot (q + 1)^{2} \cdot (q -
1)^{4} \cdot q^{5} \cdot (q^{2} + q + 1)\cdot \\
& & \bigg(q^{6} + 2 q^{5} + 25
q^{4} + 288 q^{3} - 1692 q^{2} + 3574 q - 3290\bigg) \\
c_8 & = & \frac{1}{5040} \cdot (q - 3) \cdot (q - 2) \cdot (q +
1)^{2} \cdot (q - 1)^{4} \cdot q^{5} \cdot (q^{2} + q + 1) \cdot \bigg(q^{4}
+ 6 q^{3} - 31 q^{2} + 69 q - 105\bigg) \\
c_9 & = & \frac{1}{362880} \cdot (q - 2) \cdot (q + 1)^{2} \cdot (q
- 1)^{4} \cdot q^{5} \cdot (q^{2} + q + 1) \cdot \\
& & \bigg(q^{6} + 2 q^{5} - 73
q^{4} + 344 q^{3} - 838 q^{2} + 1754 q - 2030\bigg).\end{aligned}$$
\[Rem2\]
1. Each $c_k$ is a polynomial in $q$ of degree $20$, except $c_8$ which is a polynomial degree $19$. The $q^{20}$ coefficient of each $c_k$ is the proportion of elements in the symmetric group $S_9$ that have exactly $k$ fixed points. These ‘main terms’ follow from recent work of Entin [@Entin], which we discuss in more detail below.
2. One could consider the analogous problem, counting the number of pairs $(f,g)$ with $f\in \F_q[x,y,z]_d$ and $g\in \F_q[x,y,z]_e$ that have a given number of common zeros in $\Proj^2(\F_q)$. On the way to proving Theorem \[ThmIntCubics\] we give analogous, but far simpler, answers in the cases $(d,e) = (2,2)$ and $(d,e) = (3,2)$.
In forthcoming work, we investigate the case where $e=2$ and $d$ is arbitrary, and obtain polynomial formulas. The case of two cubics seems to be a kind of boundary. We do not expect any case where $d,e \ge 3$ and $(d,e) \neq (3,3)$ for which the number of polynomials with exactly $de$ common zeros in $\Proj^2(\F_q)$ is a polynomial in $q$. We discuss this further in Section \[sec:further\].
Relationship to Previous Work
-----------------------------
The main terms that occur in the statement of Theorem \[ThmIntCubics\], and in the analogous results for $(d,e) \in \{(2,2), (3,2)\}$, follow from recent work of Entin [@Entin]. Two projective plane curves defined over $\F_q$ that intersect transversely, one of degree $d$ and one of a degree $e$, give rise to a permutation in $S_{de}$ corresponding to the action of Frobenius, $\Fr_q$, on the $de$ intersection points of the two curves. The following result is a version of [@Entin Corollary 1.3].
\[ThmEntin\] Let $d$ and $e$ be positive integers and let $k \in [0,de]$ be an integer. Let $c_k(d,e)$ denote the number of pairs $(f,g)$ with $f \in \F_q[x,y,z]_d$ and $g \in \F_q[x,y,z]_e$ that have exactly $k$ common zeros in $\Proj^2(\F_q)$. Then $$c_k(d,e) = \frac{\pi(k,de)}{(de)!} q^{\binom{d+2}{2}\binom{e+2}{2}}\left(1+\text{O}_{de}(q^{-1/2})\right),$$ where $\pi(k,de)$ is the number of permutations in $S_{de}$ with exactly $k$ fixed points.
The approach taken in [@Entin] is to show that the monodromy group relevant to this problem is the symmetric group $S_{de}$. The Frobenius action gives rise to all possible permutations in this group, and applying the Chebotarev density theorem gives this quantitative result. Because of the use of Chebotarev density, this approach cannot give precise quantitative statements like the one in Theorem \[ThmIntCubics\], and cannot distinguish between polynomial formulas and non-polynomial formulas for these kinds of counting problems.
The statement of Theorem \[ThmEntin\] does not include any assumptions about transversality since [@Entin Corollary 1.3] also implies that the number of pairs of polynomials defining curves with non-transversal intersection can be absorbed into the error term.
Theorem \[ThmIntCubics\] fits into a body of literature on error-correcting codes that come from families of genus $1$ curves. For an introduction to these ideas, see the surveys of Hurt [@Hurt], and Schoof [@SchoofSurvey]. There is an extensive discussion of classical algebraic geometry codes arising from elliptic curves in [@TVN Section 4.4.2]. In Section \[sec:background\] we review results about the projective Reed-Muller code whose codewords come from plane cubic curves. Using results of Deuring, Waterhouse, and Schoof [@Deuring; @Schoof; @Waterhouse], Elkies computes the weight enumerator of this code in [@Elkies]. Kaplan studies the weight enumerator of the dual of this code, and the Reed-Muller that comes from affine plane cubic curves in [@KaplanAGCT]. van der Geer, Schoof, and van der Vlugt [@vdGSvdV], and Schoof and van der Vlugt [@SvdV], study Zetterberg and Melas codes whose codewords come from families of elliptic curves in characteristics $2$ and $3$. Finally, Kaplan and Petrow study the quadratic residue weight enumerator of a certain Reed-Solomon code in [@KaplanPetrowTAMS]. Computing this weight enumerator involves counting isomorphism classes of elliptic curves with a fixed number of $\F_q$-points and with some additional structure related to the $2$-torsion of $E(\F_q)$.
There are many cases in which results from algebraic geometry and number theory are used to study families of error-correcting codes. For just one reference, see the book [@TVN]. It is much less common to find examples where techniques from coding theory are used to prove new results in arithmetic geometry. This is the perspective we adopt in this paper, as one of the main inputs into the proof of Theorem \[ThmIntCubics\] is a generalization of a classical theorem of MacWilliams about weight enumerators of linear codes and their duals. The main idea for this project is inspired by work of Elkies [@Elkies].
Outline of the Paper
--------------------
In the next section we recall some coding-theoretic background and rephrase Question \[Q2\] as a problem about the second weight enumerator of the projective Reed-Muller code whose codewords correspond to plane cubic curves. We recall some previous results about this code and its dual. In Section \[sec:dual\_coef\], we explain how low-weight coefficients of weight enumerators of duals of Reed-Muller codes are related to interpolation problems in algebraic geometry. We carefully analyze small collections of points that fail to impose independent conditions on conics and cubics. In Section \[sec:twoconics\], we prove analogues of Theorem \[ThmIntCubics\] for intersections of plane conics, and in Section \[sec:con\_cub\], we prove the analogue of Theorem \[ThmIntCubics\] for the intersection of a conic and a cubic. In Section \[sec:large\_int\], we count pairs of cubic curves that share a common component. In Section \[sec:pf\_thm1\], we complete the proof of Theorem \[ThmIntCubics\]. Finally, in Section \[sec:further\] we discuss questions for further study.
[ There is a large computational component to the results of this paper. All computations were done in the computer algebra system Sage. We have made the programs used for this project available at the website of the first author:\
<https://www.math.uci.edu/~nckaplan/research_files/intersections_of_cubics_sage_worksheets/>.]{}
In Section \[sec:dual\_coef\], we use a theorem of Kaplan to compute the number of collections of $9$ points in $\Proj^2(\F_q)$ such that there are two cubics intersecting at these points that do not share a common component. The result that we cite contains the additional restriction that the characteristic of $\F_q$ is not $2$ or $3$. In Appendix \[AppendixA\], we compute the relevant quantity in a different way that works for any $\F_q$, which shows that the first part of [@KaplanAGCT Theorem 3] does hold in characteristics $2$ and $3$.
Weight Enumerators of Reed-Muller Codes and their Duals {#sec:background}
=======================================================
In this section we recall some coding theory background and express Question \[Q2\] as a problem about the second weight enumerator of a projective Reed-Muller code.
A subset $C \subseteq \F_q^n$ is called a *code* of length $n$ over $\F_q$. It is a *linear code* if it is a linear subspace of $\F_q^n$. For $x= (x_1, x_2, \ldots, x_n)$ and $y= (y_1, y_2,\ldots, y_n)$ in $\F_q^n$, the *Hamming distance* between $x$ and $y$ is defined by $$d(x,y) = \#\{i\in [1,n] \mid x_i \neq y_i\}.$$ The *Hamming weight* of $x \in \F_q^n$, denoted $\wt(x)$, is $$\wt(x) = \#\{i\in [1,n] \mid x_i \neq 0\}.$$ The *Hamming weight enumerator* of $C$ is defined by $$\begin{aligned}
W_C(X,Y) & = & \sum_{c \in C} X^{n-\wt(c)} Y^{\wt(c)} = \sum_{i=0}^n A_i X^{n-i} Y^i,\end{aligned}$$ where $$A_i = \#\{c\in C \mid \wt(c) = i\}.$$
The main problems we study are not about zeros of individual polynomials, but about common zeros of pairs of polynomials. We make extensive use of the following variation of the Hamming weight enumerator.
Let $C \subseteq \F_q^n$ be a code. The *second weight enumerator* of $C$ is $$\begin{aligned}
W^{[2]}_C(X,Y) & = & \sum_{c_1, c_2 \in C} X^{n-\wt(c_1, c_2)} Y^{\wt(c_1, c_2)} = \sum_{i=0}^n A^{[2]}_i X^{n-i} Y^i,\end{aligned}$$ where $\wt(c_1, c_2)$ is the number of coordinates in which $c_1$ and $c_2$ do not simultaneously vanish.
The *support* of an element $x \in\F_q^n$, denoted $\supp(x)$, is the set of coordinates in which $x$ is nonzero. That is, $\wt(x) = |\supp(x)|$. If $S \subseteq \F_q^n$ is a linear subspace, its support, denoted $\supp(S)$, is the set of coordinates in which at least one element of $S$ is nonzero. The weight of $S$ is $\wt(S) = |\supp(S)|$. For $x_1,\ldots, x_k \in \F_q^n$, let $\wt(x_1,\ldots, x_k)$ be the weight of the subspace spanned by $x_1,\ldots, x_k$. This is consistent with the definition of $\wt(c_1, c_2)$ given above. We also write $\supp(x_1,\ldots, x_k)$ for the union of the supports of $x_1,\ldots, x_k$, which is the same as the support of the subspace spanned by $x_1,\ldots, x_k$.
There is a variation of the second weight enumerator for two codes that are not necessarily the same.
Let $C_1,C_2 \subseteq \F_q^n$. The second weight enumerator of $C_1, C_2$ is $$\begin{aligned}
W^{[2]}_{C_1, C_2}(X,Y) & = & \sum_{\substack{c_1 \in C_1\\ c_2 \in C_2}} X^{n-\wt(c_1, c_2)} Y^{\wt(c_1, c_2)} = \sum_{i=0}^n A^{[2]}_i X^{n-i} Y^i.\end{aligned}$$ In the case $C_1 = C_2$, we write $W_C^{[2]}(X,Y)$ instead of $W_{C,C}^{[2]}(X,Y)$.
It is often useful to divide up the second weight enumerator of a code $C \subseteq \F_q^n$ by the dimension of the subspace of $\F_q^n$ spanned by the pair $c_1, c_2$. There are $q^2-1$ ordered pairs of vectors that span a chosen one-dimensional subspace of $\F_q^n$ and $(q^2-1)(q^2-q)$ choices for an ordered pair that span a chosen two-dimensional subspace. Therefore, $$\label{EqSupp}
W^{[2]}_C(X,Y) = X^n + (q^2-1) W^{(1)}_C(X,Y) + (q^2-1)(q^2-q) W^{(2)}_C(X,Y),$$ where $$W^{(r)}_C(X,Y) = \sum_{S \subseteq C} X^{n-\wt(S)} Y^{\wt(S)},$$ and the sum is taken over all $r$-dimensional subspaces of $C$. We see that $$W_C(X,Y) = X^n + (q-1) W^{(1)}_C(X,Y).$$
Reed-Muller codes
-----------------
We introduce the main class of codes that we study in this paper. For ease of notation, let $N = |\Proj^n(\F_q)| = (q^{n+1}-1)/(q-1)$. Let $p_1, p_2, \ldots, p_N$ be an ordering of the elements of $\Proj^n(\F_q)$, and let $p_1', p_2',\ldots, p_N'$ be a choice of affine representative for these points.
We define the evaluation map: $$\begin{aligned}
\operatorname{ev} \colon \F_q[x_0,x_1,\ldots, x_n]_d & \mapsto & \F_q^N \\
f & \mapsto & \left(f(p'_1),\ldots, f(p'_N)\right).\end{aligned}$$ It is not difficult to see that $\ev(\alpha f + g) = \alpha\cdot \ev(f) + \ev(g)$, so the image of this map is a linear subspace of $\F_q^N$. As long as there does not exist a degree $d$ polynomial vanishing at every point in $\Proj^n(\F_q)$, which is the case for $q > d$, this map is injective, so the image of this map has dimension $\binom{n+d}{d}$. For the rest of the paper, we write $C_{n,d}$ for $\ev(\F_q[x_0,x_1,\ldots, x_n]_d)$, and refer to it as the *projective Reed-Muller code* of order $d$ and length $N$.
1. This definition depends on an ordering of the points of $\Proj^n(\F_q)$ and on a choice of affine representatives for these projective points. These Reed-Muller codes satisfy a strong form of equivalence. In particular, the weight enumerators that we study in this paper do not depend on these choices. See [@HeijnenPellikaan Section 1.7] or [@KaplanAGCT Remark 1].
2. As in [@KaplanAGCT], the definition of projective Reed-Muller code given here is not the same as the one given by Lachaud [@Lachaud], but it equivalent to it if one makes a standard choice of affine representatives.
3. Throughout this paper we focus on the code $C_{2,3}$, which is $10$-dimensional when $q>2$. Whenever we refer to $C_{2,3}$ we will assume that $q>2$, even if we do not explicit state this assumption.
The classical, or affine, Reed-Muller code is defined as follows. Let $\F_q[x_1,\ldots, x_n]_{\le d}$ denote the $\binom{n+d}{d}$-dimensional vector space of polynomials in $\F_q[x_1,\ldots, x_n]$ with degree at most $d$. Let $p_1, p_2, \ldots, p_{q^n}$ be an ordering of the elements of $\F_q^n$. We define the evaluation map: $$\begin{aligned}
\operatorname{ev} \colon \F_q[x_1,\ldots, x_n]_{\le d} & \mapsto & \F_q^{q^n} \\
f & \mapsto & \left(f(p_1),\ldots, f(p_{q^n})\right).\end{aligned}$$ The image of this map is a linear subspace of $\F_q^{q^n}$, and the evaluation map is injective when $q-1 \ge d$. We write $C^{A}_{n,d}$ for $\ev(\F_q[x_1,\ldots, x_n]_{\le d})$, and refer to it as the *affine Reed-Muller code* of order $d$ and length $q^n$.
1. The affine Reed-Muller code $C_{n,d}^A$ is monomially equivalent to $C_{n,d}$ punctured at the set of $(q^n-1)/(q-1)$ coordinates corresponding to a choice of hyperplane at infinity. See [@KaplanAGCT Section 1] for details.
2. Coordinates of codewords of Reed-Muller codes and their duals correspond to points in projective space. Throughout this paper, we will use the term ‘support’, to refer both to subsets of coordinates of codewords, and also to the underlying collections of points.
We now can state the main problem of this paper in this language of weight enumerators of codes.
Let $$W^{[2]}_{C_{2,3}}(X,Y) = \sum_{i=0}^{q^2+q+1} A^{[2]}_i X^{q^2+q+1-i} Y^i.$$ What are the coefficients $A^{[2]}_{q^2+q+1-9}, A^{[2]}_{q^2+q+1-8}, \ldots, A^{[2]}_{q^2+q+1}$?
The answer to this question is almost equivalent to the statement of Theorem \[ThmIntCubics\]. One difference is that there is a contribution to $W^{[2]}_{C_{2,3}}(X,Y)$ from pairs of codewords $(c_1, c_2)$ such that $c_1, c_2$ span a linear subspace of $\F_q^{q^2+q+1}$ of dimension at most $1$. Another difference is that there is a contribution from pairs $(c_1, c_2)$ for which the corresponding cubic curves are not equal, but share a common component.
The dual code of a linear code and the MacWilliams theorem
----------------------------------------------------------
For $x= (x_1, x_2, \ldots, x_n)$ and $y= (y_1, y_2,\ldots, y_n)$ in $\F_q^n$, let $$\langle x,y \rangle = \sum_{i=1}^n x_i y_i \in \F_q.$$ Let $C \subseteq \F_q^n$ be a linear code. The *dual code* of $C$, denote $C^{\perp}$, is $$C^\perp = \{y \in \F_q^n \mid \langle x,y \rangle = 0\ \forall x\in \F_q^n\}.$$
A theorem of MacWilliams says that the weight enumerator of a linear code determines the weight enumerator of its dual.
\[MacWilliamsThm\] Let $C \subseteq \F_q^n$ be a linear code. Then $$W_{C^\perp}(X,Y) = \frac{1}{|C|} W_C(X+(q-1)Y, X-Y).$$
There is an extensive literature about variations of Theorem \[MacWilliamsThm\]. See for example [@MacWilliamsSloane Chapter 5]. We will make extensive use of the following MacWilliams theorem for the second weight enumerator of $C_1, C_2 \subseteq \F_q^n$. For a proof see [@Shiromoto].
\[MacWilliamsThm2\] Let $C_1, C_2 \subseteq \F_q^n$ be linear codes. We have $$W^{[2]}_{C^\perp_1, C^\perp_2}(X,Y) = \frac{1}{|C_1| \cdot |C_2|}W^{[2]}_{C_1, C_2}(X+(q^2-1)Y, X-Y).$$
Weight enumerators of Reed-Muller codes
---------------------------------------
Since each nonzero linear form on $\Proj^n$ defines a hyperplane, it is easy to see that $$W_{C_{n,1}}(X,Y) = X^{\frac{q^{n+1}-1}{q-1}} + (q^{n+1}-1) X^{\frac{q^{n}-1}{q-1}} Y^{q^n}.$$ It is also straightforward to compute $W_{C_{1,d}}(X,Y)$ for any $d$, for example by noting that $C_{1,d}$ is an MDS code [@MacWilliamsSloane Chapter 11]. Aubry considers codes from projective quadric hypersurfaces in [@Aubry], and Elkies computes the weight enumerator of $W_{C^\perp_{n,2}}(X,Y)$ in [@Elkies]. Elkies also computes the weight enuemerator of the code from plane cubics, $W_{C_{2,3}}(X,Y)$, and the weight enumerator of the code from cubic surfaces, $W_{C_{3,3}}(X,Y)$ [@Elkies]. These are the only cases for which $W_{C_{n,d}}(X,Y)$ is known exactly. Many authors have studied minimum distances and other invariants of affine and projective Reed-Muller codes. For example, see [@DGM; @vanderGeerSchoofvanderVlugtC; @HeijnenPellikaan; @Lachaud; @Sorensen].
We recall expressions for $W_{C_{2,2}}(X,Y)$ and $W_{C_{2,3}}(X,Y)$ since we will need them later. A nonzero $f \in \F_q[x,y,z]_2$ defines either a double line, a pair of $\F_q$-rational lines, a pair of Galois-conjugate lines defined over $\F_{q^2}$ but not over $\F_q$, or a smooth conic. A double line and a smooth conic each have $q+1\ \F_q$-rational points. A pair of rational lines has $2q+1$ rational points. A pair of Galois-conjugate lines has $1$ rational point. Since there are $q^2+q+1\ \F_q$-rational lines and $(q^2+q+1)(q^2-q)/2$ pairs of Galois-conjugate lines, it is straightforward to prove the following result.
\[WC22prop\] We have $$\begin{aligned}
W_{C_{2,2}}(X,Y) & = & X^{q^2+q+1} + \frac{(q^2+q+1)q(q-1)^2}{2} X Y^{q^2+q} \\
& & + (q^3-q^2+1)(q^2+q+1)(q-1) X^{q+1} Y^{q^2} \\
& & + \frac{(q^2+q+1)(q+1)q(q-1)}{2} X^{2q+1} Y^{q^2-q}.\end{aligned}$$
The irreducible cubic factor in the $X^{q+1} Y^{q^2}$ coefficient of this weight enumerator comes from the fact that it is equal to the sum of two terms that factor nicely. This is a common phenomenon that we will see throughout this paper.
We also recall $W_{C_{2,2}^A}(X,Y)$. One can think of an affine conic as a projective conic minus the points of the line at infinity $L$. For example, a smooth affine conic has either $q-1, q$, or $q+1\ \F_q$-rational points depending on $\#(C\cap L)(\F_q)$, where $C$ is the corresponding projective conic. A straightforward calculation gives the following result.
We have $$\begin{aligned}
W_{C^A_{2,2}}(X,Y) & = & X^{q^2} + \frac{(q-1)(q^3-q+2)}{2} Y^{q^2} + \frac{(q-1)^2 q^3}{2} X Y^{q^2-1} \\
& & + \frac{(q-1)^2 q^3 (q+1)}{2} X^{q-1} Y^{q^2-q+1} + (q^3-q)(q^2-q+2) X^{q} Y^{q^2-q} \\
& & + \frac{(q-1)^3 q^3}{2} X^{q+1} Y^{q^2-q-1} + \frac{(q-1)(q+1)q^3}{2} X^{2q-1} Y^{q^2-2q+1} \\
& & + \frac{q(q+1)(q-1)^2}{2} X^{2q} Y^{q^2-2q}.\end{aligned}$$
We recall the necessary background to state the formula for $W_{C_{2,3}}(X,Y)$ given by Elkies [@Elkies]; see also [@KaplanAGCT]. We write $$W_{C_{2,3}}(X,Y) = W^{\text{sing}}_{C_{2,3}}(X,Y) + W^{\text{smooth}}_{C_{2,3}}(X,Y),$$ where $W^{\text{sing}}_{C_{2,3}}(X,Y)$ is the contribution to $W_{C_{2,3}}(X,Y)$ from cubic polynomials, including the zero polynomial, that define singular cubic curves, and $W^{\text{smooth}}_{C_{2,3}}(X,Y)$ is the contribution to $W_{C_{2,3}}(X,Y)$ from cubic polynomials that define smooth cubic curves. An expression for $W^{\text{sing}}_{C_{2,3}}(X,Y)$ is given as [@KaplanAGCT Lemma 2].
We have $$\begin{aligned}
& & W^{\text{sing}}_{C_{2,3}}(X,Y) = X^{q^2+q+1} +
\frac{(q^3-1)(q^3-q)}{6} X^{3q+1} Y^{q^2-2q} \\
& + &
\frac{(q^3-1) (q^4+q^3)}{6} X^{3q} Y^{q^2-2q+1} +
\frac{(q^3-1)(q^3-q^2)(q^2-q)}{2} X^{2q+2} Y^{q^2-q-1} \\
& + &
(q^3-1)(q^2+q)(q^2-q+1) X^{2q+1} Y^{q^2-q}
+
\frac{(q^6-q^3)(q^2-1) }{2} X^{2q} Y^{q^2-q+1} \\
& + &
\frac{(q^3-1)(q^6-q^5)}{2} X^{q+2} Y^{q^2-1}
+
\frac{(q^3-1)(2q^5-q^3-q+2)}{2} X^{q+1} Y^{q^2} \\
& + &
\frac{(q^3-1)(q^3-q)(q^3-q^2)}{2} X^q Y^{q^2+1}
+
\frac{(q^3-1)(q^3-q)}{3} X Y^{q^2+q} \\
& + &
\frac{(q-1)(q^3-q) (q^3-q^2)}{3} Y^{q^2+q+1}.\end{aligned}$$
In order to describe the contribution from $W_{C_{2,3}}(X,Y)$ from smooth cubic curves, we need to count the number of smooth cubic curves with a given number of $\F_q$-points. This leads to counting elliptic curves defined over $\F_q$ with a given number of $\F_q$-points. We closely follow the presentation in [@KaplanAGCT].
Let $E$ be an elliptic curve defined over $\F_q$. When we mention an elliptic curve $E$ we always implicitly mean the isomorphism class of $E$. With this convention, let ${\mathcal{C}}=\{E/\F_q\}$ denote the set of $\F_q$-isomorphism classes of elliptic curves defined over $\F_q$. We have $$\sum_{E\in {\mathcal{C}}} \frac{1}{\#\Aut_{\F_q}(E)} = q,$$ so the finite set ${\mathcal{C}}$ is a probability space where a singleton $\{E\}$ occurs with probability $${\mathbb{P}}_q(\{E\}) = \frac{1}{q \# \Aut_{\F_q}(E)}.$$ Let $t_E\in \Z$ denote the trace of the Frobenius endomorphism associated to $E$. We have $t_E=q+1 -\#E(\F_q)$ and by Hasse’s theorem $t^2_E \leq 4q$. For an integer $t$, let ${\mathcal{C}}(t)$ be the subset of ${\mathcal{C}}$ for which $t_E = t$. The following is [@KaplanAGCT Proposition 1].
\[WE\_C23\_Smooth\] Let $q \ge 3$. Then $$W_{C_{2,3}}^{\text{smooth}}(X,Y) = (q^3-1)(q^3-q)(q^3-q^2) q \sum_{t^2 \le 4q} {\mathbb{P}}_q({\mathcal{C}}(t)) X^{q+1-t} Y^{q^2+t}.$$
We now need only give an expression for $ {\mathbb{P}}_q({\mathcal{C}}(t))$. We recall some terminology related to class numbers of quadratic imaginary fields.
For $d<0$ with $d \equiv 0,1\pmod{4}$, let $h(d)$ be the class number of the unique quadratic order of discriminant $d$. Let [$$\begin{split}
h_w(d) =
\begin{cases} h(d)/3, &\text{ if } d = -3, \\
h(d)/2, & \text{ if } d=-4, \\
h(d) & \text{ if } d < 0,\ d \equiv 0,1 \pmod{4}, \text{ and } d\neq -3,-4, \\
0 & \text{otherwise},
\end{cases}
\end{split}$$]{} and for $\Delta \equiv 0,1\pmod 4$ let [$$\begin{split}
\label{HKcn}H(\Delta) = \sum_{d^2 \mid \Delta} h_w\left(\frac{\Delta}{d^2}\right)
\end{split}$$]{} be the *Hurwitz-Kronecker class number*. For $a\in \Z$ and $n$ a positive integer, the *Kronecker symbol* $\Big(\frac{a}{n}\Big)$ is defined to be the completely multiplicative function in $n$ such that if $p$ is an odd prime ${\left(\frac{a}{p}\right)}$ is the quadratic residue symbol and if $p=2$ [$$\begin{split}
\left(\frac{a}{2}\right) = \begin{cases} 0 & \text{ if } 2 \mid a, \\
1 & \text{ if } a \equiv \pm 1 \pmod 8, \\ -1 & \text{ if } a \equiv \pm5 \pmod 8 .
\end{cases}
\end{split}$$]{}
The following is a weighted version of [@Schoof Theorem 4.6], which builds on earlier work of Deuring and Waterhouse [@Deuring; @Waterhouse].
\[S46withweights\] Let $t\in \Z$. Suppose $q = p^v$ where $p$ is prime and $v\geq 1$. Then if $q$ is not a square $$\begin{aligned}
{3}
{\mathbb{P}}_q({\mathcal{C}}(t)) = & \frac{1}{2q}H(t^2-4q) \quad && \text{ if } t^2 < 4q \text{ and } p\nmid t,\\
= & \frac{1}{2q}H(-4p)\quad && \text{ if } t=0, \\
= & \frac{1}{4q} \quad && \text{ if } t^2=2q \text{ and } p =2, \\
= & \frac{1}{6q} && \text{ if } t^2=3q \text{ and } p =3, \end{aligned}$$ and if $q$ is a square $$\begin{aligned}
{3}
{\mathbb{P}}_q({\mathcal{C}}(t)) = & \frac{1}{2q}H(t^2-4q)\quad && \text{ if } t^2 < 4q \text{ and } p\nmid t,\\
= & \frac{1}{4q}\left(1 - {\left(\frac{-4}{p}\right)}\right) \quad && \text{ if } t=0, \\
= & \frac{1}{6q}\left(1 - {\left(\frac{-3}{p}\right)}\right) \quad && \text{ if } t^2 = q, \\
= & \frac{p-1}{24q}\quad && \text{ if } t^2 = 4q,\end{aligned}$$ and ${\mathbb{P}}_q({\mathcal{C}}(t)) = 0$ in all other cases.
Combining Proposition \[WE\_C23\_Smooth\] and Lemma \[S46withweights\] gives an expression for $W_{C_{2,3}}^{\text{smooth}}(X,Y)$ in terms of class numbers of orders in imaginary quadratic fields.
We move on to the computation of the weight enumerators for the dual codes. We apply the Theorem \[MacWilliamsThm\] to $W_{C_{2,2}}(X,Y)$ to determine the low-weight coefficients of $W_{C_{2,2}^\perp}(X,Y)$.
\[WC22Dualprop\] Let $$W_{C_{2,2}^\perp}(X,Y) = \sum_{i=0}^{q^2+q+1} B_i X^{q^2+q+1 - i } Y^i.$$ Then $B_0 = 1,\ B_1 = B_2 = B_3 = 0$, and $$\begin{aligned}
B_4 & = & (q-1)(q^2+q+1) \binom{q+1}{4}\\
B_5 & = & \left((q^2-1) - \binom{5}{4}(q-1)\right)(q^2+q+1) \binom{q+1}{5}\\
B_6 & = & \left((q^3 - 1) - \binom{6}{5} (q^2-1)+\binom{6}{4} (q-1)\right) (q^2+q+1) \binom{q+1}{6} \\
& & + (q-1) (q^5-q^2) \binom{q+1}{6} + (q-1) \binom{q^2+q+1}{2} \binom{q}{3}^2.\end{aligned}$$
Applying Theorem \[MacWilliamsThm\] to $W_{C^A_{2,2}}(X,Y)$ gives the following result.
Let $$W_{(C^{A}_{2,2})^\perp}(X,Y) = \sum_{i=0}^{q^2} B_i X^{q^2 - i } Y^i.$$ Then $B_0 = 1,\ B_1 = B_2 = B_3 = 0$, and $B_4 = (q-1)(q^2+q)\binom{q}{4}$.
Applying Theorem \[MacWilliamsThm\] to our expression for $W_{C_{2,3}}(X,Y)$ determines the low-weight coefficients of $W_{C_{2,3}^\perp}(X,Y)$. The following result is given in a different form as the first part of [@KaplanAGCT Theorem 3].
\[C23LowWeightDual\] Let $\F_q$ be a finite field of size $q>2$, and $$W_{C_{2,3}^\perp}(X,Y) = \sum_{i=0}^{q^2+q+1} B_i X^{q^2+q+1 - i } Y^i.$$ Then $B_0 = 1,\ B_1 = B_2 = B_3 = B_4 = 0$, and $$\begin{aligned}
B_5 & = & (q-1)(q^2+q+1) \binom{q+1}{5}\\
B_6 & = & \left((q^2-1) - \binom{6}{5}(q-1)\right)(q^2+q+1) \binom{q+1}{6}\\
B_7 & = & \left((q^3 - 1) - \binom{7}{6} (q^2-1)+\binom{7}{5} (q-1)\right) (q^2+q+1) \binom{q+1}{7} \\
B_8 & = & \left((q^4 - 1) - \binom{8}{7} (q^3-1)+\binom{8}{6} (q^2-1) - \binom{8}{5}(q-1)\right) (q^2+q+1) \binom{q+1}{7} \\
& & + (q-1) (q^5-q^2) \binom{q+1}{8} + (q-1) \binom{q^2+q+1}{2} \binom{q}{3}^2 \\
B_9 & = & (q-1)I_9(q) +(q^5-q^2)\dbinom{q+1}{9}\left((q^2-1)-9(q-1)\right)\\
& & +(q^2+q+1)\dbinom{q+1}{9}(q^5 - 9q^4 + 36q^3 - 84q^2 + 126q - 70) \\
& & +2\dbinom{q^2+q+1}{2}\dbinom{q}{5}\dbinom{q}{4}(q^2-6q+5)+\dbinom{q^2+q+1}{2}\dbinom{q}{4}^2(q^2-3q+2).\end{aligned}$$ where $$I_9(q) =\frac{1}{9!}(q^6 + 2q^5 - 73q^4 + 344q^3 - 838q^2 + 1754q - 2030)(q^2 + q + 1)(q + 1)(q - 1)^2(q - 2)q^4.$$
We will explain the form of these dual code coefficients in the next section.
In [@KaplanAGCT Theorem 3] there is an additional restriction that the characteristic of $\F_q$ is not $2$ or $3$. It seems that the proof of this result does work without any changes for all $q>2$. In the next section, we give a different proof of the formulas for $B_i$ when $i \le 8$ and also for $B_9$, except for the computation of $I_9(q)$. In Appendix \[AppendixA\] we compute $I_9(q)$ in a different way that is independent of the characteristic of $\F_q$, which completes a separate proof of the formula for $B_9$.
Dual Code Coefficients and Configurations of Points that Fail to Impose Independent Conditions {#sec:dual_coef}
==============================================================================================
Statement of Results
--------------------
The goal of this section is to analyze the supports of low-weight codewords of $C_{2,2}^\perp$ and $C_{2,3}^\perp$ and prove the following three results.
The first result will be used in Section \[sec:twoconics\] to prove an analogue of Theorem \[ThmIntCubics\] for intersections of plane conics.
\[lemmalowweightconics\] Let $$W^{[2]}_{C_{2,2}^\perp}(X,Y) = \sum_{i=0}^{q^2+q+1} B^{[2]}_i X^{q^2+q+1 - i } Y^i.$$ We have $B^{[2]}_0 = 1,\ B^{[2]}_1 = B^{[2]}_2 = B^{[2]}_3 = 0$, and $B^{[2]}_4 = (q^2-1)(q^2+q+1)\binom{q+1}{4}$.
The next result will be used in Section \[sec:con\_cub\] to prove an analogue of Theorem \[ThmIntCubics\] for the intersection of a conic and a cubic.
\[lemmalowweightconiccubic\] Let $$\begin{aligned}
W_{C_{2,2}^\perp}(X,Y) & = & \sum_{i=0}^{q^2+q+1} B^{2,2}_i X^{q^2+q+1 - i } Y^i, \\
W_{C_{2,3}^\perp}(X,Y) & = & \sum_{i=0}^{q^2+q+1} B^{2,3}_i X^{q^2+q+1 - i } Y^i,\\
W^{[2]}_{C_{2,2}^\perp,C_{2,3}^\perp}(X,Y) & = & \sum_{i=0}^{q^2+q+1} B^{[2]}_i X^{q^2+q+1 - i } Y^i.\end{aligned}$$ Then $B^{[2]}_0 = 1,\ B^{[2]}_1 = B^{[2]}_2 = B^{[2]}_3 = 0$, and $$\begin{aligned}
B^{[2]}_4 & = & B^{2,2}_4\\
B^{[2]}_5 & = & B^{2,2}_5 + B^{2,3}_5 + (q^3-1) (q^2-1) \binom{q+1}{5} \\
B^{[2]}_6 & = &B^{2,2}_6 + B^{2,3}_6 + (q^3-1)(q^2-1) (q^2+q-5) \binom{q+1}{6}.\end{aligned}$$
The final result plays a key role in the proof of Theorem \[ThmIntCubics\]. Let $$\begin{aligned}
g_3(5) & = & (q-1)^2\\
g_3(6) & = & (q^2+2q-5) (q-1)^2 \\
g_3(7) & = & (q^4-2q^3 - 4 q^2 - 12 q + 15)(q-1)^2 \\
g_3(8) & = & (q^5+3q^4 - 2 q^3- 14q^2 - 7q + 35)(q-1)^3 \\
g_3(9) & = & (q^8+2q^7 - 6 q^6 - 14 q^5 + 14 q^4 + 40 q^3 - 112 q + 70) (q-1)^2.\end{aligned}$$ In Proposition \[gn\_prop\] we will give a general formula for $g_3(q)$.
\[lemmalowweightcubics\] Let $$\begin{aligned}
W^{[2]}_{C_{2,3}^\perp}(X,Y) & = & \sum_{i=0}^{q^2+q+1} B^{[2]}_i X^{q^2+q+1 - i } Y^i \\
W_{C_{2,3}^\perp}(X,Y) & = & \sum_{i=0}^{q^2+q+1} B^{2,3}_i X^{q^2+q+1 - i } Y^i.\end{aligned}$$ Then $B^{[2]}_0= 1,\ B^{[2]}_1 = B^{[2]}_2 = B^{[2]}_3 = B^{[2]}_4=0$, and $$\begin{aligned}
B^{[2]}_5 & = & 2 B^{2,3}_5 + (q^2+q+1)\dbinom{q+1}{5}g_3(5)\\
B^{[2]}_6 & = & 2 B^{2,3}_6 + (q^2+q+1)\dbinom{q+1}{6}g_3(6)\\
B^{[2]}_7 & = & 2 B^{2,3}_7 + (q^2+q+1)\dbinom{q+1}{7}g_3(7)\\
B^{[2]}_8 & = & 2 B^{2,3}_8 + (q^2+q+1)\dbinom{q+1}{8}g_3(8) \\
& & +(q^5-q^2)\dbinom{q+1}{8} (q-1)^2+\dbinom{q^2+q+1}{2} \dbinom{q}{4}^2 (q-1)^2\\
B^{[2]}_9 & = & 2 B^{2,3}_9 + (q^2+q+1)\dbinom{q+1}{9}g_3(9)+ 2\dbinom{q^2+q+1}{2}\dbinom{q}{4}\dbinom{q}{5}(q^4-8q^2+12q-5)\\
& & + \dbinom{q^2+q+1}{2}\dbinom{q}{4}^2(q^4-5q^2+6q-2)\\
& & +(q^5-q^2)\dbinom{q+1}{9}(q^4-11q^2+18q-8)+I_9(q-1)^2,\end{aligned}$$ where $I_9(q)$ is given in Theorem \[C23LowWeightDual\].
Supports of dual codewords and points failing to impose independent conditions {#sec:supports}
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For any subset $S\subseteq \mathbb{P}^n$ one can consider the subspace of $\F_q[x_0,\ldots, x_n]_d$ defined by $$\Gamma_{S}=\left\{f\in \F_q[x_0,\ldots, x_n]_d \mid f(p)=0,\ \forall p\in S\right\}.$$ It is a well-known fact that $\Gamma_{S}$ has codimension at most $\# S$. If equality holds for this codimension then we say that $S$ *imposes independent conditions* on degree $d$ forms, or equivalently, on degree $d$ hypersurfaces, in $\mathbb{P}^n$; otherwise we say that $S$ *imposes dependent conditions* or *fails to impose independent conditions*. If $S$ fails to impose independent conditions, there exist $b_p \in \F_q$, not all zero, such that $\sum_{p\in S} b_p f(p)=0$ for all $f \in \F_q[x_0,\ldots, x_n]_d$. In our setup this produces a nonzero codeword in $C_{n,d}^\perp$ supported on the coordinates corresponding to $S$. We conclude that supports of nonzero codewords in $C_{n,d}^{\perp}$ come from sets of points in $\mathbb{P}^n$ that fail to impose independent conditions on degree $d$ hypersurfaces.
The following classical result, which can be found in [@EGH], is at the core of our analysis of low-weight codewords of $C_{2,d}^\perp$. It gives a precise description for small sets of point that fail to impose independent condition on degree $d$ plane curves.
\[EGHlemma\] Let $\Omega=\{p_1,\ldots, p_n\}\subset \mathbb{P}^2$ be any collection of $n\leq 2d+2$ distinct points. The points of $\Omega$ fail to impose independent conditions on curves of degree $d$ if only if either $d+2$ points of the points of $\Omega$ are collinear or $n=2d+2$ and $\Omega$ is contained in a conic.
This lemma implies that there are no nonzero codewords of weight less then $d+2$ in $C_{2,d}^{\perp}$.
Using Lemma \[EGHlemma\] we deduce the following geometric characterizations for the support of low-weight codewords of $C_{2,2}^\perp$ and $C_{2,3}^\perp$.
\[supportconiclemma\] Let $c \in C_{2,2}^{\perp}$ be nonzero with $\wt(c) = k$. Then $k\geq 4$ and moreover:
- If $k \in \{4, 5\}$ the support of $c$ consists of $k$ collinear points.
- If $k=6$ then the support of $c$ is either:
- $6$ collinear points,
- $6$ points on a smooth conic, or
- $6$ points on two lines, with $3$ on each, and not including the intersection point of the two lines.
Lemma \[EGHlemma\] implies that a nonzero codeword of $C_{2,2}^\perp$ has weight at least $4$. When $k=4$ the $4$ points of the support must be collinear.
[**k=5**]{}: Let $\{P_1,\ldots, P_5\}$ be the points of $\Proj^2(\F_q)$ in the support of $c$. Lemma \[EGHlemma\] implies that at least $4$ of these $5$ points are collinear. Without loss of generality suppose that $P_1,P_2,P_3,P_4$ are collinear. There is a $c' \in C_{2,2}^\perp$ with $\wt(c') = 4$ supported on $P_1, P_2, P_3,P_4$. There exists an $\alpha \in \F_q^*$ such that the coordinate corresponding to $P_1$ in $c - \alpha c'$ is equal to $0$. Therefore, $c - \alpha c'$ is a nonzero codeword of weight at most $4$ supported on $P_2,P_3,P_4,P_5$. By the result for $k=4$, these points are collinear.
[**k=6**]{}: Let $\{P_1,\ldots, P_6\}$ be the support of $c$. By Lemma \[EGHlemma\] we need only show that if at least $4$ of these points are collinear, then they are all collinear.
Without loss of generality, suppose that $P_1,P_2, P_3, P_4$ are collinear, and $P_6$ is not on this same line. As in the case $k=5$, there is a $c' \in C_{2,2}^\perp$ with $\wt(c') = 4$ supported on $P_1, P_2, P_3,P_4$, and there is an $\alpha \in \F_q^*$ such that the coordinate corresponding to $P_1$ in $c - \alpha c'$ is equal to $0$. We see that $c - \alpha c'$ is a nonzero codeword of weight at most $5$ whose support is not given by a collinear set of points. This contradicts the $k=5$ case of the statement.
\[supportcubiclemma\] Let $c \in C_{2,3}^{\perp}$ be nonzero with $\wt(c) = k$. Then $k\geq 5$ and moreover:
- If $k \in \{5,6,7\}$ the support of $c$ consists of $k$ collinear points.
- If $k=8$ then the support of $c$ is either:
- $8$ collinear points,
- $8$ points on a smooth conic, or
- $8$ points on two lines, with $4$ on each, and not including the intersection point of the two lines.
- If $k=9$ then the support of $c$ is either:
- $9$ collinear points,
- $9$ points on a smooth conic, or
- $9$ points on two lines, with $5$ points on the first line and $4$ on the second line, and not including the intersection points of the two lines,
- $9$ points on two lines consisting of the intersection point of the two lines together with $4$ additional points on each,
- $9$ points that are exactly the intersection points of two cubics , i.e., the two cubics share no common component.
\[Chaslesremark\] The fact that a collection of $9$ points that are the intersection points of two cubics fails to impose independent conditions on cubics is due to Chasles, although is colloquially known as the Cayley-Bacharach theorem [@EGH Theorem CB3]. We give a different form of this result in Proposition \[Hart4.5\] in Appendix \[AppendixA\]
Lemma \[EGHlemma\] implies that a nonzero codeword of $C_{2,3}^\perp$ has weight at least $5$. When $k=5$ the $k$ points of the support must be collinear.
[**k=6**]{}: Let $\{P_1,\ldots, P_6\}$ be the points of $\Proj^2(\F_q)$ in the support of $c$. Lemma \[EGHlemma\] implies that $5$ of these points are collinear. Without loss of generality suppose that $P_1,\ldots, P_5$ are collinear. There is a $c' \in C_{2,3}^\perp$ with $\wt(c') = 5$ supported on $\{P_1 \ldots P_5\}$. There exists an $\alpha \in \F_q^*$ such that the coordinate corresponding to $P_1$ in $c - \alpha c'$ is equal to $0$. Therefore, $c - \alpha c'$ is a nonzero codeword of weight at most $5$ supported on $\{P_2, \ldots, P_6\}$. By the result for $k=5$, these points are collinear.
[**k=7**]{}: Let $\{P_1,\ldots, P_7\}$ be the support of $c$. Lemma \[EGHlemma\] implies that $5$ of these points are collinear. Without loss of generality suppose that $P_1,\ldots, P_5$ are collinear. There is a $c' \in C_{2,3}^\perp$ with $\wt(c') = 5$ supported on $\{P_1 \ldots P_5\}$. There exists an $\alpha \in \F_q^*$ such that the coordinate corresponding to $P_1$ in $c - \alpha c'$ is equal to $0$. Therefore, $c - \alpha c'$ is a nonzero codeword of weight at most $6$ supported on $\{P_2, \ldots, P_7\}$. By the result for $k=6$, these points are collinear.
[**k=8**]{}: Let $\{P_1,\ldots, P_8\}$ be the support of $c$. By Lemma \[EGHlemma\] we need only show that if at least $5$ of these points are collinear, then they are all collinear.
Without loss of generality, suppose that $P_1,\ldots, P_5$ are collinear, and $P_8$ is not on this same line. As in the case $k=7$, there is a $c' \in C_{2,3}^\perp$ with $\wt(c') = 5$ supported on $\{P_1, \ldots, P_5\}$, and there is an $\alpha \in \F_q^*$ such that the coordinate corresponding to $P_1$ in $c - \alpha c'$ is equal to $0$. We see that $c - \alpha c'$ is a nonzero codeword of weight at most $7$ whose support is not given by a collinear set of points. This contradicts the $k=7$ case of the statement.
[**k=9**]{}: Let $\{P_1,\ldots, P_9\}$ be the support of $c$. We break this argument into two cases. In the first case, suppose that all the subsets of $8$ of these points impose independent conditions on cubics. We will show that in this case, there must be distinct cubic curves that intersect exactly at these $9$ points. In the second case, we suppose that there exists a subset of $8$ of these points that imposes dependent conditions on cubics. We will show that the $9$ points must be in one of the other configurations given in the statement.
[**Case I**]{}: Take any subset of $8$ points out of the $9$ points, say $\{P_1,\ldots, P_8\}$. They impose independent conditions on cubics, so there is a $2$-dimensional vector space of cubic polynomials that vanish on these points. Let $f_1$ and $f_2$ be a basis for this vector space. Since $\{P_1,\ldots, P_9\}$ fails to impose independent conditions on cubics, the dimension of the vector space of cubic polynomials vanishing at these points is at least $2$. We conclude that this dimension is exactly $2$, and $\{f_1, f_2\}$ is a basis for this space. Therefore, $\{P_1,\ldots, P_9\}$ is contained in the intersection of the cubic curves defined by $f_1$ and $f_2$. If these two cubics shared a common component, it would have to be an $\F_q$-rational line or a smooth conic. In the first case, at least $5$ of the points would have to lie on the shared line. In the second case, at least $8$ of the points would have to lie on the shared conic. This would contradict the assumption that every subset of $8$ of the $9$ points imposes independent conditions on cubics.
[**Case II**]{}: Suppose there is a subset of $8$ points from $\{P_1,\ldots, P_9\}$ that fails to impose independent conditions on cubics. Without loss of generality, suppose this subset is $\{P_1,\ldots, P_8\}$. By Lemma \[EGHlemma\], either $8$ of these points lie on a conic, or at least $5$ of the points are collinear.
Suppose that $\{P_1,\ldots, P_8\}$ lie on a smooth conic. There is a codeword $c' \in C_{2,3}^\perp$ of weight $8$ supported on $\{P_1, \ldots, P_8\}$, and there is an $\alpha \in \F_q^*$ such that the coordinate corresponding to $P_1$ in $c - \alpha c'$ is equal to $0$. Therefore, $c - \alpha c'$ is a nonzero codeword of weight at most $8$ supported on $\{P_2, \ldots, P_9\}$. It is clear that no $4$ points of $\{P_2,\ldots, P_9\}$ are collinear. By the result for $k=8$, these points must all lie on a smooth conic.
We next prove that if $5$ of the points are collinear, then all $9$ points lie on the union of two lines. Suppose $P_1,\ldots, P_5$ are collinear. There is a codeword $c' \in C_{2,3}^\perp$ of weight $5$ supported on $\{P_1, \ldots, P_5\}$, and there is an $\alpha \in \F_q^*$ such that the coordinate corresponding to $P_1$ in $c - \alpha c'$ is equal to $0$. Therefore, $c - \alpha c'$ is a nonzero codeword of weight at most $8$ supported on $\{P_2, \ldots, P_9\}$. Since $P_2,\ldots, P_5$ are collinear, the result for $k=8$ implies that $P_2,\ldots, P_9$ lie on two lines, and therefore, $P_1,\ldots, P_9$ lie on two lines.
We now prove that if $6$ of the points are collinear, then all $9$ are collinear. Suppose $P_1,\ldots, P_6$ are collinear. There is a codeword $c' \in C_{2,3}^\perp$ of weight $5$ supported on $\{P_1, \ldots, P_5\}$, and there is an $\alpha \in \F_q^*$ such that the coordinate corresponding to $P_1$ in $c - \alpha c'$ is equal to $0$. Therefore, $c - \alpha c'$ is a nonzero codeword of weight at most $8$ supported on $\{P_2, \ldots, P_9\}$. Since $P_2,\ldots, P_6$ are collinear, the result for $k=8$ implies that $P_2,\ldots, P_9$ are collinear, and therefore, $P_1,\ldots, P_9$ are collinear.
Counting Codewords with Given Support
-------------------------------------
As we can see in the statements of Lemma \[supportconiclemma\] and Lemma \[supportcubiclemma\], many of the low-weight codewords of $C_{2,2}^\perp$ and $C_{2,3}^\perp$ come from collections of collinear points. In this section we count the number of weight $k$ codewords supported on a set of collinear points.
\[Vdmdim\] Let $V_{d,m}$ the vector space of codewords in $C_{2,d}^\perp$ supported on a collection of $m$ collinear points. Then $\dim(V_{d,m})=\max(m-d-1,0)$.
Let $P_1,\ldots, P_m$ be the collection of collinear points defining $V_{d,m}$.
The statement for $m\leq d+1$ is immediate from Lemma \[EGHlemma\]. So let us assume $m\geq d+2$. We prove this proposition by induction on $m$.
Let $m=d+2$. Suppose that $c_1, c_2 \in V_{d,m}$ are linearly independent. There is an $\alpha \in \F_q$ such that the coordinate corresponding to $P_1$ in $c_1 - \alpha c_2$ is equal to $0$. Since $c_1 - \alpha c_2$ is a codeword supported on a set of at most $d+1$ collinear points, Lemma \[EGHlemma\] implies that it must be the zero codeword, which contradicts the assumption that $c_1, c_2$ were linearly independent.
For the induction step, suppose that $P_{m+1}$ is on the same line as $\{P_1,\ldots, P_m\}$. Obviously $V_{d,m} \subseteq V_{d,m+1}$. Let ${\varphi}\colon V_{d,m+1} \to \F_q$ be the map that takes an element of $V_{d,m+1}$ to the coordinate corresponding to $P_{m+1}$. It is clear that $\ker({\varphi}) = V_{d,m}$. We need only show that ${\varphi}$ is surjective, which is equivalent to saying that ${\varphi}$ is nonzero. By Lemma \[EGHlemma\], there is a nonzero $c \in C_{2,d}^\perp$ supported on $P_1,\ldots, P_{d+1}, P_{m+1}$, so $\wt(c) = d+2$. This implies ${\varphi}(c) \neq 0$. We conclude that $V_{d,m+1}/V_{d,m}\cong \mathbb{F}_q$, and thus $$\dim(V_{d,m+1})=\dim(V_{d,m})+1=m-d-1+1=(m+1)-d-1.$$
\[fdmprop\] Let $m \ge d+2$. The number of $c\in V_{d,m}$ with $\wt(c)=m$ is $$f_d(m)=\sum_{i=0}^{m-d-2} (q^{m-d-1-i}-1)\dbinom{m}{i}(-1)^{i}.$$
The proof follows from an inclusion-exclusion argument. Let $P_1,\ldots, P_m$ be the collinear points defining $V_{d,m}$. Let $A_i$ be the set of codewords for which the coordinate corresponding to $P_i$ is zero. By Proposition \[Vdmdim\], $$f_d(m)=q^{m-d-1}-\left|\bigcup_{i=1}^{m} A_i\right|.$$ We have $$\left|\bigcup_{i=1}^{m} A_i\right|=\sum_{\substack{S\subseteq \{1,2,\ldots m\} \\ S\neq \emptyset }} (-1)^{|S|-1} \left|\bigcap_{j\in S} A_j\right|.$$ There is a natural identification $$\bigcap_{j\in S} A_j \cong V_{d, m-|S|},$$ so applying Proposition \[Vdmdim\] completes the proof.
To prove the results stated at the beginning of Section \[sec:dual\_coef\], we need to count pairs of codewords $c_1, c_2$ such that $\wt(c_1, c_2)$ is fixed.
\[gn\_prop\] Let $m \ge d+2$. The number of (ordered) pairs $(c_1,c_2)$ with $c_1, c_2 \in V_{d,m}$ both nonzero with $\wt(c_1,c_2)=m$ is $$g_d(m)=\sum_{\substack{d+2\leq a,b\leq m\\ a+b\geq m}} \dbinom{m}{a}\dbinom{a}{b+a-m} f_d(a)f_d(b).$$
This explains the values of $g_3(m)$ given before the statement of Lemma \[lemmalowweightcubics\].
Let $P_1,\ldots, P_m$ be the collection of collinear points defining $V_{d,m}$. Suppose $c_1,c_2 \in V_{d,m}$ satisfy $\wt(c_1,c_2) = m$. Let $A \subseteq \{P_1,\ldots, P_m\}$ be the support of $c_1$, and $B \subseteq \{P_1,\ldots, P_m\}$ be the support of $c_2$. By assumption $A\cup B = \{P_1,\ldots, P_m\}$.
Suppose $|A| = a$ and $|B| = b$, so $d+2 \le a,b \le m$ and $a+b \ge m$. There are $\binom{m}{a}$ choices for $A$. We see that $B$ must contain the $m-a$ points of $\{P_1,\ldots, P_m\} \setminus A$, and also $b-(m-a)$ points of $A$. Applying Proposition \[fdmprop\] completes the proof.
We now turn from counting codewords supported on a set of collinear points, to counting codewords supported on a set of noncollinear points.
\[VSconiclemma\] Let $S$ be a set of $6$ noncollinear points that fail to impose independent conditions on conics and let $V_S$ be the vector space of codewords in $C_{2,2}^{\perp}$ supported on $S$.
- If $S$ consists of $6$ points on a smooth conic, $V_S$ is $1$-dimensional and only contains nonzero codewords of weight $6$.
- If $S$ is a collection of $6$ points on two $\F_q$-rational lines, with $3$ on each line and not including the intersection point, then $V_S$ is $1$-dimensional and only contains nonzero codewords of weight $6$.
We prove the two statements together. For each such set $S$, pick one of the points $P \in S$ and consider the natural map ${\varphi}_P \colon V_S \rightarrow \F_q$ that takes a codeword to its value at the coordinate corresponding to $P$.
Suppose there is a $c \in V_S$ with ${\varphi}_P(c) = 0$. Then $\wt(c) \le 5$, and Lemma \[supportconiclemma\] implies that $c$ is the all zero codeword. Therefore, ${\varphi}_P$ is injective. We also see that ${\varphi}_P$ is surjective, since Lemma \[supportconiclemma\] implies that $S$ fails to impose independent conditions on conics, so there is a nonzero codeword supported on $S$.
We now explain the form of the coefficients of $W_{C_{2,2}^\perp}(X,Y)$ that were given in Proposition \[WC22prop\].
- Lemma \[supportconiclemma\] and Proposition \[fdmprop\] imply that every $c \in C_{2,2}^\perp$ with $\wt(c) = 4$ is supported on $4$ collinear points, and given any $4$ collinear points there are exactly $q-1$ weight $4$ codewords supported on them.
- Similarly, every $c \in C_{2,2}^\perp$ with $\wt(c) = 5$ is supported on $5$ collinear points, and given any $5$ collinear points there are exactly $q^2-1 - \binom{5}{4}(q-1)$ weight $5$ codewords supported on them.
- Lemma \[supportconiclemma\] implies that every $c \in C_{2,2}^\perp$ with $\wt(c) = 6$ is supported either on a set of $6$ collinear points, on a set if $6$ points on a smooth conic, or on a set contained in two $\F_q$-rational lines, with $3$ on each line and not including the intersection point. Applying Proposition \[fdmprop\] and Lemma \[VSconiclemma\] along with some elementary counting implies that the number of $c \in C_{2,2}^\perp$ with $\wt(c) = 6$ is $$(q^2+q+1) \binom{q+1}{6} f_2(6) + (q^5-q^2) \binom{q+1}{6}(q-1) + \binom{q^2+q+1}{2} \binom{q}{3}^2 (q-1).$$
We now prove the analogue of Lemma \[VSconiclemma\] for sets of noncollinear points that fail to impose independent conditions on cubics. We consider codewords of weight $8$ and weight $9$ separately.
\[VScubic8lemma\] Let $S$ be a set of $8$ noncollinear points that fail to impose independent conditions on cubics and let $V_S$ be the vector space of codewords in $C_{2,3}^\perp$ supported on $S$.
- If $S$ consists of $8$ points on a smooth conic, $V_S$ is $1$-dimensional and only contains nonzero codewords of weight $8$.
- If $S$ is a collection of $8$ points on two $\F_q$-rational lines, with $4$ on each line and not including the intersection point, then $V_S$ is $1$-dimensional and only contains nonzero codewords of weight $8$.
The proof of this lemma is very similar to the proof of Lemma \[VSconiclemma\], so we omit it.
We use this result to explain the form of the $B_8$ coefficient given in Theorem \[C23LowWeightDual\]. If $c \in C_{2,3}^\perp$ has $\wt(c) = 8$ then $S = \supp(c)$ is either a set of $8$ collinear points, or one of the two configurations given in the statement of Lemma \[VScubic8lemma\]. Proposition \[fdmprop\] together with this lemma imply that the number of these weight $8$ codewords is $$(q^2+q+1)\binom{q+1}{8} f_3(8) + (q^5-q^2)\binom{q+1}{8}(q-1) + \binom{q^2+q+1}{2} \binom{q}{4}^2 (q-1).$$
We next give the analogous statement for codewords of $C_{2,3}^\perp$ of weight $9$.
\[VScubic9lemma\] Let $S$ be a set of $9$ noncollinear points that fail to impose independent conditions on cubics and let $V_S$ be the vector space of codewords in $C_{2,3}^{\perp}$ supported on $S$.
- If $S$ consists of $9$ points on a smooth conic, $V_S$ is $2$-dimensional and only contains nonzero codewords of weight $8$ and $9$.
- If $S$ is a collection on $9$ points on two lines with at least $4$ points on each, $V_S$ is $2$-dimensional and only contains codewords of weight $5, 8$ and $9$.
- If $S$ is a collection of $9$ points that are exactly the intersection points of two cubics, i.e., the two cubics share no common component, then $V_S$ is $1$-dimensional and only contains nonzero codewords of weight $9$.
In each of the cases described above, we choose a subset $S' \subset S$ of size $8$.
- If $S$ is a collection of $9$ points on a smooth conic, $S'$ is $8$ points on a smooth conic.
- If $S$ is a collection on $9$ points on two lines with at least $4$ points on each, then choose $S'$ to be a subset with $4$ points on each line, not including the intersection point.
- In the final case in the statement of the lemma, we choose $S' \subset S$ to be any subset of $8$ points. By Lemma \[supportcubiclemma\], $S'$ imposes independent conditions on cubics.
In each case, let $P$ denote the point of $S \setminus S'$.
Let $V_{S'}$ denote the vector space of codewords of $C_{2,3}^\perp$ supported on $S'$. Clearly $V_{S'} \subseteq V_S$. Consider the map ${\varphi}_P \colon V_S \rightarrow \F_q$ that takes a codewords to the value of the coordinate corresponding to $P$. It is clear that $\ker({\varphi}_P) = V_{S'}$. Lemma \[VScubic8lemma\] implies that in the first two cases, $\dim(V_{S'}) = 1$, and in the final case $V_{S'}$ consists of only the zero codeword, so ${\varphi}_P$ is an isomorphism.
We use this result to explain the form of the $B_9$ coefficient given in Theorem \[C23LowWeightDual\]. If $c \in C_{2,3}^\perp$ has $\wt(c) = 9$ then $S = \supp(c)$ is either a set of $9$ collinear points, or one of the three configurations given in the statement of Lemma \[VScubic8lemma\].
- Suppose $S$ consists of $9$ points on a smooth conic. Lemma \[VScubic8lemma\] implies that $V_S$ is $2$-dimensional. We claim that there are $9(q-1)$ nonzero codewords of weight less than $9$ in $V_S$. For any collection of $8$ points of $S$, there are $q-1$ codewords of weight $8$ supported on them. Lemma \[supportcubiclemma\] implies that these are the only nonzero codewords of weight less than $9$ in $V_S$.
- Suppose $S$ consists of $9$ points on two lines, with $5$ points on one and $4$ points on the other, not including the intersection point. Lemma \[VScubic8lemma\] implies that $V_S$ is $2$-dimensional. We claim that there are $6(q-1)$ nonzero codewords of weight less than $9$ in $V_S$.
Let $S'$ be a subset of size $8$ that we get from removing one point from the line containing $5$ points of $S$. There are $q-1$ elements of $V_S$ of weight $8$ supported on $S'$. There are also $q-1$ codewords of weight $5$ in $V_S$ supported on the $5$ collinear points of $S$. Lemma \[supportcubiclemma\] implies that these are the only nonzero codewords of weight less than $9$ in $V_S$.
- Suppose $S$ consists of $9$ points on two lines, the intersection point together with $4$ points on each. Lemma \[VScubic8lemma\] implies that $V_S$ is $2$-dimensional. We claim that there are $3(q-1)$ nonzero codewords of weight less than $9$ in $V_S$.
For each of the two subsets of $5$ collinear points in $S$ there are $q-1$ codewords of weight $5$ supported on these points. There are also $q-1$ codewords of weight $8$ supported on the subset of $8$ points of $S$ not including the intersection point. Lemma \[supportcubiclemma\] implies that these are the only nonzero codewords of weight less than $9$ in $V_S$.
Proposition \[fdmprop\] together with the observations above imply that the number of $c \in C_{2,3}^\perp$ with $\wt(c) = 9$ is $$\begin{aligned}
& & (q^2+q+1)\dbinom{q+1}{9} f_3(9)+ (q^5-q^2)\binom{q+1}{9}\left(q^2-1-9(q-1)\right) \\
& + & 2\binom{q^2+q+1}{2}\binom{q}{5}\binom{q}{4}(q^2-1 - 6(q-1))\\
& + & \binom{q^2+q+1}{2}\binom{q}{4}^2(q^2-1-3(q-1)) + (q-1) I_9(q),\end{aligned}$$ where $I_9(q)$ is the number of collections of $9$ points in $\Proj^2(\F_q)$ that are the intersection points of two cubics that do not share a common component. Comparing this expression to the one given in Theorem \[C23LowWeightDual\] proves the following.
\[I9cor\] We have that $I_9(q)$ equals $$\frac{1}{9!}(q^6 + 2q^5 - 73q^4 + 344q^3 - 838q^2 + 1754q - 2030)(q^3-1)(q + 1)(q - 1)(q - 2)q^4.$$
The computation of the $X^{q^2+q+1-9}Y^9$ coefficient of $W_{C_{2,3}^\perp}(X,Y)$ given in [@KaplanAGCT Theorem 3] has the additional restriction that the characteristic of $\F_q$ is not $2$ or $3$. In Appendix \[AppendixA\], we compute $I_9(q)$ in a different way that is independent of the characteristic of $\F_q$, which shows that Corollary \[I9cor\] holds in these additional cases.
Proofs of Lemmas \[lemmalowweightconics\], \[lemmalowweightconiccubic\], and \[lemmalowweightcubics\]
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Lemma \[supportconiclemma\] implies that $C_{2,2}^\perp$ has no nonzero codewords of weight less than $4$ and that every codeword of weight $4$ is supported on a set of $4$ collinear points. Proposition \[fdmprop\] implies that there are $q-1$ weight $4$ codewords supported on any chosen set of $4$ collinear points.
If $c_1,c_2 \in C_{2,2}^\perp$ and $\wt(c_1, c_2) = 4$, then at least one of $c_1, c_2$ must have weight $4$, and the other codeword is supported on the same set of $4$ collinear points. There are $(q^2+q+1) \binom{q+1}{4}$ ways to choose the support of $(c_1, c_2)$ and $q^2-1$ choices of an ordered pair of codewords, where at least one is nonzero, supported on these points.
Lemma \[supportconiclemma\] implies that $C_{2,2}^\perp$ has no nonzero codewords of weight less than $4$ and Lemma \[supportcubiclemma\] implies that $C_{2,3}^\perp$ has no nonzero codewords of weight less than $5$. Therefore, it is clear that $B_0^{[2]} = 1$ and $B_1^{[2]} = B_2^{[2]} = B_3^{[2]} = 0$. For each $i \in [4,6]$, the $B^{2,2}_i$ term given in the formula for $B_i^{[2]}$ accounts for pairs $(c_2, 0)$ where $c_2 \in C_{2,2}^\perp$ with $\wt(c_2) = i$. For each $i \in [5,6]$, the $B^{2,3}_i$ term given in the formula for $B_i^{[2]}$ accounts for pairs $(0, c_3)$ where $c_3 \in C_{2,3}^\perp$ with $\wt(c_3) = i$. Therefore, we need only consider the contribution from pairs $(c_2, c_3)$ where $c_2 \in C_{2,2}^\perp$ and $c_3 \in C_{2,3}^\perp$ are both nonzero. In this case, if $\wt(c_2, c_3) \le 6$, then $\wt(c_3)$ is equal to $5$ or $6$. Lemma \[supportcubiclemma\] implies that $c_3$ is supported on a set of $5$ or $6$ collinear points. We see that $\supp(c_2,c_3)$ must be a set of $5$ or $6$ collinear points.
We first consider the case where $\wt(c_2,c_3) = 5$, which implies that $\supp(c_2,c_3)$ is a set of $5$ collinear points. Given a collection of $5$ collinear points Proposition \[fdmprop\] implies that there are $q^2-1$ nonzero $c_2 \in C_{2,2}^\perp$ supported on these points and $q-1$ nonzero $c_3 \in C_{2,3}^\perp$ supported on these points. Every such pair has $\wt(c_2,c_3) = 5$. Putting this together completes the calculation of $B_5^{[2]}$.
We now consider the case where $\wt(c_2,c_3) = 6$, which implies that $\supp(c_2,c_3)$ is a set of $6$ collinear points. Proposition \[fdmprop\] implies that there are $q^3-1$ nonzero $c_2 \in C_{2,2}^\perp$ supported a collection of $6$ collinear points and $q^2-1$ nonzero $c_3 \in C_{2,3}^\perp$ supported on these same points. We subtract the number of pairs where $\wt(c_2, c_3) = 5$. We see that the contribution to $B_6^{[2]}$ from pairs where $c_2, c_3$ are both nonzero is $$(q^2+q+1)\binom{q+1}{6}\left((q^3-1)(q^2-1)- \binom{6}{5} (q^2-1)(q-1)\right).$$ This completes the proof.
Lemma \[supportcubiclemma\] implies that $C_{2,3}^\perp$ has no nonzero codewords of weight less than $5$ and that every nonzero codeword of weight at most $7$ is supported on a collinear set of points. Therefore, it is clear that $B_0^{[2]} = 1$ and $B_1^{[2]} = B_2^{[2]} = B_3^{[2]} = 0 = B_4^{[2]} = 0$. For each $i \in [5,9]$, the $2B^{2,2}_i$ term given in the formula for $B_i^{[2]}$ accounts for pairs $(c, 0)$ and $(0,c)$ where $c \in C_{2,3}^\perp$ with $\wt(c) = i$. Therefore, we need only consider the contribution from pairs $(c_1,c_2)$ with $c_1,c_2 \in C_{2,3}^\perp$ both nonzero and $\wt(c_1, c_2) \in [5,9]$. Proposition \[gn\_prop\] implies that the number of such pairs where $\supp(c_1,c_2)$ is a set of $k$ collinear points is $g_3(k)$.
We need only count pairs with $\wt(c_1,c_2) = 8$ and $\wt(c_1,c_2) = 9$ where $\supp(c_1,c_2)$ is not a set of collinear points. We first consider pairs with $\wt(c_1, c_2) = 8$ and divide the count into cases based on $\supp(c_1, c_2)$. Lemma \[supportcubiclemma\] implies that if $c\in C_{2,3}^\perp$ has weight at most $8$ then $\supp(c)$ is either a set of collinear points, $8$ points on a smooth conic, or $8$ points on two $\F_q$-rational lines, $4$ on each, and not including the intersection point.
- If $\wt(c_1,c_2) = 8$ and the support of one of these codewords is a set of $8$ points on a smooth conic, then the other codeword is also supported on these same $8$ points. Lemma \[VScubic8lemma\] implies that given $8$ points on a smooth conic, there are $q-1$ nonzero $c \in C_{2,3}^\perp$ supported on them. This case contributes $(q-1)^2 (q^5-q^2) \binom{q+1}{8}$ to $B_8^{[2]}$.
- If $\wt(c_1,c_2) = 8$ and the support of one of these codewords is a set of $8$ points on of two $\F_q$-rational lines, $4$ on each, and not including the intersection point, then the other codeword is also supported on these $8$ points. Lemma \[VScubic8lemma\] implies that given such a collection of $8$ points, there are $q-1$ nonzero $c \in C_{2,3}^\perp$ supported on them. This case contributes $(q-1)^2 \binom{q^2+q+1}{2} \binom{q}{4}^2$ to $B_8^{[2]}$.
This completes the analysis of $B_8^{[2]}$.
We now consider pairs with $\wt(c_1,c_2) = 9$. Lemma \[supportcubiclemma\] implies that if $c\in C_{2,3}^\perp$ has $\wt(c) \le 9$ then $\supp(c)$ is either a set of collinear points, $8$ or $9$ points on a smooth conic, $8$ or $9$ points on two $\F_q$-rational lines, with at least $4$ points on each, or $9$ points that are the intersection of two cubics that do not share a common component. In this last case, every subset of $8$ of the points imposes independent conditions on cubics.
- If $\wt(c_1,c_2) = 9$ and the support of one of these codewords is contained in $9$ points of a smooth conic, then $\supp(c_1,c_2)$ is a set of $9$ points of a smooth conic. Lemma \[VScubic9lemma\] implies that given $9$ points of a smooth conic, there are $q^2-1$ nonzero $c \in C_{2,3}^\perp$ supported on them. Therefore, there are $(q^2-1)^2$ pairs of nonzero $c_1,c_2\in C_{2,3}^\perp$ with support contained in these $9$ points, and $\binom{9}{8}(q-1)^2$ of these pairs actually have $\wt(c_1,c_2) = 8$. Therefore, this case contributes $(q^5-q^2)\binom{q+1}{9} \left((q^2-1)^2 - 9(q-1)^2\right)$ to $B_9^{[2]}$.
- If $\wt(c_1,c_2) = 9$ and one of these codewords is supported on a collection of $9$ points that are the intersection of two cubics that do not share a common component, then the other codeword must also be supported on these $9$ points. Lemma \[VScubic9lemma\] implies that given $9$ such points, there are $q-1$ nonzero $c\in C_{2,3}^\perp$ supported on them. Therefore, the contribution to $B_9^{[2]}$ from this case is $(q-1)^2 I_9(q)$, where $I_9(q)$ is given in Corollary \[I9cor\].
- In every other case where $\wt(c_1,c_2) = 9,\ \supp(c_1) \cup \supp(c_2)$ is $9$ points contained in two $\F_q$-rational lines, with at least $4$ points on each. We divide this count into two cases.
1. Suppose that $\supp(c_1,c_2)$ is a set of $9$ points on two $\F_q$-rational lines, with $5$ points on one line and $4$ points on the other line, not including the intersection point. There are $(q^2+q+1)(q^2+q)\binom{q}{5} \binom{q}{4}$ ways to choose $9$ points of this type. Lemma \[VScubic9lemma\] implies that given $9$ such points, there are $q^2-1$ nonzero $c \in C_{2,3}^\perp$ supported on them. Therefore, there are $(q^2-1)^2$ pairs of nonzero $c_1,c_2\in C_{2,3}^\perp$ with support contained in these $9$ points. We see that there are $(q-1)^2$ pairs where $c_1, c_2$ are supported on the $5$ collinear points. There are $\binom{5}{4}(q-1)^2$ pairs where $c_1, c_2$ are each supported on the same subset of $4$ of the $5$ collinear points together with the $4$ points of the other line. Noting that $$(q^2-1)^2 - 6(q-1) = (q^4-8q^2+12q-5)$$ completes the analysis in this case.
2. Suppose that $\supp(c_1,c_2)$ is a set of $9$ points on two $\F_q$-rational lines consisting of the intersection point together with $4$ other points on each line. There are $\binom{q^2+q+1}{2} \binom{q}{4}^2$ ways to choose $9$ points of this type. Lemma \[VScubic9lemma\] implies that given $9$ such points, there are $q^2-1$ nonzero $c \in C_{2,3}^\perp$ supported on them. Therefore, there are $(q^2-1)^2$ pairs of nonzero $c_1,c_2\in C_{2,3}^\perp$ with support contained in these $9$ points. These $9$ points contain two subsets of $5$ collinear points. For each of these subsets, there are $(q-1)^2$ pairs with $c_1, c_2$ supported on these $5$ points. There are also $(q-1)^2$ pairs where $c_1, c_2$ are both supported on the subset of $8$ of these $9$ points that does not include the intersection point. Noting that $$(q^2-1)^2 - 3(q-1)^2 = (q^4-5q^2+6q-2)$$ completes the analysis in this case.
Intersections of Two Conics {#sec:twoconics}
===========================
The goal of this section is to prove a version of Theorem \[ThmIntCubics\] for intersections of projective conics. The proof is much less intricate than the proof of Theorem \[ThmIntCubics\], but contains many of the same ideas.
\[ThmIntConics\] We have $$\begin{aligned}
W^{[2]}_{C_{2,2}}(X,Y) & = & X^{q^2+q+1} + \frac{(q-1)q(q+1)^2(q^2+q+1)}{2} X^{2q+1} Y^{q^2-q} \\
& + & (q-1)^2 q^3 (q+1)(q^2+q+1) X^{q+2} Y^{q^2-1} \\
& + & (q-1)(q+1)(q^2+q+1)(2q^3-q^2-q+1) X^{q+1} Y^{q^2} \\
& + & \frac{(q-1)^4 q^4 (q+1)^2 (q^2+q+1)}{24} X^4 Y^{q^2+q-3}\\
& + & \frac{(q-1)^3 q^4 (q+1)^2(q^2+q+1)}{2} X^3 Y^{q^2+q-2} \\
& + & \frac{(q-1)^2 q^3 (q+1)^2 (q^2+q+1) (q^3-2q^2+7q-4)}{4} X^2 Y^{q^2+q-1} \\
& + & \frac{(q^3-q)(q^3-1)(2q^6+q^5-2q^4+5q^3+6q^2-6q+3)}{6} X Y^{q^2+q}\\
& + & \frac{(q-1)^3 q^4 (q+1)(q^2+q+1)(3q^2+1)}{8} Y^{q^2+q+1}.\end{aligned}$$
Starting from the expression for $W_{C_{2,2}^\perp}(X,Y)$ given in Lemma \[lemmalowweightconics\], we prove Theorem \[ThmIntConics\] in two steps.
1. We determine the contribution to $W^{[2]}_{C_{2,2}}(X,Y)$ from conics that share a common component. This determines all but $5$ coefficients of $W^{[2]}_{C_{2,2}}(X,Y)$.
2. We apply Theorem \[MacWilliamsThm2\] to $W^{[2]}_{C_{2,2}}(X,Y)$. This gives $5$ linear equations that must be satisfied by the $5$ unknown coefficients. A linear algebra calculation completes the proof.
Recall from equation that $$\begin{aligned}
W^{[2]}_{C_{2,2}}(X,Y) & = & X^{q^2+q+1} + (q+1) \left(W_{C_{2,2}}(X,Y) - X^{q^2+q+1}\right) \\
& & + (q^2-1)(q^2-q) W_{C_{2,2}}^{(2)}(X,Y).\end{aligned}$$ Since we have already computed $W_{C_{2,2}}(X,Y)$ in Proposition \[WC22prop\], in order to determine $W^{[2]}_{C_{2,2}}(X,Y)$ we need only consider the contribution to $W^{[2]}_{C_{2,2}}(X,Y)$ from linearly independent pairs of elements in $C_{2,2}$, or equivalently, from distinct nonzero plane conics.
Bézout’s theorem implies that two conics that intersect in more than $4$ points share a common component. If two distinct conics share a common component, that component must be an $\F_q$-rational line. We have two possibilities: either one of the conics is a double line $L$ and the other conic is $L$ together with another $\F_q$-rational line, or both conics are pairs of $\F_q$-rational lines, with one of those lines in common. In this last case we have two pairs of lines $\{L, L'\}$ and $\{L,L''\}$, and the number of $\F_q$-intersection points of these pairs depends on whether $L' \cap L''$ lies on $L$ or not. Adding these cases together proves the following.
\[LemmaC22\] Let $$\begin{aligned}
W^{[2],\text{com}}_{C_{2,2}}(X,Y) & = & (q^2+q+1) (q^2+q) q^2 (q-1)^2 X^{q+2} Y^{q^2-1} \\
& & + (q^2-1)(q^2-q) (q^2+q+1)(q+1) X^{q+1} Y^{q^2}.\end{aligned}$$ Then $$\begin{aligned}
W^{[2]}_{C_{2,2}}(X,Y) & = & (q+1) W_{C_{2,2}}(X,Y) -qX^{q^2+q+1} + W^{[2],\text{com}}_{C_{2,2}}(X,Y) \\
&& + c_4 X^4 Y^{q^2+q-3} + c_3 X^3 Y^{q^2+q-2} +c_2 X^2 Y^{q^2+q-1} \\
& & + c_1 X Y^{q^2+q} + c_0 Y^{q^2+q+1},\end{aligned}$$ for some values $c_0, c_1, c_2, c_3, c_4$.
We are now ready to prove Theorem \[ThmIntConics\].
Recall the expression for $W^{[2]}_{C_{2,2}^\perp}(X,Y)$ given in Lemma \[lemmalowweightconics\]. We apply Theorem \[MacWilliamsThm2\] to the expression given in Lemma \[LemmaC22\]. Let $M$ be the $5 \times 5$ matrix with rows and columns labeled from $0$ to $4$ and $(i,j)$ entry equal to the $X^{q^2+q+1-i} Y^i$ coefficient of $(X+(q^2-1)Y)^j (X-Y)^{q^2+q+1-j}$, and let $\mathbf{c}$ be the column vector with entries $c_0,\ldots, c_4$. Then $M \cdot \mathbf{c}$ must be equal to the column vector with $5$ entries, labeled from $0$ to $4$, whose $i$^th^ entry is the $X^{q^2+q+1-i} Y^i$ coefficient of $$\begin{aligned}
& & q^{12} W_{C_{2,2}^\perp}^{[2]}(X,Y) - \bigg( (q+1) W_{C_{2,2}}(X+(q^2-1)Y,X-Y)\\
& & -q(X+(q^2-1)Y)^{q^2+q+1}+ W_{C_{2,2}}^{[2],\text{com}}(X+(q^2-1)Y,X-Y)\bigg).\end{aligned}$$ A linear algebra computation in Sage shows that we must have $$\begin{aligned}
c_0 & = & \frac{1}{8} (q + 1) (q - 1)^3 q^4 (3q^2 + 1) (q^2 + q + 1)
\\
c_1 &= & \frac{1}{6} (q + 1) (q - 1)^2 q^2 (q^2 + q + 1) (2q^5 + q^4 - 2q^3 + 5q^2 + 6q - 6) \\
c_2 &= & \frac{1}{4} (q - 1)^2 (q + 1)^2 q^3 (q^2 + q + 1) (q^3 - 2q^2 + 7q- 4) \\
c_3 &= & \frac{1}{2} (q + 1)^2 (q - 1)^3 q^4 (q^2 + q + 1) \\
c_4 & = & \frac{1}{24} (q + 1)^2 (q - 1)^4 q^4 (q^2 + q + 1).\end{aligned}$$
In an argument that depends on a calculation, it is natural to be a little skeptical. As a double-check, we give a separate argument for the calculation of $c_4$.
We compute $c_4$ using a more geometric argument. If two conics intersect in $4$ points and do not share a common component, then no $3$ of the intersection points are collinear. Let $S$ be a set of $4$ points no $3$ collinear. There are $(q^2+q+1)(q^2+q)q^2(q-1)^2/4!$ ways to choose $4$ such points. Let $\Gamma_S$ be as in the beginning of Section \[sec:supports\]. Since $|S| = 4$, Lemma \[EGHlemma\] implies that these points impose independent conditions on conics. Therefore, $|\Gamma_S| = q^2$. There are $(q^2-1)^2$ pairs of nonzero quadratic polynomials vanishing on these two points. In $(q^2-1)(q-1)$ of these cases, the two polynomials define the same conic. For every other pair, the two polynomials define conics that intersect in $S$ but do not share a common component.
Intersections of Affine Conics
------------------------------
Our strategy for computing $W^{[2]}_{C^A_{2,2}}(X,Y)$ is just like our strategy for computing $W^{[2]}_{C_{2,2}}(X,Y)$. We first count affine conics that share a common component. Two affine conics that share a common component, must have a common $\F_q$-rational line. Considering the possibilities for this pair of conics gives the analogue of Lemma \[LemmaC22\].
\[LemmaC22A\] Let $$\begin{aligned}
W^{[2],\text{com}}_{C^A_{2,2}}(X,Y) & = & (2q + 1)(q + 1)^2 (q - 1)^2 q^2 X^{q} Y^{q^2-q} \\
& & + (q + 1)^2 (q - 1)^3 q^3 X^{q+1} Y^{q^2-q-1}.\end{aligned}$$ Then $$\begin{aligned}
W^{[2]}_{C^A_{2,2}}(X,Y) & = & (q+1) W_{C^A_{2,2}}(X,Y) -qX^{q^2} + W^{[2],\text{com}}_{C^A_{2,2}}(X,Y)\\
& & + c_4 X^4 Y^{q^2-4} + c_3 X^3 Y^{q^2-3} +c_2 X^2 Y^{q^2-2} \\
& & + c_1 X Y^{q^2-1} + c_0 Y^{q^2},\end{aligned}$$ for some values $c_0, c_1, c_2, c_3, c_4$.
An analysis of low-weight codewords of $\left(C_{2,2}^A\right)^\perp$ analogous to the one for $C_{2,2}^\perp$ shows that $\left(C_{2,2}^A\right)^\perp$ has no nonzero codewords of weight less than $4$, and has ${(q-1)(q^2+q)\binom{q}{4}}$ codewords of weight $4$, exactly $q-1$ for each set of $4$ collinear points in the affine plane over $\F_q$. It is then easy to show that if we write $$W^{[2]}_{\left(C^A_{2,2}\right)^\perp}(X,Y) = \sum_{i=0}^{q^2} B^{[2]}_i X^{q^2 - i } Y^i,$$ then $B^{[2]}_0 = 1,\ B^{[2]}_1 = B^{[2]}_2 = B^{[2]}_3 = 0$, and $B^{[2]}_4 = (q^2-1) (q^2+q) \binom{q}{4}$.
We apply Theorem \[MacWilliamsThm2\] to the expression given in Lemma \[LemmaC22A\]. Let $M$ be the $5 \times 5$ matrix with rows and columns labeled from $0$ to $4$ and $(i,j)$ entry equal to the $X^{q^2-i} Y^i$ coefficient of $(X+(q^2-1)Y)^j (X-Y)^{q^2-j}$, and let $\mathbf{c}$ be the column vector with entries $c_0,\ldots, c_4$. Then $M \cdot \mathbf{c}$ must be equal to the column vector with $5$ entries, labeled from $0$ to $4$, whose $i$^th^ entry is the $X^{q^2-i} Y^i$ coefficient of $$\begin{aligned}
& & q^{12} W^{[2]}_{\left(C^A_{2,2}\right)^\perp}(X,Y)- \bigg( (q+1) W_{C^A_{2,2}}(X+(q^2-1)Y,X-Y)\\
& & -q(X+(q^2-1)Y)^{q^2}+ W_{C^A_{2,2}}^{[2],\text{com}}(X+(q^2-1)Y,X-Y)\bigg).\end{aligned}$$ A linear algebra computation in Sage proves the following.
\[thmaffineconic\] Using the notation of Lemma \[LemmaC22A\], $$\begin{aligned}
c_0 & = & \frac{3}{8} q(q+1)(q-1)^2 \left(q^8 + \frac{8}{9} q^7 + \frac{7}{3} q^6 - \frac{19}{9} q^5 + \frac{14}{3} q^4 + \frac{59}{9} q^3 - \frac{8}{3} q^2 + \frac{8}{3}\right)
\\
c_1 &= & \frac{1}{3} (q+1)(q-1)^2 q^3 \left(q^6 + 2 q^5 - \frac{5}{2} q^4 - \frac{29}{2} q^3 + \frac{15}{2} q^2 - \frac{27}{2} q + 3\right) \\
c_2 &= & \frac{1}{4} (q+1)^2 (q-1)^3 q^4 \left(q^3 - 2q^2 +14 q -11 \right) \\
c_3 &= & \frac{2}{3} \left(q - \frac{5}{4}\right) (q+1)^2 (q-1)^4 q^4 \\
c_4 & = & \frac{1}{24} (q+1)^2(q-1)^4 q^4 (q^2-3q+3).\end{aligned}$$
We will use this result in Section \[sec:large\_int\] when we count pairs of cubics that share an $\F_q$-rational line
Intersections of a Conic and a Cubic {#sec:con_cub}
====================================
In this section we explain how to use the strategy of the previous section to determine the number of pairs of a homogeneous quadratic polynomial $f_2(x,y,z)$ and a homogeneous cubic polynomial $f_3(x,y,z)$ that have a given number of common zeros in $\Proj^2(\F_q)$. Throughout the rest of this section suppose that $q>2$.
Bézout’s theorem implies that a conic and a cubic that intersect in more than $6$ points must share a common component. We determine the contribution to $W^{[2]}_{C_{2,2},C_{2,3}}(X,Y)$ from pairs that share a common component, leaving us with $7$ unknown coefficients of this weight enumerator.
\[LemmaC2C3\] Let $$\begin{aligned}
W^{[2],\text{com}}_{C_{2,2},C_{2,3}}(X,Y) & = & \frac{q^2-q}{2} (q-1)^2 (q^2+q+1)^2 X Y^{q^2+q} \\
& & + \frac{1}{2}q(q+1)(q^3-1)^2 X^{2q+1} Y^{q^2-q} \\
& & + \frac{1}{2} (q^2 + q + 1)(q + 1)(q - 1)^3 q^5 X^{q+3} Y^{q^2-2} \\
& & + 2(q^2 + q + 1)(q + 1)(q - 1)^2 q^5 X^{q+2} Y^{q^2-1} \\
& & + \frac{1}{2} (q^2+q+1)(q-1)^2(q^7+5q^5+4q^4+2q^3+2q+2) X^{q+1} Y^{q^2}.\end{aligned}$$ Then $$\begin{aligned}
W^{[2]}_{C_{2,2},C_{2,3}}(X,Y)& = & W_{C_{2,2}}(X,Y) + W_{C_{2,3}}(X,Y) - X^{q^2+q+1} +W^{[2],\text{com}}_{C_{2,2},C_{2,3}}(X,Y)\\
& & + c_6 X^6 Y^{q^2+q-5} + c_5 X^5 Y^{q^2+q-4} + c_4 X^4 Y^{q^2+q-3} + c_3 X^3 Y^{q^2+q-2} \\
& & + c_2 X^2 Y^{q^2+q-1} + c_1 X Y^{q^2+q} + c_0 Y^{q^2+q+1},\end{aligned}$$ for some values $c_0, c_1, c_2, c_3, c_4, c_5, c_6$.
If a conic and a cubic share a common component, that component must either be: a pair of Galois-conjugate lines defined over $\F_{q^2}$ but not over $\F_q$, a smooth conic, a pair of distinct $\F_q$-rational lines, or a single $\F_q$-rational line.
- If a conic and a cubic share a pair of Galois-conjugate lines defined over $\F_{q^2}$ but not over $\F_q$, then the conic is this pair of lines, and the cubic is this pair of lines together with an additional $\F_q$-rational line. The only $\F_q$-rational point in the intersection of such a conic and cubic, is the single rational point of the pair of Galois-conjugate lines. There are $(q^2+q+1)(q^2-q)/2$ pairs of Galois-conjugate lines, $q^2+q+1$ choices for the additional line of the cubic, and we include a factor of $(q-1)^2$ to account for scaling.
- There are $q^5-q^2$ smooth conics defined over $\F_q$. The number of nonzero cubic polynomials vanishing on a given conic is $q^3-1$.
- There are $\binom{q^2+q+1}{2}$ pairs of distinct $\F_q$-rational lines. The number of nonzero cubic polynomials vanishing on a given pair of lines is $q^3-1$.
- There are $(q-1)(q^2+q+1)$ quadratic polynomials that define a double line. There are $q^6-1$ nonzero cubic polynomials vanishing on a given line.
In every other case in which a conic and a cubic share a common component, the conic factors as a product of a pair of distinct $\F_q$-rational lines, and the cubic contains one, but not both, of those lines. The number of $\F_q$-points of the intersection depends on how the cubic intersects the second line.
We focus on the particular case where the conic is defined by the polynomial $xz$, and the cubic contains the line $x = 0$ but not $z=0$. At the end of the calculation we multiply by $(q^2+q+1)(q^2+q)$ to account for the choice of a pair of lines and the choice of the shared line.
A cubic polynomial that vanishes on the line $x=0$ is of the form $$f_3(x,y,z) = x(a_0 x^2 + a_1 xy + a_2 xz + a_3 y^2 + a_4 yz + a_5 z^2).$$ Since the cubic does not contain the line $z=0$ at least one of $a_0, a_1, a_3$ is nonzero. We want to count the number of zeros of this polynomial on the line $z=0$ away from the point $[0:1:0]$. Therefore, we need only count the number of zeros of $a_0 + a_1 y + a_3 y^2$ on the affine line with coordinate $y$. Note that this does not depend on $a_2, a_4, a_5$, which leads to a factor of $q^3$ at the end of our count.
There are $q^3-1$ choices of $(a_0,a_1, a_3)$:
1. There are $(q-1) \binom{q}{2}$ choices that give a quadratic polynomial with distinct $\F_q$-rational roots.
2. There are $(q-1) q$ choices that give a quadratic polynomial with a double root.
3. If $a_3 = 0$ and $a_1 \neq 0$ we get a linear polynomial with a single $\F_q$-rational root.
4. If $a_3 = a_1 = 0$ and $a_0 \neq 0$ we get a constant polynomial with no $\F_q$-rational roots.
5. There are $(q-1) \frac{q^2-q}{2}$ choices that give an irreducible quadratic polynomial.
We complete the proof by combining all of these observations. For example note that $$\begin{aligned}
& & (q-1)(q^5-q^2)(q^3-1)+(q-1)(q^2+q+1)(q^6-1) \\
& &+(q-1)(q^2+q+1)(q^2+q)q^3\left((q-1)\frac{q^2-q}{2} + (q-1)\right) \\
& & = \frac{1}{2}(q^2+q+1)(q-1)^2 \left(q^7+5q^5+4q^4+2q^3+2q+2\right).\end{aligned}$$
We are now ready to prove the following result.
Using the notation of Lemma \[LemmaC2C3\], $$\begin{aligned}
c_0 & = & \frac{53}{144} (q-1)^3 q^6 (q^2+q+1) \left(q^5 + q^4 + \frac{9}{53} q^3 + \frac{27}{53} q^2 + \frac{58}{53} q - \frac{32}{53}\right)
\\
c_1 &= & \frac{11}{30} q^4 (q-1)^2 (q^2+q+1) (q+1) \left(q^7 + \frac{1}{44} q^6 + \frac{5}{11} q^5 + \frac{20}{11} q^4 - \frac{31}{11} q^3 +\frac{159}{44} q^2 + \frac{15}{11}q - \frac{30}{11}\right) \\
c_2 &= & \frac{3}{16} (q-1)^2 (q+1)^2 q^5 (q^2+q+1) \left(q^5 - \frac{2}{9} q^4 + \frac{35}{9}q^3 - \frac{70}{9} q^2 + \frac{160}{9} q - \frac{32}{3} \right) \\
c_3 &= & \frac{1}{18} (q+1)(q-1)^3 q^5 (q^2+q+1) \left(q^5+\frac{9}{2} q^4 + \frac{3}{2} q^3 + \frac{39}{2} q^2 + \frac{79}{2} q - 9\right) \\
c_4 & = & \frac{1}{48} (q+1)(q-1)^3 q^6 (q^2+q+1) \left(q^4 + 13 q^2 + 26q - 48\right)\\
c_5 &= & \frac{1}{24} (q+1)(q-1)^4 q^6 (q^2+q+1) (q^2+2q-5) \\
c_6 & = & \frac{1}{720} (q-2)(q+1)(q-1)^4q^6 (q^2+q+1)(q^2+3q-8).\end{aligned}$$
We apply Theorem \[MacWilliamsThm2\] to the expression given in Lemma \[LemmaC2C3\]. Let $M$ be the $7 \times 7$ matrix with rows and columns labeled from $0$ to $6$ and $(i,j)$ entry equal to the $X^{q^2+1+1-i} Y^i$ coefficient of $(X+(q^2-1)Y)^j (X-Y)^{q^2+q+1-j}$, and let $\mathbf{c}$ be the column vector with entries $c_0,\ldots, c_6$. Then $M \cdot \mathbf{c}$ must be equal to the column vector with $7$ entries, labeled from $0$ to $6$, whose $i$^th^ entry is the $X^{q^2+q+1-i} Y^i$ coefficient of $$\begin{aligned}
& & q^{16} W^{[2]}_{C_{2,2}^\perp, C_{2,3}^\perp}(X,Y)- \bigg( W_{C_{2,2}}(X+(q^2-1)Y,X-Y) + W_{C_{2,3}}(X+(q^2-1)Y,X-Y) \\
& & -(X+(q^2-1)Y)^{q^2}+ W_{C_{2,2},C_{2,3}}^{[2],\text{com}}(X+(q^2-1)Y,X-Y)\bigg).\end{aligned}$$ The $X^{q^2+q+1-i}Y^i$ coefficient of $W_{C_{2,3}}(X+(q-1)Y,X-Y)$ is a polynomial in $q$ for each $i \in [0,9]$ [@KaplanAGCT Section 4]. For the same reason, for each $i \in [0,9]$ the $X^{q^2+q+1-i}Y^i$ coefficient of $W_{C_{2,3}}(X+(q^2-1)Y,X-Y)$ is also a polynomial in $q$.
A linear algebra calculation in Sage completes the proof.
Just as we did following the proof of Theorem \[ThmIntConics\], we include a separate argument for the computation of one of the weight enumerator coefficients.
We compute $c_6$ using a more geometric argument. If a conic and a cubic intersect in $6$ points and do not share a common component, then no $4$ of these intersection points are collinear. By Lemma \[EGHlemma\], such a collection of points imposes independent conditions on cubics. If $3$ of the points are collinear, then the conic must contain this line. If a conic contains $6$ points with no $4$ on a line, and $3$ of those points are collinear, then the remaining $3$ points must also be collinear and none of the $6$ points is the intersection point of the two lines.
Suppose $S$ is a collection of $6$ points no $4$ collinear. Since $S$ imposes independent conditions on cubics, there are $q^4-1$ nonzero cubic polynomials vanishing on $S$. We divide the count into two cases.
1. Suppose $S$ is a set of $6$ points on a smooth conic. There are $q-1$ nonzero quadratic polynomials vanishing on $S$, so there are $(q-1)(q^4-1)$ pairs of a nonzero quadratic polynomial and a nonzero cubic polynomial such that the corresponding conic and cubic contain $S$ in their intersection. In $(q-1)(q^3-1)$ of these cases, the cubic consists of this conic together with an additional line.
2. Suppose $S$ is a set of $6$ points on two $\F_q$-rational lines, with $3$ points on each, not including the intersection point. There are $q-1$ nonzero quadratic polynomials vanishing on $S$, so there are are $(q-1)(q^4-1)$ pairs of a nonzero quadratic polynomial and a nonzero cubic polynomial such that the corresponding conic and cubic contain $S$ in their intersection. By Bézout’s theorem, a cubic that contains a line and also contains $3$ collinear points that do not lie on that line, must contain the second line as well. Therefore, there are $(q-1)(q^3-1)$ pairs such that the conic and cubic share a common component.
Noting that $$\begin{aligned}
&(q^5-q^2) \binom{q+1}{6} (q-1)\left(q^4-1-(q^3-1)\right) + \binom{q^2+q+1}{2}\binom{q}{3}^2 (q-1)\left(q^4-1-(q^3-1)\right) \\
& = \frac{1}{720} (q-2)(q+1)(q-1)^4q^6 (q^2+q+1)(q^2+3q-8)\end{aligned}$$ completes the proof.
Cubic Curves that Share a Common Component {#sec:large_int}
==========================================
The goal of this section is to prove the analogue of Lemmas \[LemmaC22\], \[LemmaC22A\], and \[LemmaC2C3\].
\[LemmaC23\] Let $W_{C_{2,3}}^{[2],\text{com}}(X,Y)$ denote the contribution to $W_{C_{2,3}}^{[2]}(X,Y)$ from pairs of nonzero polynomials $(f,g)$ that define distinct cubic curves that share a common component. Then $$\begin{aligned}
W_{C_{2,3}}^{[2],\text{com}}(X,Y) & = & a_1 X Y^{q^2+q} + a_2 X^2 Y^{q^2+q-1} + a_{2q+1} X^{2q+1} Y^{q^2-q} \\
& &+ a_{2q+2} X^{2q+2} Y^{q^2-q-1} + \sum_{i=q+1}^{q+5} a_{i} X^{i} Y^{q^2+q+1 - i},\end{aligned}$$ where $$\begin{aligned}
a_1 & = & \frac{1}{2}(q-1)^2 (q^2+q+1) (q^2-q) (q+1)q \\
a_2 & = & \frac{1}{2}(q-1)^2 (q^2+q+1) (q^2-q) (q^2+q)(q+1)q \\
a_{2q+1} & = & \frac{1}{2}(q^2+q+1)(2q+1)(q-1)^2q^2(q+1)^2 \\
a_{2q+2} & = & \frac{1}{2}(q^2+q+1)(q-1)^2(q^2+q)^2(q^2-q) \\
a_{q+1} & = & \cfrac{1}{24} (9q^8 + 8q^7 + 21q^6 - 19q^5 + 66q^4 + 59q^3 - 48q^2 +24)(q^2 + q + 1)(q + 1)(q - 1)^2 q \\
a_{q+2} &= & \frac{1}{6}(2q^5 + 2q^4 - 7q^3 + 42q^2 - 33q + 6)(q^2 + q + 1)(q + 1)^2(q - 1)^2q^3 \\
a_{q+3} &= & \frac{1}{4}(q^3 - 2q^2 + 14q - 11)(q^2 + q + 1)(q + 1)^2(q - 1)^3q^4 \\
a_{q+4} &= & \frac{1}{6}(q^2 + q + 1)(4q - 5)(q + 1)^2(q - 1)^4 q^4 \\
a_{q+5} & = & \frac{1}{24}(q-1)^4q^4(q+1)^2(q^2+q+1)(q^2-3q+3).\end{aligned}$$
Bézout’s theorem then implies the following result.
\[CorC23\] We have $$\begin{aligned}
W_{C_{2,3}}^{[2]}(X,Y)& = & (q+1) W_{C_{2,3}}(X,Y) -qX^{q^2+q+1} + W_{C_{2,3}}^{[2],\text{com}}(X,Y) \\
& & + \sum_{j=0}^9 c_j X^j Y^{q^2+q+1-j},\end{aligned}$$ for some values $c_0, c_1, \ldots, c_9$.
Since we only consider pairs of polynomials $(f,g)$ that define distinct cubic curves, we may assume that $g$ is not a scalar multiple of $f$. Bézout’s theorem implies that two distinct cubic curves that intersect in more than $9$ points share a common component. There are several possibilities for two distinct cubics that share a common component:
- The two cubics share a common pair of Galois-conjugate lines defined over $\F_{q^2}$ but not over $\F_q$. In this case, each cubic consists of this pair of conjugate lines together with an additional $\F_q$-rational line.
- The two cubics share a common smooth conic. In this case, both cubics consist of this conic together with an $\F_q$-rational line.
- The two cubics share a common pair of distinct $\F_q$-rational lines. In this case, each cubic consists of these two lines together with another line, where we note that this third line might make one of the common lines into a double line.
- The two cubics share a unique $\F_q$-rational line. In this case there are many possibilities for how each cubic decomposes. The important fact to note is that the number of intersection points of such a pair of cubics is in $[q+1,q+5]$.
The proof consists of case analysis.
1. We compute the contribution from polynomials that define cubic curves that share a pair of Galois-conjugate lines defined over $\F_{q^2}$ but not over $\F_q$. There are $(q^2+q+1) (q^2-q)/2$ such pairs of lines. Each cubic also contains an additional $\F_q$-rational line. Since the cubics are distinct, these lines must be different. Such a pair of cubics either intersects in exactly $1\ \F_q$-point, which is the case when the two rational lines pass through the $\F_q$-point of the pair of Galois-conjugate lines, or $2\ \F_q$-points, which is the case when the two rational lines intersect at any of the $q^2+q$ other points of $\Proj^2(\F_q)$. There are $(q+1)q$ ordered pairs of distinct lines containing a chosen point. We include a factor of $(q-1)^2$ to account for scaling each polynomial. This completes the computation of $a_1$ and $a_2$.
2. We compute the contribution from polynomials that define cubic curves that share a pair of $\F_q$-rational lines. There are $\binom{q^2+q+1}{2}$ such pairs. Each cubic consists of these two common lines and another line, where that line may be one of the shared common lines. These two additional lines intersect in a unique $\F_q$-point. There are $2q+1$ points on the two shared lines, and $q^2-q$ points not on these lines. Once we choose the intersection point, there are $(q+1)q$ ordered pairs of distinct lines containing this point. We include a factor of $(q-1)^2$ to account for scaling each polynomial. This completes the computation of $a_{2q+1}$ and $a_{2q+2}$.
3. We compute the contribution from polynomials that define cubic curves that share a smooth conic. There are $q^5-q^2$ choices for the conic. Each cubic contains this common conic and an additional line. These lines intersect in a unique $\F_q$-point. There are $q+1$ points on the conic and $q^2$ points not on the conic. There are $(q+1)q$ ordered pairs of distinct lines containing a chosen intersection point. We include a factor of $(q-1)^2$ to account for scaling each polynomial. So the contribution in this case is $$(q^5-q^2) (q+1)q(q-1)^2 \left( (q+1) X^{q+1} Y^{q^2} + q^2 X^{q+2} Y^{q^2-1} \right).$$
4. We compute the contribution from polynomials that define cubic curves that share a unique rational line. There are $q^2+q+1$ choices for the line. Each cubic consists of this common line together with a plane conic that may be reducible. We need to count the number of $\F_q$-rational intersection points of these conics not on the common line. This is equivalent to counting the number of $\F_q$-rational intersection points of the pair of affine conics we get from taking the common line as the ‘line at infinity’ in $\Proj^2$. That is, the contribution from this case is $$(q^2+q+1) \left(\sum_{i=0}^4 c_i X^{q+1+i} Y^{q^2-i}\right),$$ where the values of $c_0,c_1, c_2, c_3$, and $c_4$ are given in Theorem \[thmaffineconic\]. We do not need to include a factor of $(q-1)^2$ to account for scaling each polynomial because this scaling is already included in the result of Theorem \[thmaffineconic\].
Combining these cases completes the proof of the theorem.
As an additional check, we explain how to compute the contribution from this final case in a different way. Two cubics that share a unique $\F_q$-rational line consist of that common line together with a pair of projective plane conics. These conics intersect in between $0$ and $4\ \F_q$-points. If this pair of cubics intersect in $q+5$ rational points, then the conics must intersect in $4\ \F_q$-points, and the shared line cannot contain any of them. We first choose the pair of conics and then choose the line.
The number of pairs of projective conics intersecting in $4$ points is given as $c_4$ at the end of the proof of Theorem \[ThmIntConics\]. If two conics that do not share a component intersect in $3$ or $4\ \F_q$-points, then no $3$ of these intersection points are collinear. Therefore, the number of lines not containing any of these $4$ intersection points is $q^2+q+1 - (4(q+1)- \binom{4}{2})$. This gives another proof that $$a_{q+5} = \frac{1}{24} (q+2+q+1) (q+1)^2(q-1)^4 q^4 (q^2-3q+2).$$ We can also verify the other computations from this final case using this method but do not give the details here.
Intersections of Two Cubics: The Proof of Theorem \[ThmIntCubics\] {#sec:pf_thm1}
==================================================================
We are now ready to prove Theorem \[ThmIntCubics\].
We recall the expression for the low-weight coefficients of $W_{C_{2,3}^\perp}^{[2]}(X,Y)$ given in Lemma \[lemmalowweightcubics\]. We apply Theorem \[MacWilliamsThm2\] to the expression given in Corollary \[CorC23\]. Let $M$ be the $10 \times 10$ matrix with rows and columns labeled from $0$ to $9$, and $(i,j)$ entry equal to the $X^{q^2+q+1- i}Y^i$ coefficient of $(X+(q^2-1)Y)^j (X-Y)^{q^2+q+1-j}$, and let $\mathbf{c}$ be the column vector with entries $c_0,\ldots, c_9$. Then $M \cdot \mathbf{c}$ must be equal to the column vector with $10$ entries, labeled from $0$ to $9$, whose $i$^th^ entry is the $X^{q^2+q+1-i} Y^i$ coefficient of $$\begin{aligned}
& & q^{20} W_{C_{2,3}^\perp}^{[2]}(X,Y) - \bigg((X+(q^2-1)Y)^{q^2+q+1}\\
& & + (q+1) \left(W_{C_{2,3}}(X+(q^2-1)Y,X-Y) -(X+(q^2-1)Y)^{q^2+q+1}\right)\\
& & + W_{C_{2,3}}^{[2],\text{com}}(X+(q^2-1)Y,X-Y)\bigg).\end{aligned}$$ A linear algebra computation in Sage completes the proof.
Further Questions {#sec:further}
=================
One could try to follow the strategy of the proof of Theorem \[ThmIntCubics\] to study intersections of curves of higher degree.
\[Questionde\] Let $d$ and $e$ be positive integers and let $k \in [0,de]$ be an integer. How many of the $\left(q^{\binom{d+2}{2}}- 1\right) \left(q^{\binom{e+2}{2}}-1\right)$ ordered pairs $(f,g)$ with $f \in \F_q[x,y,z]_d$ and $g \in \F_q[x,y,z]_e$, both nonzero, define curves that intersect in exactly $k\ \F_q$-points and do not share a common component?
In this paper, we answer this question when $(d,e) \in \{(2,2),(3,2),(3,3)\}$. As mentioned in Remark \[Rem2\], in forthcoming work we study the case where $e = 2$ and $d$ is arbitrary, and get a polynomial formula for each $k$. For any $(d,e)$, the main term as $q\to \infty$ is due to Entin [@Entin]; see Theorem \[ThmEntin\] in the introduction. We believe that the case $(d,e) = (3,3)$ is a kind of boundary for this problem. In this final section, we discuss two approaches to Question \[Questionde\].
We consider analogues of the main steps of the proof of Theorem \[ThmIntCubics\] for general $(d,e)$.
1. *Determine the contribution to $W_{C_{2,d},C_{2,e}}^{[2]}(X,Y)$ from pairs $(f,g)$ that define a degree $d$ curve and a degree $e$ curve that do share a common component.*
If we are able to complete this step, this leaves $de+1$ unknown coefficients of $W_{C_{2,d},C_{2,e}}^{[2]}(X,Y)$.
2. *Determine the $de+1$ lowest-weight coefficients of $W_{C_{2,d}^\perp,C_{2,e}^\perp}^{[2]}(X,Y)$ by counting configurations of up to $de$ points that fail to impose independent conditions on curves of degree $d$, and on curves of degree $e$.*
If we are able to complete this step, we get $de+1$ linear conditions that must be satisfied by the $de+1$ unknown coefficients of $W_{C_{2,d},C_{2,e}}^{[2]}(X,Y)$. We would hope that these conditions uniquely determine these coefficients.
We briefly discuss two difficulties with this approach.
Traces of Hecke Operators and Coefficients of $W_{C_{2,3}^\perp}(X,Y)$
----------------------------------------------------------------------
Suppose that $e = 3$ and $d$ is arbitrary. We want to determine the $3d+1$ lowest-weight coefficients of $W_{C_{2,d}^\perp,C_{2,3}^\perp}^{[2]}(X,Y)$. The contribution to the $X^{q^2+q+1-j} Y^j$ coefficient of this weight enumerator from pairs of the form $(0,c)$ where $c \in C_{2,3}^\perp$ is the $X^{q^2+q+1-j} Y^j$ coefficient of $W_{C_{2,3}^\perp}(X,Y)$.
Theorem \[C23LowWeightDual\] states that for each $j \le 9$, the $X^{q^2+q+1-j} Y^j$ coefficient of $W_{C_{2,3}^\perp}(X,Y)$ is a polynomial in $q$. Once $j \ge 10$, the corresponding coefficient is not a polynomial in $q$, but involves a contribution from the trace of the Hecke operator $T_q$ acting on spaces of cusp forms of weight at most $j+2$ for $\operatorname{SL}_2(\Z)$. See [@KaplanAGCT Section 4] for a more detailed discussion of how these non-polynomial terms arise in these dual code coefficients. For a precise statement of the $X^{q^2+q-9}Y^{10}$ coefficient of $W_{C_{2,3}^\perp}(X,Y)$ when $q\ge 5$ is prime see [@KaplanAGCT Theorem 3]. We conclude that when $d \ge 4$, one should not expect the first $3d+1$ lowest-weight coefficients of $W_{C_{2,d}^\perp,C_{2,3}^\perp}^{[2]}(X,Y)$ to be polynomial in $q$. When both $d,e$ are greater than $3$ we expect similar, but even more complicated, behavior for these dual code coefficients.
Counting Points that Fail to Impose Independent Conditions on Curves of Degree $d$
----------------------------------------------------------------------------------
For the rest of this section, suppose that $k < \binom{d+2}{2}$.
A key to answering the kinds of questions about intersections of curves that we study in this paper is understanding the number of collections of $k$ points in $\Proj^2(\F_q)$ that fail to impose independent conditions on curves of degree $d$. This question has been extensively studied (see [@EGH; @Harris; @Miranda]), but we do not have a complete characterization for every degree $d$ and every $k$. Let $X_{d,k}$ be the space $\{(C,p_1,\ldots, p_k)\}$ where $C$ is a plane curve of degree $d$ and $p_1,\ldots, p_k$ are distinct points of $C$. Let $\Conf^k(\Proj^2)$ denote the configuration space of unordered $k$-tuples of distinct points in $\Proj^2$. We have a projection map $\pi \colon X_{d,k} \rightarrow \Conf^k(\Proj^2)$. Since a generic set of $k$ points in $\Proj^2$ will impose independent conditions on degree $d$ curves, the fiber over a point of $\Conf^k(\Proj^2)$ is generically isomorphic to $\Proj^{\binom{d+2}{d}-k}$.
Consider $\mathcal{F}_{k,d} \subset \Conf^k(\Proj^2)$, the subset consisting of collections of $k$ distinct points that fail to impose independent conditions on degree $d$ curves. We can define $\mathcal{F}_{k,d}$ algebraically as follows. Choose an affine representative for each point of $\Proj^2$ and consider the standard set of $\binom{d+2}{2}$ monomials, $x^i y^j z^k$ where $i+j+k = d$. By evaluating this basis of monomials, an ordered collection of $k$ points in $\Proj^2$ gives $\binom{d+2}{2} \times k$ matrix $A$. The set of points imposes independent conditions on degree $d$ curves if and only if the rank of the matrix is equal to $k$. Note that the rank does not depend on a choice of ordering of the points. In this way, we see that $\mathcal{F}_{k,d}$ can be defined by the simultaneous vanishing of a set of $k \times k$ minors.
1. What can we say about $\#\mathcal{F}_{k,d}(\F_q)$? In cases where we cannot get an exact answer, what can we say about asymptotic behavior as $q \to \infty$?
2. Can only obtain information about the singular cohomology of $\mathcal{F}_{k,d}$? Can one use this information along with tools from étale cohomology to deduce consequences for $\#\mathcal{F}_{k,d}(\F_q)$?
A better understanding of the cohomology of $\mathcal{F}_{k,d}$ would likely shed light on the question of when the kinds of counting problems we discuss in this paper have polynomial or non-polynomial answers.
Acknowledgements {#acknowledgements .unnumbered}
================
The first author is supported by NSF Grant DMS-1802281. Part of this work grew out of the PhD thesis of the first author. He thanks Noam Elkies for his guidance and support throughout this project. The authors thank Igor Dolgachev for helpful comments related to an earlier draft of this paper.
The number of collections of $9$ points in $\Proj^2(\F_q)$ that are the intersection of two cubics {#AppendixA}
==================================================================================================
The goal of this appendix is to prove the following result.
\[I9count\] Let $I_9(q)$ be the number of collections of $9$ points in $\Proj^2(\F_q)$ such that there exist two cubic curves that do not share a common component intersecting at these $9$ points. Then, $$I_9(q) =\frac{1}{9!}(q^6 + 2q^5 - 73q^4 + 344q^3 - 838q^2 + 1754q - 2030)(q^2 + q + 1)(q + 1)(q - 1)^2(q - 2)q^4.$$
In Section \[sec:dual\_coef\] we analyzed codewords of $C_{2,3}^\perp$ of weight $9$. Comparing these calculations to an earlier result of Kaplan proved this result, except that [@KaplanAGCT Theorem 3] contains the additional assumption that the characteristic of $\F_q$ is not $2$ or $3$. In this appendix we give a proof of Theorem \[I9count\] that works in all characteristics, and moreover, makes no reference to results from coding theory.
Suppose $\{P_1,\ldots, P_9\}$ is a collection of $9$ points such that there exist two cubic curves that do not share a common component intersecting at these $9$ points. No $4$ of these $9$ points lie on a line, since a cubic that contains $4$ collinear points contains that line. No $7$ of these $9$ points lie on a conic, since a cubic that contains $7$ points on a conic contains that conic.
We next recall a version of what is often called the Cayley-Bacharch theorem, but is more accurately due to Chasles.
[@Hartshorne Corollary 4.5]\[Hart4.5\] Given $8$ distinct points $P_1,\ldots, P_8$ in the plane, no $4$ collinear, and no $7$ lying on a conic, there is a uniquely determined $P_9$ (possibly an infinitely near point) such that every cubic through $P_1,\ldots, P_8$ also passes through $P_9$.
We are particularly interested in the case where the point $P_9$ is distinct from the first $8$ points.
\[prop\_infnear\] Using the notation of Proposition \[Hart4.5\], $P_9$ is infinitely near to one of the first $8$ points if and only if one of the following cases holds after relabeling the points:
1. $P_1,\ldots, P_8$ lie on an absolutely irreducible singular cubic with one of the points being the singular point.
2. $P_1,\ldots, P_6$ lie on a smooth conic, and the line containing $P_7$ and $P_8$ also contains exactly one of the first $6$ points.
3. $P_1,P_2, P_3$ are collinear, and $P_4,P_5,P_6$ are collinear, where none of these $6$ points is the intersection point of these two lines, and the line containing $P_7$ and $P_8$ contains exactly one of the first $6$ points.
We thank Igor Dolgachev for helpful suggestions related to the argument below.
We first show that if $P_1\ldots, P_8$ are in one of the three configurations in the statement, then $P_9$ is infinitely near to one of the $P_i$.
1. Suppose $P_1,\ldots, P_8$ are as in the first case of the statement let $C$ be an absolutely irreducible cubic containing them with a singular point at one of the points. Without loss of generality, suppose the singular point is at $P_1$. Let $C'$ be any other cubic containing $P_1,\ldots, P_8$. The cubics $C$ and $C'$ already intersect at multiplicity $9$, so it is not possible for $C$ and $C'$ to intersect at any points not in $\{P_1,\ldots, P_8\}$, or for $C$ and $C'$ to intersect at multiplicity greater than $1$ at any of $P_2,\ldots, P_8$.
2. Suppose $P_1,\ldots, P_8$ are in the second case of the statement and let $C$ be the union of the conic through $P_1,\ldots, P_6$ and the line containing $P_7$ and $P_8$. Let $C'$ be any other cubic containing $P_1,\ldots, P_8$. The rest of the argument is identical to the previous case.
3. Suppose $P_1,\ldots, P_8$ are in the third case of the statement and let $C$ be the be the union of the lines through $P_1, P_2, P_3$, through $P_4,P_5,P_6$, and through $P_7,P_8$. Let $C'$ be any other cubic containing $P_1,\ldots, P_8$. The rest of the argument is identical to the previous case.
Let $8$ be distinct points $P_1,\ldots, P_8$ in the plane, no $4$ collinear, and no $7$ lying on a conic. By Proposition \[Hart4.5\], there is a uniquely determined $P_9$ such that every cubic through $P_1,\ldots, P_8$ also passes through $P_9$. Suppose that $P_9$ is infinitely near to one of the first $8$ points. Without loss of generality, suppose that $P_9$ is infinitely near to $P_1$. We now prove that $P_1,\ldots, P_8$ are in one of the three configurations given in the statement.
The space of cubic polynomials containing $8$ points in the plane has dimension at least $2$. Let $f,g \in \F_q[x,y,z]_3$ be linearly independent polynomials that define cubic curves vanishing at $P_1,\ldots, P_8$ with tangent direction at $P_1$ specified by $P_9$. Then it is clear that there exists $\lambda \in \F_q^*$ such that $f-\lambda g$ is a nonzero cubic containing $P_1,\ldots, P_8$ that is singular at $P_1$.
Suppose that $S = \{P_1,\ldots, P_8\}$ is a set of distinct points in the plane with no $4$ collinear and no $7$ on a conic and that there is a cubic containing these points with a singular point at $P_1$. If this cubic is absolutely irreducible we are in the first case of the proposition.
Suppose that the cubic is reducible. Every reducible cubic defined over $\F_q$ either factors as a smooth conic defined over $\F_q$ together with an $\F_q$-rational line, or as a union of three lines.
1. If the cubic factors as a smooth conic $C$ and a line $L$, then $P_1 \in C \cap L$. It is not possible to have $3$ of the remaining $7$ points of $S$ lie on $L$ and it is not possible to have $6$ of the remaining points of $S$ on $C$. Therefore, $|C\cap S| = 6,\ |L \cap S| = 3$, and $P_1$ is the only point of $S$ in $C \cap L$. After rearranging the labels of the points, we see that we are in the second case of the proposition.
2. Suppose the cubic factors as the union of $3$ lines, $L_1, L_2, L_3$. Since the union of these $3$ points contains at least $8\ \F_q$-points and none of these lines contains more than $3\ \F_q$-points, we see that the $L_i$ are distinct and $\F_q$-rational. The only singular points of the cubic are the intersection points of pairs of the lines. Without loss of generality, $P_1$ is the intersection point of $L_1$ and $L_2$. We see that each of $L_1$ and $L_2$ contains at most $2$ additional points from $S$, so $L_1 \cup L_2$ contains at most $5$ points of $S$. Since $L_3$ contains at most $3$ points from $S$, we see that $|(L_1\cup L_2) \cap S| = 5$ and $L_3$ contains exactly the $3$ reamaining points of $S$. After rearranging the labels of the points, we see that we are in the third case of the proposition.
\[J8prop\] Let $J_8(q)$ be the number of collections of $8$ points in $\Proj^2(\F_q)$ such that:
1. No $4$ points are collinear and no $7$ points lie on a conic;
2. It is not the case that the $8$ points lie on an absolutely irreducible singular cubic with the singular point at one of the $8$;
3. It is not the case that $6$ of the points lie on a smooth conic and the line containing the last $2$ contains exactly one of the other $6$;
4. It is not the case that $6$ of the points lie on two $\F_q$-lines, with $3$ points on each and not containing the intersection point, and the line containing the last $2$ contains exactly one of the other $6$.
Then $I_9(q) = J_8(q)/9$.
This follows from Proposition \[Hart4.5\], Proposition \[prop\_infnear\], and the observation that if $P_1,\ldots, P_9$ are $9$ points such that there are two cubics containing them that do not share a common component, then applying Proposition \[Hart4.5\] to any subset of $8$ of the points gives this same set of $9$.
\[no4no7\] The number of collections of $8$ points in $\Proj^2(\F_q)$ with at least $7$ on a smooth conic is $$C_{\ge 7}(q) = (q^5-q^2)\left(\binom{q+1}{8} + \binom{q+1}{7} q^2\right).$$ The number of collections of $8$ points in $\Proj^2(\F_q)$ with at least $4$ on a line is $$\begin{aligned}
L_{\ge 4}(q) & = & (q^2+q+1)\bigg(\binom{q+1}{8} + \binom{q+1}{7} q^2 + \binom{q+1}{6} \binom{q^2}{2} + \binom{q+1}{5} \binom{q^2}{3} \\
& + & \binom{q+1}{4} \binom{q^2}{4}\bigg) - (q^2+q+1)(q^2+q) \binom{q}{4} \binom{q}{3} - \binom{q^2+q+1}{2} \binom{q}{4}^2\\
& - & \binom{q^2+q+1}{2} \binom{q}{3}^2(q^2-q).\end{aligned}$$ The number of collections of $8$ points in $\Proj^2(\F_q)$ with no $4$ points collinear and no $7$ points on a conic is: $$\binom{q^2+q+1}{8} - C_{\ge 7}(q) - L_{\ge 4}(q).$$
The number of subsets of $8$ points in $\Proj^2(\F_q)$ is $\binom{q^2+q+1}{8}$. There are $q^5-q^2$ smooth plane conics, so the computation of $C_{\ge 7}(q)$ is clear. There are $q^2+q+1\ \F_q$-rational lines in $\Proj^2(\F_q)$. The number of subsets of $8$ points in $\Proj^2(\F_q)$ with at least $5$ points on a line is therefore, $$(q^2+q+1)\left(\binom{q+1}{8} + \binom{q+1}{7} q^2 + \binom{q+1}{6} \binom{q^2}{2} + \binom{q+1}{5} \binom{q^2}{3}\right).$$ There are $(q^2+q+1) \binom{q+1}{4} \binom{q^2}{4}$ ways to choose a subset $S$ of $8$ points in the plane with exactly $4$ points on a chosen line $L$ and $4$ other points. However, it may be the case that this collection of $8$ points contains $5$ collinear points, or two subsets of $4$ collinear points. The first case occurs if and only if when the $4$ other points lie on a line $L'$ and $L \cap L'$ is one of the $4$ chosen points of $L$. The number of ways to choose $8$ such points is the number of choices of $L$ times the number of choices of $L'$ times the number of ways to choose $3$ additional points of $L$ and $4$ additional points of $L'$.
The set $S$ contains exactly two subsets of $4$ collinear points if and only if one of the following conditions holds:
1. $S$ consists of $4$ points each on two lines $L, L'$, not containing the intersection point.
2. $S$ consists the intersection point of two lines $L, L'$ and $3$ additional points on each line, along with $1$ additional point not on $L \cup L'$.
There are $\binom{q^2+q+1}{2} \binom{q}{4}^2$ ways to choose a subset of the first type. Since two $\F_q$-rational lines contain $2q+1\ \F_q$-points, there are $q^2-q\ \F_q$-points not contained in two chosen lines. Therefore, there are $\binom{q^2+q+1}{2} \binom{q}{3}^2 (q^2-q)$ subsets of the second type.
It is clear that if a collection of $8$ points has at least $4$ on a line, it cannot have $7$ on a smooth conic.
\[absirred\] The number of collections of $8$ points in $\Proj^2(\F_q)$ that lie on an absolutely irreducible singular cubic with one of the $8$ points as the singular point is $$\begin{aligned}
& & (q^2+q+1)(q^3-q)q^2\binom{q}{7}+(q^2+q+1)(q^3-q)\frac{q^3-q^2}{2}\binom{q-1}{7}\\
& & +(q^2+q+1)(q^3-q)\frac{q^3-q^2}{2}\binom{q+1}{7}.\end{aligned}$$
There are $3$ kinds of absolutely irreducible singular plane cubics:
1. Cuspidal cubics, which have $q+1\ \F_q$-points;
2. Split nodal cubics, which have $q\ \F_q$-points;
3. Non-split nodal cubics, which have $q+2\ \F_q$-points.
For a count for each kind of cubic, see [@KaplanAGCT Lemma 1]. We divide each term given there by $q-1$ to account for the fact that we count plane curves rather than cubic polynomials.
\[6conic2line\] The number of collections of $6$ points on a smooth conic together with $2$ points not on that conic such that the line through those $2$ contains exactly one of the other $6$ points is $$(q^5-q^2)\binom{q+1}{6}6(q-5)\binom{q-1}{2}+(q^5-q^2)\binom{q+1}{6}6 \binom{q}{2}.$$
Suppose $P_1,\ldots, P_6$ are points of a smooth conic. There are $6(q-4)\ \F_q$-rational lines through exactly $1$ of these $6$ points. There are $6$ lines tangent to the conic and $6(q-5)$ lines not tangent to the conic. A tangent line contains $q$ points not on the conic and a non-tangent line contains $q-1$ points not on the conic.
\[332lemma\] The number of collections two collinear triples $P_1,P_2, P_3$ and $P_4,P_5,P_6$, where the intersection point is not included among these $6$ points, and two additional points $P_7,P_8$ such that the line containing them passes through exactly $1$ of the first $6$ points is $$\binom{q^2+q+1}{2}\binom{q}{3}^2 3(q-3)\binom{q-1}{2}.$$
The number of choices for two collinear triples $P_1,P_2, P_3$ and $P_4,P_5,P_6$, where the intersection point is not included among these $6$ points is $$\binom{q^2+q+1}{2}\binom{q}{3}^2.$$ Suppose $P_1, P_2, P_3$ lie on the line $L$ and $P_4,P_5,P_6$ lie on the line $L'$. There are $q+1$ lines through each of the $6$ points. There is $1$ line that contains two additional points from these $6$ and $3$ lines that contain $1$ additional point from these $6$. So, there are $q-3$ lines that do not contain any additional points from this set of $6$. For each such line, there are $q-1$ points that are not contained in $L \cup L'$.
We divide by $2$ to account for the fact that the choice of which line to label $L$ and which line to label $L'$ was arbitrary.
Combining Proposition \[J8prop\] with Lemmas \[no4no7\], \[absirred\], \[6conic2line\], and \[332lemma\] completes the proof of Theorem \[I9count\]. As a consequence, we see that Theorem \[C23LowWeightDual\] does hold for any $q>2$, even when the characteristic of $\F_q$ is $2$ or $3$.
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---
abstract: 'An example of a cocomplete abelian category that is not complete is constructed.'
address: 'School of Mathematics, University of Bristol, Bristol BS8 1TW, UK'
author:
- Jeremy Rickard
bibliography:
- 'mybib.bib'
title: A cocomplete but not complete abelian category
---
Introduction {#sec:intro}
============
In this paper we answer a question that was posted on the internet site MathOverflow by Simone Virili [@MO]. The question asked whether there is an abelian category that is cocomplete but not complete. It seemed that there must be a standard example, or an easy answer, but despite receiving a fair amount of knowledgeable interest, after many months the question still had no solution.
Recall that a complete category is one with all small limits, and that for an abelian category, which by definition has kernels, completeness is equivalent to the existence of all small products. Similarly, because of the existence of cokernels, cocompleteness for abelian categories is equivalent to the existence of all small coproducts.
A common, but wrong, first reaction to the question is that there are easy natural examples, with something such as the category of torsion abelian groups coming to mind: an infinite direct sum of torsion groups is torsion, but an infinite direct product of torsion groups may not be. Nevertheless, this category does have products: the product of a set of groups is simply the torsion subgroup of their direct product. In categorical terms, the category of torsion abelian groups is a coreflective subcategory of the category of all abelian groups (which certainly has products), with the functor sending a group to its torsion subgroup being right adjoint to the inclusion functor.
Similar, but more sophisticated, considerations doom many approaches to finding an example.
Recall that an $\mathtt{(AB5)}$ category, in the terminology of Grothendieck [@grothendieck], is a cocomplete abelian category in which filtered colimits are exact, and that a Grothendieck category is an $\mathtt{(AB5)}$ abelian category with a generator. The favourite cocomplete abelian category of the typical person tends to be a Grothendieck category, such as a module category or a category of sheaves, but it is well known that every Grothendieck category is complete (for a proof see, for example, [@kashiwara_schapira Prop. 8.3.27]).
More generally, any locally presentable category is complete [@adamek_rosicki Cor. 1.28], and most standard constructions of categories preserve local presentability.
The construction {#sec:main}
================
First we shall fix a chain of fields $\{k_\alpha\mid\alpha\in\mathbf{On}\}$ indexed by the ordinals such that $k_\beta/k_\alpha$ is an infinite degree field extension whenever $\alpha<\beta$. For example, we could take a class $\{x_\alpha\mid\alpha\in\mathbf{On}\}$ of variables indexed by $\mathbf{On}$ and let $k_\alpha=\mathbb{Q}(X_{<\alpha})$ be the field of rational functions in the set of variables $X_{<\alpha}=\{x_\gamma\mid\gamma<\alpha\}$.
We generally adopt the convention that categories are locally small (i.e., the class of morphisms between two objects is always a set). However, we’ll start by defining a “category” $\mathbf{C}$ which is not locally small.
An object $V$ of $\mathbf{C}$ consists of a $k_\alpha$-vector space $V_\alpha$ for each ordinal $\alpha$, together with a $k_\alpha$-linear map $v_{\alpha,\beta}:V_\alpha\to V_\beta$ for each pair $\alpha<\beta$ of ordinals, such that $v_{\alpha,\gamma}=v_{\beta,\gamma}v_{\alpha,\beta}$ whenever $\alpha<\beta<\gamma$. (When we denote an object by an upper case letter such as $V$, we will always use, without further comment, the corresponding lower case letter for the linear maps $v_{\alpha,\beta}$.)
A morphism $\varphi:V\to W$ of $\mathbf{C}$ consists of a $k_\alpha$-linear map $\varphi_\alpha:V_\alpha\to W_\alpha$ for each ordinal $\alpha$, such that $\varphi_\beta v_{\alpha,\beta}=w_{\alpha,\beta}\varphi_\alpha$ whenever $\alpha<\beta$.
Composition is defined in the obvious way. If $\vartheta:U\to V$ and $\varphi:V\to W$ are morphisms, then $(\varphi\vartheta)_\alpha=\varphi_\alpha\vartheta_\alpha$.
It is straightforward to check that $\mathbf{C}$ is an additive (in fact, $k_0$-linear) category, but not locally small: for example, if $V$ is the object with $V_\alpha=k_\alpha$ for every $\alpha$ and $v_{\alpha,\beta}=0$ for all $\alpha<\beta$, then a morphism $\varphi:V\to V$ has $v_\alpha:k_\alpha\to k_\alpha$ multiplication by some scalar $\lambda_\alpha\in k_\alpha$ for some arbitrary choice of $\left\{\lambda_\alpha\mid\alpha\in\mathbf{On}\right\}$, so the class of endomorphisms of $V$ is a proper class.
$\mathbf{C}$ is a (not locally small) abelian category with (small) products and coproducts in which (small) filtered colimits are exact.
It is straightforward to check that the obvious “pointwise” constructions give kernels, cokernels, products and coproducts. For example, if $\varphi:V\to W$ is a morphism, then the kernel of $\varphi$ is the object $U$ with $U_\alpha$ the kernel of $\varphi_\alpha:V_\alpha\to W_\alpha$ and $u_{\alpha,\beta}:U_\alpha\to U_\beta$ the natural map between the kernels of $\varphi_\alpha$ and $\varphi_\beta$ induced by $v_{\alpha,\beta}$.
Since all these constructions are “pointwise”, it is also straightforward to check that every monomorphism is the kernel of its cokernel and every epimorphism is the cokernel of its kernel (so that $\mathbf{C}$ is abelian), and that small filtered colimits are exact, as the verification reduces to the corresponding facts for the category of $k_\alpha$-vector spaces.
The (locally small) category that we’re really interested in is a full subcategory $\mathbf{G}$ of $\mathbf{C}$.
An object $V$ of $\mathbf{C}$ is [**$\alpha$-grounded**]{} if, for every $\beta>\alpha$, $V_\beta$ is generated as a $k_\beta$-vector space by the image of $v_{\alpha,\beta}$. The full subcategory of $\mathbf{C}$ consisting of the $\alpha$-grounded objects is denoted by $\alpha$-$\mathbf{G}$.
An object $V$ of $\mathbf{C}$ is [**grounded**]{} if it is $\alpha$-grounded for some ordinal $\alpha$. The full subcategory of $\mathbf{C}$ consisting of the grounded objects is denoted by $\mathbf{G}$.
$\mathbf{G}$ is a (locally small) $\mathtt{(AB5)}$ abelian category that is not complete.
Let $V$ be an $\alpha$-grounded object. If $\varphi: V\to W$ is a morphism then, for $\beta>\alpha$, $\varphi_\beta$ is determined by $\varphi_\alpha$, and so the class of morphisms $\varphi: V\to W$ is a set. Hence $\mathbf{G}$ is locally small.
Let $\varphi:V\to W$ be a morphism between $\alpha$-grounded objects. Clearly the kernel and cokernel of $\varphi$ are also $\alpha$-grounded. Hence $\mathbf{G}$ is closed under kernels and cokernels in $\mathbf{C}$ and so it is an exact abelian subcategory of $\mathbf{C}$.
If $\left\{V^i\mid i\in I\right\}$ is a set of grounded objects, then there is some ordinal $\alpha$ so that every $V^i$ is $\alpha$-grounded. Then $\bigoplus_{i\in I}V^i$ is also $\alpha$-grounded. Thus $\mathbf{G}$ is cocomplete and the inclusion functor $\mathbf{G}\hookrightarrow\mathbf{C}$ preserves coproducts, and hence all colimits. Exactness of filtered colimits in $\mathbf{G}$ therefore follows from the same property for $\mathbf{C}$.
Thus $\mathbf{G}$ is a locally small $\mathtt{(AB5)}$ abelian category.
For an ordinal $\alpha$, let $M^\alpha$ be the object with $$M^\alpha_\beta=
\begin{cases}
0 &\mbox{if }\beta<\alpha\\
k_\beta&\mbox{if }\beta\geq\alpha,
\end{cases}$$ and $m^\alpha_{\beta,\gamma}:k_\beta\to k_\gamma$ the inclusion map for $\alpha\leq\beta<\gamma$. Then $M^\alpha$ is $\alpha$-grounded.
If $W$ is any object of $\mathbf{C}$ then a morphism $\varphi:M^\alpha\to W$ is determined by $\varphi_\alpha: M^\alpha_\alpha=k_\alpha\to W_\alpha$, so $\mathbf{C}(M^\alpha,W)\cong W_\alpha$ and $M^\alpha$ represents the functor $W\mapsto W_\alpha$ from $\mathbf{C}$ to $k_\alpha$-vector spaces. Also, if $\alpha<\beta$ then the obvious morphism $\varphi:M^\beta\to M^\alpha$, with $\varphi_\gamma$ the identity map $k_\gamma\to k_\gamma$ for $\gamma\geq\beta$, induces a commutative square $$\xymatrix{
\mathbf{C}(M^\alpha,W)\ar[r]^-*[@]{\sim}\ar[d]_{\mathbf{C}(\varphi,W)}
&W_\alpha\ar[d]^{w_{\alpha,\beta}}\\
\mathbf{C}(M^\beta,W)\ar[r]^-*[@]{\sim}&W_\beta\\
}$$
We shall show that in $\mathbf{G}$ there is no product of a countably infinite collection of copies of $M^\alpha$, so $\mathbf{G}$ is not complete.
Suppose that $P$ is the product (in $\mathbf{G}$) of countably many copies of $M^\alpha$. Then for any ordinal $\beta\geq\alpha$, $$\mathbf{G}(M^\beta,P)\cong\mathbf{G}(M^\beta,M^\alpha)^\mathbb{N}
\cong (M^\alpha_\beta)^\mathbb{N}\cong k_\beta^\mathbb{N}$$ as $k_\beta$-vector spaces.
But if $\alpha<\beta<\gamma$ then $k_\gamma^\mathbb{N}$ is not generated as a $k_\gamma$-vector space by $k_\beta^\mathbb{N}$, since $k_\gamma/k_\beta$ is an infinite field extension. So $P$ cannot be $\beta$-grounded for any $\beta$.
We have already noted that products do exist in $\mathbf{C}$, with the obvious pointwise construction. However, the product of $\alpha$-grounded objects need not be $\beta$-grounded for any $\beta$.
Products also exist in $\alpha$-$\mathbf{G}$, since although the pointwise product may not be $\alpha$-grounded, $\alpha$-$\mathbf{G}$ is a coreflective subcategory of $\mathbf{C}$. We can $\alpha$-ground an object $V$ of $\mathbf{C}$ by replacing $V_\beta$ by $v_{\alpha,\beta}(V_\alpha)$ for $\beta>\alpha$, and the product of a set of objects in $\alpha$-$\mathbf{G}$ is obtained by $\alpha$-grounding the product in $\mathbf{C}$.
Each category $\alpha$-$\mathbf{G}$ is a cocomplete and complete abelian category (in fact, a Grothendieck category, with $\bigoplus_{\beta\leq\alpha}M^\beta$ a generator). The inclusion functors $\alpha$-$\mathbf{G}\to\beta$-$\mathbf{G}$ are exact and preserve coproducts, but do not preserve products, which explains why their union $\mathbf{G}$ has coproducts but does not have products (or at least not in an obvious way). Thanks to Zhen Lin Low for making this observation in a comment on MathOverflow [@MO].
Of course, an example of a complete abelian category that is not cocomplete can be constructed by taking the opposite category of $\mathbf{G}$.
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank MathOverflow and its community for introducing me to this and many other interesting questions, and for their comments on this question in particular.
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abstract: |
Microarray experiments generate a high data volume. However, often due to financial or experimental considerations, e.g. lack of sample, there is little or no replication of the experiments or hybridizations. These factors combined with the intrinsic variability associated with the measurement of gene expression can result in an unsatisfactory detection rate of differential gene expression (DGE). Our motivation was to provide an easy to use measure of the success rate of DGE detection that could find routine use in the design of microarray experiments or in post-experiment assessment.
In this study, we address the problem of both random errors and systematic biases in microarray experimentation. We propose a mathematical model for the measured data in microarray experiments and on the basis of this model present a t-based statistical procedure to determine DGE. We have derived a formula to determine the success rate of DGE detection that takes into account the number of microarrays, the number of genes, the magnitude of DGE, and the variance from biological and technical sources. The formula and look-up tables based on the formula, can be used to assist in the design of microarray experiments. We also propose an ad hoc method for estimating the fraction of non-differentially expressed genes within a set of genes being tested. This will help to increase the power of DGE detection.
The functions to calculate the success rate of DGE detection have been implemented as a Java application, which is accessible at
http://www.le.ac.uk/mrctox/microarray\_lab/Microarray\_Softwares/Microarray\_Softwares.htm. Supplementary information at ftp://alcyone.mrc.le.ac.uk/ Pub/twg1/BioInf03-0661suppl.pdf
author:
- |
Shu-Dong Zhang and Timothy W. Gant\
MRC Toxicology Unit, Hodgkin Building, Lancaster Road, University of Leicester, Leicester, UK
bibliography:
- 'bioinfo.bib'
title: A statistical framework for the design of microarray experiments and effective detection of differential gene expression
---
Introduction {#sec-introduction}
============
Whole genome sequencing and the related development of microarrays have given researchers unprecedented power to simultaneously determine the expressions of many thousands of genes . However, a statistical challenge facing microarray analysis is to identify differential gene expression (DGE) with a high rate of success and low rate of false positives. Such a method is required because of the number of gene expressions being simultaneously determined, and the variation associated with each can give rise to an unacceptably large number of false positives or low successful detection rate. The variations associated with gene expression experiments can be categorized into two sets. First, there are inter-individual differences between members of a population, thus sufficient biological individuals should be included in the experiments in order to account for the biological variation. Second, there are always technical errors arising from the experimental procedure, which may be further sub-categorized into random errors and systematic biases. Unlike random errors, which can be reduced by making multiple measurements, systematic biases cannot be reduced by simply doing more measurements, correct experimental designs must be employed to negate them.
One of the most serious sources of systematic bias in microarray experiments (for dual label hybridizations) is the imbalance in the measured fluorescence intensities between the two fluorescent channels [@Dudoit-etal2002; @Yang-etal2002; @Dobbin-etal2003]. A manifestation of this systematic bias is that when two identical mRNA samples are labelled with different fluorescent dyes and hybridized to the same microarray slide, one channel has a higher average fluorescence level than the other. To complicate matters further the imbalance of the two channels is not uniform, but varies from feature to feature. A feature is the area of fluorescence on a microarray corresponding to one gene and where hybridization of the labelled nucleic acids derived from this gene has taken place [@Cheung-etal1999]. To correct the labelling dye imbalance, different methods of normalizing the microarray data by adjusting the measured fluorescence levels have been proposed . These methods can be roughly classified into two categories. First, global normalization, in which the fluorescence levels of all the features are globally (uniformly) adjusted (by shifting or re-scaling) to fulfill some assumptions about the relative expressions of the genes, e.g. most genes are not differentially expressed between the two samples . However, because global normalization adjusts the fluorescence levels of all features uniformly, it cannot account for the different magnitudes of imbalances from feature to feature, so a second type of normalization method is often employed to take account of this variation. This normalization method adjusts the fluorescence level according to some local properties of the feature spot, e.g. the overall brightness of the spot [@Yang-etal2002], and usually involves fitting the measured data with a non-linear smoothed curve. The fluorescence level is then adjusted according the smoothed curve, which is assumed to describe the dependence of the imbalance on spot fluorescence intensity. But the fluorescence imbalances between the two channels are more complicated than can be described by a smoothed curve. Due to irregular intrinsic fluorescence of the microarray slide and possibly some gene-specific effect [@Tseng-etal-2001; @Zhou-etal-2002], it is unlikely that the fluorescence imbalance can be corrected for all features by the intensity-dependent normalization. To correct the fluorescence imbalance of each feature a simple method is to reverse the labelling dyes when hybridizing some microarrays. Kerr [*et al*]{} [@Kerr-etal-2000] first proposed an ANOVA model for microarray data, and showed that ANOVA methods can be used to normalize the data and estimate real changes in gene expression. Taking biological variations into account, Dobbin [*et al*]{} [@Dobbin-etal2003] have addressed the problem of statistical design of reverse dye microarrays to minimize variance with a given number of microarray slides. We have taken the analysis further to address the problem of identifying DGEs with a desired detection power and controlled number of false positives. A model and statistical testing procedure are presented in the following sections to assist research workers in the selection of an appropriate number of microarrays for an experiment in order to achieve the desired detection power, or alternatively in assessing the detection power achievable when the experiment has been done.
The model
=========
The experimental situation analyzed here is one where there are two sample groups. One of the groups might have been subjected to an event such as chemical exposure, the other being a suitable control, or the two groups might be normal and tumor tissues or different organs. For convenience the two groups will be designated as the [*treated*]{} and [*control*]{} groups.
In cells, the amount of mRNA corresponding to a particular gene is taken to correspond to the expression level of that gene. A microarray is a means to translate the level of mRNA for many genes, which cannot be measured directly, into fluorescence that can be measured directly. The model presented in this paper is designed for experiments where each gene is spotted only once on each microarray, and each individual sample is hybridized only once using one microarray. For the purpose of introduction consider one single feature spot on the microarray. We assume that the log-intensity fluorescence of this feature takes additive contributions from the following sources: the amount of corresponding mRNA in the biological sample, an effect from the quality of the feature spot, an effect from the labelling fluorochrome (including the efficiency of labelling with the fluorochrome, and possible pre-existing intrinsic fluorescence in favor of this fluorochrome), and the random measurement error. Therefore we have the following model $$G_{v,i,s,c}=I_{v,i}+A_s+D_c+\epsilon _{v,i,s,c}, \label{eq-G_visc}$$ where $G_{v,i,s,c}$ is the log-intensity (base 2 logarithms are utilized throughout this paper) of fluorescence of the feature spot;
$I_{v,i}$ is the expression level of the gene in the $i$th individual sample of group $v$ ($v=t$ for the treated group, or $v=c$ for the control group, and $i=1\cdots n$ where $n$ is the number of individuals in each group). $I_{v,i}$ is assumed to be independently and normally distributed with a mean $E_v$ and a variance $\sigma _v^2$, denoted by $I_{v,i}\sim
N(E_v,\sigma_v^2)$;
$A_s$ is the effect of feature spot quality, which is assumed to be fixed for microarray slide $s$ and independent of fluorescent label used;
$D_c$ is the effect of the fluorescent label $c$ ($c=g$ for green dye, and $c=r$ for red dye), which is assumed to be fixed with label $c$ and independent of microarray slide;
$\epsilon _{v,i,s,c}$ is the random error term which is assumed to be independently and normally distributed with a mean $0$ and a variance $\sigma _{\epsilon}^2$, denoted by $\epsilon _{v,i,s,c}
\sim N(0,\sigma _{\epsilon}^2)$.
Note that for each of the features on the microarray the log-intensity is described in the same form Eq.(\[eq-G\_visc\]). Although the equations are in the same form for each feature the actual values of $E_v$, $\sigma_v^2$, $A_s$, $D_c$, $\sigma
_{\epsilon}^2$ will be feature dependent.
$E_v$ is the mean expression level of the gene in the sample group $v$, so in comparing a gene’s expression between the treated and control groups, the quantity of interest is $E_t-E_c$, the magnitude of differential expression. The effects of feature spot quality $A_s$ and fluorescent dye $D_c$ are not of interest and therefore need to be eliminated by a suitable experimental design.
Experimental setup
==================
Let’s introduce the notation $(c_j,t_i)$ to represent a microarray as a result of the following hybridization: individual sample $c_j$ labelled with [*green*]{} dye and individual sample $t_i$ labelled with [*red*]{} dye. Here $c$ and $t$ indicate the sample group while the subscripts index different individuals in each group. As a convention, the first sample in the parenthesis is always labelled with green dye and the second with red dye.
Consider the microarray $(c_j,t_i)$. Here an individual $j$ is taken from the control group ($v=c$) and an individual $i$ from the treated group ($v=t$). RNA is extracted from both and converted to labelled cDNA using fluorescent labels [*green*]{} and [*red*]{} respectively. These are then simultaneously hybridized to the microarray $a$. This method of labelling (control sample with [*green*]{} and treated sample with [*red*]{}) is referred to as [*forward labelling*]{}. As a result of this experiment we can derive from Eq.(\[eq-G\_visc\]) $$\begin{aligned}
G_{c,j,a,g}=I_{c,j}+A_a+D_{g}+\epsilon _{c,j,a,g},\nonumber \\
%\label{eq-G_control_green}
G_{t,i,a,r}=I_{t,i}+A_a+D_{r}+\epsilon _{t,i,a,r}. \nonumber
%\label{eq-G_treat_red}\end{aligned}$$ The difference $F_a$ between the two fluorescence log-intensities is therefore $$\begin{aligned}
F_a=G_{t,i,a,r}-G_{c,j,a,g} =I_{t,i}-I_{c,j} \nonumber
\\+D_{r}-D_{g}
+\epsilon
_{t,i,a,r}-\epsilon_{c,j,a,g}, \label{eq-forward}\end{aligned}$$ and $F_a$ is normally distributed with an expected value (mean) $E_t-E_c+D_{r}-D_{g}$ and a variance $\sigma _t^2+\sigma
_c^2+2\sigma _{\epsilon} ^2$. Note that taking the difference of $G_{c,j,a,g}$ and $G_{t,i,a,r}$ causes the spot effect $A_a$ to be cancelled out and it does not therefore contribute to $F_a$. However, there is still the labelling fluor effect $D_{r}-D_{g}$ to consider. To eliminate this effect microarrays with reverse labelling are required.
Consider the microarray $(t_{i'},c_{j'})$, where another two individuals, $i'$ from the treated and $j'$ from the control groups are hybridized to another microarray $b$. On this occasion the individual from the control group $j'$ is labelled with $red$ and the individual from the treated group $i'$ with $green$. This method of labelling (control sample with $red$ and treated sample with $green$) is referred to as [*reverse labelling*]{}. From this microarray $b$ we get: $$\begin{aligned}
G_{t,i',b,g}=I_{t,i'}+A_b+D_{g}+\epsilon _{t,i',b,g},\nonumber\\
%\label{eq-G_treat_green}
G_{c,j',b,r}=I_{c,j'}+A_b+D_{r}+\epsilon _{c,j',b,r}.\nonumber
%\label{eq-G_control_red}\end{aligned}$$ The difference of the two log-intensities is $$\begin{aligned}
B_b=G_{t,i',b,g}-G_{c,j',b,r} =I_{ti'}-I_{cj'}\nonumber
\\+D_{g}-D_{r}
+\epsilon
_{t,i',b,g}-\epsilon_{c,j',b,r}, \label{eq-reverse}\end{aligned}$$ and $B_b$ is normally distributed with an expected value $E_t-E_c+D_{g}-D_{r}$ and a variance $\sigma _t^2+\sigma
_c^2+2\sigma _{\epsilon} ^2$.
The quantity $F_a$ (or $B_b$) is the difference of two log-intensities and is therefore equivalent to the logarithm of the ratio of two intensities. Thus $F_a$ (or $B_b$) is often called the [*log-ratio*]{} of a gene. The variance of the log-ratio of a gene, $\sigma _T^2=\sigma _c^2+\sigma _t^2+2\sigma
_{\epsilon} ^2$, is the sum of the biological variance of the control $\sigma _c^2$, the treated $\sigma _t^2$, and the measurement variances associated with them $2\sigma _{\epsilon}
^2$. Hereafter $\sigma_T^2$ is referred to as the [*total variance*]{} of the log-ratio of the gene.
From Eqs.(\[eq-forward\]) and (\[eq-reverse\]) it is clear that by combining measurements from both forward and reverse labelled microarrays, it is possible to eliminate the fluorescent label bias. One simple way of doing this is to take the average of Eqs.(\[eq-forward\]) and (\[eq-reverse\]). The expected value of this average is then $E_t-E_c$, which is the quantity of interest. The above arguments therefore show that to eliminate the spot effect $A_s$, we need to hybridize the control and treated samples onto the same microarray slide. To cancel out the fluorescent label effect $D_c$ we need to do both forward labelled and reverse labelled microarrays. A general formalism is presented in the following sections to deal with situations where the number of forward labelled microarrays and the number of reverse labelled microarrays are not necessarily the same.
We will consider the following experiment: $$\begin{aligned}
(c_1,t_1), (c_2,t_2), \cdots,
(c_{n_f-1},t_{n_f-1}),(c_{n_f},t_{n_f}) \nonumber \\
(t_{n_f+1},c_{n_f+1}), (t_{n_f+2},c_{n_f+2}), \cdots,
(t_{n_f+n_r},c_{n_f+n_r}). \nonumber\end{aligned}$$ In this experiment there are in total $n_f+n_r$ microarrays, $n_f$ of them are forward labelled, and the rest $n_r$ are reverse labelled. In relation to similar studies by other authors[@Jin-etal2001; @Kerr-etal2002; @Callow-etal2000] using replicated microarrays, this study focuses on a special case of microarray experiment designs, i.e., direct comparison between two groups with biological but no technical replicates in each group. It is a special case of the balanced block design as described by . They have showed that the balanced block design is the most efficient experimental setup when comparing two classes with a given number of microarrays. The limitation of this experimental setup, as Dobbin and Simon pointed out for the balanced block design, is that it is not suitable for clustering analysis.
Detecting DGE {#sec-method}
==============
Hypothesis test
---------------
For each gene printed on the microarrays, we want perform a statistical test to determine whether this gene is differentially expressed to a significant degree in the treated group compared to the control group. The null hypothesis is that the gene has the same expression level in the two groups: $$\begin{aligned}
\mbox{Null hypothesis } H_0: E_c=E_t \label{eq-null}\\
\mbox{Alternative hypothesis } H_1: E_c\neq
E_t\label{eq-alternative}\end{aligned}$$ From each of the $n_f$ forward labelled microarrays an intra-array log-ratio $F_i$ between the treated sample and the control sample is obtained, and similarly from each of the $n_r$ reverse labelled microarrays a log-ratio $B_j$. Each $F_i$ has an expected value $E_t-E_c+D_{r}-D_{g}$, so the average $\overline{F}=\sum
_{i=1}^{n_f}F_i/n_f$ has the same expected value. Similarly the average $\overline{B}=\sum _{j=1}^{n_r}B_j/n_r$ has an expected value $E_t-E_c-D_{r}+D_{g}$. Averaging $\overline{F}$ and $\overline{B}$ gives $$R=\frac{\overline{F}+\overline{B}}{2}={1\over 2n_f}\sum
_{i=1}^{n_f}F_i+{1\over 2n_r}\sum _{j=1}^{n_r}B_j, \label{eq-R}$$ which will have an expected value $E_t-E_c$, so $R$ is an unbiased estimator of our quantity of interest. Also $R$ is normally distributed with a variance $$\sigma_R^2
%=\frac{\sigma _c^2+\sigma _t^2+2\sigma _{\epsilon}^2}{4}\left(\frac{1}{n_f}+\frac{1}{n_r}\right)
=\frac{\sigma_T^2}{4}\left(\frac{1}{n_f}+\frac{1}{n_r}\right).
\label{eq-sigma_R^2}$$ When the total number of microarrays $n_f+n_r$ is fixed, the variance of $R$ is minimized at $n_f=n_r$, so whenever possible, equal numbers of forward and reverse labelled microarrays should be combined. The variances $\sigma _c^2$, $\sigma _t^2$, and $\sigma_{\epsilon}^2$ are unknowns, but fortunately there is no need to estimate them individually. For the purposes of identifying differential gene expression, estimating $\sigma_T^2$ as a whole is sufficient and $\sigma_T^2$ can be estimated using its un-biased estimator $$s^2={1\over n_f+n_r-2}\left[\sum _{i=1}^{n_f}(F_i-\overline{F})^2+
\sum _{j=1}^{n_r}(B_j-\overline{B})^2\right] \label{eq-s^2}$$ and $(n_f+n_r-2)s^2/\sigma_T^2$ will follow the $\chi^2$ distribution with $n_f+n_r-2$ degrees of freedom, independent of $\overline{F}$ and $\overline{B}$[@Brownlee1965], thus $s^2$ is independent of $R$. Note that in order to estimate $E_t-E_c$ and $\sigma_T^2$ properly it is necessary that $n_f \ge 1$, $n_r\ge 1$, and $n_f+n_r> 2$. In other words there must be at least one forward and one reverse labelled microarray, and at least three microarrays in total. It is then apparent that $$t={R-(E_t-E_c)\over
s\sqrt{\frac{1}{4}\left(\frac{1}{n_f}+\frac{1}{n_r}\right)}}
\label{eq-t}$$ is distributed as the Student’s $t$ distribution with $n_f+n_r-2$ degrees of freedom. In testing the null hypothesis Eq.(\[eq-null\]), we insert $E_t=E_c$ into Eq.(\[eq-t\]) and thus our test statistic $t_0$ is defined as, $$t_0={R\over
s\sqrt{\frac{1}{4}\left(\frac{1}{n_f}+\frac{1}{n_r}\right)}}
\label{eq-t_0}$$ Note that there is now no unknown quantity in Eq.(\[eq-t\_0\]). Under the null hypothesis that $E_t=E_c$, $t_0$ follows the Student’s distribution with $n_f+n_r-2$ degrees of freedom. Based on the value of $t_0$ the p-value of the test can be calculated. If the p-value calculated is larger than some pre-set threshold $P_{th}$, the null hypothesis is accepted that the gene has the same level of expression in both the control and treated groups. If the calculated p-value is smaller than the threshold $P_{th}$, it is declared that the test for this gene is [*positive*]{}, in the sense that its expression level in the treated group is different from that in the control group. Then depending on the sign of $t_0$ the gene is either designated as up ($t_0>0$) or down regulated ($t_0<0$).
Setting the threshold p-value {#sec-setting_threshold}
-----------------------------
A t test is performed for each gene, which is then declared as differentially expressed, or not, according to the above criteria. By adjusting the value of threshold $P_{th}$ a control can be exerted on the number of false DGE calls made. By definition, p-value is the probability of observing a value of the statistic as extreme or more extreme than the observed value, under the condition that the null hypothesis is true. For each gene whose null hypothesis is true (we call each such gene a [*null gene*]{}), its p-value is uniformly distributed in $(0,1)$. Therefore the probability that a null gene’s p-value is smaller than $P_{th}$ is just $P_{th}$. Suppose that in a total number $N$ genes, $N_0$ are null genes. When every gene on the microarray is tested, the number of false DGE calls $O_{fp}$ will has an expected value $N_0P_{th}$. So if one decides to tolerate an expected number $N_{fp}$ false DGEs the threshold p-value should be set at $P_{th}=N_{fp}/N_0$. However, in reality only $N$ is known and not $N_0$ and therefore, it is necessary to make an estimation of $N_0$ or $N_0/N$. Some methods for estimating $N_0/N$ are discussed in Sec. \[sec-estimatingN0/N\].
Once the threshold value $P_{th}$ is set, the ability to detect genuine DGE, i.e. a gene with $E_t\neq E_c$, depends on the following factors: the magnitude of differential expression $E_t-E_c$, the total variance in one microarray experiment $\sigma
_T^2$, and the number of forward and reverse labelled microarrays. Among these factors, the ones over which experimental control is exercised are $n_f$ and $n_r$. In general the larger $n_f$ and $n_r$, the more powerful will be the statistical testing. The key question is therefore, how many forward and reverse labelled microarrays are required in order to achieve a desired power of DGE detection with control on the number of false DGE calls? Based on the standard normal Z test, several authors have presented results on calculating the number of microarrays needed to achieve given statistical power while controlling false positive rate [@Wernisch2002; @Dobbin-etal2003]. These results would be applicable if we knew $\sigma _T^2$ for each gene. In reality though the variances cannot be assumed known, and more often than not, the number of microarrays used to estimate the variances is rather small. It is therefore necessary to use $t$-based test rather than the standard normal test. Other authors have also presented approximate formulas [@McShane-etal2003; @Simon-etal2004] for calculating the power of the traditional two-sample $t$ test with equal variance. In this paper we present an exact formula for calculating the power of the t-based statistical test developed here.
Determination of the threshold t-value
--------------------------------------
When the numbers of forward and reverse labelled microarrays are given, setting $P_{th}$ is equivalent to setting a threshold, say $|\xi|$, for the statistics $t_0$ defined in Eq.(\[eq-t\_0\]). With this threshold t-value, our criteria for claiming a DGE is as follows: If $t_0>|\xi|$, the gene is claimed as up-regulated ($E_t-E_c>0$); if $t_0<-|\xi|$, it is claimed as down-regulated ($E_t-E_c<0$). So the rate at which false positive claims are made is $$\begin{aligned}
P_{th}=\int _{-\infty}^{-|\xi|}\rho _{n_f+n_r-2}(t_0)dt_0
+\int_{|\xi|}^{\infty}\rho _{n_f+n_r-2}(t_0)dt_0 \nonumber \\
=2\int _{-\infty}^{-|\xi|}\rho _{n_f+n_r-2}(t_0)dt_0
=2T_{n_f+n_r-2}(-|\xi|), \label{eq-P_th}\end{aligned}$$ where $\rho_r(x)$ is the probability density function (PDF) of the Student’s distribution with $r$ degrees of freedom, and $T_r(.)$ is the cumulative probability distribution function (CDF) for the Student’s t distribution. It is therefore apparent that the threshold t-value $|\xi|$ can be obtained by solving the equation $2T_{n_f+n_r-2}(-|\xi|)=P_{th}$ with a given false positive rate $P_{th}$.
Successful detection rate {#sec-S}
-------------------------
The successful detection rate is the rate at which DGE is correctly identified (either up-regulated or down-regulated). If a gene has $E_t-E_c=\mu>0$, the successful detection rate for this gene is the probability that $t_0>|\xi|$ is observed. On the other hand, if a gene has $E_t-E_c=\mu <0$, the successful detection rate equals the probability that $t_0<-|\xi|$ is observed. It can be shown (see supplementary information I) that in both cases, the rate at which the genes behavior is correctly identified, i.e. $\mu>0$ or $\mu<0$, can be described by the following equation $$\begin{aligned}
S\left(n_f,n_r,\frac{|\mu|}{\sigma _T},|\xi|\right)
=\int _0^{\infty}p_{n_f+n_r-2}(Y)\times \nonumber \\
\Phi\left[-|\xi|\sqrt{Y\over n_f+n_r-2}+2\left(\frac{|\mu|}{\sigma
_T}\right)\sqrt{n_fn_r\over n_f+n_r}\right]dY,
\label{eq-S}\end{aligned}$$ where $p_r(Y)$ is the PDF for the $\chi ^2$ distribution with $r$ degrees of freedom, and $\Phi(.)$ is the CDF for the standard normal distribution.
Therefore the successful detection rate $S$ is a function of $n_f$, $n_r$, $|\mu|/\sigma _T$, and $|\xi|$, where $|\xi|$ can be obtained by solving Eq.(\[eq-P\_th\]) at a given $P_{th}$. Eventually, $S$ is a function of $P_{th}$, $n_f$, $n_r$, and $|\mu|/\sigma _T$.
Usage of the $S$ function
-------------------------
We have implemented the calculation of the $S$ function as a Java application, which is accessible through the URL given in the abstract. Two look-up tables also are provided in the supplementary for some typical results of $S$ for quick reference. Experiment designers can use these to find the value of $S$ at given parameters $n_f$, $n_r$, $|\mu|/\sigma_T$, and $P_{th}$, thus get some general idea of what percentage of truly DGEs can be detected by their experimental design.
The applicability of the $S$ function can be seen from two perspectives. First, for the user who has not carried out any microarray experiments on their system before, the total variances ($\sigma_T^2$) will be completely unknown. In this situation the $S$ function can serve as a post-experiment assessment to inform the user of the detection rate in their experiment based on the observed values of $R$ and $s^2$ from the measurements. For example, $3$ forward and $3$ reverse labelled microarrays, with $5000$ genes printed on each microarray, were used in a experiment. The tolerance for false positives is set at $N_{fp}=2$, and for simplicity the threshold p-value is set as $P_{th}=2/5000$. If most genes have an $s^2$ around $1$, then the typical value of $\sigma_T^2$ for the set of genes is $1$. We can now ask: for genes with two-fold differential expression and typical variance, what percentage of them can be correctly detected by this experiment? Remembering that a two-fold differential expression corresponds to $\mu=E_t-E_c=1$ or $\mu=E_t-E_c=-1$, we have $|\mu|=1$ and $\sigma_T=1$. Using the $S$ calculator or the look-up tables (Supplementary Table I) we find that the successful detection rate for $n_f=3$, $n_r=3$, $P_{th}=2/5000$, and $|\mu|/\sigma_T=1$ is $9.08\times 10^{-3}$, which means that in this experiment only $0.908\%$ of genes with two-fold DGE and with typical variance $1$ can be detected, the remaining $99\%$ are missed. If the same question was asked about genes with four-fold DGE and one decides to tolerate $N_{fp}=8$ false positive claims and the threshold is set at $P_{th}=8/5000$, then the successful detection rate for $P_{th}=8/5000$, $n_f=3$, $n_r=3$, and $|\mu|/\sigma_T=2$ is $0.217$, which means $21.7\%$ of them are successfully detected. If the detection rate is unsatisfactory, then more forward and reverse microarray datasets need to be added.
Second, if there is some general knowledge of total variance from previous experiments or other sources, then a target for the detection rate can be set. In this case, the $S$ function will assist in the determination of how many forward and reverse microarrays are required in the experiment. For example, if from previous experience we know that the typical value of the total variance for the set of genes under consideration is $\sigma_T^2=0.25$, which gives $\sigma_T=0.5$; A microarray experiment is now designed to identify DGEs between the treated and the control with a tolerance of $8$ false positive claims out of $5000$ genes being tested with $P_{th}=8/5000$ for simplicity; The pre-set target is that after this experiment no less than $60\%$ of genes with two-fold DGE and with typical variance should be detected; How many forward and reverse labelled microarrays are needed? As before, two-fold DGE corresponds to $|\mu|=1$, so one has $|\mu|/\sigma_T=2$. Using the look-up tables (Supplementary Table II, in the $|\mu|/\sigma_T=2$ panel and $P_{th}=8/5000$ column), one finds that the row $n_f=n_r=4$ gives a detection rate $S=0.605$ which is closest to meet the target. Therefore $4$ forward and $4$ reverse labelled microarrays are required in this experiment.
Controlling false positives
===========================
Procedures
----------
In this section, we explore further on how to effectively control false positives in a multiple test situation. Generally speaking, all different multiple-testing methods eventually amount to effectively setting a threshold p-value, and then rejecting all the null hypothesis with p-value below this threshold. For example, the classical Bonferroni multiple-testing procedure controls family-wise error rate at $\alpha$ by setting the threshold $P_{th}=\alpha /N$, where $N$ is the total number of hypothesis tested. In this study, we aim to control the number false positives such that the expectation of $O_{fp}$ equals $N_{fp}$, our pre-set target. As discussed in Section \[sec-setting\_threshold\], to achieve this, we should set $P_{th}=N_{fp}/N_0$, which requires an estimation of $N_0$ or $N_0/N$, the fraction of null genes in the set.
We present three procedures here for setting $P_{th}$ to control false positives:
Procedure A: Suppose we have made an estimation of $N_0/N$ as $c$, then set $P_{th}=N_{fp}/(cN)$. The method for calculating $c$ will be discussed below.
Procedure B: Set $P_{th}=N_{fp}/N$. This can be seen as using $c=1$ as the crudest estimation of $N_0/N$.
Procedure C: Suppose genes are sorted by their ascending p-values, so that $p_1\le p_2 \le p_3 \le \cdots \le p_N$, where $p_i$ is the p-value for gene $i$. Set $P_{th}=p_{i^*}$, where $i^*$ is the largest index satisfying $p_i[N-i+\min (i,N_{fp})]\leq N_{fp}$. This can be seen as estimating $N_0/N$ by $c=[N-i^*+\min
(i^*,N_{fp})]/N$. The idea behind this is that if gene $i^*$ and all genes indexed below it are to be declared DGEs, these genes should not contribute to the fraction of null genes. Thus this represents some improvement over the crudest estimation $c=1$.
We have performed simulations to compare the performances of the three procedures. Procedure A allows us to achieve the highest rate of DGE detection among the three, and the observed false positives $O_{fp}$ matches our preset target $N_{fp}$ statistically. Procedure B does not estimate $N_0/N$ effectively, and it is the most conservative procedure. So Procedure A is recommended over C and B (See Supplementary for details on simulation procedures and data).
Benjamini and Hochberg proposed the FDR approach to control the false discovery rate (FDR) at $q$ by setting $P_{th}=i^*q/N$, where $i^*$ is the largest index satisfying $p_i\leq iq/N$. The false discovery rate was defined as the expectation of the ratio of false to total positives, i.e., $q\equiv E( O_{fp}/i^*)$. When the FDR procedure controls false discovery rate at $q$, the observed false discovery rate $O_{fp}/i^*$ should have value around $q$, i.e., $q\approx
O_{fp}/i^*$, which gives $P_{th}=i^*q/N\approx O_{fp}/N$. The expectation of the threshold p-value under the FDR procedure is therefore $E(P_{th})\approx E(O_{fp}/N)=N_{fp}/N$. It is thus clear that the FDR procedure of is on average equivalent to Procedure B in this section.
Estimating $N_0/N$ {#sec-estimatingN0/N}
------------------
Pounds and Morris recently proposed the use of a beta-uniform mixture (BUM) function to approximate the distribution of p-values from a set of genes tested, and estimate the fraction of null genes in the set. Here we propose another method to estimate $N_0/N$, which does not requires the BUM form of distribution of p-values. The aim was to achieve a more accurate estimation of the fraction of null genes. As in , we wanted to extract a uniform density from the observed distribution of p-values. To achieve this, the genes were first sorted by their ascending p-values, so that $p_1\le p_2 \le p_3 \le \cdots \le p_N$, where $p_i$ is the p-value for gene $i$. Then an empirical cumulative distribution of p-values can be easily obtained by plotting $i/N$ versus $p_i$. The idea was to find a straight line tangent to the cumulative distribution curve with minimum slope. Taking into account that the cumulative distribution curve is a non-decreasing function ending at the point $(1.0,1.0)$, the minimum slope was found as follows. Each point $(p_i,i/N)$ on the cumulative distribution plot was connected with the ending point $(1.0,1.0)$ with a straight line, and the slope of the line calculated as $c_i=(1.0-i/N)/(1.0-p_i)$. Then the minimum of $c_i$ at a given range of p-value, say $P_l\le p_i \le P_u$, was found $$c_{min}=\min_i ( c_i \mid P_l\le p_i \le P_u ). \label{eq-cmin}$$ $c_{min}$ can be used as our estimation of the fraction of null genes in the set.
We have carried out simulations to test the performance of Eq.(\[eq-cmin\]), and found that it tends to underestimate the true value of $N_0/N$. Instead, using median slope as the estimation of $N_0/N$ gives more accurate results than the minimum slope. We thus use the following equation to estimate the fraction of null genes $$c_{mid}=\mbox{median} ( c_i \mid P_l\le p_i \le P_u ).
\label{eq-cmedian}$$
In a recent paper , Storey and Tibshirani used a natural cubic spline to fit the data of $c_i$ as a function of $p_i$ for a given range of p-values, then took the value of the spline at $p=1$ as the estimation of $N_0/N$. We compared the Storey-Tibshirani method with Eq.(\[eq-cmedian\]), an advantage of the latter is that it is computationally much simpler than the Storey-Tibshirani method. As can be seen from Table \[tab-summary-N1h-N5k\], both our method and the Storey-Tibshirani method become more accurate as $N$ and/or $N_0/N$ increases, and in all the cases our method gives slightly better results, as indicated by the coefficient of variation.
As for the values of $P_l$ and $P_u$, a practical guidance for choosing them is to set $P_l$ a value between $0.4$ and $0.5$, and $P_u$ between $0.9$ and $0.95$. In fact, Eq.(\[eq-cmedian\]) gives quite robust results with respect to changing the values of $P_l$ and $P_u$ within the recommended range. For a set of simulation tests with true null fraction $0.8$, using $(P_l,P_u)=(0.4,0.95)$ gives $c_{mid}=0.800\pm 0.023$, while using $(P_l,P_u)=(0.5,0.9)$ gives $c_{mid}=0.800\pm 0.024$.
The method here to estimate $N_0/N$ does not depend on the specific form of statistical tests being used, as long as the p-values pertaining to the tests are obtained. But similar to the BUM method and the the Storey-Tibshirani method, the method we are proposing here also implicitly assumes that the multiple test statistics are independent, or at least the true null statistics are independent. In the context of microarray experiments, this would require that the null genes’ expressions are independent of each other. This may be not realistic, thus the estimation of the fraction of null genes based on these methods will be less accurate. An extreme example is when all the null genes in each biological sample behave in a concerted manner, and all the non-null genes express in a synchronized way, then the p-values we observe will be concentrated on two separate points, one for all the null genes and one for the non-null genes. Such a situation will defy all the methods for estimating $N_0/N$ discussed here. Estimating the fraction of null genes with possibly strong inter-gene dependence is an important issue, and probably a very difficult one, especially without specifying their structure of interdependence beforehand. This is beyond the scope of current study, and is an issue worth of future investigation and continuous efforts. Until further statistical advances are made in this respect, the method we proposed in this paper can serve as an approximation for estimating the fraction of null statistics.
Discussion
==========
The data volume generated by microarray studies combined with the intrinsic variability of the system demands that rigorous statistical analysis be applied to the data to avoid the problem of false positives and/or low successful rate in DGE detection. In this study we have taken into account all the major variables associated with microarray data. The procedure proposed in this paper deals with fluorescent label bias often present in microarray experiments. A t statistic has been derived for hypothesis testing based on a model that describes each gene individually with its own set of parameters. An advantage of this design is that if there exists any fluorescent biases ($D_{g}\neq
D_{r}$) for some genes they will be corrected by the reverse labelling procedure. For genes with no fluorescent bias (for example, some genes may have $D_{g}= D_{r}$) the method will perform equally satisfactorily.
In this work, we have adopted the normality assumption, which leads to the test statistic $t_0$ following the Students’ t distribution under the null hypothesis. Thus the successful detection rate $S$ can be calculated in closed form. While the normality assumption seems reasonable with common technologies, especially for the measurement error $\epsilon _{visc}$, large scale replicate experiments have not yet been performed to make a precise assessment . If normality is not met, $R$ defined in Eq.(\[eq-R\]) will continue to be an unbiased estimator of the quantity of interest but $t_0$ will not follow Students’ t distribution. In this case some non-parametric methods [@Efron-etal-2001; @Tusher-etal-pnas2001; @Pan-etal-2001; @Pan-bioinfo2003] could be employed. While those methods can be readily applied to microarrays with a common reference design, where the systematic dye bias subtracts out in the calculation of the test statistic, the application of those methods to the direct comparison design needs to be further developed and investigated. If non-parametric methods have to be used the rate of successful detection cannot be as readily calculated as in Eq.(\[eq-S\]).
In the published literature it is a common practice to apply some form of normalization (global or local) to remove systematic biases before the statistical analysis of microarray data. Here we are proposing to remove much of the systematic bias by experimental means, i.e. by a dye-swapping procedure. Since the model deals with the fluorescent bias for each gene individually, no other local normalization procedure (e.g. LOWESS [@Quackenbush2002]) should be applied before the statistical testing procedure given here. However, some form of global normalization is appropriate, such as that utilized by Pollack [*et al*]{} [@Pollack1999], or that described in , where the log-ratios in a microarray dataset are globally shifted so that the most probable value of log-ratio becomes $0$. The purpose of global normalization is to adjust the effect of global factors that could generally affect the fluorescence, such as a difference between the overall concentrations of two mRNAs, and possibly the difference of photo-amplifier voltages used between the two fluorescent channels when the microarray image was scanned. All the local feature-specific bias is looked after by the reverse labelling and statistical testing procedure proposed here.
Finally a word for the overworked bench researcher facing the prospect of multiple hybridizations in order to achieve a reasonably high level of $S$ without having to contend with an unsatisfactory false positive rate. What can be regarded as reasonable? This depends on the desired outcome of the experiment. If for example the interest is in defining genes which might give rise to differential susceptibility, then there will be a desire to have a high value of $S$ in order not to miss any potential candidate genes. There would be two ways of achieving this, either by increasing the number of hybridizations or by accepting a higher false positive rate. In an experiment such as the one described then the candidate genes will probably be verified by other methods downstream. Therefore the balance is driven by the need to achieve a high $S$ and the decision is between whether it is more economical to use more microarrays, or put more resource into downstream verification. Where no downstream verification of DGEs identified in a microarray experiment are proposed then it is essential to maintain a low value of false positive rate, at the expense of $S$ if the total number of microarrays is limiting. This study does not seek to put a figure on the number of microarrays that should be hybridized in an experiment. Rather a framework is provided for the experiment designer to decide on the number of microarrays to hybridize taking into account the system, availability of sample, downstream analysis primarily and the objective of the experiment.
Acknowledgement {#acknowledgement .unnumbered}
===============
We wish to acknowledge the support of the microarray team of the MRC Toxicology Unit particularly Reginald Davies, David J. Judah, JinLi Luo and Joan Riley. We thank Andy Smith and Michael Festing for critical readings of the manuscript and helpful discussions. We also thank anonymous referees for very helpful and constructive comments.
--------- ----------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
$N_0/N$ mean stdev cv mean stdev cv mean stdev cv mean stdev cv
0.2 $c_{mid}$ 0.186 0.040 0.200 0.205 0.017 0.085 0.197 0.013 0.067 0.201 0.009 0.044
0.2 $\pi_0$ 0.158 0.109 0.544 0.209 0.061 0.307 0.178 0.042 0.212 0.203 0.017 0.087
0.8 $c_{mid}$ 0.767 0.112 0.140 0.807 0.047 0.059 0.805 0.031 0.038 0.800 0.023 0.029
0.8 $\pi_0$ 0.724 0.284 0.355 0.785 0.097 0.121 0.792 0.064 0.080 0.807 0.060 0.075
--------- ----------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
|
---
abstract: 'In this paper we study a non strictly systems of conservation law by stochastic perturbation. We show the existence and uniqueness of the solution. We do not assume that $BV$-regularity for the initial conditions. The proofs are based on the concept of entropy solution and in the characteristics method (in the influence of noise). This is the first result on the regularization by noise in hyperbolic systems of conservation law.'
author:
- 'Christian Olivera[^1]'
title: ' Regularization by noise in (2x 2) hyperbolic systems of conservation law. '
---
*Key words and phrases. Stochastic partial differential equation, Continuity equation, Hyperbolic Systems, Entropy solution , Regularization by noise.*
[**MSC2010 subject classification:**]{} 60H15, 35R60, 35L02, 60H30, 35L40.
Introduction {#Intro}
============
A large number of problems in physics and engineering are modeled by systems of conservation laws
$$\label{trasports}
\partial_t u(t, x) + div ( f(u(t,x)) ) = 0 \, ,$$
here $u=u(t,x)$ is called the conserved quantity, while $F$ is the flux. Examples for hyperbolic systems of conservation laws include the shallow water equations of oceanography, the Euler equations of gas dynamics, the magnetohydrodynamics (MHD) equations of plasma physics, the equations of nonlinear elastodynamics and the Einstein equations of general relativity. When smooth initial data are considered, it is well known that the solution can develop shocks within finite time. Therefore, global solutions can only be constructed within a space of discontinuous functions. Moreover, when discontinuities are present, weak solutions may not be unique. A central issue is to regain uniqueness by imposing appropriate selection criteria. The well-posedness theorems within the class of entropy solutions, for the scalar case, were established by Kruzkov(see [@Krus]). It is well known that the main techniques of abstract functional analysis do not apply to hyperbolic systems. Solutions cannot be represented as fixed points of continuous transformations, or in variational form, as critical points of suitable functionals. For the above reasons, the theory of hyperbolic conservation laws has largely developed by ad hoc methods. We refer to [@Bress], [@Bian] [@Dafermos] and [@Serre]. The well-posedness general system of conservation laws has been established only for initial data with sufficiently small total variation, see [@Bress] and the references therein.
We consider the following systems of conservation law
$$\label{deter}
\left \{
\begin{aligned}
&\partial_t v(t, x) + Div \big( F(v) \big )= 0 \, \\
\\[5pt]
&\partial_t u(t, x) + Div (v u ) = 0 \, .
\end{aligned}
\right .$$
We point that in the $L^{1}\cap L^{\infty}$ setting this systems ill-posedness since the classical DiPerna-Lions-Ambrossio theory of uniqueness of distributional solutions for transport/ continuity equation does not apply when the drift has $L^{1}\cap L^{2}$ regularity, see [@ambrisio] and [@DL]. Also see [@ambrisio2] and [@lellis] for new developments in the theory. We point under strong assumption on the coefficients and initial conditions P. Le Floch in [@Lef2] solved this problem using Volpert multiplication of distributions. In contrast with its deterministic counterpart, the singular stochastic continuity/transport equation with multiplicative noise is well-posed. The addition of a stochastic noise is often used to account for numerical, empirical or physical uncertainties. In [@AttFl11; @Beck; @Fre1; @Fre2; @FGP2; @MNP14; @Moli], well-posedness and regularization by linear multiplicative noise for continuity/transport equations have been obtained. We refer to [@Moli] for more details on the literature.
In this paper we study the influence of the noise in the hyperbolic systems (\[deter\]). More precisely, we consider following stochastic systems of conservation law
$$\label{trasport}
\left \{
\begin{aligned}
&\partial_t v(t, x) + Div \big( ( F(v(t,x)) \big )= 0 \, ,
\\[5pt]
&\partial_t u(t, x) + Div \big( (v(t,x) + \frac{d B_{t}}{dt}) \cdot u(t, x) \big )= 0 \, ,
\\[5pt]
&v|_{t=0}= v_{0}, \ u_{t=0}=u_{0}\ \, .
\end{aligned}
\right .$$
Here, $(t,x) \in [0,T] \times {\mathbb R}$, $\omega \in \Omega$ is an element of the probability space $(\Omega, {\mathbb P}, {\ensuremath {\mathcal{F}}})$ and $B_{t}$ is a standard Brownian motion in $\mathbb{R}$. The stochastic integration is to be understood in the Stratonovich sense. The Stratonovich form is the natural one for several reasons , including physical intuition related to the Wong-Zakai principle.
The main issue of this paper is to prove existence and uniqueness of entropy-weak solutions for the stochastic systems of the conservation law (\[trasport\]). We do not assume $BV$- regularity for the initial conditions. We use the entropy formulation of conservation law and we employ the stochastic characteristics in order to obtain a unique solution to the one-dimensional stochastic equation with a bounded measurable drift coefficient. We adapted the ideas in [@Moli] and [@Ol] in our context where the drift term in the continuity equation depend on time and it is bounded and integrable.
Throughout of this paper, we fix a stochastic basis with a $d$-dimensional Brownian motion $\big( \Omega, \mathcal{F}, \{\mathcal{F}_t: t \in [0,T] \}, \mathbb{P}, (B_{t}) \big)$.
One Example
-----------
We consider the systems
$$\label{deter3}
\left \{
\begin{aligned}
&\partial_t v(t, x) + Div \big( \frac{1}{2} v^{2}(t,x) \big )= 0 \, ,
\\[5pt]
&\partial_t u(t, x) + Div \big( ( v(t,x) u(t, x) \big )= 0 \, ,
\\[5pt]
&v|_{t=0}= v_{0}, \ u_{t=0}=u_{0}\ \, ,
\end{aligned}
\right .$$
here $v$ is the velocity and $u$ the density of the particles. This system has applications in cosmology, the model describes the evolution of matter in the last stage of the expansion of the universe as cold dust moving under gravity alone and the laws are governed by the system (\[deter3\]). Clearly the eigenvalues are equal $\lambda_1 = \lambda_2 = v$. Thus the system (\[deter3\]) is not strictly hyperbolic. The first equation of (\[deter3\])-Burgers equation is known to develop singularities in finite time even if the initial data $v_{0}$ is smooth, and it is not at all obvious to solve the second equation. One question that remains is a well-posedness theory and large time behaviour of solution. In [@Jose] and [@Tan] the authors proved existence of weak solutions via $\delta-$ shock for Riemann initial condition. Another approach of nonconservative product can be found in the work of J.F. Colombeau [@Colom] .
Scalar case.
------------
We point that recently there has been an interest in studying the effect of stochastic forcing on nonlinear conservation laws driven by space-time white noise, see [@chen2; @Debu; @Feng; @Hof]. For other hand, in [@lions] and [@lions2] the authors introduced the theory of pathwise solutions to study the stochastic conservation law driven by continuous noise.
Possible extensions.
--------------------
We point that our approach can be apply to other class of non-coupled systems like
$$\left \{
\begin{aligned}
&\partial_t v(t, x) =P(v) \, \\
\\[5pt]
&\partial_t u(t, x) + Div (f(v) u ) = 0 \, .
\end{aligned}
\right .$$
where the $P$ is some differential operator. For instance for the systems of the Hamilton-Jacobi and the continuity equations, see [@ST]. The problem of coupled systems is much more complicated and we shall consider it in future investigations.
Hypothesis
----------
We assume the following conditions
\[hyp1\] The flux $F$ satisfies $$\label{cond3-1}
F\in C^{1}$$ and the initial condition holds $$\label{weight}
v_{0}\in L^{\infty}({\mathbb R})\cap L^{1}({\mathbb R}), u_0 \in L^{2}({\mathbb R})\cap L^{1}({\mathbb R}).$$
Existence
=========
Definition of solutions
-----------------------
Let $\eta\in C^{1}({\mathbb R})$ be a convex function. If there exist $q\in C^{1}({\mathbb R})$ such that for all $v$ $$\eta^{\prime}(v) F^{\prime}(v)= q^{\prime}(v)$$ then $\eta,q$ is called an entropy-entropy flux pair of the conservation law
$$\partial_t v(t, x) + Div \big( f(v) \big )=0, \ v(t,0)=v_{0}(x).$$
\[defisoluH\] The stochastic process $v \in L^{\infty}([0,T]\times {\mathbb R})\cap L^{\infty}([0,T],L^{1}({\mathbb R}))$ and $u\in L^\infty([0,T], L^{2}(\Omega \times {\mathbb R})) \cap L^1([0,T] \times \Omega \times {\mathbb R}) $ are called a entropy weak solution of the stochastic hyperbolic systems when:
- $v$ is entropy solution of the conservation law
$$\partial_t v(t, x) + Div \big( F(v) \big )=0, \ v(t,0)=v_{0}(x).$$
That is, if for every entropy flux pair $\eta, q$ we have
$$\partial_t \eta(v) + Div (q(v))\leq 0$$
in the sense of distribution.
- For any $\varphi \in C_0^{\infty}({\mathbb R})$, the real valued process $\int u(t,x)\varphi(x) dx$ has a continuous modification which is an $\mathcal{F}_{t}$-semimartingale, and for all $t \in [0,T]$, we have $\mathbb{P}$-almost surely $$\label{DISTINTSTR}
\begin{aligned}
\int_{{\mathbb R}} u(t,x) \varphi(x) dx = &\int_{{\mathbb R}} u_{0}(x) \varphi(x) \ dx
+ \int_{0}^{t} \!\! \int_{{\mathbb R}} u(s,x) \, v(t,x) \partial_x \varphi(x) \ dx ds
\\[5pt]
& + \int_{0}^{t} \!\! \int_{{\mathbb R}} u(s,x) \ \partial_x \varphi(x) \ dx \, {\circ}{dB_s} \, .
\end{aligned}$$
\[lemmaito\] Using the same idea as in Lemma 13 [@FGP2], one can write the problem (\[DISTINTSTR\]) in Itô form as follows, a stochastic process $u\in L^\infty([0,T], L^{2}(\Omega \times {\mathbb R}) ) \cap L^1([0,T] \times \Omega \times {\mathbb R}) $ is solution of the SPDE (\[DISTINTSTR\]) iff for every test function $\varphi \in C_{0}^{\infty}(\mathbb{R})$, the process $\int u(t, x)\varphi(x) dx$ has a continuous modification which is a $\mathcal{F}_{t}$-semimartingale and satisfies the following Itô’s formulation
$$\begin{aligned}
\int_{{\mathbb R}} u(t,x) \varphi(x) dx = &\int_{{\mathbb R}} u_{0}(x) \varphi(x) \ dx
+ \int_{0}^{t} \!\! \int_{{\mathbb R}} u(s,x) \, v(t,x) \partial_x \varphi(x) \ dx ds
\\[5pt]
& + \int_{0}^{t} \!\! \int_{{\mathbb R}} u(s,x) \ \partial_x \varphi(x) \ dx \, dB_s \, + \frac{1}{2} \int_{0}^{t} \!\! \int_{{\mathbb R}} u(s,x) \ \partial_x^{2} \varphi(x) \ dx \, ds.
\end{aligned}$$
Existence.
----------
We shall prove existence of solutions under hypothesis \[hyp1\].
Assume that hypothesis \[hyp1\] holds. Then there exists entropy-weak solution of the hyperbolic systems .
[*Step 1: Conservation law .*]{} According to the classical theory of conservation law, see for instance [@Dafermos], we have that there exists a unique entropy solution of the conservation law
$$\partial_t v(t, x) + Div \big( F(v) \big )=0, \ v(t,0)=v_{0}(x).$$
If the the initial condition $v_{0}\in L^{1}({\mathbb R})\cap L^{\infty}({\mathbb R})$ then the solution $v \in L^{\infty}([0,T]\times {\mathbb R})\cap L^{\infty}([0,T],L^{1}({\mathbb R}))$.
[*Step 2: Primitive of $v$.*]{} It easy to see that for any test function $\varphi \in C_{0}^{\infty}(\mathbb{R})$ we have
$$\begin{aligned}
\int_{{\mathbb R}} v(t,x) \varphi(x) dx = &\int_{{\mathbb R}} v_{0}(x) \varphi(x) \ dx
+ \int_{0}^{t} \!\! \int_{{\mathbb R}} F(v(s,x)) \partial_x \varphi(x) \ dx ds.
\end{aligned}$$
since any entropy solution is also a weak solution.
Let $\{\rho_\varepsilon\}_\varepsilon$ be a family of standard symmetric mollifiers. Then we obtain
$$\begin{aligned}
\int_{{\mathbb R}} v(t,y) \rho_\varepsilon(x-y) dy = &\int_{{\mathbb R}} v_{0}(y) \rho_\varepsilon(x-y) dy
+ \int_{0}^{t} \!\! \int_{{\mathbb R}} F(v(s,y)) \partial_y \rho_\varepsilon(x-y) dy ds.
\end{aligned}$$
Integrating we get
$$\begin{aligned}
\int_{0}^{z} v_\varepsilon(t,x) dx = &\int_{0}^{z} v_{0}^\varepsilon(x) dz
+ \int_{0}^{t} (F(v)\ast \rho_\varepsilon)(z) ds.
\end{aligned}$$
We denoted $\int_{0}^{z} v_\varepsilon(t,x) dx=\bar{v}_\varepsilon(t,x)$.
[*Step 3: Regularization.*]{} We define the family of regularized coefficients given by $$v^{\epsilon}(t,.) = (v(t,x) \ast_{x} \rho_\varepsilon)(t,.).$$
Clearly we observe that, for every $\varepsilon>0$, any element $v^{\varepsilon}$, $u_0^\varepsilon$ are smooth (in space) and with bounded derivatives of all orders. We observe that to study the stochastic continuity equation (SCE) (\[DISTINTSTR\]) is equivalent to study the stochastic transport equation given by (regularized version): $$\label{STE-reg}
\left \{
\begin{aligned}
&d u^\varepsilon (t, x) + \nabla u^\varepsilon (t, x) \cdot \big( v^\varepsilon (t,x) dt +
\circ d B_{t} \big) +{{\rm div}}b^{\varepsilon}(x) \,u^\varepsilon (t,x) dt = 0\, ,
\\[5pt]
&u^\varepsilon \big|_{t=0}= u_{0}^\varepsilon
\end{aligned}
\right .$$ Following the classical theory of H. Kunita [@Ku Theorem 6.1.9] we obtain that
$$u^{\varepsilon}(t,x) = u_{0}^{\varepsilon} (\psi_t^{\varepsilon}(t,x)) J\psi_t^{\varepsilon}(t,x),$$ is the unique solution to the regularized equation , where $\phi_t^{\varepsilon}$ is the flow associated to the following stochastic differential equation (SDE): $$d X_t = v^\varepsilon (t,X_t) \, dt + d B_t \, , \hspace{1cm} X_0 = x \,,$$ and $\psi_t^{\varepsilon}$ is the inverse of $\phi_t^{\varepsilon}$.
[*Step 4: Itô Formula .*]{} Applying the Itô formula to $\bar{v}_\varepsilon(t,X_t^{\epsilon})$ we deduce
$$\bar{v}_\varepsilon(t,X_t^{\epsilon})= \int_{0}^{X_t^{\epsilon}} u_{0}^{\epsilon}(x) dx
+ \int_{0}^{t} (F(v)\ast \rho_\varepsilon)(s,X_s^{\epsilon}) ds + \int_{0}^{t} v_\varepsilon^{2}(s,X_s^{\epsilon}) ds$$
$$+ \int_{0}^{t} v_\varepsilon(s,X_s^{\epsilon}) dB_{s}
+ \frac{1}{2} \int_{0}^{t} (\partial_{x}v_\varepsilon\big)(s,X_s^{\epsilon}) ds$$
[*Step 5: Boundeness.*]{} We observe that
$$\|\bar{v}_\varepsilon(t,X_t^{\epsilon})\|_{L^{\infty}(\Omega\times [0,T]\times {\mathbb R})}
\leq \| v\|_{L^{\infty}( [0,T],L^1({\mathbb R}))},$$
$$\|\int_{0}^{X_t^{\epsilon}} v_{0}^{\epsilon}(x) dx\|_{L^{\infty}(\Omega\times [0,T]\times {\mathbb R})}
\leq \| v_{0}\|_{L^{1}( {\mathbb R})},$$
$$\|\int_{0}^{t} (F(v)\ast \rho_\varepsilon)(s,X_s^{\epsilon}) ds\|_{L^{\infty}(\Omega\times [0,T]\times {\mathbb R})}
\leq C \| F(v)\|_{L^{\infty}},$$
$$\| \int_{0}^{t} v_\varepsilon^{2}(s,X_s^{\epsilon}) ds\|_{L^{\infty}(\Omega\times [0,T]\times {\mathbb R})}
\leq C \| v\|_{L^{2}([0,T], L^{\infty}({\mathbb R}))}^{2}.$$
[*Step 6 : Estimation on Jacobain.*]{}
We denote
$$\begin{aligned}
\mathcal{E}\bigg(\int_0^t v_{\epsilon}(s,X_s)dB_s\bigg)=
\exp\bigg\{\int_0^t v_{\epsilon}(s,X_s^{\epsilon})dB_s-\frac{1}{2} \int_0^{t} v_{\epsilon}^{2}(s,X_s^{\epsilon})ds \bigg\},\end{aligned}$$
We note that $\partial_x X_{t}$ satisfies
$$\partial_x X_{t}=\exp\bigg\{ \int_{0}^{t} (\partial_x v_{\epsilon})(s,X_{s}) \ ds \bigg\}.$$
From steps 4-5 we have
$${\mathbb E}| \partial_x X_{t}|^{-1}\leq C {\mathbb E}\mathcal{E}\bigg(\int_0^t v_{\epsilon}(s,X_s)dB_s\bigg).$$
We observe that the processes $ \mathcal{E}\bigg(\int_0^t v_{\epsilon}(s,X_s)dB_s\bigg)$, is martingale with expectation equal to one. The we conclude that
$${\mathbb E}| \partial_x X_{t}|^{-1}\leq C .$$
[*Step 7: Passing to the limit .*]{}
Making the change of variables $y=\psi_t^{\varepsilon}(x)$ we have that
$$\begin{aligned}
\int_{{\mathbb R}} {\mathbb E}[|u^{\varepsilon}(t,x)|^2]\, dx & = \int_{{\mathbb R}} |u_{0}^{\varepsilon} (y)|^2 {\mathbb E}| J\phi_t^{\varepsilon}|^{-1} dy.\end{aligned}$$
From step 6 we have
$$\label{eq0}
\int_{{\mathbb R}} {\mathbb E}[|u^{\varepsilon}(t,x)|^2]\, dx \leq C .$$
Therefore, the sequence $\{u^{\varepsilon}\}_{\varepsilon>0}$ is bounded in $L^\infty([0,T], L^{2}(\Omega \times {\mathbb R}) ) \cap L^1([0,T] \times \Omega \times {\mathbb R})$. Then there exists a convergent subsequence, which we denote also by $u^{\varepsilon}$, such that converge weakly in $L^\infty([0,T], L^{2}(\Omega \times {\mathbb R}) )$ to some process $u\in L^\infty([0,T], L^{2}(\Omega \times {\mathbb R})) \cap L^1([0,T] \times \Omega \times {\mathbb R}) $.
Now, if $u^{\varepsilon}$ is a solution of , it is also a weak solution, that is, for any test function $\varphi\in C_0^{\infty}({\mathbb R})$, $u^{\varepsilon}$ satisfies (written in the Itô form): $$\begin{aligned}
\int_{{\mathbb R}} u^{\varepsilon}(t,x) \varphi(x) dx = &\int_{{\mathbb R}} u^{\varepsilon}_{0}(x) \varphi(x) \ dx + \int_{0}^{t} \!\! \int_{{\mathbb R}} u^{\varepsilon}(s,x) \, v^{\varepsilon}(s,x) \partial_x \varphi(x) \ dx ds \\
& + \int_{0}^{t} \!\! \int_{{\mathbb R}} u^{\varepsilon}(s,x) \ \partial_x \varphi(x) \ dx \, dB_s \, + \frac{1}{2} \int_{0}^{t} \!\! \int_{{\mathbb R}} u^{\varepsilon}(s,x) \ \partial_x^{2} \varphi(x) \ dx \, ds\,.\end{aligned}$$ Thus, for prove existence of the SCE is enough to pass to the limit in the above equation along the convergent subsequence found. This is made through of the same arguments of [@FGP2 theorem 15].
Uniqueness.
===========
In this section, we shall present a uniqueness theorem for the SPDE (\[trasport\])
\[uni2\] Under the conditions of hypothesis \[hyp1\], uniqueness holds for entropy -weak solutions of the hyperbolic problem .
[*Step 1: Set of solutions.*]{} The uniqueness of the conservation law
$$\partial_t v(t, x) + Div \big( F(v) \big )=0, \ v(t,0)=v_{0}(x).$$
follows from the classical theory of entropy solutions.
[*Step 2:* ]{} We remark that the set of solutions of equation (\[DISTINTSTR\]) is a linear subspace of $ L^{\infty}([0,T]\times R, L^{2}(\Omega)) \cap L^1([0,T] \times \Omega \times {\mathbb R})$, because the stochastic continuity equation is linear, and the integrability conditions is a linear constraint. Therefore, it is enough to show that a $u$ with initial condition $u_0= 0$ vanishes identically.
[*Step 1: Primitive of the solution.*]{} We define $V(t,x)=\int_{-\infty}^{x} u(t,y) \ dy $. We consider a nonnegative smooth cut-off function $\eta$ supported on the ball of radius 2 and such that $\eta=1$ on the ball of radius 1. For any $R>0$, we introduce the rescaled functions $\eta_R (\cdot) = \eta(\frac{.}{R})$. Let be $\varphi\in C_0^{\infty}({\mathbb R})$, we have
$$\int_{{\mathbb R}} V(t,x) \varphi(x) \eta_R (x) dx = - \int_{{\mathbb R}} u(t,x) \theta(x) \eta_R (x) dx
-\int_{{\mathbb R}} V(t,x) \theta(x) \partial_x \eta_R (x) dx\,,$$
where $\theta(x) =\int_{-\infty}^{x} \varphi(y) \ dy$. By definition of the solution $u$, taking as test function $ \theta(x) \eta_R (x)$ we deduce that
$$\begin{aligned}
\label{DISTINTSTRTR}
\int_{{\mathbb R}} & V(t,x) \ \eta_R (x) \varphi(x) dx = - \int_{0}^{t} \!\! \int_{{\mathbb R}} \partial_x V(s,x) \, v(s,x) \eta_R (x) \varphi(x) \ dx ds \nonumber\\[5pt]
& - \int_{0}^{t} \!\! \int_{{\mathbb R}} \partial_x V(s,x) \ \eta_R (x) \varphi(x) \ dx \, {\circ}{dB_s} - \int_{0}^{t} \!\! \int_{{\mathbb R}} \partial_x V(s,x) \, v(s,x) \partial_x \eta_R (x) \theta(x) \ dx ds\nonumber\\[5pt]
&- \int_{0}^{t} \!\! \int_{{\mathbb R}} \partial_x V(s,x) \ \partial_x \eta_R (x) \theta(x) \ dx \, {\circ}{dB_s}-\int_{{\mathbb R}} V(t,x) \theta(x) \partial_x \eta_R (x) dx.\end{aligned}$$
Since $V\in L^{\infty}([0,T],L^{1}({\mathbb R}) )$ taking the limit as $R\rightarrow \infty$ we get
$$\label{DISTINTSTRT}
\begin{aligned}
\int_{{\mathbb R}} V(t,x) \varphi(x) dx = \\[5pt]
- \int_{0}^{t} \!\! \int_{{\mathbb R}} \partial_x V(s,x) \, v(s,x) \varphi(x) \ dx ds
- \int_{0}^{t} \!\! \int_{{\mathbb R}} \partial_x V(s,x) \ \varphi(x) \ dx \, {\circ}{dB_s} .
\end{aligned}$$
[*Step 2: Smoothing.*]{} Let $\{\rho_{\varepsilon}(x)\}_\varepsilon$ be a family of standard symmetric mollifiers. For any $\varepsilon>0$ and $x\in{\mathbb R}^d$ we use $\rho_\varepsilon(x-\cdot)$ as test function and we obtain $$\begin{aligned}
\int_{{\mathbb R}} V(t,y) \rho_\varepsilon(x-y) \, dy = &\, - \int_{0}^{t} \int_{{\mathbb R}} \big( v(s,y) \partial_y V(s,y) \big) \rho_\varepsilon(x-y) \ dy ds
\\[5pt]
& - \int_{0}^{t} \!\! \int_{{\mathbb R}} \partial_y V(s,y) \, \rho_\varepsilon(x-y) \, dy \circ dB_s
\end{aligned}$$
We put $V_\varepsilon(t,x)= (V\ast \rho_\varepsilon)(x)$, $v_\varepsilon(t,x)= (v \ast \rho_\varepsilon)(t,x)$ and $(vV)_\varepsilon(t,x)= (v.V\ast \rho_\varepsilon)(x)$. Then have
$$V_{\varepsilon}(t,x) + \int_{0}^{t} v\varepsilon(s,x) \partial_x V_{\varepsilon}(s,x) \, ds + \int_{0}^{t} \partial_{x} V_{\varepsilon}(s,x) \, \circ dB_s$$ $$=\int_{0}^{t} \big(\mathcal{R}_{\epsilon}(V,v) \big) (x,s) \, ds ,$$
where $ \mathcal{R}_{\epsilon}(V,v) = v_\varepsilon \ \partial_x V_\varepsilon - (v\partial_x V)_\varepsilon $.
[*Step 3: Method of Characteristic.*]{}
We consider the stochastic flow
$$d X_t^{\epsilon} = v^\varepsilon (t,X_t^{\epsilon}) \, dt + d B_t \, , \hspace{1cm} X_0 = x \,.$$
Using the same arguments that in steps 3-5-6 of the existence proof we have
$$\label{est3}
{\mathbb E}|JX_{t-s}^{\epsilon}|^{2}\leq C.$$
Applying the Itô-Wentzell-Kunita formula to $ V_{\varepsilon}(t,X_{t}^{\epsilon})$ , see Theorem 8.3 of [@Ku2], we have
$$V_{\varepsilon}(t,X_{t}^{\epsilon}) = \int_{0}^{t} \big(\mathcal{R}_{\epsilon}(V,v) \big) (X_s^{\epsilon},s) ds .$$
Hence
$$V_{\varepsilon}(t,x) =\int_{0}^{t} \big(\mathcal{R}_{\epsilon}(V,v) \big) (X_{t-s}^{-1,\epsilon},s) ds .$$
Multiplying by the test functions $\varphi$ and integrating in ${\mathbb R}$ we obtain
$$\int V_{\varepsilon}(t,x) \ \varphi(x) dx =
\int_{0}^{t} \int \big(\mathcal{R}_{\epsilon}(V,v) \big) (X_{t-s}^{-1,\epsilon},s) \ \ \varphi(x) \ \, dx \ ds .$$
Doing the change of variable we obtain
$$\int_{0}^{t} \int \big(\mathcal{R}_{\epsilon}(V,v) \big) (X_{t-s}^{-1,\epsilon},s) \ \varphi(x) \ \, dx \ ds =
\int_{0}^{t} \int \big(\mathcal{R}_{\epsilon}(V,v) \big) (x,s) \ JX_{t-s}^{\epsilon} \varphi(X_{t-s}^{\epsilon}) \ \, dx \ ds .$$
[*Step 4: Convergence of the commutator.*]{} Now, we observe that $\mathcal{R}_{\epsilon}(V,b)$ converge to zero in $L^{2}([0,T]\times {\mathbb R})$. In fact, we have
$$(v \ \partial_x V)_{\varepsilon} \rightarrow v \ \partial_x V \ in \ L^{2}([0,T]\times {\mathbb R}),$$
and by the dominated convergence theorem we obtain
$$v_{\epsilon} \partial_x V_{\varepsilon} \rightarrow v \ \partial_x V \ in \ L^{2}([0,T]\times {\mathbb R}).$$
[*Step 5: Conclusion.*]{} From step 3 we have
$$\label{conv}
\int V_{\varepsilon}(t,x) \ \varphi(x) dx =
\int_{0}^{t} \int \big(\mathcal{R}_{\epsilon}(V,v) \big) (x,s) \ JX_{t-s}^{\epsilon} \varphi(X_{t-s}^{\epsilon}) \ \, dx \ ds ,$$
Using Hölder’s inequality we obtain
$${\mathbb E}\bigg|\int_{0}^{t} \int \bigg(\mathcal{R}_{\epsilon}(V,v) \bigg) (x,s) \ JX_{t-s}^{\epsilon} \varphi(X_{t-s}^{\epsilon}) \ \, dx \ ds \bigg|$$
$$\leq \bigg({\mathbb E}\int_{0}^{t} \int |\big(\mathcal{R}_{\epsilon}(V,v) \big) (x,s)|^{2} \ \, dx \ ds \bigg)^{\frac{1}{2}}
\bigg({\mathbb E}\int_{0}^{t} \int | JX_{t-s}^{\epsilon} \varphi(X_{t-s}^{\epsilon})|^{2} \ \, dx \ ds \bigg)^{\frac{1}{2}}$$
From step 4 we deduce
$$\bigg({\mathbb E}\int_{0}^{t} \int |\big(\mathcal{R}_{\epsilon}(V,v) \big) (x,s)|^{2} \ \, dx \ ds \bigg)^{\frac{1}{2}}\rightarrow 0.$$
From estimation (\[est3\]) we obtain
$$\bigg({\mathbb E}\int_{0}^{t} \int | JX_{t-s}^{\epsilon} \varphi(X_{t-s}^{\epsilon})|^{2} \ \, dx \ ds \bigg)^{\frac{1}{2}}$$ $$\leq C \bigg(\int_{0}^{t} \int_{{\mathbb R}} |\varphi(x)|^{2} \ \, dx \ ds \bigg)^{\frac{1}{2}}\leq C \ \int_{{\mathbb R}} |\varphi(x)|^{2} \ \, dx.$$
Passing to the limit in equation (\[conv\]) we conclude that $V=0$. Then we deduce that $u=0$.
Acknowledgements {#acknowledgements .unnumbered}
================
Christian Olivera C. O. is partially supported by CNPq through the grant 460713/2014-0 and FAPESP by the grants 2015/04723-2 and 2015/07278-0.
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[^1]: Departamento de Matemática, Universidade Estadual de Campinas, Brazil. E-mail: [*[email protected]*]{}.
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---
abstract: 'Absolute negative mobility (ANM) is a highly counterintuitive phenomenon, where the sign of the induced current is opposite to the applied forces. We show that ANM can occurr in a one-dimensional interacting Hamiltonian system when its equilibrium state is perturbed by coupled forces. This seemingly paradoxical result is possible due to the self-organization occurring in the system in response to the applied forces.'
author:
- Jiao Wang
- Giulio Casati
- Giuliano Benenti
title: Absolute negative mobility in an interacting Hamiltonian system
---
Absolute negative mobility (ANM) is arguably the most counterintuitive transport phenomenon, in that a system responds to an applied static force by moving the current against that force. It has been pointed out that ANM cannot take place around a thermal equilibrium state [@Vandenbroeck01; @Hanggi02], otherwise it could be exploited to construct a perpetuum mobile of the second kind, with a single heat bath performing work against the force. In nonequilibrium systems, however, there is no fundamental law that forbids ANM, and indeed ANM has been investigated in a variety of nonequilibrium setups, e.g., in relation to particle separation [@Hanggi12; @Hanggi19], self-propulsion [@Hanggi14], tracer dynamics in a steady laminar flow [@Vulpiani16], and also experimentally in semiconductor superlattices [@Keay95], microfluidic systems [@Anselmetti05], and Josephson junctions [@Nagel08], subject to ac electric fields.
The above argument of a perpetuum mobile only applies when there is a single flow in response to a single driving force acting on the system. Indeed, it has been shown recently that, at least in an abstract stochastic model, ANM can take place by perturbing an equilibrium state by means of two forces [@Mukamel]. More precisely, the effect was obtained for a stochastic dynamics, with a tracer particle subject to two driving forces and moving on a discrete ring populated by neutral particles, which in turn obey a symmetric exclusion process. This interesting result, though not in contradiction with thermodynamics, appears highly counterintuitive. Indeed, it would imply that a system at equilibrium exposed to two thermodynamic forces, could, under appropriate conditions, exhibit ANM against both forces. As a matter of fact, such type of ANM has not been shown possible in any physical system. In particular, in the model of hard Brownian disks in a narrow planar chain, of which the above stochastic model serves as a toy model, ANM was not found [@Mukamel]. This raises the basic question whether, besides abstract stochastic models, a purely dynamical, Hamiltonian system that exhibits ANM when perturbing its equilibrium state, exists.
In this letter we give a positive answer to this question by considering a one-dimensional (1D), two-masses interacting gas model. It is found that by perturbing its equilibrium state with biases in temperature and chemical potential, it is possible to have one flow (either particle or energy) *against both biases*. This seemingly paradoxical result is possible due to a negative Onsager cross-coefficient for thermodiffusion, which in turn is rooted in the surprising property of our model of adapting its structure in response to external gradients, with separation of the two species of particles of different masses [@PRL2017].
Our model can be viewed as a classical version of the Lieb-Liniger model [@LL] for a diatomic gas. A schematic drawing is provided in Fig. \[fig1\], where the two species of particles of mass $\mathcal{M}_1$ and $\mathcal{M}_2$ are denoted as bullets and rods, respectively, for visualization purposes. The masses are confined to move in a 1D box of length $L$, with the Hamiltonian $$H=\sum_{i} \frac{p_i^2}{2m_{i}}+\sum_{i<j}V(x_{i}-x_{j}).
\label{eq1}$$ Here $m_i\in\{\mathcal{M}_1,\mathcal{M}_2\}$, $p_i$, and $x_i$ are, respectively, the mass, momentum, and position of the $i$th particle, and the potential $V(x)$ is a step function, i.e., $V(x)=h$ for $x \le |r|$ and $V(x)=0$ otherwise, with $h\ge 0$ being the potential barrier.
![Schematic drawing of the 1D, two-mass, interacting gas model shown to own ANM property in this work. It consists of a diatomic gas of particles, represented for visualization purposes by bullets and rods, respectively. The system is coupled to two reservoirs at its two ends, with which the rods exchange energy while the bullets exchange both energy and particles.[]{data-label="fig1"}](fig1.eps){width="8.5cm"}
In this work we restrict ourselves to the limiting case $r\to 0$. In this limit, all particles move freely and the interactions between them occur only when two neighboring particles meet. At such a moment, either the two particles simply pass through each other without changing their velocities – if their total energy in the frame of the center of mass is larger than the potential barrier $h$, or they collide elastically, otherwise.
Note that for $h=0$, all particles move freely and the system is integrable. For $h=\infty$, the system reduces to the nonintegrable hard-core two-mass gas [@Giulio86], a paradigm for the 1D transport problem that has been intensively studied. Interestingly, for a finite nonzero value of $h$, the system’s transport behavior is very rich [@Kurchan; @PRL2017]. In particular, it is found that as the temperature gradient applied to the system increases, the system may undergo a nonequilibrium phase transition, through which the system self-organizes its structure to abruptly enhance its heat conductivity as an adaptive response [@PRL2017].
In the studies of [@Kurchan; @PRL2017], two heat baths of different temperatures, $T_L$ and $T_R$, are brought into contact with the system at its two ends. When a particle hits a bath, it is reflected back with a new velocity, $v$, sampled from the distribution [@bath] $$P_{\alpha}(v)={m|v|\beta_\alpha}e^{-{mv^2\beta_\alpha}/2},
\label{eq2}$$ where $m$ is the mass of the particle, $\beta_\alpha=1/(k_B T_\alpha)$ ($\alpha=L, R$), and $k_B$ is the Boltzmann constant. In this way the system exchanges energy with the baths and a stationary heat current will form asymptotically. In this setup, the heat current always flows from the hot bath to the cold bath [@Kurchan; @PRL2017], ANM is thus ruled out.
We therefore investigate another setup where the particle current also plays a role (see Fig. \[fig1\]): two reservoirs of one kind of particles, say, the ‘bullets’, are introduced to replace the heat baths. The reservoirs are modeled as infinite 1D ideal gases [@reservoir], which are allowed to exchange both energy and (bullet) particles with the system: when a bullet particle hits a reservoir, it will be absorbed; meanwhile, the reservoirs emit bullet particles into the system randomly with rates $\gamma_\alpha$ ($\alpha=L, R$). The injection rate $\gamma_\alpha$ represents how many particles are injected from reservoir $\alpha$ per unit time, which is related to the temperature, $T_\alpha$, and the chemical potential, $\mu_\alpha$, of reservoir $\alpha$ by [@rate] $$\gamma_{\alpha}=\frac{\rho_0}{\sqrt{2\pi \mathcal{M}_1 \beta_0}}
\frac{\beta_0}{\beta_\alpha}e^{\mu_\alpha \beta_\alpha-\mu_0 \beta_0}.
\label{eq3}$$ Here $\beta_0=1/(k_B T_0)$ and $T_0$, $\rho_0$, and $\mu_0$ are, respectively, the temperature, particle number density, and chemical potential of a reference state. An injected particle owns a random velocity given by Eq. (\[eq2\]), and the time interval between two neighboring injections of a reservoir obeys the Poisson distribution; i.e., $P(t)=\gamma e^{-\gamma t}$. Instead the rod particles only exchange energy with the reservoirs; i.e., when a rod particle hits a reservoir, it will be reflected back with a new velocity randomly chosen from the distribution of Eq. (\[eq2\]) [@note]. As the reservoirs do not exchange rod particles with the system, the total number of rod particles is conserved.
The system is subject to the thermodynamic forces $\mathcal{F}_\rho=\mu_L\beta_L-\mu_R\beta_R$ and $\mathcal{F}_u=\beta_R-\beta_L$, conjugated to the energy and particle currents, $J_\rho$ and $J_u$, respectively. We set $\mathcal{F}_\rho>0$ and $\mathcal{F}_u>0$, so that a negative current signals ANM. Note that this phenomenon should not be confused with thermodiffusion, where the two thermodynamic forces have opposite sign instead and, for instance, the motion of particle against a chemical potential difference is possible thanks to a temperature difference.
For numerical studies, we set $T_L=T+\Delta T/2$, $\mu_L=\mu+\Delta \mu/2$, $T_R=T-\Delta T/2$, and $\mu_R=\mu-\Delta \mu/2$. We mainly focus on the case of $T=1$ and $\mu=1.5$ as an illustrating example, but it has been checked and verified that the ANM effect does not depend on this particular choice. Other parameter values adopted are: $k_B=1$, $\mathcal{M}_1=1$, $\mathcal{M}_2=0.5$ (as explained below, to have ANM it is crucial that $\mathcal{M}_1>\mathcal{M}_2$, that is, the particles exchanged with the reservoirs are the heavier ones); and for the reference state, $\rho_0=1$, $T_0=1$, and $\mu_0=0$. The potential barrier is set to be $h=1$ throughout, unless explicitly stated otherwise. The number of rods in the system is set to be half of the expected particle number of a 1D ideal gas at the equilibrium state of $T$ and $\mu$; i.e., $N_{\mathcal{M}_2} = \rho L/2$ with $\rho=\rho_0(\sqrt{\beta_0/\beta})e^{\beta\mu-\beta_0\mu_0}$. For a given set of $\Delta T$ and $\Delta \mu$, the system is evolved with an effective event-driven algorithm [@algorithm] till the stationary state is reached. Then the system is evolved further to evaluate the particle and the energy current. Finally the dependence of the currents on the thermodynamic forces is analyzed. For all numerical results obtained, the relative errors are smaller than 0.5%, hence the error bars (smaller than the data symbols) are omitted in figures.
![(a) and (b): The particle current $J_\rho$ (a) and the energy current $J_u$ (b) as a function of the thermodynamic force $\mathcal{F}_u$ for the studied 1D, two-mass interacting system with $\mathcal{F}_\rho=0$. The negative particle current $J_\rho$ signifies ANM. The system size is $L=20$. (c) and (d): The same as (a) and (b), respectively, but for the currents as a function of the thermodynamic force $\mathcal{F}_\rho$ (with $\mathcal{F}_u=0$). The effect of ANM is signified by the negative energy current $J_u$ instead.[]{data-label="fig2"}](fig2.eps){width="8.0cm"}
We start seeking ANM in the simpler cases when only one thermodynamic force acts. First of all, we set $\mathcal{F}_\rho=0$ to see how the currents depend on the force $\mathcal{F}_u$. A typical result is shown in Fig. \[fig2\](a) and (b), where we can see that while $J_u$ monotonically increases with $\mathcal{F}_u$ and keeps positive, $J_\rho$ becomes more negative with increasing $\mathcal{F}_u$ first, then turns to increase after a turning point. As $J_\rho$ is negative over the whole range of $\mathcal{F}_u$ investigated, i.e., the particle current flows from the low to the high temperature reservoir, this is a clear evidence of ANM. Note that for small $\mathcal{F}_u$ ($<0.3$ in this case) both currents depend on $\mathcal{F}_u$ linearly, suggesting that the system is in the linear response regime, therefore the results shown in Fig. \[fig2\](a) indicate that in our system, ANM may happen in both the linear response regime and beyond.
Similarly, a parallel study reveals how the currents depend on the thermodynamic force $\mathcal{F}_\rho$ with $\mathcal{F}_u=0$ \[see Fig. \[fig2\](c) and (d)\]. Again, ANM is observed, but in this case it is in the energy current $J_u$ instead; i.e., the energy flows from the reservoir of the lower chemical potential to that of the higher one (note that here the two reservoirs are at the same temperature due to $\mathcal{F}_u=0$).
-0.2cm ![The particle current $J_\rho$ (a) and the energy current $J_u$ (b) as a function of the thermodynamic forces $\mathcal{F}_u$ and $\mathcal{F}_\rho$. ANM occurs in $J_\rho$ when the thermodynamic forces fall in the area above the white dashed line in (a) but in $J_u$ when fall in the area below the white dashed line in (b). The system size is $L=20$.[]{data-label="fig3"}](fig3.eps "fig:"){width="8.0cm"}
In order to have an overall grasp of how the currents depend on the thermodynamic forces, we thoroughly compute the currents for various $\mathcal{F}_\rho$ and $\mathcal{F}_u$. The results are summarized in Fig. \[fig3\]. An area of ($\mathcal{F}_\rho$, $\mathcal{F}_u$) for ANM in the particle current $J_\rho$ can be recognized in Fig. \[fig3\](a) (above the white dashed line), while that for ANM in the energy current $J_u$ can be found in Fig. \[fig3\](b) (below the white dashed line). The two ANM areas do not overlap, as expected since simultaneous ANM in both currents would lead to negative entropy production rate, thus violating the second law of thermodynamics.
Next, we study the role the interactions play in generating ANM. To this end, we investigate how an ANM current depends on the potential barrier. For example, in Fig. \[fig4\](a), the results for $J_\rho$ with $\mathcal{F}_u>0$ and $\mathcal{F}_\rho=0$ is given. It shows that as $h\to 0$, i.e., the interactions tends to vanish, $J_\rho$ increases and turns to be positive, indicating that interactions are necessary to obtain ANM. On the other hand, in the limit $h\to \infty$, when the particles tend to collide elastically without passing through each other, the ANM current decays. Therefore, allowing the particles passing through each other is a crucial element of the interactions for inducing ANM in our system [@note2].
![The dependence on the interaction potential barrier $h$ of (a) the particle current $J_\rho$ and (b) the Onsager cross-coefficient $\mathcal{L}_{\rho u}$. The thermodynamic forces are $\mathcal{F}_\rho=0$ and $\mathcal{F}_u=0.1$, respectively, and the system size is $L=160$. []{data-label="fig4"}](fig4.eps){width="8.0cm"}
In the linear response regime, the currents are related to the thermodynamic forces as [@Callen; @Groot] $$\left(
\begin{array}{c}
J_\rho\\
J_u
\end{array}
\right) = \left(
\begin{array}{cc}
\mathcal{L}_{\rho \rho} & \mathcal{L}_{\rho u} \\
\mathcal{L}_{u \rho} & \mathcal{L}_{u u}
\end{array}
\right) \left(
\begin{array}{c}
\mathcal{F}_\rho /L\\
\mathcal{F}_u /L
\end{array}
\right) ,
\label{eq:lresponseSM}$$ where $\mathcal{L}_{ij}$ ($i,j=\rho, u$) are the Onsager kinetic coefficients. For the general case where the currents are positive for positive forces, $\mathcal{L}_{ij}>0$. However, thermodynamics does not forbid the cross-coefficients to be negative. As shown in Fig. \[fig4\](b), our model exhibits such unusual feature, and it is when $\mathcal{L}_{\rho u}<0$ that ANM occurs in the linear response regime.
The fact that $\mathcal{L}_{\rho u}<0$ can be understood as follows. If we set $\mathcal{F}_\rho=0$ and $\mathcal{F}_u>0$, then the probability for two particles to cross each other is higher when the light particle (a rod) is closer to the hot end and the heavy particle (a bullet) is closer to the cold end. In this case, the relative velocity of the two particles is on average higher than in the opposite configuration, hence favors crossing. This creates an unbalance in the particle density for the two species, with the rods staying preferably closer to the cold side and the bullets close to the hot side (such an unbalance develops to a phase separation in the far-from-equilibrium regime). As only bullets exchange with the reservoir, we therefore have an average flow of bullets from the cold to the hot reservoir, i.e. $J_\rho<0$, which in turn implies $\mathcal{L}_{\rho u}<0$ \[see Eq. (\[eq:lresponseSM\])\] in the linear response regime.
Now we state two properties of ANM in our model. First, numerical data on the size dependence of ANM (see the Supplementary Material) show that, in both the linear response regime and beyond, ANM first becomes stronger and then saturates with increasing of the system size. These data provide a clear indication of the existence and relevance of ANM in our model in the limit of large system size.
Second, for a large system size, a phase separation occurs at large driving, causing the ANM effect dramatically enhanced. As an example, we show in Fig. \[fig5\](a) the particle current $J_\rho$ driven by $\mathcal{F}_u$ with $\mathcal{F}_\rho=0$ for a large system size $L=1280$. It can be seen that it features a turning point at $\mathcal{F}_u^{(c)}
\approx 0.4$. For $\mathcal{F}_u<\mathcal{F}_u^{(c)}$, $|J_{\rho}|$ increases with $\mathcal{F}_u$ slowly; but for $\mathcal{F}_u >
\mathcal{F}_u^{(c)}$, when the phase separation happens (see below), it increases with $\mathcal{F}_u$ in a much sharper way. This abrupt change is even more evident in the rescaled particle current $\tilde{\mathcal{L}}_{\rho u} \equiv J_\rho L/\mathcal{F}_u$ \[the inset of Fig. \[fig5\](a)\]. This quantity, which tends to $\mathcal{L}_{\rho u}$ in the linear response regime $\mathcal{F}_u \to 0$, first decays (in absolute value) for $\mathcal{F}_u < \mathcal{F}_u^{(c)}$ and then rapidly increases when $\mathcal{F}_u > \mathcal{F}_u^{(c)}$.
-0.05cm ![(a) The particle current $J_\rho$ as a function of the thermodynamic force $\mathcal{F}_u$ with $\mathcal{F}_\rho=0$. The system size is $L=1280$. The inset shows $\tilde{\mathcal{L}}_{\rho u}\equiv J_\rho L / \mathcal{F}_u$. (b) The average mass of particles (of both species) passing a given point $x$ along the system. The five curves correspond to the middle five points shown in panel (a).[]{data-label="fig5"}](fig5.eps "fig:"){width="8cm"}
To illustrate the phase separation at $\mathcal{F}_u=\mathcal{F}_u^{(c)}
\approx 0.4$, we show in Fig. \[fig5\](b) the averaged particle mass, $\langle m(x) \rangle$, for all the particles that pass a certain position $x$. We can see that, for small driving force, $\langle m(x) \rangle$ is almost uniform except that at the left end it is a bit higher. Hence the two types of particles are overall uniformly mixed. As the driving force is strengthened, the left end lifts higher, suggesting that bullets (of mass $\mathcal{M}_1=1$) tend to accumulate at the left end and rods (of mass $\mathcal{M}_2=0.5$) are pushed to the right. When the driving force reaches the critical value, the left end is exclusively occupied by the bullets. For even stronger driving force, a domain of pure bullet particles emerges at the left end, and the rods are brought out of contact with the left reservoir completely, indicating that a critical transition in system’s structure has occurred. The change in structure is a self-organization behavior to adaptively respond to the external forces, which recalls that observed in the system’s heat conduction behavior [@PRL2017].
In summary, we have shown that ANM can happen in a 1D, interacting Hamiltonian system, either in the particle or the energy current. The effect is observed both within linear response and beyond and is rooted in the self-organization in response to the applied forces, up to complete phase separation in the far-from-equilibrium regime, where ANM is greatly enhanced. Our results raise the question of what the general conditions for ANM are and whether other mechanisms different from self-organization exist. It might be also interesting to explore possible new effects that negative Onsager cross-coefficients may induce [@Lepri_note; @Lepri2012]. For instance, one could in principle design a thermoelectric circuit [@Benenti2017; @note3] with only one kind of electric carriers rather than alternating $p$- and $n$-doped semiconductors as in a thermocouple. The opposite response to a temperature difference could be obtained by alternating a channel with standard response to applied thermodynamic forces and a channel that exhibits ANM.
We acknowledge support by NSFC (Grants No. 11535011 and No. 11335006) and by the INFN through the project QUANTUM. The computational resources utilized in this study were provided by Shanghai Supercomputer Center.
Supplementary Material: Absolute negative mobility in an interacting Hamiltonian system
=======================================================================================
Here we provide more analysis of ANM in our model. For all the numerical results presented, the parameters adopted in simulations are the same as in the main text.
I. Dependence on system size
----------------------------
In Fig. \[figS1\], the dependence of the currents $J_\rho$ and $J_u$ on the thermodynamic forces $\mathcal{F}_\rho$ and $\mathcal{F}_u$ is shown for various system sizes. It can be seen that, in general, the currents drop as the system size increases. But, however, the relevance of ANM is not decreasing. Indeed, as shown in Fig. \[figS2\], at a given force $\mathcal{F}_u$ or $\mathcal{F}_\rho$, as the system size increases, both the negative ratio $J_\rho/J_u$ (at $\mathcal{F}_\rho=0$) and $J_u/J_\rho$ (at $\mathcal{F}_u=0$) would increase first (in absolute value) and then tend to saturate.
![The same as Fig. 2 in the main text, but for five different system sizes. In each panel, the black, red, blue, orange, and magenta curve are for, respectively, the system size of $L=20$, 40, 80, 160, and 320.[]{data-label="figS1"}](SMfig1.eps){width="8.4cm"}
![(a) Ratio $J_\rho/J_u$ versus the thermodynamic force $\mathcal{F}_u$ (at $\mathcal{F}_\rho=0$) and (b) $J_u/J_\rho$ versus $\mathcal{F}_\rho$ (at $\mathcal{F}_u=0$) for different system sizes. Legends in panel (a) also apply to panel (b).[]{data-label="figS2"}](SMfig2.eps){width="8.4cm"}
II. Thermoelectric implications
-------------------------------
In the linear response regime, the currents are related to the thermodynamic forces as [@Callen; @Groot] $$\left(
\begin{array}{c}
J_\rho\\
J_u
\end{array}
\right) = \left(
\begin{array}{cc}
\mathcal{L}_{\rho \rho} & \mathcal{L}_{\rho u} \\
\mathcal{L}_{u \rho} & \mathcal{L}_{u u}
\end{array}
\right) \left(
\begin{array}{c}
\mathcal{F}_\rho /L\\
\mathcal{F}_u /L
\end{array}
\right) ,
\label{eq:lresponse}$$ where $\mathcal{L}_{ij}$ ($i,j=\rho, u$) are the Onsager kinetic coefficients.
In our system, if the bullet particles are charge carriers, then the electric conductivity $\sigma$, the thermal conductivity $\kappa$, and the Seebeck coefficient $S$ of the system are connected with $\mathcal{L}_{ij}$ ($i,j=\rho, u$) as $$\sigma=\frac{e^2}{T}\,\mathcal{L}_{\rho\rho},
~\kappa=\frac{1}{T^2}\frac{\det\mathcal{L}}{\mathcal{L}_{\rho\rho}},
~S=\frac{1}{eT}\left(\frac{\mathcal{L}_{\rho u}}{\mathcal{L}_{\rho\rho}}-\mu\right),$$ where $e$ is the charge of a bullet particle and ${\det\mathcal{L}}$ denotes the determinant of the matrix of Onsager kinetic coefficients. The thermoelectric figure of merit $ZT$ can be expressed in terms of these transport coefficients as [@Benenti2017] $$ZT=\frac{\sigma S^2}{\kappa}\,T.$$ Thermodynamics imposes $ZT\ge 0$, with the efficiency of heat to work conversion $\eta=0$ when $ZT=0$ and $\eta\to\eta_C$ when $ZT\to\infty$, $\eta_C$ being the Carnot efficiency.
![The dependence of the Onsager kinetic coefficients on the system size. The Onsager kinetic coefficients are evaluated based on Eq. (\[eq:lresponse\]) by computing the currents with ($\mathcal{F}_\rho$, $\mathcal{F}_u$) being (0, 0.04) and (0.04, 0), respectively.[]{data-label="figS3"}](SMfig3.eps){width="8.4cm"}
![The dependence of the transport coefficients and $ZT$ on the system size in the linear response regime based on the computed Onsager kinetic coefficients presented in Fig. \[figS3\].[]{data-label="figS4"}](SMfig4.eps){width="8.4cm"}
In Fig. \[figS3\], the system size dependence of the Onsager kinetic coefficients is provided. It can be seen that the absolute value of all Onsager coefficients increases with the system size. Moreover, the cross-coefficient $\mathcal{L}_{\rho u}$ is negative ($\mathcal{L}_{u \rho}=\mathcal{L}_{\rho u}$ due to the Onsager reciprocal relations [@Callen]). For simulations, the charge $e$ is set to be unity throughout.
The size dependence of the transport coefficients and $ZT$ is shown in Fig. \[figS4\]. It can be seen that while the Seebeck coefficient is negative and saturates to a certain value, the electric conductivities $\sigma$ and the heat conductivity $\kappa$ keep growing, but at different rates so that $ZT$ increases monotonically. Extrapolation of these results to the thermodynamic limit would imply that the Carnot efficiency would be achieved in that limit, similar to other momentum conserving systems [@Benenti2013; @Benenti2014; @Chen2015; @Luo2018].
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The existence of ANM does not depend on the coupling between the rod particles and the reservoirs. For example, ANM is verified to exist even if the rod particles are insulated from the reserviors by setting their reflected velocity unchanged but the direction.
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****\
ABSTRACT :\
In this article, we study the quotients of numerical semigroups, generated by two coprime positive numbers, denoted $\frac{<a,b>}{d}$. We give formulae for the usual invariants of these semigroups, expressed in terms of continued fraction expansions and Ostrowski-like numeration of some rationals, simply related to inputs $a,b,d$. So, we obtain quadratic complexity algorithms to compute these invariants. As a consequence, we will show that, for these numerical semigroups, the type is always lower than the embedding dimension and deduce Wilf’s property. We also consider the “ reverse problem” : given $I$ a finite set of integers, we obtain an expression of all possible triplets $(a,b,d)$, such that $I$ is the set of minimal generators of $\frac{<a,b>}{d}$.\
Introduction
============
basics about numerical semigroups {#subsec:intro}
---------------------------------
We give here some definitions and elementary results without proofs. See **\[2\],\[8\]** and **\[11\]** ( ) for more details. First, we will use some classical notations :\
- we denote ${\mathbb{N}}$ the set of all non negative integers, ${\mathbb{N}}^*$ the set of positive integers and ${\mathbb{Z}}$ the set of integers.\
- for every couple of integers $p,q$, the notation ${ \{ p,..,q \} }$ means the set of integers $k$ such that $p\leqslant k\leqslant q$. We extend this notation to ${ \{ p,..,\infty \} }=\{ k\in {\mathbb{Z}}, k\geqslant p \}$.\
- for an element $x$ and two subsets $I$ and $J$ of an additive group $G$, we will use the classical notations : $$I+J=\{ i+j, i\in I,j\in J\} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}-I=\{ -i,i\in I\} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}x+J=\{ x+j,j\in J\}$$ - we will also denote for an integer $n$, a subset $K$ of ${\mathbb{Z}}$, an element $x$ of $G$ and a subset $J$ of $G$ : $$nJ=\{ nj,j\in J \} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}xK=\{ kx, k\in K \}$$
$\bullet$ A *numerical semigroup* $S$ is a cofinite submonoid of $({\mathbb{N}},+)$, where ${\mathbb{N}}$ denotes the set of non negative integers. We can also define a numerical semigroup with a system of generators. Let $B$ be a non empty subset of ${\mathbb{N}}$ and $<B>$ the set of all non negative integer combinations of finite subsets of $B$. Then $<B>$ is a numerical semigroup ( *generated by $B$*) if and only if the elements of $B$ are coprime ( not necesseraly pairwise coprime). In the case of finite $B$ ( which is the general case), say $B=\{ n_1,n_2,\cdots,n_r \}$, then : $$<B> = \left\{ \sum_{k=1}^r x_kn_k {\hspace*{0.2cm}}; {\hspace*{0.2cm}}x_1,x_2,\cdots,x_r\in {\mathbb{N}}\right\}$$ *Let $S$ be a numerical semigroup*.\
$\bullet$ An element $s$ of $S$ is *irreducible* in $S$ if and only if there is no positive elements $s'$ and $s"$ of $S$ such that $s=s'+s"$. The set of irreducible elements of $S$ will be denoted ${\mathrm{Irr}}(S)$. It is well known that ${\mathrm{Irr}}(S)$ is the minimal set of generators of $S$ : that is to say ${\mathrm{Irr}}(S)$ generates $S$ and every set of generators of $S$ contains ${\mathrm{Irr}}(S)$. ${\mathrm{Irr}}(S)$ is finite and its cardinality is called the *embedding dimension of $S$*, denoted by $e(S)$.\
${\mathrm{Irr}}(S)$ is also the set of minimal elements of $S\backslash \{ 0 \}$, for the order $\leqslant_S $ induced by $S$ : $$\forall x,y\in {\mathbb{Z}}, (x\leqslant_S y \Leftrightarrow y-x\in S )$$ The smallest positive element of $S$ is called the *multiplicity* of $S$ and is denoted $m(S)$. It is also the lowest element of ${\mathrm{Irr}}(S)$.\
$\bullet$ The finite set ${\mathbb{N}}\backslash S$ is often named the *set of gaps of $S$* and is denoted $G(S)$. Its cardinality, named the *genus* of $S$, is denoted $g(S)$. The greatest element of $G(S)$ is called the *Frobenius number* of $S$ and is denoted $f(S)$, with the convention that $f({\mathbb{N}})=-1$. It is in general difficult to compute this Frobenius number, but a lot of investigations have been made ( **\[11\]** ).\
The number $f(S)+1$ is sometimes called the *conductor* of $S$ and denoted $c(S)$. The set of elements of $S$ lower than $f(S)$ will be denoted by $S_0$, so that we have the following partition of ${\mathbb{N}}$ : $${\mathbb{N}}= G(S) \cup S_0 \cup { \{ c(S) ,.., +\infty \} }$$ So, we have : $$\#S_0=c(S)-g(S)$$ $\bullet$ Note that if $s\in S$, then $f(S)-s\not\in S$, so that : $$f(S)-S_0\subset G(S){\hspace*{0.2cm}}\text{ so } {\hspace*{0.2cm}}\# S_0 \leqslant g(S) {\hspace*{0.2cm}}\text{ and } {\hspace*{0.2cm}}\frac{g(S)}{c(S)}\geqslant \frac{1}{2}$$ A particular case if when we have an equality : then $S$ is said *symmetric*.\
$\bullet$ The Frobenius number of $S$ is a particular case of the notion of *pseudo-Frobenius numbers* ( PF numbers in this paper) of $S$, that can be defined as the maximal elements of ${\mathbb{Z}}\backslash S$ for the order $\leqslant_S $. ${\mathrm{PF}}(S)$ will denote the set of these elements : it is a subset of $G(S)$ [^1]and $f(S)$ is its greatest element ( for the usual order on ${\mathbb{Z}}$). The cardinality of ${\mathrm{PF}}(S)$ is the *type* of $S$, denoted $t(S)$.\
${\mathrm{PF}}(S)$ is also the minimal subset of ${\mathbb{Z}}$ that satisfies the following “ symmetry property” : $${\mathbb{Z}}\backslash S={\mathrm{PF}}(S)-S=\bigcup_{p\in {\mathrm{PF}}(S)}(p-S)$$ If we restrict to ${\mathbb{N}}$, then we obtain : $$G(S)\subset {\mathrm{PF}}(S)-S_0=\bigcup_{p\in {\mathrm{PF}}(S)}(p-S_0)$$
The above inclusion proves that : $ g(S)\leqslant t(S) \# S_0$ and then : $$\frac{g(S)}{c(S)}\leqslant 1-\frac{1}{t(S)+1}$$
$\bullet$ *Wilf’s property* for a numerical semigroup $S$ is the following : $$\frac{g(S)}{c(S)}\leqslant 1-\frac{1}{e(S)}$$ *Wilf’s conjecture* claims that this inequality holds for every numerical semigroup ( **\[17\]**). It has been proven in many cases ( **\[5\],\[6\],\[7\],\[10\],\[15\]** ), but the general case still remains open. According to the previous inequalities, it would be sufficient to prove that $t(S)<e(S)$. Unfortunately, it is not true for all numerical semigroups ( see example after Theorem 11 in **\[7\]**).\
Note that the ratio $g(S)/c(S)$ could be named : the “ density of gaps of $S$”.\
$\bullet$ The *quotient of $S$ by a positive integer $d$* is defined by : $$S/d= \{ x \in {\mathbb{N}}, dx\in S \}$$ It is a numerical semigroup and we remark that $S/d={\mathbb{N}}$ if and only if $d\in S$. We also have an obvious calculation rule, when we consider quotients of a quotient : $$\forall d,d'\in {\mathbb{N}}^*, {\hspace*{0.2cm}}\frac{S/d}{d'}=\frac{S}{dd'}$$ Many investigations have been made to relate invariants of $S/d$ to invariant of $S$ ( see introduction of **\[1\]** for details and further references).\
In this paper, we will study the most simple case, that is, when $S$ is generated by two integers : let $a$ and $b$ be two coprime integers greater than one. Since Sylvester in 1882 ( **\[16\]**), we know that $<a,b>$ is symmetric and we have simple formulae for $f$ and $g$ : $$f(<a,b>)=ab-a-b {\hspace*{0.2cm}};{\hspace*{0.2cm}}g(<a,b>)= \frac{(a-1)(b-1)}{2}$$
Curiously, $<a,b>$ is very simple, but its quotients are not in general ! It was an open problem to express the invariants of $S= \frac{<a,b>}{d}$, in terms of $a,b,d$ ( **\[4\]**). A recent paper ( **\[1\]**) gives an expression of $g(<a,b>/d)$ with a sum of integral parts, using Hilbert series. We will obtain another formulation for $g(<a,b>/d)$ in Theorem \[theo:10\], which could easily be relied to the previous one.\
$\bullet$ Another motivation, for the study of these numerical semigroups, lies in the fact that they can be defined by two alternative ways :\
- let $a,b,c$ be positive integers, then the set of integers $x$ that verify $ax$ mod $ b \leqslant cx$ ( where mod denotes the remainder ) is a numerical semigroup, named *proportionally modular*. In **\[13\]**, it is shown that : a numerical semigroup is proportionally modular if and only if it exists three pairwise coprime positive integers $a,b,d$ such that $S= \frac{<a,b>}{d}$. Furthermore, it is proven in **\[12\]**, that we can choose $b=a+1$. In **\[13\]**, they build a set of generators of these $\frac{<a,b>}{d}$, but it is not always minimal...\
- an equivalent definition is : for $q$ and $q'$ two rationals such that $q<q'$, ${\mathbb{N}}\cap <[q,q']>$ defines a numerical semigroup, that is proportionally modular ( **\[9\]**).\
$\bullet$ A question at the end of **\[1\]** is about the invariants of $S/d$, when $S$ is generated by an arithmetic sequence. We prove in our paper that, in that case, $S$ is itself a quotient of $<a,b>$ for some $a$ and $b$, which is a particular case of a more general result exposed in **\[9\]**( ) : a numerical semigroup $S$ is of the form $\frac{<a,b>}{d}$, for some $a,b,d$ if and only if ${\mathrm{Irr}}(S)$ can be enumerated by what we call a *modular-convex* sequence of positive integers $(n_k)_{k\in{ \{ 0,..,r \} }}$ : $$\forall k\in { \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}\frac{n_{k-1}+n_{k+1}}{n_k} \in { \{ 2,..,\infty \} }$$ We give a new proof of this result and we also give explicit formulae for all possible $a,b$ and $d$ : Theorems \[theo:1\] , \[theo:2\] , \[theo:3\] in section \[subsec:solutions\] . When $S$ is generated by an arithmetic sequence ( which corresponds to the case when $ \frac{n_{k-1}+n_{k+1}}{n_k}=2$ for all $k$), we obtain ( see Corollary \[coro:1\] at the end of \[subsec:solutions\] ) : for $a,q,r$ positive integers such that $a$ and $q$ are coprime $$<a,a+q,a+2q,\cdots,a+rq>= \frac{<a,a^2+q(ar+1)>}{ar+1}$$ This is a particular case of more general results and formulae, but it can be proven directly with elementary considerations. Combining this with Theorems \[theo:4\] ,\[theo:5\] , \[theo:6\] , \[theo:7\] , \[theo:8\] , \[theo:9\] and \[theo:10\] ( see section \[sec:invariants\]), that give expressions of invariants of $\frac{<a,b>}{d}$ and our remark about $(S/d)/d'=S/dd'$, we obtain the answer to the above question...
overview of the article
-----------------------
First, in section \[sec:representation\], following an idea of **\[13\]**, we use a representation of a numerical semigroup $S$ as a subset of the cylinder ${\mathbb{Z}}^2/w{\mathbb{Z}}$, where $w=(b,-a)$ is a non null vector of ${\mathbb{Z}}^2$, via an additive map $(x,y)\to ax+by$, where $a$ and $b$ are two coprime elements of $S$. For the numerical semigroups we are interseted in, that is $\frac{<a,b>}{d}$, this will give a “ lattice-stair case shape” representation : the intersection of a lattice and a regular stair case shape subset of ${\mathbb{Z}}^2$ ( modulo $w{\mathbb{Z}}$).\
This approach leads in \[subsec:relations\] to simple relations between $G(S),{\mathrm{Irr}}(S),{\mathrm{PF}}(S)$ and elements or minimal sets ( for the product order on ${\mathbb{Z}}^2$) of the above lattice in some rectangles. We remark in \[subsec:minimal\] that these minimal sets can be parametrized by “ double inductive” sequences of points, that are convex and unimodular : we name them “ modular-convex” ( see above).\
In \[sec:reverse\], we relate these sequences with a kind of continued fraction expansion and use them to solve the “ reverse problem” : we obtain in \[subsec:solutions\] a complete expression of all triplets of pairwise coprime positive integers $a,b,d$ such that $1<a<b$ and ${\mathrm{Irr}}(\frac{<a,b>}{d})$ is a given finite subset of ${\mathbb{N}}$.\
In order to solve the “ direct problem” ( express invariants of $\frac{<a,b>}{d}$ in terms of $a,b,d$), we need another parametrization of the above minimal sets by Kronecker sequences $(\{ n\alpha -\beta \},n)_n$. So, we use some general considerations about usual Diophantine approximation ( \[subsec:CFE\] and \[subsec:semicv\]), as well as some more unusual ( \[subsec:alphanum\] ) Ostrowski-like numeration ( detailed in **\[3\]** ).\
Finally, we give in section \[sec:invariants\]. all formulae about the usual invariants of numerical semigroups $\frac{<a,b>}{d}$. We end this paper with the remarkable property : $e(S)-t(S)\geqslant 1$ for every numerical semigroup $S$ of this type $\frac{<a,b>}{d}$. This inequlity implies Wilf’s property for these numerical semigroups, as mentioned in \[subsec:intro\].
some notations and definitions
------------------------------
$\bullet$ For any real $x$, we denote respectively $\lfloor x \rfloor, \lceil x \rceil$ and $\{ x \}$ the floor, ceil and fractional part of $x$.\
$\bullet$ For a vector $u$ of ${\mathbb{Z}}^2$, we will denote $x(u)$ and $y(u)$ its coordinates.\
We will denote $\leqslant$ the natural partial product order on ${\mathbb{Z}}^2$ : $u\leqslant v$ if and only if $ (x(u)\leqslant x(v)$ and $ y(u)\leqslant y(v))$.\
Then, for a subset $U$ of ${\mathbb{Z}}^2$, $\min(U)$ and $\max(U)$ are respectively the sets of *minimal* and *maximal* elements of $U$, for this product order on ${\mathbb{Z}}^2$.\
$\bullet$ If $M$ is a submonoid of $({\mathbb{R}}^n,+)$, we will denote : $$M^0 = \{ u\in M, -u\in M\} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}M^*=M\backslash M^0$$ $${\mathrm{Irr}}(M)=\{ u\in M^*, \forall v,v'\in M^*, u\not = v+v' \}$$
\[lem:1\] let $\psi:{\mathbb{Z}}^p \to {\mathbb{R}}^n$ be an additive map ( $n$ and $p$ are positive integers) and $M$ be a submonoid of $\psi({\mathbb{Z}}^p)$. Then : $$(\psi^{-1}(M))^0=\psi^{-1}(M^0) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}(\psi^{-1}(M))^*=\psi^{-1}(M^*) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\mathrm{Irr}}(\psi^{-1}(M))=\psi^{-1}({\mathrm{Irr}}(M))$$
first two results are well known and almost obvious. Let $u\in{\mathrm{Irr}}(\psi^{-1}(M))$. We denote $m=\psi(u)$. Suppose that $m=n+n'$, with $n,n'\in M^*$. Then, we have $v,v'\in \psi^{-1}(M), w\in \ker(\psi)$, such that $u=v+v'+w$ and $\psi(v)=n,\psi(v')=n'$. But, if we denote $v"=v'+w$, then $u=v+v"$ and $\psi(v")=n'$. So, $v,v"\in (\psi^{-1}(M))^*$ : contradiction with our hypothesis on $u$. So, $m\in {\mathrm{Irr}}(M)$.\
Conversely, let $m\in {\mathrm{Irr}}(M)$ and $u\in \psi^{-1}(M^*)$, such that $\psi(u)=m$. Suppose that $u=v+v'$, with $v,v'\in (\psi^{-1}(M))^*$, then $m=n+n'$, where $n=\psi(v)$ and $n'=\psi(v')$. So, $n,n'\in M^*$ : contradiction with our hypothesis on $m$. So, $u\in {\mathrm{Irr}}(\psi^{-1}(M))$.
$\bullet$ A subset $U$ of ${\mathbb{Z}}^2$ is said to be *$w$-invariant* if and only if $U+w=U$.
\[lem:2\] if $U$ is a $w$-invariant subset of ${\mathbb{Z}}^2$, for $w\in{\mathbb{Z}}^2$, then $\min(U),\max(U),{\mathbb{Z}}^2\backslash U$ are $w$-invariant. ( then $\max({\mathbb{Z}}^2\backslash U)$ is also $w$-invariant)
Suppose that $w+U=U$. Let $u\in\min(U)$ and $u'=u+w$, then $u'\in U$, since $U$ is $w$-invariant. If there is $u"\in U$ such that $u"\leqslant u'$, then $v=u"-w\in U$ ( $U$ is $w$-invariant) and $v\leqslant u$. So that, $v=u$ since $u$ is minimal in $U$. Then $u"=u'$ and $u'$ is minimal in $U$. This proves that $w+\min(U)\subset \min(U)$. But $U$ is also $(-w)$-invariant, then $-w+\min(U)\subset \min(U)$ and finally $\min(U)$ is $w$-invariant. The other results are proven in the same way.
Representations in ${\mathbb{Z}}^2$ {#sec:representation}
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$\bullet$ one of the simple ideas of this paper is to use Lemma \[lem:1\] in the particular case, when $M$ is a numerical semigroup ( so $n=1$) and $p=2$ : so we have a numerical semigroup $S$ and a *representation* $\psi$ of $S$ in ${\mathbb{Z}}^2$, that is a map $\psi : {\mathbb{Z}}^2\to {\mathbb{R}}$, such that $S\subset \psi({\mathbb{Z}}^2)$.\
Hence, if we denote $M=\psi^{-1}(S)$, we just have to study ${\mathrm{Irr}}(M)$ to easily deduce results on ${\mathrm{Irr}}(S)$. For that aim, we will consider minimal elements of $M^*$, for the product order on ${\mathbb{Z}}^2$. So, we need to have simple relations between ${\mathrm{Irr}}(M)$ and $\min(M^*)$.\
**Question :** for what kind of submonoids $M$ of $({\mathbb{Z}}^2,+,\leqslant)$, do we have : ${\mathrm{Irr}}(M)=\min(M^*)$ ?\
We can give an elementary exemple : ${\mathrm{Irr}}({\mathbb{N}}^2)=\min(({\mathbb{N}}^2)^*)=\{ (1,0);(0,1)\}$.\
We will not answer the question, but only give a partial result, that will be sufficient for our purpose.
\[lem:3\] let $L$ be a sublattice of ${\mathbb{Z}}^2$ and $w\in L$, such that $w,-w\not\in {\mathbb{N}}^2$.\
Then, for $M=(L\cap {\mathbb{N}}^2)+w{\mathbb{Z}}$, we have : $$M^0=w{\mathbb{Z}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\mathrm{Irr}}(M)=\min(M^*)$$
First, $w{\mathbb{Z}}\subset M^0$. On the other hand, if $m\in M^0$, then $m=m'+k'w$ and $-m=m"+k"w$, where $m',m"\in L\cap{\mathbb{N}}^2$ and $k',k"\in{\mathbb{Z}}$. Then, $m'+m"=jw$, where $j\in {\mathbb{Z}}$. So, $jw \geqslant 0$ : contradiction with our hypothesis on $w$, if $j\not = 0$. Hence, $j=0$ and $m'+m"=0$. We deduce that $m'=m"=0$, for $m',m"\in{\mathbb{N}}^2$. So, $m\in w{\mathbb{Z}}$.\
We remark that $M\subset L$, since $w\in L$.\
Let $u\in {\mathrm{Irr}}(M)$, then $u\in M^*$. We suppose that there exists $v\in M^*$ such that $u>v$. Then $v'=u-v >0$ and $v'\in L$, for $u,v\in L$. So, $v'\in M^*$, for $M^0=w{\mathbb{Z}}$ and $w,-w\not\in {\mathbb{N}}^2$ : no elements of $M^0$ is positive. So $u=v+v'$, with $v,v'\in M^*$ : contradiction with our hypothesis on $u$. So, $u\in \min(M^*)$.\
Conversely : if $u\in \min(M^*)$. Suppose that $u=v+v'$ and $v,v'\in M^*$. Then, we can find $m,m'\in L\cap ({\mathbb{N}}^2)^*$ and $k,k'\in {\mathbb{Z}}$, such that : $v=m+kw$ and $v'=m'+k'w$. Now, we denote $v"=v'+kw$ and we have $u=m+v"$, where $m>0$ and $v"\in M^*$, for $v'=v" $ ( mod $M^0$). So, $u>v"$ : contradiction with our hypothesis on $u$. So, $u\in {\mathrm{Irr}}(M)$.
$\bullet$ For $a$ and $b$ two coprime positive integers, we define two maps by : $${\varphi}: \begin{cases} {\mathbb{Z}}^2\to {\mathbb{Z}}\\ (x,y)\to ax+by \end{cases} {\hspace*{0.2cm}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\hspace*{0.2cm}}\psi :\begin{cases} {\mathbb{Z}}^2 \to (1/d) {\mathbb{Z}}\\ (x,y)\to (ax+by)/d\end{cases}$$
These maps are additive group morphisms. They are strictly increasing, surjective and their kernel is $w{\mathbb{Z}}$, where $w=(-b,a)$. Let $S=<a,b>$ be the numerical semigroup generated by $a$ and $b$ and $S'=\frac{S}{d}$. We denote $L$ the lattice defined by $L=\{ (x,y)\in{\mathbb{Z}}^2, ax+by=0 $ ( mod $d$)$\}$. We will prove in Lemma \[lem:5\] : $${\varphi}^{-1}(S)={\mathbb{N}}^2+w{\mathbb{Z}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}\psi^{-1}(S')=L\cap ({\mathbb{N}}^2+w{\mathbb{Z}})= (L\cap {\mathbb{N}}^2)+w{\mathbb{Z}}$$ So, this representation of $\frac{<a,b>}{d}$ in ${\mathbb{Z}}^2$ is of the above type and will allow us ( see Lemma \[lem:1\] and \[lem:3\]) to claim that : $$\psi^{-1}\left({\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)\right)=\min(L\cap (({\mathbb{N}}^2)^*+w{\mathbb{Z}}))$$ This is detailed in this section and in \[subsec:irreducible\].
a lattice-stair case shape representation
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$\bullet$ Let $S$ be a numerical semigroup and $d$ a positive integer. We have then : $$d ( S/d)=\{ s\in S, s=0 ( \text{ mod } d )\}=S\cap d{\mathbb{Z}}$$ Let $a$ and $b$ be two coprime integers. With the two maps defined just above ( previous page), we have : $$\psi^{-1}(S/d)={\varphi}^{-1}( d ( S/d))={\varphi}^{-1}(S) \cap L$$ where $L$ is the lattice : $$L=\{ (x,y)\in{\mathbb{Z}}^2, ax+by =0 ( \text{ mod } d )\}={\varphi}^{-1}(d{\mathbb{Z}})=\psi^{-1}({\mathbb{Z}})$$ and :
$$\psi^{-1}({\mathbb{N}})=L\cap H_+ {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\psi^{-1}(G(S/d))=(H_+\backslash {\varphi}^{-1}(S))\cap L$$ where $H_+$ denotes the halfplane $\{ (x,y)\in{\mathbb{Z}}^2, ax+by\geqslant 0\}$.\
$\bullet$ Now we suppose that $d$ is coprime with $a $ and $b$. With elementary arithmetic, we find that, in every square $[x,x+d[\times [y,y+d[$, $L$ has exactly one point on every line and one point on every column of ${\mathbb{Z}}^2$.\
$L$ is generated by $w=(b,-a), v_2=(0,d)$ and $v_1=(d,0)$ ( but it is not a basis) : indeed these three vectors are in $L$ and since $a$ and $d$ are coprime, for every $y$ in ${\mathbb{Z}}$, we can find $x\in{\mathbb{Z}}$ such that $(x,y)\in w{\mathbb{Z}}+v_2{\mathbb{Z}}$. So with the former remark, we obtain, adding multiples of $v_1$, all the points of $L$.\
$\bullet$ We will study the simplest case, namely when $S$ is generated by two coprime integers $a,b$. We begin with some remarks.
\[lem:4\].\
(i) the general case for $\frac{<a,b>}{d} $ is when $a,b,d$ are pairwise coprime. Indeed, for any integer $d'$ coprime with $b$ : $$\frac{<d'a,b>}{dd'}=\frac{<a,b>}{d}$$ (ii) if $d$ divides $f(<a,b>)=ab-a-b$, then $\frac{<a,b>}{d}$ is symmetric and $f(\frac{<a,b>}{d})=\frac{f(<a,b>)}{d}$.
\(i) let $k$ be an integer : $$kdd' \in <d'a,b> \Leftrightarrow \exists x,y\in{\mathbb{N}}, kdd'=xd'a+yb \Leftrightarrow \exists x,z\in{\mathbb{N}}, kd=xa+zb \Leftrightarrow kd\in <a,b>$$ as if $d'$ divides $yb$, then it divides $y$, because $d'$ and $b$ are coprime.\
(ii) we denote $f=\frac{f(<a,b>}{d}$ and suppose that $f$ is an integer. Then $df\not\in <a,b>$, so $f$ is a gap of $\frac{<a,b>}{d}$. In addition, $dn\in <a,b>$ for all integer $n$ such that $n>f$, because $dn> f(<a,b>)$. So, $f=f(\frac{<a,b>}{d})$.\
If $n$ is a gap of $\frac{<a,b>}{d}$, then $dn$ is a gap of $<a,b>$, so $df-dn\in <a,b>$, for $<a,b>$ is symmetric. Hence, $f-n\in \frac{<a,b>}{d}$. Conversely, if $s\in \frac{<a,b>}{d}$ and $s<f$, then $f-s$ is a gap of $\frac{<a,b>}{d}$. So, $\frac{<a,b>}{d}$ is symmetric.
Thus, we will suppose that $d$ is a positive integer, coprime with $a$ and $b$. We will also suppose that $d\not = a $ and $d\not = b$. Indeed : if, more generally, $d\in <a,b>$ then $kd\in <a,b>$ for all non negative integer $k$ and then $\frac{<a,b>}{d} ={\mathbb{N}}$.\
$\bullet$ The representation of $S=<a,b>$ and $S'=S/d$ are simple : $S$ is represented in ${\mathbb{Z}}^2$ by a “ full stair-case shape” and $S'$ by its intersection with the lattice $L$.\
This is proven by the following Lemma :
\[lem:5\] let $a,b,d$ be three positive pairwise coprime integers.\
Let $S=<a,b>$ and $S'=\frac{<a,b>}{d}$. We denote $w=(b,-a)$.\
Let $T$ be the triangle, “ upper-half” of the rectangle ${ \{ 1,..,b-1 \} }\times { \{ -(a-1),..,-1 \} }$ : $$T=\{ (x,y)\in{\mathbb{Z}}^2, x\in { \{ 1,..,b-1 \} }, y\in { \{ -a+1,..,-1 \} }, ax+by>0 \}$$
\(i) $({\mathbb{N}}^2+w{\mathbb{Z}}, T+w{\mathbb{Z}})$ is a partition of $H_+=\{ (x,y)\in{\mathbb{Z}}^2, ax+by\geqslant 0\}$.\
(ii) $${\varphi}^{-1}(S)= {\mathbb{N}}^2+w{\mathbb{Z}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\varphi}^{-1}(G(S))= T+w{\mathbb{Z}}$$
\(iii) $$\psi^{-1}(S')=(L\cap {\mathbb{N}}^2)+w{\mathbb{Z}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}\psi^{-1}(G(S'))= ( L\cap T )+w{\mathbb{Z}}$$ $\psi$ is bijective from $L\cap T$ to $G(S')$.
\(i) if $u\in ({\mathbb{N}}^2+w{\mathbb{Z}}) \cap (T+w{\mathbb{Z}})$, then we have $u=u_1+k_1w=u_2+k_2w$, where $u_1\in T,u_2\in{\mathbb{N}}^2,k_1,k_2\in{\mathbb{Z}}$. So $u_1-u_2=kw$, where $k\in {\mathbb{Z}}$. Note $u_1=(x_1,y_1)$ and $u_2=(x_2,y_2)$. We have $x_1\in { \{ 1,..,b-1 \} }, x_2\in{\mathbb{N}}, y_1\in { \{ -a+1,..,-1 \} }, y_2\in{\mathbb{N}},k\in{\mathbb{Z}}$ such that : $x_1-x_2=kb$ and $y_2-y_1=ka$. So $ka>0$, thus $k\geqslant 1$ and $x_2<0$ : contradiction... This proves that $ ({\mathbb{N}}^2+w{\mathbb{Z}}) \cap (T+w{\mathbb{Z}})=\emptyset$.\
On the other hand, if $(x,y)\in H_+$, then $\frac{x}{b}\geqslant -\frac{y}{a}$. We consider two cases :\
Case 1 : if $\lfloor x/b \rfloor = \lfloor -y/a \rfloor$ and $x/b,y/a$ are not integers, then if we denote $k$ this common integer, we have : $x=kb+x'$ and $y=-ka+y'$ , with $(x',y')\in T$. , so $(x,y)\in T+w{\mathbb{Z}}$.\
Case 2 : if $x/b=-y/a$ and are integers, then $(x,y)\in w{\mathbb{Z}}\subset({\mathbb{N}}^2+w{\mathbb{Z}}) $.\
Case 3 : if $\lfloor x/b \rfloor > \lfloor -y/a \rfloor$, then if we denote $k=\lfloor x/b \rfloor$, we have $x=kb+x'$ and $y=-ka+y'$, with $(x',y')\in {\mathbb{N}}^2$, so $(x,y)\in {\mathbb{N}}^2+w{\mathbb{Z}}$.\
(ii) We have ${\varphi}({\mathbb{N}}^2)=S$ and use (i) for $G(S)$.\
(iii) with the general remarks at the beginning of the section and the fact that $L$ is $w$-invariant : $$\psi^{-1}(S')=L\cap ({\mathbb{N}}^2+w{\mathbb{Z}})=(L\cap {\mathbb{N}}^2)+w{\mathbb{Z}}$$ the same arguments rules for $G(S')$.\
Now, $\psi$ is bijective from $T\cap L$ to $G(S')$, since for every $u\in T, k\in {\mathbb{Z}}^*, u+kw\not\in T$.
$\bullet$ For now in this section, we adopt the following hypothesis and notations :\
- we suppose that $a,b,d$ are pairwise coprime.\
- we denote $L'=L\backslash \{ (0,0)\}$ and $S'=\frac{<a,b>}{d}$.\
- the symbol $A\overset{\psi} {\simeq}B$ means that $A$ and $B$ are in bijective correspondence via $\psi$.\
- we use the usual product order on ${\mathbb{Z}}^2$ ( partial order).
\[lem:6\].\
(i) $$S'\backslash \{ 0 \} \overset{\psi} {\simeq} ({\mathbb{N}}\times { \{ 0,..,a-1 \} })\cap L'$$ (ii) $${\mathrm{Irr}}(S') \subset \psi(\min(({ \{ 0,..,d \} }\times { \{ 0,..,a-1 \} })\cap L')$$ (iii) $$d<b \Rightarrow {\mathrm{Irr}}(S')\overset{\psi} {\simeq} \min(({ \{ 0,..,d \} }\times { \{ 0,..,a-1 \} })\cap L')$$
**Remark :** ${\mathrm{Irr}}(S')$ is the minimal set of $S'\backslash \{ 0 \} $ for the order $\leqslant_S$ on ${\mathbb{Z}}$. The result (i) would suggest that, since $\psi$ is increasing, then ${\mathrm{Irr}}(S')$ is via $\psi$ in one to one correspondence with $\min(E)$, where $E= ({\mathbb{N}}\times { \{ 0,..,a-1 \} })\cap L'$..., but this is not true in general, because $\psi$ is not an order isomorphism on these sets. We can have $u,u'\in E$, with $\psi(u)\leqslant_{S'} \psi(u')$ and $u\not \leqslant u'$ : it is the case if $u'\geqslant u+w$ and $y(u')<y(u)$.\
— (i) $\psi$ is injective on $({\mathbb{N}}\times { \{ 0,..,a-1 \} })\cap L'$, because for two different elements $u,u'$ of this set, $u'-u \not\in w{\mathbb{Z}}$ ( as $| x(u'-u) |<a$). Moreover, let $u\in ({\mathbb{N}}^2\cap L')+w{\mathbb{Z}}$, then : $$u=kw+u' \text{ with } k=-\left\lfloor \frac{y(u)}{a}\right\rfloor \text{ and } y(u')\in { \{ 0,..,a-1 \} }$$ in addition, $u=jw+u"$ for some integer $j$ and $u"\in {\mathbb{N}}^2$, so : $u'=u"+(j-k)w$. We deduce $y(u')\geqslant (k-j)a$, so $k-j\leqslant 0$, and $x(u')\geqslant (j-k)b\geqslant 0$. Then, $u'\in ({\mathbb{N}}\times { \{ 0,..,a-1 \} })\cap L$. In addition, $u\not\in w{\mathbb{Z}}$, so $u"\not\in w{\mathbb{Z}}$, hence $u'\not = (0,0)$.\
— (ii) Let $s\in {\mathrm{Irr}}(S')$ and $u\in ({\mathbb{N}}\times { \{ 0,..,a-1 \} })\cap L'$ such that $s=\psi(u)$ ( see (i)). If $x(u)>d$, then $u=(d,0)+u'$ with $u'\in ({\mathbb{N}}\times { \{ 0,..,a-1 \} })\cap L'$, so $s=a+s'$, with $s'=\psi(u')\in S'$ and $s'\not = 0$, which is in contradiction with $s \in {\mathrm{Irr}}(S')$, since $a\in S'$. We have proven that $x(u)\leqslant d$, and so that $u\in ({ \{ 0,..,d \} }\times { \{ 0,..,a-1 \} })\cap L'$.\
Now, we want to prove that $u$ is minimal in $M=({ \{ 0,..,d \} }\times { \{ 0,..,a-1 \} })\cap L'$. If it is not, then we have $u'\in M$, such that $u'<u$. Let $u"=u-u'$, then $u"\in L'$ and $0\leqslant x(u')\leqslant x(u) \leqslant d$, so $x(u")\in { \{ 0,..,d \} }$, and with similar arguments, $y(u")\in{ \{ 0,..,a-1 \} }$. So $u"\in M$ and $s=s'+s"$, with $s'=\psi(u'),s"=\psi(u")\in S'\backslash \{ 0 \}$. It is impossible since $s\in {\mathrm{Irr}}(S')$.\
— (iii) Reversely : let $u\in \min(M)$ and $s=\psi(u)$. Then, $s\in S'$ and $s\not = 0$. We would like to prove that $s$ is irreducible in $S'$ : if it was not, then we would have $s_1$ and $s_2$ in $S'$ such that $s=s_1+s_2$ and $s_1,s_2$ non null. Let $u_1$ and $u_2$ be the unique elements in $({\mathbb{N}}\times { \{ 0,..,a-1 \} })\cap L'$ ( see (i)), such that $s_1=\psi(u_1)$ and $s_2=\psi(u_2)$. Then, $\psi(u)=\psi(u_1+u_2)$ and $u_1+u_2\in ({\mathbb{N}}\times { \{ 0,..,2a-2 \} })\cap L'$. So, $u=u_1+u_2$ or $u=u_1+u_2+w$. In the last case, we have $x(u)\geqslant b$, so that, if $d<b$, we obtain a contradiction. In that case $d<b$, we can conclude that $u=u_1+u_2$ and then $u_1,u_2\in M$, since they have non negative $x$. Then, $u$ is not minimal in this set, as $u_1>0$ and $u_2>0$ : contradiction. So $\psi(S')=\min(M)$ and we conclude with (i).
relations between invariants of $\frac{<a,b>}{d}$ and a lattice in a rectangle {#subsec:relations}
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$\bullet$ We begin with the genus $g(S')$. At the beginning of the previous subsection, we have seen in Lemma 5 that $T\cap L$ and $G(S')$ ( see notations in this Lemma) are in bijective correspondence via $\psi$. We will use another property with the following Proposition :
\[prop:1\] let $R_0$ be the rectangle : ${ \{ 1,..,b-1 \} }\times { \{ -(a-1),..,-1 \} }$ and $L$ the lattice $L=\{ (x,y)\in {\mathbb{Z}}^2, ax+by=0[d] \}$. $$g\left(\frac{<a,b>}{d}\right)= \frac{1}{2}\#(L\cap R_0)$$
The above remark proves that $g(S')= \# (T\cap L)$. In addition, the symetry of ${\mathbb{Z}}^2$ , $\sigma: (x,y)\to (b-x,-a-y)$ carries $T$ onto $T'$ such that $(T,T')$ is a partition of $R_0$.\
Indeed, $T=\{ (x,y)\in R_0, ax+by>0 \}$, so $T'=\{ (x,y)\in R_0, ax+by<0 \}$. Moreover, there is no point $(x,y)$ in $R_0$ such that $ax+by=0$, since $a$ and $b$ are coprime. So, the result is proven.
$\bullet$ Next proposition is our main result to express minimal generators of $\frac{<a,b>}{d}$ :
\[prop:2\] let $L$ be the lattice $L=\{ (x,y)\in {\mathbb{Z}}^2, ax+by=0[d] \}$.\
$\blacktriangleright$ Case 1 : if $d<a<b$, then $a,b\in {\mathrm{Irr}}\left(\frac{<a,b>}{d}\right) $ and : $${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)\backslash\{a,b\}\overset{\psi} {\simeq}\min(L \cap { \{ 1,..,d-1 \} }^2)$$ $\blacktriangleright$ Case 2 : if $a<d<b$, then $a\in {\mathrm{Irr}}\left(\frac{<a,b>}{d}\right),b\not\in {\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)$ and : $${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)\backslash\{a\}\overset{\psi} {\simeq} \min(L \cap ({ \{ 1,..,d-1 \} }\times{ \{ 1,..,a-1 \} }))$$ $\blacktriangleright$ Case 3 : if $a<b<d$, then : $${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)\overset{\psi} {\simeq} \min(L\cap { \{ 1,..,x_1 \} }\times { \{ 0,..,a-1 \} })$$ where $x_0=\min\{ x(u), u\in ({\mathbb{N}}\times { \{ 1,..,a-1 \} })\cap L \}$ and $x_1=\min(d,x_0+b-1)$.\
so $a\in{\mathrm{Irr}}\left(\frac{<a,b>}{d}\right) \Leftrightarrow x_0>d-b$.
**Remark :** if $d>\max(a,b)$ and $x_0+b>d$, then the result of Case 3 is the same as for Case 2.\
We can summarize all these cases as follows : there exists two positive integers $c$ and $c'$ such that : ${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)\overset{\psi} {\simeq} \min(L'\cap { \{ 0,..,c \} }\times { \{ 0,..,c' \} } )$, where $L'= L\backslash \{ (0,0)\}$.\
For case 1, $c=c'=d$, for case 2, $c=d$ and $c'=a-1$, for case 3, $c=x_1$ and $c'=a-1$.\
We use Lemma \[lem:6\] (iii).\
$\blacktriangleright$ Case 1 : if $d<a<b$ : then, $(d,0)$ and $(0,d)\in \min(L' \cap ({ \{ 0,..,d \} }\times{ \{ 0,..,a-1 \} }))$ ( remind that $L$ has exactly one point on every ${\mathbb{Z}}$-line or row of length $d$), so $a,b\in {\mathrm{Irr}}(S')$ and we have also $ {\mathrm{Irr}}(S')\backslash\{a,b\}\overset{\psi} {\simeq}\min(L \cap { \{ 1,..,d-1 \} }^2)$, since $(0,d)$ and $(d,0)$ are not comparable with all elements of this set. Moreover, if $u\in { \{ 0,..,d \} }\times { \{ d,..,a-1 \} }$, then $(0,d)<u$.\
$\blacktriangleright$ Case 2 : if $a<d<b$ : then $(d,0)\in \min(L' \cap ({ \{ 0,..,d \} }\times{ \{ 0,..,a-1 \} }))$, but $(0,d)$ is not in this set ! We conclude in the same way as above.\
$\blacktriangleright$ Case 3 : if $a<b<d$ : this case is more intricate.\
— we have seen in Lemma \[lem:6\] (ii), that : if $s\in {\mathrm{Irr}}(S')$, then $s=\psi(u)$, with $u\in \min(({ \{ 0,..,d \} }\times { \{ 0,..,a-1 \} })\cap L')$.\
We have $x(u)\geqslant x_0$, with the definition of $x_0$, for if $y(u)=0$, then $u=(d,0)$ and $x_0<d$ ( see property of $L$ again). Let us prove that “$x(u)\leqslant x_0+b-1$ ”. If $x(u)\geqslant x_0+b$, we denote $u'=u-w$, so $u_0\leqslant u'$, where $u_0=(x_0,y_0)\in L' \cap ({ \{ 0,..,d \} }\times{ \{ 0,..,a-1 \} })$. So, if we denote $s_0=\psi(u_0)$, we obtain $s_0\leqslant_{S'} s=\psi(u')$. But, $s_0\in S'$ and $s_0\not = 0$ : contradiction with $s\in {\mathrm{Irr}}(S')$ !\
So, $u\in ({ \{ x_0,..,x_1 \} }\times { \{ 0,..,a-1 \} })\cap L)$ and is minimal in it, since it is minimal in a bigger set.\
— on the other hand, suppose that $u\in \min( ({ \{ x_0,..,x_1 \} }\times { \{ 0,..,a-1 \} })\cap L)$ and $s=\psi(u)$, then $s\in S'$ and $s\not=0$. Suppose that $s=s_1+s_2$, with $s_1$ and $s_2$ in $S'$ and $s_1,s_2$ non null. Let $u_1$ and $u_2$ be the unique elements in $({\mathbb{N}}\times { \{ 0,..,a-1 \} })\cap L'$, such that $s_1=\psi(u_1)$ and $s_2=\psi(u_2)$. Then, $\psi(u)=\psi(u_1+u_2)$ and $u_1+u_2\in ({\mathbb{N}}\times { \{ x_0,..,2a-2 \} })\cap L'$. So, $u=u_1+u_2$ or $u=u_1+u_2+w$. In the last case, we have $x(u)\geqslant b+x_0>x_1$ : contradiction. So, $u=u_1+u_2$ and $u$ is not minimal in $({ \{ x_0,..,x_1 \} }\times { \{ 0,..,a-1 \} })\cap L)$, since $u_1,u_2$ are in this set and are not null ( $x_0\leqslant x(u_1),x(u_2)$ and $x(u_1)+x(u_2)=x(u)\leqslant x_1$).\
We have proved that : if $u\in \min( ({ \{ x_0,..,x_1 \} }\times { \{ 0,..,a-1 \} })\cap L)$, then $\psi(u)\in {\mathrm{Irr}}(S')$, so the result is obtained with Lemma \[lem:6\] (i), since $\psi$ is bijective over this set.\
To finish : $(0,d)\not\in { \{ x_0,..,x_1 \} }\times { \{ 0,..,a-1 \} })\cap L)$ and $(d,0)\in { \{ x_0,..,x_1 \} }\times { \{ 0,..,a-1 \} })\cap L$ if and only if $x_1\geqslant d$. In that case, it is minimal in this set !
$\bullet$ Now, in the same way as for ${\mathrm{Irr}}(S')$, the set ${\mathrm{PF}}(S')$ can be related with maximal points of a certain rectangle of $L$ :
\[lem:7\] we denote $R$ the rectangle of ${\mathbb{Z}}^2$ defined by : $ R= { \{ b-d,..,b-1 \} }\times { \{ -d,..,-1 \} } $. We denote the numerical semigroup $S=\frac{<a,b>}{d}$, the lattice $L=\{ (x,y)\in {\mathbb{Z}}^2, ax+by=0[d] \}$ and the triangle $T=\{ (x,y)\in{\mathbb{Z}}^2, x\in { \{ 1,..,b-1 \} }, y\in { \{ -a+1,..,-1 \} }, ax+by>0 \} $. Then : $${\mathrm{PF}}(S)\overset{\psi} {\simeq}\max(L\cap R)\cap T$$ We can precise if we distinguish several cases :\
$\blacktriangleright$ Case 1 : if $d<a<b$ : $${\mathrm{PF}}(S) \overset{\psi} {\simeq} \max(L\cap R)$$ $\blacktriangleright$ Case 2 : if $a<d<b$ : $${\mathrm{PF}}(S) \overset{\psi} {\simeq} \max(L\cap { \{ b-d,..,b-1 \} }\times { \{ -(a-1),..,-1 \} })\cap T$$ $\blacktriangleright$ Case 3 : if $a<b<d$ : $${\mathrm{PF}}(S) \overset{\psi} {\simeq} \max(L\cap { \{ 1,..,b-1 \} }\times { \{ -(a-1),..,-1 \} })\cap T$$
**Remark :** this result will be improved in next subsection, after we have proved that these maximal points are in $T$ ( see Proposition 4). But now, we can use the following argument : let $C$ be a square of ${\mathbb{Z}}^2$, say $C={ \{ x_0,..,x_0+c-1 \} }\times \inter{y_0,y_0+c-1}$, and $E$ a subset of $C$, that contains exactly one point on every row and one point on every line. Then, minimal points of $E$ are all under ( or on) the diagonal $x+y=x_0+y_0+c-1$ : these points $(x,y)$ verify : $x+y\leqslant x_0+y_0+c-1$. The proof is obvious, with reducio ad absurdum for instance.\
We will use similar arguments as in proof of the previous result. As a reminder : $G(S')\overset{\psi} {\simeq}L\cap T$ ( see Lemma \[lem:5\] (iii)).\
— Let $s\in {\mathrm{PF}}(S')$, then $s\in G(S')$, so we have $u\in L\cap T$ such that $s=\psi(u)$.\
We claim : $u\in L\cap R$ : indeed, if $u\not\in L\cap R$ ( only possible if $d<\max(a,b)$), then $u+(d,0)$ or $u+(0,d)\in L\cap T$ and so $s+a$ or $s+b\in G(S')$, which contradicts our hypothesis on $s$. Suppose that $u$ is not maximal in $L\cap R$, then we have $u'\in L\cap R$ such that $u<u'$. Let $u"=u'-u$, then $u"\in {\mathbb{N}}^2\cap L'$ so $s"=\psi(u")\in S'$. But, $s"\not = 0$, since $u">0$. Now $s+s"=\psi(u')\not\in S'$ : contradiction with our hypothesis on $s$.\
— Conversely, let $u\in \max(L\cap R)\cap T$ and $s=\psi(u)$. Then $u\in L\cap T$, so $s\in G(S')$. Suppose that $s\not\in {\mathrm{PF}}(S')$, then we have $s'\in S'$, such that $s'\not = 0$ and $s"=s+s'\not\in S'$. Let $u'\in L'\cap {\mathbb{N}}^2$, such that $\psi(u')=s'$ and $u"\in L\cap T$, such that $\psi(u")=s"$ ( indeed $s"\in G(S')$, for $s">0$). Then, $\psi(u+u')=\psi(u")$, so $u+u'=u"+kw$, for some integer $k$. With first coordinate in ${\mathbb{Z}}^2$, we obtain : $$1\leqslant x(u+u')=x(u"+kw)<(k+1)b$$ so $k\geqslant 0$. With second coordinate, we obtain : $$-a<y(u+u')=y(u"+kw)< -ak$$ so $k\leqslant 0$. We deduce : $k=0$ and so $u<u"$, since $u'>0$ : it is in contradiction with our hypothesis.\
We have proved the first assertion.\
$\blacktriangleright$ Case 1 : if $d<a<b$ : with the remark above, $\max(R\cap L)$ is a subset of the upper-half triangle of the square $R\cap L$. Then this triangle is a subset of $T$ and the result is proven.\
$\blacktriangleright$ Case 2 : if $a<d<b$ : let denote $R'={ \{ b-d,..,b-1 \} }\times { \{ -(a-1),..,-1 \} })$. Then, $\max(R\cap L)\cap T\subset \max(R'\cap L)\cap T $, for $R\cap T\subset R'\subset R$. On the other hand, no element of $R\backslash R'$ is greater than any element of $R'$ ( look at $y$), so the result is proven.\
$\blacktriangleright$ Case 3 : if $a<b<d$ : same arguments as in Case 2 with $R"= { \{ 1,..,b-1 \} }\times { \{ -(a-1),..,-1 \} })$.
minimal points of a lattice in a rectangle {#subsec:minimal}
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$\bullet$ Proposition \[prop:2\] and Lemma \[lem:5\] are about minimal points of a lattice in a rectangle ( in ${\mathbb{R}}^2$ with the product order). Indeed, if we use an “half turn symmetry” in ${\mathbb{R}}^2$, such as $u\to v-u$, then maximal points are turned into minimal points and vice versa...
\[lem:8\] Let $L$ be a lattice of ${\mathbb{R}}^2$ and $R={ \{ x_0,..,x_0+c \} }\times { \{ y_0,..,y_0+c' \} }\backslash \{ (x_0,y_0)\}$, where $x_0,y_0$ are two reals and $c,c'$ are two positive reals.\
if $(u_k)_{k\in { \{ 1,..,e \} }}$ is $x$-monotonous or $y$-monotonous and describes $\min(L\cap R)$, then :\
(i) both $(x(u_k))_k$ and $(y(u_k))_k$ are monotonous : one is increasing and the other is decreasing.\
(ii) $$\forall i\in{ \{ 2,..,e-1 \} }, u_{i+1}+u_{i-1} \geqslant 2u_i$$
that is to say : $(u_i-u_{i-1})_i$ is an increasing sequence in ${\mathbb{Z}}^2$ ( with the product order).\
(iii) we have equality in (ii) for a fixed $i\in { \{ 2,..,e-1 \} }$ if and only if $2u_i-u_{i-1} \in R$.
**Remark 1 :** we could name *order-convex* the property (ii).\
**Remark 2 :** the curve connecting the $(u_k)_k$ is the “ under border” of the convex hull of $L\cap R$ in ${\mathbb{R}}^2$.\
**Remark 3 :** (iii) means that $(u_i-u_{i-1})_i$ is constant as long as possible : the curve connecting the $(u_k)_k$ remains straight as long as it does not get out of the rectangle.\
\(i) if only one of $(x(u_k))_k$ and $(y(u_k))_k$ is monotonous, then we can find $i$ such that $u_i$ and $u_{i+1}$ are comparable, which is not possible !\
(ii) we remark that the conclusion is unchanged if we reverse the indexation of the sequence, so that we can suppose that $(x(u_k))_k$ is decreasing. Thus, $(y(u_k))_k$ is increasing.\
Let $i\in{ \{ 2,..,e-1 \} }$ and we denote $v_i=u_{i+1}+u_{i-1}-u_i$. Then, $v_i\in L\cap R$, for : $$x(u_{i+1})< x(v_i)< x(u_{i-1}) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}y(u_{i-1})<y(v_i)<y(u_{i+1})$$ So, there exists $k\in{ \{ 1,..,e \} }$, such that $u_k\leqslant v_i$. But, the previous inequalities show that $k$ can not be different from $i$, so $u_i\leqslant v_i$...q.e.d.\
(iii) if we have equality, then $2u_i-u_{i-1}=u_{i+1}\in R$. Conversely : if $2u_i-u_{i-1}\in R$, then if we denote $t_i$ that element, we have $u_{i+1}\geqslant t_i$ ( see proof of (ii)) and $t_i\in L\cap R$. But $u_{i+1}$ is minimal in $L\cap R$, so $u_{i+1}=t_i$.
$\bullet$ Now, we are able to propose a statement that improves Lemma \[lem:7\] :
\[prop:3\] $a,b,d$ are three pairwise coprime positive integers.\
We denote $\psi: {\mathbb{Z}}^2\to (1/d){\mathbb{Z}}; (x,y)\to (ax+by)/d$ and $L=\{ (x,y), ax+by=0[d] \}=\psi^{-1}({\mathbb{Z}})$.\
The symbol $\overset{\psi} {\simeq}$ means that both sets are in bijective correspondance via $\psi$. $${\mathrm{PF}}\left(\frac{<a,b>}{d}\right) \overset{\psi} {\simeq} \max(L\cap { \{ b-\delta,..,b-1 \} }\times { \{ -\delta',..,-1 \} })$$ where $\delta=\min(d,b-1)$ and $\delta'=\min(d,a-1)$.
we denote $R_1= { \{ b-\delta,..,b-1 \} }\times { \{ -\delta',..,-1 \} })$. If we compare this statement with Lemma \[lem:7\], we have just to show that $\max(L\cap R_1) \subset T$, with notations of Lemma \[lem:7\]. This has been done in the case $d<\min(a,b)$.\
If $a<d<b$, then $R_1={ \{ b-d,..,b-1 \} }\times { \{ -(a-1),..,-1 \} }$. We denote $R_0={ \{ b-d,..,b \} }\times { \{ -a,..,-1 \} }$ and we claim that $\max(L\cap R_1)=\max(L\cap R_0)\backslash\{ (b,-a)\}$. Indeed, $(b,-a)$ is a point of $L$ and it is the only one in $R_0$ on the row $y=b$. So, $(b,-a)\in \max(L\cap R_0)$ and $\max(L\cap R_1)\subset\max(L\cap R_0)$, since there is no point in $L\cap (R_0\backslash R_1)$ that is greater than a point of $R_1$. Our equality is proven. By now, our former Lemma proves ( via an half-turn ) that $\max(L\cap R_0)$ can be enumerated by an “order-concave” sequence of points : $\max(L\cap R_0)= \{ u_k , k\in{ \{ 0,..,t \} }\}$, such that $u_0=(b,-a)$ and $(u_k-u_{k-1})_k$ is decreasing. We deduce that : $\max(L\cap R_1)=\{ u_k , k\in{ \{ 1,..,t \} }\}\subset T$, for $u_t=(x_1,-1)$ the unique point of $L$ that lies on the ${\mathbb{Z}}^2$-segment of length $d$ : $x\in { \{ b-d,..,b-1 \} }, y=-1$.\
If $d>\max(a,b)$, then $R_1={ \{ 1,..,b-1 \} }\times { \{ -(a-1),..,-1 \} }$. We use similar arguments as in previous case with $R_0={ \{ 0,..,b-1 \} }\times { \{ -a,..,0 \} }$, as we remark that $(0,0)$ and $(b,-a)$ are points of $L$, that are maximal points of $L\cap R_0$ ( no point of $L$ on the ${\mathbb{Z}}^2$-segment $x\in { \{ 1,..,b \} },y=0$). Moreover, no point of $R_1$ is greater than one of these two points.
$\bullet$ In order to study the minimal generators of $\frac{<a,b>}{d}$, we consider a particular case of Lemma \[lem:8\] :
\[lem:9\].\
let $L$ be a lattice of ${\mathbb{R}}^2$ and $R={ \{ 0,..,c \} }\times { \{ 0,..,c' \} }\backslash \{ (0,0) \}$, where $c,c'$ are positive reals.\
We can describe $\min(L\cap R)$ by a sequence $(u_i)_{i\in{ \{ 1,..,e \} }}$, such that $(u_1,u_2)$ is a ${\mathbb{Z}}$-basis of $L$ ( if $e\geqslant 2$), $(x(u_i))_i,(y(u_i))_i$ are monotonous and : $$\forall i\in{ \{ 2,..,e-1 \} },\exists k_i\in{ \{ 2,..,+\infty \} }, u_{i+1}+u_{i-1}=k_iu_i {\hspace*{0.2cm}}(1)$$ moreover : $$\forall i\in{ \{ 2,..,e-1 \} },k_i= \min(n\in{\mathbb{N}}, nu_i\geqslant u_{i-1}) {\hspace*{0.2cm}}(2)$$
Let us consider $(u_k)_{k\in { \{ 1,..,e \} }}=\min(L\cap R)$, such that $(x(u_k))_k$ is decreasing and $e=\#(\min(L\cap R))$. Then, $(y(u_k))_k$ is increasing ( see arguments of the proof of the previous Lemma).\
If $e\geqslant 2$, then $(u_1,u_2)$ is a ${\mathbb{Z}}$-basis of $L$, because there is no point of $L$ in the interior ( in ${\mathbb{R}}^2$, or ${\mathbb{Q}}^2$) of the triangle based on $(0,u_1,u_2)$, hence by symmetry $u\to u_1+u_2-u$, there are no points in the interior of the parallelogram based on $(0,u_1,u_2,u_1+u_2)$.\
Now, let us prove the induction formula. Let $i\in{ \{ 2,..,e-1 \} }$. Let $j$ be the grestest integer such that $u_{i+1}+u_{i-1}\geqslant ju_i$ and $\rho=u_{i+1}+u_{i-1}- ju_i$. Then, $\rho\geqslant (0,0)$ and $j\geqslant 2$, according to the previous Lemma. If $\rho=(0,0)$, our result is proven. Else, we have $x(\rho)<x(u_i)$ or $y(\rho)<y(u_i)$. Suppose that $x(\rho)<x(u_i)$. We also have : $$y(\rho)\leqslant y(u_{i+1})+y(u_{i-1})-2y(u_i)< y(u_{i+1})-y(u_i)<y(u_{i+1})$$ Hence, $\rho\in L\cap R$ and it exists $k\in{ \{ 1,..,e \} }$, such that $u_k\leqslant \rho$. But $x(\rho)<x(u_k)$ if $k\leqslant i$ and $y(\rho)<y(u_k)$ if $k>i$, so this is impossible. We use similar arguments if $y(\rho)<y(u_i)$, proving that $x(\rho)<x(u_{i-1})$.\
Conclusion : $\rho=(0,0)$ and our formula is proven.\
We end this proof with the expression of $k_i$ as a minimum. Let $i\in { \{ 1,..,e-1 \} }$. As $u_{i+1}\geqslant 0$, then $k_iu_i\geqslant u_{i-1}$. If there exists a positive integer $n<k_i$ such that $nu_i\geqslant u_{i-1}$, then : we denote $w=nu_i-u_{i-1}$ , so $u_{i+1}=w+(k_i-n)u_i>w$. So, $0\leqslant w < u_{i+1}$ and $w\in L$. In addition, $w\not =0$, since $u_{i-1}\not = nu_i$ ( this would imply $u_i\leqslant u_{i-1}$). Hence, $w\in R\cap L$ and $w< u_{i+1}$ : contradiction.
**Remark :** with hypothesis of previous Lemma, we can also conclude that $(u_i,u_{i+1})$ is a ${\mathbb{Z}}$-basis of $L$ if $i\in{ \{ 1,..,e-1 \} }$ ( obvious induction with formula (1)).\
This has a direct consequence on ${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)$, when we use Proposition \[prop:2\] :\
Let $a,b,d$ be three pairwise coprime positive integers, then we can find a finite sequence of positive integers $(m_k)_{k\in { \{ 1,..,e \} }}$, such that $m_1$ and $m_2$ are coprime, ${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)=\{ m_k, k\in{ \{ 1,..,e \} }\}$ and : $$\forall i\in{ \{ 2,..,e-1 \} }, \exists k_i\in{ \{ 2,..,+\infty \} },m_{i+1}+m_{i-1}=k_im_i$$
As proven in **\[9\]**, the converse is also true, if in addition we suppose that $m_i\not\in <m_k,k\not = i >$ for all $i\in{ \{ 1,..,e \} }$ ! We will explore that in next section.
The reverse problem {#sec:reverse}
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$\bullet$ Let us consider the set $A$ of triplets of positive integers $(a,b,d)$ such that $1<a<b$ and $a,b,d$ are pairwise coprime. We also consider the following map : $$\mathcal{I} : \begin{cases} A \to {\mathcal{P}}({\mathbb{N}}) \\
(a,b,d) \to {\mathrm{Irr}}\left(\frac{<a,b>}{d}\right) \end{cases}$$
In **\[9\]**, we can find a precise description of the range of $\mathcal{I} $ : it is the set $B$ of finite subset of ${\mathbb{N}}$, that can be enumerated by a sequence $(n_k)_{k\in{ \{ 1,..,e \} }}$ , such that : $$\begin{cases} (i) {\hspace*{0.2cm}}\forall k\in{ \{ 1,..,e \} },n_k\not\in < n_i,i\not = k > \\
(ii) {\hspace*{0.2cm}}\gcd(n_1,n_2)=1 \\
(iii){\hspace*{0.2cm}}\forall i\in{ \{ 2,..,e-1 \} }, \exists c_i\in{\mathbb{N}}, n_{i-1}+n_{i+1}-2n_i=c_in_i \end{cases}$$ We have proved half of this result at the end of previous section. We will prove the other half, giving the preimage of each element of $B$.\
**Remark 1 :** we mention that $< P >$ is the ${\mathbb{N}}$-span of $P$ in a monoid, so (i) could be named “ ${\mathbb{N}}$-independance of the $(n_i)_i$” and is obvious for a minimal set of generators of a semigroup. This implies that the $(n_k)_k$ are pairwise distincts.\
Property (ii) could be generalized ( as a consequence of (iii)) : $$\forall i\in{ \{ 1,..,e-1 \} }, \gcd(n_i,n_{i+1})=1$$ Property (iii) implies that $(n_i)_i$ is convex and that, in a sense, “the second derivative of $(n_i)_i$” is a ( integral) multiple of $(n_i)_i$ : if we denote $\underline{\delta}(n)=(n_i-n_{i-1})_i$ and $\overline{\delta}(n)=(n_{i+1}-n_i)_i$ then $$\underline{\delta}\overline{\delta}(n) =\overline{\delta}\underline{\delta}(n)=c\times n$$ where $c$ is a ${\mathbb{N}}$-valued sequence.\
**Remark 2 :** If the sequence $(n_k)_{k\in{ \{ 1,..,e \} }}$ satisfies conditions (i),(ii) and (iii), then, for every $k_1,k_2\in{ \{ 1,..,e \} }$, such that $k_1\leqslant k_2$, the sequence $(n_k)_{k\in{ \{ k_1,..,k_2 \} }}$ satisfies (i),(ii) and (iii).\
**Question :** given such a finite subset $I$ of ${\mathbb{N}}$, how many indexations of its elements do exist such that the above conditions (i),(ii),(iii) are satisfied ?\
If $I$ contains more that one element, then it exists as least two indexations : $(n_k)_{k\in{ \{ 1,..,e \} }}$ and $(n_{e+1-k})_{k\in{ \{ 1,..,e \} }}$. In some cases, there exist other indexations : for example, if $I=\{ 7,11,59 \}$, we have 4 indexations of $I$ that satisfy (i),(ii) and (iii) : $$(59,11,7) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}(7,11,59) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}(11,7,59) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}(59,7,11)$$ As a consequence, we can find 2 triplets $(a,b,d)$ of pairwise coprime integers such that $d<a<b$ and ${\mathrm{Irr}}(\frac{<a,b>}{d})= I$ ( see Thoerem \[theo:1\] in 3.4) : $${\mathrm{Irr}}\left(\frac{<7,59>}{6}\right)= \{ 7,11,59 \} = {\mathrm{Irr}}\left(\frac{<11,59>}{10}\right)$$
modular-convex sequences {#subsec:modular}
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$\bullet$ Let us denote $\Delta$ the operator of discrete second derivative of ${\mathbb{R}}^n$-valued sequences ( $n\in{\mathbb{N}}^*$) : $$\Delta : u \to v \text{ such that } \forall i, {\hspace*{0.2cm}}v_i=u_{i-1}+u_{i+1}-2u_i$$ Then, we have $\Delta= \overline{\delta}\underline{\delta}=\underline{\delta} \overline{\delta}$, with notations of remark 1 above.\
In 2.4 and Lemma \[lem:9\], we have proved that the minimal set of a lattice of ${\mathbb{R}}^2$ in a rectangle $R=({ \{ 0,..,c \} }\times { \{ 0,..,c' \} })\backslash \{ (0,0)\}$ can be enumerated by a sequence $u$, satisfying the relation :
$$\Delta(u)=c\times u$$ where $c$ is a sequence valued in ${\mathbb{N}}$. In other words : $$\forall i,\exists c_i\in{\mathbb{N}}, {\hspace*{0.2cm}}u_{i+1} +u_{i-1}=(c_i+2)u_i$$
We also have a $\mathrm{SL}_2({\mathbb{Z}})$-matricial version : $$\forall i, \exists d_i \in { \{ 2,..,+\infty \} },{\hspace*{0.2cm}}\begin{pmatrix} u_{i+1} \\ u_i \end{pmatrix} = \begin{pmatrix} d_i & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} u_{i} \\ u_{i-1} \end{pmatrix}$$
For convenience, we will name such sequences with the very unperfect term “ modular-convex”, for this mix of unimodular matrices and convexity-like property :
\[def:1\] a ${\mathbb{R}}^n$ valued sequence $(u_i)_{i\in J}$ is **modular-convex** if and only if it exists a ${\mathbb{N}}$-valued sequence $(c_i)_{i\in\overset{\sim}{J}}$, such that $\Delta(u)=c\times u$. ( $J$ is an interval of integers and $\overset{\sim}{J}$ is $J$ without its finite extremities )
**Remark 1 :** a numerical modular-convex sequence is not always convex. But, we have a simple result : if $(x_i)_{i\in{ \{ 0,..,r \} }}$ is a numerical modular-convex sequence such that $x_1\geqslant \max(0,x_0)$, then it is convex. Indeed, by obvious induction, we prove that $x_i$ is non negative for $i>0$ and $(x_i-x_{i-1})_{i\geqslant 1}$ is non decreasing and non negative.\
$\bullet$ Now, we take interest in these sequences and will show that they are fully given by three terms : first, second and last one. In addition, the coefficients $(d_i)_i$ are related with some continued fraction development :
\[prop:4\] let $u,v,w \in {\mathbb{R}}^n$ such that $u,v$ are linearly independant ( so $n\geqslant 2$) $$\exists r\in{\mathbb{N}}^*, \exists (u_i)_{i\in{ \{ 0,..,r \} }} \text{ modular-convex, such that } \begin{cases} u_0=u \\ u_1=v \\ u_r=w \end{cases} \Leftrightarrow \exists p,q \in{\mathbb{N}}, \begin{cases} 0<p<q \\ \gcd(p,q)=1 \\ w=-pu+qv \end{cases}$$ in that case, the sequence $(u_i)_i$ is unique and the integers $c=(c_i)_i$ such that $\Delta(u)=c\times u$ are given by the following algorithm, where we denote $d_i=c_i+2$ : $$t_0=\dfrac{p}{q }, {\hspace*{0.2cm}}\forall i\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}d_i=\left\lceil \frac{1}{t_i} \right\rceil {\hspace*{0.2cm}}; {\hspace*{0.2cm}}t_{i+1}=\left\{ - \frac{1}{t_i}\right\} \text{ and } t_r=0 \text{ ( stop condition )}$$
**Remark 2 :** the above coefficients $(d_i)_{i\in{ \{ 1,..,r-1 \} }}$ are the coefficients of the *ceiling continued fraction expansion* of $\frac{q}{p}$. That is to say : $$\frac{q}{p}= d_1-\dfrac{1}{d_2-\dfrac{1}{d_3-\cdots }}$$ Unlike the usual continued fraction algorithm, that is of quadratic complexity in terms of the binary size of inputs, this one is exponential in worst cases : indeed, if we take $q=n$ and $p=n-1$, then we obtain a sequence of $n-1$ digits all equal to 2, for the ceiling continued fraction expansion of $1/n$.\
There exists a relation between this two types of continued fraction expansion, but it is not obvious and will be explicited in \[subsec:CFE\].\
However, we can easily give some relation for *ceiling convergents* : we denote $\lceil d_1,d_2,\cdots,d_r \rceil$, the ceiling continued fraction expansion of $\frac{q}{p}$, as above. We also define, for $i\in{ \{ 1,..,r \} }$, the *ceiling convergent* $\frac{q_i}{p_i}=\lceil d_1,d_2,\cdots,d_i\rceil$. Hence : $$\forall i\in{ \{ 1,..,r \} },{\hspace*{0.2cm}}p_i=d_ip_{i-1}-p_{i-2}, {\hspace*{0.2cm}}q_i=d_iq_{i-1}-q_{i-2}$$ with $p_{-1}=-1, {\hspace*{0.2cm}}p_0=0, {\hspace*{0.2cm}}q_{-1}=0, {\hspace*{0.2cm}}q_0=1$. The proof of this result is similar to the proof for classical partial quotients...\
\(i) We suppose that we have a modular-convex sequence $(u_i)_{i\in{ \{ 0,..,r \} }}$ in ${\mathbb{R}}^n$ such that $u_0=u,u_1=v$ and $u_r=w$. Let us denote $(-x_i,y_i)$ the coordinates of $u_i$ in $(u,v)$, for it is clear that all $u_i$ are integer combinations of $(u,v)$. We also denote $(d_i)_{i\in{ \{ 1,..,r-1 \} }}$, such that : $$\forall i\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}u_{i+1}=d_iu_i-u_{i-1}$$ Then, all $d_i$ are integers greater than one. We have $x_0=-1,x_1=0,x_2=1$ and $y_0=0,y_1=1,y_2=d_1$. Yet, $(x_i)_i$ and $(y_i)_i$ are modular-convex and $x_1\geqslant \max(x_0,0)$ and $y_1\geqslant \max(y_0,0)$, so ( see Remark 1) they are convex. In addition, $x_1-x_0>0$ and $y_1-y_0>0$, so $(x_i)_{i\geqslant 0}$ and $(y_i)_{i\geqslant 0}$ are increasing sequences. Likewise, if we denote $\epsilon_i=y_i-x_i$ for $i\in{ \{ 0,..,r \} }$, then $(\epsilon_i)_i$ is modular-convex, so convex and non decreasing, since $\epsilon_0=\epsilon_1 > 0$. Thus, $\epsilon_i>0$ for all $i$. So, $0<p<q$ and half of (i) is proven.\
With same arguments, we prove that $(d_1x_i-y_i)_{i\geqslant 2}$ is a non negative and increasing sequence, for $d_1x_2-y_2=0$ and $d_1x_3-y_3=1$. Similarly, we prove that $(y_i-(d_1-1)x_i)_{i\geqslant 1}$ is a positive and increasing sequence, for $y_1-(d_1-1)x_1=1y_2-(d_1-1)x_2$. So : $$\forall i\in{ \{ 2,..,r \} }, {\hspace*{0.2cm}}d_1=\lceil y_i/x_i \rceil$$ Thus, in particular, $d_1= \lceil y_r/x_r \rceil = \lceil q/p \rceil$. Now, we can do the same with $(u_1,u_2)$ instead of $(u_0,u_1)$, for $u_1$ and $u_2$ are linearly independant and all $u_i, i\in{ \{ 1,..,r \} }$ are in the lattice generated by these two vectors, and so on...\
If we denote, for $k\in{ \{ 1,..,r \} },(-X_k,Y_k)$ the coordinates of $u_r$ in the basis $(u_{k-1},u_k)$ of the lattice $L$ generated by $(u_0,u_1)$, then : $X_1=x_r=p,Y_1=y_r=q$ and $X_r=0,Y_r=1$. We also have the following induction relations : $$\forall k\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}X_k=Y_{k+1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}Y_k=d_kY_{k+1}-X_{k+1}$$ We deduce, by obvious decreasing induction on $k$, that $\gcd(X_k,Y_k)=1$ for all $k$, so $\gcd(p,q)=1$.\
Now, we have proved that $d_1=\lceil Y_1/X_1 \rceil$ and more generally that $d_k=\lceil Y_k/X_k\rceil$ for all $k\in{ \{ 1,..,r-1 \} }$. So, if we denote $t_k= \frac{X_k}{Y_k}$, then $t_r=0$ and : $$\forall k\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}d_k=\left\lceil \frac{1}{t_k}\right \rceil {\hspace*{0.2cm}}; {\hspace*{0.2cm}}t_{k+1}=d_k-\frac{1}{t_k}=\left\{ -\frac{1}{t_k}\right\}$$
\(ii) Now, we suppose that $w=qv-pu$, with $p,q$ two coprime positive integers, such that $p<q$. We define a finite sequence $(d_i)_{i\in{ \{ 1,..,r-1 \} }}$, as we use the algorithm detailed in Proposition 4. Indeed, we obtain $t_r=0$ for a certain positive integer $r$, because if we denote $t_k=\frac{X_k}{Y_k}$ the reduced fraction ( for $k\geqslant 1$), then : $$\forall k\geqslant 1, {\hspace*{0.2cm}}0 \leqslant X_{k+1}<Y_{k+1}=X_k$$ for $t_{k+1}\in [0,1[$. Since, for all $k$, we have $t_k\in [0,1[$, then $1/t_k>1$, so $d_k\geqslant 2$. We define the sequence $(u_i)_{i\in{ \{ 1,..,r \} }}$ in $L$ by : $$u_0=u, {\hspace*{0.2cm}}u_1=v, {\hspace*{0.2cm}}\forall i\in { \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}u_{i+1}=d_iu_i-u_{i-1}$$ This sequence is modular-convex and $u_r=w$, according to (i) above.
irreducible elements of certain submonoids of ${\mathbb{Z}}$ or ${\mathbb{Z}}^2$ {#subsec:irreducible}
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$\bullet$ The following result is a complement to Lemma \[lem:9\] and is important for next subsection : as usual, we consider the product order on ${\mathbb{Z}}^2$ and denote $C^*={\mathbb{N}}^2\backslash \{ (0,0)\}$ the *positive cone*.
\[lem:10\] let $d$ and $m$ be two coprime integers such that $0<m<d$. Let $u_0=(d,0),u_1=(m,1)$ and $L=\mathrm{Span}_{{\mathbb{Z}}}(u_0,u_1)$. We denote $(d_i)_{i\in{ \{ 1,..,r-1 \} }}$ the coefficients of the ceiling continued development of $m/d$ and $(u_i)_{i\in{ \{ 0,..,r \} }}$ the modular-convex sequence defined by $u_0,u_1$ and $u_r=(0,d)$ ( see Proposition \[prop:4\], for $(0,d)=-mu_0+du_1$). Reminder : $(x(u_i))_i$ is decreasing and $(y(u_i))_i$ is increasing.\
Let $a,b$ be two integers such that $0<a<b$ and $w=(b,-a)\in L$. We denote ${\mathbf{\mathcal{S}}}^*=C^*+ w{\mathbb{Z}}$.\
(i) $$\min(L\cap C^*)= \{ u_i, i\in{ \{ 0,..,r \} }\}$$ (ii) $$\min(L\cap {\mathbf{\mathcal{S}}}^*)\subset \min(L\cap C^*)+w{\mathbb{Z}}$$ (iii)\
$\blacktriangleright $ Case 1 : if $d<a<b$ : $$\min(L\cap {\mathbf{\mathcal{S}}}^*)= \{ u_i, i\in{ \{ 0,..,r \} }\}+w{\mathbb{Z}}$$ $\blacktriangleright $ Case 2 : if $a<d<b$ : $$\min(L\cap {\mathbf{\mathcal{S}}}^*)= \{ u_i, i\in{ \{ 0,..,s \} }\}+w{\mathbb{Z}}$$ where $y(u_s)<a\leqslant y(u_{s+1})$.\
$\blacktriangleright $ Case 3 : if $d<a<b$ : $$\min(L\cap {\mathbf{\mathcal{S}}}^*)= \{ u_i, i\in{ \{ \sigma,..,s \} }\}+w{\mathbb{Z}}$$ where $y(u_s)<a\leqslant y(u_{s+1})$ and $x(u_{\sigma})<b+x(u_s)\leqslant x(u_{\sigma-1})$.
**Remark 1 :** as we have seen in Lemma \[lem:2\], $ \min(L\cap {\mathbf{\mathcal{S}}}^*)$ is $w$-invariant, since ${\mathbf{\mathcal{S}}}^*$ and $L$ are $w$-invariant. So, we can choose representatives of “classes modulo $w$” of minimal points in the strip : $${\mathcal{A}}=\{ (x,y), x\in{\mathbb{N}}, y\in { \{ 0,..,a-1 \} }\}$$ We remark that the $u_i$ that appear above are in ${\mathcal{A}}$.\
**Remark 2 :** we could summarize the above three cases in one formulation : $$\min(L\cap {\mathbf{\mathcal{S}}}^*)= \{ u_i, i\in{ \{ \sigma,..,s \} }\}+w{\mathbb{Z}}$$ where $y(u_s)<a\leqslant y(u_{s+1})$ and $x(u_{\sigma})<b+x(u_s)\leqslant x(u_{\sigma-1})$. With additional border conditions : $x_{-1}=y_{r+1}=+\infty$.\
\(i) we remark that for all $x,y\in{\mathbb{Z}},xu_0+yu_1=(xd+ym,y)$, so : on every “ horizontal segment” of ${\mathbb{Z}}^2$ of length $d$, there is a unique point of $L$. On the other hand, since $m$ and $d$ are coprime, $xd+ym=0$ only if $y$ is a multiple of $d$ : so on every “ vertical segment” of ${\mathbb{Z}}^2$ of length $d$, there is a unique point of $L$. We deduce that $u_0,u_1,u_r\in \min(L\cap C^*)$ and so $\min(L\cap C^*)=\min(L\cap R)$, with $R=({ \{ 0,..,d \} }^2)^*$.\
Now, according to Lemma \[lem:9\], $\min(L\cap R)$ can be enumerated by a modular-convex sequence $(v_i)_{i\in{ \{ 0,..,r' \} }}$ such that $(x(v_i))_i$ is decreasing and $(y(v_i)_i$ is increasing. So, $v_0=u_0,v_1=u_1$ and $v_{r'}=u_r$. Now, using Proposition \[prop:4\], we deduce that $r'=r$ and $v_i=u_i$ for all $i\in{ \{ 0,..,r \} }$.\
(ii) let $u\in \min(L\cap {\mathbf{\mathcal{S}}}^*)$ and $u\in {\mathcal{A}}$. Then, $u\in L\cap C^*$, since ${\mathcal{A}}\subset C^*$. If there exists $v\in L\cap C^*$ such that $v<u$, then $v\in L\cap {\mathbf{\mathcal{S}}}^*$, which contradicts the fact that $u$ is minimal in $L\cap {\mathbf{\mathcal{S}}}^*$. So, $u\in \min(L\cap C^*)$. We have proved that : $$\min(L\cap {\mathbf{\mathcal{S}}}^*)\cap {\mathcal{A}}\subset \min(L\cap C^*)$$ We deduce (ii) with Remark 1 above.\
(iii)\
$\blacktriangleright $ Case 1 : if $d<a<b$. Let $i\in{ \{ 0,..,r \} }$, then $u_i\in {\mathcal{A}}$. Moreover : $$0\leqslant x(u_i)\leqslant d<b {\hspace*{0.2cm}}\text{ and } {\hspace*{0.2cm}}0\leqslant y(u_i) \leqslant d <a$$ Now, we suppose that we have $v\in L\cap {\mathbf{\mathcal{S}}}^*$ such that $v<u_i$. Then, there exists an integer $k$, such that $v'=v-kw\in C^*$. But, $k$ can not be positive, for we would have $x(v')=x(v)-kb<0$ and $k$ can not be negative, for we would have $y(v')=y(v)+ka<0$. Hence $k=0$ and $v\in C^*$, which contradicts (i)...\
So, $u\in \min(L\cap {\mathbf{\mathcal{S}}}^*)$. We conclude with (ii).\
$\blacktriangleright $ Case 2 : if $a<d<b$. We use the same arguments as in Case 1, if $i\in{ \{ 0,..,s \} }$. If, $i\in{ \{ s+1,..,r \} }$, then $u_i\not\in{\mathcal{A}}$. In addition, $x(u_i)\geqslant 0$ and $y(u_i)\geqslant a$, so $u_i>(-b+d,a)=u_0-w$. But, $u_0-w\in L\cap {\mathbf{\mathcal{S}}}^*$, so $u_i\not\in \min(L\cap {\mathbf{\mathcal{S}}}^*)$.\
$\blacktriangleright $ Case 3 : if $a<b<d$. If $i\in { \{ s+1,..,r \} }$, then $u_i\not\in {\mathcal{A}}$. If $i\in { \{ 0,..,\sigma -1 \} }$, let $v=u_s+w\in L\cap {\mathbf{\mathcal{S}}}^*$. Then $v<u_i$, for $x(v)=b+x(u_s)\leqslant x(u_i)$ and $y(v)=y(u_s)-a<0\leqslant y(u_i)$. So $u_i\not\in \min(L\cap {\mathbf{\mathcal{S}}}^*)$.\
Now, let $i\in { \{ \sigma,..,s \} }$. We suppose that there exists $v\in L\cap {\mathbf{\mathcal{S}}}^*$ such that $v<u_i$. Then, we claim that $v\in C^*$. Indeed : we have an integer $k$ such that $v'=v-kw\in {\mathcal{A}}$.\
—– if $k<0$, then $y(v)=y(v')-ka\geqslant a$, which contradicts $v<u_i$, for $y(u_i)<a$.\
—– if $k>0$, then $$0\leqslant x(v')\leqslant x(v)-kb\leqslant x(u_i)-b<x(u_s)$$ But, $v'\in L\cap C^*$, so there exists $j\in{ \{ 0,..,r \} }$, such that $v'\geqslant u_j$ ( see (i)). The above inequality proves that $j>s$ and so $y(v')\geqslant y(u_j)\geqslant a$, which contradicts $v'\in {\mathcal{A}}$.\
So, $v\in L\cap C^*$ and $v<u_i$ : it is impossible according to (i).
$\bullet$ the previous Lemma is important, because these minimal sets are related with minimal generators and irreducible elements ( see beginning of section 2.) in the following meaning :\
Let $M$ be a submonoid of $({\mathbb{R}}^n,+)$ such that $0$ is the unique invertible element of $M$. $\mathrm{Gen}(M)$ denotes the intersection of all sets of generators ( as a submonoid) of $M$. The elements of $\mathrm{Gen}(M)$ are the *minimal generators* of $M$.
\[lem:11\] let $L$ be a lattice of ${\mathbb{Z}}^2$. $$\min(L\cap ({\mathbb{N}}^2)^*)={\mathrm{Gen}}(L\cap {\mathbb{N}}^2)={\mathrm{Irr}}(L\cap {\mathbb{N}}^2)$$
we denote $C^*=({\mathbb{N}}^2)^*$.\
- “ ${\mathrm{Irr}}\subset \min $” : if $u\in L\cap C^*$ is not minimal in $L\cap C^*$, then there exists $v\in L\cap C^*$ such that $v<u$. Note $v'=u-v$, then $v'\in L\cap C^*$ and $u=v+v'$, so $u\not \in {\mathrm{Irr}}(L\cap {\mathbb{N}}^2)$.\
- “ $\min \subset {\mathrm{Gen}}$” : if $m\in \min(L\cap C^*)$ and $(u_k)_{k\in I}$ is a system of generators of $L\cap {\mathbb{N}}^2$, say $L\cap {\mathbb{N}}^2=<u_k,k\in I>$. Then , $m=\sum_{k\in I}\alpha_ku_k$, where $\alpha_k\in{\mathbb{N}}$ for all $k$. Yet, $m\not = (0,0)$, thus there exists a $k\in I$, such that $\alpha_k\geqslant 1$ and so $m\geqslant u_k$, since all $u_i\geqslant (0,0)$. Yet $u_k\in L\cap C^*$, so $m=u_k$ and $m$ belongs to every set of generators of $L\cap {\mathbb{N}}^2$.\
- “ ${\mathrm{Gen}}\subset {\mathrm{Irr}}$” : suppose that $u\in L'\cap {\mathbb{N}}^2$ and $u\not\in {\mathrm{Irr}}(L\cap {\mathbb{N}}^2)$. Then, we can find $u_1$ and $u_2$ in $L'\cap {\mathbb{N}}^2$, such that $u=u_1+u_2$. So that we can replace $u$ by $(u_1,u_2)$ in any system of generators of $L\cap {\mathbb{N}}^2$ that would contain $u$ : $u\not\in {\mathrm{Gen}}(L\cap {\mathbb{N}}^2)$.
$\bullet$ We now state a result that will we improved later ( see Theorem \[theo:5\]). It gives an expression of ${\mathrm{Irr}}(\frac{<a,b>}{d})$ in terms of $a,b,d$ and is a consequence of Lemma \[lem:8\] and Proposition \[prop:2\].
\[prop:5\] let $a,b,d$ be three pairwise coprime positive integers such that $d\not \in <a,b>$ and $d>1$. We denote $m\in { \{ 1,..,d-1 \} }$, such that $am+b=0[d]$. We denote $\lceil d_1,\cdots,d_{r-1}\rceil$ the ceiling continued fraction expansion of $\frac{d}{m}$ ( see the algorithm of Proposition \[prop:4\]) and define sequences of integers, by double induction : $$y_0=0, y_1=1, {\hspace*{0.2cm}}\forall i\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}y_{i+1}=d_iy_i-y_{i-1}$$ $$x_0=d, x_1=m, {\hspace*{0.2cm}}\forall i\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}x_{i+1}=d_ix_i-x_{i-1}$$ in addition, we denote $y_{r+1}=x_{-1}=+\infty$. $$n_0=a, n_1=\frac{am+b}{d}, {\hspace*{0.2cm}}\forall i\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}n_{i+1}=d_in_i-n_{i-1}$$ Then $x_r=0,{\hspace*{0.2cm}}y_r=d,{\hspace*{0.2cm}}n_r=b,{\hspace*{0.2cm}}(x_i)_i$ is decreasing and $(y_i)_i$ is increasing. In addition :\
$\blacktriangleright$ Case 1 : if $d<a<b$ $${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)=\{ n_i, i\in { \{ 0,..,r \} } \}$$
$\blacktriangleright$ Case 2 : if $a<d<b$ $${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)=\{ n_i, i\in { \{ 0,..,s \} } \}$$ where $s$ is such that $y_s<a\leqslant y_{s+1}$.\
$\blacktriangleright$ Case 3 : if $a<b<d$ $${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)=\{ n_i, i\in { \{ \sigma,..,s \} } \}$$ where $s$ is such that $y_s<a\leqslant y_{s+1}$ and $\sigma$ is such that $x_{\sigma}<x_s+b\leqslant x_{\sigma-1}$.
If we denote $q_i/p_i$ the reduced fraction of $\lceil d_1,d_2,\cdots,d_i\rceil$ ( see the remark below Proposition \[prop:4\]), then, with obvious induction : $$\forall i \in{ \{ 0,..,r \} }, {\hspace*{0.2cm}}x_i=-dp_{i-1}+mq_{i-1}, {\hspace*{0.2cm}}y_i=q_{i-1}, {\hspace*{0.2cm}}n_i=-ap_{i-1}+\tau q_{i-1}$$ where $\tau=\frac{am+b}{d}$. So $x_r=-dp_{r-1}+mq_{r-1}=-dm+md=0, {\hspace*{0.2cm}}y_r=q_{r-1}=d$ and $n_r=-ap_{r-1}+\tau q_{r-1}=-am+\tau d =b$. Now, by obvious induction, $(y_i)_i$ is increasing and $(x_i)_i$ is decreasing.\
Let us consider now ( with the product order in ${\mathbb{Z}}^2$) : $$L=\{ (x,y)\in{\mathbb{Z}}^2, ax+by=0[d] \} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}M=\min(L\cap {\mathbb{N}}^2\backslash \{ (0,0)\} )$$ But $(d,0)$ and $(0,d)$ are in $M$, so $M=\min (L\cap (({ \{ 0,..,d \} })^2)^*)$. So, according to Lemma \[lem:7\], $M$ can be enumerated by a finite modular-convex sequence $(u_i)_{i\in{ \{ 0,..,r \} }}$ such that $(x(u_i))_i$ is decreasing and $(y(u_i))_i$ is increasing. We have $u_0=(d,0)$ and $u_r=(0,d)$. Moreover, $(m,1)\in L$, so $(m,1)\in M$ and $u_1=(m,1)$. Now, $u_r=-mu_0+du_1$ and $0<m<d$. In addition, $\gcd(m,d)=1$, since $\gcd(b,d)=1$. With Proposition \[prop:4\], $(u_i)_i$ satisfy the induction relation : $$\forall i\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}u_{i+1}=d_iu_i-u_{i-1}$$ So, $x_i=x(u_i)$ and $y_i=y(u_i)$ for $i\in{ \{ 0,..,r \} }$. We deduce : $n_i=\psi(u_i)$ for $i\in{ \{ 0,..,r \} }$, where $\psi : (x,y)\to \frac{ax+by}{d}$.\
At the end, results for $ {\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)$ are direct consequences of Proposition \[prop:2\] ...
${\mathbb{N}}$-independance of sequences of integers
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$\bullet$ Now, we are almost ready to give the solutions of the “ reverse problem”, that is to say : to provide all triplets of coprime positive integers $a,b,d$ such that $0<a<b$ and ${\mathrm{Irr}}(\frac{<a,b>}{d})$ is a given finite subset of ${\mathbb{N}}$. At the beginning of this section, we have reminded the necessary conditions for a finite subset $I$ of ${\mathbb{N}}$ to be the set of minimal generators of some $\frac{<a,b>}{d}$. One of them is of course, that elements of $I$ are ${\mathbb{N}}$-independant. Let us precise this notion and give a useful criterion :\
First, we remind a notation : for a subset $A$ of a monoid $M$, $<A>$ denotes the submonoid of $M$ generated by $A$.
\[def:2\] elements of a subset $A$ of a monoid $M$ are *${\mathbb{N}}$-independant* if and only if : $$\forall a\in A, a\not\in <A\backslash \{ a \}>$$
**Remark :** we will name also “ $A$ is ${\mathbb{N}}$-independant” such a definition.\
For our purpose, we deal with the particular case of the monoid ${\mathbb{N}}$. Let $A$ be a subset of ${\mathbb{N}}$. We claim that : $$A \text{ is ${\mathbb{N}}$-independant } \Leftrightarrow {\mathrm{Irr}}(<A>)=A$$
**Question :** given a finite set $A$ of positive integers, do we have an efficient algorithm to know if $A$ is ${\mathbb{N}}$-independant or not ?\
It is difficult to answer this question in general, but in the particular case when $A$ can be enumerated by a modular-convex sequence, we have a simple criterion, stated in Proposition \[prop:6\]. But, first, we need an arithmetic Lemma, that will be also useful later ( see Theorem \[theo:2\] and \[theo:3\]) :
\[lem:12\] let $a,b,c$ be three non null integers such that $b$ and $c$ are coprime.\
There exists an infinity of positive integer $k$, such that $a$ and $kc-b$ are coprime.
for any prime divisor $p$ of $a$, we consider : $$K_p=\{ k\in{\mathbb{Z}}, kc=b[p]\}$$ if $c=0[p]$, then $K_p$ is empty, since $b\not =0[p]$ ( $b$ and $c$ are coprime). Else, $K_p$ is a class of congruence modulo $p$, say : $K_p=\{ k\in{\mathbb{Z}},k=r_p[p]\}$. So, if we denote ${\mathcal{P}}(a)$ the finite set of prime divisor of $a$ that does not divide $c$ : $$\forall k\in{\mathbb{Z}}, ( \gcd(a,kc-b)=1 \Leftrightarrow \forall p\in {\mathcal{P}}(a) , k\not = r_p [p] )$$ With chinese remainder Lemma, we obtain that $k$ is such that $kc-b$ and $a$ are coprime if and only if $k$ take certain values modulo $q=\prod_{p\in{\mathcal{P}}(a)}p$.
Now, we propose a result that gives a criterion for a modular-convex sequence of positive integers to be ${\mathbb{N}}$-independant, but also the description of ${\mathrm{Irr}}(<A>)$ for $A$ enumerated by such a sequence.\
We will use what we have named the “ ceiling continued fraction expansion ” of a rational $x$, denoted $\lceil d_1,d_2,\cdots,d_r \rceil$, as well as $\left(\frac{q_i}{p_i}\right)_{i>0}$ the finite sequence of “ ceiling partial quotients” of $x$, defined by ( see \[subsec:modular\]): $$p_{-1}=-1,p_0=0, {\hspace*{0.2cm}}\forall i\in{ \{ 1,..,r \} }, p_i=d_ip_{i-1}-p_{i-2}$$ $$q_{-1}=0,q_0=1, {\hspace*{0.2cm}}\forall i\in{ \{ 1,..,r \} }, q_i=d_iq_{i-1}-q_{i-2}$$
\[prop:6\] let $(n_i)_{i\in{ \{ 0,..,r \} }}$ be a modular-convex sequence of positive integers such that $n_0$ and $n_1$ are coprime. We denote $(d_i)_i$ the sequence of integers such that : $$\forall i\in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}d_i\in{\mathbb{N}}, {\hspace*{0.2cm}}d_i\geqslant 2, {\hspace*{0.2cm}}n_{i+1}=d_in_i-n_{i-1}$$ We also denote $\frac{q_i}{p_i}$ the reduced fraction of $\lceil d_1,d_2,\cdots,d_i \rceil$ for all $i\in{ \{ 1,..,r-1 \} }$ ( see \[subsec:modular\]). More explicitly, $(q_i)_{i\in{ \{ -1,..,r-1 \} }}$ is the following sequence :\
$$q_{-1}=0, q_0=1, {\hspace*{0.2cm}}\forall i\in{ \{ 0,..,r-1 \} }, {\hspace*{0.2cm}}q_i=d_iq_{i-1}-q_{i-2}$$ (i) $$\{ n_i,i\in{ \{ 0,..,r \} }\} \text{ is ${\mathbb{N}}$-independant } \Leftrightarrow q_{r-1}<\min(n_0,n_r)$$ (ii) $${\mathrm{Irr}}( < n_i,i\in{ \{ 0,..,r \} }>)= \{ n_i, i\in{ \{ \sigma,..,s \} } \}$$ where $s$ is such that $y_s<n_0\leqslant y_{s+1}$ and $\sigma$ is such that $x_{\sigma}<x_s+n_r\leqslant x_{\sigma-1}$, with $ x_{-1}= y_{r+1}=+\infty$ and : $$\forall i\in{ \{ 0,..,r \} }, {\hspace*{0.2cm}}y_i=q_{i-1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}x_i=p_{r-1}q_{i-1}-q_{r-1}p_{i-1}$$
**Remark 1 :** as we have already seen in \[subsec:modular\], the algorithm that computes the “ ceiling continued fraction expansion” of a fraction is of exponential complexity in worst cases. But here, the entries are the $(d_i)_i$, so computing $q_{r-1}$ is very fast...\
**Remark 2 :** again, we have already mentioned that, for a modular-convex sequence $(n_j)_{j\in{ \{ 0,..,r \} }} $, we have $\gcd(n_{i-1},n_{i})=\gcd(n_0,n_1)$ for all $i\in{ \{ 1,..,r \} }$, so $\gcd((n_j)_{j\in{ \{ 0,..,r \} }})=\gcd(n_0,n_1)$.\
A bit more general statement of Proposition 6 would propose the criterion : $$\{ n_i,i\in{ \{ 0,..,r \} }\} \text{ is ${\mathbb{N}}$-independant } \Leftrightarrow q_{r-1}<\dfrac{\min(n_0,n_r)}{\gcd(n_0,n_1)}$$
**Remark 3 :** The proof will appear to be an application of Lemma \[lem:10\]. It is quite direct, if the first and last term of the sequence are coprime, but more intricate when they are not. We will use an argument of limit in that last case...\
\(i) is a direct consequence of (ii), for $y_r=q_{r-1}=x_0$ and $x_r=0$. ( see below)\
(ii) let $u_0=(q_{r-1},0),u_1=(p_{r-1},1)$ and $(u_i)_{i\in{ \{ 0,..,r \} }}$ defined by : $$\forall i\in{ \{ 1,..,r-1 \} },u_{i+1}=d_iu_i-u_{i-1}$$ Then $x(u_i)=x_i$ and $y(u_i)=y_i$ for all $i\in{ \{ 0,..,r \} }$, by obvious induction. So, $u_r=(0,q_{r-1})$. We also remark that : $$\forall i\in{ \{ 0,..,r \} }, {\hspace*{0.2cm}}x_i=p_{r-1}q_{i-1}-q_{r-1}p_{i-1}= \begin{vmatrix} p_{r-1} & p_{i-1} \\ q_{r-1} & q_{i-1} \end{vmatrix}$$ We denote $a=n_0$ and $b=n_r$ and define a map $\psi$ from ${\mathbb{Z}}^2$ to $(1/q_{r-1}){\mathbb{Z}}$ : $$\forall x,y\in{\mathbb{Z}},{\hspace*{0.2cm}}\psi(x,y)=\frac{ax+by}{q_{r-1}}$$ We have $\psi(u_0)=n_0$ and $\psi(u_r)=n_r$. We claim that : $\psi(u_i)=n_i$ for all $i$. Indeed, if we denote $\epsilon_i=\psi(u_i)-n_i$ for all $i$, then $\epsilon_0=0$ and : $$\forall i\in{ \{ 1,..,r-1 \} },\epsilon_{i+1}=d_i\epsilon_i-\epsilon_{i-1}$$ Then, by obvious induction, $\epsilon_i=\epsilon_1y_i$ for all $i\in{ \{ 0,..,r \} }$. But, $\epsilon_r=0$ and $y_r=q_{r-1}$, so $\epsilon_1=0$.\
We deduce : $$\forall i\in{ \{ 0,..,r \} }, {\hspace*{0.2cm}}\psi(u_i)=n_i$$ Now, we denote $L=\mathrm{Span}_{{\mathbb{Z}}}(u_0,u_1)$. With Lemma \[lem:10\] and \[lem:11\], we obtain : $${\mathrm{Gen}}(L\cap{\mathbb{N}}^2)=\min(L\cap ({\mathbb{N}}^2)^*)=\{ u_i,i\in { \{ 0,..,r \} }\}$$ Let $S=<n_i, ,i\in { \{ 0,..,r \} }>$. Then, $S=\psi(L\cap {\mathbb{N}}^2)$.\
We also remark that, if we denote $w=(b,-a)$, then $w=n_1u_0-au_1 \in L$, for $n_1=\frac{ap_{r-1}+b}{q_{r-1}}$.\
$\blacktriangleright$ Case 1 : if $\gcd(n_0,n_r)=1$. Then, $\ker(\psi)=w{\mathbb{Z}}$, so : $$\psi^{-1}(S)= ( L\cap {\mathbb{N}}^2)+w{\mathbb{Z}}= L\cap ({\mathbb{N}}^2+w{\mathbb{Z}})$$ So with Lemma \[lem:1\] and \[lem:3\], $\psi^{-1}({\mathrm{Irr}}(S))=\min(L\cap {\mathbf{\mathcal{S}}}^*)$, where ${\mathbf{\mathcal{S}}}^*=({\mathbb{N}}^2)^*+w{\mathbb{Z}}$. We conclude with Lemma \[lem:10\].\
$\blacktriangleright$ Case 2 : if $\gcd(n_0,n_r)>1$. According to Lemma \[lem:12\], there exist an infinite number of integer $d_r$, such that $d_r\geqslant 2$ and $n_0,n_{r+1}$ are coprime, if we denote $n_{r+1}=d_rn_r-n_{r-1}$. For these values of $n_{r+1}$, the sequence $(n_j)_{j\in{ \{ 0,..,r+1 \} }}$ satisfy the assumptions of Case 1. We denote : $$q_r=d_rq_{r-1}-q_{r-2} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}p_r=d_rp_{r-1}-p_{r-2}$$ So, we obtain : $${\mathrm{Irr}}( < n_i,i\in{ \{ 0,..,r+1 \} }>)= \{ n_i, i\in{ \{ \sigma,..,s' \} } \}$$ where $s'$ is such that $y'_{s'}<n_0<y'_{s'+1}$ and $\sigma$ is such that $x'_{\sigma}<x'_{s'}+n_{r+1}<x'_{\sigma-1}$, with $ x'_{-1}= y'_{r+2}=+\infty$ and $(x'_i)_i, ( y'_i)_i$ defined by : $$\forall i\in{ \{ -1,..,r+1 \} }, {\hspace*{0.2cm}}y'_i=q_{i-1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}x'_i=\begin{vmatrix} p_r & p_{i-1}\\ q_r & q_{i-1} \end{vmatrix}$$ So, $y'_i=y_i$ for all $i\in{ \{ 0,..,r \} }$, while $y_{r+1}=+\infty$ and $y'_{r+1}=q_r$. So, we just have to choose $d_r$ large enough so that $q_r >n_0$, to obtain $s'=s$ ( $s$ being defined as in Proposition \[prop:5\]). Again, we can choose $d_r$ large enough such that $n_{r+1}\in <n_0,n_1>$. Then, $ < n_i,i\in{ \{ 0,..,r+1 \} }>= < n_i,i\in{ \{ 0,..,r \} }>$.\
Now, we denote $(X_i)_i$ the sequence defined by : $$\forall i\in { \{ 0,..,r-1 \} }, {\hspace*{0.2cm}}X_i=\begin{vmatrix} p_{r-2} & p_{i-1}\\ q_{r-2} & q_{i-1} \end{vmatrix}$$ Then, we can write : $$\forall i\in{ \{ 0,..,r-1 \} },{\hspace*{0.2cm}}x'_i=tx_i-X_i {\hspace*{0.2cm}}\text{ where } t=d_r$$ We also have : $n_{r+1}=tn_r-n_{r-1}$. So, the conditions on $(x'_i)_i$, mentioned above, become : $$\begin{cases} n_{r-1} + X_s-X_{\sigma}< t(n_r+x_s-x_{\sigma}) \hspace{2.7cm} (1) \\
t(n_r+x_s-x_{\sigma -1} ) \leqslant n_{r-1}+X_s-X_{\sigma -1} \hspace{2cm} (2)\end{cases}$$ Since $t$ can be as large as we want, condition (1) is equivalent to : $$n_r+x_s-x_{\sigma}>0 \text{ or } \begin{cases} n_r+x_s-x_{\sigma}=0 \\ n_{r-1} + X_s-X_{\sigma}< 0 \end{cases}$$ Suppose now that $x_{\sigma}-x_s=n_r=b$. We have $u_{\sigma},u_s\in L$ and $w\in L $ ( see remark just before Case 1), so $v=u_{\sigma}-u_s-w \in L$. But $x(v)=0$, so $y(v)=kq_{r-1}$, for some integer $k$ ( there is only one point of $L$ on each vertical line of ${\mathbb{Z}}^2$ of length $q_{r-1}$). So, $n_{\sigma}-n_s=kn_r$ : contradiction with the fact that $n_s,n_{\sigma}\in {\mathrm{Irr}}( < n_i,i\in{ \{ 0,..,r \} }>)$. We have proved that $n_r+x_s-x_{\sigma}\not = 0$, so condition (1) and (2) ( that we treat with similar arguments) are equivalent ( for $t$ large enough) to : $$x_{\sigma} < x_s+n_r < x_{\sigma-1} \Leftrightarrow x_{\sigma} < x_s+n_r \leqslant x_{\sigma-1}$$
solutions {#subsec:solutions}
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$\bullet$ We remind our “ reverse problem” : if $r$ is a positive integer and $(n_i)_{i\in { \{ 0,..,r \} }}$ is a finite sequence of positive integers satisfying conditions (i),(ii),(iii) detailed at the beginning of section \[sec:reverse\], what are the triplets $(a,b,d)$ of coprime positive integers, such that $a<b$ and ${\mathrm{Irr}}(\frac{<a,b>}{d})=\{ n_i,i \in{ \{ 0,..,r \} } \}$ ?\
We also recall some notations : we have defined by induction in \[subsec:modular\] the notation $\lceil d_0,d_1,d_2,\cdots,d_{i}\rceil$, as follows : $$\lceil d_0,d_1,d_2,\cdots,d_{i}\rceil=d_0-\frac{1}{\lceil d_1,d_2,\cdots,d_{i}\rceil}$$ This is a variant of the usual convergents of continued fraction and has been named : “ ceiling convergents”. The $(d_i)_i$ are integers such that $d_i\geqslant 2$ for all positive $i$.\
To every real $\alpha$, we can associate such a sequence of $(d_i)_i$ which is infinite if $\alpha$ is irrational and finite else ( plus an infinite $(+\infty)$-tail). Then, $\alpha$ is the limit of the sequence of above “ ceiling convergents”. The algorithm to obtain these coefficients has been detailed for rationals in \[subsec:modular\] and is generalized in \[subsec:CFE\] to the case of irrationals.\
**Remark :** in the proof of next three Theorems, we will have integers $a,b,d,m,n_1$ such that $b=dn_1-am, \gcd(d,m)=1$ and $\gcd(a,n_1)=1$. We will deduce then : $$\gcd(a,b)=\gcd(a,d)=\gcd(b,d)$$
$\bullet$ First, we look for solutions $(a,b,d)$ such that $d<a<b$. In that case, the situation is very constraint and we obtain a finite number of solution ( one in general) :
\[ Reverse-problem, solutions $(a,b,d)$ such that $d<a<b$\]. \[theo:1\]\
let $I$ be a finite subset of ${\mathbb{N}}\backslash\{ 0,1\}$.\
$\blacklozenge$ (i) if $I$ is ${\mathbb{N}}$-independant and can be enumerated by a modular-convex sequence $(n_k)_{k\in{ \{ 0,..,r \} }}$ , such that : $ \gcd(n_0,n_r)=1 \text{ and } n_0<n_r $, then $I={\mathrm{Irr}}(\frac{<a,b>}{d})$, where $a=n_0,b=n_r$ and $d\in{ \{ 1,..,a-1 \} }$ such that $dn_1=b[a]$.\
$\blacklozenge$ (ii) there are as many triplets of coprime integers $(a,b,d)$ such that $1<d<a<b$ and $I={\mathrm{Irr}}(\frac{<a,b>}{d})$, as there are sequences of integers $(n_k)_k$ that verify conditions of (i).
\(i) we denote $a=n_0,b=n_r$ and $(d_i)_{i\in{ \{ 1,..,r-1 \} }}$ the sequence of integers such that : $$\forall i \in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}d_i\geqslant 2, {\hspace*{0.2cm}}n_{i+1}=d_in_i-n_{i-1}$$ Let $m,d$ be coprime positive integers such that : $$1-\frac{m}{d}=\lceil 1,d_1,d_2,\cdots,d_{r-1}\rceil {\hspace*{0.2cm}}\text{ that is : } {\hspace*{0.2cm}}\frac{d}{m}=\lceil d_1,d_2,\cdots,d_{r-1}\rceil$$ Since $I$ is ${\mathbb{N}}$-independant, we can use Proposition \[prop:6\] and claim that : $d<\min(a,b)$.\
In addition, by obvious induction ( see the proof of Proposition \[prop:5\], for example) : $$\forall i\in{ \{ 0,..,r \} },{\hspace*{0.2cm}}n_i=q_{i-1}n_1-p_{i-1}n_0 {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\text{ where } {\hspace*{0.2cm}}\frac{q_i}{p_i}=\lceil d_1,d_2,\cdots,d_{i}\rceil$$ so $b=n_r=dn_1-mn_0$ and $am+b=dn_1=0[d]$. In addition, $a$ and $b$ are coprime, so $a,b,d$ are pairwise coprime ( see remark above). Yet, $m\in{ \{ 1,..,d-1 \} }$ and $d\not\in <a,b>$, so we can apply Proposition \[prop:5\] ( Case 1) and we obtain our result.\
(ii) direct consequence of (i) and Proposition \[prop:5\] ( Case 1).
**Examples :** at the beginning of this section, we have seen an example of a ${\mathbb{N}}$-independant set of integers, namely $I=\{ 7,11,59\}$, that can be enumerated by two different modular-convex sequences verifying conditions of Theorem \[theo:1\] (i) : $(7,11,59)$ ( $d_1=6$) and $(11,7,59)$ ( $d_1=10$), so we have : $${\mathrm{Irr}}(\frac{<7,59>}{6})={\mathrm{Irr}}(\frac{<11,59>}{10})=\{ 7,11,59 \}$$ What about $I=\{ 10,17,24 \}$ ? $I$ is ${\mathbb{N}}$-independant ( esay to check) and is enumerated by the modular-convex sequence $(10,17,24)$ ( $d_1=2$), but $10$ and $24$ are not coprime. We also remark that $(17,10,24)$ is not modular-convex. So, there is no enumeration of $I$ by a sequence verifying conditions of (i) : for all triplets $(a,b,d)$ of pairwise coprime positive integers such that $d<a<b$, we can claim that $I\not = {\mathrm{Irr}}(\frac{<a,b>}{d})$.\
$\bullet$ Secondly, we consider solutions such that $a<d<b$. In that case, we can extend in one direction the sequence $(n_i)_i$ in a modular-convex sequence. We then obtain an infinite number of solutions :
\[ Reverse-problem, solutions $(a,b,d)$ such that $a<d<b$\]. \[theo:2\]\
let $I$ be a finite subset of ${\mathbb{N}}\backslash\{ 0,1\}$ that is ${\mathbb{N}}$-independant and can be enumerated by a modular-convex sequence $(n_k)_{k\in{ \{ 0,..,r \} }}$, such that $n_0$ and $n_1$ are coprime.\
We denote $(d_i)_{i\in{ \{ 1,..,r-1 \} }}$ the sequence of integers such that : $$\forall i \in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}d_i\geqslant 2, {\hspace*{0.2cm}}n_{i+1}=d_in_i-n_{i-1}$$ $\blacklozenge$ (i) Let $\rho$ be an integer such that $\rho>r$ and $(d_i)_{i\in { \{ r,..,\rho-1 \} }}$ a sequence of integers bigger than $1$.\
We denote $d$ and $m$ the coprime positive integers such that $ \frac{d}{m}=\lceil d_1,d_2,\cdots,d_{\rho-1}\rceil$ and $\frac{q_i}{p_i}$ the reduced fraction of $\lceil d_1,d_2,\cdots,d_i\rceil$ for all $i\in{ \{ 1,..,\rho-1 \} }$.\
We denote $a=n_0$ and $b=dn_1-mn_0$. Then :
$$\begin{cases} \gcd(a,b)=1 \\ a\leqslant q_r \\ d<b \end{cases} \Rightarrow \begin{cases} a<d<b \\ {\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)=I \end{cases}$$ $\blacklozenge$ (ii) all triplets of pairwise coprime $(a,b,d)$ such that ${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)= I$ and $a<d<b$ can be obtained as in (i).\
$\blacklozenge$ (iii) there is an infinity of pairwise coprime $(a,b,d)$ such that ${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)=I,a<d<b$ and $\rho=r+1$ in (i).
\(i) we suppose that $a$ and $b$ are coprime, $a\leqslant q_r$ and $d<b$. We have $d=q_{\rho-1}$ and $(q_i)_i$ is an increasing sequence, so : $ a\leqslant q_r \leqslant d<b $.\
Yet, $a\not =d$, for $\gcd(a,b)=1$. So $a<d<b$. We use Proposition \[prop:5\] ( case 2) and obtain : $${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)= \{ n_i, i\in{ \{ 0,..,s \} }\}$$ where $s$ is the index such that $y_s<a\leqslant y_{s+1}$ and $y_i=q_{i-1}$ for all $i$ ( obvious induction on $i$). But, $a\leqslant q_r$ by hypothesis and $q_{r-1}<a$, because $I$ is ${\mathbb{N}}$-independant ( see Proposition \[prop:6\]). So $s=r$ and we obtain our result (i).\
(ii) a direct consequence of Proposition \[prop:5\] ( case 2).\
(iii) using (i), the question is to find what values of $d_r$, integer greater than $1$, are such that : $$\gcd(a,d_rn_r-n_{r-1})=1 {\hspace*{0.2cm}}(1) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}a< d_rq_{r-1}-q_{r-2}<d_rn_r-n_{r-1} {\hspace*{0.2cm}}(2)$$ Indeed, if we choose $\rho=r+1$, then $d=q_r=d_rq_{r-1}-q_{r-2}$ and $b=n_{r+1}=d_rn_r-n_{r-1}$. But, $q_{r-1}<n_r$, for $I$ is ${\mathbb{N}}$-independant ( see Proposition \[prop:6\]). So, condition (2) is valid for $d_r$ large enough. Now, $n_{r-1}$ and $n_r$ are coprime, since $n_0,n_1$ are coprime ( see remark 1 at the beginning of \[sec:reverse\]). So, with Lemma \[lem:12\], condition (1) is valid for an infinite number of positive integers $d_r$... that proves our result.
**Examples :** we go back to our previous examples at the end of Theorem \[theo:1\].\
We take $I=\{ 7,11,59 \}$, we denote $a=7$ and we search integers $d_2$ such that $d_2\geqslant 2,b=59d_2-11, d=6d_2-1$ and $\gcd(a,b)=1, a<d<b$. We remark that the condition $a<d<b$ will always be verified here and that the condition $\gcd(a,b)=1$ is equivalent to $d_2\not = -1[7]$. So : $$\forall k\in { \{ 2,..,+\infty \} }, k\not = -1[7] \Rightarrow {\mathrm{Irr}}\left(\frac{<7,59k-11>}{6k-1}\right)= \{ 7,11,59 \}$$ Same arguments with the enumeration of $I$ by $(11,7,59)$, that is also modular-convex, gives : $$\forall k\in { \{ 2,..,+\infty \} }, k\not = -1[11] \Rightarrow {\mathrm{Irr}}\left(\frac{<11,59k-7>}{10k-1}\right)= \{ 7,11,59 \}$$ We consider now $I=\{ 10,17,24 \}$ and obtain with similar arguments : $$\forall k\in { \{ 6,..,+\infty \} }, k\not = 3 [5] \Rightarrow {\mathrm{Irr}}\left( \frac{<10,24k-17>}{2k-1}\right)=\{ 10,17,24 \}$$ What do we obtain for $k=2,4,5$, that is for values of $k$ such that $b>a>d$ ? Using Proposition 5, we obtain ( $I=\{ 10,17,24 \}$) : $${\mathrm{Irr}}\left( \frac{<10,31>}{3}\right)=I\cup \{ 31 \} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\mathrm{Irr}}\left( \frac{<10,79>}{7}\right)=I\cup \{79 \} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\mathrm{Irr}}\left( \frac{<10,103>}{9}\right)=I\cup \{ 103\}$$
$\bullet$ Finally, we search the solutions such that $a<b<d$. In that case, we can extend in both directions the sequence $(n_i)_i$ in a modular-convex sequence. We then obtain an infinite number of solutions :
. \[theo:3\]\
let $I$ be a finite subset of ${\mathbb{N}}\backslash\{ 0,1\}$ that is ${\mathbb{N}}$-independant and can be enumerated by a modular-convex sequence $(n_k)_{k\in{ \{ 0,..,r \} }}$, such that $n_0$ and $n_1$ are coprime.\
We denote $(d_i)_{i\in{ \{ 1,..,r-1 \} }}$ the sequence of integers such that : $$\forall i \in{ \{ 1,..,r-1 \} }, {\hspace*{0.2cm}}d_i\geqslant 2, {\hspace*{0.2cm}}n_{i+1}=d_in_i-n_{i-1}$$ $\blacklozenge$ (i) Let $\rho$ and $\nu$ be integers such that $\rho > r$ and $\nu < 0$. Let $(d_i)_{i\in { \{ \nu+1,..,\rho-1 \} }}$ be an extension of $(d_i)_{i\in{ \{ 1,..,r-1 \} }}$ in a sequence of integers bigger than $1$, that defines an extension of $(n_i)_{i\in{ \{ 0,..,r \} }}$ in a modular-convex sequence $(n_i)_{i\in{ \{ \nu,..,\rho \} }}$, such that $n_i>0$ for all $i$.\
We denote $d$ and $m$ the coprime positive integers such that $ \frac{d}{m}=\lceil d_{\nu+1},\cdots,d_{\rho-1}\rceil$ and $\frac{q_i}{p_i}$ the reduced fraction of $\lceil d_{\nu+1},d_{\nu+2},\cdots,d_i\rceil$ for all $i\in{ \{ \nu+1,..,\rho-1 \} }$. Moreover $q_{\nu-1}=0,q_{\nu}=1$ and $p_{\nu-1}=-1,p_{\nu}=0$.\
We denote $a=n_{\nu}$ and $b=dn_{\nu+1}-mn_{\nu}$ and $x_i=mq_{i-1}-dp_{i-1}$ for all $i\in { \{ \nu,..,\rho \} }$.\
Then, $n_{\rho}=b$ and :
$$\begin{cases} \gcd(a,b)=1 \\ a\leqslant q_r \\ a<b<d \\ x_0<x_r+b \leqslant x_{-1} \end{cases} \Rightarrow {\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)=I$$ $\blacklozenge$ (ii) all triplets of pairwise coprime $(a,b,d)$ such that ${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)= I$ and $a<b<d$ can be obtained as in (i).\
$\blacklozenge$ (iii) there is an infinity of pairwise coprime $(a,b,d)$ such that ${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)=I$, $a<b<d$ such that $\nu=-1,\rho=r+1$ in (i).
\(i) by obvious induction, we obtain : $$\forall i\in { \{ \nu,..,\rho \} }, {\hspace*{0.2cm}}n_i=\frac{ax_i+by_i}{d} {\hspace*{0.2cm}}\text{ where } {\hspace*{0.2cm}}y_i=q_{i-1}$$ so $n_{\rho}=b$, for $x_{\rho}=dm-md=0$ and $y_{\rho}=q_{\rho-1}=d$.\
Now, we suppose that $a$ and $b$ are coprime, $a\leqslant q_r$ and $a<b<d$. Then, $a,b,d$ are pairwise coprime and using Proposition \[prop:5\] Case 3, with indices in range ${ \{ \nu,..,\rho \} }$ instead of ${ \{ 0,..,r \} }$, we obtain : $${\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)= \{ n_i, i\in{ \{ \sigma,..,s \} }\}$$ where $s,\sigma\in { \{ \nu,..,\rho \} }$ such that $y_s<a\leqslant y_{s+1}$ and $x_{\sigma}< x_s+b \leqslant x_{\sigma-1}$.\
But, $a\leqslant q_r$ by hypothesis and $q_{r-1}<a$, because $I$ is ${\mathbb{N}}$-independant ( see Proposition \[prop:6\]). So $s=r$. Our hypothesis $x_0<x_r+b \leqslant x_{-1}$ gives $\sigma=0$, so we obtain result (i).\
(ii) a direct consequence of Proposition \[prop:5\] ( case 3), with a shift on indices of $(n_i)_i$ so that $\sigma=0$.\
(iii) so, we restrict ourselves to the extension of $(d_i)_i$ and $(n_i)_i$ by one term in both directions : that is to say, we choose $\rho=r+1$ and $\nu=-1$. Then, we search $d_0$ and $d_r$ two integers such that $d_0,d_r\geqslant 2$ and : $$n_{r+1},n_{-1}\in{\mathbb{N}}^* {\hspace*{0.2cm}}\mathbf{(1)} {\hspace*{0.2cm}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}\gcd(n_0,n_{r+1})=1 {\hspace*{0.2cm}}\mathbf{(2)} {\hspace*{0.2cm}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}d>\max(n_{-1},n_{r+1}) {\hspace*{0.2cm}}\mathbf{(3)} {\hspace*{0.2cm}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}x_0<x_r+n_{r+1} \leqslant x_{-1} {\hspace*{0.2cm}}\mathbf{(4)} {\hspace*{0.2cm}}$$ where : $$n_{r+1}=d_rn_r-n_{r-1}, {\hspace*{0.2cm}}n_{-1}=d_0n_0-n_1 {\hspace*{0.2cm}}\text{ and } {\hspace*{0.2cm}}x_{-1}=d {\hspace*{0.2cm}}, {\hspace*{0.2cm}}x_0=m$$ $d/m$ being the reduced fraction of $\lceil d_0,d_1,\cdots,d_r\rceil$ and $q_i/p_i$ being the reduced fraction of $\lceil d_0,d_1,\cdots,d_i\rceil$. In addition, $p_{-1}=0$ and $q_{-1}=1$. So, $p_r=m$ and $q_r=d$.\
We can simplify condition **(4)**. Indeed, we have here $x_r=1$, for $x_r=D_r$ with following notation and result : $$\forall i\in{ \{ \nu,..,\rho \} }, {\hspace*{0.2cm}}D_i=\begin{vmatrix} q_{i-1} & q_i\\ p_{i-1} & p_i \end{vmatrix} {\hspace*{0.2cm}}\text{ then } D_{i+1}=D_i$$ yet $D_{\nu}=1$, so $x_r=1$. Condition **(4)** can be rewritten : $m\leqslant n_{r+1}< d$.\
Now, $d$ and $m$ depends on $d_0$ and $d_r$ : we denote $Q_i/P_i$ the reduced fraction of $\lceil d_1,\cdots,d_i\rceil$, for all $i\in{ \{ 1,..,r \} }$. We can also set $P_{-1}=-1, P_0=0$ and $Q_{-1}=0, Q_0=1$. By obvious induction, we deduce : $$\forall i\in{ \{ -1,..,r \} }, {\hspace*{0.2cm}}p_i=Q_i {\hspace*{0.2cm}}; {\hspace*{0.2cm}}q_i=d_0Q_i-P_i$$ So : $$m= d_rQ_{r-1}-Q_{r-2} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}d=d_0( d_rQ_{r-1}-Q_{r-2})-(d_rP_{r-1}-P_{r-2})$$ Let us summarize :\
- condition **(1)** is satisfied if $d_0$ and $d_r$ are large enough.\
- condition **(2)** is satisfied for an infinite number of $d_0$, when $d_r$ is fixed ( for $\gcd(n_{1},n_0)=1$) and for an infinite number of $d_r$, when $d_0$ is fixed ( for $\gcd(n_{r-1},n_r)=1$), according to Lemma \[lem:12\].\
- condition **(3)** reduces to $d>n_{-1}$ if condition **(4)** is satisfied. But, if $d_r$ is large enough so that : $ d_rQ_{r-1}-Q_{r-2}>n_0$, then $d>n_{-1}$, if $d_0$ is large enough ( see expressions of $d$ and $n_{-1}$ above)\
- condition **(4)** is equivalent to $m\leqslant n_{r+1}<d$. But, since $(n_0,\cdots,n_r)$ is ${\mathbb{N}}$-independant, we have $Q_{r-1}<n_r$ ( see Proposition \[prop:6\] (i)), so $m<n_{r+1}$ for $d_r$ large enough. Now, if $d_r$ is fixed, then $n_{r+1}, P_r$ and $Q_r$ are fixed and for $d_0$ large enough, we will have $d>n_{r+1}$, because : $$d= d_0Q_r-P_r$$
Conclusion : we first choose ( and fix) $d_r$ large enough so that $n_{r+1}>0, {\hspace*{0.2cm}}d_rQ_{r-1}-Q_{r-2}>n_0$ and $n_{r+1}\geqslant m$. Then, we can choose an infinity of integers $d_0\geqslant 2$ such that $n_{-1}>0,\gcd(n_0,n_{r+1})=1, \\ d>n_{-1}$ and $d>n_{r+1}$. That proves (iii).
**Examples :** again, we take our two examples :\
First, $I=\{ 7,11,59 \}$, enumerated by the modular-convex sequence $(7,11,59)$ ( $d_1=6$).\
We denote $k=d_2$ and $j=d_0$. Then $d=6kj-k-j, {\hspace*{0.2cm}}m=6k-1, {\hspace*{0.2cm}}b=59k-11$ and $a=7j-11$. If $k$ and $j$ are two integers bigger than $1$ and such that $d > \max(a,b) \geqslant m>0$ and $\gcd(a,b)=1$, then ${\mathrm{Irr}}(\frac{<a,b>}{d})=I$. So : $$\forall k,j\in{\mathbb{N}}, {\hspace*{0.2cm}}\begin{cases} j \geqslant 10 , {\hspace*{0.2cm}}k\geqslant 2 \\ \gcd(7j-11,59k-11)=1 \end{cases} \Rightarrow {\mathrm{Irr}}\left( \frac{<7j-11,59k-11>}{6kj-k-j}\right)=\{ 7,11,59 \}$$
On the other hand, $I=\{ 7,11,59 \}$ is enumerated by the modular-convex sequence $(11,7,59)$ ( $d_1=10$).\
We denote $k=d_2$ and $j=d_0$. Then $d=10kj-k-j, {\hspace*{0.2cm}}m=10k-1, {\hspace*{0.2cm}}b=59k-7$ and $a=11j-7$. If $k$ and $j$ are two integers bigger than $1$ and such that $d > \max(a,b) \geqslant m>0$ and $\gcd(a,b)=1$, then ${\mathrm{Irr}}(\frac{<a,b>}{d})=I$. So : $$\forall k,j\in{\mathbb{N}}, {\hspace*{0.2cm}}\begin{cases} j \geqslant 6 , {\hspace*{0.2cm}}k\geqslant 2 \\ \gcd(11j-7,59k-7)=1 \end{cases} \Rightarrow {\mathrm{Irr}}\left( \frac{<11j-7,59k-7>}{10kj-k-j}\right)=\{ 7,11,59 \}$$
Now, we consider $I=\{ 10,17,24 \}$, enumerated by the modular-convex sequence $(10,17,24)$ ( $d_1=2$).\
We denote $k=d_2$ and $j=d_0$. Then $d=2kj-k-j, {\hspace*{0.2cm}}m=2k-1, {\hspace*{0.2cm}}b=24k-17$ and $a=10j-17$. If $k$ and $j$ are two integers bigger than $1$ and such that $d > \max(a,b) \geqslant m>0$ and $\gcd(a,b)=1$, then ${\mathrm{Irr}}(\frac{<a,b>}{d})=I$. So : $$\forall k,j\in{\mathbb{N}}, {\hspace*{0.2cm}}\begin{cases} j \geqslant 13, {\hspace*{0.2cm}}k\geqslant 6 \\ \gcd(10j-17,24k-17)=1 \end{cases} \Rightarrow {\mathrm{Irr}}\left( \frac{<10j-17,24k-17>}{2kj-k-j}\right)=\{ 10,17,24 \}$$
$\bullet$ Now, we end this section with a very particular case : when $I$ is an arithmetic sequence.
\[coro:1\] let $a$ and $k$ be two coprime integers greater than $1$ and $r\in { \{ 1,..,a-1 \} }$. Then : $$< a+jk, j\in { \{ 0,..,r \} }>=\frac{<a,a^2+dk>}{ar+1}$$ If $a$ and $r$ are coprime, wa have a simpler result : $$< a+jk, j\in { \{ 0,..,r \} }>=\frac{<a,a+rk>}{r}$$
first, the sequence $( a+jk)_{ j\in{ \{ 0,..,r \} } }$ is ${\mathbb{N}}$-independant. Indeed, with Proposition \[prop:6\] and its notations, we have $d_i=2$ for all $i\in{ \{ 0,..,r-1 \} }$, so $q_i=i+1$, by obvious induction. Hence, we obtain $q_{r-1}=r<a<a+rk$.\
If $a $ and $r$ are coprime, then $a$ and $b$ are coprime, when we set $b=a+rk$. Moreover, if we denote $d=r$ and $n_1=a+k$, then $d\in{ \{ 1,..,a-1 \} }$ and : $$dn_1=ar+rk=b \text{ mod } a$$ So, we can apply Theorem \[theo:1\] and obtain the result in that case.\
In the general case, we use Theorem \[theo:2\] (i) and its notations, with $\rho=r+1$ and $d_r=a+1$. We obtain $d=ar+1$ and $b=a^2+dk$. In addition, $a$ and $b$ are coprime and $a\leqslant q_r=d<b$.
**Remark :** as mentioned in the introduction, this result can be proven directly...
Diophantine approximation
=========================
two kinds of continued fraction expansions {#subsec:CFE}
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All results given in this subsection are well known and we just want to recall some notations and simple facts. For all reals $x$, $\lfloor x \rfloor$ denotes its floor ,$\lceil x \rceil $ its ceiling and $\{ x \}$ its fractional part. We have $x=\lfloor x \rfloor + \{ x \}$.\
$\bullet$ We have met in \[subsec:modular\] the notion of “ ceiling continued fraction expansion of a rational”. We will precise the definition and remind some results about usual continued fraction expansion.\
One usually uses the Gauss map $T$, defined on $]0,1[$, by $T(x)=\{ 1/x \}$ to define the usual continued fraction expansion of an irrational. We will rather use here a variant :\
Let $F$ and $C$ be the following maps defined on ${\mathbb{R}}$ : $$F : \begin{cases} {\mathbb{R}}\cup \{ +\infty\} \to ]1,+\infty] \\ x\to +\infty \text{ if } x\in{\mathbb{Z}}\cup \{ +\infty\} \\ x \to \frac{1}{\{ x \}} \text{ else }\end{cases} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\hspace*{0.2cm}}C : \begin{cases} {\mathbb{R}}\cup \{ +\infty\} \to ]1,+\infty] \\ x\to +\infty \text{ if } x\in{\mathbb{Z}}\cup \{ +\infty\} \\ x \to \frac{1}{1-\{ x \}} \text{ else }\end{cases}$$ We use the natural convention : $\lfloor +\infty \rfloor = \lceil +\infty \rceil=+\infty$. By obvious induction, we can prove that : if $\alpha$ is not rational, then $F^k(\alpha)\not = +\infty$ and $C^k(\alpha)\not = +\infty$ for every $k\in{\mathbb{N}}$. It is a consequence of euclidean algorithm on integers that : if $\alpha$ is rational, then $F^k(\alpha)=C^k(\alpha)=+\infty$ for some positive integer $k$.\
Now, given a real $\alpha$, we define two sequences ( named *partial quotients* ) by : $$\forall k\in{\mathbb{N}}, {\hspace*{0.2cm}}a_k=\lfloor F^k(\alpha) \rfloor {\hspace*{0.2cm}}; {\hspace*{0.2cm}}d_k= \lceil C^k(\alpha) \rceil$$ If, $\alpha$ is an irrational, then $a_k$ and $d_k$ are positive integers for all $k\in{\mathbb{N}}$ and $d_k\geqslant 2$ if $k>0$.\
If, $\alpha$ is a rational, then $a_k=d_k=+\infty$ for $k$ large enough.\
$[a_k]_{k\in{\mathbb{N}}}$ is the *usual continued fraction expansion* of $x$ ( we could also name it the “ floor continued fraction expansion” of $x$) and $\lceil d_k \rceil _{k\in{\mathbb{N}}}$ is the “ ceiling continued fraction expansion” of $x$.\
We usually define the *( usual or ceiling) convergents* of $\alpha$ relative to these continued fraction expansions by : $$\forall i\in {\mathbb{N}}, {\hspace*{0.2cm}}\frac{p_i}{q_i}=[a_0,a_1,\cdots,a_i] {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\frac{p'_i}{q'_i}=\lceil d_0,d_1,\cdots,d_i\rceil$$ if necessary, we will precise $ \frac{p_i(\alpha)}{q_i(\alpha)}$. The above brackets are inductively defined by ( with the usual convention : $1/\infty =0$) : $$[a_0,a_1,\cdots,a_i]=a_0+\frac{1}{[a_1,\cdots,a_i]} {\hspace*{0.2cm}};{\hspace*{0.2cm}}\lceil d_0,d_1,\cdots,d_i\rceil =d_0-\frac{1}{\lceil d_1,\cdots,d_i\rceil }$$ Thus, we can express these with finite fraction superpositions : $$[a_0,a_1,\cdots,a_i]=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{a_3+\cdots }}} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\lceil d_0,d_1,\cdots,d_i\rceil = d_0-\dfrac{1}{d_1-\dfrac{1}{d_2-\dfrac{1}{d_3-\cdots }}}$$
The sequences of ( usual or ceiling) convergents of $\alpha$ converge towards $\alpha$.\
For example : if $\alpha= \frac{1+\sqrt{5}}{2}$ ( the so-called “ golden ratio”), then : $a_k=1$ for all $k\in{\mathbb{N}}$, $d_0=2$ and $d_k=3$ for all $k\in{\mathbb{N}}^*$. Indeed : for these expansions we have : $$x=1+\frac{1}{x} \text{ and } x=2-\frac{1}{1+x}$$
With these parallel definitions, we could expect similar properties...but, this is not the case :
First, a useful Lemma that relates iterates of $F$ and $C$ :
\[lem:13\] let $\alpha$ be a real, but not an integer. We denote $a=\lfloor F(\alpha) \rfloor$. $$\forall i\in{ \{ 1,..,a \} }, {\hspace*{0.2cm}}C^i(\alpha)= 1+ \frac{1}{F(\alpha)-i}$$ So : $$C^{a}(\alpha)=1+F^2(\alpha)$$
Second result is a direct consequence of the first result, which is obtained by finite induction on $i$ :\
- it is true for $i=1$, since, if we denote $\epsilon=\{ \alpha \}$, then : $$C(\alpha)= \frac{1}{1-\epsilon} {\hspace*{0.2cm}}\text{ and } {\hspace*{0.2cm}}1+\frac{1}{F(\alpha)-1}=1+ \frac{1}{\frac{1}{\epsilon}-1}=1+ \frac{\epsilon}{1-\epsilon}= \frac{1}{1-\epsilon}$$ - suppose it is true for $i-1$, with $i\in{ \{ 2,..,a \} }$, then we remark that $C^{i-1}(\alpha)\in ]1,2]$ for $F(\alpha)\geqslant i$. The special case when $i=a=F(\alpha)$ leads to $C^{i-1}(\alpha)=2$ and so $C^i(\alpha)=+\infty=1+\frac{1}{0}$ ( with usual convention). Now, we suppose that $F(\alpha)>a$, so that $C^{i-1}(\alpha)\in ]1,2[$. Then : $$C^i(\alpha)=\frac{1}{1-\{ C^{i-1}(\alpha)\}}=\frac{1}{2-C^{i-1}(\alpha)}=\frac{F(\alpha)-(i-1)}{F(\alpha)-i}=1+ \frac{1}{F(\alpha)-i}$$
The following Lemma states the relations between the two kinds of partial quotients seen above, for the case of irrationals. The case of rationals will be treated later, for we will choose an alternate version of usual “ last partial quotient” for a rational.
\[lem:14\] let $\alpha$ be an irrational and $[a_k]_{k\in{\mathbb{N}}},\lceil d_k \rceil_{k\in{\mathbb{N}}}$ its usual and ceiling continued fraction expansions.\
(i) $\lceil d_k \rceil_{k\in{\mathbb{N}}^*}$ is obtained from $[a_k]_{k\geqslant 1}$ by the following substitution of patterns : $$\forall i\in{\mathbb{N}}^*, {\hspace*{0.2cm}}(a_{2i-1},a_{2i}) \to ((2,)^{a_{2i-1}-1},a_{2i}+2)$$ (ii) for the converse : we denote $(k_i)_{i\in{\mathbb{N}}}$ an increasing sequence of non negative integers such that $k_0=0$ and $(d_{k_i})_{i\in{\mathbb{N}}^*}$ is the subsequence of values of $(d_k)_{k\in{\mathbb{N}}^*}$ such that $d_k\not = 2$. Then : $$\forall i\in {\mathbb{N}}^*,{\hspace*{0.2cm}}\begin{cases} a_{2i-1}=k_i-k_{i-1} \\ a_{2i}=d_{k_i}-2 \end{cases}$$
\(ii) is a direct consequence of (i). For (i), we use previous Lemma : we will explain the process for the first step, namely $i=1$. First, we recall that $C^i(\alpha)$ and $F^j(\alpha)$ are never integer since $\alpha$ is not rational.\
Then, as mentioned in the proof of Lemma \[lem:13\], $C^i(\alpha)\in ]1,2[$ for all $i\in { \{ 1,..,a_1-1 \} }$, so $d_i=2$ for these values of $i$.\
Later, $C^{a_1}(\alpha)=1+F^2(\alpha)$ and $F^2(\alpha)$ is not an integer, so : $$\lceil C^{a_1}(\alpha)\rceil = 2+\lfloor F^2(\alpha) \rfloor$$ which proves that $d_{a_1}=a_2+2$. Finally : $$\{ C^{a_1}(\alpha) \}=\{ F^2(\alpha) \}$$ so, the process continues at the next step...
$\bullet$ There are two ways for usual continued fraction expansion of rationals : those ending with an integer greater than $1$ ( and an infinite sequence of $\infty$) or those ending with $1$ ( and an infinite sequence of $\infty$).\
For example, $5/7$ has two usual continued fraction expansions : $[0,1,2,2]$ and $[0,1,2,1,1]$ ( the ending $\infty$ sequence has been omitted). We will choose the second way and denote CFE this kind of usual continued fraction expansion.
The set of sequences representing CFE of this type, for reals, is then :
$${\mathcal{C}}= \{ (t_k)\in {\mathbb{Z}}\times {\mathbb{N}}^* \times (\overline{{\mathbb{N}}^*})^{{\mathbb{N}}}, \forall k\geqslant 2, ( t_k= \infty \Rightarrow ( t_{k+1}=\infty \text{ and } t_{k-1}\in \{ \infty ,1 \} ) \}$$
Since we will be interested by natural order on reals and especially on rationals, we mention the corresponding order on CFE sequences : if we define ${\varphi}$ by $${\varphi}: \begin{cases} {\mathcal{C}}\to {\mathbb{R}}\\ (t_k)_{k\in{\mathbb{N}}} \to [t_k]_{k\in{\mathbb{N}}} \end{cases}$$ this map is bijective and increasing, with the Alternate Lexicographic Order ( ALO) on ${\mathcal{C}}$ defined by : $$(t_k)_{k\in{\mathbb{N}}}\leqslant_A (t'_k)_{k\in{\mathbb{N}}} \Leftrightarrow (\forall k\in{\mathbb{N}}, t_k=t'_k ) \text{ or } \exists j\in{\mathbb{N}}, \begin{cases} \forall k<j, t_k=t'_k \\ (-1)^jt_j< (-1)^j t'_j \end{cases}$$
$\bullet$ Now is the time to give a “ rational version” of Lemma \[lem:14\] :
\[lem:15\] let $\alpha$ be a rational and $[a_0,\cdots,a_r,1], \lceil d_0,d_1,\cdots,d_{\rho}\rceil$ its usual and ceiling continued fraction expansions. ( $\infty$-tails are not mentioned)\
(i) $r$ and $\rho$ are related by the following formula : $$\rho = \sum_{i=1}^{\lfloor (r+1)/2 \rfloor } a_{2i-1}$$ (ii) $\lceil d_k \rceil_{k\in{ \{ 1,..,\rho \} }}$ is obtained from $[a_k]_{k\in { \{ 1,..,r \} }}$ by the following substitution of patterns : $$\forall i\in{ \{ 1,..,\lfloor r/2 \rfloor \} }, {\hspace*{0.2cm}}(a_{2i-1},a_{2i}) \to ((2,)^{a_{2i-1}-1},a_{2i}+2)$$ if $r$ is odd, then we add $d_{\rho}=2$.\
(iii) for the converse : let $(k_i)_{i\in{ \{ 0,..,s \} }}$ be an increasing sequence of non negative integers such that $k_0=0, k_{s}=\rho$ and $(d_{k_i})_{i\in{ \{ 1,..,s -1 \} }}$ is the subsequence of values of $(d_k)_{k\in{ \{ 1,..,\rho-1 \} }}$ such that $d_k\not = 2$. Then $s=\lfloor (r+1)/2 \rfloor$ and : $$\forall i\in { \{ 1,..,s \} }, {\hspace*{0.2cm}}\begin{cases} a_{2i-1}=k_i-k_{i-1} \\ a_{2i}=d_{k_i}-2 {\hspace*{0.2cm}}( \text{ if } 2i\leqslant r )\end{cases}$$
\(i) and (ii) are direct consequences of (iii), that we prove as in the proof of Lemma \[lem:14\], adding the following fact. If $F(\alpha)$ is an integer, then $r=1,a_1=F(\alpha)-1$ and $C^{a_1}(\alpha)=2$ ( see Lemma \[lem:13\], with $a=a_1+1$). So, $d_{a_1}=2,\rho=a_1,s=1$ and $a_1=k_1-k_0$. If $F^2(\alpha)$ is an integer, then $a_1=\lfloor F(\alpha)\rfloor,a_2=F^2(\alpha)-1$ and $C^{a_1}(\alpha)=a_2+2$ ( see Lemma \[lem:13\]). So, $r=2,d_{a_1}=a_2+2, \rho=a_1$ and $s=1$.\
As in the proof of Lemma \[lem:14\], this can be generalized to the case : $F^j(\alpha)$ is an integer for a positive integer $j$.
semi-convergents and best rational in an interval {#subsec:semicv}
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$\bullet$ Let $\alpha$ be a real with CFE $[a_k]_{k\in{\mathbb{N}}}$ and $(p_k/q_k)_k$ its convergents sequence.\
A *semi-convergent* of $\alpha$ is any rational of the form $\frac{mp_k+p_{k-1}}{mq_k+q_{k-1}}$, with $m\in { \{ 0,..,a_k \} }$ and $k\in {\mathbb{N}}$ such that $a_k<\infty$ ( we take $m>0$ if $k=0$ to avoid $1/0$ !). So, convergents are particular semi-convergents.
\[lem:16\] Let $\alpha$ be a real with CFE $[a_k]_{k\in{\mathbb{N}}}$. Semi-convergents of $\alpha$ are exactly the rationals with CFE $[a_0,\cdots,a_{s-1},b_s,1]$, such that $s\in{\mathbb{N}},b_s\in{ \{ 1,..,a_s \} }$ and $a_{s+1}<\infty$.
**Remark :** it would not be difficult to deduce that the ceiling convergents of $\alpha$ are exactly the semi-convergents of $\alpha$ that are greater or equal than $\alpha$. ( see Lemma \[lem:15\] and order on CFE)\
Consequence of the definition and the well known fact : $ \forall m\in {\mathbb{N}}^*, [a_0,\cdots,a_{s-1},m]=\frac{mp_{s-1}+p_{s-2}}{mq_{s-1}+q_{s-2}}$.
$\bullet$ Now, we would like to precise the CFE of reals in $\overset{\longleftrightarrow}{[\theta, \theta']}$ ( denotes the set of reals that are between $\theta$ and $\theta'$, even if $\theta > \theta'$), where $\theta$ and $\theta'$ are two different reals and find the rationals in this interval with the lowest reduced denominator. :
\[def:3\] let $x$ be a real. We name *CFE-depth* of $x$ the non negative integer, denoted $\mu(x)$ and defined by : $\mu(x)=+ \infty$ if $x$ is irrational and $\mu(x)=s$, if $x=[a_0,a_1,\cdots,a_s,1]$ is the CFE of $x$.
We remark that : $$\mu(x)=0 \Leftrightarrow x\in{\mathbb{Z}}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}\forall n\in{\mathbb{Z}}, \mu(x+n)=\mu(x) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\forall x\not\in {\mathbb{Z}}, \mu(T(x))=\mu(x)-1$$
We denote $\theta = [t_k]_{k\in{\mathbb{N}}}$ and $\theta' = [t'_k]_{k\in{\mathbb{N}}}$ and will abreviate $t$ and $t'$ these CFE-sequences. We denote $r$ the smallest integer $k$ such that $t_k\not = t'_k$. If we suppose that $t_r<t'_r$, then we have $r\leqslant \min(\mu(\theta),\mu(\theta'))+2$, when $\theta$ or $\theta'$ is rational ( if they are both irrationals, $r$ is finite ! ). Indeed, the extremal case when $r=\mu(\theta)+2$ for example corresponds to $\theta=[t_0,\cdots,t_{r-2},1]$ and $\theta'=[t_0,\cdots,t_{r-2},1,t'_r,...]$, with $t'_r<\infty$.
We remark that, all integers in $\overset{\longleftrightarrow}{[\theta, \theta']}$ minimize the denominator of their reduced fraction : it is $1$ !! So, we can suppose that $\lfloor \theta \rfloor = \lfloor \theta' \rfloor$ and even that $\theta, \theta'\in [0,1[$.\
The following result claims that, in that case, there is only one rational in $\overset{\longleftrightarrow}{[\theta, \theta']}$, that minimizes the value of its denominator : it is usually named the “ *best rational*” in $\overset{\longleftrightarrow}{[\theta, \theta']}$
\[prop:7\] let $\theta$ and $\theta'$ be two different reals in $[0,1[$ and $\theta = [t_k]_{k\in{\mathbb{N}}},\theta' = [t'_k]_{k\in{\mathbb{N}}}$ their respective CFE. We denote $r$ the lowest integer $k$ such that $t_k\not = t'_k$.\
**(i)** there is a unique rational in $\overset{\longleftrightarrow}{[\theta, \theta']}$ that minimizes the denominator. We denote it $\gamma$.\
- if $r\leqslant \min(\mu(\theta),\mu(\theta'))$, then $\gamma= [t_0,\cdots,t_{r-1},\min(t_r,t'_r),1]$.\
- else, $\mu(\theta)<\mu(\theta')$ ( up to swap) and $\gamma=\theta$.\
**(ii)** in both cases, $\mu(\gamma)\leqslant \min(\mu(\theta),\mu(\theta'))$ and $\gamma = [t_0,\cdots,t_{s-1},\min(t_s,t'_s),1]$, where $s=\mu(\gamma)\leqslant r$, so $\forall k\in{ \{ 0,..,s-1 \} },t_k=t'_k$.\
**(iii)** the best rational in $\overset{\longleftrightarrow}{[\theta, \theta']}$ is the common semi-convergent of $\theta$ and $\theta'$ with the greatest denominator.
see **\[3\]**( ) 1.4 Proposition 1.
**Remark :** as a direct consequence of (iii) : $\theta$ is the best rational in $\overset{\longleftrightarrow}{[\theta, \theta']} $ if and only if $\theta$ is a semi-convergent of $\theta'$.\
$\bullet$ Let $\alpha$ be a real, $[a_k]_{k\in {\mathbb{N}}^*}$ its CFE and $r=\mu(\alpha)$, the CFE-depth of $\alpha$. So, we denote $[a_0,a_1,\cdots,a_r,1]$ the CFE of $\alpha$ if $\alpha$ is rational. We also denote $(p_n/q_n)_n$ the usual sequence of convergents of $\alpha$.\
We consider the usual notion of best rational approximation of a real $\alpha$ : for $p,q$ two integers, $p/q$ is said a *best rational approximation* of $\alpha$ if and only if : $$\forall q'\in { \{ 1,..,q \} }{\hspace*{0.2cm}}, {\hspace*{0.2cm}}\forall p'\in{\mathbb{Z}}{\hspace*{0.2cm}}, {\hspace*{0.2cm}}\left|\dfrac{p'}{q'}-\alpha \right| \geqslant \left|\dfrac{p}{q}-\alpha \right|$$ It is well known that best rational approximation of a real are exactly its reduced convergents.\
Now, we can consider two sided similar definitions : for $p,q$ two integers, $p/q$ is said a *best left rational approximation* of $\alpha$ if and only if : $$\forall q'\in { \{ 1,..,q \} }{\hspace*{0.2cm}}, {\hspace*{0.2cm}}\forall p'\in{\mathbb{Z}}{\hspace*{0.2cm}}, {\hspace*{0.2cm}}\dfrac{p'}{q'} \leqslant \dfrac{p}{q} \leqslant \alpha \text{ or } \dfrac{p'}{q'}>\alpha$$ $p/q$ is said a *best right rational approximation* of $\alpha$ if and only if : $$\forall q'\in { \{ 1,..,q \} }{\hspace*{0.2cm}}, {\hspace*{0.2cm}}\forall p'\in{\mathbb{Z}}{\hspace*{0.2cm}}, {\hspace*{0.2cm}}\dfrac{p'}{q'} \geqslant \dfrac{p}{q} \geqslant \alpha \text{ or } \dfrac{p'}{q'}<\alpha$$
Here is a corollary of Proposition \[prop:7\] :
\[coro:2\] .\
(i) best left rational approximations of $\alpha$ are the semi-convergents of $\alpha$, that are lower than $\alpha$.\
(ii) best right rational approximations of $\alpha$ are the semi-convergents of $\alpha$, that are greater than $\alpha$.
\(i) we remark that $p/q$ is a best left rational approximation of $\alpha$ if and only if $p/q$ is the best rational in $[p/q, \alpha]$ and use the remark below Proposition \[prop:7\]. Same arguments for (ii).
If we denote $(p_k/q_k)_k$ the reduced convergents of $\alpha$, then :\
- its best left rational approximations are : $$\frac{p_{2i}+mp_{2i+1}}{q_{2i}+mq_{2i+1}} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}i\in { \{ 0,..,(\mu(\alpha)-1)/2 \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}m\in { \{ 0,..,a_{2i+2} \} }$$ - its best right rational approximations are : $$\frac{p_{2i-1}+mp_{2i}}{q_{2i-1}+mq_{2i}} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}i\in { \{ 1,..,\mu(\alpha)/2 \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}m\in { \{ 0,..,a_{2i+1} \} }$$
\[prop:8\] .\
(i) let $\alpha$ and $\alpha'$ be two reals such that $0<\alpha<\alpha'<1$. We denote $\gamma$ the best rational in $]\alpha, \alpha']$ and $q$ the denominator of its reduced fraction. Then $$q = \max \{ N\in{\mathbb{N}}, \forall n\in { \{ 0,..,N-1 \} }, \lfloor n\alpha \rfloor = \lfloor n\alpha' \rfloor \}$$ (ii) let $\alpha$ be a real in $[0,1)$ and $p/q$ a reduced fraction, with $q\in{\mathbb{N}}^*$, such that $\alpha$ is not the best left strict convergent of $p/q$. $$p/q \text{ is a semi-convergent of } \alpha \Leftrightarrow \forall k\in { \{ 0,..,q-1 \} }, \lfloor k\alpha \rfloor = \lfloor kp/q \rfloor$$
see proof of Proposition 8 in 4.2 of **\[3\]** ( )
$\alpha$-numeration for a rational $\alpha$ {#subsec:alphanum}
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$\bullet$ Let $\alpha$ be a rational in $[0,1[$ and $\alpha=[0,a_1,\cdots,a_r,1]$ its CFE. We will denote $(p_k/q_k)_{0\leqslant k \leqslant r }$ its convergents, so that $\alpha= p_{r+1}/q_{r+1}$.
\[def:4\] a sequence $d$ in ${\mathbb{N}}^r$ is said *$\alpha$-admissible* if and only if : $$\forall j\in{ \{ 1,..,r \} },\begin{cases} d_j\in{ \{ 0,..,a_j \} }\\ d_j=0 \Rightarrow ( \forall i\geqslant j, d_i=0 )\text{ or } d_{j-1}=a_{j-1} \end{cases}$$
We will denote $E_{\alpha}$ the set of $\alpha$-admissible sequences.\
**Remark :** for $j=1$, the second condition reduces to $d_1=0\Rightarrow \forall i\geqslant 1,d_i=0$. So to say, $d=(0,\cdots,0)$ is the only element of $E_{\alpha}$ , whose first coordinate is $0$.\
We consider the **reversed lexicographic order ( RLO)** denoted $ \underset{R}{\leqslant}$ on ${\mathbb{N}}^r$ : $$d \underset{R}{\leqslant} d' \Leftrightarrow d=d' \text{ or } \exists j\in { \{ 1,..,r \} }, \begin{cases} d_j<d'_j \\
\forall i\in { \{ j+1,..,r \} }, d_i=d'_i \end{cases}$$ It is a total order on $E_{\alpha}$.
\[prop:9\] the map $\Psi_{\alpha}$ is an order isomorphism from $(E_{\alpha} ,\leqslant_R)$ to ${ \{ 0,..,q_{r+1}-1 \} },\leqslant)$. $$\Psi_{\alpha} : \begin{cases} E_{\alpha} \to { \{ 0,..,q_{r+1}-1 \} } \\ d\to \sum\limits_{j=1}^rd_jq_{j-1} \end{cases}$$
see **\[3\]** 2.2. ( )
The following algorithm explains the inverse function of $\Psi_{\alpha}$. We will denote $m_k=q_k+q_{k-1}$ for any $k\in{ \{ 0,..,r \} }$. So $m_r=q_{r+1}$.
\[algo:1\] let $n\in { \{ 0,..,m_r-1 \} }$.\
With the following algorithm, we have $d\in E_{\alpha}$ and $\Psi_{\alpha}(d)=n$.
**Remark :** as a direct consequence : $E_{\alpha}$ has $q_{r+1}$ elements.\
$\bullet$ Now, we will deal with $\alpha$-numeration for elements of $U_{\alpha}= \{ \{ k\alpha\}, k\in{\mathbb{N}}\} $. Since, $\alpha=\frac{p_{r+1}}{q_{r+1}}$ and this fraction is reduced, we have $U_{\alpha}= \{ \frac{n}{q_{r+1}}, n\in { \{ 0,..,q_{r+1}-1 \} }\}$. So, this set is very simple, but we will focus on the map $k\to \{ k\alpha \}$, with the order point of view :\
We consider the **alternate lexicographic order ( ALO)** denoted $ \underset{A}{\leqslant}$ on ${\mathbb{N}}^r$ : $$d \underset{A}{\leqslant}d' \Leftrightarrow d=d' \text{ or } \exists j\in { \{ 1,..,r \} }, \begin{cases} (-1)^{j-1}d_j<(-1)^{j-1}d'_j \\
\forall i\in { \{ 1,..,j-1 \} }, d_i=d'_i \end{cases}$$ It is another total order on $E_{\alpha}$.\
We define also : $$\forall i\in{ \{ -2,..,r \} }, {\hspace*{0.2cm}}\delta_i=(-1)^i( q_{i}\alpha - p_i)$$ We have, with $a_0=0$ here : $$\delta_{-2}=\alpha {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\delta_{-1}=1 {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\delta_0=\{ \alpha \} =\alpha {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\forall i\in{ \{ 0,..,r \} }, {\hspace*{0.2cm}}\delta_i=-a_i\delta_{i-1}+\delta_{i-2}$$ Let $T$ be the Gauss map : $]0,1[\to [0,1[, x\to \{ 1/x \}$.\
By induction on $i$, with the fact that : $a_i=\left\lfloor \frac{1}{T^{i-1}(\alpha)} \right\rfloor$ if $i\leqslant r-1$, we obtain : $$\forall i\in{ \{ 0,..,r-1 \} }, \frac{\delta_i}{\delta_{i-1}}=T^i(\alpha)$$
Beware : for $i=r$, $T^{r-1}(\alpha)=[0,a_r,1]=\frac{1}{a_r+1}$, so : $$\frac{\delta_r}{\delta_{r-1}}= \frac{\delta_{r-2}-a_r\delta_{r-1}}{\delta_{r-1}}= \frac{1}{T^{r-1}(\alpha)}-a_r=1$$ So : $\delta_r=\delta_{r-1}$. We also have ( see \[...\]) $ \delta_r=\delta_{r-1}=\frac{1}{q_{r+1}}$.\
To summarize this : $$\forall i\in { \{ 0,..,r-1 \} }, 0<\delta_i<\delta_{i-1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\delta_r=\delta_{r-1}=\frac{1}{q_{r+1}}$$
\[prop:10\] .\
(i) the map $\Lambda_{\alpha} $ ( defined below) is an order isomorphism, with ALO on $E_{\alpha}$ : $$\Lambda_{\alpha} : \begin{cases} E_{\alpha} \to \left\{ \frac{n}{q_{r+1}}, n\in { \{ 0,..,q_{r+1}-1 \} }\right\} \\ d\to \sum\limits_{j=1}^rd_j(-1)^{j-1}\delta_{j-1} \end{cases}$$
\(ii) we have : $$\forall n\in { \{ 0,..,q_{r+1}-1 \} }, \{ n\alpha \}= \Lambda_{\alpha}(\Psi_{\alpha}^{-1}(n))$$
see **\[3\]** 2.2 ( ).
**Remarks :** result (ii) means that the map $n\to \{n\alpha \}$ ( with $0\leqslant n < q_{r+1}$), is, from the order point of view, the “ same thing” as the identity $(E_{\alpha},RLO)\to (E_{\alpha},ALO)$.\
We can sum up these formulae : $ \forall n\in { \{ 0,..,q_{r+1}-1 \} }$, with $d=\Psi_{\alpha}^{-1}(n)$ : $$n=\sum_{j=1}^r d_jq_{j-1} {\hspace*{0.2cm}};{\hspace*{0.2cm}}\lfloor n\alpha\rfloor =\sum_{j=1}^r d_jp_{j-1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\{ n\alpha \}=\sum_{j=1}^r (-1)^{j-1}d_j\delta_{j-1}$$
The following algorithm expresses the inverse function of $\Lambda_{\alpha}$.
\[algo:2\] let $\beta\in \{ n/q_{r+1}, n\in{\mathbb{N}}\}$.\
Applying the algorithm below ( with $\beta_0=\beta$), we have :\
(i) $b\in E_{\alpha}$.\
(ii) $\beta = \Lambda_{\alpha}(b)$.
minimal points of $(\{ n\alpha - \beta \},n )_n$
------------------------------------------------
In this subsection, we prove several Lemma about minimal points of this sequence, in different cases, from the simplest to the more intricate. These results will be useful, when we will study in section 5, ${\mathrm{Irr}}$ and ${\mathrm{PF}}$ of $\frac{<a,b>}{d}$, since points of $L$ in a rectangle of vertice $d$ can be parametrized by such sequences ( see \[subsec:relations\]).\
$\bullet$ First, we study minimal points in ${\mathbb{R}}^2$, with the product order, of the sequence $(\{ n\alpha \},n)_{1\leqslant n \leqslant N }$, where $N$ is a positive integer.\
With Proposition \[prop:10\], we can affirm that these points have the property that if $n=\Psi_{\alpha}(d)$, then for any other $d'\in E_{\alpha}$, we have $d<d'$ for ALO or for RLO !\
The following Lemma gives the result for the particular case $N=q-1$, where $q$ is the denominator of the reduced fraction of $\alpha$. We denote, as usual, $[a_0,a_1,\cdots,a_r,1]$ the CFE of $\alpha$.\
\[lem:17\] the minimal points in ${\mathbb{R}}^2$ of the set $\{ (\{ n\alpha \},n) , 1\leqslant n \leqslant q-1 \}$ are for the following integers $n$, given by three equivalent formulations :\
(i) with the $\alpha$-numeration of $n$ ( the $0$-tail is not written) : $$(1){\hspace*{0.2cm}}; {\hspace*{0.2cm}}(1,j) , j\in { \{ 1,..,a_2 \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}(1,a_2,0,j), j\in{ \{ 1,..,a_4 \} } ; \cdots ; (1,a_2,0,\cdots,0,j) , j\in{ \{ 1,..,a_{2\lfloor r/2 \rfloor} \} }$$ we can sum up this with : $(1)$ and sequences $ (d_k)_{k\in{ \{ 1,..,r \} }}$ such that $d_1=1$ and : $$\forall i\in{ \{ 1,..,k-1 \} }, d_{2i}=a_{2i}, d_{2i+1} =0, {\hspace*{0.2cm}}d_{2k}\in { \{ 1,..,a_{2k} \} }, \forall i> 2k, d_i=0, {\hspace*{0.2cm}}k\in{ \{ 1,..,\lfloor r/2 \rfloor \} }$$ (ii) with denominators of convergents of $\alpha$ : $$q_0 {\hspace*{0.2cm}}; {\hspace*{0.2cm}}q_{2k-2}+jq_{2k-1}, j\in{ \{ 1,..,a_{2k} \} }, k\in { \{ 1,..,\lfloor r/2 \rfloor \} }$$
\(iii) with diophantine approximation : these are the denominators of the reduced semiconvergents of $\alpha$, that are lower than $ \alpha$.
- let us verify that these points are minimal : for $(1)$, it is obvious. We denote $A_{p}=(a_2,0,a_4,0,\cdots,a_{2p-2},0)$, for any positive integer $p$ ( with $A_1=()$, the empty sequence).\
Let $d=(1,A_p,j)$ for $j\in { \{ 1,..,a_{2p} \} }$ and $p\leqslant r/2$. If $d'<d$ for ALO and $d'\not = 0$, then $d'=(1,A_p,j')$ with $j'>j$, or $d'=(1,A_p,j,0,k)$, with $k\in{ \{ 1,..,a_{2p+2} \} }$. In both cases, we have $d'>d$ for RLO.\
- now, let us verify that no other point is minimal : if $d'$ is not among these $\alpha$-numeration of integers $n$, then let $k$ be the least integer such that $d'_k$ is different from all the $d_k$ of (i). Then, we claim that $k$ is odd. Indeed, if $k$ were even, we would have $d'=(1,A_p,d'_k,...)$, with $k=2p+2$ and $d'_k=0$. So $d'=(1,A_p)$, according to our definition of $\alpha$-admissible sequences, but this is in the list (i).\
So, $k$ is odd and if we denote $d=(1,A_p,d_{k-1})$, with $k=2p+1$, then $d<d'$ for ALO and RLO.\
- (ii) and (iii) are direct consequences of (i)
$\bullet$ We will also need a more general case : but it is sufficient to sort among the former values of $n$, those which are lower than $N$ :
\[lem:18\] let $N\in{ \{ 1,..,q-1 \} }$. We denote $(N_1,N_2,\cdots,N_s)$ the $\alpha$-numeration of $N$, with $N_s>0$. The minimal points in ${\mathbb{R}}^2$ of the set $\{ (\{ n\alpha \},n) , 1\leqslant n \leqslant N\}$ are for the following values of $n$ :\
(i) with the $\alpha$-numeration of $n$ ( the $0$-tail is not written) : $$(1){\hspace*{0.2cm}}; {\hspace*{0.2cm}}(1,j) , j\in { \{ 1,..,a_2 \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}(1,a_2,0,j), j\in{ \{ 1,..,a_4 \} } ; \cdots ; (1,a_2,0,\cdots,a_{2\lfloor s/2 \rfloor-2},0,j) , j\in{ \{ 1,..,\mu \} }$$ where $\mu=N_s$ if $s$ is even, $\mu=a_{s-1}$ if $s$ is odd.\
we can sum up this with : $(1)$ and $(d_k)_{k\in{ \{ 1,..,r \} }} $, such that $ d_1=1$ and : $$\forall i\in{ \{ 1,..,k-1 \} }, d_{2i}=a_{2i}, d_{2i+1} =0 {\hspace*{0.2cm}}, {\hspace*{0.2cm}}d_{2k}\in { \{ 1,..,a'_{2k} \} }, \forall i> 2k, d_i=0, {\hspace*{0.2cm}}k\in{ \{ 1,..,\lfloor s/2 \rfloor \} }$$ where $a'_{2k}=a_{2k}$ if $2k<s$, and $a'_s=N_s$.\
(ii) with denominators of convergents of $\alpha$ : $$q_0 {\hspace*{0.2cm}}; {\hspace*{0.2cm}}q_{2k-2}+jq_{2k-1}, j\in{ \{ 1,..,a'_{2k} \} }, k\in { \{ 1,..,\lfloor s/2 \rfloor \} }$$ where $a'_{2k}=a_{2k}$ if $2k<s$, and $a'_s=N_s$.
It is a consequence of the former Lemma, when we remember ( see Proposition \[prop:9\]) that $\Psi_{\alpha}$ is an increasing map from $E_{\alpha}$ to ${ \{ 0,..,q-1 \} }$, with RLO. So, we just have to select, among the $d$ of the previous results those which are lower than $(N_1,N_2,\cdots,N_s)$ for RLO. Indeed, no $(\{ k\alpha \},k)$, with $k>N$ is lower than any $(\{ n\alpha \},n)$, with $n\leqslant N$...
$\bullet$ We generalize a bit more again , adding another condition :
\[lem:19\] let $N\in{ \{ 1,..,q-1 \} }$ and $(N_1,N_2,\cdots,N_s)$ its $\alpha$-numeration of $N$, with $N_s>0$.\
Let $\beta \in \{ \{ n\alpha\},n\in{\mathbb{N}}\}$, and $(b_1,b_2,\cdots, b_{s'})$ its $\alpha$-numeration, with $b_{s'}>0$. We suppose $\beta\not = 0$.\
The minimal points in ${\mathbb{R}}^2$ of the set $\{ (\{ n\alpha \},n) \text{ such that }, 1\leqslant n \leqslant N\ \text{ and } \{ n\alpha \}\leqslant \beta \}$ are :
$\blacktriangleright$ Case 1 : if $b_1\geqslant 2$ or $b_2=0$, that is to say $\alpha\leqslant \beta$, these are the same as in former Lemma !
$\blacktriangleright$ Case 2 : else ( $b_1=1$ and $b_2>0$).\
We denote $t=\min\{ i>0, b_{2i+1}>0 \}$ and $t=\lfloor s'/2 \rfloor$ if $\forall i>0, b_{2i+1}=0$ ( $s' $ is even in that case).\
(i) with the $\alpha$-numeration of $n$ : $$(1,a_2,0,a_4,\cdots,a_{2t-2},0,j) {\hspace*{0.2cm}}, {\hspace*{0.2cm}}j\in { \{ b_{2t},..,a'_{2t} \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\text{ and }$$ $$(1,a_2,0,a_4,\cdots,a_{2k-2},0,j){\hspace*{0.2cm}}; {\hspace*{0.2cm}}j \in { \{ 1,..,a'_{2k} \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}k\in{ \{ t+1,..,\lfloor s/2 \rfloor \} }$$
where $a'_{2k}=a_{2k}$ if $2k<s$ ; $a'_s=N_s$.\
(ii) with denominators of convergents of $\alpha$ : $$q_{2k-2}+jq_{2k-1}, {\hspace*{0.2cm}}j\in{ \{ c_{2k},..,a'_{2k} \} }, {\hspace*{0.2cm}}k\in { \{ t,..,\lfloor s/2 \rfloor \} }$$ with the same notations as in (i) and : $c_{2k}= \begin{cases} b_{2k} \text{ if } k=t \\
1 \text{ else }\end{cases}$.
**Remark :** if $t>\lfloor s/2 \rfloor$, then there is no minimal element, so the set is empty : there is no integer $n$ such that $n\in { \{ 1,..,N \} }$ and $\{ n\alpha \} \leqslant \beta$.\
Let denote $E=\{ (\{ n\alpha \},n), n\in{ \{ 1,..,N \} }\}$ and $E'=\{ (\{ n\alpha \},n)\in E, \{ n\alpha \}\leqslant \beta \}$. We claim that : $$\min(E')=\min(E)\cap E'$$ Indeed, The second subset is obviously included in the first one and no element of $E\backslash E'$ is lower than any element of $E'$.\
So, we just have to select among minimal elements $ (\{ n\alpha \},n)$ of $E$, whose $\alpha-$numeration, say $d$, is given by former Lemma, those that verify : $\{ n\alpha \}\leqslant \beta$, that is to say $d\leqslant_A b$.\
$\blacktriangleright$ Case 1 : if $b_1>1$ or $b_2=0$, that is to say $\alpha\leqslant \beta$, then every $d$ of former Lemma verify the condition $d\leqslant_A b$.\
$\blacktriangleright$ Case 2 : else, we have $b_1=1$ and $b_2>0$.\
$\blacktriangleright \blacktriangleright$ subcase 1 : suppose that $b_{2j+1}>0$ for some $j>0$. With notation of our Lemma, we have : $$\forall i\in{ \{ 1,..,t-1 \} }, b_{2i}=a_{2i} \text{ and } b_{2i+1}=0$$ Let $d$ such that $ d_1=1$ and : $$\forall i\in{ \{ 1,..,k-1 \} }, d_{2i}=a_{2i}, d_{2i+1} =0 {\hspace*{0.2cm}}, {\hspace*{0.2cm}}d_{2k}\in { \{ 1,..,a'_{2k} \} }, \forall i> 2k, d_i=0$$ where $k\in{ \{ 1,..,\lfloor s/2 \rfloor \} }$, with $a'_{2k}=a_{2k}$ if $2k<s$, and $a'_s=N_s$ ( see former Lemma).\
If $k<t$, then $d>_A b$ : indeed, $d_i=b_i$ for all $i<2k+2$, for $d_{2k}$ must be $a_{2k}=b_{2k}$, while $d_{2k+2}=0< b_{2k+2}$, because $b_{2k+1}=0$ and $b_{2t+1}>0$.\
If $k=t$, then $d_i=b_i$ for all $i<2k$, so $d\leqslant_A b$ if and only if $d_{2k}\geqslant b_{2k}$, for $d_{2k+1}=0<b_{2k+1}$.\
If $k>t$, then $d\leqslant_A b$, for $d_i=b_i$ for all $i<2t$ and $d_{2t}=a_{2t}\geqslant b_{2t} $ and $ d_{2t+1}=0< b_{2t+1}$.\
$\blacktriangleright \blacktriangleright$ subcase 2 : suppose that $b_{2i+1}=0$ for all $i>0$. Then, $s'$ can not be odd, so : $$b=(1,a_2,0,a_4,\cdots,0,a_{s'-2},0,b_{s'})$$ The same arguments as above prove that a minimal element of former Lemma is in $E'$ if and only if $d_i=b_i$ for all $i<s'$ and $d_{s'}\geqslant b_{s'}$. The formula still rules.
$\bullet$ Now, we will need another generalization : we study the minimal points in ${\mathbb{R}}^2$, with the product order, of the sequence $(\{ n\alpha -\beta \},n)_{n_0\leqslant n < q }$, where $n_0=0$ or $1$, $q$ is a positive integer and $\alpha=\frac{p}{q},\beta= \frac{p'}{q}$, with $p,p'\in { \{ 1,..,q-1 \} }$ and $p,q$ coprime.\
First, we remark that : $$\forall x\in {\mathbb{R}}, \{ x-\beta \} = \begin{cases} \{ x \} -\beta \in [0,1-\beta [ \text{ if } \{ x \} \geqslant \beta \\
\{ x \} + 1- \beta \in [ 1- \beta , 1 [ \text{ if } \{ x \} < \beta \end{cases}$$
If the sequence begins with the index $i$ at $0$, the point $(1-\beta,0)$ obtained for $n=0$ is obviously a minimal point of the sequence. So, the other minimal points $(\{ n\alpha \},n)$ must verify $\{ n\alpha \}\geqslant \beta$. We have the same situation if the sequence begins at $i=1$ and if $\alpha \geqslant \beta$.\
If the sequence begins at $i=1$ and if $\alpha < \beta$, then the “ first” minimal points will be such that $\{ n\alpha\}<\beta$ and the following such that $\{ n\alpha \}\geqslant \beta$. So, the lowest integer $n$ such that $\{ n\alpha \}\geqslant \beta$ is important.\
The case $\alpha=\beta$ is obvious ( only one or two minimal points for $n=0$ and $n=1$).\
\[lem:20\] let $q$ be a positive integer and $\alpha=\frac{p}{q},\beta= \frac{p'}{q}$, with $p,p'\in { \{ 1,..,q-1 \} }$ and $p,q$ coprime. We denote $\alpha=[0,a_1,\cdots,a_r,1]$ the CFE and $(b_1,b_2,\cdots,b_s)$ the $\alpha$-numeration of $\beta$, where $b_s>0$.\
We denote $t= \begin{cases} (s+1)/2 \text{ if } \forall i>0,b_{2i}=0 \\ \min\{ i\in {\mathbb{N}}^*, b_{2i}\not = 0 \} \text{ else } \end{cases} $ ( $s$ is odd in the first case).\
Minimal points of $(\{ n\alpha -\beta \},n)_{1\leqslant n < q }$ are obtained for following $n$, given by their $\alpha$-numeration : $(1)$ ( first ), $(b_i)_i$ ( last) and : $$(1,a_2,0,\cdots,a_{2k-2},0,j) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}j\in{ \{ 1,..,a_{2k} \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}k\in{ \{ 1,..,t-1 \} }$$ $$(b_1,b_2, \cdots,b_{2k-1}, j){\hspace*{0.2cm}}; {\hspace*{0.2cm}}j\in{ \{ 0,..,b_{2k}-1 \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}k\in{ \{ t,..,\lfloor s/2 \rfloor \} }$$
**Remark 1 :** if $b_2>0$, that is to say if $t=1$, then the first shape of $ \alpha$-numeration of $n$ is absent. In addition, if $b_1=1$ and $b_2>0$, then we count two times the first minimal point for $n=1$.\
**Remark 2 :** if $b_{2i}=0$ for all $i$, then the second shape of $ \alpha$-numeration of $n$ is absent, except for $(b_i)_i$.\
**Remark 3 :** if the sequence begins at $i=0$ instead of $i=1$, then the point $(1-\beta,0)$ obtained for $n=0$ is obviously a minimal point of the sequence. So, the other minimal points $(\{ n\alpha \},n)$ must verify $\{ n\alpha \}\geqslant \beta$. They are obtained for $n$ with $\alpha$-numeration : $$(b_1,b_2, \cdots,b_{2k-1}, j){\hspace*{0.2cm}}; {\hspace*{0.2cm}}j\in{ \{ 0,..,b_{2k}-1 \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}k\in{ \{ t,..,\lfloor s/2 \rfloor \} }$$
As long as $\{ n\alpha \}<\beta$, minimal points of our sequence are obtained for the same values of $n$ than those of $(\{n\alpha \},n)_n$, because $\{ n\alpha-\beta \}=\{n\alpha \}+1-\beta$.\
We remark that ( see definition of $E_{\alpha}$) : $$\forall i\in{ \{ 1,..,t-1 \} }, b_{2i}=0 {\hspace*{0.2cm}}; {\hspace*{0.2cm}}b_{2i-1}=a_{2i-1}$$ Now, we need the $\alpha$-numeration, say $\nu$, of $n_1$, the least integer $n$ such that $\{ n\alpha \} \geqslant \beta$. If $ \alpha \geqslant\beta$ then $n_1=1$. Else, with Proposition \[prop:9\], $\nu$ is the minimum of elements $d=(d_1,\cdots,d_r)$ of $E_{\alpha}$ for RLO, such that $d \geqslant_A b$, where $b=(b_k)_{k\in { \{ 1,..,r \} }}$ and $\geqslant_A$ means for ALO.\
We claim that $\nu=(b_1,b_2,\cdots,b_{2t-1})$. Indeed, the condition $d \geqslant_A b$ implies that : $$\forall i\in{ \{ 1,..,t-1 \} }, d_{2i}=0=b_{2i} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}d_{2i-1}=a_{2i-1}=b_{2i-1} \text{ and } d_{2t-1}\geqslant b_{2t-1}$$ But, the above $\delta$ satisfies $\nu\geqslant_A b$ and $\nu$ is minimal ( for RLO) among these one. Furthermore : $$\forall n<n_1, \{ n\alpha - \beta \}\in [ 1- \beta,1[ {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\{ n_1\alpha - \beta \}\in [ 0,1- \beta[$$ Note that this is also true if $b_{2i}=0$ for all $i>0$ and that, in that case, we have $\nu=b$ and so $\{ \nu\alpha \}=\beta$ and no minimal point after this one...\
We return to the general case : we have seen just above that the lowest integer $n$ such that $\{ n\alpha \}\geqslant \beta$ is $n_1=(b_1,b_2,\cdots,b_{2t-1})_{\alpha}$. So, for $n<n_1$, we use values of $n$ of Lemma \[lem:18\] : we obtain first part of our result ( that involves the $(a_i)$), for $b_{2t-1}>0$ : indeed $b_{2t}>0$ and $b_{2t-2}=0$.\
Now, if $n\geqslant n_1$, let denote $d=(d_1,\cdots,d_r)$ its $\alpha$-numeration. Then, the minimality condition for $(\{ n\alpha - \beta \},n)$ is equivalent to : $d\geqslant_A b$ and $ d$ is minimal among these ( elements of $E_{\alpha}$ greater than $b$ for ALO) for the product of orders (ALO,RLO). So, $d$ must verify $(b_1,\cdots,b_s)\leqslant_A d <_A (b_1,\cdots,b_{2t-1})$. It gives the successive values of $d$ : $(b_1,\cdots,b_{2t-1},j)$ as $j\in { \{ 0,..,b_{2t}-1 \} }$. Later, if $b_{2t+2}\not = 0$, the first ( for RLO) $d$ that verify $d\geqslant_A b$ and that is lower ( for ALO) than $(b_1,b_2,\cdots,b_{2t-1},b_{2t}-1)$ is $(b_1,b_2,\cdots,b_{2t-1},b_{2t},b_{2t+1})$. Then, we have $(b_1,b_2,\cdots,b_{2t-1},b_{2t},b_{2t+1},j)$ with $j$ taking successive values of ${ \{ 1,..,b_{2t+2}-1 \} }$...and so on.
number of integers $k$ such that $\{ k\alpha \}\leqslant \beta $ and $k\leqslant \nu$ {#subsec:counting}
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We consider two rationals $\alpha=\frac{p}{q}$ and $\beta=\frac{m}{q}$ in $[0,1[$ and a positive integer $\nu$, such that : $$\gcd(p,q)=1 {\hspace*{0.2cm}}; {\hspace*{0.2cm}}0<m<q {\hspace*{0.2cm}}; {\hspace*{0.2cm}}0< \nu < q$$ We denote $n=(n_k)_k$ and $b=(b_k)_k$ the respective $\alpha$-numeration of $\nu$ and $\beta$. We denote $\sigma$ the usual shift on sequences.\
We will also use the two total orders on finite sequences of reals : RLO, denoted $\leqslant_R$ and ALO, denoted $\leqslant_A$ ( see \[subsec:alphanum\]).\
We also denote : $$C(\alpha,\beta,\nu)= \#\{ k\in { \{ 0,..,\nu \} }, \{ k\alpha \}\leqslant \beta \}$$
We will denote $[a_0,a_1,\cdots,a_r,1]$ the CFE of $\alpha$ ( with $a_0=0$).\
We also use the following notations : $$\alpha_0=\alpha {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\forall k\in { \{ 1,..,r \} },\alpha_k=\left\{ \frac{1}{\alpha_{k-1}}\right\}$$ $$\nu_0=\nu {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\forall k\in { \{ 1,..,r-2 \} }, \nu_k= \begin{cases} \lfloor \nu_{k-1}\alpha_{k-1}\rfloor \text{ if } n_{k}\not = 0 \text{ or } n_{k+1}=0 \\
\lfloor \nu_{k-1}\alpha_{k-1}\rfloor +1 \text{ else } \end{cases}$$ $$\beta_0=\beta {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\forall k\in { \{ 1,..,r \} }, \beta_k= \frac{1}{\alpha_{k-1}}(b_k\alpha_{k-1}-\beta_{k-1})$$
\[prop:11\] we denote $n=(n_k)_k$ the $\alpha$-numeration of $\nu$ and $b=(b_k)_k$ the $\alpha$-numeration of $\beta$. We denote $s$ the minimum of the lengths of $n$ and $b$, when we drop the eventual infinite “ $0$-tail”. So, $n_s$ or $b_s$ is not null, but $\sigma^s(n)$ or $\sigma^s(b)$ is the null sequence.\
$$C(\alpha,\beta,\nu)=D+ \sum_{i=1}^{s } (-1)^{i-1}[ b_i \nu_i +\tau_i+\epsilon_i-\epsilon'_i]$$ where : $$D = \textbf{1}_{ n\leqslant_A b}+ \textbf{1}_{ b\leqslant_R n}-\textbf{1}_{ n=b} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\tau_i= \begin{cases} 1 \text{ if } n_in_{i+1}=0 \text{ and } \sigma^i(n)\not = (0) \\ \min(b_i,n_i) \text{ else} \end{cases}$$ $$\epsilon'_i= \begin{cases} 1 \text{ if } \sigma^i(b) <_R \sigma^i(n) \\ 0 \text{ else } \end{cases}{\hspace*{0.2cm}}; {\hspace*{0.2cm}}\epsilon_i= \begin{cases} 1 \text{ if } b_i< n_i \text{ and } \sigma^i(b) <_A \sigma^i(n) \\ 0 \text{ else } \end{cases}$$
see **\[3\]** 4.4. ( )
Invariants of $\frac{<a,b>}{d}$ {#sec:invariants}
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Let $a$ and $b$ be two coprime positive integers. We want to study the invariants of $ \frac{<a,b>}{d}$.\
We remind that we can suppose, without loss of generality, that $a,b,d$ are pairwise coprime. In addition, we can also suppose that $d\not\in <a,b>$, because $ \frac{<a,b>}{d}={\mathbb{N}}$ if $d\in <a,b>$.
multiplicity
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The multiplicity of $ \frac{<a,b>}{d}$ is the smallest positive integer $k$ such that $kd\in <a,b>$. We could deduce a formula from the result concerning the irreducible elements, because the multiplicity is the smallest element among these one. But, the following theorem is much more interesting, for it gives a very simple expression of $m( \frac{<a,b>}{d})$ in terms of “ best rational in an interval”...\
\[lem:21\] let $n$ be a positive integer and $a,b$ be two coprime positive integers. We suppose that $n=xa-yb=ab(\frac{x}{b}-\frac{y}{a})$, with $x,y$ integers. Then : $$n\in <a,b>\backslash ( a { \{ 0,..,b-1 \} }) \Leftrightarrow \left\lfloor \frac{x}{b} \right\rfloor \not = \left\lfloor \frac{y}{a} \right\rfloor$$
- we suppose that $ n\in <a,b>\backslash (a{ \{ 0,..,b-1 \} })$, then we can find two integers $i,j$, such that $n=ai+bj$, $j$ is positive and $i$ is non negative. Indeed, if $j=0$, then $n=ai$ and $i\geqslant b$, so $n=ai'+bj'$, with $i'=i-b$ and $j'=a$. So : $(x-i)a=(y+j)b$. But, $a$ and $b$ are coprime, so it exists an integer $k$ such that : $$x-i=kb \text{ and } y+j=ka$$ So : $\frac{y}{a}<k\leqslant \frac{x}{b}$ and we deduce $\left\lfloor \frac{x}{b} \right\rfloor \not = \left\lfloor \frac{y}{a} \right\rfloor$.\
- we suppose that $\left\lfloor \frac{x}{b} \right\rfloor \not = \left\lfloor \frac{y}{a} \right\rfloor$. We have $\frac{n}{ab}= \frac{x}{b}- \frac{y}{a} $ and $n$ is positive, so, there exists an integer $k$ such that : $\frac{y}{a}<k\leqslant \frac{x}{b}$. Now, we denote $i=x-kb$ and $j=ka-y$. Then, $i\in{\mathbb{N}},j\in {\mathbb{N}}^*$ and $n=ax-by=ai+bj$. So, $ n\in <a,b>$.\
Now, suppose that $n\in a{ \{ 0,..,b-1 \} }$. Then we can use our first argument with $j=0$ and $i\in { \{ 0,..,b-1 \} }$. We obtain : it exists an integer $k$ such that : $$x-i=kb \text{ and } y=ka$$ So, $\frac{y}{a}=k=\left\lfloor \frac{y}{a} \right\rfloor$. But $\frac{x}{b}=k+\frac{i}{b}\in [k,k+1[$, so $ \left\lfloor \frac{x}{b}\right\rfloor = \left\lfloor \frac{y}{a} \right\rfloor$ : contradiction.
\[theo:4\] let $a,b,d$ be three pairwise coprime positive integers such that $d\not\in <a,b>$ and $d\geqslant 2$.\
We can find $\alpha' \in \frac{1}{b}{\mathbb{Z}}$ and $\alpha\in \frac{1}{a}{\mathbb{Z}}$ such that $\alpha'-\alpha=\frac{d}{ab}$. Then :\
$$m\left(\frac{<a,b>}{d}\right) = \text{ the denominator of the reduced best rational in } [\alpha,\alpha']$$
We can write $\alpha'=\frac{x}{b}$ and $\alpha=\frac{y}{a}$, with $x$ and $y$ are integers such that $d=ax-by$ ( basic arithmetic, since $a$ and $b$ are coprime). We remark that : since $d\not\in <a,b>$, then, with our previous Lemma, $\lfloor \alpha \rfloor= \lfloor \alpha' \rfloor $. We denote $ p/q$ the reduced best rational in $[\alpha,\alpha']$. We have two cases : if $p/q=\alpha$, then $q=a$ for $y$ and $a$ are coprime ( $ d$ and $a$ are coprime). So $qd\in <a,b>$. Else, we have $qd=ab(q\alpha -a\alpha')$ and $\lfloor q\alpha \rfloor \not = \lfloor q\alpha' \rfloor$ ( see Proposition \[prop:8\]), since $p/q$ is the best rational in $]\alpha,\alpha']$ and $\alpha < \alpha'$. So our previous Lemma gives : $qd \in <a,b>$. In both cases, we have $qd\in <a,b>$.\
Now, for $k\in { \{ 1,..,q-1 \} }$, $kd=akx-bky$ and Proposition \[prop:8\] gives : $\lfloor k\alpha \rfloor = \lfloor k\alpha' \rfloor$ ( there is no rational $i/k$ in $]\alpha,\alpha']$). So : $kd\not\in <a,b>$ or $kd\in a{\mathbb{N}}$. If $kd\in a{\mathbb{N}}$, then $k\in a{\mathbb{N}}$, for $a$ and $d$ are coprime. But $q\leqslant a$, for $ \alpha=\frac{y}{a}$, so $k<a$, contradiction.
minimal generators
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$\bullet$ We begin with a result that describes precisely the minimal generators of $ \frac{<a,b>}{d}$, namely ${\mathrm{Irr}}(\frac{<a,b>}{d})$ and so its cardinality $e(\frac{<a,b>}{d})$. The case $d>\max(a,b)$ is the most intricate...
\[theo:5\] let $a,b,d$ be three pairwise coprime positive integers such that $d\not\in <a,b>$.\
Let $m\in { \{ 1,..,d-1 \} }$ such that $am+b = 0 [d]$. Let $\alpha=m/d$ and $[0,a_1,\cdots,a_r,1]$ its CFE.\
Let $(p_k/q_k)_k$ its reduced convergents and for $k\in { \{ -1,..,r+1 \} },{\hspace*{0.2cm}}\delta_k=(-1)^k(q_k\alpha-p_k)$, as defined in \[subsec:alphanum\].\
We denote $\tau=\frac{am+b}{d}$ and $\mu_i=\tau q_i-ap_i$ for all $i\in { \{ 0,..,r \} }$.\
We denote $(N_1,\cdots,N_s)$ the $\alpha$-numeration of $a-1$ ( with $N_s>0$).\
We denote $x_0=\begin{cases} d(\delta_{s-2}-N_s\delta_{s-1}) \text{ if } s \text{ is even}\\ d\delta_{s-1} \text{ else} \end{cases} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\hspace*{0.2cm}}\nu= \begin{cases} N_s \text{ if } s \text{ is even} \\ 0 \text{ else }\end{cases}$ .\
All sets below are written with pairwise distincts elements.\
$\blacktriangleright$ Case 1 : $d<a<b$. $$e\left( \frac{<a,b>}{d}\right)=3+ \sum_{k=1}^{\lfloor r/2 \rfloor }a_{2k}$$ $${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\left\{ a {\hspace*{0.2cm}}; {\hspace*{0.2cm}}b {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\tau {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\mu_{2k-2}+j\mu_{2k-1}, {\hspace*{0.2cm}}j\in{ \{ 1,..,a_{2k} \} }, k\in { \{ 1,..,\lfloor r/2 \rfloor \} } \right\}$$
$\blacktriangleright$ Case 2 : $a<d<b$ or ($a<b<d$ and $x_0+b>d$).
$$e\left( \frac{<a,b>}{d}\right)=2+ \sum_{k=1}^{\lfloor \frac{s-1}{2} \rfloor }a_{2k}+ \nu$$
$${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\left\{ a {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\tau {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\mu_{2k-2}+j\mu_{2k-1}, {\hspace*{0.2cm}}\begin{cases} j\in{ \{ 1,..,a_{2k} \} } \text{ if } k\in { \{ 1,..,\lfloor \frac{s-1}{2} \rfloor \} } \\
j\in{ \{ 1,..,N_s \} } \text{ if } k=s/2 \end{cases}\right\}$$
$\blacktriangleright$ Case 3 : $a<b<d$ and $m< x_0+b \leqslant d $.\
Same results as in Case 2, except that $a\not\in {\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)$ and so $ e\left( \frac{<a,b>}{d}\right)=1+ \sum\limits_{k=1}^{\lfloor \frac{s-1}{2} \rfloor }a_{2k}+ \nu$.\
$\blacktriangleright$ Case 4 : $a<b<d $ and $x_0+b\leqslant m$.\
We denote $\beta=\frac{x_0+b-1}{d}$ and $(b_1,b_2,\cdots, b_{s'})$ its $\alpha$-numeration, where $b_{s'}>0$.\
We also denote $t=\min\{ i>0, b_{2i+1}>0 \}$ and $t=s'/2$ if $\forall i>0, b_{2i+1}=0$ ( $s'$ is even in that case).\
Then , $a\not\in {\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)$ and : $${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\left\{ \mu_{2k-2}+j\mu_{2k-1} {\hspace*{0.2cm}}, {\hspace*{0.2cm}}j\in{ \{ c_{2k},..,a'_{2k} \} } {\hspace*{0.2cm}}, {\hspace*{0.2cm}}k\in { \{ t,..,\lfloor s/2 \rfloor \} } \right\}$$ where $c_{2k}=\begin{cases} b_{2k} \text{ if } k= t \\ 1 \text{ else } \end{cases}$ and ${\hspace*{0.2cm}}a'_{2k}= \begin{cases} a_{2k} \text{ if } {\hspace*{0.2cm}}2k<s \\ N_s \text{ if } {\hspace*{0.2cm}}2k=s \end{cases}$.\
$$e\left( \frac{<a,b>}{d}\right)=1+a_{2t}-b_{2t}+\sum_{k=t+1}^{\lfloor \frac{s-1}{2} \rfloor }a_{2k}+\nu$$
**Remark :** the hypothesis of Case 4 is equivalent to $\beta<\alpha$, that is to say $b_1=1$ and $b_2>0$.
as in section \[sec:representation\], we denote $S'= \frac{<a,b>}{d}$ and $\psi : \begin{cases} {\mathbb{Z}}^2\to \frac{1}{d}{\mathbb{Z}}\\ (x,y)\to \frac{ax+by}{d} \end{cases}$ and $L=\psi^{-1}({\mathbb{Z}})$.\
We can describe $({ \{ 1,..,d-1 \} })^2\cap L$ with the finite sequence $(d\{n\alpha \},n)_{n\in { \{ 1,..,d-1 \} }}$. Indeed :\
For all $\forall n\in { \{ 1,..,d-1 \} }, d\{n\alpha \}\in{ \{ 1,..,d-1 \} }$, for $\{ n\alpha \}\not = 0$ if $n\not = 0 [d]$, because $m$ and $d$ are coprime ( $b$ and $d$ are coprime). Moreover : $$\forall n\in { \{ 1,..,d-1 \} },ad\{n\alpha \}+bn=(am+b)n-ad\lfloor n\alpha \rfloor =0[d]$$ In addition, each $d\times d$ square in $L$ contains a unique point on each line and on each row...\
Now, let us prove that, if $a<d$, then : $x_0=\min\{ x(u), u\in L\cap ( {\mathbb{N}}\times { \{ 1,..,a-1 \} })\}$, in order to use Proposition \[prop:2\]. First, we can replace, in this formula, ${\mathbb{N}}$ by ${ \{ 0,..,d-1 \} }$, for $L$ is $(d,0)$-invariant. We have seen above that $L\cap ( { \{ 0,..,d-1 \} }\times { \{ 1,..,a-1 \} })$ is enumerated by $(d\{n\alpha \},n)_{n\in { \{ 1,..,a-1 \} }}$, because $a<d$. So, if we denote $x'_0=\min(L\cap ( { \{ 0,..,d-1 \} }\times { \{ 1,..,a-1 \} }))$, then $x'_0=d\min ( \{n\alpha\}, n\in{ \{ 1,..,a-1 \} })$. Using our $\alpha$-numeration and Lemma \[lem:18\], we obtain that this minimum is obtained for the following value of $n$ : $$n=\begin{cases} q_{s-3}+a_{s-1}q_{s-2}=q_{s-1} \text{ if } s \text{ is odd } \\ q_{s-2}+N_sq_{s-1} \text{ if } s \text{ is even } \end{cases}$$ This implies that $x_0=x'_0$ and our formula...\
$\blacktriangleright$ Case 1 : with Proposition \[prop:2\] : $${\mathrm{Irr}}(S')=\{ a,b \}\cup \psi(\min(({ \{ 1,..,d-1 \} })^2\cap L))$$ But, with remarks at the beginning of this proof and Lemma \[lem:17\], we have $ \min( ({ \{ 1,..,d-1 \} })^2\cap L))$ enumerated by $(d\{ \alpha \},1)$ and the $(d\{n\alpha \},n)$ with : $$n=(1,a_2,0,\cdots,a_{2k-2},0,j)_{\alpha}, {\hspace*{0.2cm}}k\in{ \{ 1,..,\lfloor r/2 \rfloor \} }, j\in{ \{ 1,..,a_{2k} \} }$$ We obtain ( see Proposition \[prop:10\] and formulae around) : $$n=q_{2k-2}+jq_{2k-1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\{ n\alpha \}= \delta_{2k-2}-j\delta_{2k-1}$$ But $\psi$ is into on $({ \{ 1,..,d-1 \} })^2$ since $d<a<b$, so we deduce the formula for $e\left(\frac{<a,b>}{d}\right)$. Moreover, we have $\psi(d\{ \alpha \},1)=a\{ \alpha \}+\frac{b}{d}=\tau$ and $ \forall k\in{ \{ 1,..,r \} }, j\in{ \{ 1,..,a_{2k} \} }$ : $$n=q_{2k-2}+jq_{2k-1} \Rightarrow
\psi((d\{ n\alpha \},n))=a(\delta_{2k-2}-j\delta_{2k-1})+\frac{b}{d}(q_{2k-2}+jq_{2k-1})$$
$\blacktriangleright$ Case 2 : if $a<d<b$, the arguments are essentially the same as in Case 1. The only difference is a limitation on the value of $n$ : with Proposition \[prop:2\], we have $n\in{ \{ 1,..,a-1 \} }$ instead of $n\in{ \{ 1,..,d-1 \} }$. With Lemma \[lem:18\], we obtain the result.\
If $a<b<d$ and $x_0+b>d$ : with Proposition \[prop:2\] and its notations, we have $x_1=d$ and $a\in {\mathrm{Irr}}(S')$. In addition : $${\mathrm{Irr}}(S')=\{ a \}\cup \psi(\min(({ \{ 1,..,d-1 \} }\times { \{ 1,..,a-1 \} })\cap L))$$ for $a$ is obtained with the point $(d,0)$ in $L$. So, the results are the same as in the case $a<d<b$.\
$\blacktriangleright$ Case 3 : if $a<b<d$ and $m< x_0+b \leqslant d $. With Proposition \[prop:2\] again, we have $x_1=x_0+b-1$ and $a\not\in {\mathrm{Irr}}(S')$. Likewise : $${\mathrm{Irr}}(S')= \psi(\min(({ \{ 1,..,x_0+b-1 \} }\times { \{ 1,..,a-1 \} })\cap L))$$ But $({ \{ 1,..,x_0+b-1 \} }\times { \{ 1,..,a-1 \} })\cap L$ is enumerated by the $(d\{n\alpha \},n)$, such that $\{n\alpha \}\leqslant \beta$ and $n\leqslant a-1$. Our hypothesis $ m< x_0+b$ means that $\alpha\leqslant \beta$, so the point $(d\{ \alpha \},1)$ is in $({ \{ 1,..,x_0+b-1 \} }\times { \{ 1,..,a-1 \} })\cap L$. We deduce that the condition “$\{n\alpha \}\leqslant \beta$” is dispensable for the minimal points of $({ \{ 1,..,x_0+b-1 \} }\times { \{ 1,..,a-1 \} })\cap L$. Thus, we are in the same condition as in previous Case, except for $a$...\
$\blacktriangleright$ Case 4 : if $a<b<d $ and $x_0+b\leqslant m$. Let us remark that $\beta<\alpha$, hence $b_1=1$ and $b_2>0$. We can take up the arguments of Case 3, except that the condition “$\{n\alpha \}\leqslant \beta$” is important. So, we can conclude with Lemma \[lem:19\].
$\bullet$ What are the extremal possible values of $ e\left( \frac{<a,b>}{d}\right)$ and when does it happen ?\
\[ Extremal cases\] \[prop:12\]
We keep the notations of previous theorem.\
$\blacktriangleright$ Case 1 : if $d<a<b$, then : $$3\leqslant e\left( \frac{<a,b>}{d}\right)\leqslant d+1$$ $ \triangleright {\hspace*{0.2cm}}e\left( \frac{<a,b>}{d}\right)=3\Leftrightarrow d / (a+b) $. In that case, ${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)= \{ a,b, \frac{a+b}{d} \} $.\
$\triangleright {\hspace*{0.2cm}}e\left( \frac{<a,b>}{d}\right) = d+1 \Leftrightarrow d /( b-a )$. In that case, ${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\left\{ a+ \frac{n(b-a)}{d}, n\in{ \{ 0,..,d \} } \right\} $.\
$\blacktriangleright$ Case 2 : if $a<d<b$ or ($a<b<d$ and $x_0+b>d$) , then : $$2\leqslant e\left( \frac{<a,b>}{d}\right)\leqslant a$$ $ \triangleright {\hspace*{0.2cm}}e\left( \frac{<a,b>}{d}\right)=2\Leftrightarrow \frac{m}{d}<\frac{1}{a-1}$. In that case, ${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\{ a, \tau \}$.\
$\triangleright {\hspace*{0.2cm}}e\left( \frac{<a,b>}{d}\right)=a \Leftrightarrow \frac{m}{d}>1-\frac{1}{a-1}$. In that case, ${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\left\{ a+n(\tau-a), n\in{ \{ 0,..,a-1 \} }\right\} $.\
$\blacktriangleright$ Case 3 : if $a<b<d$ and $m< x_0+b \leqslant d $, then : $$e\left( \frac{<a,b>}{d}\right)\leqslant a-1$$ $\triangleright {\hspace*{0.2cm}}e\left( \frac{<a,b>}{d}\right)=a-1 \Leftrightarrow \frac{m}{d}>1-\frac{1}{a-1}$. In that case, ${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\left\{ a+n(\tau-a), n\in{ \{ 1,..,a-1 \} }\right\} $.\
$\blacktriangleright$ Case 4 : if $a<b<d $ and $x_0+b\leqslant m$, then : $$e\left( \frac{<a,b>}{d}\right)\leqslant a-2$$ $ \triangleright {\hspace*{0.2cm}}e\left( \frac{<a,b>}{d}\right)=a-2 \Leftrightarrow \begin{cases} \{ 2\alpha \}\leqslant \beta \\
\alpha \geqslant 1- \frac{\{ 2 \alpha \}}{a-3} \end{cases}$. In that case, ${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\left\{ a+n(\tau-a), n\in{ \{ 2,..,a-1 \} }\right\} $.\
**Remark 1 :** for all of these cases, ${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)$ has elements in arithmetic progression, with expression $a+n(\tau-a)$, but the range of $n$ is different for each fo these cases.\
**Remark 2 :** if $a<d<b$ and $d$ divides $a+b$, then $m=1$, so $\frac{m}{d}<\frac{1}{a-1}$ and ${\mathrm{Irr}}\left( \frac{<a,b>}{d}\right)=\{ a, \tau \}$, this is consistent with respect to Case 1.\
Similarly, if $a<d<b$ and $d$ divides $b-a$, then $m=d-1$, so $ \frac{m}{d}>1-\frac{1}{a-1}$ and $e$ is maximal, as in Case 1.\
**Examples :**\
We take $a=151$ and $b=503$. Then : $${\mathrm{Irr}}\left( \frac{<a,b>}{6}\right)=\{ 109,151,503 \} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\mathrm{Irr}}\left( \frac{<a,b>}{32}\right)= \{ 151, 162, 173, 184, 195, 206, 217, 228,... 503 \}$$ $${\mathrm{Irr}}\left( \frac{<a,b>}{218}\right)=\{ 3, 151 \} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\mathrm{Irr}}\left( \frac{<a,b>}{176}\right)= \{ 151, 153, 155, 157, 159, 161, 163, 165, ... 451 \}$$
$\blacktriangleright$ Case 1 : the formula of Theorem \[theo:5\] (i) proves that $3\leqslant e\left( \frac{<a,b>}{d}\right)$. Proposition \[prop:2\] gives the other inequality, as there are at most $d-1$ minimal elements in ${ \{ 0,..,d-1 \} }^2$.\
— Now, $ e\left( \frac{<a,b>}{d}\right) =3$ if and only if $a_{2k}=0$ for all $k>0$, with notations of Theorem \[theo:5\]. But, the $a_i$ are positive integers for positive indices $i$, so it means that $r=0$ or $1$. This is equivalent to the fact that $\alpha=[a_0,a_1,1]$ or $\alpha=[a_0,1]$. But, $\alpha\not\in {\mathbb{Z}}$, since $d$ does not divide $b$, so : $$e\left( \frac{<a,b>}{d}\right) =3 \Leftrightarrow \frac{1}{\{\alpha \}}\in {\mathbb{N}}^* \Leftrightarrow \frac{d}{m}\in{\mathbb{N}}^*$$ But, $d$ and $m$ are coprime, so $ e\left( \frac{<a,b>}{d}\right) =3$ is equivalent to $ m=1 [d]$, that is to say $ a+b=0[d] $. In that case, we have 3 minimal generators of $ \frac{<a,b>}{d}$ : $a,b$ and $\tau$.\
— Now, $ e\left( \frac{<a,b>}{d}\right)$ is maximal if and only if all points $(d\{ n\alpha \},n)_{n\in { \{ 1,..,d-1 \} }}$ are minimal in $L\cap { \{ 1,..,d-1 \} }^2$ ( see proof of Theorem \[theo:5\]). This means that the sequence $(\{ n \alpha \})_{n\in{ \{ 1,..,d-1 \} }}$ is decreasing ( with notations of Theorem \[theo:2\]). This is equivalent to $d\{ \alpha \}=d-1$, since all these points have different $x$ and different $y$ in ${ \{ 1,..,d-1 \} }$. In that case, $d \{ n\alpha\}=d-n$ for all $n\in { \{ 1,..,d-1 \} }$. Hence, all the points above are pairwise not comparable. We can conclude : $$e\left( \frac{<a,b>}{d}\right)=d+1 \Leftrightarrow m=d-1 [d] \Leftrightarrow d / (b-a)$$ In that case, the minimal generators are : $a,b$ and the $\frac{1}{d}\left(a(d-n)+bn\right)$ where $n\in{ \{ 1,..,d-1 \} }$.\
$\blacktriangleright$ Case 2 : if $a<d<b$ or ($a<b<d$ and $x_0+b>d$), again, with Theorem \[theo:5\] and Proposition \[prop:2\], we obtain both inequalities.\
— we have with same arguments as in Case 1 : $$e(S')=2 \Leftrightarrow s=1 \Leftrightarrow a-1\leqslant a_1 \Leftrightarrow a-1 \leqslant \lfloor 1/\alpha \rfloor \Leftrightarrow a-1 \leqslant d/m$$ In that case, according to Theorem \[theo:5\], ${\mathrm{Irr}}(S')=\{ a , \tau \}$.\
— we have $e(S')=a$ if and only if all points $(d\{ n\alpha \},n)_{n\in { \{ 1,..,a-1 \} }}$ are minimal in $L\cap { \{ 1,..,d-1 \} }\times { \{ 1,..,a-1 \} }$ ( see proof of Theorem \[theo:5\]). This means that the sequence $(\{ n \alpha \})_{n\in{ \{ 1,..,a-1 \} }}$ is decreasing, that is to say $u_{n+1}-u_n=\alpha-1$ if we denote $u_0=1$ and $u_n=\{ n\alpha \}$. This is equivalent to , $(a-1)(1-\alpha)< 1$ and gives our condition. In that case, we obtain : $$\forall n\in{ \{ 1,..,a-1 \} }, \{n\alpha \}= \alpha - (n-1)(1-\alpha)= n\alpha - (n-1)$$ We deduce easily the expression of irreducible elements of $S'$.\
$\blacktriangleright$ Case 3 : if $a<b<d$ and $m< x_0+b \leqslant d $... see Theorem \[theo:5\] and Case 2.\
$\blacktriangleright$ Case 4 : if $a<b<d $ and $x_0+b\leqslant m$. We use Proposition \[prop:2\], with the fact that there is no point in $L\cap { \{ 1,..,x_1 \} }\times { \{ 0,..,a-1 \} }$, on the lines $y=0$ and $y=1$. So it remains $a-2$ lines for possible minimal points...\
— we have $e(S')=a-2$ if and only if each of the $a-2$ remaining lines contains one minimal point. This is equivalent to the fact that the sequence $(\{ n\alpha \})_{2\leqslant n\leqslant a-1}$ is decreasing and $\leqslant \beta$. This gives our conditions, after similar considerations as in Case 2. The result on ${\mathrm{Irr}}(S')$ in that case is obtained as in Case 2.
pseudo-Frobenius numbers
------------------------
We will distinguish several cases to make the statements more readable and in our proofs, we will use different “ half-turn” in ${\mathbb{Z}}^2$ to compute maximal points of a lattice in a rectangle ( see \[subsec:minimal\]) :\
$\bullet$ Let us begin with special cases, when ${\mathrm{PF}}$ is reduced to a single element or is obtained from ${\mathrm{Irr}}$ via a symmetry :
\[ Easy cases\] \[theo:6\].\
let $a,b,d$ be 3 pairwise coprime positive integers such that $d\not\in <a,b>$ and $a<b$.\
$\blacktriangleright$ Case 1 : if $(a-1)(b-1)=1[d]$, then $\frac{<a,b>}{d}$ is symmetric ( its type is one) and : $$f\left(\frac{<a,b>}{d}\right)= \frac{f(<a,b>)}{d}=\frac{ab-a-b}{d}$$ $\blacktriangleright$ Case 2 : if $a=1[d]$, then : $${\mathrm{PF}}\left(\frac{<a,b>}{d}\right)= \frac{(a-1)b}{d}-\left({\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)\backslash\{ b\} \right)$$
$\blacktriangleright$ Case 3 : if $b=1[d]$, then : $${\mathrm{PF}}\left(\frac{<a,b>}{d}\right)= \frac{(b-1)a}{d}-\left({\mathrm{Irr}}\left(\frac{<a,b>}{d}\right)\backslash\{ a\} \right)$$ For cases 2 and 3, we obtain : $t=e-1$ for $\frac{<a,b>}{d}$.
**Remark :** Case 1 is not compatible with Case 2 or Case 3. But we can have $a=b=1[d]$ : in that case, we have $d<\min(a,b)$ and $b-a=0[d]$, so ${\mathrm{Irr}}(<a,b>/d)=\left\{ a+ \frac{n(b-a)}{d}, n\in{ \{ 0,..,d \} } \right\} $. We obtain, if we denote $f=ab-a-b$ : $${\mathrm{PF}}\left(\frac{<a,b>}{d}\right)=\left\{ \frac{1}{d}[f-nb-(d-1-n)a], {\hspace*{0.2cm}}n\in{ \{ 0,..,d-1 \} }\right\}$$
we will denote $S'=\frac{<a,b>}{d}$ and, as in Lemma \[lem:7\] : $\psi$ $\begin{cases} {\mathbb{Z}}^2\to (1/d) {\mathbb{Z}}\\ (x,y)\to\frac{ax+by}{d}\end{cases}$ and : $$L=\{ (x,y)\in{\mathbb{Z}}^2, ax+by=0[d]\} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}R= { \{ b-d,..,b-1 \} }\times { \{ -d,..,-1 \} }$$
$\blacktriangleright$ Case 1 : if $(a-1)(b-1)=1[d]$, then $(b-1,-1)$ is an element of $L$. So, it is the maximum of $L\cap R$. We conclude that $\frac{<a,b>}{d}$ has only one ${\mathrm{PF}}$-number, which is $\psi(b-1,-1)$...\
$\blacktriangleright$ Case 2 : if $a=1[d]$, then $(b,-1)$ is in $L$, but not in $R$. Let us consider $\sigma : (x,y)\to (b-x,-1-y)$, that maps $L\cap R$ onto $L\cap R'$, where $R'={ \{ 1,..,d \} }\times { \{ 0,..,d-1 \} }$. So, $\max(L\cap R)= \sigma(\min(L\cap R'))$.\
But $L\cap R'=(L\cap R")\backslash \{ (0,d) \}$, if we denote $R"={ \{ 0,..,d \} }^2\backslash \{ (0,0) \}$. This implies that $\min(L\cap R')=\min(L\cap R")\backslash \{ (0,d) \}$, for $(0,d)$ is a minimal point of $L\cap R"$. In addition, $d<a<b$, so ${\mathrm{Irr}}(S')=\psi(\min(L\cap R"))=\psi(\min(L\cap R"))\cup \{ b\}$ ( see Proposition \[prop:2\]). Now, we obtain, with Lemma \[lem:7\] : $${\mathrm{PF}}(S')=\psi(\max(L\cap R))= \frac{ab-b}{d}-({\mathrm{Irr}}(S')\backslash \{ b\})$$ $\blacktriangleright$ Case 3 : if $b=1[d]$, then $(b-1,0)$ is in $L$, but not in $R$. Let us consider $\sigma' : (x,y)\to (b-1-x,-y)$. We have $d<b$. If $d<a$, we argue as in Case 2.\
Else : ${\mathrm{PF}}(S')=\psi(\max(L\cap A))$, where $A={ \{ b-d,..,b-1 \} }\times { \{ -(a-1),..,-1 \} }$ ( see Proposition \[prop:3\]). Yet, $\sigma'(L\cap A)=L\cap A'$, where $ A'={ \{ 0,..,d-1 \} }\times { \{ 1,..,a-1 \} }$. But, ${\mathrm{Irr}}(S')=\psi(\min(L\cap A'_2))$, where $A'_2= { \{ 1,..,d \} }\times { \{ 0,..,a-1 \} }$ ( see Proposition \[prop:2\]). Now $(d,0)$ is a minimal point of $L\cap A'_2$ and there is no point in $L\cap A'$ with $x=0$, so : $\min(L\cap A')=\min(L\cap A'_2)\backslash \{ (d,0)\}$ and we conclude as above...
$\bullet$ In the previous proof, we have used the half-turn $u\to v-u$ in ${\mathbb{Z}}^2$, with $v=(b,-1)$ or $v=(b-1,0)$. Now, we will use it with $v=(b-1,-1)$.
. \[theo:7\]\
let $a,b,d$ be three pairwise coprime positive integers, such that $d<\min(a,b)$ and $d$ does not divide $ab-a-b$. Let $m\in { \{ 1,..,d-1 \} }$ such that $ am+b=0[d]$.\
We denote $\alpha = \frac{m}{d}$, $[0,a_1,a_2,\cdots,a_r,1]$ its CFE and $(p_k/q_k)_k$ its convergents.\
We denote $\tau=\frac{am+b}{d}$, $\beta=1-\{ \frac{b-1+m}{d}\}$ and $(b_1,b_2,\cdots,b_{r})$ the $\alpha$-numeration of $\beta$.\
We denote $B=\{ i\in { \{ 1,..,\lfloor r/2\rfloor \} }, b_{2i}\not = 0 \}$ , $\mu_i=\tau q_i-ap_i$ for all $i\in { \{ 0,..,r \} }$ and : $$f=ab-a-b=f(<a,b>) {\hspace*{0.2cm}};{\hspace*{0.2cm}}f_1 = \frac{f}{d}-a(1-\beta) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}f_2= \frac{f}{d}-\frac{b}{d}\sum_{i=1}^rb_iq_{i-1}$$ Then :
$${\mathrm{PF}}\left(\frac{<a,b>}{d}\right)= \{ f_1,f_2 \}\cup \left\{ \frac{f}{d}+a\beta-j\mu_{2k-1}- \sum_{i=1}^{2k-1}b_i\mu_{i-1},{\hspace*{0.2cm}}j\in{ \{ 0,..,b_{2k}-1 \} }, k\in B\right\}$$
$$t\left(\frac{<a,b>}{d}\right)
= 2+ \sum\limits_{k=1}^{\lfloor r/2 \rfloor}b_{2k}$$
**Remark 1 :** if $d$ does not divide $f$ and $d<a<b$, then the type of $\frac{<a,b>}{d}$ is at least equal to 2, so $\frac{<a,b>}{d} $ is not symmetric.\
let $S'=\frac{<a,b>}{d}$. We use Lemma \[lem:7\] ( or Proposition \[prop:3\]) : $${\mathrm{PF}}(S')= \psi(\max(R\cap L ))$$ where $\psi : \begin{cases} {\mathbb{Z}}^2\to \frac{1}{d}{\mathbb{Z}}\\ (x,y)\to \frac{ax+by}{d} \end{cases} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}L=\psi^{-1}({\mathbb{Z}})$ and $R= { \{ b-d,..,b-1 \} }\times { \{ -d,..,-1 \} } $.\
But, we can describe $R\cap L$ with the finite sequence $(b-1-g(n),-n-1)_{n\in{ \{ 0,..,d-1 \} }}$, where $g(n)$ is the integer of ${ \{ 0,..,d-1 \} }$ such that $a(b-1-g(n))-b(n+1)=0[d]$. We obtain : $\forall n\in{ \{ 0,..,d-1 \} },$ $$a(b-1-g(n))-b(n+1)=0[d] \Leftrightarrow ag(n)=a(b-1+m+mn)[d] \Leftrightarrow g(n)=b-1+m+mn [d]$$ Indeed, $d$ and $a$ are coprime. So : $$\forall n\in{ \{ 0,..,d-1 \} } , {\hspace*{0.2cm}}g(n)= d \left\{ \frac{b-1+m+mn}{d} \right\}$$ We remark that : $$g(0)=0 \Leftrightarrow b-1+m=0[d] \Leftrightarrow a(b-1+m)=0 [d] \Leftrightarrow ab-a-b=0[d]$$ Since $f\not = 0 [d]$, then $g(0)\not = 0$ and : $$\max((b-1-g(n),-n-1)_{n\in{ \{ 0,..,d-1 \} }})= \sigma ( \min((g(n),n)_{n\in{ \{ 0,..,d-1 \} }}))$$ Where $\sigma : \begin{cases} {\mathbb{Z}}^2\to {\mathbb{Z}}^2 \\ (x,y)\to (b-1-x,-1-y) \end{cases}$.\
We use now Lemma \[lem:20\] ( and Remark 3 below it), that gives minimal points of $(\{ n\alpha - \beta \},n)_{n\in{ \{ 0,..,d-1 \} }}$, where $\alpha= \frac{m}{d}$ and $\beta=\{ - \frac{b-1+m}{d}\}=1-\{ \frac{b-1+m}{d}\} \in ]0,1[$, since $d$ does not divide $b-1+m$ ( see above).\
For a general $n$ ( corresponding to a minimal point of $(\{ n\alpha - \beta \},n)_{n\in{ \{ 0,..,d-1 \} }}$), we obtain the PF-number of $S'$ : $$\frac{f-bn}{d}-a\{n\alpha-\beta \}$$ For $n=0$, we have $g(0)=d(1-\beta)$ ( this corresponds to the trivial minimal point $(1-\beta,0)$). This gives the PF-number for $S'$ : $f_1=\frac{f}{d}-a(1-\beta)$.\
For $n$, whose $\alpha$-numeration is $(b_k)_k$ ( this means that $n$ is the integer in ${ \{ 0,..,d-1 \} }$, such that $\{ n\alpha \}=\beta$), we obtain $\{ n\alpha-\beta\}=0$ and so this gives the PF-number for $S'$ : $f_2=\frac{f-nb}{d}$, where $n=\sum\limits_{i=1}^rb_iq_{i-1}$.\
For the general case $n$, whose $\alpha$-numeration is $(b_1,\cdots,b_{2k-1},j)$, with $j\in{ \{ 0,..,b_{2k}-1 \} }$ and $k\in B$, we obtain : $$\frac{f}{d}-\frac{b}{d}\left( \sum_{i=1}^{2k-1}b_iq_{i-1}+jq_{2k-1}\right)-a\left( \sum_{i=1}^{2k-1}b_i(q_{i-1}\alpha-p_{i-1})+j(q_{2k-1}\alpha-p_{2k-1})\right)+a\beta$$ Indeed, $\{n\alpha - \beta \}=\{ n\alpha \}-\beta$, becasue $\{ n\alpha \}<\beta$ for these values of $n$. ( see Lemma \[lem:20\]). This gives our formula...\
For the type $t(S')$, we remark that $\psi$ is injective over $R$, if $d<\min(a,b)$ ( see Proposition \[prop:3\]).
$\bullet$ To be complete, we end with the case $d>\min(a,b)$, that is somwhat different : in the proof we use an half-turn $u\to v-u$, with $v=(b,0)$.
\[ The case $d>\min(a,b)$\]. \[theo:8\]\
let $a,b,d$ be 3 pairwise coprime integers, such that $1<\min(a,b)<d$ and $d$ does not divide $ab-a-b$.\
Let $m\in { \{ 1,..,d-1 \} }$ such that $ am+b=0[d]$. We denote $\alpha = \frac{m}{d}$, $[0,a_1,a_2,\cdots,a_r,1]$ its CFE and $(p_k/q_k)_k$ its convergents.\
We denote $\tau=\frac{am+b}{d}$ ( a positive integer), $\beta=\{ a\alpha \}=1- \left\{\frac{b}{d}\right\},(b_1,b_2,\cdots,b_s)$ the $\alpha$-numeration of $\beta$, where $b_s>0$ and $\mu_i=\tau q_i-ap_i$ for all $i\in { \{ 0,..,r \} }$.\
We also denote $B=\{ i\in { \{ 1,..,\lfloor s/2\rfloor \} }, b_{2i}\not = 0 \}$ and $k_0= \begin{cases} (s+1)/2 \text{ if } B=\emptyset \\ \min(B) \text{ else } \end{cases} $ ( $s$ is odd in the first case). Last notations :
$${\mathcal{F}}_0 = \left\{ a\left\lfloor \frac{b}{d}\right \rfloor - \tau \right\} \text{ if } \alpha < \beta {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\mathcal{F}}_0=\emptyset \text{ else }$$ $${\mathcal{F}}_1= \left\{ a\left\lfloor \frac{b}{d}\right \rfloor - \mu_{2k-2}-j\mu_{2k-1} , j\in{ \{ 1,..,a_{2k} \} }, k\in { \{ 1,..,k_0-1 \} }\right\}$$
$${\mathcal{F}}_2= \left\{ a\left\lfloor \frac{b}{d}\right \rfloor + a -j\mu_{2k-1}- \sum_{i=1}^{2k-1}b_i\mu_{i-1}, j\in{ \{ 0,..,b_{2k}-1 \} }, k\in B \right\}$$
$\blacktriangleright $ Case 1 : if $ a<d<b $ $${\mathrm{PF}}\left(\frac{<a,b>}{d}\right)= {\mathcal{F}}_0\sqcup {\mathcal{F}}_1\sqcup {\mathcal{F}}_2$$ $$t\left(\frac{<a,b>}{d}\right) = \sum_{k=1}^{k_0-1}a_{2k}+\sum\limits_{k=k_0}^{\lfloor s/2 \rfloor}b_{2k}+ \textbf{1}_{\alpha<\beta}$$
$\blacktriangleright $ Case 2 : if $ d>\max(a,b) $ $${\mathrm{PF}}\left(\frac{<a,b>}{d}\right)= {\mathcal{F}}_2 {\hspace*{0.2cm}}; {\hspace*{0.2cm}}t\left(\frac{<a,b>}{d}\right) =\sum\limits_{k=k_0}^{\lfloor s/2 \rfloor}b_{2k}$$
**Remark :** $(b_k)_k$ is also the $\alpha$-numeration of $a$. Moreover, if $d>\max(a,b)$, we can not have $b_{2i}=0$ for all $i$, because ${\mathrm{PF}}(<a,b>/d)$ can not be empty !\
If $a<d<b$ and $\alpha>\beta$, that is if $b_1=1$ and $b_2>0$, then $k_0=1$ and ${\mathcal{F}}_1$ is empty.\
We suppose that $a<b$, so $a<d$. This proof will be similar to the proof of Theorem \[theo:7\]. We mention the notations used in this one : $S'=\frac{<a,b>}{d}$ and with Proposition \[prop:3\] : $${\mathrm{PF}}(S')= \psi(\max(R\cap L )) {\hspace*{0.2cm}}\text{ where } {\hspace*{0.2cm}}\psi : \begin{cases} {\mathbb{Z}}^2\to \frac{1}{d}{\mathbb{Z}}\\ (x,y)\to \frac{ax+by}{d} \end{cases}$$ and $ L=\psi^{-1}({\mathbb{Z}}) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}R= { \{ b-\delta,..,b-1 \} }\times { \{ -(a-1),..,-1 \} } $, where $\delta=\min(b-1,d)$.\
The difference with the proof of Theorem \[theo:7\] lies in $R$ and in the parametrization of $R\cap L$ : we will use the sequence $(b-h(n),-n)_{n\in{ \{ 1,..,a-1 \} }}$, where $h(n)$ is the integer of ${ \{ 1,..,d \} }$ such that $a(b-h(n))-bn=0[d]$. But, in the case $d>\max(a,b)$, we will ask that $h(n)<b$ !!\
$\blacktriangleright $ Case 1 : $a<d<b$. We obtain : $$\forall n\in{ \{ 1,..,a-1 \} }, {\hspace*{0.2cm}}ah(n)=ab-nb=(n-a)am[d]\Leftrightarrow h(n)=nm-am[d]$$ Yet $a<d$ and $(b,-a)$ is a point of $L$, so there is no point of $L$ among the $(b,-n)_{n\in{ \{ 1,..,a-1 \} }}$. Hence, $h(n)\not=0[d]$ for $n\in{ \{ 1,..,a-1 \} }$. So : $$\forall n\in{ \{ 1,..,a-1 \} },{\hspace*{0.2cm}}h(n)=d\left\{ (n-a)\alpha \right\}=d\left\{ n\alpha-\beta \right\}$$ So, $L\cap R$ is the image of $U=(h(n),n)_{n\in{ \{ 1,..,a-1 \} }}$ by the map $\sigma : (x,y)\to (b-x,-y)$ and : $$\max(L\cap R)= (b,0)-\min(U)$$ If $(h(n),n)$ is a minimal point of $U$, then we obtain the following PF number of $S'$ : $$\psi(b-h(n),-n)= \frac{b(a-n)}{d}- a\{ n\alpha - \beta \}$$ Now, we use Lemma \[lem:20\] and first remark that $n=a$, that is $n=(b_1,\cdots,b_s)_{\alpha}$, is excluded. Secondly, we distinguish two kinds of minimal point of $(\{ n\alpha - \beta \},n)_{n\in { \{ 1,..,a-1 \} }}$ :\
- those that satisfy $\{ n\alpha \}<\beta$, which corresponds to ( see Proof of Theorem \[theo:7\]) : $$n=q_{2k-2}+ j q_{2k-1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\{ n\alpha \}= \delta_{2k-2}-j\delta_{2k-1}, {\hspace*{0.2cm}}j\in{ \{ 1,..,a_{2k} \} }, k\in { \{ 1,..,k_0-1 \} }$$ For these minimal points, we have $\{ n\alpha - \beta \}=\{ n\alpha \}+1-\beta= \{ n\alpha \}+ \frac{b}{d}-\lfloor b/d \rfloor$. We deduce the formula for the elements of ${\mathcal{F}}_1$.\
- those that satisfy $\{ n\alpha \}\geqslant \beta$, which corresponds to : $$n=\sum_{i=1}^{2k-1}b_iq_{i-1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}\{ n\alpha \}= \sum_{i=1}^{2k-1}b_i(q_{i-1}\alpha - p_{i-1}), {\hspace*{0.2cm}}j\in{ \{ 0,..,b_{2k}-1 \} }, k\in { \{ k_0,..,\lfloor s/2 \rfloor \} }$$ For these minimal points, we have $\{ n\alpha - \beta \}=\{ n\alpha \}-\beta= \{ n\alpha \}+ \frac{b}{d}-\lfloor b/d \rfloor-1$. We deduce the formula for the elements of ${\mathcal{F}}_2$.\
Now, if we want to conclude for PF-numbers of $S'$, we just have to remark that Lemma \[lem:20\] also gives a minimal point of $U$ for $n=1$, which is $(d\{ \alpha-\beta\},1)$. If $\alpha > \beta$, that point is in ${\mathcal{F}}_2$ and already counted. Else, it gives a new PF-number : $ a\lfloor b/d \rfloor - \tau$.\
For the type, we just have to count elements of ${\mathrm{PF}}(S')$, keeping in mind that the sets ${\mathcal{F}}_0,{\mathcal{F}}_1,{\mathcal{F}}_2$ are pairwise disjoints and that the parametrization of their elements is injective.\
$\blacktriangleright $ Case 2 : if $d>\max(a,b)$. We keep above arguments with the additional condition that $h(n)<b$ : this gives the condition $\{ n\alpha - \beta \}< 1-\beta$, that is $\{n\alpha \}\geqslant \beta$. We conclude with Lemma \[lem:20\].
**Remark :** unlike in the case $d<\min(a,b)$, we can find $a,b,d$ such that $a<d<b$, $d$ does not divide $f$ and $\frac{<a,b>}{d}$ is symmetric : for example, with $a=11, b=89$ and $d=20$, we obtain $\frac{<a,b>}{d}=<5,11>$.\
the Frobenius number
--------------------
$\bullet$ Once we can compute PF-numbers of a numerical semigroup, we know that the Frobenius number is the greatest of them. In the expressions of PF-numbers that appear in Theorems \[theo:7\] and \[theo:8\], we have two parameters $j$ and $k$ that have values in a finite interval of integers. Moreover, we remark that, if $k$ is fixed, then these expressions on $j\to H_k(j)$ are affine, so monotonous. Hence, the maximum is obtained for an extremal value of $j$ : that is to say $0$ or $1$ or $a_{2k}$ or $b_{2k}-1$ ( if $b_{2k}\not = 0$ !). Remark that $(b_k)_k$ is not the same for Theorem \[theo:7\] and \[theo:8\].\
We denote $B=\{ i\in { \{ 1,..,\lfloor r/2 \rfloor \} }, b_{2i}\not = 0 \}$ and obtain :
$\blacktriangleright $ if $d<\min(a,b)$ $$f\left(\frac{<a,b>}{d}\right)=\max\left( f_1,f_2, \frac{f}{d}+a\beta-j\mu_{2k-1}- \sum_{i=1}^{2k-1}b_i\mu_{i-1}, {\hspace*{0.2cm}}j=0, j =b_{2k}-1, k\in B\right)$$
$\blacktriangleright $ if $a<d<b$ $$f\left(\frac{<a,b>}{d}\right)=\max( a\left\lfloor \frac{b}{d}\right \rfloor - \tau , a\left\lfloor \frac{b}{d}\right \rfloor - \mu_{2k-2}-j\mu_{2k-1} , {\hspace*{0.2cm}}j=1,j=a_{2k}, k\in { \{ 1,..,k_0-1 \} },$$ $$, a\left\lfloor \frac{b}{d}\right \rfloor + a -j\mu_{2k-1}- \sum_{i=1}^{2k-1}b_i\mu_{i-1},{\hspace*{0.2cm}}j=0, j=b_{2k}-1, k\in B )$$
$\blacktriangleright $ if $d>\max(a,b)$ : $$f\left(\frac{<a,b>}{d}\right)=\max\left( a\left\lfloor \frac{b}{d}\right \rfloor +a-j\mu_{2k-1}- \sum_{i=1}^{2k-1}b_i\mu_{i-1},{\hspace*{0.2cm}}j=0, j =b_{2k}-1, k\in B\right)$$
Now, for which $k$ do we obtain a maximum ? To answer this question, we think of our representation ( $\psi : (x,y)\to (ax+by)/d$) of $S=\frac{<a,b>}{d}$ in ${\mathbb{Z}}^2$ and remember that the points representing ${\mathrm{PF}}(S)$ ( see Lemma \[lem:8\] and Proposition \[prop:3\]) can be parametrized by an $x$-decreasing sequence $(u_k)_{k\in{ \{ 1,..,t \} }}$ that is order-concave in ${\mathbb{Z}}^2$ ( with the partial product order), which means that : $(u_k-u_{k-1})_k$ is decreasing. But Theorem \[theo:7\] and \[theo:8\] give the PF-numbers of $S$, say $(\psi(u_i))_i$, respecting the previous order ( see proof of these theorems). Consequence : if we denote $(h_i)_i$ that sequence of integers, then $(h_i-h_{i-1})_i$ is decreasing, for $\psi$ is increasing ( $a$ and $b$ are positive). So, $h_i$ is the Frobenius number of $S$ for the unique $i$ such that $h_i-h_{i-1}>0$ and $h_{i+1}-h_i<0$.\
Yet, the slope of our affine functions $H_k$ is $-\mu_{2k-1}$, see notations in Theorem \[theo:7\] and \[theo:8\]. So, $(\mu_{2k-1})_k$ is increasing and we will discuss the sign of $\mu_{2k-1}$. We recall that : $$\forall i\in{ \{ 0,..,r \} }, \mu_i=\tau q_i-ap_i=aq_i\left(\alpha+\frac{b}{ad}-\frac{p_i}{q_i}\right)$$ where $a,b,d$ are three positive pairwise coprime integers, $\alpha=\frac{m}{d}$, $m\in{ \{ 1,..,d-1 \} }$ coprime with $d$, such that $am+b=0[d]$ and $(p_i/q_i)_{i\in{ \{ 0,..,r \} }}$ are the usual convergents of $\alpha$. So, for all integer $k\in{ \{ 1,..,\lfloor (r+1)/2 \rfloor \} }$ : $$\mu_{2k-1}>0 {\hspace*{0.2cm}}\Leftrightarrow {\hspace*{0.2cm}}\frac{p_{2k-1}}{q_{2k-1}}-\alpha < \frac{b}{ad} {\hspace*{0.2cm}}\Leftarrow {\hspace*{0.2cm}}q_{2k-1}^2> \frac{ad}{b}$$ But, we will need more precise notations...
\[theo:9\] let $a,b,d$ be three pairwise coprime integers.\
Let $m\in { \{ 1,..,d-1 \} }$ such that $ am+b=0[d]$. We denote $\alpha = \frac{m}{d}$, $[0,a_1,a_2,\cdots,a_r,1]$ its CFE and $(p_k/q_k)_k$ its convergents. We denote $\tau=\frac{am+b}{d}, \mu_i=\tau q_i-ap_i$ for $i\in { \{ -1,..,r \} }$ , $f=f(<a,b>)=ab-a-b$.
$$A_-=\left\{ i, b_{2i}\not = 0 \text{ and } \mu_{2i-1}<0 \right\} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}A_+=\left\{ i, b_{2i}\not = 0 \text{ and } \mu_{2i-1}>0 \right \}$$ $$k_1=\max(A_-) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}k_2=\min(A_+)$$
$\blacktriangleright$ Case 1 : if $d$ divides $f$ then $f\left(\frac{<a,b>}{d}\right)=\frac{f}{d}$.\
$\blacktriangleright$ Case 2 : if $d<\min(a,b)$, then we denote $\beta=1-\left\{\frac{b+m-1}{d}\right\}=(b_1,\cdots,b_r)_{\alpha}$.
$$\text{ if } A_-=\emptyset {\hspace*{0.2cm}}, {\hspace*{0.2cm}}f\left(\frac{<a,b>}{d}\right)=\frac{f}{d}+a\beta-a$$ $$\text{ if } A_+=\emptyset {\hspace*{0.2cm}}, {\hspace*{0.2cm}}f\left(\frac{<a,b>}{d}\right)=\frac{f}{d}-\frac{b}{d}\sum_{i=1}^rb_iq_{i-1}$$ $$\text{ else } {\hspace*{0.2cm}}, {\hspace*{0.2cm}}f\left(\frac{<a,b>}{d}\right)=\frac{f}{d}+a\beta-\min\left(-\mu_{2k_1-1}+\sum_{i=1}^{2k_1}b_i\mu_{i-1},\sum_{i=1}^{2k_2-1}b_i\mu_{i-1}\right)$$
$\blacktriangleright$ Case 3 : if $a<d<b$, we denote $1-\left\{\frac{b}{d}\right\}=(b_1,\cdots,b_r)_{\alpha}$. $$\gamma = \min \left( i\in{ \{ 1,..,\lfloor (r+1)/2 \rfloor \} },+\mu_{2i-1}>0 \right) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}k_0= \begin{cases} k_0=\lfloor r/2\rfloor \text{ if } b_{2i}=0 \text{ for all integer } i \\ \min(i,b_{2i}\not = 0) \text{ else} \end{cases}$$
$$\text{ if } \gamma < k_0 {\hspace*{0.2cm}}, {\hspace*{0.2cm}}f\left(\frac{<a,b>}{d}\right)=a\left\lfloor \frac{b}{d}\right\rfloor-\mu_{2\gamma-2}$$
$$\text{ if } \gamma > k_0 {\hspace*{0.2cm}}, {\hspace*{0.2cm}}f\left(\frac{<a,b>}{d}\right)=a\left\lfloor \frac{b}{d}\right\rfloor+a-\min\left(-\mu_{2k_1-1}+\sum_{i=1}^{2k_1}b_i\mu_{i-1},\sum_{i=1}^{2k_2-1}b_i\mu_{i-1}\right)$$
$$\text{ if } \gamma = k_0 {\hspace*{0.2cm}}, {\hspace*{0.2cm}}f\left(\frac{<a,b>}{d}\right)=a\left\lfloor \frac{b}{d}\right\rfloor-\min\left(\mu_{2k_0-2},-a+\sum_{i=1}^{2k_0-1}b_i\mu_{i-1}\right)$$
$\blacktriangleright$ Case 4 : if $d>\max(a,b)$, , we denote $1-\left\{\frac{b}{d}\right\}=(b_1,\cdots,b_r)_{\alpha}$. Then : $$\text{ else } {\hspace*{0.2cm}}, {\hspace*{0.2cm}}f\left(\frac{<a,b>}{d}\right)=a\left\lfloor \frac{b}{d}\right\rfloor+a-\min\left(-\mu_{2k_1-1}+\sum_{i=1}^{2k_1}b_i\mu_{i-1},\sum_{i=1}^{2k_2-1}b_i\mu_{i-1}\right)$$
**Question :** could we find a simpler formula for $f(\frac{<a,b>}{d})$ ? If we refer to our proof and notations of Theorem \[theo:4\], we can claim that : $$f\left(\frac{<a,b>}{d}\right)=\max\{ k\in{\mathbb{N}}\backslash (a { \{ 0,..,b-1 \} }), \lfloor k\alpha \rfloor = \lfloor k\alpha' \rfloor \}$$ But, it is not so easy to compute this max...\
$\blacktriangleright$ Case 1 : has already been shown at Theorem \[theo:6\].\
$\blacktriangleright$ Case 2 : we use and refer to Theorem \[theo:7\]. We remind the result of this Theorem : if we denote $B=\{ i\in { \{ 1,..,\lfloor r/2\rfloor \} }, b_{2i}\not = 0 \}, {\hspace*{0.2cm}}f_1 = \frac{f}{d}-a(1-\beta) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}f_2= \frac{f}{d}-\frac{b}{d}\sum\limits_{i=1}^rb_iq_{i-1}$, then :
$${\mathrm{PF}}\left(\frac{<a,b>}{d}\right)= \{ f_1,f_2 \}\cup \left\{ \frac{f}{d}+a\beta-j\mu_{2k-1}- \sum_{i=1}^{2k-1}b_i\mu_{i-1},{\hspace*{0.2cm}}j\in{ \{ 0,..,b_{2k}-1 \} }, k\in B\right\}$$
$\blacktriangleright\blacktriangleright$ subcase 1 : if $A_-=\emptyset$, then $\mu_{2k-1}>0$ for all $k$ in $B$, with notations of Theorem \[theo:7\]. So, the Frobenius is obtained for the “first” pseudo-Frobenius in the list, namely $f_1$.\
$\blacktriangleright\blacktriangleright$ subcase 2 : if $A_+=\emptyset$, then $\mu_{2k-1}<0$ for all $k$ in $B$, with notations of Theorem \[theo:7\]. So, the Frobenius is obtained for the “last” pseudo-Frobenius in the list, namely $f_2$.\
$\blacktriangleright\blacktriangleright$ subcase 3 : if both $A_-$ and $A_+$ are non empty, then the Frobenius is obtained for one of the two following pseudo-Frobenius, listed in Theorem \[theo:7\] : $$f_-=\frac{f}{d}+a\beta- ( b_{2k_1}-1)\mu_{2k_1-1}-\sum_{i=1}^{2k_1-1}b_i\mu_{i-1} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}f_+=\sum_{i=1}^{2k_2-1}b_i\mu_{i-1}$$ $f_-$ being the greater of PF-numbers parametrized by $k$ such that $\mu_{2k-1}<0$ ( $k=k_1$ and $j=b_{2k-1}-1$) and $f_+$ being the greater of PF-number parametrized by $k$ such that $\mu_{2k-1}>0$ ( $k=k_2$ and $j=0$). Our formula is a direct consequence.\
$\blacktriangleright$ Case 3 : we use and refer to Theorem \[theo:8\]. We remind the principle notations and result in that case :
$${\mathcal{F}}_0 = \left\{ a\left\lfloor \frac{b}{d}\right \rfloor - \tau \right\} \text{ if } \alpha < \beta {\hspace*{0.2cm}}; {\hspace*{0.2cm}}{\mathcal{F}}_0=\emptyset \text{ else }$$ $${\mathcal{F}}_1= \left\{ a\left\lfloor \frac{b}{d}\right \rfloor - \mu_{2k-2}-j\mu_{2k-1} , j\in{ \{ 1,..,a_{2k} \} }, k\in { \{ 1,..,k_0-1 \} }\right\}$$
$${\mathcal{F}}_2= \left\{ a\left\lfloor \frac{b}{d}\right \rfloor + a -j\mu_{2k-1}- \sum_{i=1}^{2k-1}b_i\mu_{i-1}, j\in{ \{ 0,..,b_{2k}-1 \} }, k\in { \{ k_0,..,\lfloor s/2 \rfloor \} }\right\}$$
$${\mathrm{PF}}\left(\frac{<a,b>}{d}\right)= {\mathcal{F}}_0\sqcup {\mathcal{F}}_1\sqcup {\mathcal{F}}_2$$
$\blacktriangleright\blacktriangleright$ subcase 1 : if $\gamma < k_0$, then the Frobenius of $S$ is among the elements of ${\mathcal{F}}_1$, for $\mu_{2k-1}>0$ for $k\geqslant k_0-1$. It is obtained for one of the following pseudo-Frobenius listed in ${\mathcal{F}}_1$ : $$f_-= a\left\lfloor \frac{b}{d}\right\rfloor- \mu_{2\gamma -4}-a_{2\gamma -2}\mu_{2\gamma -3} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}f_+= a\left\lfloor \frac{b}{d}\right\rfloor- 0- \mu_{2\gamma-2}$$ $f_-$ being the greater of PF-numbers parametrized by $k$ such that $\mu_{2k-1}<0$ ( $k=\gamma -1$ and $j=a_{2\gamma -2}$) and $f_+$ being the greater of PF-number parametrized by $k$ such that $\mu_{2k-1}>0$ ( $k=\gamma $ and $j=0$). But, $f_+=f_-$ and we deduce our formula.\
$\blacktriangleright\blacktriangleright$ subcase 2 : if $\gamma > k_0$, then the Frobenius of $S$ is among the elements of ${\mathcal{F}}_2$, becasue $\mu_{2k-1}<0$ for $k= k_0$. Now, the arguments are the same as in Case 1 subcase 3...\
$\blacktriangleright\blacktriangleright$ subcase 3 : if $\gamma = k_0$, then $\mu_{2k_0-3}<0$ and $\mu_{2k_0-1}>0$. So, the Frobenius of $S$ is one of the following integers : $$f_-= a\left\lfloor \frac{b}{d}\right\rfloor-\mu_{2k_0-2} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}f_+= a\left\lfloor \frac{b}{d}\right\rfloor-a- \sum_{i=1}^{2k_0-1}b_i\mu_{i-1}$$ $f_-$ being the greater of PF-numbers parametrized by $k$ such that $\mu_{2k-1}<0$ ( $k=k_0 -1$ and $j=a_{2k_0 -2}$) and $f_+$ being the greater of PF-number parametrized by $k$ such that $\mu_{2k-1}>0$ ( $k=k_0 $ and $j=0$).\
$\blacktriangleright$ Case 4 : the arguments are the same as in Case 3 subcase 2, for we only consider ${\mathcal{F}}_2$...
genus
-----
\[theo:10\] Let $a,b,d$ be three pairwise coprime positive integers and $m$ an integer such that $b=am$ mod $d$. $$g\left(\frac{<a,b>}{d}\right)
= \frac{g(<a,b>)}{d}+ \frac{1}{2}\left[C(\alpha,\beta,\nu)-1- \beta\nu\right]$$
where $\alpha = \left\{ \frac{m}{d} \right \}, \beta = \left\{ \frac{b-1}{d} \right \},\nu= d \left\{ \frac{a-1}{d} \right \}$ and $C=\#\{ k\in{ \{ 0,..,\nu \} }, \{ k\alpha \}\leqslant \beta \}$.\
( see Proposition \[prop:11\] in \[subsec:counting\] for an expression of $C$).\
In particular, we obtain : $$a=1 {\hspace*{0.2cm}}[d] \text{ or } b =1 {\hspace*{0.2cm}}[d] \Rightarrow g\left(\frac{<a,b>}{d}\right)
= \frac{g(<a,b>)}{d}= \frac{(a-1)(b-1)}{2d}$$
According to Proposition \[prop:1\], we have to count points in $R\cap L$, where : $$R={ \{ 1,..,b-1 \} }\times { \{ -(a-1),..,-1 \} } {\hspace*{0.2cm}}; {\hspace*{0.2cm}}L=\{ (x,y)\in{\mathbb{Z}}^2, ax+by=0 \text{ mod } m \}$$ But, we know that $L$ contains exactly one point one every “ horizontal” or “ vertical” segment of length $d$. So, if we denote $N=\# ( R\cap L)$, we have : $$N= (b-1)\left\lfloor \frac{a-1}{d} \right\rfloor + d\left\{ \frac{a-1}{d} \right\}\left\lfloor \frac{b-1}{d} \right\rfloor + N'$$ where the first term counts points of $L$ in the sub-rectangle ${ \{ 1,..,b-1 \} }\times { \{ -d\left\lfloor \frac{a-1}{d} \right\rfloor,..,-1 \} }$. The second term counts points of $L$ in the sub-rectangle ${ \{ 1,..,d\left\lfloor \frac{b-1}{d} \right\rfloor \} }\times { \{ -(a-1),..,-d\left\lfloor \frac{a-1}{d} \right\rfloor-1 \} }$. And finally, $N'$ counts points of $L$ in the subrectangle : ${ \{ d\left\lfloor \frac{b-1}{d} \right\rfloor +1,..,b-1 \} }\times { \{ -(a-1),..,-d\left\lfloor \frac{a-1}{d} \right\rfloor-1 \} }$. But, since $L$ is invariant via translation of $(d,0)$ and $(0,d)$, $N'$ is the number of points of $L$ in the rectangle ${ \{ 1,..,{ \} }(b-1)\%d}\times { \{ {,..,- \} }(a-1)\%d}{-1}$, where $\%$ denotes the remainder of the euclidean division.\
To express $N'$, we use the following parametrization of points of $L$ in a rectangle : $$L\cap{ \{ 0,..,d-1 \} }\times { \{ {,..,- \} }(a-1)\% d}{-1}=\{ (x_k,-k) , k\in{ \{ 1,..,{ \} }(a-1)\% d}, \text{ such that } ax_k-bk=0 \text{ mod } d \}$$ but : $$ax_k-bk=0 \text{ mod }d \Leftrightarrow
ax_k=amk \text{ mod } d \Leftrightarrow x_k=mk \text{ mod } d$$ indeed $a$ and $d$ are coprime. But, $x_k\in { \{ 0,..,d-1 \} }$, so $x_k = d \left\{ \frac{km}{d}\right\}$. Now, we count points with $x_k\in{ \{ 1,..,{ \} }(b-1)\%d}$, so the condition on $x_k$ is expressed by $0< \{ \frac{km}{d} \} \leqslant \{ \frac{b-1}{d}\}$. It gives : $$N'= C(\alpha,\beta,\nu)-1$$ ( $-1$ because we omit $k=0$), with $\alpha = \left\{ \frac{m}{d} \right \}, \beta = \left\{ \frac{b-1}{d} \right \},\nu= d \left\{ \frac{a-1}{d} \right \}$ and $C$ defined as above.\
If $d$ divides $a-1$ or $b-1$, then $\beta$ or $\nu$ is null, so $C(\alpha,\beta,\nu)=1$ and $\beta\nu=0$.
**Remark 1 :** that expression of $g$ is not symmetric in terms of $a$ and $b$, even if the product $\beta\nu$ is. But it proves that : $$C(\alpha,\beta,\nu)=C(\alpha',\beta',\nu')$$ where $\alpha'=\{\frac{m'}{d}\}$ and $m'$ is the inverse of $m$ mod $d$, $\beta'= \frac{\nu}{d}, \nu'=d\beta$.\
**Remark 2 :** $C(\alpha,\beta,\nu)-1$ is the number of integer $k$ such that $k\in{ \{ 1,..,\nu \} }$ and $\{ k\alpha \}\leqslant \beta$. But $\beta$ is the probability that a real in $[0,1[$ lies in $[0,\beta]$. So, if we choose randomly $\nu$ reals in $[0,1[$, then $\beta\nu$ is the expectation of the number of these reals that lie in $[0,\beta]$.\
Hence, the term $C(\alpha,\beta,\nu)-1-\beta\nu$ can be interpreted as a “signed deviation from the random mean value”...\
**Remark 3 :** this expression of $g$ and especially that of $f$ is too intricate to help us about Wilf’s property...
comparison of $t$ and $e$, Wilf’s property
------------------------------------------
$\bullet$ We have already mentionned that minimal generators and pseudo-Frobenius numbers of a numerical semigroup $S$ have similar properties : they are respectively minimal and maximal sets of $S\backslash \{ 0 \}$ and ${\mathbb{Z}}\backslash S$ for the order induced by $S$. That kind of symmetry could imply some relation between their finite cardinality. Unfortunately, this is not the case : we have $t(S)<e(S)$, if $e(S)\leqslant 3$, but for $e(S)\geqslant 4, t(S)$ can be as large as we want ( **\[7\]** ). Nonetheless, we know that $t(S)<m(S)$, considering rests modulo $m(S)$.\
So, this is remarkable that $t(S)<e(S)$ when $S$ is a proportionally modular semigroup, that is to say $S=\frac{<a,b>}{d}$, for some three pairwise coprime integers $a,b,d$ :
\[theo:11\] if $a,b,d$ are 3 pairwise coprime positive integers such that $d\not\in <a,b>$, then :
$$t\left(\frac{<a,b>}{d}\right)< e\left(\frac{<a,b>}{d}\right)$$ So, Wilf’s property holds for these numerical semigroups !
**Remark :** before presenting a proof, we want to underline that, depending on the respective positions of $a,b,d$, this proof will be either obvious or laborious ( never difficult nonetheless). We are however convinced that this inequality, between $e$ and $t$, could be deduced from a deeper result about minimal points in a lattice of ${\mathbb{Z}}^2$. This will be perhaps the subject of a next article...\
To make this 3-pages proof readable, here is a summary : since we use Theorem \[theo:5\], \[theo:6\], \[theo:7\] and \[theo:8\], we must distinguish several cases.\
$\blacktriangleright$ Case 1 : if $d$ divides $a-1, b-1$ or $ab-a-b$ ( the frame of Theorem \[theo:6\]). Obvious case because ${\mathrm{Irr}}(S)$ and ${\mathrm{PF}}(S)$ are symmetric ( except for one element ), if $S=\frac{<a,b>}{d}$.\
$\blacktriangleright$ Case 2 : if $d<\min(a,b)$ ( the frame of Theorem \[theo:7\]), it is a consequence of definition of $\alpha$-numeration ( $b_i\leqslant a_i$ with notations of this Theorem).\
$\blacktriangleright$ Case 3 : if $a<d<b$, the proof is more technical and we will study two subcases depending on $\alpha$-numeration of $a-1$ and $a$ : a general subcase, where these numerations have the same length and an exceptional subcase. But, we keep some room to manoeuvre.\
$\blacktriangleright$ Case 4,5,6 : if $d>\max(a,b)$. Proofs are becoming longer, more intricate and we show that some exceptional subcases are impossible. Case 6 fills almost half of the proof...\
We denote $S=\frac{<a,b>}{d}$. We will use some previous Theorems and their notations, as well as our $\alpha$-numeration. We refer the reader to the statements of these theorems and to section \[subsec:alphanum\].\
$\blacktriangleright$ Case 1 : we suppose that we are in one of the three “ easy cases” of Theorem \[theo:6\] : if $d$ divides $ab-a-b$, then $t(S)=1$ and $e(S)\geqslant 2$. If, $d$ divides $a-1$ or $b-1$, then Theorem \[theo:6\] proves that $t(S)=e(S)-1$.\
$\blacktriangleright$ Case 2 : we suppose that $d<\min(a,b)$.\
With Theorems \[theo:5\] and \[theo:7\], we obtain : $$e\left( \frac{<a,b>}{d}\right)=3+ \sum_{k=1}^{\lfloor r/2 \rfloor }a_{2k} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}t\left(\frac{<a,b>}{d}\right)=2+\sum\limits_{k=1}^{\lfloor r/2 \rfloor}b_{2k}$$ But, $\forall k\in { \{ 1,..,\lfloor r/2 \rfloor \} }, b_{2k}\leqslant a_{2k}$, so the inequality is true.\
We notice, that $e(S)=t(S)+1$ if and only if : $ \forall k\in { \{ 1,..,\lfloor r/2 \rfloor \} }, b_{2k}= a_{2k}$.\
$\blacktriangleright$ Case 3 : we suppose that $a<d<b$.\
We use Theorems \[theo:5\] and \[theo:8\] with the common notations $\alpha, (a_k)_k,k_0$. We also denote $(b_k)_k$, the $\alpha$-numeration of $a$, ${\hspace*{0.2cm}}(N_k)_k$ the $\alpha$-numeration of $a-1$ and $s=\max(i,N_i\not = 0)$.\
$\blacktriangleright\blacktriangleright$ Subcase 1 : if $b_k=0$ for all $k>s$. $$e(S)-t(S)= 2+\nu- \chi_{\alpha < \beta}- c_s +\sum_{k=k_0}^{\lfloor (s-1)/2 \rfloor}(a_{2k}-b_{2k})$$
Where $c_s=b_s$ if $s$ is even and $0$ else. The sum above is non negative ( see Case 2) and, since $(b_k)_k$ is the successor of $(N_k)_k$ in $E_{\alpha}$ for RLO, we have $N_s=b_s$ or $b_s-1$. Hence, $\nu-c_s=0$, except if $s$ is even and $N_s=b_s-1$ ( see below). So, in the general case, we have $e(S)-t(S)\geqslant 1$.\
Now, what does happen if $s$ is even and $N_s=b_s-1$ ? This is verified if and only if $(b_k)_k$ is the lowest element in $(E_{\alpha},RLO)$ such that $b_s$ is fixed and $b_k=0$ for $k>s$, that is to say $b_{s-2k}=a_{s-2k}$ and $b_{s-2k+1}=0$ for $k>1$ ( except $b_1=1$). In that case, we obtain $c_{s}=N_s+1=\nu+1$ and the sum is null so : $e(S)-t(S)=1-\chi_{\alpha<\beta}=1$, for $\beta<\alpha$ ( $b_1=1$ and $b_2>0$). So, we have $e(S)-t(S)=1$.\
$\blacktriangleright\blacktriangleright$ Subcase 2 : if $(N_1,N_2,\cdots,N_{s})$ is the greatest element in $(E_{\alpha},RLO)$ ( see \[subsec:alphanum\]) of that form, that is to say if $(N_k)_k=(a_1,a_2,a_3,\cdots,a_{s})$. Then $b_{s+1}=1$ and $b_{s-2k}=0$ for all $k\geqslant 0$ ( except $b_1=1$ if $s$ is odd) and $b_{s-2k+1}=a_{s-2k+1}$ if $k\in { \{ 1,..,\lfloor s/2 \rfloor \} }$ . Then $$e(S)-t(S)= 2+\nu- \chi_{\alpha < \beta}- c_{s+1}+\sum_{k=k_0}^{\lfloor (s-1)/2 \rfloor}(a_{2k}-b_{2k})$$ where $c_{s+1}=1$ if $s$ is odd and $c_{s+1}=0$ else. We make a distinction according to the parity of $s$ :\
$\blacktriangleright \blacktriangleright\blacktriangleright $ if $s$ is even : then $b_{2k}=0$ for all $k$, $c_{s+1}=0$ and $\nu=N_{s}=a_{s}$, so $e(S)-t(S)\geqslant 2$.\
$\blacktriangleright \blacktriangleright\blacktriangleright $ if $s$ is odd : then $b_{2k}=a_{2k}$ for all $2k< s$, $c_{s+1}=1$ and $\nu=0$. In addition, $\alpha >\beta$, for $b_1=1$ and $b_2>0$ ( we remark that if $s=1$, then $b_2=1$). So $e(S)-t(S)=1$.\
$\blacktriangleright$ Case 4 : if $d>\max(a,b)$ and $x_0+b>d$.\
We use again Theorem \[theo:5\] and \[theo:8\] with their notations. For the notation $(b_k)_k$, we will use it for the $\alpha$-numeration of $a$ ( that is also the $\alpha$-numeration of $1-\frac{b}{d}$). Moreover $s=\max(i,N_i>0)$. $$t(S)=\sum_{k=k_0}^{\lfloor r/2\rfloor}b_{2k} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}e(S)=2+\nu+\sum_{k=1}^{\lfloor (s-1)/2\rfloor}a_{2k}$$ As in the previous case, we distinguish same two subcases and similar arguments :\
$\blacktriangleright\blacktriangleright$ Subcase 1 : if $b_k=0$ for all $k>s$. $$e(S)-t(S)= 2+\nu- c_s +\sum_{k=1}^{k_0-1}a_{2k}+\sum_{k=k_0}^{\lfloor (s-1)/2 \rfloor}(a_{2k}-b_{2k})$$ Where $c_s=b_s$ if $s$ is even and $0$ else. If $s$ is odd, then $\nu=0=c_s$. If $s$ is even, then $\nu=N_s$ and $c_s=b_s \in \{ N_s,N_s+1 \}$. In both cases, we have $\nu-c_s\geqslant -1$, so $e(S)-t(S)\geqslant 1$.\
$\blacktriangleright\blacktriangleright$ Subcase 2 : see Case 3 subcase 2.\
$\blacktriangleright$ Case 5 : if $d>\max(a,b)$ and $m<x_0+b\leqslant d$.\
We are in the same situation as in Case 4, except that $e(S)$ is one unity less. So, we have to look carefully at this case.\
$\blacktriangleright\blacktriangleright$ Subcase 1 : if $b_k=0$ for all $k>s$. $$e(S)-t(S)= 1+\nu- c_s +\sum_{k=1}^{k_0-1}a_{2k}+\sum_{k=k_0}^{\lfloor (s-1)/2 \rfloor}(a_{2k}-b_{2k})$$ Where $c_s=b_s$ if $s$ is even and $0$ else. We remark that if $k_0>1$, then we can easily conclude, thanks to the term $a_2\geqslant 1$ and the Case 4.\
If $s$ is odd, then $\nu=0=c_s$ and $e(S)-t(S)\geqslant 1$.\
If $s$ is even : if $b_s=N_s$, we also have $e(S)-t(S)\geqslant 1$. If $b_s=N_s+1$, then $b_{2k}=a_{2k}$ and $b_{2k+1}=0$ for all $k<s/2$, except $b_1=1$. We will show that this is not possible. Indeed, we have ( see \[subsec:alphanum\] ) : $$1-\frac{b}{d}= \delta_0-\sum_{i=1}^{\frac{s}{2}-1}a_{2i}\delta_{2i-1}-(N_s+1)\delta_{s-1}= \cdots = \delta_{s-2}-(N_s+1)\delta_{s-1}$$ yet : $$x_0=d(\delta_{s-2}-N_s\delta_{s-1})$$ so : $$x_0+b=d(1+\delta_{s-1})>d$$ which contradicts our hypothesis.\
$\blacktriangleright\blacktriangleright$ Subcase 2 : again we refer to Case 3 Subcase 2. If $s$ is even, then with same arguments, $e(S)-t(S)\geqslant 1$. If $s$ is odd, we will show that it is not possible : indeed, we would have $$(b_k)_k=(1,a_2,0,a_4,0,\cdots,0,a_{s-1},0,1)$$ so : $$1-\frac{b}{d}= \delta_{s-1}-\delta_s$$ yet, $x_0=d\delta_{s-1}$, so : $x_0+b=d+d\delta_s>d$ : contradiction.\
$\blacktriangleright $ Case 6 : if $d>\max(a,b)$ and $x_0+b \leqslant m$.\
Again, we use Theorem \[theo:5\] and \[theo:8\]. Since some common notations name different things in each theorem, we will precise this : we use $k_1$ instead of $t$ in Theorem \[theo:5\] and we do not use $s$ in Theorem \[theo:8\], since it names $s=\max(i,N_i>0)$ in Theorem \[theo:5\]. We use $(b'_k)_k$ instead of $(b_k)_k$ in Theorem \[theo:5\], for $(b_k)_k$ names the $\alpha$-numeration of $a$ in Theorem \[theo:8\].\
So $\alpha=m/d$, $\beta=\frac{x_0+b-1}{d}$ and $(b'_k)_k$ is the $\alpha$-numeration of $\beta$. Again, $(b_k)_k$ is the $\alpha$-numeration of $a$.\
**N.B. :** the hypothesis $x_0+b \leqslant m$ is equivalent to $\beta<\alpha$, that is to say $b'_1=1$ and $b'_2>0$.\
Moreover : $$\sigma = \max(k,b_k>0) {\hspace*{0.2cm}}; {\hspace*{0.2cm}}s'=\max(k, b'_k>0)$$ We mention that, since $(b_k)_k$ is the successor of $(N_k)_k$ for ALO in $E_{\alpha}$, then $\sigma=s$ in the general case or $\sigma=s+1$ in one exceptional case ( see Case 3 subcase 2).\
We have with Theorems \[theo:5\] and \[theo:8\] : $$e(S)= 1+a_{2k_1}-b'_{2k_1}+\nu+\sum_{k=k_1+1}^{\lfloor (s-1)/2\rfloor}a_{2k} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}t(S)=\sum_{k=k_0}^{\lfloor r/2\rfloor}b_{2k}$$
We underline the definitions of $k_0$ and $k_1$ : $$k_0=\begin{cases} \frac{\sigma+1}{2} \text{ if } b_{2i}=0 \text{ for all } i \\ \min(i>0,b_{2i}>0) \text{ else } \end{cases} {\hspace*{0.2cm}}; {\hspace*{0.2cm}}k_1 =\begin{cases} \frac{s'}{2} \text{ if } b'_{2i+1}=0 \text{ for all } i>0 \\ \min(i>0,b'_{2i+1}>0) \text{ else } \end{cases}$$
**Step 1** : we will prove that $k_0> k_1$.\
First, we remark that $b'_{2i+1}=0$ if $i\in{ \{ 1,..,k_1-1 \} }$ and $b'_{2i}=a_{2i}$ if $i<k_1$. In addition, $b'_{2k_1}>0$ if $k_1>1$. Indeed : if $b'_{2i+1}$ for all $i>0$, then $2k_1=s'$. Else, $b'_{2k_1+1}>0$ and $b'_{2k_1-1}=0$. Yet : $(b'_k)_k=(1,a_2,0,a_4,0,\cdots,0,a_{2k_1-2},0,b'_{2k_1},...)$ and so : $$k_1>1 \Rightarrow (b'_k)_k <_A (1,a_2,0,a_4,0,\cdots,0,a_{2k_1-2})$$ Then : $$\frac{x_0+b-1}{d} <\delta_0-\sum_{i=1}^{k_1-1}a_{2i}\delta_{2i-1}= \delta_{2k_1-2}$$ But, this inequality is also true if $k_1=1$, since $\delta_0=\alpha$ ( see N.B. above).\
On the other hand, using similar arguments for $(b_k)_k$, we obtain : $$(b_k)_k \leqslant_A(a_1,0,a_3,\cdots,0,a_{2k_0-3},0,b_{2k_0-1})$$ Indeed : $b_{2i}=0$ and $b_{2i-1}=a_{2i-1}$ for $i\in{ \{ 1,..,k_0-1 \} }$. In addition, if $b_{2i}>0$ for some $i$, then $b_{2k_0}>0$ and we have a strict inequality. Else, $2k_0-1=\sigma$ and we have an equality !\
So : $$1-\frac{b}{d}\leqslant b_{2k_0-1}\delta_{2k_0-2}+\sum_{i=0}^{k_0-2}a_{2i+1}\delta_{2i}=1-\delta_{2k_0-3} +b_{2k_0-1}\delta_{2k_0-2}$$ Hence : $$1-x_0+d \delta_{2k_1-2} > b \geqslant d(\delta_{2k_0-3}-b_{2k_0-1}\delta_{2k_0-2})$$ But we have $x_0-1\geqslant 0$ and $$\delta_{2k_0-3}-b_{2k_0-1}\delta_{2k_0-2}=(a_{2k_0-1}-b_{2k_0-1})\delta_{2k_0-2}+\delta_{2k_0-1}\geqslant \delta_{2k_0-1}$$ So : $$\delta_{2k_1-2}> \delta_{2k_0-1}$$ Now, $(\delta_i)_i$ is non increasing, so $k_0 > k_1$.\
**Step 2 :**\
$\blacktriangleright\blacktriangleright$ Subcase 1 : if $b_k=0$ for all $k>s$. That is to say, if $\sigma=s$.\
$$e(S)-t(S)= 1+\nu- c_s+a_{2k_1}-b'_{2k_1} +\sum_{i=k_1+1}^{k_0-1}a_{2i}+\sum_{i=k_0}^{\lfloor (s-1)/2 \rfloor}(a_{2i}-b_{2i})$$ where $c_s=b_s$ if $s$ is even and $0$ else. If $s$ is odd, then $\nu=0=c_s$ and $e(S)-t(S)\geqslant 1$. If $s$ is even, then $\nu=N_s$ and $c_s=b_s \in\{ N_s,N_s+1 \}$. If, $b_s=N_s$, same conclusion. If $b_s=N_s+1$, then, as in Case 5, subcase 1, we obtain $x_0+b>d$, which contradicts our hypothesis.\
$\blacktriangleright\blacktriangleright$ Subcase 2 : if $(N_k)_k=(a_1,a_2,a_3,\cdots,a_{s})$. That is to say, if $\sigma=s+1$. Then $b_{s+1}=1$ and $b_{s-2k}=0$ for all $k\geqslant 0$ ( except $b_1=1$ if $s$ is odd) and $b_{s-2k+1}=a_{s-2k+1}$ if $k\in{ \{ 1,..,\lfloor s/2 \rfloor \} }$.\
$$e(S)-t(S)= 1+\nu- c_{s+1}+a_{2k_1}-b'_{2k_1} +\sum_{i=k_1+1}^{k_0-1}a_{2i}+\sum_{i=k_0}^{\lfloor (s-1)/2 \rfloor}(a_{2i}-b_{2i})$$ where $c_{s+1}=1$ if $s$ is odd and $c_{s+1}=0$ else. If, $s$ is even, then $\nu-c_{s+1}=N_s>0$, so $e(S)-t(S)\geqslant 1$. If $s$ is odd, this is not possible ( see Case 5, subcase 2).
[aa]{}
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**All suggestions, corrections, critics, remarks, questions... are welcome and can be made at the following email adress :** [email protected]
[^1]: except when $S={\mathbb{N}}$, where $G(S)=\emptyset$ and ${\mathrm{PF}}(S)=\{ -1 \}$
|
---
abstract: |
We apply a general method developed recently for the derivation of the diagonal representation of an arbitrary matrix valued quantum Hamiltonian to the particular case of Bloch electrons in an external electromagnetic field. We find the diagonal representation as a series expansion to the second order in $\hbar .$ This result is the basis for the determination of the effective in-band Hamiltonian of interacting Bloch electrons living in different energy bands. Indeed, the description of effects such as magnetic moment-moment interactions mediated by the magnetic part of the full electromagnetic interaction requires a computation to second order in $\hbar
$. It is found that the electronic current is made of two contributions: the first one comes from the velocity and the second one is a magnetic moment current similar to the spin current for Dirac particles. This last contribution is responsible for the interaction between magnetic moments similarly to the spin-spin interaction in the Breit Hamiltonian for Dirac electrons in interaction.
address: |
$^1$Institut Fourier, UMR 5582 CNRS-UJF UFR de Mathématiques, Université Grenoble I, BP74, 38402 Saint Martin d’Hères, Cedex, France\
$^2$ Laboratoire de Physique Moléculaire et des Collisions, ICPMB-FR CNRS 2843, Université Paul Verlaine-Metz, 57078 Metz Cedex 3, France
author:
- 'Pierre Gosselin$^{1}$ and Hervé Mohrbach$^{2}$'
title: 'Bloch electrons interacting with an external electromagnetic field and Bloch electrons in interaction. '
---
Introduction:
=============
The properties of electrons in solids are usually described in the framework of Bloch theory of electrons in a periodic potential. In particular, the study of the dynamics and transport properties of Bloch electrons perturbated by external fields led to important results for the understanding of metals, semiconductors and insulators properties [MERMIN]{}. For weak fields such that interband transitions called Zener tunnelling are negligible, the dynamics of a Bloch electron in a given $n$th band is usually based on the following semiclassical equations of motion $$\begin{aligned}
\dot{\mathbf{r}} &=&\partial \mathcal{E}(\mathbf{k})/\hbar \partial \mathbf{k%
} \notag \\
\hbar \dot{\mathbf{k}} &=&-e\mathbf{E}-e\dot{\mathbf{r}}\times \mathbf{B(r)}
\label{blochequations}\end{aligned}$$where $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic fields respectively and $\mathcal{E}(\mathbf{k})=\mathcal{E}_{0}(\mathbf{k})-%
\mathbf{m(k).B}$ is the energy of the band including a correction due to the orbital magnetic moment $\mathbf{m(k)}$. In the band energy $\mathcal{E}_{0}$ of the unperturbated crystal, the electron momentum $\mathbf{K}$ has been substituted by the gauge covariant momentum $\mathbf{k}=\mathbf{K}+e\mathbf{A%
}(\mathbf{R})/\hbar $. This substitution has been first justified by Peierls in the context of the tight binding model and for this reason is called the Peierls substitution [@PEIERLS]. A full justification was later given by Kohn [@KOHN]. A simpler version of the proof was later provided by Blount, Roth and Wannier and Fredkin who could derived an approximate band energy operator as an asymptotic series expansion in the fields strength as well as general expressions for the first few terms in this series [Blount]{}[@ROTH][@WANNIER]. Their methods, although different, are all based on some approximate unitary transformation of the initial Hamiltonian of Bloch electrons in an electromagnetic field which eliminates the interband matrix elements and leads to an effective diagonal-in-band energy operator. The principal advantage of the Blount’s method is that it is an application of a general diagonalization scheme applicable to any kind of matrix valued Hamiltonian whereas the two other ones are specific to the solid states. In particular, Blount has also considered the case of a Dirac particle in an electromagnetic field whose Hamiltonian was diagonalized to the second order in the fields strength and first order in their first derivatives. Later on, Weigert and Littlejohn developer a systematic method to diagonalize general quantum Hamiltonian in a series expansion in $\hbar $ [@LITTLEJOHN] instead in fields strength. an obvious advantage of this kind of expansion is obviously that it can be valid for strong external field but also that the semiclassical limit is readily obtained. Unfortunately, the method [@LITTLEJOHN] involves formal series expansion in terms of symbols of operators which makes the method very complicated for practical applications. It is worth mentioning that recently a variant of the Foldy Wouthuysen transformation valid for strong fields and based also on an expansion in $\hbar $ of the Dirac Hamiltonian was presented [SILENKO]{}.
This is not the end story with regards to the Bloch equations $\left( \ref%
{blochequations}\right) $. Indeed, Karplus, Luttinger and Kohn [@KARPLUS] predicted very early a spontaneous Hall effect in ferromagnetic materials due to a corrective term to the velocity in Eq. $\left( \ref{blochequations}%
\right) $, known as the anomalous velocity. Later, Adams and Blount [ADAMS]{}, by interpreting this term as resulting from the noncommutativity of the intraband coordinate operators, derived new semiclassical equations of motion for Bloch electrons with an anomalous velocity. However these equations turn out to be correct only for external electric fields. It is only recently that the correct equations of motion of Bloch electrons in the presence of both electric and magnetic fields and including anomalous velocity were derived by Chang and Niu [@NIU]. Indeed, using a time-dependent variational principle in a Lagrangian formulation and a description of the electron in terms of wave packets, they found the following new equations of motion in the presence of electromagnetic fields $$\begin{aligned}
\dot{\mathbf{r}} &=&\partial \mathcal{E}(\mathbf{k})/\hbar \partial \mathbf{k%
}-\dot{\mathbf{k}}\times \Theta (\mathbf{k}) \notag \\
\hbar \dot{\mathbf{k}} &=&-e\mathbf{E}-e\dot{\mathbf{r}}\times \mathbf{B(r)}
\label{niuequations}\end{aligned}$$The correction term to the velocity $-\dot{\mathbf{k}}\times \Theta $ is the anomalous velocity which is due to the presence of a Berry curvature $\Theta
(\mathbf{k})$ of electronic Bloch state in the given $n$th band, associated to the electron motion in the $n$th energy band. For crystals with simultaneous time-reversal and spatial inversion symmetry, the Berry curvature and the magnetic moment vanish identically throughout the Brillouin zone. This is the case of most applications in solid state physics, but there are situations where these symmetries are not simultaneously present like in GaAs where inversion symmetry is broken or in ferromagnets which break time reversal symmetries. In the same way, the presence of a strong magnetic field, the magnetic bloch bands corresponding to the unperturbated system breaks the time inversion symmetries. In all these cases, the dynamical and transport properties must be described by the full equations of motion given by Eq. $\left( \ref{niuequations}\right) $. Notice that even for crystals with simultaneous time-reversal and spatial inversion symmetry, energy bands degeneracies can lead to a non vanishing curvature and magnetization; a typical example is provided by the graphene [@NIUGRAPHENE] This case is due to the presence of a topological Berry phase associated to bands degeneracies [@BERRY]. This particular situation is not considered in this paper.
The method developed in [@NIU] is by construction limited to the semiclassical level, but the description of phenomena such as the electromagnetic interaction of Bloch electrons, as will be discussed in this paper, requires a theory which goes beyond the semiclassical approximation. Recently we came back to the initial considerations of Blount and others with regards to the diagonalization procedure for an arbitrary matrix valued Hamiltonian $H$ (applicable to any kind of quantum system which has an energy band spectrum) in the presence of external fields and a first original method based on a differential equation of the diagonal in-band energy operator with respect to Planck constant $\hbar $ was proposed in [@SERIESPIERRE]. In this approach, where $\hbar $ is promoted as a (formal) running parameter noted $\alpha $, one has to diagonalize $H$ at the scale $\alpha $ where it is assumed that the canonical dynamical operators satisfy the algebra $\left[ R_{\alpha }^{i},P_{\alpha }^{j}\right]
=i\alpha \delta _{ij}$. Relating two diagonalization processes for close values of $\alpha ,$ leads to a differential equation of the required diagonal Hamiltonian $\varepsilon _{\alpha }$ with respect to $\alpha $. This differential equation has to be supplemented by an additional equation which is the consequence of the unitarity condition of the matrix $U_{\alpha
}$ diagonalizing $H$ at the scale $\alpha .$ The resolution of this differential equation can then be performed by a systematic series expansion in $\hbar $, and in this way, at least in principle, an exact diagonalization of arbitrary Hamiltonians can be achieved. This approach reveals that the diagonal energy operator is most naturally expressible in terms of covariant (noncanonical ) coordinates $\mathbf{r}=\mathbf{R+}\emph{A%
}_{\mathbf{R}}$ and momentum operators $\mathbf{p}=\mathbf{P+}\emph{A}_{%
\mathbf{P}}$ which are both corrected by Berry connections terms $\emph{A}_{%
\mathbf{R/P}}$ and which satisfy a non-commutative algebra. Particle motion in this noncanonical phase space is obviously drastically modified (as in Eq. $\left( \ref{niuequations}\right) $). Particularly interesting is the fact that, in the semiclassical limit, which is often enough to get physical insight to the problem considered, the diagonal Hamiltonian is obtained by a straightforward integration of this differential equation. In this limiting case, the differential approach turns out to be so powerful that the general diagonal representation for an arbitrary matrix valued Hamiltonian in terms of covariant operators and commutators between Berry connections could be given (actually, this general formula was first derived by a direct diagonalization procedure in [@PIERRESEMIDIAG]). This result allowed us to deduce effective semiclassical Hamiltonians and to predict new phenomena in various physical situations. First, the study of Bloch electron in magnetic Bloch bands [@PIERREEUROPHYS] showed that besides the position operator which get a Berry-phase contribution (as already shown by [@NIU]), the momentum in the band energy $\mathcal{E}(\mathbf{k})$ also has to be replaced by a new Berry-dependent momentum operator $\mathbf{k}=\mathbf{K}+e%
\mathbf{A}/\hbar -\emph{A}_{\mathbf{P}}$ instead of the Peierls substitution. It turns out that this result is essential for the correct derivation of the full equations of motion for Bloch electrons Eq. $\left( %
\ref{niuequations}\right) $. Likewise, for electrons in graphene in a magnetic field, it was also observed that it is in terms of $\mathbf{k}$ that the semiclassical quantification of the orbit has to be achieved [PIERREGRAPHENE]{}.
At the semiclassical level other systems were also investigated, like Dirac electrons in electromagnetic [@SERIESPIERRE] and gravitational fields [@PIERREELECGRAVIT] with the discovery of a spin-magnetotorsion coupling. The study of the photon in a static gravitational field when polarization effect are taken into account predicts the gravitational birefringence phenomenon where an helicity dependent anomalous velocity deviates the photon from the usual Einsteinien geodesics [PIERREPHOTONGRAVIT]{}. This kind of polarization effects which are called spin Hall effect of light have been recently observed [@SPINHALLSCIENCE]. Despite these results, the applicability of the differential approach beyond the semiclassical turns out to be very complicated. Even the deduction of the diagonal representation of a generic Hamiltonian at second order in $\hbar $ is a prohibitively difficult problem, although a solution was found for simple practical applications like a photon in an homogeneous isotropic media and Bloch electron in uniform electric field [SERIESPIERRE]{}. But clearly, one can not expect to use this method in the case of several Bloch electrons in electromagnetic interaction, a problem we would like to consider in this paper. However, very recently, a new general and powerful method for the diagonalization of an arbitrary matrix valued Hamiltonian has been proposed by Gosselin and Mohrbach (referred as GM) [@DIAGOEXACT].It leads to a particularly compact and elegant exact expression for the required diagonal energy operator.** **This approach is therefore particularly well adapted to problems in solid state physics and to Dirac particles in external fields. This last case can be considered intuitively as a simple two bands versions (particles and anti-particles) of Bloch electrons in a crystal. The philosophy behind this approach consists in mapping the initial quantum system to a classical one which can be diagonalized and then to return to the full quantum system. This method is not based on a differential equation for the diagonal energy operator with respect to $%
\hbar $, but it also requires the introduction of new mathematical objects like non-commuting operators which evolve with $\hbar $ promoted as a running variable. This new mathematical construction leads us to define a differential calculus on a non-commutative space showing some similarities with the stochastic calculus as both stress the role of second order terms. This approach allows us to write both the diagonal Hamiltonian $\varepsilon
\left( \mathbf{x}\right) $ and the transforming matrix $U\left( \mathbf{x}%
\right) $ (where $\mathbf{x=(r,p)}$ is the phase space of the covariant dynamical operators) as a result of the application of integro-differential operators on $\varepsilon _{0}\left( \mathbf{X}_{0}\right) $ and $%
U_{0}\left( \mathbf{X}_{0}\right) $ respectively, i.e. $\varepsilon \left(
\mathbf{x}\right) =\widehat{O}\left( \varepsilon _{0}\left( \mathbf{X}%
_{0}\right) \right) $ and $U\left( \mathbf{x}\right) =\widehat{N}\left(
U_{0}\left( \mathbf{X}_{0}\right) \right) $. Matrices with the subscribe $0$ correspond to the operators replaced by classical commuting variables $%
\mathbf{X}_{0}\mathbf{=}\left( \mathbf{R}_{0}\mathbf{,P}_{0}\right) .$ The only requirement of the method is the knowledge of $U_{0}\left( \mathbf{X}%
_{0}\right) $ at $\alpha =0$ which gives the diagonal form $\varepsilon
_{0}\left( \mathbf{X}_{0}\right) .$ Generally, these equations do not allow to find directly $\varepsilon \left( \mathbf{x}\right) $, $U\left( \mathbf{x}%
\right) $, however, they allow us to produce the solutions for $\varepsilon
\left( \mathbf{x}\right) $ and $U\left( \mathbf{x}\right) $ recursively in a series expansion in $\hbar .$ But contrary to the procedure in [SERIESPIERRE]{}, it turns out that the expansion in a series of $\hbar $ is much more easier to obtain than by the successive integration of the differential equation and therefore more convenient for getting higher order contributions. Remarkably, it was also found that the exact expression for $%
\varepsilon \left( \mathbf{x}\right) $ is actually an exact solution of the differential equation of [@SERIESPIERRE]. This result obviously places the present approach on a firm base[@DIAGOEXACT]. Another interesting feature of this approach is the confirmation of the fundamental role played by Berry curvatures in these systems since the method results in an effective diagonal Hamiltonian with Berry phase corrections as well as noncommutative coordinates and momentum covariant operators as in previous approaches [@SERIESPIERRE][@PIERRESEMIDIAG]**).**
Although similar in spirit to Blount’s method and in particular to Weigert and Littlejohn one [@LITTLEJOHN], the approach proposed in [DIAGOEXACT]{} is essentially different as it is based on a very new mathematical formulation. (The general method of [@LITTLEJOHN] leads also to a diagonal in-band energy representation as a formal series expansion written in terms of symbols of operators which makes the method very complicated for practical applications). In our opinion this new approach is more tractable for applications and, as an illustration of this statement, a general in-band energy for any arbitrary Hamiltonian to the second order in $\hbar $ was achieved in [@DIAGOEXACT] (higher order expressions becomes again very cumbersome but could in principle be computed). This expression will be the starting point for a straightforward study of a single Bloch electron in an external electromagnetic field. Another purpose of the present work is the adaptation of GM’s results to the case of Bloch electrons in interaction. Note that both problems can be transposed to the case of Dirac electrons which are actually treated by the same method in the another paper [@PIERREDIRAC]. It is obviously the Coulomb (electrostatic) interaction between Bloch (Dirac) electrons that dominates over the magnetic one, so that a first order diagonalization seems to be sufficient. But, in the presence of non vanishing electronic magnetic moments, other effects like moment-moment interactions mediated by the magnetic part of the full electromagnetic self-interaction are expected. This comes out by analogy with the spin-spin interaction in the Breit Hamiltonian of non-relativistic Dirac particles [@BREIT] which is recover in the non-relativistic limit [@PIERREDIRAC]. Obviously these kind of interactions are of second order in $\hbar $ and a diagonalization procedure which goes to this order is necessary.
Before starting, two points are worth mentioning. As already noted in [LITTLEJOHN]{} but also in [@PIERRESEMIDIAG][@SERIESPIERRE], there is certain latitude in finding $U$ which reflects a kind of gauge invariance of the method. Because of this freedom two equivalent diagonalizing operators leads to two different forms for the diagonal Hamiltonian but of course to the same eigenvalues. This is similar to the Schrodinger equation in a magnetic field where the Hamiltonian and the wave function gauge dependence combines itself to give gauge independent energy levels. In our case it turns out that this freedom is only present at the second order in $\hbar $, because the first order diagonalization is performed with the zero order matrix $U_{0}\left( \mathbf{X}_{0}\right) $ which can be uniquely defined. Actually this gauge dependence can be included in the gauge covariant dynamical operator $\mathbf{x=(r,p)}$. Then the diagonal in-bands energy operator is uniquely defined when it is written in terms of $\mathbf{x}$ instead of the canonical operators $\mathbf{X.}$ As a particular gauge choice can be made on the ground of simplicity and convenience, here as in previous works [@SERIESPIERRE][@DIAGOEXACT] the reality condition of diagonal elements (the anti-hermitian diagonal elements are setting to zero) of $U$ is imposed.
The second point we would like to mention is that, in order to simplify the expressions, only time independent electromagnetic fields are considered, but results can be easily extended to include time dependent interactions. In addition, as the ultimate goal is to consider Bloch electrons in interaction through an internal electromagnetic field, the time dependence of the vector potential can be safely neglected as all retarded effects in the electromagnetic interaction are negligible owing to the fact that Bloch electrons are non-relativistic.
The paper is organized as follows. In the next section we give a reminder of the diagonalization procedure of [@DIAGOEXACT] for an arbitrary matrix valued Hamiltonian. We provide some detailed formulas for the diagonalized energy operator at the second order in $\hbar $ as well as for the Berry phases at this order. Section 3 applies this formalism to the case of a Bloch electron in an external electromagnetic field. The diagonalized Hamiltonian at the second order in $\hbar $ is written in terms of the transformed dynamical variables and magnetization operators. In section 4, we consider the case of $P$ Bloch electrons interacting through an internal electromagnetic field. Diagonalizing the matter part of the Hamiltonian to the second order in $\hbar $ and solving for the electromagnetic field yields the $P$ particles effective Hamiltonian. Last section is for the conclusion.
Diagonalization of an arbitrary matrix valued Hamiltonian
=========================================================
To start with, an outline of the approach developed by Gosselin and Mohrbach to diagonalize formally an arbitrary matrix valued Hamiltonian is given, with the notations of GM [@DIAGOEXACT]. Let consider an arbitrary quantum mechanical system whose state space is a tensor product $L^{2}\left(
\mathcal{R}^{3}\right) \otimes \mathcal{V}$ with $\mathcal{V}$ some internal space. In other words, the Hamiltonian of this system can be written as a matrix $H\left( \mathbf{R,P}\right) $ of size $\dim \mathcal{V}
$ whose elements are operators depending on a couple of canonical variables $\mathbf{X}=\left( \mathbf{R,P}\right) .$ The archetype example is usually the Dirac Hamiltonian with $\mathcal{V}=C^{4}$, but as shown in [PIERRESEMIDIAG]{} the following set up fits with all system presenting an energy band spectrum, as for Bloch electron (where $\mathcal{V}$ correspond to the energy band indices), a system which is the main concern of the present paper.
In [@DIAGOEXACT], a method to find an unitary matrix $U\left( \mathbf{X}%
\right) $ to diagonalize any arbitrary matrix-valued quantum Hamiltonian $%
H\left( \mathbf{X}\right) $ such that $\varepsilon \left( \mathbf{X}\right)
=UH\left( \mathbf{X}\right) U^{+}$ is the diagonal in-band energy operator was achieved.
As explained in the introduction, the principle of this method is to link continuously an Hamiltonian in which the variables are considered as classical (i.e. $\hbar =0$) to the true Hamiltonian we aim at diagonalizing (that is at scale $\hbar $). The idea is to diagonalize the Hamiltonian for $%
\hbar =0$, which appears in general to be much easier, and then to come back to scale $\hbar $ to obtain the required Band Hamiltonian. To perform this program, we have to proceed in an indirect way. Actually, we first need to introduce a family of canonical variables $\left( \mathbf{R}_{\alpha }%
\mathbf{,P}_{\alpha }\right) $ indexed by a continuous parameter $\alpha
\in \left[ 0,\hbar \right] $, such that the commutators are given by $\left[
\mathbf{R}_{\alpha }^{i}\mathbf{,P}_{\alpha }^{i}\right] =i\alpha $ and then, to introduce for any arbitrary function $F\left( \mathbf{R}_{\alpha }%
\mathbf{,P}_{\alpha }\right) $ both a notion of differentiation and integration describing the variation of $F\left( \mathbf{R}_{\alpha }\mathbf{%
,P}_{\alpha }\right) $ as $\alpha $ varies. These notions have to take into account the fact that the commutation relations are depending on $\alpha $. It leads us naturally to introduce the notion of infinitesimal non commutative canonical variables $\left( d\mathbf{R}_{\alpha }\mathbf{,dP}%
_{\alpha }\right) $ as well as a form of differential calculus presenting some formal analogy with the non commutative stochastic calculus. The introduction of these differentials notion, will allow to connect ultimately our Hamiltonian at scales $0$ and $\hbar $.
More precisely now, we introduce a space of non commuting infinitesimal operators $dX_{\alpha }^{i}\equiv \left\{ dR_{\alpha }^{i},dP_{\alpha
}^{i}\right\} $ $\ i=1,2,3$ indexed by a continuous parameter $\alpha $, that satisfy the following infinitesimal Heisenberg algebra with a reversed sign $\left[ dR_{\alpha }^{i},dP_{\alpha \prime }^{j}\right] =-id\alpha
\delta _{\alpha ,\alpha \prime }\delta _{ij}\ $and $\left[ dR_{\alpha
}^{i},dR_{\alpha \prime }^{j}\right] =\left[ dP_{\alpha }^{i},dP_{\alpha
\prime }^{j}\right] =0.$ From it, we define a set of running coordinate and momentum operators by writing the following formal sums $R_{\alpha
}^{i}=R^{i}-\int_{\alpha }^{\hbar }dR_{\lambda }^{i}$, and $P_{\alpha
}^{i}=P^{i}-\int_{\alpha }^{\hbar }dP_{\lambda }^{i}$ with the choice of convention $dR_{\alpha }^{i}=R_{\alpha }^{i}-R_{\alpha -d\alpha }^{i}$ and $%
dP_{\alpha }^{i}=P_{\alpha }^{i}-P_{\alpha -d\alpha }^{i}$, so that the running operators satisfy $\left[ R_{\alpha }^{i},P_{\alpha }^{j}\right]
=i\alpha \delta _{ij}$ and $\left[ R_{\alpha }^{i},R_{\alpha }^{j}\right] =%
\left[ P_{\alpha }^{i},P_{\alpha }^{j}\right] =0.$ For $\alpha =\hbar $ we recover the usual canonical operators $R^{i}\equiv R_{\hbar }^{i}$and $%
P^{i}\equiv P_{\hbar }^{i}$ which evidently satisfy the canonical Heisenberg algebra. The differential of an arbitrary function $F\left( \mathbf{X}%
_{\alpha },\alpha \right) $ where $X_{\alpha }^{i}\equiv \left\{ R_{\alpha
}^{i},P_{\alpha }^{i}\right\} $ on this space is given by
$$\begin{aligned}
dF\left( \mathbf{X}_{\alpha },\alpha \right) &=&\sum_{i=1}^{6}\nabla
_{X_{\alpha }^{i}}F\left( \mathbf{X}_{\alpha },\alpha \right) dX_{\alpha
}^{i}-\frac{1}{4}\sum_{i,j=1}^{6}\nabla _{X_{\alpha }^{i}}\nabla _{X_{\alpha
}^{j}}F\left( \mathbf{X}_{\alpha },\alpha \right) \left( dX_{\alpha
}^{i}dX_{\alpha }^{j}+dX_{\alpha }^{i}dX_{\alpha }^{j}\right) \notag \\
&&+\left( \frac{\partial F\left( \mathbf{X}_{\alpha },\alpha \right) }{%
\partial \alpha }+\left\langle F\left( \mathbf{X}_{\alpha },\alpha \right)
\right\rangle \right) d\alpha \label{dalphaY}\end{aligned}$$
with $i$,$j=1..6$. We also assume that $X_{\alpha }^{i}\equiv R_{\alpha
}^{i} $ for $i=1,2,3$ and $X_{\alpha }^{i}\equiv P_{\alpha }^{i}$ for $%
i=4,5,6$ .The notation $\left\langle F\left( \mathbf{X}_{\alpha },\alpha
\right) \right\rangle $ (which in [@SERIESPIERRE] was corresponding to the operation $-\frac{i}{2}Asym\nabla _{R_{i}}\nabla _{P^{i}}F\left( \mathbf{%
X}_{\alpha },\alpha \right) $) is defined as a specific procedure on a series expansion of $F$ in the variables $R_{\alpha }^{i}$, $P_{\alpha }^{i}$ in the following way : let $F$ be a sum of monomials of the kind $%
M_{1}\left( \mathbf{R}_{\alpha }\right) M_{2}\left( \mathbf{P}_{\alpha
}\right) M_{3}\left( \mathbf{R}_{\alpha }\right) ....$ the $M_{i}$ being arbitrary monomials in $R_{\alpha }$ or $P_{\alpha }$ alternatively. Let the operator $\nabla _{R_{i}}\nabla _{P^{i}}$ acts on such an expression by deriving all combinations of one monomial in $\mathbf{R}_{\alpha }$ and one monomial in $\mathbf{P}_{\alpha }$. For each of these combinations, insert a $dR_{\alpha }^{i}$ at the place where the derivative $\nabla _{R_{i}}$ is acting and in a same manner a $dP_{\alpha }^{j}$ at the place where the derivative $\nabla _{P^{i}}$ is acting. This leads to an expression with two kind of terms, one kind being proportional to the $dR_{\alpha
}^{i}dP_{\alpha }^{j},$ and the second proportional to $dP_{\alpha
}^{j}dR_{\alpha }^{i}.$ Then rewrite this expression in terms of $dR_{\alpha
}^{i}dP_{\alpha }^{j}+dP_{\alpha }^{j}dR_{\alpha }^{i}$ and $dR_{\alpha
}^{i}dP_{\alpha }^{j}-dP_{\alpha }^{j}dR_{\alpha }^{i}=-i\delta ^{ij}d\alpha
$. Then $\left\langle F\left( \mathbf{X}_{\alpha },\alpha \right)
\right\rangle $ is defined as minus the contributions of terms proportional to $-i\delta ^{ij}d\alpha $ in the computation in the procedure just considered. This definition implies a procedure which is clearly dependent of the symmetrization chosen for the expansion of $F$.
To make the definition of $\left\langle F\left( \mathbf{X}_{\alpha },\alpha
\right) \right\rangle $ clearer, consider some important practical examples. If the function $F$ has the following form $F=\frac{1}{2}\left(
A\left( \mathbf{R}_{\alpha }\right) B\left( \mathbf{P}_{\alpha }\right)
+B\left( \mathbf{P}_{\alpha }\right) A\left( \mathbf{R}_{\alpha }\right)
\right) $ which corresponds to a frequent choice of symmetrization in $%
R_{\alpha }$ and $P_{\alpha }$, then $\left\langle F\left( \mathbf{X}%
_{\alpha },\alpha \right) \right\rangle =\frac{i}{4}\left[ A\left( \mathbf{R}%
_{\alpha }\right) ,B\left( \mathbf{P}_{\alpha }\right) \right] $. Another choice of symmetrization leads in general to a different result. For instance, if we rewrite the same function $F$ in a fully symmetrized form in $R_{\alpha }$ and $P_{\alpha }$ (that is invariant by all permutations in $%
R_{\alpha }$ and $P_{\alpha }$) which is also often used, we have now have a different result since $\left\langle F\left( \mathbf{X}_{\alpha },\alpha
\right) \right\rangle =0$.
Nevertheless, this dependence of $\left\langle F\left( \mathbf{X}_{\alpha
},\alpha \right) \right\rangle $ in the symmetrization choice is not astonishing at all. Actually changing the symmetrization of a function $%
F\left( \mathbf{X}_{\alpha },\alpha \right) $ introduces some explicit terms in $\alpha $ which changes also the term $\partial _{\alpha }Fd\alpha $ present in the differential Eq. $\left( \ref{dalphaY}\right) $. As a consequence, neither the partial derivative with respect to $\alpha $, nor the bracket are invariant by a change of form. But, what is invariant is the sum $\partial _{\alpha }F+\left\langle F\right\rangle $. This assertion is shown in [@DIAGOEXACT].
Now, what really matters for us in Eq. $\left( \ref{dalphaY}\right) $ is this invariant term proportional to $d\alpha $ this is why we define an expectation operation $\mathcal{E}\left( .\right) $ so that $$\mathcal{E}\left( dF\left( \mathbf{X}_{\alpha },\alpha \right) \right) =%
\mathcal{E}\left( \left( \frac{\partial F\left( \mathbf{X}_{\alpha },\alpha
\right) }{\partial \alpha }+\left\langle F\left( \mathbf{X}_{\alpha },\alpha
\right) \right\rangle \right) d\alpha \right) \label{epsidf}$$
Combining the expectation operator as well as the differential allows to relate a function evaluated at the physical scale $\hbar $, $F\left( \mathbf{%
X}_{\hbar },\hbar \right) $ to that same function evaluated at the scale $0$, $F\left( \mathbf{X}_{0},0\right) $ that is when the canonical variables are considered as classical ones. Actually by integration of the previous relation one has: $$F\left( \mathbf{X}_{\hbar },\hbar \right) =\mathcal{E}\left\{ F\left(
\mathbf{X}_{0},0\right) +\int_{0}^{\hbar }\left( \left( \frac{\partial
F\left( \mathbf{X}_{\alpha },\alpha \right) }{\partial \alpha }+\left\langle
F\left( \mathbf{X}_{\alpha },\alpha \right) \right\rangle \right) d\alpha
\right) \right\}$$ (we use a property that the expectation operator satisfies $\mathcal{E}%
\left( F\left( \mathbf{X}_{\hbar },\hbar \right) \right) =F\left( \mathbf{X}%
_{\hbar },\hbar \right) $).
Here recall that we denote by $\mathbf{X}_{\alpha }$ the dynamical variables when $\mathbf{P}$ and $\mathbf{R}$ are considered as commuting classical variables that is when $\hbar =\alpha $. This last formula is the starting point of the method. Iterating this relation, by successive differentiations and integrations, one can relate a full quantum function $F\left( \mathbf{X}%
_{\hbar },\hbar \right) $ to the same function evaluated with classical variables $\mathbf{X}_{0}$. This will prove very useful in a problem of diagonalization of a matricial Hamiltonian since in that case, the diagonalization when the canonical variables $\mathbf{X}_{0}$ commute reduces to the diagonalization of an usual matrix of finite size. Thus diagonalizing our Hamiltonian at scale $0$, that is finding an $F\left(
\mathbf{X}_{0},0\right) $ is in general an easier task.
However, this not the end of the story. Having found a way to relate $%
F\left( \mathbf{X}_{\hbar },\hbar \right) $ to its classical counterpart is not enough since we want ultimately to recover expressions of interest evaluated at $\mathbf{X}_{\hbar }$ not at the $\hbar =0$ scale. This kind of coming back process after a classicaldiagonalization is performed by an other operation defined in [DIAGOEXACT]{}. It allows in the previous integral relation to replace, inside the expectation, $\mathbf{X}_{0}$ and $\mathbf{X}_{\alpha }$ by $\mathbf{X}%
_{\hbar }$ at the price of a modification of the expression inside the integral. Define the exponentiated Bracket plus Shift operator (EBS) between $\alpha _{2}$ and $\alpha _{1}$ as acting on any function $F\left( \mathbf{X}%
_{\alpha _{2}},\alpha _{2}\right) $ to yield an other function depending on $%
\left( \mathbf{X}_{\alpha _{1}},\alpha _{2}\right) $, $\alpha _{1}>\alpha
_{2}$ : $$\begin{aligned}
\exp \left( -\left\langle .\right\rangle _{\alpha _{2}\rightarrow \alpha
_{1}}^{S}\right) &\equiv &T\exp \left( -\int_{\alpha _{2}}^{\alpha _{1}}S_{%
\mathbf{X}_{\alpha _{1}}}\left\langle .\right\rangle _{\alpha }S_{\mathbf{X}%
_{\alpha }}d\alpha \right) \\
&=&\sum \int_{\alpha _{2}<\beta _{n}<...\beta _{1}<\alpha _{1}}\left[ S_{%
\mathbf{X}_{\alpha _{1}}}\left\langle .\right\rangle _{\beta _{n}}S_{\mathbf{%
X}_{\beta _{n}}}\right] ...\left[ \left\langle .\right\rangle _{\beta
_{1}}S_{\mathbf{X}_{\beta _{1}}}\right] d\beta _{1}...d\beta _{n}\end{aligned}$$ where the Shift operation $S_{\mathbf{X}_{\alpha _{1}}}$ sets the dynamical variables $\mathbf{X}_{\alpha }$ to $\mathbf{X}_{\alpha _{1}}$ and satisfies $S_{\mathbf{X}_{\alpha }}S_{\mathbf{X}_{\beta }}=S_{\mathbf{X}%
_{\alpha }}$ whatever the values of $\alpha $ and $\beta $. Apart from the repeated application of the Bracket $\left\langle .\right\rangle _{\alpha }$ the EBS operation has the virtue to shift progressively the variables from $%
\mathbf{X}_{0}$, the classical variables to $\mathbf{X=X}_{\hbar }$ the full quantum variables. Actually its main property is the following : $$\mathcal{E}F\left( \mathbf{X}_{\alpha _{2}},\alpha _{2}\right) =\mathcal{E}%
\exp \left( -\left\langle .\right\rangle _{\alpha _{2}\rightarrow \alpha
_{1}}^{S}\right) F\left( \mathbf{X}_{\alpha _{2}},\alpha _{2}\right)$$ so that it can of course be specialized to : $$\mathcal{E}F\left( \mathbf{X}_{0},0\right) =\mathcal{E}\exp \left(
-\left\langle .\right\rangle _{0\rightarrow \hbar }^{S}\right) F\left(
\mathbf{X}_{0},0\right)$$ These two formulas can be understood intuitively as follows. The EBS operation changes the function (by the action of the bracket defined above) but also changes progressively the variables from $\mathbf{X}_{\alpha _{2}}$ to $\mathbf{X}_{\alpha _{1}}$ (through the shift operator). As a consequence, and despite the appearances, the expression in the right hand side $\mathcal{E}\exp \left( -\left\langle .\right\rangle _{\alpha
_{2}\rightarrow \alpha _{1}}^{S}\right) F\left( \mathbf{X}_{\alpha
_{2}},\alpha _{2}\right) $ is a function of $\mathbf{X}_{\alpha _{1}}$. Moreover, both the EBS operation and the shift of variable compensate each over to produce the equality with the left hand side.
With this mathematical construction in hand it was possible to write the solution of our diagonalization procedure for a general matrix valued Hamiltonian through an unitary transformation $U$ as the solution of the following system of integro-differential equations. Introducing $U_{0}\left(
\mathbf{X}_{0}\right) $ the diagonalization matrix when $\hbar =0$, such that the classical energy matrix obtained as $\varepsilon _{0}\left( \mathbf{%
X}_{0}\right) =U_{0}H_{0}\left( \mathbf{X}_{0}\right) U_{0}^{+}$ is a diagonal matrix, we could write : $$\begin{aligned}
\varepsilon \left( \mathbf{X}\right) &=&\mathcal{E}\left( \left[ \mathcal{T}%
\exp \left[ \int_{0<\alpha <\hbar }e^{-\left\langle .\right\rangle _{\alpha
\rightarrow \hbar }^{S}}O_{\alpha }e^{-\left\langle .\right\rangle
_{0\rightarrow \alpha }^{S}}d\alpha \right] \right] \varepsilon _{0}\left(
\mathbf{X}_{0}\right) \right) \label{EE} \\
U\left( \mathbf{X}\right) &=&\mathcal{E}\left( \left[ \mathcal{T}\exp \left[
\int_{0<\alpha <\hbar }e^{-\left\langle .\right\rangle _{\alpha \rightarrow
\hbar }^{S}}N_{\alpha }e^{-\left\langle .\right\rangle _{0\rightarrow \alpha
}^{S}}d\alpha \right] \right] U_{0}\left( \mathbf{X}_{0}\right) \right)
\label{U}\end{aligned}$$ $\mathcal{T}$ is the notation for the time ordered product, the operator $%
\exp \left( -\left\langle .\right\rangle _{\alpha _{2}\rightarrow \alpha
_{1}}^{S}\right) $ acts as explained above, and $O_{\alpha }$ and $N_{\alpha
}$ act in the following way :
The operation $O_{\alpha }\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right)
=\left( \frac{\partial }{\partial \alpha }+\left\langle .\right\rangle
\right) \varepsilon _{\alpha }\left( \mathbf{X}_{\alpha }\right) $ can also be written as $O_{\alpha }\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right)
=\left( T_{\alpha }+M_{\alpha }\right) \varepsilon _{0}\left( \mathbf{X}%
_{\alpha }\right) $ with a translation operator $T$ and a magnetization $M$ operator (this terminology is explained in [@SERIESPIERRE]) where $$\begin{aligned}
T_{\alpha }\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right) &=&\mathcal{P}%
_{+}\left\{ \frac{1}{2}\left( \mathcal{A}_{\alpha }^{R_{l}}\nabla
_{R_{l}}\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right) +\nabla
_{R_{l}}\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right) \mathcal{A}%
_{\alpha }^{R_{l}}+\mathcal{A}_{\alpha }^{P_{l}}\nabla _{P_{l}}\varepsilon
_{0}\left( \mathbf{X}_{\alpha }\right) +\nabla _{P_{l}}\varepsilon
_{0}\left( \mathbf{X}_{\alpha }\right) \mathcal{A}_{0}^{P_{l}}\right)
\right\} \notag \\
M_{\alpha }\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right) &=&\frac{i}{4}%
\left\{ \mathcal{P}_{+}\left\{ \left[ \varepsilon _{0}\left( \mathbf{X}%
_{\alpha }\right) ,\mathcal{A}_{\alpha }^{R_{l}}\right] \mathcal{A}_{\alpha
}^{P_{l}}-\left[ \varepsilon _{0}\left( \mathbf{X}_{\alpha }\right) ,%
\mathcal{A}_{\alpha }^{P_{l}}\right] \mathcal{A}_{\alpha }^{R_{l}}\right\}
+H.C.\right\} \notag \\
&&+\mathcal{P}_{+}\left[ U_{\alpha }\left( \left( \frac{\partial }{\partial
\alpha }+\left\langle .\right\rangle \right) H\left( \mathbf{X}_{\alpha
}\right) \right) U_{\alpha }^{+}\right] \label{TM}\end{aligned}$$ These equations require some explanations. First, the operators $\mathcal{A}%
_{\alpha }^{\mathbf{X}}$ are given by $$\begin{aligned}
\mathcal{A}_{\alpha }^{\mathbf{R}}\left( \mathbf{X}_{\alpha }\right) &=&i%
\left[ U_{\alpha }\left( \mathbf{X}_{\alpha }\right) \mathbf{\nabla }_{%
\mathbf{P}}U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right) \right]
\label{Ber} \\
\text{and }\mathcal{A}_{\alpha }^{\mathbf{P}}\left( \mathbf{X}_{\alpha
}\right) &=&-i\left[ U_{\alpha }\left( \mathbf{X}_{\alpha }\right) \mathbf{%
\nabla }_{\mathbf{R}}U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right) %
\right] \notag\end{aligned}$$ and $\mathcal{P}_{+}$ and $\mathcal{P}_{-}$ are respectively the projectors on the diagonal and off-diagonal elements of matrices. From Eq. $\left( \ref%
{TM}\right) $ one can deduce the following relation $$\mathcal{E}\left( \mathcal{T}\exp \left[ \int_{0<\alpha <\hbar }T_{\alpha
}d\alpha \right] \varepsilon _{0}\left( \mathbf{X}_{0}\right) \right) =%
\mathcal{E}\varepsilon _{0}\left( \mathbf{x}\right)$$ where $\mathbf{x=}\left( \mathbf{r},\mathbf{p}\right) $ and $\mathbf{r}$ and $\mathbf{p}$ are new covariant coordinate and momentum operators defined in the following way : $$\begin{aligned}
\mathbf{r} &\equiv &\mathbf{R+}\emph{A}^{\mathbf{R}} \notag \\
\mathbf{p} &\equiv &\mathbf{P+}\emph{A}^{\mathbf{P}} \label{RP1}\end{aligned}$$ The Berry connections terms being defined as : $$\begin{aligned}
\mathbf{\emph{A}^{\mathbf{R}}} &=&\int_{0<\alpha <\hbar }\mathcal{P}_{+}%
\left[ \mathcal{A}_{\alpha }^{\mathbf{R}}\right] d\alpha +\int_{0<\alpha
<\hbar }\frac{1}{2}\left[ \left[ \mathcal{P}_{+}\left[ \mathcal{A}_{\alpha
}^{\mathbf{X}}\right] .\mathbf{\nabla }_{\mathbf{X}}\int_{0<\alpha
_{1}<\alpha }\mathcal{P}_{+}\left[ \mathcal{A}_{\alpha _{1}}^{\mathbf{R}}%
\right] \right] +H.C.\right] d\alpha _{1}d\alpha +... \\
\mathbf{\emph{A}^{\mathbf{P}}} &=&\int_{0<\alpha <\hbar }\mathcal{P}_{+}%
\left[ \mathcal{A}_{\alpha }^{\mathbf{P}}\right] d\alpha +\int_{0<\alpha
<\hbar }\frac{1}{2}\left[ \left[ \mathcal{P}_{+}\left[ \mathcal{A}_{\alpha
}^{\mathbf{X}}\right] .\mathbf{\nabla }_{\mathbf{X}}\int_{0<\alpha
_{1}<\alpha }\mathcal{P}_{+}\left[ \mathcal{A}_{\alpha _{1}}^{\mathbf{P}}%
\right] \right] +H.C.\right] d\alpha _{1}d\alpha +...\end{aligned}$$ and $\mathbf{X}$ denotes the vector $\left( \mathbf{R,P}\right) $. In fact in [@DIAGOEXACT] we show that $\mathbf{\emph{A}^{\mathbf{R}}}$ and $%
\mathbf{\emph{A}^{\mathbf{P}}}$ have to be corrected by a third order in $%
\hbar $ terms $\delta \mathbf{\emph{A}^{\mathbf{R}}}$ and $\delta \mathbf{%
\emph{A}^{\mathbf{P}}}$ but these terms will always be neglected here.
Therefore, the operator $T$ as a part of $O$ naturally leads to the emergence of the covariant dynamical coordinates which turn out the be the physical dynamical variables of particles as shown in several situations [@PIERRESEMIDIAG][@SERIESPIERRE][@PIERREPHOTONGRAVIT][PIERREELECGRAVIT]{}.
Second, using the gauge setting to zero the anti-hermitian diagonal elements of $U_{\alpha }$ [@DIAGOEXACT] we have $$\begin{aligned}
N_{\alpha }U_{\alpha } &=&\left[ \left( \frac{\partial }{\partial \alpha }%
+\left\langle .\right\rangle \right) U_{\alpha }\right] =-\left[
.,\varepsilon _{\alpha }\right] ^{-1}.\left[ \mathcal{P}_{-}\left\{ \frac{1}{%
2}\left( \mathcal{A}_{\alpha }^{R_{l}}\nabla _{R_{l}}\varepsilon _{\alpha
}+\nabla _{R_{l}}\varepsilon _{\alpha }\mathcal{A}_{\alpha }^{R_{l}}+%
\mathcal{A}_{\alpha }^{P_{l}}\nabla _{P_{l}}\varepsilon _{\alpha }+\nabla
_{P_{l}}\varepsilon _{\alpha }\mathcal{A}_{\alpha }^{P_{l}}\right) \right\}
\right. \notag \\
&&\left. +\frac{i}{4}\mathcal{P}_{-}\left\{ \left[ \varepsilon _{\alpha },%
\mathcal{A}_{\alpha }^{R_{l}}\right] \mathcal{A}_{\alpha }^{P_{l}}-\left[
\varepsilon _{\alpha },\mathcal{A}_{\alpha }^{P_{l}}\right] \mathcal{A}%
_{\alpha }^{R_{l}}\right\} +H.C.\right] -\frac{i}{4}\mathcal{P}_{-}\left\{ %
\left[ \mathcal{A}_{\alpha }^{R_{l}},\mathcal{A}_{\alpha }^{P_{l}}\right]
U_{\alpha }\right\} \label{NU}\end{aligned}$$ where the inverse of the commutator operation $\left[ ,\varepsilon _{\alpha }%
\right] $ has the following properties $$\begin{aligned}
\left[ \left[ .,\varepsilon _{\alpha }\right] ^{-1}.M,\varepsilon _{\alpha }%
\right] &=&\left[ .,\varepsilon _{\alpha }\right] ^{-1}.\left[ M,\varepsilon
_{\alpha }\right] =M\text{\ \ \ for \ \ }\left[ M,\varepsilon _{\alpha }%
\right] \neq 0\text{ } \notag \\
\left[ .,\varepsilon _{\alpha }\right] ^{-1}.M &=&0\text{ \ \ if \ }\left[
M,\varepsilon _{\alpha }\right] =0 \label{NU2}\end{aligned}$$ for an arbitrary matrix valued operator $M$. It means that the operator $%
\left[ .,\varepsilon _{\alpha }\right] ^{-1}$ acts on the space of endomorphism of the state space in the following way : it is zero when acting on the kernel of the operator $\left[ .,\varepsilon _{\alpha }\right]
$ whose action is to compute the commutator with $\varepsilon _{\alpha }$, and is the inverse of $\left[ .,\varepsilon _{\alpha }\right] $ on the complementary subspace of the kernel.
Note that, as it will appear clearly later on, having both $\varepsilon
\left( \mathbf{X}\right) $ and $U\left( \mathbf{X}\right) $ at order $n$ in $%
\hbar $, and reinserting in the exponential of Eqs. $\left( \ref{EE}\right)
\left( \ref{U}\right) $ allows us to find $\varepsilon \left( \mathbf{X}%
\right) $ and $U\left( \mathbf{X}\right) $ at order $n+1$ in $\hbar $. A needed assumption for this procedure to work is that the diagonalization $%
\varepsilon _{0}\left( \mathbf{X}_{0}\right) =U_{0}H_{0}\left( \mathbf{X}%
_{0}\right) U_{0}^{+}$ is explicitly known when $\hbar =0$, i.e. when $%
\mathbf{P}$ and $\mathbf{R}$ are treated as commuting variables**.**
We end up this section with a technical remark that will be important for the sequel. As shown in our formula $\left( \ref{U}\right) $ the solutions for the diagonalization process depends on $\varepsilon _{0}\left( \mathbf{X}%
_{0}\right) $, $U_{0}\left( \mathbf{X}_{0}\right) $. While the final results do not depend on the way variables are symmetrized (that is the order we write the products of components of $\mathbf{X}$, one has to start with an initial symmetrization for the diagonalized energy at the zeroth order as well as for $U_{0}\left( \mathbf{X}_{0}\right) $ or equivalently the Berry phase. Since this detail will be important only at the second order in $%
\hbar $ while considering the Bloch electron in an electromagnetic field, we do not mention any choice for the moment.
Eqs. $\left( \ref{U}\right) \left( \ref{NU}\right) $ show that a diagonal Hamiltonian representation can be found to any desired order in $\hbar $, and in the following we will carry it out until the second order. Let start first with the first order.
Covariant dynamical operator algebra
------------------------------------
From equations Eq. $\left( \ref{RP1}\right) $ we readily deduce the following non trivial algebra between the dynamical operators $$\begin{aligned}
\left[ r_{i},r_{j}\right] &=&i\hbar ^{2}\Theta _{ij}^{rr}=i\hbar ^{2}\left(
\nabla _{P_{i}}\emph{A}_{R_{j}}-\nabla _{P_{j}}\emph{A}_{R_{i}}\right)
+\hbar ^{2}\left[ \emph{A}_{R_{j}},\emph{A}_{R_{i}}\right] \notag \\
\left[ p_{i},p_{j}\right] &=&i\hbar ^{2}\Theta _{ij}^{pp}=-i\hbar ^{2}\left(
\nabla _{R_{i}}\emph{A}_{P_{j}}-\nabla _{R_{j}}\emph{A}_{P_{i}}\right)
+\hbar ^{2}\left[ \emph{A}_{P_{i}},\emph{A}_{P_{j}}\right] \notag \\
\left[ p_{i},r_{j}\right] &=&-i\hbar \delta _{ij}+i\hbar ^{2}\Theta
_{ij}^{pr}=-i\hbar \delta _{ij}-i\hbar ^{2}\left( \nabla _{R_{i}}\emph{A}%
_{R_{j}}+\nabla _{P_{j}}\emph{A}_{P_{i}}\right) +\hbar ^{2}\left[ \emph{A}%
_{P_{i}},\emph{A}_{R_{j}}\right] \label{commalgebra}\end{aligned}$$ where the terms $\Theta _{ij}$ are Berry curvatures definitions. Of course these non trivial commutation relations also give new contributions to the equations of motion and thus lead to new phenomena [@PIERREEUROPHYS][PIERRESEMIDIAG]{}[@SERIESPIERRE][@PIERREPHOTONGRAVIT][PIERREELECGRAVIT]{}. The commutation relations are valid to any order in $%
\hbar ,$ but in practice we can compute them as well as the energy $%
\varepsilon \left( \mathbf{X}\right) $ in a series expansion in $\hbar .$
Diagonal representation at the first order in $\hbar $
------------------------------------------------------
In this section we derive by a straightforward application of Eq. $\left( %
\ref{EE}\right) $ the semiclassical effective diagonal in-bands Hamiltonian$\varepsilon $ for one particle. Using the fact that at this level $%
\int_{0}^{\hbar }O_{\alpha }d\alpha \varepsilon _{0}\left( \mathbf{X}%
_{\alpha }\right) =\hbar \left( T_{\hbar }+M_{\hbar }\right) \varepsilon
_{0}\left( \mathbf{X}\right) $ we have $$\varepsilon =\varepsilon _{0}\left( \mathbf{X}\right) +\hbar \left( T_{\hbar
}+M_{\hbar }\right) \varepsilon _{0}\left( \mathbf{X}\right) +O(\hbar ^{2})
\label{E}$$ From the definition of $T$ and $M$ in Eq. $\left( \ref{TM}\right) $ we see that we must determine the quantities $\mathcal{A}_{\hbar }^{\mathbf{X}%
}=\left( \mathcal{A}_{\hbar }^{\mathbf{R}},\mathcal{A}_{\hbar }^{\mathbf{P}%
}\right) $ Eq. $\left( \ref{Ber}\right) $ defined previously. At this level of the approximation and due to the factor $\hbar $ in the previous equation it is enough to know this quantity at the zeroth order in $\hbar $. Skipping the $\hbar $ index, they are thus simply given by
$$\begin{aligned}
\mathcal{A}_{\hbar }^{\mathbf{X}} &=&\mathcal{A}_{0}^{\mathbf{R}%
}=iU_{0}\left( \mathbf{X}\right) \mathbf{\nabla }_{\mathbf{P}%
}U_{0}^{+}\left( \mathbf{X}\right) \\
\mathcal{A}_{\hbar }^{\mathbf{P}} &=&\mathcal{A}_{0}^{\mathbf{P}%
}=-iU_{0}\left( \mathbf{X}\right) \mathbf{\nabla }_{\mathbf{R}%
}U_{0}^{+}\left( \mathbf{X}\right)\end{aligned}$$
where as before $U_{0}$ is the matrix that would diagonalize the Hamiltonian if $\mathbf{R}$ and $\mathbf{P}$ where commuting variables (that is the matrix $U_{0}$ which diagonalizes the classical energy when $\hbar =0$). It means that, interestingly, at this first order in $\hbar $, the diagonalizing matrix $U$ is the matrix defined at the order $\hbar ^{0}$ in which the classical variables $\mathbf{X}_{0}$ have been replaced by the quantum operator $\mathbf{X.}$ This is the clue of the method: at each order $\hbar ^{n}$ we just need to know the matrix $U$ at the order $\hbar ^{n-1}.$ Therefore, the semiclassical diagonalization is within the present approach simple to achieve (and simpler than any other method we are aware for) as it only requires the determination of the matrix diagonalizing the classical energy.
Introducing also the covariant dynamical operators $\mathbf{x}=\mathbf{X+}%
\emph{A}^{\mathbf{X}}$ with the Berry connection $\emph{A}^{\mathbf{X}%
}=\hbar \mathcal{P}_{+}\left[ \mathcal{A}_{\hbar }^{\mathbf{X}}\right] $, as well as the integrated non projected Berry connections $\mathcal{A}^{\mathbf{%
X}}=\int_{0}^{\hbar }\mathcal{A}_{\alpha }^{\mathbf{X}}d\alpha =\hbar
\mathcal{A}_{\hbar }^{\mathbf{X}}$ at this order, the diagonal energy Eq. $%
\left( \ref{E}\right) $ can be written as $$\varepsilon =\varepsilon _{0}\left( \mathbf{x}\right) +\frac{i}{2\hbar }%
\mathcal{P}_{+}\left\{ \left[ \varepsilon _{0}\left( \mathbf{X}\right) ,%
\mathcal{A}^{R^{i}}\right] \mathcal{A}^{P_{i}}-\left[ \varepsilon _{0}\left(
\mathbf{X}\right) ,\mathcal{A}^{P_{i}}\right] \mathcal{A}^{R^{i}}\right\}
+O(\hbar ^{2}) \label{SE}$$ where summation over $i$ is assumed. Here $\varepsilon _{0}\left( \mathbf{x}%
\right) $ corresponds to the classical diagonal energy operators in which the classical variables $\mathbf{X}_{0}$ have been replaced by the full noncomutative covariant operators which satisfy a non commutative algebra as we will see later when considering the practical examples. The second contribution which is of order $\hbar $, gives for instance for Bloch electrons in a magnetic field $\mathbf{B}$ as we shall see later, the coupling between $\mathbf{B}$ and the magnetic moment operator .
Note also that this general diagonal energy for an arbitrary one particle system was already derived in previous works by different methods [PIERRESEMIDIAG]{}[@SERIESPIERRE] and turns out to be very useful for the study of several different physical systems [@PIERREEUROPHYS][PIERREELECGRAVIT]{}[@PIERREPHOTONGRAVIT][@PIERREGRAPHENE].
Diagonal representation at the second order in $\hbar $
-------------------------------------------------------
The Hamiltonian diagonalization at this order requires the matrix $U$ at the first order: $U\left( \mathbf{X}\right) =U_{0}\left( \mathbf{X}\right)
+\hbar U_{1}\left( \mathbf{X}\right) U_{0}\left( \mathbf{X}\right) $ where $%
U_{1}\left( \mathbf{X}\right) $ is determined from the relation $\hbar
U_{1}\left( \mathbf{X}\right) =\int_{0}^{\hbar }N_{\alpha }d\alpha
U_{0}\left( \mathbf{X}\right) $ as a consequence of Eq. $\left( \ref{U}%
\right) .$ With expression $\left( \ref{NU}\right) $ we readily obtain :
$$\begin{aligned}
U_{1}\left( \mathbf{X}\right) &=&\left[ .,\varepsilon _{0}\right] ^{-1}.%
\left[ \mathcal{P}_{-}\left\{ \frac{1}{2}\left( \mathcal{A}%
_{0}^{R_{l}}\nabla _{R_{l}}\varepsilon _{0}\left( \mathbf{X}\right) +\nabla
_{R_{l}}\varepsilon _{0}\left( \mathbf{X}\right) \mathcal{A}_{0}^{R_{l}}+%
\mathcal{A}_{0}^{P_{l}}\nabla _{P_{l}}\varepsilon _{0}\left( \mathbf{X}%
\right) +\nabla _{P_{l}}\varepsilon _{0}\left( \mathbf{X}\right) \mathcal{A}%
_{0}^{P_{l}}\right) \right\} \right. \notag \\
&&\left. -\frac{i}{2}\left\{ \left[ \varepsilon _{0}\left( \mathbf{X}\right)
,\mathcal{A}_{0}^{R_{l}}\right] \mathcal{A}_{0}^{P_{l}}-\left[ \varepsilon
_{0}\left( \mathbf{X}\right) ,\mathcal{A}_{0}^{P_{l}}\right] \mathcal{A}%
_{0}^{R_{l}}\right\} \right] -\frac{i}{4}\left[ \mathcal{A}_{0}^{R_{l}},%
\mathcal{A}_{0}^{P_{l}}\right] \label{U1}\end{aligned}$$
At the same order the (non-diagonal) Berry connections $\mathcal{A}_{\alpha
}^{\mathbf{X}}=\left( \mathcal{A}_{\alpha }^{\mathbf{R}},\mathcal{A}_{\alpha
}^{\mathbf{P}}\right) $ are again given by $$\begin{aligned}
\mathcal{A}_{\alpha }^{\mathbf{R}}\left( \mathbf{X}_{\alpha }\right) &=&i%
\left[ U_{\alpha }\left( \mathbf{X}_{\alpha }\right) \mathbf{\nabla }_{%
\mathbf{P}}U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right) \right] =\frac{1%
}{\alpha }U_{\alpha }\left( \mathbf{X}_{\alpha }\right) \mathbf{R}_{\alpha
}U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right) \\
\text{and }\mathcal{A}_{\alpha }^{\mathbf{P}}\left( \mathbf{X}_{\alpha
}\right) &=&-i\left[ U_{\alpha }\left( \mathbf{X}_{\alpha }\right) \mathbf{%
\nabla }_{\mathbf{R}}U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right) %
\right] =\frac{1}{\alpha }U_{\alpha }\left( \mathbf{X}_{\alpha }\right)
\mathbf{P}_{\alpha }U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right) \notag\end{aligned}$$ where now $U\left( \mathbf{X}_{\alpha }\right) $ is the transformation to the first order in $\alpha $ i.e. $U_{0}\left( \mathbf{X}_{\alpha }\right)
+\alpha U_{1}\left( \mathbf{X}_{\alpha }\right) ,$ in which $\mathbf{X}$ is replaced by the running operator $\mathbf{X}_{\alpha }$. Using the hermiticity of $\mathcal{A}_{\alpha }^{\mathbf{X}}$, so that one has $%
\mathcal{A}_{\alpha }^{\mathbf{X}}=\mathcal{A}_{\alpha }^{\mathbf{X}}+\left(
\mathcal{A}_{\alpha }^{\mathbf{X}}\right) ^{+}$ we can expand $\mathcal{A}^{%
\mathbf{X}}$ as : $$\mathcal{A}_{\alpha }^{\mathbf{X}}=\left( \frac{1}{2}\left( \left[ 1+\alpha
U_{1}\left( \mathbf{X}_{\alpha }\right) \right] U_{0}\left( \mathbf{X}%
_{\alpha }\right) \right) \frac{\mathbf{X}_{\alpha }}{\alpha }\left(
U_{0}^{+}\left( \mathbf{X}_{\alpha }\right) \left[ 1+\alpha U_{1}^{+}\left(
\mathbf{X}_{\alpha }\right) \right] \right) +\frac{1}{2}H.C.\right) -\frac{%
\mathbf{X}_{\alpha }}{\alpha }$$ (the $\frac{1}{\alpha }$ factor reminds that in our definition of $\mathcal{A%
}_{\alpha }^{\mathbf{X}}$ the Gradient with respect to $\mathbf{X}_{\alpha }$ is normalized, i.e. divided by $\alpha $). After some recombinations, the previous expression can be written in a more convenient form : $$\begin{aligned}
\mathcal{A}_{\alpha }^{\mathbf{X}} &=&\frac{1}{2\alpha }U_{0}\left( \mathbf{X%
}_{\alpha }\right) \left[ \mathbf{X}_{\alpha }\mathbf{,}U_{0}^{+}\left(
\mathbf{X}_{\alpha }\right) \right] +H.C. \\
&&+\frac{1}{2}\left[ U_{1}\left( \mathbf{X}_{\alpha }\right) \left[
U_{0}\left( \mathbf{X}_{\alpha }\right) \mathbf{X}_{\alpha }U_{0}^{+}\left(
\mathbf{X}_{\alpha }\right) \right] +\left[ U_{0}\left( \mathbf{X}_{\alpha
}\right) \mathbf{X}_{\alpha }U_{0}^{+}\left( \mathbf{X}_{\alpha }\right) %
\right] U_{1}^{+}\left( \mathbf{X}_{\alpha }\right) \right] +H.C.-\mathbf{X}%
_{\alpha }\end{aligned}$$ using now the fact that at the lowest order $U_{0}\left( \mathbf{X}_{\alpha
}\right) \left[ \frac{\mathbf{X}_{\alpha }}{\alpha },U_{0}^{+}\left( \mathbf{%
X}_{\alpha }\right) \right] =\mathcal{A}_{0}^{\mathbf{X}}\left( \mathbf{X}%
_{\alpha }\right) $, one has : $$\begin{aligned}
\mathcal{A}_{\alpha }^{\mathbf{X}} &=&\frac{1}{2\alpha }\left[ U_{0}\left(
\mathbf{X}_{\alpha }\right) \left[ \mathbf{X}_{\alpha }\mathbf{,}%
U_{0}^{+}\left( \mathbf{X}_{\alpha }\right) \right] +H.C.\right] \\
&&+\frac{1}{2}\left[ U_{1}\left( \mathbf{X}_{\alpha }\right) \mathcal{A}%
_{0}^{\mathbf{X}}\left( \mathbf{X}_{\alpha }\right) +\mathcal{A}_{0}^{%
\mathbf{X}}\left( \mathbf{X}_{\alpha }\right) U_{1}^{+}\left( \mathbf{X}%
_{\alpha }\right) +\left[ \mathbf{X}_{\alpha },U_{1}^{+}\left( \mathbf{X}%
_{\alpha }\right) \right] +H.C.\right]\end{aligned}$$ decomposing $U_{1}^{+}\left( \mathbf{X}_{\alpha }\right) $ into Hermitian and antihermitian part we are thus led to : $$\begin{aligned}
\mathcal{A}_{\alpha }^{\mathbf{X}} &=&\frac{1}{2\alpha }\left( U_{0}\left(
\mathbf{X}_{\alpha }\right) \left[ \mathbf{X}_{\alpha }\mathbf{,}%
U_{0}^{+}\left( \mathbf{X}_{\alpha }\right) \right] +H.C.\right) +\left[
\mathbf{X}_{\alpha }\mathbf{+}\mathcal{A}_{0}^{\mathbf{X}}\mathbf{,}\mathit{%
ah}\left( U_{1}^{+}\left( \mathbf{X}_{\alpha }\right) \right) \right] \\
&&+\left( \mathcal{A}_{0}^{\mathbf{X}}H\left( U_{1}^{+}\left( \mathbf{X}%
_{\alpha }\right) \right) +\mathit{h}\left( U_{1}^{+}\left( \mathbf{X}%
_{\alpha }\right) \right) \mathcal{A}_{0}^{\mathbf{X}}\right)\end{aligned}$$ where $\mathit{ah}\left( Z\right) $ and $\mathit{ah}\left( Z\right) $ denote the antihermitian and Hermitian part of an operator $Z$ respectively. Now with Eq. $\left( \ref{U1}\right) $ and after the integration, we are led for $\mathcal{A}^{\mathbf{X}}=\hbar \mathcal{A}_{0}^{\mathbf{X}}+\hbar ^{2}%
\mathcal{A}_{1}^{\mathbf{X}}$ to the following expression : $$\mathcal{A}_{\alpha }^{\mathbf{X}}=\mathcal{A}_{0}^{\mathbf{X}}\left(
\mathbf{R+}\frac{\alpha }{2}\mathcal{A}_{0}^{R_{l}},\mathbf{P+}\frac{\alpha
}{2}\mathcal{A}_{0}^{P_{l}}\right) -\left[ \mathcal{B},\mathbf{X}_{\alpha }+%
\mathcal{A}_{0}^{\mathbf{X}}\right] \label{Conn2}$$ where we introduced the notations $$\mathcal{A}_{0}^{\mathbf{X}}\left( \mathbf{R+}\frac{\alpha }{2}\mathcal{A}%
_{0}^{R_{l}},\mathbf{P+}\frac{\alpha }{2}\mathcal{A}_{0}^{P_{l}}\right)
\equiv \mathcal{A}_{0}^{\mathbf{X}}+\frac{\alpha }{4}\left\{ \mathcal{A}%
_{0}^{R_{l}}\nabla _{R_{l}}\mathcal{A}_{0}^{\mathbf{X}}+\mathcal{A}%
_{0}^{P_{l}}\nabla _{R_{l}}\mathcal{A}_{0}^{\mathbf{X}}+H.C.\right\}
\label{conn2A}$$ and $$\begin{aligned}
\mathcal{B} &=&\left[ .,\varepsilon _{0}\right] ^{-1}.\left( \mathcal{P}%
_{-}\left\{ \frac{1}{2}\mathcal{A}_{0}^{R_{l}}\nabla _{R_{l}}\varepsilon
_{0}\left( \mathbf{X}\right) +\frac{1}{2}\mathcal{A}_{0}^{P_{l}}\nabla
_{P_{l}}\varepsilon _{0}\left( \mathbf{X}\right) +H.C.\right\} \right.
\notag \\
&&\left. -\frac{i}{4}\left\{ \left[ \varepsilon _{0}\left( \mathbf{X}\right)
,\mathcal{A}_{0}^{R_{l}}\right] \mathcal{A}_{0}^{P_{l}}-\left[ \varepsilon
_{0}\left( \mathbf{X}\right) ,\mathcal{A}_{0}^{P_{l}}\right] \mathcal{A}%
_{0}^{R_{l}}+H.C.\right\} \right) \notag \\
&=&\left[ .,\varepsilon _{0}\right] ^{-1}.\left( \mathcal{P}_{-}\left\{
\frac{1}{2}\mathcal{A}_{0}^{R_{l}}\nabla _{R_{l}}\varepsilon _{0}\left(
\mathbf{X}\right) +\frac{1}{2}\mathcal{A}_{0}^{P_{l}}\nabla
_{P_{l}}\varepsilon _{0}\left( \mathbf{X}\right) +H.C.\right\} \right.
\notag \\
&&\left. -\frac{i}{4}\left\{ \mathcal{P}_{-}\mathcal{A}_{0}^{R_{l}}\mathcal{P%
}_{+}\mathcal{A}_{0}^{P_{l}}-\mathcal{P}_{-}\mathcal{A}_{0}^{P_{l}}\mathcal{P%
}_{+}\mathcal{A}_{0}^{R_{l}}+H.C.\right\} \right) \label{conn2B}\end{aligned}$$ These formula, although abstruse will be useful in our next sections. Having found the matrices $U\left( \mathbf{X}\right) $ and $\mathcal{A}_{\alpha }^{%
\mathbf{X}}$ at the required order we can now determine the series expansion for the diagonalized Hamiltonian to the second order in $\hbar $. From equation Eq. $\left( \ref{EE}\right) $ we can write
$$\varepsilon \left( \mathbf{X}\right) =\varepsilon _{0}\left( \mathbf{X}%
\right) +\int_{0}^{\hbar }O_{\alpha }d\alpha \varepsilon _{0}\left( \mathbf{X%
}_{\alpha }\right) +\int_{0}^{\hbar }O_{\alpha _{1}}\int_{0}^{\alpha
_{1}}O_{\alpha _{2}}d\alpha _{2}d\alpha _{1}\varepsilon _{0}\left( \mathbf{X}%
_{\alpha _{2}}\right) -\frac{\hbar }{2}\left\langle \varepsilon _{0}\left(
\mathbf{X}\right) \right\rangle$$
where the last contribution is given by $$-\frac{1}{2}\hbar \left\langle \varepsilon _{0}\left( \mathbf{X}\right)
\right\rangle =\frac{i}{4}\hbar Asym\left\{ \nabla _{P_{l}}\nabla
_{R_{l}}\varepsilon _{0}\left( \mathbf{X}\right) \right\}$$ The first contribution $\int_{0}^{\hbar }O_{\alpha }d\alpha \varepsilon
_{0}\left( \mathbf{X}\right) $ can be expanded as before as: $$\begin{aligned}
\int_{0}^{\hbar }O_{\alpha }d\alpha \varepsilon _{0}\left( \mathbf{X}\right)
&=&\int_{0}^{\hbar }\mathcal{P}_{+}\left\{ \frac{1}{2}\left( \mathcal{A}%
_{\alpha }^{R_{l}}\nabla _{R_{l}}\varepsilon _{0}\left( \mathbf{X}_{\alpha
}\right) +\nabla _{R_{l}}\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right)
\mathcal{A}_{\alpha }^{R_{l}}+\mathcal{A}_{\alpha }^{P_{l}}\nabla
_{P_{l}}\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right) +\nabla
_{P_{l}}\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right) \mathcal{A}%
_{\alpha }^{P_{l}}\right) \right\} d\alpha \\
&&+\int_{0}^{\hbar }\mathcal{P}_{+}\left\{ \frac{i}{4}\left\{ \left[
\varepsilon _{0}\left( \mathbf{X}_{\alpha }\right) ,\mathcal{A}_{\alpha
}^{R_{l}}\right] \mathcal{A}_{\alpha }^{P_{l}}-\left[ \varepsilon _{0}\left(
\mathbf{X}_{\alpha }\right) ,\mathcal{A}_{\alpha }^{P_{l}}\right] \mathcal{A}%
_{\alpha }^{R_{l}}\right\} +H.C.\right. \\
&&\left. +\left[ U_{\alpha }\left( \left( \frac{\partial }{\partial \alpha }%
+\left\langle .\right\rangle \right) H\left( \mathbf{X}_{\alpha }\right)
\right) U_{\alpha }^{+}\right] \right\} d\alpha\end{aligned}$$ where the Berry connections have to be expanded to the first order, whereas the second order contribution $$\int_{0}^{\hbar }O_{\alpha _{1}}\int_{0}^{\alpha _{1}}O_{\alpha _{2}}d\alpha
_{2}d\alpha _{1}\varepsilon _{0}\left( \mathbf{X}\right) -\frac{\hbar }{2}%
\left\langle \varepsilon _{0}\left( \mathbf{X}\right) \right\rangle$$ has to be expanded to the zeroth order in the Berry connections. Notice that due to the integration process, the squared terms in $\mathcal{A}_{0}^{%
\mathbf{R}}$, $\mathcal{A}_{0}^{\mathbf{P}}$ as well as the first order terms in the Berry phase get a $\frac{1}{2}$ factor.
The consequence is that these contributions can be recombined to yield : $$\begin{aligned}
\varepsilon \left( \mathbf{X}\right) &=&\varepsilon _{0}\left( \mathbf{x}%
\right) +\frac{i}{2}\mathcal{P}_{+}\left\{ \left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}%
^{P_{l}}-\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}%
^{P_{l}}\right] \mathcal{\hat{A}}^{R_{l}}-\left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\left[ \mathcal{\hat{A}}^{R_{l}},\mathcal{\hat{A}}^{P_{l}}%
\right] \right] \right\} \notag \\
&&-\frac{1}{8}\mathcal{P}_{+}\left\{ \left[ \left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}%
^{P_{l}}-\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}%
^{P_{l}}\right] \mathcal{\hat{A}}^{R_{l}},\mathcal{\hat{A}}^{R_{l}}\right]
\mathcal{\hat{A}}^{P_{l}}\right. \notag \\
&&\left. -\left[ \left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{%
\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}^{P_{l}}-\left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{P_{l}}\right] \mathcal{\hat{%
A}}^{R_{l}},\mathcal{\hat{A}}^{P_{l}}\right] \mathcal{\hat{A}}%
^{R_{l}}\right\} -\frac{\hbar }{2}\left\langle \varepsilon _{0}\left(
\mathbf{x}\right) \right\rangle \label{E2}\end{aligned}$$ with : $$\begin{aligned}
\mathcal{\hat{A}}^{R_{l}} &=&\frac{1}{2}\left[ 1-\frac{1}{2}\left( \mathcal{P%
}_{+}\mathcal{A}_{0}^{\mathbf{X}}.\mathbf{\nabla }_{\mathbf{X}}\right) %
\right] \mathcal{A}^{R_{l}}\left( \mathbf{x}\right) +H.C. \\
\mathcal{\hat{A}}^{P_{l}} &=&\frac{1}{2}\left[ 1-\frac{1}{2}\left( \mathcal{P%
}_{+}\mathcal{A}_{0}^{\mathbf{X}}.\mathbf{\nabla }_{\mathbf{X}}\right) %
\right] \mathcal{A}^{P_{l}}\left( \mathbf{x}\right) +H.C.\end{aligned}$$ and where we have denoted again $\mathcal{A}^{\mathbf{X}}\left( \mathbf{X}%
\right) =\int_{0}^{\hbar }\mathcal{A}_{\alpha }^{\mathbf{X}}\left( \mathbf{X}%
\right) d\alpha $. Remark that the last term $-\frac{\hbar }{2}\left\langle
\varepsilon _{0}\left( \mathbf{x}\right) \right\rangle $ in the expression for $\varepsilon \left( \mathbf{X}\right) $ is of order $\hbar ^{2}$ since $%
\left\langle \varepsilon _{0}\left( \mathbf{x}\right) \right\rangle $ involves some commutators and is thus of order $\hbar $. As explained above, one can, at each order of the expansion, safely replace the canonical operators $\mathbf{X=}\left( \mathbf{R,P}\right) $ by the covariant ones $\mathbf{x=}\left( \mathbf{r,p}\right) $ which are given by the following expression to the second order (dropping once again the $\hbar
$ index) :
$$\mathbf{x}=\mathbf{X+}\emph{A}^{\mathbf{X}}\equiv \mathbf{X+}\hbar \emph{A}%
_{0}^{\mathbf{X}}+\frac{\hbar }{2}^{2}\emph{A}_{1}^{\mathbf{X}}
\label{rpnew}$$
with : $\emph{A}^{\mathbf{X}}=\left( \emph{A}^{\mathbf{R}},\emph{A}^{\mathbf{%
P}}\right) $ and (as before we drop the index $\hbar $) $\mathbf{X}=\left(
\mathbf{R},\mathbf{P}\right) =\mathbf{X}_{\hbar }=\left( \mathbf{R}_{\hbar },%
\mathbf{P}_{\hbar }\right) $: $$\begin{aligned}
\mathbf{\emph{A}^{\mathbf{R}}} &=&\int\limits_{0<\alpha <\hbar }\mathcal{P}%
_{+}\left[ \mathcal{A}_{\alpha }^{\mathbf{R}}\right] d\alpha
+\int\limits_{0<\alpha <\hbar }\frac{1}{2}\left[ \left[ \frac{1}{2}\left(
\mathcal{P}_{+}\left[ \mathcal{A}_{\alpha }^{\mathbf{X}}\right] .\mathbf{%
\nabla }_{\mathbf{X}}+H.C.\right) \int\limits_{0<\alpha _{1}<\alpha }%
\mathcal{P}_{+}\left[ \mathcal{A}_{\alpha _{1}}^{\mathbf{R}}\right] \right]
+H.C.\right] d\alpha _{1}d\alpha \\
&=&\mathcal{P}_{+}\left[ \hbar \mathcal{A}_{0}^{\mathbf{R}}\left( \mathbf{R+}%
\frac{\hbar }{4}\mathcal{A}_{0}^{R_{l}},\mathbf{P+}\frac{\hbar }{4}\mathcal{A%
}_{0}^{P_{l}}\right) -\frac{\hbar ^{2}}{2}\left[ \mathcal{B},\mathbf{R}+%
\mathcal{A}_{0}^{\mathbf{R}}\right] \right] +\frac{\hbar ^{2}}{4}\left(
\mathcal{P}_{+}\left[ \mathcal{A}_{0}^{\mathbf{X}}\right] .\mathbf{\nabla }_{%
\mathbf{X}}\mathcal{P}_{+}\left[ \mathcal{A}_{0}^{\mathbf{R}}\right]
+H.C.\right) \\
\mathbf{\emph{A}^{\mathbf{P}}} &=&\int\limits_{0<\alpha <\hbar }\mathcal{P}%
_{+}\left[ \mathcal{A}_{\alpha }^{\mathbf{P}}\right] d\alpha
+\int\limits_{0<\alpha <\hbar }\frac{1}{2}\left[ \left[ \frac{1}{2}\left(
\mathcal{P}_{+}\left[ \mathcal{A}_{\alpha }^{\mathbf{X}}\right] .\mathbf{%
\nabla }_{\mathbf{X}}+H.C.\right) \int\limits_{0<\alpha _{1}<\alpha }%
\mathcal{P}_{+}\left[ \mathcal{A}_{\alpha _{1}}^{\mathbf{P}}\right] \right]
+H.C.\right] d\alpha _{1}d\alpha +... \\
&=&\mathcal{P}_{+}\left[ \hbar \mathcal{A}_{0}^{\mathbf{P}}\left( \mathbf{R+}%
\frac{\hbar }{4}\mathcal{A}_{0}^{R_{l}},\mathbf{P+}\frac{\hbar }{4}\mathcal{A%
}_{0}^{P_{l}}\right) -\frac{\hbar ^{2}}{2}\left[ \mathcal{B},\mathbf{P}+%
\mathcal{A}_{0}^{\mathbf{P}}\right] \right] +\frac{\hbar ^{2}}{4}\left(
\mathcal{P}_{+}\left[ \mathcal{A}_{0}^{\mathbf{X}}\right] .\mathbf{\nabla }_{%
\mathbf{X}}\mathcal{P}_{+}\left[ \mathcal{A}_{0}^{\mathbf{P}}\right]
+H.C.\right)\end{aligned}$$ Remark ultimately, that had we chosen the variables $\mathbf{X}$ rather than $\mathbf{x}$ to express our Hamiltonian, we would have rather written at our order of approximation : $$\begin{aligned}
\mathcal{\hat{A}}^{R_{l}} &=&\frac{1}{2}\left[ 1+\frac{1}{2}\left( \mathcal{P%
}_{+}\mathcal{A}_{0}^{\mathbf{X}}.\mathbf{\nabla }_{\mathbf{X}}\right) %
\right] \mathcal{A}^{R_{l}}\left( \mathbf{X}\right) +H.C. \\
\mathcal{\hat{A}}^{P_{l}} &=&\frac{1}{2}\left[ 1+\frac{1}{2}\left( \mathcal{P%
}_{+}\mathcal{A}_{0}^{\mathbf{X}}.\mathbf{\nabla }_{\mathbf{X}}\right) %
\right] \mathcal{A}^{P_{l}}\left( \mathbf{X}\right) +H.C.\end{aligned}$$ However, as explained before, the transformed variables $\mathbf{x}$ fit better to write the Hamiltonian since they enter directly in $\varepsilon
_{0}\left( \mathbf{x}\right) $.
Eq $\left( \ref{E2}\right) $ is the desired series expansion to the second order in $\hbar $ of the diagonal Hamiltonian. In the next section it will be the angular stone for the computation of the effective in-bands Hamiltonian of a Bloch electron in an external electromagnetic field. Once this will be achieved the case of several Bloch electrons will be investigated.
Bloch electron in an electromagnetic field
==========================================
To start, an outline of the first order diagonalization for the special case of an electron in an crystal lattice perturbated by the presence of an external electromagnetic field considered in refs. [@PIERREEUROPHYS] and [@PIERRESEMIDIAG] is given.
The Hamiltonian of an electron in an crystal lattice perturbated by the presence of an external electromagnetic field is $$H=\frac{\left( \mathbf{P-}e\mathbf{A}\right) ^{2}}{2m}+V_{p}\left( \mathbf{R}%
\right) +eV\left( \mathbf{R}\right)$$ ($e<0$) where $V_{p}\left( \mathbf{R}\right) $ the periodic potential, $%
\mathbf{A}$ and $V$ the vector and scalar potential respectively. Our purpose is this section is to compute the diagonal in-bands energy Hamiltonian for this system to the second order in a series expansion in $%
\hbar .$ This can be done by using the general results of the previous section, in particular with Eqs. $\left( \ref{SE}\right) \left( \ref{E2}%
\right) $. The major difficulty to find the diagonal representation relies on the fact that in presence of an electromagnetic field, the lattice translation operators $\mathbf{T}$ do not commute any more (see [@NIU] and references therein). To deal with this problem it is convenient to express the total magnetic field as the sum of a constant field $\mathbf{B}%
_{0}$ and small nonuniform part $\delta \mathbf{B}(\mathbf{R})$. The Hamiltonian can be written $H=H_{0}+eV(\mathbf{R})$, with $H_{0}$ the magnetic contribution ($V$ being the electric potential) which reads $$H_{0}=\frac{1}{2m}\left( \mathbf{P}-e\mathbf{A}(\mathbf{R})-e\delta \mathbf{A%
}(\mathbf{R})\right) ^{2}+V_{p}\left( \mathbf{R}\right) \label{Hmagnetic}$$ where $\mathbf{A}(\mathbf{R})$ and $\delta \mathbf{A}(\mathbf{R})$ are the vectors potential of the homogeneous and inhomogeneous magnetic field, respectively. $V_{p}\left( \mathbf{R}\right) $ is the periodic potential. The large constant part $\mathbf{B}_{0}$ is chosen such that the magnetic flux through a unit cell is a rational fraction of the flux quantum $h/e$. The advantage of such a decomposition is that for $\delta \mathbf{A}(\mathbf{%
R})=0$ the magnetic translation operators are commuting quantities allowing to exactly diagonalize the Hamiltonian and to treat $\delta \mathbf{A}(%
\mathbf{R})$ as a small perturbation. The state space of the Bloch electron is spanned by the basis vector $\left\vert n,\mathbf{k}\right\rangle
=\left\vert \mathbf{k}\right\rangle \otimes \left\vert n\right\rangle $ with $n$ corresponding to a band index and $\mathbf{k}$ a common eigenvalue of the translation operators. In this representation $\mathbf{K}\left\vert n,%
\mathbf{k}\right\rangle =\mathbf{k}\left\vert n,\mathbf{k}\right\rangle $ and the position operator is $\mathbf{R=}i\partial /\partial \mathbf{K}$ which implies the canonical commutation relation $\left[ \mathbf{R}_{i}%
\mathbf{,K}_{j}\right] =i\delta _{ij}$. Note that here $\mathbf{R}$ and $%
\mathbf{K}$, rather than $\mathbf{R}$ and $\mathbf{P}$ will play the role of canonical variables.
The diagonalization of the Hamiltonian in Eq.$\left( \ref{Hmagnetic}\right) $ is first derived for $\delta \mathbf{A}=0$ by diagonalizing simultaneously $%
H_{0}$ and the magnetic translation operators $\mathbf{T}$. Start with an arbitrary basis of eigenvectors of $\mathbf{T}$. As explained in [PIERRESEMIDIAG]{}, in this basis $H_{0}$ can be seen as a square matrix with operators entries and is diagonalized through a unitary matrix $U(\mathbf{K}%
) $ which depends only on $\mathbf{K}$ (since $U$ should leave $\mathbf{K}$ invariant, i.e., $U\mathbf{K}U^{+}=\mathbf{K}$), such that in-bands energy matrix is $\varepsilon =UHU^{+}=\varepsilon _{0}(\mathbf{K})+V(U\mathbf{R}%
U^{+})$, where $\varepsilon _{0}(\mathbf{K})$ is the unperturbated ($\delta
\mathbf{A=0}$) diagonal energy matrix made of the magnetic bands elements $%
\varepsilon _{0,n}(\mathbf{K})$ with $n$ the band index.
Now, to add a perturbation $\delta A(\mathbf{R)}$ that breaks the translational symmetry, we have to replace $\mathbf{K}$ in all expressions by $$\mathbf{\Pi }=\mathbf{K}-\frac{e}{\hbar }\delta A(\mathbf{R)}$$ the (band) electron momentum and as the flux $\mathbf{\delta B}$ on a plaquette is not a rational multiple of the flux quantum, we cannot diagonalize simultaneously its components $\Pi _{i}$ since they do not commute anymore. Actually $$\hbar \lbrack \Pi ^{i},\Pi ^{j}]=ie\varepsilon ^{ijk}\delta B_{k}(\mathbf{R})$$ To deal with this non-commutativity, we adapt our method to diagonalize the Hamiltonian perturbatively in $\hbar $. To start, an outline of the first order diagonalization for the special case of an electron in an crystal lattice perturbated by the presence of an external electromagnetic field considered in refs. [@PIERREEUROPHYS] and [@PIERRESEMIDIAG] is now given.
Semiclassical diagonalization: Generalized Peierls substitution
---------------------------------------------------------------
Following section II, the diagonalization at the lowest order is just obtained by replacing $U(\mathbf{K})$ by $U\left( \mathbf{\Pi }\right) .$ This last matrix would actually diagonalize the Hamiltonian if $\mathbf{R}$ in $\delta A(\mathbf{R)}$ was a parameter commuting with $\mathbf{K}$). Note that a subtlety arises here (and that we will find again later) in the application in our method. We do not consider, at this level that $\mathbf{R}
$ and $\mathbf{K}$ commute, but only that the $\mathbf{R}$ in $\delta A(%
\mathbf{R)}$ and $V\left( \mathbf{R}\right) $ commutes with $\mathbf{K}$. In other words we assume that $\mathbf{R}$ has been replaced by a parameter in these potentials. The reason of this difference with our general set up comes from the fact that the initial diagonalization is not performed for a function of, say, $\mathbf{P}$ alone, but both of $\mathbf{P}$ and $\mathbf{R%
}$ through the periodic potential. However this difference does not alter our method which allows to recover the contributions of the electromagnetic potential as a series of $\hbar $.
As consequence of our procedure, the non projected Berry connections are $%
\mathcal{A}_{0}^{R_{i}}=iU\nabla _{K_{i}}U^{+}$ and $\mathcal{A}%
_{0}^{K_{l}}=e\nabla _{R_{l}}\delta A_{k}\mathbf{(R)}\mathcal{A}_{0}^{R_{k}}$. However, it turns out be more relevant to replace $\mathbf{K}$ by the covariant momentum $\mathbf{\Pi }$ in the physical expressions so that instead of $\mathcal{A}_{0}^{K_{l}}$ we better consider the quantity $%
\mathcal{A}_{0}^{\Pi _{l}}=e\mathcal{A}_{0}^{R_{k}}B^{m}\varepsilon _{kml}.$ Remark that the Berry connections just defined are non diagonal and are matrices whose index correspond to interband transitions.
The physical dynamical variables for the $n$-th band dynamics to the first order in $\hbar $ imply a projection on the $n$-th band. For the intraband coordinate operator $\mathbf{\mathbf{r\equiv r}}_{n}=\mathcal{P}_{n}(U(%
\mathbf{\Pi })\mathbf{R}U^{+}(\mathbf{\Pi }))$ we obtain $$\mathbf{\mathbf{r}}\simeq \mathbf{R}+\emph{A}_{0}^{\mathbf{R}}(\pi )+O(\hbar
) \label{rn}$$ and for covariant intraband momentum we obtain in the same manner $\mathbf{%
\pi \equiv \pi }_{n}=\mathbf{\Pi }+\mathcal{A}_{0}^{\mathbf{\Pi }}$ which writes also $$\mathbf{\pi }\simeq \mathbf{\Pi }+e\emph{A}_{0}^{\mathbf{R}}(\pi )\times
\delta \mathbf{B}(\mathbf{r})/\hbar +O(\hbar ) \label{pn}$$ with $\emph{A}_{0}^{\mathbf{R}}=\hbar \mathcal{P}_{n}\left( \mathcal{A}_{0}^{%
\mathbf{R}}\right) $ and $\mathcal{P}_{n}$ the projection on the $n$-th band. Remember that previously $\mathcal{P}_{+}$ was the projection on the diagonal elements of a matrix. Using now the general expression for the semiclassical Hamiltonian Eq. $\left( \ref{SE}\right) $, we obtain the desired semiclassical $n$th-band Hamiltonian (dropping the index $n$) $%
\varepsilon \left( \mathbf{\pi ,r}\right) $ as : $$\varepsilon \left( \mathbf{\pi ,r}\right) =\varepsilon _{0}\left( \mathbf{%
\pi }\right) +eV\left( \mathbf{\mathbf{r}}\right) -\mathcal{M}(\mathbf{\pi }%
).\delta \mathbf{B}(\mathbf{\mathbf{r})} \label{E1bloch}$$ where $\varepsilon _{0}\left( \mathbf{\pi }\right) $ is the unperturbated $n$th-magnetic band energy in which $\mathbf{K}$ has been replaced by $\mathbf{%
\pi },$ a procedure that we can adequately call the *generalized* Peierls substitution and which was introduced for the first time in [PIERREEUROPHYS]{}. The second term in Eq. $\left( \ref{E1bloch}\right) $ constitutes the electrostatic potential, and the third is the coupling between $\delta \mathbf{B}$ and the magnetic moment (or magnetization) which is given: $$\mathcal{M}(\mathbf{\pi })=\mathcal{P}_{n}(\frac{-ie}{2\hbar }\left[
\varepsilon _{0}(\mathbf{\pi }),\hbar \mathcal{A}_{0}^{\mathbf{R}}(\mathbf{%
\pi })\right] \times \hbar \mathcal{A}_{0}^{\mathbf{R}}(\mathbf{\pi }))$$ It can explicitly check that this expression of the magnetization is the same than previous expression found with different approaches [@NIU][Landau]{}.
### Dynamical operators algebra and equations of motion
From the dynamical operators a new algebra has to be considered . Indeed we have $$\begin{aligned}
\left[ r^{i},r^{j}\right] &=&i\Theta (\mathbf{\pi })^{ij} \notag \\
\hbar \left[ \pi ^{i},\pi ^{j}\right] &=&ie\varepsilon ^{ijk}\delta B_{k}(%
\mathbf{r})+ie^{2}\varepsilon ^{ipk}\delta B_{k}\varepsilon ^{jql}\delta
B_{l}\Theta ^{pq}/\hbar \notag \\
\left[ r^{i},\hbar \pi ^{j}\right] &=&i\hbar \delta ^{ij}-ie\varepsilon
^{jlk}\delta B_{k}(\mathbf{r})\Theta ^{il}(\mathbf{\pi }) \label{CC}\end{aligned}$$ with $\Theta ^{ij}(\mathbf{\pi })=\partial ^{i}\emph{A}^{j}(\mathbf{\pi }%
)-\partial ^{j}\emph{A}^{i}(\mathbf{\pi })$ the Berry curvature. Whereas the term of order $\delta B^{2}$ in $\left[ \pi ^{i},\pi ^{j}\right] $ is usually negligible, it turns out that all terms in $\left[ r^{i},\hbar \pi
^{j}\right] $ are essential for the correct computation of the semiclassical equations of motion which are $$\begin{aligned}
\overset{\cdot }{\mathbf{r}} &=&\partial \varepsilon /\hbar \partial \mathbf{%
\pi }-\dot{\mathbf{\pi }}\times \Theta (\mathbf{\pi }) \notag \\
\hbar \dot{\mathbf{\pi }} &=&e\mathbf{E}+e\dot{\mathbf{r}}\times \delta
\mathbf{B}(\mathbf{r})-\mathcal{M}.\partial \delta \mathbf{B}/\partial
\mathbf{r} \label{EQM}\end{aligned}$$ where we have defined the vector $\Theta ^{i}=\varepsilon ^{ijk}\Theta
_{jk}/2$. As shown in the following, the generalization Peierls substitution, not only is essential for the correct determination of the equations of motion, but also for the Bohr-Sommerfeld (BS) quantization condition.
### Bohr-Sommerfeld quantization
To underline the relevance of the generalized Peierls substitution at the level of the semiclassical quantization of energy levels for an electron motion in an external uniform magnetic, we adapt the arguments of ref. [PIERREEBS]{}. For $V=0$ the equations of motion Eq. $\left( \ref{EQM}\right) $ become $$\mathbf{\dot{r}}=D\left( \frac{\partial \varepsilon }{\hbar \partial \mathbf{%
\pi }}\right) \ \ \ \ \text{and \ }\ \hbar \overset{\cdot }{\mathbf{\pi }}%
=eD\left( \frac{\partial \varepsilon }{\hbar \partial \mathbf{\pi }}\mathbf{%
\times B}\right) \label{EQBS}$$ with $D^{-1}=1+\frac{e}{\hbar }\mathbf{B\Theta .}$ For convenience $\delta
\mathbf{B\equiv B}$ is chosen in the $z$-direction $\mathbf{B=}B\mathbf{k,}$ the energy reads $\varepsilon =\varepsilon _{0}\left( \pi \right)
-M_{z}\left( \pi \right) B.$ Consequently the orbits satisfies the conditions $\varepsilon _{0}=$const and $\pi _{z}=$const. The semiclassical quantization of energy levels can be done according to the Bohr-Sommerfeld quantization rule $$\oint K_{\perp }dR_{\perp }=2\mathcal{\pi }\left( n+1/2\right) \label{BSS}$$ where $K_{\perp }$ is the canonical pseudo-momentum in the plane perpendicular to the axis $\pi _{z}=cte.$ The integration is taken over a period of the motion and $n$ is a large integer. Now, it turns out to be convenient to choose the gauge $\widetilde{A}_{y}=BX,$ $\widetilde{A}_{x}=%
\widetilde{A}_{z}=0.$ In this gauge, one has $\pi _{z}=P_{z}=cte$, and the usual covariant momentum $\Pi _{y}=K_{y}-\frac{eB}{\hbar }X.$ In this case the BS condition reads $\oint \Pi _{x}d\Pi _{y}=\frac{-2\mathcal{\pi }eB}{%
\hbar }\left( n+1/2\right) .$ Assuming that the physically relevant variables are instead the covariant ones, writing thus $BX=B(x-\emph{A}_{x})$ the generalized covariant momentum defined as $\pi _{y}=\Pi _{y}-\frac{eB}{%
\hbar }\emph{A}_{x}$ becomes $$\pi _{y}=K_{y}-\frac{eB}{\hbar }x$$ which is formally the same relation as the one between the canonical variables, but now relating the new covariant generalized dynamical operators. This relation with the help of the equations of motion Eq. $%
\left( \ref{EQBS}\right) $ gives $\overset{\cdot }{K}_{y}=\overset{\cdot }{%
\pi }_{y}+\frac{eB}{\hbar }\overset{\cdot }{x}=0$ thus $K_{y}$ is a constant of motion so that $\oint K_{Y}dY=K_{Y}\oint dY=0$ and Eq.$\left( \ref{BSS}%
\right) $ becomes simply $\oint K_{x}dX=2\mathcal{\pi }\left( n+1/2\right) .$ Now using the definition of the generalized momentum $K_{x}=\pi _{x}-\frac{e%
\emph{A}_{y}}{\hbar }B$ and the differential of the canonical position $%
dX=dx-d\emph{A}_{x}=\frac{\hbar d\pi _{y}}{-eB}-d\emph{A}_{x},$ the Bohr-Sommerfeld condition Eq. $\left( \ref{BSS}\right) $ becomes $$\oint \pi _{x}d\pi _{y}=-2\mathcal{\pi }eB\left( n+\frac{1}{2}-\frac{1}{2%
\mathcal{\pi }}\oint \emph{A}_{\bot }d\mathbf{\pi }_{\perp }\right)
\label{pipiA}$$ where the integral is now taken along a closed trajectory $\Gamma $ in the $%
\pi $ space and $\frac{1}{2\mathcal{\pi }}\oint \emph{A}_{\bot }d\mathbf{\pi
}_{\perp }=\phi _{B}$ is the Berry phase for the orbit $\Gamma $.Integration in Eq. $\left( \ref{pipiA}\right) $ defines the cross-sectional area $\oint \pi _{x}d\pi _{y}=S_{0}(\varepsilon _{0},K_{z})$ of the orbit $%
\Gamma $ which is the intersection of the constant energy surface $%
\varepsilon _{0}\left( \pi \right) =$const and the plane $\pi _{z}=K_{z}=$const. As shown in [@PIERREEBS] $S_{0}(\varepsilon =\varepsilon
_{0}-M_{z}B,\pi _{z})$ can then be written as $$S_{0}(\mathcal{E},\pi _{z})=\frac{2\pi \left\vert e\right\vert B}{\hbar }%
\left( n+\frac{1}{2}-\phi _{B}-\frac{1}{2\mathcal{\pi }\left\vert
e\right\vert }\oint \frac{M_{z}\left( \mathbf{\pi }\right) d\kappa }{%
\left\vert \partial \varepsilon /\hbar \partial \mathbf{\pi }_{\perp
}\right\vert }\right) \label{SO}$$ here $d\kappa =\sqrt{d\pi _{x}^{2}+d\pi _{y}^{2}}$ is an elementary length of the $\pi $ orbit. We have thus succeeded to deduce the required result Eq. $\left( \ref{SO}\right) $ (first found by Roth [@ROTH] in a different way and without mentioning the Berry phase), as resulting from the generalized Peierls substitution in the BS condition. The importance in this expression of the Berry’s phase for electrons in metal in connection to band degeneracy was later discussed by Mikitik and Sharlay [@MIKITIK]. Eq. $\left( \ref{SO}\right) $ implicitly determines the energy levels $%
\varepsilon _{n}\left( K_{z}\right) $. For instance for the case where the Fermi surface is an ellipsoid of revolution characterized by two effective masses, a transverse $m_{\perp }$ and a longitudinal $m_{l}$ one has $$\varepsilon _{0}=\hbar ^{2}\left( \frac{\mathbf{\pi }_{\perp }^{2}}{%
2m_{\perp }}+\frac{K_{z}^{2}}{2m_{l}}\right)$$ and $S(\varepsilon ,K_{z})$ is a disc of radius square $\mathbf{\pi }_{\perp
}^{2}$. Therefore for $M_{z}=$const we have the following generalized relation for the Landau levels: $$\varepsilon _{n}=\frac{\left\vert e\right\vert B\hbar }{m_{\perp }}\left( n+%
\frac{1}{2}-\phi _{B}-\frac{m_{\perp }}{\left\vert e\right\vert }%
M_{z}\right) +\frac{\hbar ^{2}K_{z}^{2}}{2m_{l}}$$ which shows that both the magnetic moment and Berry’s phase can influence the energy levels. A nice illustration of this result is provided by electrons in graphene with broken inversion symmetry [@PIERREGRAPHENE].
Second order diagonalization
----------------------------
As explained in section 2, the Hamiltonian diagonalization at this order requires the computation of the Berry connections and the covariant dynamical variables. This will thus be our first task. Note that, from now on, for the sake on simplicity, we will keep the notation introduced in the previous section, $\delta \mathbf{B\equiv B}$.
### Computation of the Berry phases and dynamical variables to the second order
Starting from the general expressions for the non projected Berry phases at the second order Eqs. \[Conn2\]\[conn2A\]\[conn2B\] we have in the present case for $\mathcal{A}^{\mathbf{X}}=\int_{0}^{\hbar }\mathcal{A}%
_{\alpha }^{\mathbf{X}}d\alpha $ : $$\begin{aligned}
\mathcal{A}^{\mathbf{X}} &=&\hbar \mathcal{A}_{0}^{\mathbf{X}}\left( \mathbf{%
R+}\frac{\hbar }{4}\mathcal{A}_{0}^{R_{l}},\mathbf{K+}\frac{\hbar }{4}%
\mathcal{A}_{0}^{P_{l}}\right) \\
&&-\frac{\hbar ^{2}}{2}\left\{ \left[ .,\varepsilon _{0}\right] ^{-1}.\left(
\mathcal{P}_{-}\left\{ \frac{1}{2}\mathcal{A}_{0}^{P_{l}}\nabla
_{K_{l}}\varepsilon _{0}+\frac{1}{2}\mathcal{A}_{0}^{R_{l}}\nabla
_{R_{l}}\varepsilon _{0}+H.C.\right\} \right) \right. \\
&&\left. -\frac{i}{4}\left\{ \mathcal{P}_{-}\mathcal{A}_{0}^{R_{k}}\mathcal{P%
}_{+}\mathcal{A}_{0}^{R_{l}}+H.C.\right\} \varepsilon _{klm}eB_{m},\mathbf{X}%
+\hbar \mathcal{A}_{0}^{\mathbf{X}}\right\}\end{aligned}$$ where now $\mathbf{X=}\left( \mathbf{R,K}\right) $ and the same for $\mathbf{%
Y}$. We aim now at writing the connection $\mathcal{A}^{\mathbf{X}}$ in a more convenient form as an expansion in terms of the zero-order (actually first order in $\hbar $) $\mathcal{A}_{0}^{\mathbf{X}}.$ To do so, we first start to express the crystal momentum Berry phase $\mathcal{A}^{\mathbf{K}}$ as a function of the position Berry phase $\mathcal{A}^{\mathbf{R}}$. The computation of $\mathcal{A}^{\mathbf{K}}$ involves a commutator with $%
\mathbf{K}+\mathcal{A}_{0}^{\mathbf{K}}$ that we compute first. Notice that the first order energy operator $\varepsilon _{0}$ in Eq. $\left( \ref%
{E1bloch}\right) $ and the Berry connections at the zeroth order depend on $%
\mathbf{\Pi }=\mathbf{K-}\frac{e}{\hbar }\mathbf{A}^{\text{ }}$ as well as on $\mathcal{A}_{0}^{K_{m}}=e\nabla _{R_{m}}A_{l}\mathcal{A}_{0}^{R_{l}}$, which allows us to write the commutator of $K_{m}+\hbar \mathcal{A}%
_{0}^{K_{m}}$ with any such operator , e.g. $\left[ K_{m}+\hbar \mathcal{A}%
_{0}^{K_{m}},\cdot \right] $ in the following way : $$\left[ K_{m}+\hbar \mathcal{A}_{0}^{K_{m}},\cdot \right] =e\nabla
_{R_{m}}A_{l}\left[ R_{l}+\hbar \mathcal{A}_{0}^{R_{l}},\cdot \right]$$ Let us stress that this formula is not valid for the electric potential term appearing in the Berry phase formula since it depends exclusively on $%
\mathbf{R}$. The action of $\left[ K_{m}+\hbar \mathcal{A}_{0}^{K_{m}},%
\right] $ on this term has thus to be computed independently. The additional contribution due to the electric field and the magnetic field to the momentum Berry phase is seen easily from the formula for the to be $$-i\frac{\hbar ^{2}}{2}\left[ \left[ .,\hat{\varepsilon}_{0}\right]
^{-1}.\left( \mathcal{A}_{0}^{R_{l}}\nabla _{R_{m}}\nabla _{R_{l}}V\left(
\mathbf{R}\right) \right) -\frac{ie}{2}\left\{ \mathcal{P}_{-}\mathcal{A}%
_{0}^{\mathbf{R}}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right\}
.\nabla _{R_{m}}\mathbf{B}\right]$$
Now the first term and the last term in the crystal momentum Berry phase can also be replaced as a function of the position Berry phase contribution. But at this point we need to go back to the technicality mentioned in the first section. While the final results do not depend on the way variables are symmetrized, one has to choose an initial symmetrization for the diagonalized energy at the zeroth order as well as for the Berry phase. For the diagonalized energy, we will choose to write it as a series of symmetric monomials in the momentum $\mathbf{\Pi }$. Concerning the Berry phase for the crystal momentum the natural choice is to symmetrize initially the variables such that at the lowest order in $\hbar $ one has $\mathcal{A}%
_{0}^{\mathbf{K}}=\frac{1}{2}e\nabla _{R_{m}}A_{l}\left( \mathbf{R}\right)
\mathcal{A}_{0}^{\mathbf{R}}+\frac{1}{2}e\mathcal{A}_{0}^{\mathbf{R}}\nabla
_{R_{m}}A_{l}\left( \mathbf{R}\right) $. This will introduce some technical problem later, but it is the most simple choice for us. As a consequence, we obtain readily for the terms of interest : $$\begin{aligned}
&&\hbar \mathcal{A}_{0}^{\mathbf{K}}\left( \mathbf{R+}\frac{\hbar }{4}%
\mathcal{A}_{0}^{R_{l}}\right) +\frac{\hbar ^{2}}{4}\left( \mathcal{P}_{+}%
\left[ \mathcal{A}_{0}^{\mathbf{Y}}\right] .\mathbf{\nabla }_{\mathbf{Y}}%
\mathcal{P}_{+}\left[ \mathcal{A}_{0}^{K_{m}}\right] +H.C.\right) \\
&=&\frac{1}{2}\hbar e\nabla _{R_{m}}A_{l}\left( \mathbf{R+}\frac{1}{4}%
\mathcal{A}_{0}^{\mathbf{R}}\right) \mathcal{A}_{0}^{\mathbf{R}}\left(
\mathbf{R+}\frac{\hbar }{4}\mathcal{A}_{0}^{R_{l}},\mathbf{K+}\frac{\hbar }{4%
}\mathcal{A}_{0}^{P_{l}}\right) \\
&&+\frac{1}{2}\hbar \mathcal{A}_{0}^{\mathbf{R}}\left( \mathbf{R+}\frac{%
\hbar }{4}\mathcal{A}_{0}^{R_{l}}\right) e\nabla _{R_{m}}A_{l}\left( \mathbf{%
R+}\frac{1}{4}\mathcal{A}_{0}^{\mathbf{R}},\mathbf{P+}\frac{1}{4}\mathcal{A}%
_{0}^{\mathbf{P}}\right) \\
&&+\frac{\hbar ^{2}}{4}e\nabla _{R_{m}}A_{l}\left( \mathcal{P}_{+}\left[
\mathcal{A}_{0}^{\mathbf{R}}\right] .\mathbf{\nabla }_{\mathbf{Y}}\mathcal{P}%
_{+}\left[ \mathcal{A}_{0}^{R_{m}}\right] +H.C.\right)\end{aligned}$$ so that can now gather all these terms and compute $\mathcal{A}^{K_{m}}$ as a function of $\mathcal{A}^{R_{l}}$ : $$\begin{aligned}
\mathcal{A}^{K_{m}} &=&\frac{1}{2}\left( e\nabla _{R_{m}}A_{l}\left( \mathbf{%
R+}\frac{1}{4}\mathcal{A}_{0}^{\mathbf{R}},\mathbf{P+}\frac{1}{4}\mathcal{A}%
_{0}^{\mathbf{P}}\right) \mathcal{A}^{R_{l}}+\mathcal{A}^{R_{l}}e\nabla
_{R_{m}}A_{l}\left( \mathbf{R+}\frac{1}{4}\mathcal{A}_{0}^{\mathbf{R}},%
\mathbf{P+}\frac{1}{4}\mathcal{A}_{0}^{\mathbf{P}}\right) \right) \\
&&-i\frac{\hbar ^{2}}{2}\left[ \left[ .,\hat{\varepsilon}_{0}\right]
^{-1}.\left( \mathcal{A}_{0}^{R_{l}}\nabla _{R_{m}}\nabla _{R_{l}}V\left(
\mathbf{R}\right) \right) -\frac{ie}{2}\left\{ \mathcal{P}_{-}\mathcal{A}%
_{0}^{\mathbf{R}}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right\}
.\nabla _{R_{m}}\mathbf{B}\right]\end{aligned}$$ The potential $A_{l}$ being developer to the first order in $\hbar $. As expected, only the contribution due to the electric term is not rewritten as a function of $\mathcal{A}^{R_{l}}$. The quantity $-i\frac{\hbar ^{2}}{2}%
\left[ \left[ .,\hat{\varepsilon}_{0}\right] ^{-1}.\left( \mathcal{A}%
_{0}^{R_{l}}\nabla _{R_{m}}\nabla _{R_{l}}V\left( \mathbf{R}\right) \right) -%
\frac{ie}{2}\left\{ \mathcal{P}_{-}\mathcal{A}_{0}^{\mathbf{R}}\times
\mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right\} .\nabla _{R_{m}}\mathbf{B%
}\right] $ can be computed on the bands as being equal to : $$\begin{aligned}
&&-i\frac{\hbar ^{2}}{2}\left[ \left[ .,\hat{\varepsilon}_{0}\right]
^{-1}.\left( \mathcal{A}_{0}^{R_{l}}\nabla _{R_{m}}\nabla _{R_{l}}V\left(
\mathbf{R}\right) \right) -\frac{ie}{2}\left\{ \mathcal{P}_{-}\mathcal{A}%
_{0}^{\mathbf{R}}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right\}
.\nabla _{R_{m}}\mathbf{B}\right] _{MN} \\
&=&i\frac{\hbar ^{2}}{2}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right)
_{MN}\nabla _{R_{m}}\nabla _{R_{l}}V\left( R\right) }{\hat{\varepsilon}_{0M}-%
\hat{\varepsilon}_{0N}}-\frac{\hbar ^{2}}{4}\left\{ \mathcal{P}_{-}\mathcal{A%
}_{0}^{\mathbf{R}}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right\}
_{MN}.\nabla _{R_{m}}\mathbf{B}\text{ if }M\neq N \\
&=&0\text{ if }M=N\end{aligned}$$ so that we have for the $(M,N)$ component : $$\begin{aligned}
\mathcal{A}_{MN}^{K_{m}} &=&\frac{1}{2}\left[ \left( e\nabla
_{R_{m}}A_{l}\left( \mathbf{R+}\frac{\hbar }{4}\mathcal{A}_{0}^{\mathbf{R}%
}\right) \mathcal{A}^{R_{l}}+\mathcal{A}^{R_{l}}e\nabla _{R_{m}}A_{l}\left(
\mathbf{R+}\frac{\hbar }{4}\mathcal{A}_{0}^{\mathbf{R}}\right) \right) %
\right] _{MN} \\
&&+\left[ i\frac{\hbar ^{2}}{2}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right)
_{MN}\nabla _{R_{m}}\nabla _{R_{l}}V\left( R\right) }{\hat{\varepsilon}_{0M}-%
\hat{\varepsilon}_{0N}}-\frac{\hbar ^{2}}{4}\left\{ \mathcal{P}_{-}\mathcal{A%
}_{0}^{\mathbf{R}}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right\}
_{MN}.\nabla _{R_{m}}\mathbf{B}\right] \hat{\delta}_{MN} \\
&=&\frac{1}{2}\left[ \left( e\nabla _{R_{m}}A_{l}\left( \mathbf{R}\right)
\mathcal{A}^{R_{l}}+\mathcal{A}^{R_{l}}e\nabla _{R_{m}}A_{l}\left( \mathbf{R}%
\right) \right) \right] _{MN} \\
&&\mathbf{+}\frac{\hbar ^{2}}{8}e\nabla _{R_{k}}\nabla _{R_{m}}A_{l}\left(
\mathbf{R},\mathbf{P}\right) \left( \mathcal{A}_{0}^{R_{l}}\mathcal{A}%
_{0}^{R_{k}}+\mathcal{A}_{0}^{R_{k}}\mathcal{A}_{0}^{R_{l}}\right) \\
&&+\left[ i\frac{\hbar ^{2}}{2}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right)
_{MN}\nabla _{R_{m}}\nabla _{R_{l}}V\left( R\right) }{\hat{\varepsilon}_{0M}-%
\hat{\varepsilon}_{0N}}-\frac{\hbar ^{2}}{4}\left\{ \mathcal{P}_{-}\mathcal{A%
}_{0}^{\mathbf{R}}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right\}
_{MN}.\nabla _{R_{m}}\mathbf{B}\right] \hat{\delta}_{MN}\end{aligned}$$ where we defined the notation $\hat{\delta}_{MN}=1-\delta _{MN}$,
As explained before, the relevant variables for the physical expressions are rather the momentum variables $\mathbf{\Pi =K-A}$. Since at first order we already had $\mathcal{A}_{0}^{\mathbf{\pi }}=e\mathcal{A}_{0}^{\mathbf{R}%
}\times \mathbf{B}$ we would expect a similar formula at the second order. However, since our method has been designed to deal with quantities symmetrized as functions of $\mathbf{R}$ and $\mathbf{K}$, it appears that the momentum variable does not appear directly in our Hamiltonian. We thus postpone its introduction till the computation of the diagonalized Hamiltonian.
We can now, as a second step, concentrate on the position Berry phase. We rewrite it as : $$\begin{aligned}
\mathcal{A}^{\mathbf{R}} &=&\hbar \mathcal{A}_{0}^{\mathbf{R}}+\frac{\hbar
^{2}}{8}\left\{ \mathcal{A}_{0}^{R_{l}}\nabla _{R_{l}}\mathcal{A}_{0}^{%
\mathbf{R}}+\mathcal{A}_{0}^{K_{l}}\nabla _{K_{l}}\mathcal{A}_{0}^{\mathbf{R}%
}+H.C.\right\} \\
&&+\left[ R+\hbar \mathcal{A}_{0}^{\mathbf{R}},\frac{\hbar ^{2}}{2}\left[ .,%
\hat{\varepsilon}_{0}\right] ^{-1}.\left[ \left( \mathcal{P}_{-}\left\{
\frac{1}{2}\mathcal{A}_{0}^{R_{l}}\nabla _{R_{l}}\varepsilon _{0}\left(
\mathbf{X}\right) +\frac{1}{2}\mathcal{A}_{0}^{K_{l}}\nabla
_{K_{l}}\varepsilon _{0}\left( \mathbf{X}\right) +H.C.\right\} \right) %
\right] \right] \\
&&+\left[ R+\hbar \mathcal{A}_{0}^{\mathbf{R}},\frac{e\hbar ^{2}}{4}\left(
\mathcal{P}_{-}\mathcal{A}_{0}^{R_{n}}\mathcal{P}_{+}\mathcal{A}%
_{0}^{R_{r}}+H.C.\right) \varepsilon ^{nru}.B_{u}\right]\end{aligned}$$ We can also rewrite the third term $\left( M,N\right) $ component as : $$\begin{aligned}
&&\left( \left[ .,\hat{\varepsilon}_{0}\right] ^{-1}.\left[ \left( \mathcal{P%
}_{-}\left\{ \frac{1}{2}\mathcal{A}_{0}^{R_{l}}\nabla _{R_{l}}\varepsilon
_{0}\left( \mathbf{X}\right) +\frac{1}{2}\mathcal{A}_{0}^{K_{l}}\nabla
_{K_{l}}\varepsilon _{0}\left( \mathbf{X}\right) +H.C.\right\} \right) %
\right] \right) _{MN} \\
&=&\frac{1}{2}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right) _{MN}\nabla
_{R_{l}}\left( \varepsilon _{0M}+\varepsilon _{0N}\right) +\left( \mathcal{A}%
_{0}^{K_{l}}\right) _{MN}\nabla _{K_{l}}\left( \varepsilon _{0M}+\varepsilon
_{0N}\right) }{\hat{\varepsilon}_{0M}-\hat{\varepsilon}_{0N}}\text{ if }%
M\neq N \\
&=&0\text{ if }M=N\end{aligned}$$ Using also, for quantities depending on $\mathbf{\Pi =K-A}$ and $\mathbf{R}$ that $\left( \mathcal{A}_{0}^{R_{l}}\right) _{MP}\nabla _{R_{l}}+\left(
\mathcal{A}_{0}^{K_{l}}\right) _{MP}\nabla _{K_{l}}=-e\mathbf{B}.\left(
\left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{MP}\times \nabla _{\mathbf{\Pi }%
}\right) $ at the lowest order in $\hbar $ we now give an expanded formula for the position Berry phase at the second order : $$\begin{aligned}
\left( \mathcal{A}^{\mathbf{R}}\right) _{MN} &=&\left( \hbar \mathcal{A}%
_{0}^{\mathbf{R}}\right) _{MN}-\frac{\hbar ^{2}}{16}\sum_{P}\left\{ e\mathbf{%
B}.\left( \left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{MP}\times \nabla _{%
\mathbf{\Pi }}\right) \left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{PN}+M\leftrightarrow N\right\} \\
&&+\frac{\hbar ^{2}}{2}i\mathbf{\nabla }_{\mathbf{\Pi }}\frac{\left(
\mathcal{A}_{0}^{R_{l}}\right) _{MN}\nabla _{R_{l}}V\left( \mathbf{R}\right)
+e\mathbf{B}.\left( \left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{MN}\times
\nabla _{\mathbf{\Pi }}\right) \left( \hat{\varepsilon}_{0M}+\hat{\varepsilon%
}_{0N}\right) }{\hat{\varepsilon}_{0M}-\hat{\varepsilon}_{0N}}\hat{\delta}%
_{MN} \\
&&+\frac{\hbar ^{2}}{2}\sum_{P}\left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{MP}\hat{\delta}_{PM}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right)
_{PN}\nabla _{R_{l}}V\left( \mathbf{R}\right) +e\mathbf{B}.\left( \left(
\mathcal{A}_{0}^{\mathbf{R}}\right) _{PN}\times \nabla _{\mathbf{\Pi }%
}\right) \left( \hat{\varepsilon}_{0P}+\hat{\varepsilon}_{0N}\right) }{\hat{%
\varepsilon}_{0P}-\hat{\varepsilon}_{0N}} \\
&&-\frac{\hbar ^{2}}{2}\sum_{P}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right)
_{MP}\nabla _{R_{l}}V\left( \mathbf{R}\right) +e\mathbf{B}.\left( \left(
\mathcal{A}_{0}^{\mathbf{R}}\right) _{MP}\times \nabla _{\mathbf{\Pi }%
}\right) \left( \hat{\varepsilon}_{0M}+\hat{\varepsilon}_{0P}\right) }{\hat{%
\varepsilon}_{0M}-\hat{\varepsilon}_{0P}}\hat{\delta}_{MP}\left( \mathcal{A}%
_{0}^{\mathbf{R}}\right) _{PN} \\
&&-\frac{e\hbar ^{2}}{4}\mathbf{\nabla }_{\mathbf{\Pi }}\left( \left(
\mathcal{A}_{0}^{\mathbf{R}}\right) _{MN}\times \left( \mathcal{A}_{0}^{%
\mathbf{R}}\right) _{NN}\right) .\mathbf{B}\hat{\delta}_{MN} \\
&&+i\frac{e\hbar ^{2}}{8}\left( \sum_{P}\left( \mathcal{A}_{0}^{\mathbf{R}%
}\right) _{MP}\left( \left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{PN}\times
\left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{NN}\right) .\mathbf{B}\hat{%
\delta}_{PN}+H.C.\right) \\
&&-i\frac{e\hbar ^{2}}{8}\left( \sum_{P}\left( \left( \mathcal{A}_{0}^{%
\mathbf{R}}\right) _{MP}\times \left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{PP}\right) .\mathbf{B}\hat{\delta}_{MP}\left( \mathcal{A}_{0}^{\mathbf{R}%
}\right) _{PN}+H.C.\right)\end{aligned}$$ and its projection on Band $N$. $$\begin{aligned}
\mathcal{P}_{N}\left( \mathcal{A}_{NN}^{\mathbf{R}}\right) &=&\left(
\mathcal{A}^{\mathbf{R}}\right) _{NN}=\left( \hbar \mathcal{A}_{0}^{\mathbf{R%
}}\right) _{NN}-\frac{\hbar ^{2}}{8}\sum_{P}\left\{ e\mathbf{B}.\left(
\left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{NP}\times \nabla _{\mathbf{\Pi }%
}\right) \left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{PN}\right\} \\
&&+\frac{\hbar ^{2}}{2}\sum_{P}\left( \left( \mathcal{A}_{0}^{\mathbf{R}%
}\right) _{NP}\left( \mathcal{A}_{0}^{R_{l}}\right) _{PN}+\left( \mathcal{A}%
_{0}^{R_{l}}\right) _{NP}\left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{PN}\right) \frac{\nabla _{R_{l}}V\left( \mathbf{R}\right) }{\hat{%
\varepsilon}_{0P}-\hat{\varepsilon}_{0N}}\hat{\delta}_{PN} \\
&&-\frac{\hbar ^{2}}{2}\left( \sum_{P}\left( e\mathbf{B}.\left( \left(
\mathcal{A}_{0}^{\mathbf{R}}\right) _{NP}\times \nabla _{\mathbf{\Pi }%
}\right) \right) \frac{\left( \hat{\varepsilon}_{0P}+\hat{\varepsilon}%
_{0N}\right) }{\hat{\varepsilon}_{0P}-\hat{\varepsilon}_{0N}}\hat{\delta}%
_{PN}\left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{PN}-H.C.\right) \\
&&+i\frac{e\hbar ^{2}}{8}\sum_{P}\left( \left( \mathcal{A}_{0}^{\mathbf{R}%
}\right) _{NP}\left( \left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{PN}\times
\left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{NN}\right) -\left( \left(
\mathcal{A}_{0}^{\mathbf{R}}\right) _{NP}\times \left( \mathcal{A}_{0}^{%
\mathbf{R}}\right) _{PP}\left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{PN}\right) .\mathbf{B}\right) \hat{\delta}_{PN} \\
&&+H.C.\end{aligned}$$ For later purpose we will need also the modified Berry phase $\emph{A}_{N}^{%
\mathbf{R}}$ appearing in the dynamical Band variable $\mathbf{r}_{N}$. A direct application of our general set up of the previous section yields. $$\begin{aligned}
\emph{A}_{N}^{\mathbf{R}} &=&\mathcal{A}_{NN}^{\mathbf{R}}+\frac{\hbar ^{2}}{%
4}\left( \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{\mathbf{Y}}\right] .\mathbf{%
\nabla }_{\mathbf{Y}}\mathcal{P}_{N}\left[ \mathcal{A}_{0}^{\mathbf{X}}%
\right] +H.C.\right) \\
&=&\mathcal{A}_{NN}^{\mathbf{R}}+\frac{\hbar ^{2}}{4}\left( \mathcal{A}%
_{0NN}^{\mathbf{R}}.\mathbf{\nabla }_{\mathbf{R}}\mathcal{A}_{0NN}^{\mathbf{R%
}}+\mathcal{A}_{0NN}^{\mathbf{P}}.\mathbf{\nabla }_{\mathbf{P}}\mathcal{A}%
_{0NN}^{\mathbf{R}}+H.C.\right) \\
&=&\mathcal{A}_{NN}^{\mathbf{R}}-e\frac{\hbar ^{2}}{4}\left( \mathbf{B}%
.\left( \left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{NN}\times \nabla _{%
\mathbf{\Pi }}\right) \mathcal{A}_{0NN}^{\mathbf{R}}+H.C.\right)\end{aligned}$$ Ultimately, the projected dynamical variables follow directly from our general method of diagonalization. They are given by the canonical variables shifted by the modified projected Berry phases : $$\begin{aligned}
\mathbf{r}_{N} &=&\mathbf{R}+\emph{A}_{N}^{\mathbf{R}} \\
\mathbf{k}_{N} &=&\mathbf{K}+\emph{A}_{N}^{\mathbf{K}}\equiv \mathbf{K}+%
\mathcal{A}_{NN}^{\mathbf{K}} \\
\mathbf{\pi }_{N} &=&\mathbf{\Pi }+\mathcal{A}^{\mathbf{\pi }}=\mathbf{\Pi }+%
\frac{1}{2}\left[ e\mathcal{A}\mathbf{\times B}\left( \mathbf{R+}\frac{\hbar
}{4}\mathcal{A}_{0}^{\mathbf{R}},\mathbf{P+}\frac{\hbar }{4}\mathcal{A}_{0}^{%
\mathbf{P}}\right) +H.C.\right]\end{aligned}$$ Note that since $-i\nabla _{R_{m}}\frac{\hbar ^{2}}{4}\left[ .,\hat{%
\varepsilon}_{0}\right] ^{-1}.\left( \mathcal{A}_{0}^{R_{l}}\nabla
_{R_{l}}V\left( \mathbf{R}\right) +H.C.\right) $ has no diagonal element $$\begin{aligned}
\emph{A}_{N}^{K_{m}} &=&\frac{1}{2}\left[ \left( e\nabla _{R_{m}}A_{l}\left(
\mathbf{R}\right) \emph{A}_{N}^{R_{l}}+\emph{A}_{N}^{R_{l}}e\nabla
_{R_{m}}A_{l}\left( \mathbf{R}\right) \right) \right] _{NN} \\
&&\mathbf{+}\frac{\hbar ^{2}}{8}e\nabla _{R_{k}}\nabla _{R_{m}}A_{l}\left(
\mathbf{R},\mathbf{P}\right) \mathcal{P}_{N}\left( \mathcal{A}_{0}^{R_{l}}%
\mathcal{A}_{0}^{R_{k}}+\mathcal{A}_{0}^{R_{k}}\mathcal{A}_{0}^{R_{l}}\right)
\\
&&+\frac{\hbar ^{2}}{4}e\nabla _{R_{k}}\nabla _{R_{m}}A_{l}\left( \mathbf{R},%
\mathbf{P}\right) \left( \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{l}}\right]
\mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{k}}\right] +\mathcal{P}_{N}\left[
\mathcal{A}_{0}^{R_{k}}\right] \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{l}}%
\right] \right)\end{aligned}$$
### Band Hamiltonian
We now turn to the Problem of deriving the diagonalized Hamiltonian. As explained in section II, the effective diagonal Hamiltonian for the $Nth$ band can be written (see Eq. $\left( \ref{E2}\right) $ ) $$\begin{aligned}
H_{N}\left( \mathbf{X}\right) &=&\varepsilon _{0N}\left( \mathbf{x}\right) +%
\frac{i}{2}\mathcal{P}_{N}\left\{ \left[ \varepsilon _{0}\left( \mathbf{x}%
\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}^{P_{l}}-\left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{P_{l}}\right]
\mathcal{\hat{A}}^{R_{l}}\right\} \notag \\
&&-\frac{1}{8}\mathcal{P}\left\{ _{N}\left[ \mathcal{P}_{+}\left[ \left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right]
\mathcal{\hat{A}}^{P_{l}}-\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{\hat{A}}^{P_{l}}\right] \mathcal{\hat{A}}^{R_{l}}\right] ,\mathcal{%
\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}^{P_{l}}\right. \notag \\
&&\left. -\left[ \mathcal{P}_{+}\left[ \left[ \varepsilon _{0}\left( \mathbf{%
x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}^{P_{l}}-\left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{P_{l}}\right] %
\right] \mathcal{\hat{A}}^{R_{l}},\mathcal{\hat{A}}^{P_{l}}\right] \mathcal{%
\hat{A}}^{R_{l}}\right\} -\frac{\hbar }{2}\left\langle \varepsilon
_{0}\left( \mathbf{x}\right) \right\rangle\end{aligned}$$ Where $\mathcal{P}_{N}$ is the projection on this band. As explained in [DIAGOEXACT]{} this Hamiltonian can be rewritten in a more enlightening way as : $$H_{N}=\hat{\varepsilon}_{0}\left( \mathbf{\pi }\right) +M\left[ \varepsilon
_{0}\right] +\frac{1}{2}M\left[ M\left[ \varepsilon _{0}\right] \right]
+V\left( \mathbf{r}\right) -\frac{\hbar }{2}\left\langle \varepsilon
_{0}\left( \mathbf{x}\right) \right\rangle$$ Where the magnetization operator $M$ acts on the following way : $$M\left[ X\right] =\frac{i}{4\hbar }\mathcal{P}_{N}\left\{ \left[ X,\mathcal{%
\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}^{K_{l}}-\left[ X,\mathcal{\hat{A}}%
^{K_{l}}\right] \mathcal{\hat{A}}^{R_{l}}\right\} +H.C.$$ As consequence, the Hamiltonian is given by a series of magnetization terms acting on the classical Hamiltonian, that is the Hamiltonian obtained for classical commuting dynamical variables. We will more give detailed formula for the various terms as functions of the first order Berry phases, but we turn first to the problem of choice of variables in writing the energy operator.
#### Momentum variables. Second order generalized Peierls substitution
For the moment, the pair of canonical variables implied by our method are the non physical one $x=\left( \mathbf{%
r}_{N},\mathbf{k}_{N}\right) $ $$\begin{aligned}
\mathbf{r}_{N} &=&\mathbf{R}+\emph{A}_{N}^{\mathbf{R}} \\
\mathbf{k}_{N} &=&\mathbf{K}+\emph{A}_{N}^{\mathbf{K}}\equiv \mathbf{K}+%
\mathcal{A}_{NN}^{\mathbf{K}}\end{aligned}$$ One aims at replacing $\mathbf{k}_{N}$ by a physical momentum variable which reduces to $\mathbf{\Pi }%
=\mathbf{K-}\frac{e}{\hbar }\mathbf{A}$ at the lowest order. To do so assume the electric potential is set to $0$ since it plays no role here, and rewrite the quantity $\varepsilon _{0N}\left( \mathbf{x}\right) $ that arises in the diagonalized Hamiltonian as : $$\varepsilon _{0N}\left( \mathbf{x}\right) =\varepsilon _{0N}\left( \mathbf{k}%
_{N}-\mathbf{A}\left( \mathbf{r}_{N}\right) \right)$$ so that one would be tempted to choose $\mathbf{k}_{N}-\mathbf{A}\left(
\mathbf{r}_{N}\right) $ as the right momentum variable. However a computation of this quantity yields : $$\begin{aligned}
\mathbf{k}_{N}-e\mathbf{A}\left( \mathbf{r}_{N}\right) &=&\mathbf{K}+\emph{A}%
_{N}^{\mathbf{K}}-e\mathbf{A}\left( \mathbf{R}+\emph{A}_{N}^{\mathbf{R}%
}\right) \\
&=&\mathbf{K}-\mathbf{A}\left( \mathbf{R}\right) +\frac{1}{2}e\left[ \mathbf{%
\nabla }A_{l}\left( \mathbf{R}\right) \emph{A}_{N}^{R_{l}}+\emph{A}%
_{N}^{R_{l}}\mathbf{\nabla }A_{l}\left( \mathbf{R}\right) \right] \\
&&\mathbf{+}\frac{\hbar ^{2}}{8}e\nabla _{R_{k}}\mathbf{\nabla }A_{l}\left(
\mathbf{R},\mathbf{P}\right) \mathcal{P}_{N}\left( \mathcal{A}_{0}^{R_{l}}%
\mathcal{A}_{0}^{R_{k}}+\mathcal{A}_{0}^{R_{k}}\mathcal{A}_{0}^{R_{l}}\right)
\\
&&+\frac{\hbar ^{2}}{4}e\mathbf{\nabla }\nabla _{R_{k}}A_{l}\left( \mathbf{R}%
,\mathbf{P}\right) \left( \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{l}}%
\right] \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{k}}\right] +\mathcal{P}_{N}%
\left[ \mathcal{A}_{0}^{R_{k}}\right] \mathcal{P}_{N}\left[ \mathcal{A}%
_{0}^{R_{l}}\right] \right) \\
&&-e\left[ \nabla _{R_{k}}\mathbf{A}\left( \mathbf{R}\right) \emph{A}%
_{N}^{R_{k}}+\emph{A}_{N}^{R_{k}}\nabla _{R_{k}}\mathbf{A}\left( \mathbf{R}%
\right) \right] \\
&&-\frac{\hbar ^{2}}{4}e\nabla _{R_{k}}\nabla _{R_{l}}\mathbf{A}\left(
\mathbf{R},\mathbf{P}\right) \left( \mathcal{P}_{N}\left[ \mathcal{A}%
_{0}^{R_{l}}\right] \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{k}}\right] +%
\mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{k}}\right] \mathcal{P}_{N}\left[
\mathcal{A}_{0}^{R_{l}}\right] \right) \\
&=&\mathbf{\Pi }+e\emph{A}_{N}^{\mathbf{R}}\times \mathbf{B}\left( \mathbf{R}%
\right) +e\frac{\hbar }{4}\left[ \mathcal{P}_{N}\left[ \mathcal{A}%
_{0}^{R_{k}}\right] \left( \emph{A}_{N}^{\mathbf{R}}\times \nabla _{R_{k}}%
\mathbf{B}\left( \mathbf{R}\right) \right) +H.C.\right] \\
&&+\frac{\hbar ^{2}}{8}e\nabla _{R_{k}}\mathbf{\nabla }A_{l}\left( \mathbf{R}%
,\mathbf{P}\right) \mathcal{P}_{N}\left( \mathcal{A}_{0}^{R_{l}}\mathcal{A}%
_{0}^{R_{k}}+\mathcal{A}_{0}^{R_{k}}\mathcal{A}_{0}^{R_{l}}\right)\end{aligned}$$ While the first three terms are physical since they involve the momentum $%
\mathbf{\Pi }$ and the magnetic field, the last one is problematic since it will involve in the Hamiltonian some symmetric combinations of the vector potential of the kind $\nabla _{R_{k}}A_{l}+\nabla _{R_{l}}A_{k}$. The appearance of such terms is non physical but has nothing surprising here since we ordered all our expressions in $\mathbf{R}$ and $\mathbf{K}$, not in $\mathbf{R}$ and $\mathbf{\Pi }$. As a consequence, we cannot expect in one individual expression to have only contributions of the momentum and magnetic field operators. An other way to understand this problem is to remember our choice of symmetrization for the crystal momentum Berry phase that was easy to handle but does not fit when aiming at rewriting the Hamiltonian in terms of $\mathbf{\Pi }$. Actually, having let the derivatives of the electromagnetic potential half on the left and half on the right automatically induces commutators terms proportional to $\nabla
_{R_{k}}A_{l}+\nabla _{R_{l}}A_{k}$. However, we know that the overall process of diagonalization of the initial Hamiltonian has to depend only on $%
\mathbf{\Pi }$, $\mathbf{R}$, $\mathbf{B}\left( \mathbf{R}\right) $ so that we know that the symmetric contributions $\nabla _{R_{k}}A_{l}+\nabla
_{R_{l}}A_{k}$ to the Hamiltonian have to cancel.
This assertion will be checked explicitly for the above problematic term whose contribution (developing $\varepsilon _{0N}\left( \mathbf{x}\right) $ to the required order) is $\frac{\hbar ^{2}}{8}e\nabla _{R_{k}}\nabla
_{R_{n}}A_{l}\left( \mathbf{R},\mathbf{P}\right) \mathcal{P}_{N}\left(
\mathcal{A}_{0}^{R_{l}}\mathcal{A}_{0}^{R_{k}}+\mathcal{A}_{0}^{R_{k}}%
\mathcal{A}_{0}^{R_{l}}\right) \nabla _{\Pi _{n}}\varepsilon _{0N}\left(
\mathbf{x}\right) $, when diagonalizing the Hamiltonian. Actually we will find a counterpart to it. However, it is unnecessary in general to check the cancellation. These have to vanish by construction, so that we can discard them directly. Their local appearance in individual terms will be discussed briefly while computing the Bracket term for $\varepsilon _{0N}\left(
\mathbf{x}\right) $.
As a consequence of this discussion, our relevant variable for the momentum in the Hamiltonian will be : $$\mathbf{\pi =\Pi }+e\emph{A}_{N}^{\mathbf{R}}\times \mathbf{B}\left( \mathbf{%
R}\right) +e\frac{\hbar }{4}\left[ \mathcal{P}_{N}\left[ \mathcal{A}%
_{0}^{R_{k}}\right] \left( \emph{A}_{N}^{\mathbf{R}}\times \nabla _{R_{k}}%
\mathbf{B}\left( \mathbf{R}\right) \right) +H.C.\right]$$ and in the sequel, the relevant couple of canonical variables will be : $$\left( \mathbf{x}\right) =\left( \mathbf{r}_{N},\mathbf{\pi }_{N}\right)
=\left( \mathbf{r,\pi }\right)$$ (the index $N$ being understood).
#### Magnetization terms
At this order of approximation, and using our expressions for the Berry phases, one can directly develop the expression for $M\left[ \varepsilon _{0}%
\right] $ to obtain the following decomposition : $$\begin{aligned}
M\left[ \varepsilon _{0}\right] &=&-\frac{1}{2}\left( \mathbf{\mu }\left(
\mathbf{x}\right) \mathbf{.B}\left( \mathbf{x}\right) +\mathbf{B}\left(
\mathbf{x}\right) \mathbf{.\mu }\left( \mathbf{x}\right) \right) -\mathbf{%
\hat{\mu}}^{k}\left( \mathbf{x}\right) \mathbf{.}\nabla _{R_{k}}\mathbf{B}%
\left( \mathbf{x}\right) \\
+ &&\text{Darwin term}\end{aligned}$$ where the Darwin term refers to an expression which reveals to be of the same kind as the Darwin term in the Dirac Hamiltonian as we will see soon.
In this expression we have defined $$\begin{aligned}
\mu _{n} &=&-\frac{i}{2\hbar }\mathcal{P}_{N}\left\{ \left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{%
A}}^{R_{m}}\right\} \varepsilon ^{lmn} \\
\hat{\mu}_{l}^{k} &=&-\frac{1}{16}\mathcal{P}_{N}\left[ \left[ \left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{m}}\right]
,\nabla _{\Pi _{k}}\mathcal{\hat{A}}^{R_{n}}\right] \right] \varepsilon
^{lmn} \\
&&-\frac{i\hbar ^{2}}{8}\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{A}_{0}^{R_{k}}\right] _{NM}\left\{ \left( \mathcal{P}_{-}\mathcal{A}%
_{0}^{\mathbf{R}}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right)
_{l}\right\} _{MN}+\frac{i\hbar ^{2}}{8}\left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\left\{ \left( \mathcal{P}_{-}\mathcal{A}_{0}^{\mathbf{R}%
}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right) _{l}\right\} %
\right] _{NM}\mathcal{A}_{0MN}^{R_{k}}\end{aligned}$$ These expressions are directly obtained by expanding the magnetization term $%
M\left[ \varepsilon _{0}\right] =\frac{i}{4\hbar }\mathcal{P}_{N}\left\{ %
\left[ \varepsilon _{0},\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}%
^{K_{l}}-\left[ \varepsilon _{0},\mathcal{\hat{A}}^{K_{l}}\right] \mathcal{%
\hat{A}}^{R_{l}}\right\} $ and computing the commutators that appear when letting the magnetic field half on the right and half of the left of the relevant expressions. Note at this point that, doing so, some contributions involving $\nabla _{R_{k}}A_{l}+\nabla _{R_{l}}A_{k}$ appear that cancel the corresponding contribution arising from the second order in $\hbar $ development of $\varepsilon _{0N}\left( \mathbf{x}\right) $ as announced before.
More precisely, the derivation of $\mathbf{\mu }\left( \mathbf{x}\right) $ and $\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right) $ is as follows : starting with the term $\frac{i}{2}\mathcal{P}_{N}\left\{ \left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{%
A}}^{P_{l}}-\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}%
}^{P_{l}}\right] \mathcal{\hat{A}}^{R_{l}}\right\} $ we have to put the magnetic field half to the left and half to the right in the following way $$\begin{aligned}
&&\frac{i}{4}e\mathcal{P}_{N}\left\{ \left[ \varepsilon _{0}\left( \mathbf{x}%
\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}^{K_{l}}-\left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{K_{l}}\right]
\mathcal{\hat{A}}^{R_{l}}\right\} +H.C. \\
&=&\frac{i}{8}e\mathcal{P}_{N}\left\{ \left[ \varepsilon _{0}\left( \mathbf{x%
}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \left( \mathcal{\hat{A}}%
^{R_{n}}\nabla _{l}A_{n}+\nabla _{l}A_{n}\mathcal{\hat{A}}^{R_{n}}\right) -%
\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\left( \mathcal{\hat{A}}%
^{R_{n}}\nabla _{l}A_{n}+\nabla _{l}A_{n}\mathcal{\hat{A}}^{R_{n}}\right) %
\right] \mathcal{\hat{A}}^{R_{l}}\right\} +H.C.\end{aligned}$$ Letting the gradient of the potential to the left or to the right yields the required contribution for $\mathbf{\mu }\left( \mathbf{x}\right) $ and $%
\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right) $ is obtained by computing the commutators involving the gauge field. These terms are (we skip $%
\mathcal{P}_{N}$ for convenience and introduced a $\frac{1}{2}$ to take into account the integral over $\alpha $) : $$\begin{aligned}
&&\frac{i}{16}e\left( \left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{\hat{A}}^{R_{l}}\right] \left[ \nabla _{l}A_{n}-\nabla _{n}A_{l},%
\mathcal{\hat{A}}^{R_{n}}\right] \right) -\frac{ie}{16}\left[ \nabla
_{l}A_{n},\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}%
^{R_{n}}\right] \mathcal{\hat{A}}^{R_{l}}\right] \\
&&-\frac{i}{16}e\left[ \left( \mathcal{\hat{A}}^{R_{n}}\left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] \mathcal{\hat{A}}%
^{R_{l}}+\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}%
\right] \mathcal{\hat{A}}^{R_{n}}\mathcal{\hat{A}}^{R_{l}}\right) \right]
+H.C. \\
&=&-\frac{\hbar }{16}e\varepsilon ^{nlk}\left( \left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \nabla _{\Pi _{p}}%
\mathcal{\hat{A}}^{R_{n}}\nabla _{R_{p}}B_{k}\right) -\frac{ie}{16}\left[
\nabla _{l}A_{n},\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{%
\hat{A}}^{R_{n}}\right] \mathcal{\hat{A}}^{R_{l}}\right] \\
&&-\frac{i}{16}e\left( \left[ \mathcal{\hat{A}}^{R_{n}},\left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] \mathcal{\hat{A}}%
^{R_{l}}\right] +\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\nabla
_{l}A_{n}\right] \left( \mathcal{\hat{A}}^{R_{n}}\mathcal{\hat{A}}^{R_{l}}+%
\mathcal{\hat{A}}^{R_{l}}\mathcal{\hat{A}}^{R_{n}}\right) \right) +H.C. \\
&=&-\frac{\hbar }{16}e\varepsilon ^{nlk}\left( \left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \nabla _{\Pi _{p}}%
\mathcal{\hat{A}}^{R_{n}}\nabla _{R_{p}}B_{k}\right) -\frac{i}{16}\left( e%
\left[ \nabla _{l}A_{n},\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{\hat{A}}^{R_{n}}\right] \mathcal{\hat{A}}^{R_{l}}\right] \right) \\
&&-\frac{i}{16}e\left( \left[ \mathcal{\hat{A}}^{R_{n}},\left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] \right] \mathcal{\hat{A%
}}^{R_{l}}+\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}%
\right] \left[ \mathcal{\hat{A}}^{R_{n}},\mathcal{\hat{A}}^{R_{l}}\right]
\right. \\
&&\left. +\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}%
\right] \left( \mathcal{\hat{A}}^{R_{n}}\mathcal{\hat{A}}^{R_{l}}+\mathcal{%
\hat{A}}^{R_{l}}\mathcal{\hat{A}}^{R_{n}}\right) \right) +H.C.\end{aligned}$$ the projection of the last term $$\begin{aligned}
&&\frac{-i}{16}e\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\nabla
_{l}A_{n}\right] \mathcal{P}_{N}\left( \mathcal{\hat{A}}^{R_{n}}\mathcal{%
\hat{A}}^{R_{l}}+\mathcal{\hat{A}}^{R_{l}}\mathcal{\hat{A}}^{R_{n}}\right)
+H.C. \\
&=&-\frac{1}{8}e\nabla _{l}\nabla _{m}A_{n}\mathcal{P}_{N}\left( \mathcal{%
\hat{A}}^{R_{n}}\mathcal{\hat{A}}^{R_{l}}+\mathcal{\hat{A}}^{R_{l}}\mathcal{%
\hat{A}}^{R_{n}}\right) \nabla _{\Pi _{m}}\varepsilon _{0N}\left( \mathbf{x}%
\right)\end{aligned}$$ gives a contribution that cancel the one we have neglected in the momentum term as announced before. Moreover $$-\frac{i}{16}e\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\nabla
_{l}A_{n}\right] \left[ \mathcal{\hat{A}}^{R_{n}},\mathcal{\hat{A}}^{R_{l}}%
\right] +H.C.=-\left[ \frac{i}{16}e\left[ \varepsilon _{0}\left( \mathbf{x}%
\right) ,\nabla _{l}A_{n}\right] ,\left[ \mathcal{\hat{A}}^{R_{n}},\mathcal{%
\hat{A}}^{R_{l}}\right] \right]$$ and this term does not contribute to the diagonal part at our order of approximation.
The last term we are left to compute is then : $$\begin{aligned}
&&-\frac{\hbar }{16}e\varepsilon ^{nlk}\left( \left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \nabla _{\Pi _{p}}%
\mathcal{\hat{A}}^{R_{n}}\nabla _{R_{p}}B_{k}\right) -\left( \frac{i}{16}e%
\left[ \nabla _{l}A_{n},\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{\hat{A}}^{R_{n}}\right] \mathcal{\hat{A}}^{R_{l}}\right]
+H.C.\right) \\
&&-\left( \frac{i}{16}e\left[ \mathcal{\hat{A}}^{R_{n}},\left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] \right] \mathcal{\hat{A%
}}^{R_{l}}+H.C.\right) \\
&=&-\frac{\hbar }{16}e\varepsilon ^{nlk}\left( \left[ \left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] ,\nabla _{\Pi
_{p}}\mathcal{\hat{A}}^{R_{n}}\right] \nabla _{R_{p}}B_{k}\right) -\frac{i}{%
16}e\left[ \nabla _{l}A_{n},\left[ \left[ \varepsilon _{0}\left( \mathbf{x}%
\right) ,\mathcal{\hat{A}}^{R_{n}}\right] ,\mathcal{\hat{A}}^{R_{l}}\right] %
\right] \\
&&-\frac{i}{16}e\left[ \left[ \mathcal{\hat{A}}^{R_{n}},\left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] \right] ,\mathcal{\hat{%
A}}^{R_{l}}\right] \\
&=&-\frac{\hbar }{16}e\varepsilon ^{nlk}\left( \left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \nabla _{\Pi _{p}}%
\mathcal{\hat{A}}^{R_{n}}\nabla _{R_{p}}B_{k}\right) +\frac{\hbar }{16}%
e\nabla _{p}\nabla _{l}A_{n},\nabla _{\Pi _{p}}\left[ \left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{n}}\right] ,\mathcal{%
\hat{A}}^{R_{l}}\right] \\
&&-\frac{i}{16}e\left[ \mathcal{\hat{A}}^{R_{n}},\left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] ,\mathcal{\hat{A}}%
^{R_{l}}\right] \\
&=&-\frac{\hbar }{16}e\varepsilon ^{nlk}\left( \left[ \varepsilon _{0}\left(
\mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] \nabla _{\Pi _{p}}%
\mathcal{\hat{A}}^{R_{n}}\nabla _{R_{p}}B_{k}\right) \\
&&+\frac{\hbar }{32}e\nabla _{p}\nabla _{l}A_{n},\nabla _{\Pi _{p}}\left( %
\left[ \left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}%
^{R_{n}}\right] ,\mathcal{\hat{A}}^{R_{l}}\right] +\left[ \left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] ,\mathcal{%
\hat{A}}^{R_{n}}\right] +\left[ \left[ \mathcal{\hat{A}}^{R_{l}},\mathcal{%
\hat{A}}^{R_{n}}\right] ,\varepsilon _{0}\left( \mathbf{x}\right) \right]
\right) \\
&&-\frac{i}{32}e\left[ \left[ \mathcal{\hat{A}}^{R_{n}},\left[ \varepsilon
_{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] \right] ,\mathcal{\hat{%
A}}^{R_{l}}\right] +\left[ \left[ \mathcal{\hat{A}}^{R_{l}},\left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] \right] ,%
\mathcal{\hat{A}}^{R_{n}}\right] \\
&&+\left[ \left[ \mathcal{\hat{A}}^{R_{n}},\mathcal{\hat{A}}^{R_{l}}\right] ,%
\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}\right] %
\right]\end{aligned}$$ where we used twice the Jacobi in the last identity to rewrite $\frac{\hbar
}{16}e\nabla _{p}\nabla _{l}A_{n},\nabla _{\Pi _{p}}\left[ \left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right] ,%
\mathcal{\hat{A}}^{R_{n}}\right] $ and $-\frac{i}{16}e\left[ \mathcal{\hat{A}%
}^{R_{n}},\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,\nabla _{l}A_{n}%
\right] ,\mathcal{\hat{A}}^{R_{l}}\right] $. While isolating the contribution in the magnetic field, that is proportional to $\nabla
_{R_{k}}A_{l}-\nabla _{R_{l}}A_{k}$ and introducing $\mathcal{P}_{N}$ only the first term in the last identity remains, which yields the first term in the magnetization $\hat{\mu}_{l}^{k}$. The contribution $\nabla
_{R_{k}}A_{l}+\nabla _{R_{l}}A_{k}$ can be disgarded as discussed in the next subsection.
The two last contributions for $\hat{\mu}_{l}^{k}$ arise from the term $-%
\frac{\hbar ^{2}}{4}\left\{ \mathcal{P}_{-}\mathcal{A}_{0}^{\mathbf{R}%
}\times \mathcal{P}_{+}\mathcal{A}_{0}^{\mathbf{R}}\right\} _{MN}.\nabla
_{R_{m}}\mathbf{B}$ derived before in the expansion of $\mathcal{\hat{A}}%
^{K_{l}}$. Inserted in $\frac{i}{2}e\mathcal{P}_{N}\left\{ \left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{R_{l}}\right]
\mathcal{\hat{A}}^{K_{l}}-\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{\hat{A}}^{K_{l}}\right] \mathcal{\hat{A}}^{R_{l}}\right\} $ yields directly the terms announced.
Now, concerning the Darwin contribution, it is given by : $$\begin{aligned}
\text{Darwin term} &=&\frac{\hbar ^{2}}{4}\left[ \sum_{M}\left( \mathcal{A}%
_{0}^{R_{l}}\right) _{NM}\left( \mathcal{A}_{0}^{R_{m}}\right) _{MN}\right]
\nabla _{R_{m}}\nabla _{R_{l}}V\left( R\right) \\
&&+\frac{\hbar ^{2}}{4}\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{A}_{0}^{R_{m}}\right] _{NM}\frac{\left( \mathcal{A}%
_{0}^{R_{l}}\right) _{MN}\nabla _{R_{m}}\nabla _{R_{l}}V\left( R\right) }{%
\hat{\varepsilon}_{0M}-\hat{\varepsilon}_{0N}}\end{aligned}$$ Its derivation is straightforward and corresponds to isolate in $\frac{i}{2}e%
\mathcal{P}_{N}\left\{ \left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{\hat{A}}^{R_{l}}\right] \mathcal{\hat{A}}^{K_{l}}-\left[
\varepsilon _{0}\left( \mathbf{x}\right) ,\mathcal{\hat{A}}^{K_{l}}\right]
\mathcal{\hat{A}}^{R_{l}}\right\} $ the contribution of $\mathcal{\hat{A}}%
^{K_{l}}$ involving the potential term (see above). At our order of approximation, the $\mathcal{\hat{A}}^{\mathbf{X}}$ terms reduce to $%
\mathcal{A}_{0}^{\mathbf{X}}$.
The last important magnetization contribution in the Hamiltonian is due to the squared action of the magnetization operator $M$. This is of second order in field and can be written : $$\begin{aligned}
M\left[ M\left[ \varepsilon _{0}\right] \right] &=&\left( \frac{i\hbar }{4}%
\right) ^{2}\mathcal{P}_{N}\left\{ \left[ \mathcal{P}_{+}\left( \left[ \hat{%
\varepsilon}_{0},\mathcal{\hat{A}}_{0}^{R_{l}}\right] \mathcal{\hat{A}}%
_{0}^{P_{l}}-\left[ \hat{\varepsilon}_{0},\mathcal{\hat{A}}_{0}^{R_{l}}%
\right] \mathcal{\hat{A}}_{0}^{P_{l}}\right) ,\mathcal{\hat{A}}_{0}^{R_{m}}%
\right] \mathcal{\hat{A}}_{0}^{P_{m}}\right. \\
&&\left. -\left[ \mathcal{P}_{+}\left( \left[ \hat{\varepsilon}_{0},\mathcal{%
\hat{A}}_{0}^{R_{l}}\right] \mathcal{\hat{A}}_{0}^{P_{l}}-\left[ \hat{%
\varepsilon}_{0},\mathcal{\hat{A}}_{0}^{R_{l}}\right] \mathcal{\hat{A}}%
_{0}^{P_{m}}\right) ,\mathcal{\hat{A}}_{0}^{P_{m}}\right] \mathcal{\hat{A}}%
_{0}^{R_{m}}\right\} +H.C. \\
&=&\left( \frac{i\hbar }{2}\right) ^{2}\mathcal{P}_{N}\left\{ \left[
\mathcal{P}_{+}\left( \left[ \hat{\varepsilon}_{0},\mathcal{P}_{-}\mathcal{%
\hat{A}}_{0}^{R_{k}}\right] \mathcal{P}_{-}\mathcal{\hat{A}}%
_{0}^{R_{l}}\right) ,\mathcal{P}_{-}\mathcal{\hat{A}}_{0}^{R_{m}}\right]
\mathcal{P}_{-}\mathcal{\hat{A}}_{0}^{R_{n}}\varepsilon
_{klp}B_{p}\varepsilon _{mnq}B_{q}\right\} \\
&\equiv &\tilde{\mu}_{pq}B_{p}B_{q}\end{aligned}$$
#### Computation of $-\hbar \left\langle \protect\varepsilon %
_{0}\left( \mathbf{x}\right) \right\rangle $
The form $\ $for the diagonalized Hamiltonian given in (first equat) is taken from [@DIAGOEXACT]. As explained in this paper the computation of the energy and in particular of the term $-\frac{\hbar }{2}\left\langle
\varepsilon _{0}\left( \mathbf{x}\right) \right\rangle $ has to be performed by assuming a certain kind of symmetrization for the various expressions, that is a way to order the powers of $\mathbf{R}$ and $\mathbf{K}$ in expressions as $\varepsilon _{0}$. Its value is in fact the track of the initial choice of symmetrization in the diagonalization process. Let us remark first, that since at the lowest order (that is zeroth order in $\hbar
$), only $\hat{\varepsilon}_{0}\left( \mathbf{x}\right) $ mixes $\mathbf{R}$ and $\mathbf{K}$, the term $-\frac{\hbar }{2}\left\langle \varepsilon
_{0}\left( \mathbf{x}\right) \right\rangle $ will reduce to $-\frac{\hbar }{2%
}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right) \right\rangle $. Second, and more importantly, let us stress that the final Hamiltonian is independent of this ordering, but this last one is necessary to give a precise meaning to each expression. The bracket term $-\frac{\hbar }{2}%
\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right) \right\rangle $ has in fact to counterbalance different choices of symmetrization and provide ultimately the same diagonalized Hamiltonian.
Concerning the choice of symmetrization between the canonical variables $%
\mathbf{R}$ and $\mathbf{K}$ a symmetric form in $\mathbf{R}$ and $\mathbf{K}
$ is in general chosen, such as putting the powers of $\mathbf{K}$ half on the left and half to the right. An other choice is a complete symmetrization in $\mathbf{R}$ and $\mathbf{K}$, that is permuting this to variables in all monomials, all permutations being equally weighted. However, both this type of symmetrization does not fit here since the relevant variable is $\mathbf{%
\Pi }\mathbf{=K}-eA\left( \mathbf{R}\right) $. We thus have rather to consider a symmetrization in the gauge invariant variable $\mathbf{\Pi }$. We will compute $-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left(
\mathbf{x}\right) \right\rangle $ in two cases that might be relevant for the applications, depending on the problem at stake.
Assume first for $\hat{\varepsilon}_{0}\left( \mathbf{x}\right) $ a completely symmetrized form in the powers of components of $\mathbf{\Pi }$. That is, we will consider $\hat{\varepsilon}_{0}\left( \mathbf{x}\right) $ written as a series of terms of the kind $\Psi ^{i_{1}i_{2}i_{3}...}\Pi
_{i_{1}}\Pi _{i_{2}}\Pi _{i_{3}}...$ symmetrized in the indices $i_{j}$. This form corresponds in fact to the most general one, not taking into account any spatial symmetry for the problem
The $\left\langle .\right\rangle $ operation defined in [@DIAGOEXACT] has been recalled in the first section and can be computed explicitly for $%
\hat{\varepsilon}_{0}$, in the following way :
Start with $\Psi ^{i_{1}i_{2}i_{3}...i_{n}}\Pi _{i_{1}}\Pi _{i_{2}}\Pi
_{i_{3}}...\Pi _{i_{n}}$ , $\Psi ^{i_{1}i_{2}i_{3}...}$ being a completely symmetrized tensor.
Applying the rule given in the first section, the contribution of this term to $-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}%
\right) \right\rangle $ is obtained by deriving (that is removing) by some momentum components $\Pi _{i_{k}}$ $\Pi _{i_{l}}$ and inserting $\nabla
_{R_{i_{l}}}A^{i_{k}}$. One gets for this contribution : $$-ie\frac{\hbar }{4}\sum_{kl}\Psi ^{i_{1}i_{2}...i_{n}}\left[ \Pi
_{i_{1}}...\Pi _{i_{k-1}}\nabla _{R_{i_{l}}}A^{i_{k}}\Pi _{i_{k+1}}...\hat{%
\Pi}_{i_{l}}...\Pi _{i_{n}}-\Pi _{i_{1}}...\hat{\Pi}_{i_{k}}...\Pi
_{i_{l-1}}\nabla _{R_{i_{l}}}A^{i_{k}}\Pi _{i_{l+1}}...\Pi _{i_{n}}\right]$$ where the hat recalls that the variable is omitted from the series. Let move the gradient of the potential half on the left and half on the right to gather them by pair. We thus have : $$\begin{aligned}
&&-ie\frac{\hbar }{4}\sum_{kl}\Psi ^{i_{1}i_{2}...i_{n}}\left[ \Pi
_{i_{1}}...\Pi _{i_{k-1}}\nabla _{R_{i_{l}}}A^{i_{k}}\Pi _{i_{k+1}}...\hat{%
\Pi}_{i_{l}}...\Pi _{i_{n}}-\Pi _{i_{1}}...\hat{\Pi}_{i_{k}}...\Pi
_{i_{l-1}}\nabla _{R_{i_{l}}}A^{i_{k}}\Pi _{i_{l+1}}...\Pi _{i_{n}}\right] \\
&=&-ie\frac{\hbar }{8}\sum_{kl}\Psi ^{i_{1}i_{2}...i_{n}}\left[ \Pi
_{i_{1}}...\Pi _{i_{k-1}}\left[ \nabla _{R_{i_{l}}}A^{i_{k}}-\nabla
_{R_{i_{l}}}A^{i_{k}}\right] \Pi _{i_{k+1}}...\hat{\Pi}_{i_{l}}...\Pi
_{i_{n}}\right] \\
&&+ie\frac{\hbar }{8}\sum_{kl}\Psi ^{i_{1}i_{2}...i_{n}}\left[ \Pi
_{i_{1}}...\hat{\Pi}_{i_{k}}...\Pi _{i_{l-1}}\left[ \nabla
_{R_{i_{l}}}A^{i_{k}}-\nabla _{R_{i_{l}}}A^{i_{k}}\right] \Pi
_{i_{l+1}}...\Pi _{i_{n}}\right] \\
&&+e\frac{\hbar ^{2}}{8}\sum_{k<p<l}\nabla _{R_{i_{p}}}\left[ \nabla
_{R_{i_{l}}}A^{i_{k}}+\nabla _{R_{i_{l}}}A^{i_{k}}\right] \Pi _{i_{1}}...%
\hat{\Pi}_{i_{k}}...\hat{\Pi}_{i_{p}}...\hat{\Pi}_{i_{l}}...\Pi _{i_{n}} \\
&=&+e\frac{\hbar ^{2}}{8}\sum_{k<p<l}\nabla _{R_{i_{p}}}\left[ \nabla
_{R_{i_{l}}}A^{i_{k}}+\nabla _{R_{i_{l}}}A^{i_{k}}\right] \Pi _{i_{1}}...%
\hat{\Pi}_{i_{k}}...\hat{\Pi}_{i_{p}}...\hat{\Pi}_{i_{l}}...\Pi _{i_{n}}\end{aligned}$$ the last equality is obtained since $\Psi ^{i_{1}i_{2}...i_{n}}$ is completely symmetric so that the contribution of $\nabla
_{R_{i_{l}}}A^{i_{k}}-\nabla _{R_{i_{l}}}A^{i_{k}}$ vanishes.
As a consequence one has : $$-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right)
\right\rangle =\frac{ie}{8}\hbar \left( B_{k}\varepsilon ^{jik}\right)
\nabla _{\Pi _{i}}\nabla _{\Pi _{j}}\hat{\varepsilon}_{0}+\text{%
contributions proportional to }\mathbf{\nabla }\left( \nabla
_{j}A^{i}+\nabla _{i}A^{j}\right)$$
As a consequence, with our initial choice of symmetrization, $-\frac{\hbar }{%
2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right) \right\rangle $ contains only symmetric contributions. However, as we explained in the previous section, since by construction ultimately $\varepsilon \left(
\mathbf{X}\right) $ has to depend only on $\mathbf{\Pi }$ and $\mathbf{B}$ (since $\mathbf{B}$ appears as the commutators of the components of $\mathbf{%
\Pi }$), this symmetric contribution has to find its counterpart in our previous development of the energy operator. This counterpart is automatically the contributions we discarded before.
However, we can at this point explain a bit more how this works. Roughly, in the diagonalization process, the Bracket of the energy enters in two ways in our method since basically the trick to obtain the diagonalized Hamiltonian is to add and subtract $\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}%
\right) \right\rangle $ (see [@DIAGOEXACT], or more explicitly the differential equation in [@SERIESPIERRE]) to two different quantities that do not contain any symmetric term .
The first way comes from developing $$\begin{aligned}
&&\varepsilon _{\alpha }\left( \mathbf{X}_{\alpha +d\alpha }+d\mathbf{X}%
_{\alpha }\right) -U_{\alpha }\left( \mathbf{X}_{\alpha }\right) H_{0}\left(
\mathbf{X}_{\alpha }\right) U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right)
\\
&=&U_{\alpha }\left( \mathbf{X}_{\alpha +d\alpha }+d\mathbf{X}_{\alpha
}\right) H_{0}\left( \mathbf{X}_{\alpha +d\alpha }+d\mathbf{X}_{\alpha
}\right) U_{\alpha }^{+}\left( \mathbf{X}_{\alpha +d\alpha }+d\mathbf{X}%
_{\alpha }\right) -U_{\alpha }\left( \mathbf{X}_{\alpha }\right) H_{0}\left(
\mathbf{X}_{\alpha }\right) U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right)\end{aligned}$$ This term, yields all the relevant contributions to the diagonalized Hamiltonian we derived before. However, due to our initial choice of canonical variables $\mathbf{R}$, $\mathbf{K}$, the series expansion of $%
\left\langle UH_{0}U^{+}\right\rangle $ led us also to some contributions proportional to $\mathbf{\nabla }\left( \nabla _{j}A^{i}+\nabla
_{i}A^{j}\right) $. They arise because the Bracket operation isolates non physical terms such $\nabla _{j}A^{i}$. When we recombined them to get magnetic field contributions the $\mathbf{\nabla }\left( \nabla
_{j}A^{i}+\nabla _{i}A^{j}\right) $ appeared.
The second way comes from developing $$-\left[ \varepsilon _{\alpha }\left( \mathbf{X}_{\alpha +d\alpha }+d\mathbf{X%
}_{\alpha }\right) -\varepsilon _{\alpha }\left( \mathbf{X}_{\alpha +d\alpha
}\right) \right]$$ that will yield the contribution $-\left\langle \hat{\varepsilon}_{0}\left(
\mathbf{x}\right) \right\rangle $ including its symmetric contributions. But since $U_{\alpha }\left( \mathbf{X}_{\alpha }\right) H_{0}\left( \mathbf{X}%
_{\alpha }\right) U_{\alpha }^{+}\left( \mathbf{X}_{\alpha }\right) $ and $%
\varepsilon _{\alpha }\left( \mathbf{X}_{\alpha +d\alpha }\right) $ can be written in terms including only physical variables $\mathbf{R}$ and $\mathbf{%
\Pi }$ (both the energy operator and diagonalization process are assumed to depend on these variables), the symmetric contributions of these two terms come only from $U_{\alpha }\left( \mathbf{X}_{\alpha +d\alpha }+d\mathbf{X}%
_{\alpha }\right) H_{0}\left( \mathbf{X}_{\alpha +d\alpha }+d\mathbf{X}%
_{\alpha }\right) U_{\alpha }^{+}\left( \mathbf{X}_{\alpha +d\alpha }+d%
\mathbf{X}_{\alpha }\right) -\varepsilon _{\alpha }\left( \mathbf{X}_{\alpha
+d\alpha }+d\mathbf{X}_{\alpha }\right) $, and as such cancel by construction.
As a check, one can see that, typically, the contribution arising from $%
\left\langle UH_{0}U^{+}\right\rangle $ were proportional to $\nabla
_{i}\nabla _{l}\nabla _{j}U^{+}\nabla _{l}\left( \nabla _{j}A^{i}+\nabla
_{i}A^{j}\right) \hat{\varepsilon}_{0}\left( \mathbf{x}\right) \propto
\mathcal{A}_{0}^{R_{i}}\nabla _{l}\mathcal{A}_{0}^{R_{j}}\nabla _{l}\left(
\nabla _{j}A^{i}+\nabla _{i}A^{j}\right) $ which is exactly the kind of symmetric terms we were left with while computing the magnetization term. But precisely these contributions are by construction equal to those produced by $\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left(
\mathbf{x}\right) \right\rangle $ (since $UH_{0}U^{+}=\varepsilon _{0}\left(
\mathbf{x}\right) $). As a consequence, $-\frac{\hbar }{2}\left\langle \hat{%
\varepsilon}_{0}\left( \mathbf{x}\right) \right\rangle $ acts in fact as a compensation term, and the overall terms in $\mathbf{\nabla }\left( \nabla
_{j}A^{i}+\nabla _{i}A^{j}\right) $ is constrained to cancel whatever the form of the Hamiltonian or the Berry phases.
As a consequence of this discussion, we can neglect the symmetric term and keep definitely for the choice of full symmetrization on the momentum components : $$-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right)
\right\rangle =0$$ However, the choice of symmetrizing all powers of the momentum $\mathbf{\Pi }
$ does not fit when the system presents some symmetries. The Basic example is the Dirac Hamiltonian satisfying rotational invariance, but one can also consider fermi surfaces having some ellipsoidal form. In that case a natural choice of symmetrization can be done with respect to some or several quadratic forms in the momentum of the kind $\Pi _{i}A^{ij}\Pi _{j}$ (typically for the Dirac case $A^{ij}=\delta ^{ij}$). To inspect this case, we will thus compute $-\frac{\hbar }{2}\left\langle \hat{\varepsilon}%
_{0}\left( \mathbf{x}\right) \right\rangle $ when $\hat{\varepsilon}%
_{0}\left( \mathbf{x}\right) $ is written as a series expansion of products and powers of such quadratic forms $\Pi _{i}A^{ij}\Pi _{j}$. To do so, we develop $\hat{\varepsilon}_{0}\left( \mathbf{x}\right) $ as a symmetrized series of monomial terms of the kind $C\left( X\right) ^{i}\Pi _{i}\left(
Q\right) \Pi _{j}\left( Y\right) ^{j}D$. The $\left( X\right) ^{i}$ and $%
\left( Y\right) ^{j}$ being shortcut for $\Pi _{j}A^{ij}$ and $\Pi
_{i}B^{ij}$ with $A^{ij}$ and $B^{ij}$ some quadratic forms arising in the expansion. $C$, $D$, $\left( Q\right) $ are arbitrary. Note at this point that implicitly, $\left( X\right) ^{i}\Pi _{i}\left( Q\right) \Pi _{j}\left(
Y\right) ^{j}$ has to be considered as part of a sum of four symmetric terms obtained by permuting $\left( X\right) ^{i}$ and $\Pi _{i}$, $\Pi _{j}$ and $%
\left( Y\right) ^{j}$, that is $\frac{1}{4}\left[ \left( X\right) ^{i}\Pi
_{i}\left( Q\right) \Pi _{j}\left( Y\right) ^{j}+\Pi _{i}\left( X\right)
^{i}\left( Q\right) \Pi _{j}\left( Y\right) ^{j}+\left( X\right) ^{i}\Pi
_{i}\left( Q\right) \left( Y\right) ^{j}\Pi _{j}+\Pi _{i}\left( X\right)
^{i}\left( Q\right) \left( Y\right) ^{j}\Pi _{j}\right] $. For the sake of simplicity we only keep $\left( X\right) ^{i}\Pi _{i}\left( Q\right) \Pi
_{j}\left( Y\right) ^{j}$, the symmetrization being implicit and we skip the terms $C$ and $D$ that play no role in the sequel. $\left( X\right) ^{i}\Pi
_{i}\left( Q\right) \Pi _{j}\left( Y\right) ^{j}$ contributes to $-\frac{%
\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right)
\right\rangle $ by deriving (that is removing) $\Pi _{i}$ $\Pi _{j}$ and inserting $\nabla _{j}A^{i}$ (the sums over the indices $i$,$j$ is understood): $$\begin{aligned}
&&\frac{i}{4}e\hbar \left( \left( X\right) ^{i}\nabla _{j}A^{i}\left(
Q\right) \left( Y\right) ^{j}-\left( X\right) ^{j}\left( Q\right) \nabla
_{j}A^{i}\left( Y\right) ^{i}\right) \\
&=&\frac{i}{4}e\hbar \left( \left( X\right) ^{i}\nabla _{j}A^{i}\left(
Q\right) \left( Y\right) ^{j}-\left( X\right) ^{i}\left( Q\right) \nabla
_{i}A^{j}\left( Y\right) ^{j}\right)\end{aligned}$$ We can gather the potential terms by putting them half to the left of $Q$ and half to the right of $Q$. We can thus write this term at order $\hbar
^{2}$ (which is enough for us here given our order of approximation) $$\begin{aligned}
&&\frac{i}{4}e\hbar \left( \left( X\right) ^{i}\nabla _{j}A^{i}\left(
Q\right) \left( Y\right) ^{j}-\left( X\right) ^{i}\left( Q\right) \nabla
_{i}A^{j}\left( Y\right) ^{j}\right) \\
&=&\frac{ie}{8}\hbar \left( X\right) ^{i}\left( Q\right) \varepsilon
^{jik}B_{k}\left( Y\right) ^{j}+\frac{ie}{8}\hbar \left( X\right)
^{i}\varepsilon ^{jik}B_{k}\left( Q\right) \left( Y\right) ^{j} \\
&&-\frac{1}{8}e\hbar ^{2}\nabla _{l}\left( \nabla _{j}A^{i}+\nabla
_{i}A^{j}\right) \left( X\right) ^{i}\nabla _{l}\left( Q\right) \left(
Y\right) ^{j}+\frac{1}{4}e\hbar ^{2}\left( X\right) ^{i}\nabla _{l}\left(
Q\right) \left( Y\right) ^{j}\nabla _{l}\left( \nabla _{j}A^{i}+\nabla
_{i}A^{j}\right)\end{aligned}$$ To the order $\hbar ^{2}$, the terms $\frac{ie}{8}\hbar \left( X\right)
^{i}\left( Q\right) \varepsilon ^{jik}B_{k}\left( Y\right) ^{j}+\frac{ie}{8}%
\hbar \left( X\right) ^{i}\varepsilon ^{jik}B_{k}\left( Q\right) \left(
Y\right) ^{j}$, once summed over the symmetric monomials, involve all the possible permutations of $\varepsilon ^{jik}B_{k}$ inside the series expansion of the second derivative of $\hat{\varepsilon}_{0}$. As a consequence, we can write : $$-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right)
\right\rangle =\frac{ie}{8}\hbar \left( B_{k}\varepsilon ^{jik}\right)
\nabla _{\Pi _{i}}\nabla _{\Pi _{j}}\hat{\varepsilon}_{0}+\text{%
contributions proportional to }\mathbf{\nabla }\left( \nabla
_{j}A^{i}+\nabla _{i}A^{j}\right)$$ As explained just before, it is understood that $B_{k}$ is inserted in a completely symmetric way inside the series expansion of $\nabla _{\Pi
_{i}}\nabla _{\Pi _{j}}\hat{\varepsilon}_{0}$ (that is cyclically permuted inside the series).
Moreover, we can again discard the symmetric terms $\mathbf{\nabla }\left(
\nabla _{j}A^{i}+\nabla _{i}A^{j}\right) $.
We thus have with this choice of symmetrization : $$-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right)
\right\rangle =\frac{ie}{8}\hbar \left( B_{k}\varepsilon ^{jik}\right)
\nabla _{\Pi _{i}}\nabla _{\Pi _{j}}\hat{\varepsilon}_{0}$$
We will conclude this section by obtaining a developer form for $\frac{ie}{8}%
\hbar \left( B_{k}\varepsilon ^{jik}\right) \nabla _{\Pi _{i}}\nabla _{\Pi
_{j}}\hat{\varepsilon}_{0}$. It is performed by going back to its series expansion. Let us assume in a first time that $\hat{\varepsilon}_{0}$ is a function of $\mathbf{\Pi }^{2}$ only (rotational invariance), so that we can choose a natural symmetrization to express $\hat{\varepsilon}_{0}\left(
\mathbf{x}\right) $ as a power series of $\mathbf{\Pi }^{2}$.
The sum of monomials of the kind $$\frac{ie}{8}\hbar C\left( X\right) ^{i}\left( Q\right) \varepsilon
^{jik}B_{k}\left( Y\right) ^{j}D+\frac{ie}{8}\hbar C\left( X\right)
^{i}\varepsilon ^{jik}B_{k}\left( Q\right) \left( Y\right) ^{j}D$$ can be written (we now reintroduce our implicit permutations that let $B_{k}$ to be half on the left and half on the right of $\left( X\right) ^{i}$ and $%
\left( Y\right) ^{j}$) : $$\begin{aligned}
&&\frac{ie}{16}\hbar C\Pi _{i}\varepsilon ^{jik}B_{k}\left( Q\right) \Pi
_{j}D+\frac{ie}{16}\hbar C\Pi _{i}\left( Q\right) \varepsilon ^{jik}B_{k}\Pi
_{j}D \\
&&+\frac{ie}{16}\varepsilon ^{jik}B_{k}\hbar C\Pi _{i}\left( Q\right) \Pi
_{j}D+\frac{ie}{16}\hbar C\Pi _{i}\left( Q\right) \Pi _{j}\varepsilon
^{jik}B_{k}D\end{aligned}$$ Letting move the magnetic field half to the left and half to the right yields to the order $\hbar ^{2}$ yields : $$\begin{aligned}
&&\frac{ie}{16}\hbar C\Pi _{i}\varepsilon ^{jik}B_{k}\left( Q\right) \Pi
_{j}D+\frac{ie}{16}\hbar C\Pi _{i}\left( Q\right) \varepsilon ^{jik}B_{k}\Pi
_{j}D \\
&&+\frac{ie}{16}\varepsilon ^{jik}B_{k}\hbar C\Pi _{i}\left( Q\right) \Pi
_{j}D+\frac{ie}{16}\hbar C\Pi _{i}\left( Q\right) \Pi _{j}\varepsilon
^{jik}B_{k}D \\
&=&\varepsilon ^{jik}B_{k}\frac{ie}{8}\hbar \left[ C\Pi _{i}\left( Q\right)
\Pi _{j}D+C\Pi _{i}\left( Q\right) \Pi _{j}D\right] +\frac{ie}{8}\hbar \left[
C\Pi _{i}\left( Q\right) \Pi _{j}D+C\Pi _{i}\left( Q\right) \Pi _{j}D\right]
\varepsilon ^{jik}B_{k} \\
&&+\left( \mathbf{\nabla }\times \mathbf{B}\right) .\mathbf{\Pi }\frac{e}{8}%
\hbar ^{2}C\left( Q\right) D\end{aligned}$$ Concentrate on the first term. One has $$\begin{aligned}
&&\varepsilon ^{jik}B_{k}\frac{ie}{8}\hbar \left[ C\Pi _{i}\left( Q\right)
\Pi _{j}D+C\Pi _{i}\left( Q\right) \Pi _{j}D\right] +\frac{ie}{8}\hbar \left[
C\Pi _{i}\left( Q\right) \Pi _{j}D+C\Pi _{i}\left( Q\right) \Pi _{j}D\right]
\varepsilon ^{jik}B_{k} \\
&=&\varepsilon ^{jik}B_{k}\frac{ie}{8}\hbar \left[ C\Pi _{i}\left( Q\right)
\Pi _{j}D-C\Pi _{j}\left( Q\right) \Pi _{i}D\right]\end{aligned}$$ to the order $\hbar ^{2}$. Now, $$\begin{aligned}
&&\varepsilon ^{jik}B_{k}\frac{ie}{8}\hbar \left[ C\Pi _{i}\left( Q\right)
\Pi _{j}D-C\Pi _{j}\left( Q\right) \Pi _{i}D\right] \\
&=&\frac{ie}{8}\hbar C\left( Q\right) \left[ \Pi _{i},\Pi _{j}\right]
D\varepsilon ^{jik}B_{k} \\
&&+\frac{ie}{8}\hbar C\Pi _{i}\left[ \left( Q\right) ,\Pi _{j}\right]
D\varepsilon ^{jik}B_{k}-\frac{ie}{8}\hbar C\Pi _{j}\left[ \left( Q\right)
,\Pi _{i}\right] D\varepsilon ^{jik}B_{k}\end{aligned}$$ Summing over the symmetric monomials of the series expansion of $\hat{%
\varepsilon}_{0}$ , each of this commutator will yield a corresponding contribution to $-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left(
\mathbf{x}\right) \right\rangle $. The first commutator will yield the contribution : $$\begin{aligned}
\frac{ie}{8}\hbar C\left( Q\right) \left[ \Pi _{i},\Pi _{j}\right]
D\varepsilon ^{jik}B_{k} &=&\frac{e}{8}\hbar ^{2}C\left( Q\right) D\left(
\varepsilon ^{ijk}B_{k}\right) ^{2} \\
&\rightarrow &\frac{1}{2}\frac{e^{2}}{8}\hbar ^{2}\left( \varepsilon
^{ijk}B_{k}\right) ^{2}4\left( \nabla _{\mathbf{\Pi }^{2}}\right) ^{2}\hat{%
\varepsilon}_{0} \\
&=&\frac{e^{2}}{2}\hbar ^{2}\left( B^{2}\right) \left( \nabla _{\mathbf{\Pi }%
^{2}}\right) ^{2}\hat{\varepsilon}_{0}\end{aligned}$$ the global $\frac{1}{2}$ amounts for the fact that given our conventions the derivative with respect to $\Pi _{i}$ are always on the left of the derivatives with respect to $\Pi _{j}$. This implies a $\frac{1}{2}$ factor each $\nabla _{\Pi _{i}}\nabla _{\Pi _{j}}$ term. We have also used the fact that given the rotational invariance $\left( \nabla _{\Pi _{i}}\right) ^{2}$ and $\left( \nabla _{\Pi _{i}}\right) ^{2}$ can be replaced by $4\left(
\nabla _{\mathbf{\Pi }^{2}}\right) ^{2}$. Actually the commutator $\left[
\Pi _{i},\Pi _{j}\right] $ amounts for taking twice the derivative with respect to $\Pi _{i}$ and $\Pi _{j}$ (times $\varepsilon ^{ijk}B_{k}$) and twice the derivative with respect to $\Pi _{i}$ and $\Pi _{j}$ yield each a contribution proportional to $\nabla _{\mathbf{\Pi }^{2}}$. On the other hand the second commutator will lead to the contribution : $$\begin{aligned}
+\frac{ie}{8}\hbar C\Pi _{j}\left[ \Pi _{i},\left( Q\right) \right]
D\varepsilon ^{ijk}B_{k} &\rightarrow &\frac{1}{2}\frac{e^{2}}{24}\hbar
^{2}\left( \nabla _{\Pi _{l}}^{\prime }\nabla _{\Pi _{j}}^{\prime }2\left(
\nabla _{\mathbf{\Pi }^{2}}\right) \hat{\varepsilon}_{0}\right) \varepsilon
^{jik}B_{k}\varepsilon ^{ilm}B_{m} \\
&=&\frac{e^{2}}{48}\hbar ^{2}\left( \left( B^{2}\right) 2\nabla _{\Pi
_{j}}^{\prime }\nabla _{\Pi _{j}}^{\prime }\left( \nabla _{\mathbf{\Pi }%
^{2}}\right) ^{2}\hat{\varepsilon}_{0}-B_{k}\nabla _{\Pi _{k}}^{\prime
}B_{m}\nabla _{\Pi _{m}}^{\prime }2\left( \nabla _{\mathbf{\Pi }^{2}}\right)
\hat{\varepsilon}_{0}\right)\end{aligned}$$ with the implicit convention, recalled by the $^{\prime }$, and implied by construction and the definitions of $\left( Q\right) $, $\Pi _{i}$, $\Pi
_{j} $), that a factor $\mathbf{\Pi }^{2}$ that has been derived by $\nabla
_{\Pi _{m}}$ will not be derived again by $\nabla _{\Pi _{k}}$ (the same convention applying of course for $\nabla _{\Pi _{l}}^{\prime }\nabla _{\Pi
_{j}}^{\prime }$). Similarly $\nabla _{\Pi _{j}}^{\prime }\nabla _{\Pi
_{j}}^{\prime }$ means that a power of $\mathbf{\Pi }^{2}$ is not derived twice by $\Pi _{j}$.
The reason for the global $\frac{1}{2}$ factor is the same as before, the derivatives with respect to $\Pi _{i}$ being on the left of the derivatives with respect to $\Pi _{j}$. A $\frac{1}{3}$ factor has also arisen from the fact that the derivative with respect to $\Pi _{l}$ has to be taken between the derivatives with respect to $\Pi _{i}$ and $\Pi _{j}$. Due to the symmetrization of the variables, it amounts for only one third of the derivative with respect to $\Pi _{l}$.
Similarly, the last term will give : $$\begin{aligned}
-\frac{ie}{8}\hbar C\Pi _{j}\left[ \left( Q\right) ,\Pi _{i}\right]
D\varepsilon ^{jik}B_{k} &\rightarrow &\frac{e^{2}}{12}\hbar ^{2}\left(
\nabla _{\Pi _{l}}^{\prime }\nabla _{\Pi _{i}}^{\prime }\left( \nabla _{%
\mathbf{\Pi }^{2}}\right) \hat{\varepsilon}_{0}\right) \varepsilon
^{ijk}B_{k}\varepsilon ^{ljm}B_{m} \\
&=&\frac{e^{2}}{48}\hbar ^{2}\left( \left( B^{2}\right) 2\nabla _{\Pi
_{j}}^{\prime }\nabla _{\Pi _{j}}^{\prime }\left( \nabla _{\mathbf{\Pi }%
^{2}}\right) ^{2}\hat{\varepsilon}_{0}-B_{k}\nabla _{\Pi _{k}}^{\prime
}B_{m}\nabla _{\Pi _{m}}^{\prime }2\left( \nabla _{\mathbf{\Pi }^{2}}\right)
\hat{\varepsilon}_{0}\right)\end{aligned}$$ with the same convention as before for the derivatives.
Ultimately, we compute similarly the contribution : $$\begin{aligned}
\left( \mathbf{\nabla }\times \mathbf{B}\right) .\mathbf{\Pi }\frac{e}{8}%
\hbar ^{2}C\left( Q\right) D &\rightarrow &\frac{1}{2}\frac{e}{8}\hbar
^{2}\left( \mathbf{\nabla }\times \mathbf{B}\right) .\mathbf{\Pi }4\left(
\nabla _{\mathbf{\Pi }^{2}}\right) ^{2}\hat{\varepsilon}_{0} \\
&=&\frac{e}{4}\hbar ^{2}\left( \mathbf{\nabla }\times \mathbf{B}\right) .%
\mathbf{\Pi }\left( \nabla _{\mathbf{\Pi }^{2}}\right) ^{2}\hat{\varepsilon}%
_{0}\end{aligned}$$ Gathering all the relevant terms yields in the end : $$\begin{aligned}
-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right)
\right\rangle &=&\frac{e^{2}}{2}\hbar \left( B^{2}\right) \left( \nabla _{%
\mathbf{\Pi }^{2}}\right) ^{2}\hat{\varepsilon}_{0}+\frac{e^{2}}{12}\hbar
^{2}\left( \left( B^{2}\right) \nabla _{\Pi _{j}}^{\prime }\nabla _{\Pi
_{j}}^{\prime }\left( \nabla _{\mathbf{\Pi }^{2}}\right) ^{2}\hat{\varepsilon%
}_{0}-B_{k}\nabla _{\Pi _{k}}^{\prime }B_{m}\nabla _{\Pi _{m}}^{\prime
}\left( \nabla _{\mathbf{\Pi }^{2}}\right) \hat{\varepsilon}_{0}\right) \\
&&+\frac{e}{4}\hbar ^{2}\left( \mathbf{\nabla }\times \mathbf{B}\right) .%
\mathbf{\Pi }\left( \nabla _{\mathbf{\Pi }^{2}}\right) ^{2}\hat{\varepsilon}%
_{0}\end{aligned}$$ A direct computation shows that specializing to the case of the Dirac Hamiltonian in an electromagnetic field, will be given in [@PIERREDIRAC].
The more general case can now be treated in a very similar way. Coming back to the general form for the monomials, leads to consider again the monomials $-\frac{ie}{8}\hbar \left( X\right) ^{i}\varepsilon ^{ijk}B_{k}\left(
Q\right) \left( Y\right) ^{j}-\frac{ie}{8}\hbar \left( X\right) ^{j}\left(
Q\right) \varepsilon ^{jik}B_{k}\left( Y\right) ^{i}$. Here to alleviate the notation we assume again implicitly that $B_{k}$ is put half on the left and half on the right of $\left( X\right) ^{i}$ and $\left( Y\right) ^{j}$). the contribution to move them on the left of $\left( X\right) ^{i}$ and the right of $\left( Y\right) ^{j}$ leads to a similar contribution to $-\frac{%
\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right)
\right\rangle $ as before : $\frac{e}{8}\hbar ^{2}\left( \mathbf{\nabla }%
\times \mathbf{B}\right) .\mathbf{\nabla }_{\mathbf{\Pi }}\left( \nabla _{%
\mathbf{\Pi }^{2}}\right) ^{2}\hat{\varepsilon}_{0}$ Concerning the other terms, we recombine the monomial $-\frac{ie}{8}\hbar \left( X\right)
^{i}\left( Q\right) \left( Y\right) ^{j}\varepsilon ^{ijk}B_{k}$ with the corresponding contribution $+\frac{ie}{8}\hbar \left( Y\right) ^{j}\left(
Q\right) \left( X\right) ^{i}\varepsilon ^{ijk}B_{k}$ (the $\varepsilon
^{jik}B_{k}$ being implicitly half on the left and half on the right of the expressions): $$\begin{aligned}
&&-\frac{ie}{8}\hbar \left( X\right) ^{i}\left( Q\right) \left( Y\right)
^{j}\varepsilon ^{ijk}B_{k}+\frac{ie}{8}\hbar \left( Y\right) ^{j}\left(
Q\right) \left( X\right) ^{i}\varepsilon ^{ijk}B_{k} \\
&=&-\frac{ie}{8}\hbar \left[ \left( X\right) ^{i}\left( Q\right) \left(
Y\right) ^{j}-\left( Y\right) ^{j}\left( Q\right) \left( X\right) ^{i}\right]
\varepsilon ^{ijk}B_{k} \\
&=&-\frac{ie}{8}\hbar \left( \left( X\right) ^{i}\left( Y\right) ^{j}-\left(
Y\right) ^{j}\left( X\right) ^{i}\right) \left( Q\right) \varepsilon
^{ijk}B_{k} \\
&&-\frac{ie}{8}\hbar \left[ \left( X\right) ^{i},\left( Q\right) \right]
\left( Y\right) ^{j}\varepsilon ^{ijk}B_{k}+\frac{ie}{8}\hbar \left[ \left(
Y\right) ^{j},\left( Q\right) \right] \left( X\right) ^{i}\varepsilon
^{ijk}B_{k}\end{aligned}$$ As before, summing over the symmetric monomials of the series expansion of $%
\hat{\varepsilon}_{0}$ , each of this commutator will yield a corresponding contribution in derivative of $\hat{\varepsilon}_{0}$. The first commutator in $\left[ \left( X\right) ^{i}\left( Y\right) ^{j}-\left( X\right)
^{j}\left( Y\right) ^{i}\right] $ is computed in the following way. The rotational invariance does not exist anymore now, but $\left( X\right) ^{i}$ and $\left( Y\right) ^{j}$ are still functions of $\mathbf{\Pi }$. As such they will yield the contribution : $$\begin{aligned}
&&\frac{ie}{8}\hbar \left( Q\right) \left[ \left( X\right) ^{i}\left(
Y\right) ^{j}-\left( X\right) ^{j}\left( Y\right) ^{i}\right] \varepsilon
^{ijk}B_{k} \\
&\rightarrow &\frac{1}{2}\frac{e^{2}}{8}\hbar ^{2}\left( \varepsilon
^{ijk}B_{k}\right) \left( \varepsilon ^{lmn}B_{n}\right) \left( \nabla _{\Pi
_{l}}\right) \left( \nabla _{\Pi _{i}}\right) \left( \nabla _{\Pi
_{m}}\right) \left( \nabla _{\Pi _{j}}\right) \hat{\varepsilon}_{0}\end{aligned}$$ Once again, the global $\frac{1}{2}$ amounts for the fact that given our conventions the derivative with respect to $\Pi _{i}$ are always on the left of the derivatives with respect to $\Pi _{j}$.
Similarly, the second and third commutators will lead to the contribution : $$\begin{aligned}
&&-\frac{ie}{8}\hbar \left( X\right) ^{i}\left[ \left( Q\right) ,\left(
Y\right) ^{j}\right] \varepsilon ^{ijk}B_{k}+\frac{ie}{8}\hbar \left(
Y\right) ^{j}\left[ \left( Q\right) ,\left( X\right) ^{i}\right] \varepsilon
^{ijk}B_{k} \\
&\rightarrow &\frac{e^{2}}{48}\hbar ^{2}\left( \nabla _{\Pi _{l}}^{\prime
}\nabla _{\Pi _{i}}^{\prime }\left[ \left( \nabla _{\Pi _{n}}\right) .\left(
\nabla _{\Pi _{j}}\right) \right] \hat{\varepsilon}_{0}\right) \varepsilon
^{ijk}B_{k}\varepsilon ^{lnm}B_{m} \\
&&-\frac{e^{2}}{48}\hbar ^{2}\left( \nabla _{\Pi _{l}}^{\prime }\nabla _{\Pi
_{j}}^{\prime }\left[ \left( \nabla _{\Pi _{n}}\right) .\left( \nabla _{\Pi
_{i}}\right) \right] \hat{\varepsilon}_{0}\right) \varepsilon
^{ijk}B_{k}\varepsilon ^{lnm}B_{m} \\
&=&\frac{e^{2}}{24}\hbar ^{2}\left( \nabla _{\Pi _{l}}^{\prime }\nabla _{\Pi
_{i}}^{\prime }\left[ \left( \nabla _{\Pi _{n}}\right) .\left( \nabla _{\Pi
_{j}}\right) \right] \hat{\varepsilon}_{0}\right) \varepsilon
^{ijk}B_{k}\varepsilon ^{lnm}B_{m}\end{aligned}$$ with again the implicit convention, recalled by the $^{\prime }$ and implied by construction and the definitions of $\left( Q\right) $, $\left( X\right)
^{i}$, $\left( Y\right) ^{j}$), that the derivatives $\nabla _{\Pi
_{l}}^{\prime }$ and $\nabla _{\Pi _{i}}^{\prime }$ are not applied on the same quadratic term $\Pi _{i}A^{ij}\Pi _{j}$ in the series expansion of $%
\hat{\varepsilon}_{0}$. $\left[ \left( \nabla _{\Pi _{n}}\right) .\left(
\nabla _{\Pi _{j}}\right) \right] $ is a notation to recall that $\left(
\nabla _{\Pi _{n}}\right) \left( \nabla _{\Pi _{j}}\right) $ act both on the same quadratic term,
Gathering all the relevant terms yields ultimately for the general case : $$\begin{aligned}
-\frac{\hbar }{2}\left\langle \hat{\varepsilon}_{0}\left( \mathbf{x}\right)
\right\rangle &=&\frac{e^{2}}{16}\hbar ^{2}\left( \varepsilon
^{ijk}B_{k}\right) \left( \varepsilon ^{lmn}B_{n}\right) \left( \nabla _{\Pi
_{l}}\right) \left( \nabla _{\Pi _{i}}\right) \left( \nabla _{\Pi
_{m}}\right) \left( \nabla _{\Pi _{j}}\right) \hat{\varepsilon}_{0} \\
&&+\frac{e^{2}}{24}\hbar ^{2}\left( \nabla _{\Pi _{l}}^{\prime }\nabla _{\Pi
_{i}}^{\prime }\left[ \left( \nabla _{\Pi _{n}}\right) .\left( \nabla _{\Pi
_{j}}\right) \right] \hat{\varepsilon}_{0}\right) \varepsilon
^{ijk}B_{k}\varepsilon ^{lnm}B_{m}+\frac{e}{8}\hbar ^{2}\left( \mathbf{%
\nabla }\times \mathbf{B}\right) .\mathbf{\nabla }_{\mathbf{\Pi }}\left(
\nabla _{\mathbf{\Pi }^{2}}\right) ^{2}\hat{\varepsilon}_{0}\end{aligned}$$
#### Final form for the Band Hamiltonian
Gathering all the previous terms, and writing $\hat{\varepsilon}_{0}\left(
\mathbf{\pi }\right) $ as a completely symmetrized series in the powers of the momentum (in the lack of any a priory symmetry), we have ultimately the diagonalized Hamiltonian : $$\begin{aligned}
H_{d} &=&\hat{\varepsilon}_{0}\left( \mathbf{\pi }\right) +V\left( \mathbf{r}%
\right) -\frac{1}{2}\left( \mathbf{\mu }\left( \mathbf{x}\right) \mathbf{.B}%
\left( \mathbf{x}\right) +\mathbf{B}\left( \mathbf{x}\right) \mathbf{.\mu }%
\left( \mathbf{x}\right) \right) -\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}%
\right) \mathbf{.}\nabla _{R_{k}}\mathbf{B}\left( \mathbf{x}\right) +\mathbf{%
B.\tilde{\mu}.B} \notag \\
&&+\frac{\hbar ^{2}}{4}\left[ \sum_{M}\left( \mathcal{A}_{0}^{R_{l}}\right)
_{NM}\left( \mathcal{A}_{0}^{R_{m}}\right) _{MN}\right] \nabla
_{R_{m}}\nabla _{R_{l}}V\left( R\right) \notag \\
&&+\frac{\hbar ^{2}}{4}\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{A}_{0}^{R_{m}}\right] _{NM}\frac{\left( \mathcal{A}%
_{0}^{R_{l}}\right) _{MN}\nabla _{R_{m}}\nabla _{R_{l}}V\left( R\right) }{%
\hat{\varepsilon}_{0M}-\hat{\varepsilon}_{0N}} \notag \\
&& \label{H1P}\end{aligned}$$ The double scalar product $\mathbf{B.\tilde{\mu}.B}$ meaning that the two index tensor $\mathbf{\tilde{\mu}}$ being contracted twice with $\mathbf{B}$.
#### Example : Darwin term for the Bloch electron
We now consider more specifically what we have called the Darwin term. Its interpretation turns out to be more transparent when $B=0,$ therefore we keep here only the electrostatic potential. In this case, we have $$\mathcal{A}^{\mathbf{R}}=\hbar \mathcal{A}_{0}^{\mathbf{R}}\left( \mathbf{R+}%
\frac{\hbar }{4}\mathcal{A}_{0}^{R_{l}},\mathbf{K}\right) -e\frac{\hbar ^{2}%
}{2}\left[ \hbar \left[ .,\hat{\varepsilon}_{0}\right] ^{-1}.\left( \mathcal{%
P}_{-}\left\{ \frac{1}{2}\mathcal{A}_{0}^{R_{l}}\nabla _{R_{l}}V\left(
\mathbf{R}\right) +H.C.\right\} \right) ,\mathbf{R}+\mathcal{A}_{0}^{\mathbf{%
R}}\right]$$ and $\mathcal{A}_{0}^{K_{l}}=0$ as shown in [@PIERREEUROPHYS] but at the second order we have the following contributions and $$\left( \mathcal{A}^{\mathbf{K}}\right) _{MN}=-\frac{i}{2}\hbar ^{2}\nabla
_{R_{m}}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right) _{MN}\nabla
_{R_{l}}V\left( R\right) }{\hat{\varepsilon}_{0M}-\hat{\varepsilon}_{0N}}%
\hat{\delta}_{MN}$$ showing that the non diagonal part of $\mathcal{A}^{\mathbf{K}}$ is non null. It will lead to a magnetization term : the Darwin term. We can write : $$\begin{aligned}
\left( \mathcal{A}^{\mathbf{R}}\right) _{MN} &=&\left( \hbar \mathcal{A}%
_{0}^{\mathbf{R}}\right) _{MN}+\frac{\hbar ^{2}}{16}\left\{ \left( \mathcal{A%
}_{0}^{R_{l}}\right) _{MP}\nabla _{R_{l}}\left( \mathcal{A}_{0}^{\mathbf{R}%
}\right) _{PN}+M\leftrightarrow N\right\} +\frac{\hbar ^{2}}{2}i\mathbf{%
\nabla }_{\mathbf{K}}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right) _{MN}\nabla
_{R_{l}}V\left( \mathbf{R}\right) }{\hat{\varepsilon}_{0M}-\hat{\varepsilon}%
_{0N}}\hat{\delta}_{MN} \\
&&+\frac{\hbar ^{2}}{2}\left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{MP}\hat{%
\delta}_{PM}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right) _{PN}\nabla
_{R_{l}}V\left( \mathbf{R}\right) }{\hat{\varepsilon}_{0P}-\hat{\varepsilon}%
_{0N}}-\frac{\hbar ^{2}}{2}\frac{\left( \mathcal{A}_{0}^{R_{l}}\right)
_{MP}\nabla _{R_{l}}V\left( \mathbf{R}\right) }{\hat{\varepsilon}_{0M}-\hat{%
\varepsilon}_{0P}}\hat{\delta}_{MP}\left( \mathcal{A}_{0}^{\mathbf{R}%
}\right) _{PN}+H.C.\end{aligned}$$ and the projected variables : $$\begin{aligned}
\left( \emph{A}^{\mathbf{R}}\right) _{N} &=&\left( \mathcal{A}^{\mathbf{R}%
}\right) _{NN}=\left( \hbar \mathcal{A}_{0}^{\mathbf{R}}\right) _{MN}+\frac{%
\hbar ^{2}}{8}\left( \mathcal{A}_{0}^{R_{l}}\right) _{MP}\nabla
_{R_{l}}\left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{PN} \\
&&+\frac{\hbar ^{2}}{2}\left( \left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{NP}\left( \mathcal{A}_{0}^{R_{l}}\right) _{PN}+\left( \mathcal{A}%
_{0}^{R_{l}}\right) _{NP}\left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{PN}\right) \hat{\delta}_{PN}\frac{\nabla _{R_{l}}V\left( \mathbf{R}\right)
}{\hat{\varepsilon}_{0P}-\hat{\varepsilon}_{0N}} \\
&&+H.C. \\
\left( \emph{A}^{\mathbf{K}}\right) _{N} &=&0\end{aligned}$$ Therefore the Hamiltonian is : $$\begin{aligned}
H_{d} &=&\hat{\varepsilon}_{0}\left( \mathbf{k}\right) +V\left( \mathbf{r}%
\right) +M\left[ \varepsilon _{0}\right] \\
&=&\hat{\varepsilon}_{0}+V\left( \mathbf{r}\right) -\frac{i\hbar ^{2}}{8}%
\mathcal{P}_{+}\left\{ \left[ \left[ \hat{\varepsilon}_{0},\mathcal{A}%
_{0}^{R_{m}}\right] \left[ X\mathcal{,}K_{m}\right] \right] -\left[ \left[
\hat{\varepsilon}_{0},\left[ X\mathcal{,}K_{m}\right] \right] \mathcal{A}%
_{0}^{R_{m}}\right] \right\} +H.C.\end{aligned}$$ with : $$X=\left[ .,\hat{\varepsilon}_{0}\right] ^{-1}.\left( \mathcal{P}_{-}\left\{
\frac{1}{2}\mathcal{A}_{0}^{R_{l}}\nabla _{R_{l}}V\left( \mathbf{R}\right)
+H.C.\right\} \right)$$ which reduces to $$\begin{aligned}
H_{N} &=&\hat{\varepsilon}_{0}+V\left( \mathbf{r}\right) +\frac{\hbar ^{2}}{8%
}\left[ \sum_{M}\left( \mathcal{A}_{0}^{R_{l}}\right) _{NM}\left( \mathcal{A}%
_{0}^{R_{m}}\right) _{MN}\right] \nabla _{R_{m}}\nabla _{R_{l}}V\left(
R\right) \notag \\
&&+\frac{\hbar ^{2}}{8}\left[ \varepsilon _{0}\left( \mathbf{x}\right) ,%
\mathcal{A}_{0}^{R_{m}}\right] _{NM}\frac{\left( \mathcal{A}%
_{0}^{R_{l}}\right) _{MN}\nabla _{R_{m}}\nabla _{R_{l}}V\left( R\right) }{%
\hat{\varepsilon}_{0M}-\hat{\varepsilon}_{0N}}+H.C. \label{H1Darwin}\end{aligned}$$ This is an interaction in second derivative of the potential, Darwin type.
The effective in-bands Hamiltonian Eq. $\left( \ref{H1P}\right) $ is the desired result concerning one Bloch electron in an external electromagnetic potential and will be very useful for the latter computation of the effective Hamiltonian of several Bloch electrons in self-interaction. But at this point it necessary to compare if it is possible with previous approaches in particular with Blount’s one.
#### Comparison with previous results
In a powerful series of paper, Blount [@Blount][@Blount1] designed a method to derive the diagonalized Hamiltonian both for a Bloch electron in a constant magnetic field and the Dirac electron in an arbitrary magnetic field as series expansion in powers of the fields. In both cases, at the first order in $\hbar $ our results coincide with his ones. At the second order, the direct comparison for the Bloch electron is difficult to do, since Blount’s choice of variables (the canonical ones) differs from ours. However, the comparison in the particular case of the Dirac electron (2 bands Hamiltonian) can be performed and will be explained in detail in [PIERREDIRAC]{}. The result is that, despite some important similarities, our results differ slightly From Blount’s ones. Actually some mistake arise in his results due to his choice of variables, which induces wrong expressions for the Berry curvature, and as a consequence, in the Hamiltonian. As a consistency check we have shown in [@PIERREDIRAC], that at order $\frac{1%
}{m^{3}}$, we recover the usual Foldy-Wouthuysen formula for diagonalized Hamiltonian of the Dirac electron in an electromagnetic field. This is apparently not the case in [@Blount1] where the coefficient of the squared magnetic field is apparently incorrect.
Interacting Bloch electrons
===========================
Having found the diagonalized Hamiltonian of a one particle system, we can now focus on the multiparticles Bloch electrons in interaction. In this context we must consider the interaction mediated by the Coulomb interaction. Although this last one dominates the magnetic interactions, it is known that, for material whose electrons have a non vanishing magnetic moment, new effects such as interaction between the moments mediated by the magnetic field can have important effects. For this reason, we are now interested in investigating the diagonalization of the Hamiltonian of several electrons in self-interaction through the full electromagnetic field. However, for practical reasons and to deal with tractable formula, we will restrict ourselves to the case of relatively small interaction.
Derivation of the microscopic Hamiltonian
-----------------------------------------
To start , we will derive the microscopic classical Hamiltonian for this system, before going to the quantum version. Note that the electromagnetic field will always be considered as classical and only the particles will be treated at the quantum level. Let us introduce the non relativistic Lagrangian of $P$ particles self-interacting through the electromagnetic field and moving in a periodic potential $V_{p}$ which is the only external potential: $$L=\frac{1}{2}\sum_{\alpha }m^{\left( \alpha \right) }\left( \mathbf{\dot{R}}%
^{\left( \alpha \right) }\right) ^{2}-V_{p}\left( \mathbf{R}^{\left( \alpha
\right) }\right) -\frac{1}{2}F_{\mu \nu }F^{\mu \nu }+\int d^{3}x\mathbf{J}.%
\mathbf{A-}\int d^{3}xeA_{0}J_{0}$$ Here $F_{\mu \nu }$ is a classical electromagnetic field whose potential is $%
\left( \mathbf{A,}A_{0}\right) $, $e$ are the particles charges. The current $\mathbf{J}$ is given by $\mathbf{J}\left( \mathbf{x}\right) \mathbf{=}%
\sum_{\alpha }e\delta \left( \mathbf{x-R}^{\left( \alpha \right) }\right)
\mathbf{\dot{R}}^{\left( \alpha \right) }$ and $J_{0}$ is the density of charges, $J_{0}=\sum_{\alpha }e\delta \left( \mathbf{x-R}^{\left( \alpha
\right) }\right) $.
Choosing a gauge for the electromagnetic field, for example the Coulomb gauge (see [@Weinberg] ), some computations lead to write the particles and field Hamiltonian as $$\begin{aligned}
H &=&\sum_{\alpha =1}^{P}\mathbf{P}^{\left( \alpha \right) }\mathbf{\dot{R}}%
^{\left( \alpha \right) }-\frac{1}{2}\sum_{\alpha }m^{\left( \alpha \right)
}\left( \mathbf{\dot{R}}^{\left( \alpha \right) }\right) ^{2}-\int d^{3}x%
\mathbf{J}.\mathbf{A+}\int d^{3}xeA_{0}J_{0}+\Pi _{em}^{2}+\frac{1}{2}\left(
\mathbf{\nabla \times A}\right) ^{2} \\
&&-\frac{1}{2}\left( \Pi +\mathbf{\nabla }A_{0}\right) ^{2}+V_{p}\left(
\mathbf{R}^{\left( \alpha \right) }\right)\end{aligned}$$ with $\Pi _{em}$ the solenoidal part of the electromagnetic momentum defined by $\Pi _{em}=\dot{A}$ satisfying the constraint $\mathbf{\nabla }.\Pi
_{em}=0$ [@Weinberg]. $\mathbf{P}^{\left( \alpha \right) }$ is the usual canonical particle momentum. Using also our definition for $\mathbf{J}$, we are thus led to : $$\begin{aligned}
H &=&\sum_{\alpha =1}^{P}\frac{\left( \mathbf{P}^{\left( \alpha \right) }-e%
\mathbf{A}^{\left( \alpha \right) }\left( \mathbf{R}^{\left( \alpha \right)
}\right) \right) ^{2}}{2m^{\left( \alpha \right) }}\mathbf{+}\frac{1}{2}\int
d^{3}\mathbf{x}e\left[ A_{0}\left( \mathbf{x}\right) J_{0}\left( \mathbf{x}%
\right) +\frac{1}{2}\Pi _{em}^{2}+\frac{1}{2}\left( \mathbf{\nabla \times A}%
\right) ^{2}-\left( \mathbf{\nabla }A_{0}\right) ^{2}\right] \notag \\
&&+V_{p}\left( \mathbf{R}^{\left( \alpha \right) }\right) \notag \\
&=&\sum_{\alpha =1}^{P}\frac{\left( \mathbf{P}^{\left( \alpha \right) }-e%
\mathbf{A}^{\left( \alpha \right) }\left( \mathbf{R}^{\left( \alpha \right)
}\right) \right) ^{2}}{2m^{\left( \alpha \right) }}\mathbf{+}\frac{1}{2}\int
d^{3}\mathbf{x}eA_{0}\left( \mathbf{x}\right) \sum_{\alpha }e\delta \left(
\mathbf{x-R}^{\left( \alpha \right) }\right) \notag \\
&&+\frac{1}{2}\int d^{3}x\left[ \Pi _{em}^{2}+\frac{1}{2}\int d^{3}x\left(
\mathbf{\nabla \times A}\right) ^{2}-\left( \mathbf{\nabla }A_{0}\right) ^{2}%
\right] +V_{p}\left( \mathbf{R}^{\left( \alpha \right) }\right) \label{HP}\end{aligned}$$ Recall that $V_{p}\left( \mathbf{R}^{\left( \alpha \right) }\right) $ stands for the periodic potential. Note also that, given that the particle density is a sum of delta function centered around the particles positions, the potential $\frac{1}{2}\int d^{3}\mathbf{x}eA_{0}\left( \mathbf{x}\right)
\sum_{\alpha }e\delta \left( \mathbf{x-R}^{\left( \alpha \right) }\right) $ reduces to : $$\frac{1}{2}\int d^{3}\mathbf{x}eA_{0}\left( \mathbf{x}\right) \sum_{\alpha
}e\delta \left( \mathbf{x-R}^{\left( \alpha \right) }\right) =\sum_{\alpha
\neq \beta }\frac{e^{2}}{8\pi \left\vert \mathbf{R}^{\left( \alpha \right) }-%
\mathbf{R}^{\left( \beta \right) }\right\vert }$$ However, we will not use this solved form right now and rather keep the electromagnetic field $%
A_{0}\left( \mathbf{x}\right) $ (see below).
The previous expression is desired Hamiltonian for $P$ classical electrons in interaction and in a periodic potential. It will be the starting point for the quantification and for the computation of the effective in-bands Hamiltonian to the second order in $\hbar .$ This order is required to determine the magnetization-magnetization interaction in the same way that the spin-spin interaction in the Breit Hamiltonian for Dirac particles [BREIT]{}. The consideration of interacting electrons, each of them being constrained to be in a single band, will require the Hamiltonian diagonalization and the projection on each respective band. To do so we just have to apply the method developer for the one particle scheme.
Remark, at this point that the Hamiltonian could be further simplified by using the Coulomb gauge condition which allows to replace directly the potential $A_{0}$. Actually, as a consequence of the Coulomb gauge constraint one has $-\mathbf{\nabla }^{2}A_{0}=J_{0}$. However, for the sake of symmetry we aim at treating the electrostatic potential as an external field, as we do for the vector potential, and to replace it after transformation. We will explain later why this is in fact simpler and innocuous for the final result.
Diagonalization process
-----------------------
We will now diagonalize our Hamiltonian to the second order in $\hbar $ as before. The space of states acting is the tensor product of $P$ copies of the individual spaces $\mathcal{V}=\otimes _{\alpha =1,...,P}\mathcal{V}%
_{\alpha }$ with the Hamiltonian Eq. $\left( \ref{HP}\right) $ in which $%
V_{p}$ is the periodic potential, $\mathbf{A}^{\left( \alpha \right) }$ is the potential created by the other charges on charge $\alpha $ and $v\left(
\mathbf{R}^{\left( \alpha \right) }\right) =\int d^{3}\mathbf{x}eA_{0}\left(
\mathbf{x}\right) e\delta \left( \mathbf{x-R}^{\left( \alpha \right)
}\right) $ the electrostatic potential involving the particle $\alpha $.
The Hamiltonian $H$ is diagonalized at the second order in $\hbar $ in a straightforward way. Using our previous experience with the one electron, consider : $$U_{0}=\otimes _{\alpha =1...P}U^{\left( \alpha \right) }$$ Where $U^{\left( \alpha \right) }$ are the individual particle diagonalization matrix found in the previous section to the first order in $%
\hbar $. Note that this matrix do no act on the electromagnetic field, whose integration variable $x$ is independent from the quantum mechanical operators $\mathbf{R}^{\left( \alpha \right) }$, $\mathbf{P}^{\left( \alpha
\right) }$. This is the usual procedure for this kind of system. We first diagonalize the part of the Hamiltonian which describes the particles dynamics and leave the electromagnetic field untouched (see for instance Feynman’s book for this procedure [@FEYNMANBOOK]). This is essential in order to have each particle living in a particular energy band. It means that we assume here an adiabatic process in which the electromagnetic interaction does not cause interband jump. If we were eliminating straight the electromagnetic field in the initial Hamiltonian Eq. $\left( \ref{HP}%
\right) $ with the help of the Maxwell equations, we would of course get an equivalent Hamiltonian for particles only, but it would automatically mix all the energy bands. Then the possibility to assume an adiabatic process and to project each individual particles Hamiltonian on a specific band would be lost.
Therefore we will choose the first route and after the diagonalization one will solve the Maxwell equations which will determine the exact form of the particle currents. These last ones will then be further injected into the Hamiltonian to get the final in-bands effective Hamiltonian of the particles. Let us show how this procedure works.
From Eqs. $\left( \ref{H1P}\right) \left( \ref{HP}\right) $ the Hamiltonian in the diagonal representation of the particles part is thus directly : $$\begin{aligned}
H_{N} &=&\sum_{\alpha }\hat{\varepsilon}_{0N^{\left( \alpha \right)
}}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha \right)
}\right) +\sum_{\alpha }v\left( \mathbf{r}^{\left( \alpha \right) }\right) -%
\frac{1}{2}\sum_{\alpha }\left( \mathbf{\mu }\left( \mathbf{x}^{\left(
\alpha \right) }\right) \mathbf{.B}^{\left( \alpha \right) }\left( \mathbf{x}%
^{\left( \alpha \right) }\right) +\mathbf{B}^{\left( \alpha \right) }\left(
\mathbf{x}^{\left( \alpha \right) }\right) \mathbf{.\mu }\left( \mathbf{x}%
\right) \right) \\
&&-\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right) \mathbf{.}\nabla _{R_{k}}%
\mathbf{B}^{\left( \alpha \right) }\left( \mathbf{x}^{\left( \alpha \right)
}\right) +\mathbf{B}^{\left( \alpha \right) }\mathbf{.\tilde{\mu}.B}^{\left(
\alpha \right) } \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\left[ \sum_{M^{\left( \alpha \right)
}}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right) _{N^{\left(
\alpha \right) }M^{\left( \alpha \right) }}\left( \mathcal{A}%
_{0}^{R_{m}^{\left( \alpha \right) }}\right) _{M^{\left( \alpha \right)
}N^{\left( \alpha \right) }}\right] \nabla _{R_{m}^{\left( \alpha \right)
}}\nabla _{R_{l}^{\left( \alpha \right) }}v\left( \mathbf{r}^{\left( \alpha
\right) }\right) \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\frac{\left[ \left[ \varepsilon
_{0}^{\left( \alpha \right) }\left( \mathbf{x}\right) ,\mathcal{A}%
_{0}^{R_{m}^{\left( \alpha \right) }}\right] _{N^{\left( \alpha \right)
}M^{\left( \alpha \right) }}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha
\right) }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha \right)
}}\nabla _{R_{m}^{\left( \alpha \right) }}\nabla _{R_{l}^{\left( \alpha
\right) }}\right] v\left( \mathbf{r}^{\left( \alpha \right) }\right) }{\hat{%
\varepsilon}_{0M^{\left( \alpha \right) }}^{\left( \alpha \right) }-\hat{%
\varepsilon}_{0N^{\left( \alpha \right) }}} \\
&&+\frac{1}{2}\int d^{3}x\left[ \Pi _{em}^{2}+\frac{1}{2}\int d^{3}x\left(
\mathbf{\nabla \times A}\right) ^{2}-\left( \mathbf{\nabla }A_{0}\right) ^{2}%
\right]\end{aligned}$$ The notations of the previous section have been kept, adding only the superscript $\left( \alpha \right) $ to label the particles in Eq. $\left( %
\ref{H1P}\right) $. For example, $\mathbf{x}^{\left( \alpha \right) }=\left(
\mathbf{r}^{\left( \alpha \right) },\mathbf{k}^{\left( \alpha \right)
}\right) $ is the couple of dynamical variables for the particle $\left(
\alpha \right) $, $\mathbf{x}^{\left( \alpha \right) }=\left( \mathbf{R}%
^{\left( \alpha \right) }+\emph{A}_{N^{\left( \alpha \right) }}^{\mathbf{R}%
\left( ^{\alpha }\right) },\mathbf{K}^{\left( \alpha \right) }+\emph{A}%
_{N^{\left( \alpha \right) }}^{\mathbf{K}^{\left( \alpha \right) }}\right) $ and the Berry phases $\emph{A}_{N^{\left( \alpha \right) }}^{\mathbf{R}%
\left( ^{\alpha }\right) }$, $\emph{A}_{N^{\left( \alpha \right) }}^{\mathbf{%
K}^{\left( \alpha \right) }}$ involved in these operators have been computed in the single particle section. Notice also that $N$ is a shorthand for the multi-index $N=\left( N^{\left( \alpha \right) }\right) _{\alpha =1,...P}$ recalling that the Hamiltonian describes $P$ particles respectively in the bands $N^{\left( 1\right) }...N^{\left( P\right) }$. Ultimately $\mathbf{B}%
^{\left( \alpha \right) }$ is the field acting on particle $\alpha $.
Effective Hamiltonian for $P$ particles and particles currents:
---------------------------------------------------------------
Our aim is now to replace the electromagnetic field in the Hamiltonian as a function of the dynamical variables of the particles system. We first do so by solving the Hamiltonian equations for the electromagnetic field in the limit of relatively weak interactions.
### Dynamics for the Electromagnetic Field
To get rid of the electromagnetic field and writing an interaction Hamiltonian for the particles, we first focus on the vector potential and start with the Hamiltonian equation for the electromagnetic field in the Coulomb gauge [@Weinberg] :
$$\begin{aligned}
\mathbf{\dot{A}}\left( \mathbf{x},t\right) &=&\Pi _{em} \\
\dot{\Pi}_{em}^{i}\left( \mathbf{x},t\right) &=&-\int d^{3}y\left[ \delta
^{ij}\delta ^{3}\left( \mathbf{x-y}\right) +\frac{\partial ^{2}}{\partial
x^{i}\partial x^{j}}\frac{1}{4\pi \left\vert \mathbf{x-y}\right\vert }\right]
\mathbf{\nabla }\times \mathbf{\nabla }A^{j}\left( \mathbf{y},t\right) \\
&&+\frac{i}{\hbar }\left[ H_{mat},\dot{\Pi}_{em}^{i}\left( \mathbf{x}%
,t\right) \right] \\
&=&-\int d^{3}y\left[ \delta ^{ij}\delta ^{3}\left( \mathbf{x-y}\right) +%
\frac{\partial ^{2}}{\partial x^{i}\partial x^{j}}\frac{1}{4\pi \left\vert
\mathbf{x-y}\right\vert }\right] \mathbf{\nabla }\times \mathbf{\nabla }%
A^{j}\left( \mathbf{y},t\right) \\
&&-\int d^{3}y\left[ \delta ^{ij}\delta ^{3}\left( \mathbf{x-y}\right) +%
\frac{\partial ^{2}}{\partial x^{i}\partial x^{j}}\frac{1}{4\pi \left\vert
\mathbf{x-y}\right\vert }\right] \frac{\delta }{\delta A^{j}\left( \mathbf{%
x,t}\right) }H_{mat}\end{aligned}$$
where $H_{mat}$ denotes the matter part (that is excluding the free part of the electromagnetic field Hamiltonian). Using the gauge constraints : $%
\mathbf{\nabla }.\mathbf{\Pi }_{em}=\mathbf{\nabla }.\mathbf{A}=0$. we are led to : $$\square \mathbf{A}\left( \mathbf{\left( \mathbf{x},t\right) }\right) =-\int
d^{3}y\left[ \delta ^{ij}\delta ^{3}\left( \mathbf{x-y}\right) +\frac{%
\partial ^{2}}{\partial x^{i}\partial x^{j}}\frac{1}{4\pi \left\vert \mathbf{%
x-y}\right\vert }\right] \frac{\delta }{\delta A^{j}\left( \mathbf{y,t}%
\right) }H_{mat}$$ where $\frac{\delta }{\delta A^{i}\left( \mathbf{y,t}\right) }$ stands for the functional derivative with respect to $A^{i}\left( \mathbf{y,t}\right) $ .
### Explicit expression for the current
We aim now at writing a more explicit formula for the effective current $%
j^{j}\mathbf{\left( \mathbf{y},t\right) =-}\frac{\delta }{\delta A^{j}\left(
\mathbf{x,t}\right) }H_{mat}$. To do so, we decompose $H_{mat}$ as : $$\begin{aligned}
H_{mat} &=&\sum_{\alpha }\hat{\varepsilon}_{0N^{\left( \alpha \right)
}}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha \right)
}\right) +\sum_{\alpha }v\left( \mathbf{r}^{\left( \alpha \right) }\right) \\
&&-\frac{1}{2}\sum_{\alpha }\left( \mathbf{\mu }\left( \mathbf{x}^{\left(
\alpha \right) }\right) \mathbf{.B}^{\left( \alpha \right) }\left( \mathbf{x}%
^{\left( \alpha \right) }\right) +\mathbf{B}^{\left( \alpha \right) }\left(
\mathbf{x}^{\left( \alpha \right) }\right) \mathbf{.\mu }\left( \mathbf{x}%
\right) \right) -\sum_{\alpha }\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}%
\right) \mathbf{.}\nabla _{R_{k}}\mathbf{B}^{\left( \alpha \right) }\left(
\mathbf{x}^{\left( \alpha \right) }\right) \\
&&+\mathbf{B}^{\left( \alpha \right) }\mathbf{.\tilde{\mu}.B}^{\left( \alpha
\right) }+D^{\left( \alpha \right) }\end{aligned}$$ where $D^{\left( \alpha \right) }$ stands for the Darwin term : $$\begin{aligned}
D^{\left( \alpha \right) } &=&\frac{\hbar ^{2}}{4}\left[ \sum_{M^{\left(
\alpha \right) }}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right)
}}\right) _{N^{\left( \alpha \right) }M^{\left( \alpha \right) }}\left(
\mathcal{A}_{0}^{R_{m}^{\alpha }}\right) _{M^{\left( \alpha \right)
}N^{\left( \alpha \right) }}\right] \nabla _{R_{m}^{\left( \alpha \right)
}}\nabla _{R_{l}^{\left( \alpha \right) }}v\left( \mathbf{r}^{\left( \alpha
\right) }\right) \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\frac{\left[ \left[ \varepsilon
_{0}^{\left( \alpha \right) }\left( \mathbf{x}\right) ,\mathcal{A}%
_{0}^{R_{m}^{\left( \alpha \right) }}\right] _{N^{\left( \alpha \right)
}M^{\left( \alpha \right) }}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha
\right) }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha \right)
}}\nabla _{R_{m}^{\left( \alpha \right) }}\nabla _{R_{l}^{\left( \alpha
\right) }}\right] v\left( \mathbf{r}^{\left( \alpha \right) }\right) }{\hat{%
\varepsilon}_{0M^{\left( \alpha \right) }}^{\left( \alpha \right) }-\hat{%
\varepsilon}_{0N^{\left( \alpha \right) }}}\end{aligned}$$ The computation of $j^{j}\mathbf{\left( \mathbf{y},t\right) =-}\frac{\delta
}{\delta A^{j}\left( \mathbf{x,t}\right) }H_{mat}$ has to be performed carefully since the dependence in the gauge field in $H_{mat}$ is intricate. We aim at computing it as a function of the physical variables $\mathbf{r}^{\left( \alpha \right) }$, $\mathbf{\pi }^{\left( \alpha \right) }$ and not as a function of the initial microscopic canonical variables. However, the physical variables $%
\mathbf{r}^{\left( \alpha \right) }$, $\mathbf{\pi }^{\left( \alpha \right)
} $ depend on the value of the fields evaluated at the microscopic positions and momentum. $$\begin{aligned}
\mathbf{r}^{\left( \alpha \right) } &=&\mathbf{R}^{\left( \alpha \right) }%
\mathbf{+}\emph{A}_{N}^{\mathbf{R}^{\left( \alpha \right) }}\left( \mathbf{R}%
^{\left( \alpha \right) }\mathbf{,\Pi }^{\left( \alpha \right) }\right) \\
\mathbf{\pi }^{\left( \alpha \right) } &=&\mathbf{\Pi }^{\left( \alpha
\right) }+e\emph{A}_{N}^{\mathbf{R}^{\left( \alpha \right) }}\left( \mathbf{R%
}^{\left( \alpha \right) }\mathbf{,\Pi }^{\left( \alpha \right) }\right)
\times \mathbf{B}\left( \mathbf{R}^{\left( \alpha \right) }\right) \\
&&+e\frac{\hbar }{4}\mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{k}^{\left(
\alpha \right) }}\right] \left( \emph{A}_{N}^{\mathbf{R}^{\left( \alpha
\right) }}\left( \mathbf{R}^{\left( \alpha \right) }\mathbf{,\Pi }^{\left(
\alpha \right) }\right) \times \nabla _{R_{k}^{\left( \alpha \right) }}%
\mathbf{B}\left( \mathbf{R}^{\left( \alpha \right) }\right) +H.C.\right)\end{aligned}$$ to turn this difficulty we reintroduce the variables $\mathbf{r}^{\left(
\alpha \right) }$, $\mathbf{\pi }^{\left( \alpha \right) }$ recursively in our Berry phases.
Starting with $\mathbf{r}^{\left( \alpha \right) }$, at our order of approximation and using $\mathbf{R}^{\left( \alpha \right) }=\mathbf{\mathbf{%
r}^{\left( \alpha \right) }-}\emph{A}_{N}^{\mathbf{R}^{\left( \alpha \right)
}}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{,\pi }^{\left( \alpha
\right) }\right) $ : $$\begin{aligned}
\mathbf{r}^{\left( \alpha \right) } &=&\mathbf{R}^{\left( \alpha \right) }%
\mathbf{+}\emph{A}_{N}^{\mathbf{R}^{\left( \alpha \right) }}\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{,\pi }^{\left( \alpha \right) }\right)
-\left( \emph{A}_{N}^{\mathbf{\Pi }^{\left( \alpha \right) }}.\mathbf{\nabla
}_{\mathbf{\pi }^{\left( \alpha \right) }}\right) \emph{A}_{N}^{\mathbf{R}%
}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{,\pi }^{\left( \alpha
\right) }\right) \\
&\equiv &\mathbf{R}^{\left( \alpha \right) }\mathbf{+}\tilde{\emph{A}}^{%
\mathbf{R}^{\left( \alpha \right) }}\left( \mathbf{r}^{\left( \alpha \right)
}\mathbf{,\pi }^{\left( \alpha \right) }\right)\end{aligned}$$ (we have neglected the terms in $\mathbf{\nabla }_{\mathbf{r}^{\left( \alpha
\right) }}$ that do not appear at order $\hbar ^{2}$ and skipped the index $%
N $ for the sake of simplicity). We do the same thing for the momentum by starting to write it as a function of $\mathbf{k}^{\left( \alpha \right) }$, $\mathbf{r}^{\left( \alpha \right) }$. Given the definition of the momentum, we explained before that it was not given by $\mathbf{k}^{\left( \alpha
\right) }-e\mathbf{A}\left( \mathbf{r}^{\left( \alpha \right) }\right) $, but rather $\mathbf{k}^{\left( \alpha \right) }-e\mathbf{A}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) -\frac{\hbar ^{2}}{8}e\nabla _{R_{k}}%
\mathbf{\nabla }A_{l}\left( \mathbf{R},\mathbf{P}\right) \mathcal{P}%
_{N}\left( \mathcal{A}_{0}^{R_{l}}\mathcal{A}_{0}^{R_{k}}+\mathcal{A}%
_{0}^{R_{k}}\mathcal{A}_{0}^{R_{l}}\right) $. It was explained that this variable had a physical meaning (the subtracted term was there to compensate a non gauge invariant term). This last term cancels one term in the Berry phase for $\mathbf{K}^{\left( \alpha \right) }$. As a consequence, $$\begin{aligned}
\mathbf{\pi }^{\left( \alpha \right) } &=&\mathbf{k}^{\left( \alpha \right)
}-e\mathbf{A}\left( \mathbf{r}^{\left( \alpha \right) }\right) =\mathbf{K}%
^{\left( \alpha \right) }+\emph{A}_{N}^{\mathbf{K}^{\left( \alpha \right)
}}\left( \mathbf{R}^{\left( \alpha \right) }\mathbf{,\Pi }^{\left( \alpha
\right) }\right) -e\mathbf{A}\left( \mathbf{r}^{\left( \alpha \right)
}\right) \\
&&-\frac{\hbar ^{2}}{8}e\nabla _{R_{k}}\mathbf{\nabla }A_{l}\left( \mathbf{R}%
,\mathbf{P}\right) \mathcal{P}_{N}\left( \mathcal{A}_{0}^{R_{l}}\mathcal{A}%
_{0}^{R_{k}}+\mathcal{A}_{0}^{R_{k}}\mathcal{A}_{0}^{R_{l}}\right) \\
&=&\mathbf{K}^{\left( \alpha \right) }-e\mathbf{A}\left( \mathbf{r}^{\left(
\alpha \right) }\right) +\frac{1}{2}e\left[ \mathbf{\nabla }A_{l}\left(
\mathbf{R}^{\left( \alpha \right) }\right) \emph{A}_{N}^{R_{l}}\left(
\mathbf{R}^{\left( \alpha \right) }\mathbf{,\Pi }^{\left( \alpha \right)
}\right) +\emph{A}_{N}^{R_{l}}\left( \mathbf{R}^{\left( \alpha \right) }%
\mathbf{,\Pi }^{\left( \alpha \right) }\right) \mathbf{\nabla }A_{l}\left(
\mathbf{R}^{\left( \alpha \right) }\right) \right] \\
&&+\frac{\hbar ^{2}}{4}e\mathbf{\nabla }\nabla _{R_{k}}A_{l}\left( \mathbf{R}%
,\mathbf{P}\right) \left( \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{l}}%
\right] \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{k}}\right] +\mathcal{P}_{N}%
\left[ \mathcal{A}_{0}^{R_{k}}\right] \mathcal{P}_{N}\left[ \mathcal{A}%
_{0}^{R_{l}}\right] \right) \\
&=&\mathbf{K}^{\left( \alpha \right) }-e\mathbf{A}\left( \mathbf{r}^{\left(
\alpha \right) }\right) +\frac{1}{2}e\left[ \mathbf{\nabla }A_{l}\left(
\mathbf{R}^{\left( \alpha \right) }\right) \tilde{\emph{A}}^{R_{l}}\left(
\mathbf{r}^{\left( \alpha \right) }\mathbf{,\pi }^{\left( \alpha \right)
}\right) +\tilde{\emph{A}}^{R_{l}}\left( \mathbf{r}^{\left( \alpha \right) }%
\mathbf{,\pi }^{\left( \alpha \right) }\right) \mathbf{\nabla }A_{l}\left(
\mathbf{R}^{\left( \alpha \right) }\right) \right] \\
&&-\frac{\hbar ^{2}}{4}e\mathbf{\nabla }\nabla _{R_{k}}A_{l}\left( \mathbf{R}%
,\mathbf{P}\right) \left( \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{l}}%
\right] \mathcal{P}_{N}\left[ \mathcal{A}_{0}^{R_{k}}\right] +\mathcal{P}_{N}%
\left[ \mathcal{A}_{0}^{R_{k}}\right] \mathcal{P}_{N}\left[ \mathcal{A}%
_{0}^{R_{l}}\right] \right) \\
&\equiv &\mathbf{K}^{\left( \alpha \right) }-e\mathbf{A}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) +\frac{1}{2}e\left[ \mathbf{\nabla }%
A_{l}\left( \mathbf{R}^{\left( \alpha \right) }\right) \tilde{\emph{A}}%
^{R_{l}}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{,\pi }^{\left(
\alpha \right) }\right) +\tilde{\emph{A}}^{R_{l}}\left( \mathbf{r}^{\left(
\alpha \right) }\mathbf{,\pi }^{\left( \alpha \right) }\right) \mathbf{%
\nabla }A_{l}\left( \mathbf{R}^{\left( \alpha \right) }\right) \right] \\
&&+\hbar ^{2}e\mathbf{\nabla }\nabla _{R_{k}}A_{l}\left( \mathbf{r}^{\left(
\alpha \right) }\right) \tilde{\emph{A}}_{lk}\end{aligned}$$ Actually, as seen on their definition, the variables $\mathbf{\pi }^{\left(
\alpha \right) }$ and $\mathbf{r}^{\left( \alpha \right) }$ depend on $%
\mathbf{A}\left( \mathbf{r}^{\left( \alpha \right) }\right) $. Rather than solving these circular equations where the variables of interest appear on both side as functions of the canonical variables $\mathbf{R}^{\left( \alpha
\right) }$, $\mathbf{P}^{\left( \alpha \right) }$ we rather unsolved the dependence of the variables in $\mathbf{A}$ and keep this circularity to compute the derivatives of interest, since it will allow to express all the results as functions of the variables $\mathbf{r}^{\left( \alpha \right) }$, $\mathbf{\pi }^{\left( \alpha \right) }$.
Keeping this in mind, we start by considering a simple example that will allow to find some general rules for the computation of the functional derivatives with respect to the gauge potential. Let $G$ be a function depending on a variable $x$ (in our context $\mathbf{r}^{\left( \alpha
\right) }$, $\mathbf{\pi }^{\left( \alpha \right) }$) depending itself on a function $a$ of $x$ (here the gauge field) through the relation $x=f\left(
a\left( x\right) \right) $.
We aim at computing first $\frac{\delta }{\delta a\left( y\right) }G\left(
x\right) $ where $y$ is an independent variable (i.e. the space parameter in our context).
Start first with $\frac{\delta }{\delta a\left( y\right) }x=\frac{\delta }{%
\delta a\left( y\right) }f\left( a\left( x\right) \right) $ : $$\frac{\delta }{\delta a\left( y\right) }x=\frac{\delta }{\delta a\left(
y\right) }f\left( a\left( x\right) \right) =\frac{\delta a\left( x\right) }{%
\delta a\left( y\right) }f^{\prime }\left( a\left( x\right) \right)$$ It would be wrong to conclude that $\frac{\delta a\left( x\right) }{\delta
a\left( y\right) }$ reduces to $\delta \left( x-y\right) $ because $y$ is an independent variables but $x$ is not and depends on $a$. One rather has to consider that $a\left( x\right) $ is an infinite series of composition $%
a\left( x\right) =a\left( f\left( a\left( x\right) \right) \right) =f\left(
a\left( f\left( a\left( x\right) \right) \right) \right) =f\left( a\left(
f\left( a\left( f\left( a\left( x\right) \right) \right) \right) \right)
\right) $ and so on. A slight functional variation of $a$ propagates along all the series and has to be taken into account to write symbolically the infinite series : $$\begin{aligned}
\delta a\left( x\right) &=&\delta \left[ a(f(a(f(a(f(a(f(a...\right] =\delta
a(f(a(f(a(f(a(f(a...+a(f(\delta a(f(a(f(a(f(a... \\
&&+a(f(a(f(\delta a(f(a(f(a...+a(f(a(f(a(f(\delta
a(f(a...+a(f(a(f(a(f(a(f(\delta a...+...\end{aligned}$$ where the inserted $\delta $ acts solely on the $a$ directly on its right. The variation $\delta a\left( x\right) $ can be rewritten as : $$\delta a\left( x\right) =\delta a\left( x\right) +a(f(\delta a\left(
x\right) )+a(f(a(f(\delta a\left( x\right) )))+a(f(a(f(a(f(\delta a\left(
x\right) ))))+...$$ where now $x$ can be seen as a frozen variable (that is not depending on $a$) everywhere in the right hand side. A direct application of the chain rule yields directly : $$\frac{\delta a\left( x\right) }{\delta a\left( y\right) }=\delta \left(
x-y\right) \left[ 1+a^{\prime }\left( x\right) f^{\prime }\left( a\left(
x\right) \right) +\left( a^{\prime }\left( x\right) f^{\prime }\left(
a\left( x\right) \right) \right) ^{2}+...\right]$$ so that : $$\frac{\delta }{\delta a\left( y\right) }x=f^{\prime }\left( a\left( x\right)
\right) \frac{\delta a\left( x\right) }{\delta a\left( y\right) }=\delta
\left( x-y\right) f^{\prime }\left( a\left( x\right) \right) \left[
1+a^{\prime }\left( x\right) f^{\prime }\left( a\left( x\right) \right)
+\left( a^{\prime }\left( x\right) f^{\prime }\left( a\left( x\right)
\right) \right) ^{2}+...\right]$$ and ultimately : $$\frac{\delta }{\delta a\left( y\right) }G\left( x\right) =\delta \left(
x-y\right) G^{\prime }\left( x\right) f^{\prime }\left( a\left( x\right)
\right) \left[ 1+a^{\prime }\left( x\right) f^{\prime }\left( a\left(
x\right) \right) +\left( a^{\prime }\left( x\right) f^{\prime }\left(
a\left( x\right) \right) \right) ^{2}+...\right]$$ Of course, this expansion is formal, and assumed to converge. Moreover, in the sequel the sum will always be truncated to a finite order.
We can now translate this results in our context to compute the derivative $%
\mathbf{-}\frac{\delta }{\delta A^{j}\left( \mathbf{x,t}\right) }H_{mat}$ since it involves only a generalization to several variables. The formula are a bit more involved since the circularity depends on three variables. Actually $\mathbf{r}^{\left( \alpha \right) }$ depends on $\mathbf{\pi }%
^{\left( \alpha \right) }$ through the Berry phase, $\mathbf{\pi }^{\left(
\alpha \right) }$ depends on $\mathbf{A}$, that depends on $\mathbf{r}%
^{\left( \alpha \right) }$. products like $a^{\prime }\left( x\right)
f^{\prime }\left( a\left( x\right) \right) $ will be now replaced by products of three types of derivatives, $\frac{d\mathbf{r}^{\left( \alpha
\right) }}{d\mathbf{A}}$, $\frac{d\mathbf{\pi }^{\left( \alpha \right) }}{d%
\mathbf{A}}$, $\frac{\partial \mathbf{A}}{\partial \mathbf{r}^{\left( \alpha
\right) }}=$ $\mathbf{\nabla A}$. However, a simplification arises here. Actually, since the beginning, we have assumed that the magnetic part of the interaction between the particles is relatively weak with respect to the electrostatic potential. We will thus assume that the current is relatively weak, and the expansion will be performed only at the first order in the field. As a consequence, at this order of approximation, it is useful for the sequel to note that the composition series for $\frac{\delta A^{k}\left(
\mathbf{r}^{\left( \alpha \right) }\mathbf{,t}\right) }{\delta A^{i}\left(
\mathbf{x,t}\right) }$ reduces to $\left[ \delta _{i}^{k}+\nabla
_{l}A^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right) \frac{%
dr_{l}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left( \alpha
\right) }\right) }\right] $.
Introducing the needed indices, taking into account this approximation and going to the second order in $\hbar $ we are led to the following results for the various terms involved in $\mathbf{-}\frac{\delta }{\delta
A^{j}\left( \mathbf{x,t}\right) }H_{mat}$ : $$\begin{aligned}
&&-\sum_{\alpha }\frac{\delta \left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha
\right) }\right) +D^{\left( \alpha \right) }\right) }{\delta A^{i}\left(
\mathbf{x,t}\right) } \\
&=&-\sum_{\alpha }\frac{\partial }{\partial \pi _{j}^{\left( \alpha \right)
}\left( \mathbf{x,t}\right) }\left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha
\right) }\right) +D^{\left( \alpha \right) }\right) \left[ \frac{d\pi
_{j}^{\left( \alpha \right) }}{dA^{k}\left( \mathbf{r}^{\left( \alpha
\right) }\right) }\right] \left[ \delta _{i}^{k}+\nabla _{l}A^{k}\left(
\mathbf{r}^{\left( \alpha \right) }\right) \frac{dr_{l}^{\left( \alpha
\right) }}{dA^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }\right]
\delta \left( \mathbf{x-r}^{\left( \alpha \right) }\right)\end{aligned}$$ and $$-\sum_{\alpha }\frac{\delta }{\delta A^{i}\left( \mathbf{x,t}\right) }%
v\left( \mathbf{r}^{\left( \alpha \right) }\right) =-\sum_{\alpha }\frac{%
\partial v\left( \mathbf{r}^{\left( \alpha \right) }\right) }{\partial
r_{j}^{\left( \alpha \right) }}\frac{dr_{j}^{\left( \alpha \right) }}{%
dA^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right) }\left[ \delta
_{i}^{k}+\nabla _{l}A^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right)
\frac{dr_{l}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }\right] \delta \left( \mathbf{x-r}^{\left( \alpha
\right) }\right)$$ and also $$\begin{aligned}
&&-\frac{\delta }{\delta A^{i}\left( \mathbf{x,t}\right) }\left[ -\frac{1}{2}%
\sum_{\alpha }\left( \mathbf{\mu }\left( \mathbf{x}^{\left( \alpha \right)
}\right) \mathbf{.B}^{\left( \alpha \right) }\left( \mathbf{x}^{\left(
\alpha \right) }\right) +\mathbf{B}^{\left( \alpha \right) }\left( \mathbf{x}%
^{\left( \alpha \right) }\right) \mathbf{.\mu }\left( \mathbf{x}\right)
\right) -\sum_{\alpha }\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right)
\mathbf{.}\nabla _{R_{k}}\mathbf{B}^{\left( \alpha \right) }\left( \mathbf{x}%
^{\left( \alpha \right) }\right) \right. \\
&&\left. +\mathbf{B}^{\left( \alpha \right) }\mathbf{.\tilde{\mu}.B}^{\left(
\alpha \right) }\right] \\
&=&\left[ \frac{1}{2}\sum_{\alpha }\left( \left[ \frac{d\pi _{j}^{\left(
\alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }%
\frac{\partial }{\partial \pi _{j}^{\left( \alpha \right) }\left( \mathbf{x,t%
}\right) }+\frac{dr_{j}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) }\frac{\partial }{\partial r_{j}^{\left(
\alpha \right) }}\right] \left( \mathbf{\mu }\left( \mathbf{x}^{\left(
\alpha \right) }\right) +\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right)
\mathbf{.}\nabla _{R_{k}}\right) \right) .\mathbf{B}^{\left( \alpha \right)
}\left( \mathbf{x}^{\left( \alpha \right) }\right) \right] \\
&&\times \delta \left( \mathbf{x-r}^{\left( \alpha \right) }\right) +H.C. \\
&&+\frac{1}{2}\sum_{\alpha }\left[ \left( \mathbf{\mu }\left( \mathbf{x}%
^{\left( \alpha \right) }\right) \times \mathbf{\nabla }\right) _{p}.\left(
\delta _{i}^{p}\mathbf{+}\nabla _{l}A^{p}\left( \mathbf{r}^{\left( \alpha
\right) }\right) \frac{dr_{l}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{%
r}^{\left( \alpha \right) }\right) }\right) \delta \left( \mathbf{x-r}%
^{\left( \alpha \right) }\right) \right. \\
&&\left. +\left[ \mathbf{\hat{\mu}}^{k}\left( \mathbf{x}^{\left( \alpha
\right) }\right) \mathbf{\times }\nabla _{R_{k}}\mathbf{\nabla }\delta
\left( \mathbf{x-r}^{\left( \alpha \right) }\right) \right] _{i}\right] +H.C.
\\
&&-\frac{1}{2}\left[ \mathbf{B}\left( \mathbf{x}^{\left( \alpha \right)
}\right) \mathbf{.\tilde{\mu}}\left( \mathbf{x}^{\left( \alpha \right)
}\right) \times \mathbf{\nabla }\delta \left( \mathbf{x-R}^{\left( \alpha
\right) }\right) -\mathbf{\nabla }\delta \left( \mathbf{x-R}^{\left( \alpha
\right) }\right) \times \mathbf{\tilde{\mu}}\left( \mathbf{x}^{\left( \alpha
\right) }\right) \mathbf{.B}\left( \mathbf{x}^{\left( \alpha \right)
}\right) \right] _{i}\end{aligned}$$ These expressions involve some derivatives that are computed as follows, as directly implied by our previous remarks. $\frac{dr_{l}^{\left( \alpha
\right) }}{dA^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }$, $%
\frac{d\pi _{j}^{\left( \alpha \right) }}{dA^{k}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }$ cover the dependence of the transformed dynamical variables in $A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) $. Recall that they have to be computed such that in $A^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) $, the $\mathbf{r}^{\left( \alpha \right) }$ is frozen, so that $A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) $ is seen as an ordinary, or independent variable.
Starting with $\frac{dr_{l}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) }$ the dependence in the field comes from the Berry phase of $r_{l}^{\left( \alpha \right) }$, **$\tilde{\emph{A}%
}$**$^{R_{l}^{\left( \alpha \right) }}$ which is a function of the two variables $\mathbf{\pi }^{\left( \alpha \right) }$ and $\mathbf{r}^{\left(
\alpha \right) }$. We can thus write : $$\frac{dr_{l}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }=\left( \frac{\partial \mathbf{\tilde{\emph{A}}}%
^{R_{l}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left( \alpha \right) }%
}\frac{d\pi _{n}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }+\frac{\partial \mathbf{\tilde{\emph{A}}}%
^{R_{l}^{\left( \alpha \right) }}}{\partial r_{n}^{\left( \alpha \right) }}%
\frac{dr_{n}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }+\frac{\partial \mathbf{\tilde{\emph{A}}}%
^{R_{l}^{\left( \alpha \right) }}}{\partial A^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }\right)$$ the second term in the right hand side is of order $\hbar ^{3}$ since $\frac{%
\partial \mathbf{\tilde{\emph{A}}}^{R_{l}^{\left( \alpha \right) }}}{%
\partial r_{n}^{\left( \alpha \right) }}$ is of order $\hbar ^{2}$ and $%
\frac{dr_{n}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }$ is of order $\hbar $. At order $\hbar ^{2}$ one thus has : $$\frac{dr_{l}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }=\frac{\partial \mathbf{\tilde{\emph{A}}}%
^{R_{l}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left( \alpha \right) }%
}\frac{d\pi _{n}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }+\frac{\partial \mathbf{\tilde{\emph{A}}}%
^{R_{l}^{\left( \alpha \right) }}}{\partial A^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }$$ Practically, the partial derivative $\frac{\partial \mathbf{\tilde{\emph{A}}}%
^{R_{l}^{\left( \alpha \right) }}}{\partial A^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }$ with respect to the field is obtained by decomposing : $$\begin{aligned}
\frac{\partial \mathbf{\tilde{\emph{A}}}^{R_{l}^{\left( \alpha \right) }}}{%
\partial A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) } &=&\frac{%
\partial \mathbf{\tilde{\emph{A}}}^{R_{l}^{\left( \alpha \right) }}}{%
\partial A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }-\frac{%
\partial }{\partial A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }%
\left( \emph{A}_{N}^{\mathbf{\Pi }^{\left( \alpha \right) }}.\mathbf{\nabla }%
_{\mathbf{\pi }^{\left( \alpha \right) }}\right) \emph{A}_{N}^{R_{l}^{\left(
\alpha \right) }}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{,\pi }%
^{\left( \alpha \right) }\right) \\
&=&\frac{\partial \mathbf{\emph{A}}^{R_{l}^{\left( \alpha \right) }}}{%
\partial A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }-\left(
\mathbf{\nabla }_{\mathbf{r}^{\left( \alpha \right) }}\times \mathbf{\nabla }%
_{\mathbf{\pi }^{\left( \alpha \right) }}\right) \emph{A}_{N}^{R_{l}^{\left(
\alpha \right) }}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{,\pi }%
^{\left( \alpha \right) }\right)\end{aligned}$$ and $\frac{\partial \mathbf{\emph{A}}^{R_{l}^{\left( \alpha \right) }}}{%
\partial A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }$ is carried by the following term computed previously in the one particle case : $$\begin{aligned}
\mathcal{A}_{1N}^{\mathbf{R}^{\left( \alpha \right) }} &=&-\frac{\hbar ^{2}}{%
8}\sum_{P}\left\{ e\mathbf{B}.\left( \left( \mathcal{A}_{0}^{\mathbf{R}%
^{\left( \alpha \right) }}\right) _{NP}\times \nabla _{\mathbf{\Pi }^{\left(
\alpha \right) }}\right) \left( \mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha
\right) }}\right) _{PN}\right\} \\
&&-e\frac{\hbar ^{2}}{4}\left( \mathbf{B}.\left( \left( \mathcal{A}_{0}^{%
\mathbf{R}^{\left( \alpha \right) }}\right) _{NN}\times \nabla _{\mathbf{\Pi
}^{\left( \alpha \right) }}\right) \mathcal{A}_{0NN}^{\mathbf{R}^{\left(
\alpha \right) }}+H.C.\right) \\
&&-\frac{\hbar ^{2}}{2}\left( \sum_{P}\left( e\mathbf{B}.\left( \left(
\mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha \right) }}\right) _{NP}\times
\nabla _{\mathbf{\Pi }^{\left( \alpha \right) }}\right) \right) \frac{\left(
\hat{\varepsilon}_{0P}+\hat{\varepsilon}_{0N}\right) }{\hat{\varepsilon}%
_{0P}-\hat{\varepsilon}_{0N}}\hat{\delta}_{PN}\left( \mathcal{A}_{0}^{%
\mathbf{R}^{\left( \alpha \right) }}\right) _{PN}-H.C.\right) + \\
&&\frac{ie\hbar ^{2}}{8}\sum_{P}\left( \left( \mathcal{A}_{0}^{\mathbf{R}%
^{\left( \alpha \right) }}\right) _{NP}\left( \left( \mathcal{A}_{0}^{%
\mathbf{R}^{\left( \alpha \right) }}\right) _{PN}\times \left( \mathcal{A}%
_{0}^{\mathbf{R}^{\left( \alpha \right) }}\right) _{NN}\right) -\left(
\left( \mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha \right) }}\right)
_{NP}\times \left( \mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha \right)
}}\right) _{PP}\left( \mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha \right)
}}\right) _{PN}\right) .\mathbf{B}\right) \hat{\delta}_{PN} \\
&&+H.C.\end{aligned}$$ $\frac{\partial }{\partial \mathbf{A}\left( \mathbf{x}\right) }\mathcal{A}%
_{1N}^{\mathbf{R}^{\left( \alpha \right) }}$ is a two tensor whose components $\left( i,l\right) $ is given by : $\frac{\partial \mathbf{\emph{A%
}}^{R_{l}^{\left( \alpha \right) }}}{\partial A^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }=\varepsilon ^{ijk}X_{i}^{l\left( \alpha \right)
}\nabla _{k}$ $$\begin{aligned}
X_{i}^{l\left( \alpha \right) } &=&-\frac{\hbar ^{2}}{8}\sum_{P}\left\{
e\left( \left( \mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha \right) }}\right)
_{NP}\times \nabla _{\mathbf{\Pi }}\right) _{j}\left( \mathcal{A}%
_{0}^{R_{l}^{\left( \alpha \right) }}\right) _{PN}\right\} -e\frac{\hbar ^{2}%
}{4}\left( \left( \left( \mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha \right)
}}\right) _{NN}\times \nabla _{\mathbf{\Pi }}\right) _{j}\mathcal{A}%
_{0NN}^{R_{l}^{\left( \alpha \right) }}+H.C.\right) \\
&&-\frac{\hbar ^{2}}{2}\left( \sum_{P}e\left( \left( \mathcal{A}_{0}^{%
\mathbf{R}^{\left( \alpha \right) }}\right) _{NP}\times \nabla _{\mathbf{\Pi
}}\right) _{j}\frac{\left( \hat{\varepsilon}_{0P}+\hat{\varepsilon}%
_{0N}\right) }{\hat{\varepsilon}_{0P}-\hat{\varepsilon}_{0N}}\hat{\delta}%
_{PN}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right)
_{PN}-H.C.\right) \\
&&+i\frac{e\hbar ^{2}}{8}\sum_{P}\left( \left( \mathcal{A}%
_{0}^{R_{l}^{\left( \alpha \right) }}\right) _{NP}\left( \left( \mathcal{A}%
_{0}^{\mathbf{R}^{\left( \alpha \right) }}\right) _{PN}\times \left(
\mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha \right) }}\right) _{NN}\right)
_{j}\right. \\
&&\left. -\left( \left( \mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha \right)
}}\right) _{NP}\times \left( \mathcal{A}_{0}^{\mathbf{R}^{\left( \alpha
\right) }}\right) _{PP}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right)
}}\right) _{PN}\right) _{j}\right) \hat{\delta}_{PN}+H.C.\end{aligned}$$ and the gradient acts on $\delta \left( \mathbf{x-r}^{\left( \alpha \right)
}\right) $.
Having given some expanded formula for $\frac{dr_{l}^{\left( \alpha \right) }%
}{dA^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }$ it thus remains to compute $\frac{d\pi _{n}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) }$. To do so, recall again that the variable $\mathbf{r}^{\left( \alpha \right) }$ is frozen with respect to $\mathbf{A}$, which is thus thought as an independent variable. We use also our previous result on the Berry phase : $$\begin{aligned}
\pi _{m}^{\left( \alpha \right) } &=&K_{m}^{\left( \alpha \right) }-e\mathbf{%
A}\left( \mathbf{r}^{\left( \alpha \right) }\right) +\frac{1}{2}e\left[
\nabla _{m}A_{l}\left( \mathbf{R}^{\left( \alpha \right) }\right) \tilde{%
\emph{\ A}}^{R_{l}}+\tilde{\emph{A}}^{R_{l}}\nabla _{m}A_{l}\left( \mathbf{R}%
^{\left( \alpha \right) }\right) \right] \\
&&+e\nabla _{m}\nabla _{R_{n}}A_{r}\left( \mathbf{r}^{\left( \alpha \right)
}\right) \tilde{\emph{A}}_{nr} \\
\text{with }\mathbf{\tilde{\emph{A}}}_{n,r} &=&-\frac{\hbar ^{2}}{4}\sum_{P}%
\left[ \left( \mathcal{A}_{0}^{R_{n}^{\left( \alpha \right) }}\right)
_{PN}\left( \mathcal{A}_{0}^{R_{r}^{\left( \alpha \right) }}\right)
_{NP}+\left( \mathcal{A}_{0}^{R_{r}^{\left( \alpha \right) }}\right)
_{PN}\left( \mathcal{A}_{0}^{R_{n}^{\left( \alpha \right) }}\right) _{NP}%
\right]\end{aligned}$$ Define also $\mathbf{\emph{A}}_{m,i}=e\mathbf{\frac{d\tilde{\emph{A}}_{n,r}}{%
dA^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }}\nabla _{\mathbf{r}%
_{n}^{\left( \alpha \right) }}\nabla _{\mathbf{r}_{m}^{\left( \alpha \right)
}}A_{r}\left( \mathbf{r}^{\left( \alpha \right) }\right) +e$**$\tilde{%
\emph{\ A}}$**$_{n,i}\nabla _{\mathbf{r}_{n}^{\left( \alpha \right) }}\nabla
_{\mathbf{r}_{m}^{\left( \alpha \right) }}$ so that one has : $$\begin{aligned}
-\frac{1}{e}\frac{d\pi _{m}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) } &=&\delta _{i}^{m}-\tilde{\emph{A}}%
\mathbf{^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{m}^{\left(
\alpha \right) }}-\nabla _{\mathbf{r}_{m}^{\left( \alpha \right)
}}A^{n}\left( \mathbf{r}^{\left( \alpha \right) }\right) \frac{d\mathbf{%
\tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) }\mathbf{+\emph{A}}_{m,i} \\
&=&\delta _{i}^{m}-\tilde{\emph{A}}\mathbf{^{R_{i}^{\left( \alpha \right) }}}%
\nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}-\nabla _{\mathbf{r}%
_{m}^{\left( \alpha \right) }}A^{n}\left( \mathbf{r}^{\left( \alpha \right)
}\right) \frac{dr_{n}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) }+\mathbf{\emph{A}}_{m,i} \\
&=&\delta _{i}^{m}-\tilde{\emph{A}}\mathbf{^{R_{i}^{\left( \alpha \right) }}}%
\nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}+\mathbf{\emph{A}}_{m,i} \\
&&-\nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}A^{n}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) \times \left( \frac{\partial \tilde{\emph{A}%
}^{R_{n}^{\left( \alpha \right) }}}{\partial \pi _{r}^{\left( \alpha \right)
}}\frac{d\pi _{r}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }+\frac{\partial \tilde{\emph{A}}^{R_{n}^{\left(
\alpha \right) }}}{\partial r_{r}^{\left( \alpha \right) }}\frac{\partial
\tilde{\emph{A}}^{R_{r}^{\left( \alpha \right) }}}{\partial \pi _{q}^{\left(
\alpha \right) }}\frac{d\pi _{q}^{\left( \alpha \right) }}{dA^{i}\left(
\mathbf{r}^{\left( \alpha \right) }\right) }+\frac{\partial \tilde{\emph{A}}%
^{R_{n}^{\left( \alpha \right) }}}{\partial A^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }\right) \\
&=&\delta _{i}^{m}-\tilde{\emph{A}}\mathbf{^{R_{i}^{\left( \alpha \right) }}}%
\nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}-\nabla _{\mathbf{r}%
_{m}^{\left( \alpha \right) }}A^{n}\left( \mathbf{r}^{\left( \alpha \right)
}\right) \left( \frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha
\right) }}}{\partial \pi _{r}^{\left( \alpha \right) }}\frac{d\pi
_{r}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left( \alpha
\right) }\right) }+\frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha
\right) }}}{\partial A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }%
\right) +\mathbf{\emph{A}}_{m,i}\end{aligned}$$ to our order of approximation.
We can rewrite the last equation as : $$\frac{d\pi _{u}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }\left[ \delta _{u}^{m}-e\nabla _{\mathbf{r}%
_{m}^{\left( \alpha \right) }}A^{n}\left( \mathbf{r}^{\left( \alpha \right)
}\right) \frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{%
\partial \pi _{u}^{\left( \alpha \right) }}\right] =-e\left( \delta _{i}^{m}-%
\tilde{\emph{A}}\mathbf{^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}%
_{m}^{\left( \alpha \right) }}-\nabla _{\mathbf{r}_{m}^{\left( \alpha
\right) }}A^{n}\left( \mathbf{r}^{\left( \alpha \right) }\right) \frac{%
\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{\partial
A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }+\mathbf{\emph{A}}%
_{m,i}\right)$$ which is solved to the second order in $\hbar $ and first order in field by : $$\begin{aligned}
-\frac{1}{e}\frac{d\pi _{m}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) } &=&\delta _{i}^{m}-\tilde{\emph{A}}%
\mathbf{^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{m}^{\left(
\alpha \right) }}-\nabla _{\mathbf{r}_{m}^{\left( \alpha \right)
}}A^{n}\left( \mathbf{r}^{\left( \alpha \right) }\right) \frac{\partial
\tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{\partial A^{i}\left(
\mathbf{r}^{\left( \alpha \right) }\right) } \\
&&+e\nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}A^{n}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) \frac{\partial \tilde{\emph{A}}%
^{R_{n}^{\left( \alpha \right) }}}{\partial \pi _{u}^{\left( \alpha \right) }%
}\left( \delta _{i}^{u}-\tilde{\emph{\ A}}\mathbf{^{R_{i}^{\left( \alpha
\right) }}}\nabla _{\mathbf{r}_{u}^{\left( \alpha \right) }}\right) +\mathbf{%
\emph{A}}_{m,i} \\
&=&\delta _{i}^{m}-\tilde{\emph{A}}\mathbf{^{R_{i}^{\left( \alpha \right) }}}%
\nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}-\nabla _{\mathbf{r}%
_{m}^{\left( \alpha \right) }}A^{n}\left( \mathbf{r}^{\left( \alpha \right)
}\right) \frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{%
\partial A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) } \\
&&+e\nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}A^{n}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) \left( \frac{\partial \tilde{\emph{\ A}}%
^{R_{n}^{\left( \alpha \right) }}}{\partial \pi _{i}^{\left( \alpha \right) }%
}-\frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{\partial
\pi _{u}^{\left( \alpha \right) }}\tilde{\emph{A}}\mathbf{^{R_{i}^{\left(
\alpha \right) }}}\nabla _{\mathbf{r}_{u}^{\left( \alpha \right) }}\right) +%
\mathbf{\emph{A}}_{m,i}\end{aligned}$$ the gradient appearing at the end of the right hand side has to be understood as acting on the delta function $\delta \left( \mathbf{x-r}%
^{\left( \alpha \right) }\right) $ in the expression for $-\sum_{\alpha }%
\frac{\delta }{\delta A^{i}\left( \mathbf{x,t}\right) }\left( \hat{%
\varepsilon}_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left(
\mathbf{\pi }^{\left( \alpha \right) }\right) +D^{\left( \alpha \right)
}\right) $.
### Solution for the Electromagnetic Field
With the current at hand, it is now possible to find the expression for the electromagnetic field as a function of the particles dynamical variables. However, note that in the expression for $\mathbf{j}\left( \mathbf{x}\right)
$ the electromagnetic field appears everywhere through the Berry phases and explicitly through the magnetic field. This means that in fact reinserting $%
\mathbf{j}\left( \mathbf{x}\right) $ in the expression for $\mathbf{A}\left(
\mathbf{x}\right) $ allows only to compute this last potential perturbatively. Assuming, as before that the current is relatively weak, the right hand side can be expanded at the first order in the field. The expressions depending on the field can be put to the left of the Maxwell equation. which can be now written : $$\left[ \square +\delta O\left( \mathbf{x}\right) \right] \mathbf{A}\left(
\mathbf{x,}t\right) =\int d^{3}y\left[ \delta ^{ij}\delta ^{3}\left( \mathbf{%
x-y}\right) +\frac{\partial ^{2}}{\partial x^{i}\partial x^{j}}\frac{1}{4\pi
\left\vert \mathbf{x-y}\right\vert }\right] \mathbf{\hat{\jmath}}\left(
\mathbf{y,}t\right) _{\mid \mathbf{A}=0}$$ with : $$\begin{aligned}
\hat{\jmath}_{i}\left( \mathbf{x}\right) &=&-\left( \frac{\delta }{\delta
\mathbf{A}\left( \mathbf{x}\right) }H_{mat}\right) _{\mid \mathbf{A}=0} \\
&=&e\sum_{\alpha }\frac{\partial \left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha
\right) }\right) +D^{\left( \alpha \right) }\right) }{\partial \pi
_{j}^{\left( \alpha \right) }\left( \mathbf{x,t}\right) }\left[ \left(
\delta _{i}^{j}-\tilde{\emph{A}}\mathbf{^{R_{i}^{\left( \alpha \right) }}}%
\nabla _{\mathbf{r}_{j}^{\left( \alpha \right) }}+\mathbf{\emph{A}}_{j,i\mid
\mathbf{A}=0}\right) \right] _{\mid \mathbf{A}=0}\delta \left( \mathbf{x-r}%
^{\left( \alpha \right) }\right) \\
&&-\sum_{\alpha }\frac{\partial v\left( \mathbf{r}^{\left( \alpha \right)
}\right) }{\partial r_{j}^{\left( \alpha \right) }}\left[ \frac{\partial
\tilde{\emph{A}}^{R_{j}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left(
\alpha \right) }}\left( \delta _{i}^{n}-\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{n}^{\left( \alpha
\right) }}+\mathbf{\emph{A}}_{n,i\mid \mathbf{A}=0}\right) +\frac{\partial
\tilde{\emph{A}}^{R_{j}^{\left( \alpha \right) }}}{\partial A^{i}\left(
\mathbf{r}^{\left( \alpha \right) }\right) }\right] _{\mid \mathbf{A}%
=0}\delta \left( \mathbf{x-r}^{\left( \alpha \right) }\right) \\
&&-\sum_{\alpha }\left[ \left( \mathbf{\mu }\left( \mathbf{x}^{\left( \alpha
\right) }\right) \times \mathbf{\nabla }\right) _{i}\delta \left( \mathbf{x-r%
}^{\left( \alpha \right) }\right) +\left[ \mathbf{\hat{\mu}}^{k}\left(
\mathbf{x}^{\left( \alpha \right) }\right) \mathbf{\times }\nabla _{R_{k}}%
\mathbf{\nabla }\delta \left( \mathbf{x-r}^{\left( \alpha \right) }\right) %
\right] _{i}\right] _{\mid \mathbf{A}=0}\end{aligned}$$ The operator $\delta O\left( \mathbf{x}\right) $ will in fact yields negligible contributions in our applications. We give nevertheless its form in the appendix for the sake of completeness. Note that the correction $%
\delta O\left( \mathbf{x}\right) $ to the Dalembertian are of order $\hbar
^{2}$ due to the definition of the terms involved.
As a consequence, we can thus write for the electromagnetic field : $$A^{i}\mathbf{\left( \mathbf{x},t\right) =}\int G_{ij}\left( \mathbf{x-y,t-t}%
^{\prime }\right) \hat{\jmath}^{j}\mathbf{\left( \mathbf{y},t^{\prime
}\right) }dy^{3}dt^{\prime }$$ with : $$\begin{aligned}
&&G_{ij}\left( \mathbf{x-y,t-t}^{\prime }\right) \\
&\mathbf{=}&\int \left[ \frac{1}{\frac{\partial ^{2}}{\partial t^{2}}-%
\mathbf{\nabla }_{\mathbf{x}}^{2}-\delta O\left( \mathbf{x}\right) }\left(
\mathbf{x-z,t-t}^{\prime }\right) \left[ \delta ^{ij}\delta ^{3}\left(
\mathbf{z-y}\right) +\frac{\partial ^{2}}{\partial x^{i}\partial x^{j}}\frac{%
1}{4\pi \left\vert \mathbf{z-y}\right\vert }\right] \right] _{ij}dz\end{aligned}$$ A more tractable for the effective Green function $G_{ij}\left( \mathbf{%
x-y,t-t}^{\prime }\right) $ can be found if we move to the Fourier transform. Actually, one can write : $$\begin{aligned}
G_{ij}\left( \mathbf{x-y,t-t}^{\prime }\right) &=&\int \exp \left( i\mathbf{%
p.}\left( \mathbf{x-y}\right) -i\omega \left( t-t^{\prime }\right) \right)
\left( \delta _{ir}-\frac{p_{i}p_{r}}{\mathbf{p}^{2}}\right) \left( \frac{1}{%
\omega ^{2}-\left( \mathbf{p}^{2}-\delta O\left( \mathbf{p}\right) \right) }%
\right) _{rj}d\mathbf{p}d\omega \\
&\simeq &\int \exp \left( i\mathbf{p.}\left( \mathbf{x-y}\right) -i\omega
\left( t-t^{\prime }\right) \right) \frac{\delta _{ir}-\frac{p_{i}p_{r}}{%
\mathbf{p}^{2}}}{\omega ^{2}-\mathbf{p}^{2}}\left( \delta _{rj}+\frac{1}{2}%
\frac{\delta O_{rj}\left( \mathbf{p}\right) }{\omega ^{2}-\mathbf{p}^{2}}%
\right) d\mathbf{p}d\omega \\
&\equiv &\int \exp \left( i\mathbf{p.}\left( \mathbf{x-y}\right) -i\omega
\left( t-t^{\prime }\right) \right) G_{ij}\left( \mathbf{p,}\omega \right) d%
\mathbf{p}d\omega\end{aligned}$$ and the operators $G^{\left( 0\right) }$ $\delta O,\delta G$ are given in Fourier components by : $$\begin{aligned}
G_{ij}^{\left( 0\right) }\left( \mathbf{p,}\omega \right) &=&\frac{\delta
_{ij}-\frac{p_{i}p_{j}}{\mathbf{p}^{2}}}{\omega ^{2}-\mathbf{p}^{2}} \\
\delta O_{ij}\left( \mathbf{p}\right) &=&\frac{\delta _{ir}-\frac{p_{i}p_{r}%
}{\mathbf{p}^{2}}}{\omega ^{2}-\mathbf{p}^{2}}\delta O_{1rj}\left( \mathbf{p}%
\right) \\
G_{ij}\left( \mathbf{p,}\omega \right) &=&G_{ir}^{\left( 0\right) }\left(
\mathbf{p,}\omega \right) \left( \delta _{rj}+\frac{1}{2}\frac{\delta
O_{rj}\left( \mathbf{p}\right) }{\omega ^{2}-\mathbf{p}^{2}}\right)\end{aligned}$$ and the Fourier transform $\delta O_{1}\left( \mathbf{p}\right) $ is given in the appendix.
Now that we have given the effective photon propagator, note that we can decompose the vector potential at order $\hbar ^{2}$ at a point $\mathbf{x}$ : $$\begin{aligned}
A_{i}\left( \mathbf{x},t\right) &=&\int G_{ij}\left( \mathbf{x-y,t-t}%
^{\prime }\right) \jmath _{1}^{j}\mathbf{\left( \mathbf{y},t^{\prime
}\right) }dy^{3}dt^{\prime } \\
&&-\int dt^{\prime }\sum_{\alpha }\left[ \mu _{l}\left( \mathbf{x}^{\left(
\alpha \right) }\right) +\hat{\mu}_{l}^{k}\left( \mathbf{x}^{\left( \alpha
\right) }\right) \nabla _{R_{k}}\right] \varepsilon ^{lmj}\nabla
_{m}G_{ij}\left( \mathbf{x-x}^{\left( \alpha \right) }\left( t^{\prime
}\right) ,t-t^{\prime }\right) dt^{\prime } \\
&=&\int G_{ij}\left( \mathbf{x-y,t-t}^{\prime }\right) \jmath _{1}^{j}%
\mathbf{\left( \mathbf{y},t^{\prime }\right) }dy^{3}dt^{\prime }-\int
dt^{\prime }\mathcal{M}_{l}^{ef}\varepsilon ^{lmj}\nabla _{m}^{\mathbf{x}%
}G_{ij}\left( \mathbf{x-x}^{\left( \alpha \right) }\left( t^{\prime }\right)
,t-t^{\prime }\right)\end{aligned}$$ where $\mathcal{M}_{l}^{ef}$ is the effective magnetization differential operator : $$\mathcal{M}_{m}^{ef}=\sum_{\alpha }\left[ \mu _{m}\left( \mathbf{x}^{\left(
\alpha \right) }\right) +\hat{\mu}_{m}^{k}\left( \mathbf{x}^{\left( \alpha
\right) }\right) \nabla _{R_{k}}\right] _{\mid \mathbf{A}=0}$$ and the current $\mathbf{\jmath }_{1}$ is given by : $$\begin{aligned}
\mathbf{\jmath }_{1} &=&e\sum_{\alpha }\frac{\partial \left( \hat{\varepsilon%
}_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left( \mathbf{\pi }%
^{\left( \alpha \right) }\right) +D^{\left( \alpha \right) }\right) }{%
\partial \pi _{j}^{\left( \alpha \right) }\left( \mathbf{x,t}\right) }\left[
\left( \delta _{i}^{j}-\tilde{\emph{A}}\mathbf{^{R_{i}^{\left( \alpha
\right) }}}\nabla _{\mathbf{r}_{j}^{\left( \alpha \right) }}+\mathbf{\emph{A}%
}_{j,i\mid \mathbf{A}=0}\right) \right] \delta \left( \mathbf{x-r}^{\left(
\alpha \right) }\right) \\
&&-\sum_{\alpha }\frac{\partial v\left( \mathbf{r}^{\left( \alpha \right)
}\right) }{\partial r_{j}^{\left( \alpha \right) }}\left[ \frac{\partial
\tilde{\emph{A}}^{R_{j}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left(
\alpha \right) }}\left( \delta _{i}^{n}-\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{n}^{\left( \alpha
\right) }}+\mathbf{\emph{A}}_{n,i\mid \mathbf{A}=0}\right) +\frac{\partial
\tilde{\emph{A}}^{R_{j}^{\left( \alpha \right) }}}{\partial A^{i}\left(
\mathbf{r}^{\left( \alpha \right) }\right) }\right] \delta \left( \mathbf{x-r%
}^{\left( \alpha \right) }\right)\end{aligned}$$ In $\nabla _{m}^{\mathbf{x}}$ recalls that the gradient is taken with respect to the variable $\mathbf{x}$.
### Electromagnetic part of the Hamiltonian
We can now compute the Hamiltonian for the electromagnetic field as : $$\begin{aligned}
\frac{1}{2}\int d^{3}x\Pi _{em}^{2}+\frac{1}{2}\int d^{3}x\left( \mathbf{%
\nabla \times A}\right) ^{2} &=&\frac{1}{2}\int dy^{3}dx^{3}A^{i}\mathbf{%
\left( \mathbf{x},t\right) }\left[ \delta ^{ij}\left( \frac{\partial ^{2}}{%
\partial t^{2}}-\mathbf{\nabla }^{2}\right) +\frac{\partial ^{2}}{\partial
x^{i}\partial x^{j}}\right] A^{j}\mathbf{\left( \mathbf{x},t\right) } \\
&=&\frac{1}{2}\int dy^{3}dz^{3}dt^{\prime }dt^{\prime \prime }dx^{3}\hat{%
\jmath}^{l}\mathbf{\left( \mathbf{y},t^{\prime }\right) }G_{il}\left(
\mathbf{x-y,}t-t^{\prime }\right) \\
&&\times \left[ \delta ^{ij}\left( \frac{\partial ^{2}}{\partial t^{2}}-%
\mathbf{\nabla }^{2}\right) +\frac{\partial ^{2}}{\partial x^{i}\partial
x^{j}}\right] G_{jm}\left( \mathbf{x-z,}t-t^{\prime \prime }\right) \hat{%
\jmath}^{m}\mathbf{\left( \mathbf{z},t^{\prime \prime }\right) } \\
&\equiv &\frac{1}{2}\int dy^{3}dx^{3}dt^{\prime }dt\hat{\jmath}^{i}\mathbf{%
\left( \mathbf{x},t\right) }\hat{G}\left( \mathbf{x-y,}t-t^{\prime }\right)
\hat{\jmath}^{j}\mathbf{\left( \mathbf{y},t^{\prime }\right) }\end{aligned}$$ where the effective propagator $\hat{G}\left( \mathbf{x-y,}t-t^{\prime
}\right) $ is given through its Fourier transform : $$\hat{G}\left( \mathbf{x-y,}t-t^{\prime }\right) =\int \exp \left( i\mathbf{p.%
}\left( \mathbf{x-y}\right) -i\omega \left( t-t^{\prime }\right) \right)
\hat{G}_{ij}\left( \mathbf{p,}\omega \right) d\mathbf{p}d\omega$$ with : $$\begin{aligned}
\hat{G}_{ij}\left( \mathbf{p,}\omega \right) &=&G_{il}\left( \mathbf{p,}%
\omega \right) \left[ \left( \delta _{lm}\left( \mathbf{p}^{2}-\omega
^{2}\right) -p_{l}p_{m}\right) \right] G_{mj}\left( \mathbf{p,}\omega \right)
\\
\hat{G}_{ij}\left( \mathbf{p,}\omega \right) &=&\left( \delta _{ir}+\frac{1}{%
2}\frac{\delta O_{ir}\left( \mathbf{p}\right) }{\mathbf{p}^{2}-\omega ^{2}}%
\right) \frac{\delta _{rl}-\frac{p_{r}p_{l}}{\mathbf{p}^{2}}}{\mathbf{p}%
^{2}-\omega ^{2}}\left( \delta _{lm}\left( \mathbf{p}^{2}-\omega ^{2}\right)
-p_{l}p_{m}\right) \frac{\delta _{ms}-\frac{p_{m}p_{s}}{\mathbf{p}^{2}}}{%
\mathbf{p}^{2}-\omega ^{2}}\left( \delta _{sj}+\frac{1}{2}\frac{\delta
O_{sj}\left( \mathbf{p}\right) }{\mathbf{p}^{2}-\omega ^{2}}\right) \\
&&\frac{\delta _{ij}-\frac{p_{i}p_{j}}{\mathbf{p}^{2}}}{\mathbf{p}%
^{2}-\omega ^{2}}+\frac{1}{2}\frac{\delta O_{ir}\left( \mathbf{p}\right) }{%
\mathbf{p}^{2}-\omega ^{2}}\frac{\delta _{rj}-\frac{p_{r}p_{j}}{\mathbf{p}%
^{2}}}{\mathbf{p}^{2}-\omega ^{2}}+\frac{1}{2}\frac{\delta _{is}-\frac{%
p_{i}ps}{\mathbf{p}^{2}}}{\mathbf{p}^{2}-\omega ^{2}}\frac{\delta
O_{sj}\left( \mathbf{p}\right) }{\mathbf{p}^{2}-\omega ^{2}}\end{aligned}$$ If we neglect the retardation effect, which is legitimate since we work in the non relativistic limit, we can discard the dependence in $\omega $, so that : $$\hat{G}_{ij}\left( \mathbf{p}\right) =\frac{\delta _{ij}-\frac{p_{i}p_{j}}{%
\mathbf{p}^{2}}}{\mathbf{p}^{2}}+\frac{1}{2}\frac{\delta O_{ir}\left(
\mathbf{p}\right) }{\mathbf{p}^{2}}\frac{\delta _{rj}-\frac{p_{r}p_{j}}{%
\mathbf{p}^{2}}}{\mathbf{p}^{2}}+\frac{1}{2}\frac{\delta _{is}-\frac{p_{i}ps%
}{\mathbf{p}^{2}}}{\mathbf{p}^{2}}\frac{\delta O_{sj}\left( \mathbf{p}%
\right) }{\mathbf{p}^{2}}$$ and : $$\frac{1}{2}\int d^{3}x\Pi _{em}^{2}+\frac{1}{2}\int d^{3}x\left( \mathbf{%
\nabla \times A}\right) ^{2}=\frac{1}{2}\int dy^{3}dx^{3}\hat{\jmath}^{i}%
\mathbf{\left( \mathbf{x},t\right) }\hat{G}\left( \mathbf{x-y}\right) \hat{%
\jmath}^{j}\mathbf{\left( \mathbf{y},t\right) }$$
### Electric potential
Now, the case of the electrostatic potential is a bit peculiar. As we said before, we replace it through the potential equation in the Coulomb gauge : $%
-\mathbf{\nabla }^{2}A_{0}=J_{0}$, so that
$\int d^{3}xeA_{0}\left( \mathbf{x}\right) J_{0}\left( \mathbf{x}\right) =%
\frac{1}{2}\int d^{3}x\int d^{3}yJ_{0}\left( \mathbf{x}\right) \frac{1}{4\pi
\left| \mathbf{x-y}\right| }J_{0}\left( \mathbf{y}\right) =\int d^{3}x\left(
\mathbf{\nabla }A_{0}\right) ^{2}$
Considering $v\left( \mathbf{r}^{\left( \alpha \right) }\right) $ alone, one has first : $$\begin{aligned}
&&v\left( \mathbf{r}^{\left( \alpha \right) }\right) -\frac{1}{2}\int
d^{3}x\left( \mathbf{\nabla }A_{0}\right) ^{2} \\
&=&\int d^{3}\mathbf{x}eA_{0}\left( \mathbf{x}\right) e\delta \left( \mathbf{%
x-r}^{\left( \alpha \right) }\right) -\frac{1}{2}\int d^{3}x\left( \mathbf{%
\nabla }A_{0}\right) ^{2} \\
&=&\frac{1}{2}\int d^{3}\mathbf{x}eA_{0}\left( \mathbf{x}\right) e\delta
\left( \mathbf{x-r}^{\left( \alpha \right) }\right) \\
&=&\frac{1}{2}\int d^{3}x\int d^{3}yJ_{0}\left( \mathbf{x}\right) \frac{1}{%
4\pi \left\vert \mathbf{x-y}\right\vert }J_{0}\left( \mathbf{y}\right) \\
&=&\frac{1}{2}\sum_{\alpha \neq \beta }\frac{1}{4\pi \left\vert \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right\vert } \\
&=&\frac{1}{2}\sum_{\alpha \neq \beta }V\left( \mathbf{r}^{\left( \alpha
\right) }\mathbf{-r}^{\left( \beta \right) }\right)\end{aligned}$$ where $V\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta
\right) }\right) $ is the usual Coulomb interaction. The same token is applied to the derivatives of $v\left( \mathbf{r}^{\left( \alpha \right)
}\right) $ appearing in the Hamiltonian, so that the part involving the electrostatic potential can be written : $$\begin{aligned}
H_{\text{elec}} &=&\sum_{\alpha }v\left( \mathbf{r}^{\left( \alpha \right)
}\right) +\frac{\hbar ^{2}}{4}\sum_{\alpha }\left[ \sum_{M^{\left( \alpha
\right) }}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right)
_{N^{\left( \alpha \right) }M^{\left( \alpha \right) }}\left( \mathcal{A}%
_{0}^{R_{m}^{\alpha }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha
\right) }}\right] \nabla _{R_{m}^{\left( \alpha \right) }}\nabla
_{R_{l}^{\left( \alpha \right) }}v\left( \mathbf{r}^{\left( \alpha \right)
}\right) \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\frac{\left[ \left[ \varepsilon
_{0}^{\left( \alpha \right) }\left( \mathbf{x}\right) ,\mathcal{A}%
_{0}^{R_{m}^{\left( \alpha \right) }}\right] _{N^{\left( \alpha \right)
}M^{\left( \alpha \right) }}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha
\right) }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha \right)
}}\nabla _{R_{m}^{\left( \alpha \right) }}\nabla _{R_{l}^{\left( \alpha
\right) }}\right] v\left( \mathbf{r}^{\left( \alpha \right) }\right) }{\hat{%
\varepsilon}_{0M^{\left( \alpha \right) }}^{\left( \alpha \right) }-\hat{%
\varepsilon}_{0N^{\left( \alpha \right) }}} \\
&&-\frac{1}{2}\int d^{3}x\left( \mathbf{\nabla }A_{0}\right) ^{2} \\
&=&\frac{1}{2}\sum_{\alpha }V\left( \mathbf{r}^{\left( \alpha \right) }-%
\mathbf{r}^{\left( \beta \right) }\right) \\
&&+\frac{1}{4}\sum_{\alpha }\left[ \sum_{M^{\left( \alpha \right) }}\left(
\mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right) _{N^{\left( \alpha
\right) }M^{\left( \alpha \right) }}\left( \mathcal{A}_{0}^{R_{m}^{\alpha
}}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha \right) }}\right]
\nabla _{R_{m}^{\left( \alpha \right) }}\nabla _{R_{l}^{\left( \alpha
\right) }}\sum_{\beta \neq \alpha }V\left( \mathbf{r}^{\left( \alpha \right)
}-\mathbf{r}^{\left( \beta \right) }\right) \\
&&+\frac{1}{4}\sum_{\alpha }\frac{\left[ \left[ \varepsilon _{0}^{\left(
\alpha \right) }\left( \mathbf{x}\right) ,\mathcal{A}_{0}^{R_{m}^{\left(
\alpha \right) }}\right] _{N^{\left( \alpha \right) }M^{\left( \alpha
\right) }}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right)
_{M^{\left( \alpha \right) }N^{\left( \alpha \right) }}\nabla
_{R_{m}^{\left( \alpha \right) }}\nabla _{R_{l}^{\left( \alpha \right) }}%
\right] \sum_{\beta \neq \alpha }V\left( \mathbf{r}^{\left( \alpha \right) }-%
\mathbf{r}^{\left( \beta \right) }\right) }{\hat{\varepsilon}_{0M^{\left(
\alpha \right) }}^{\left( \alpha \right) }-\hat{\varepsilon}_{0N^{\left(
\alpha \right) }}}\end{aligned}$$ We can now come back to our previous remark about the replacing the electrostatic potential before or after diagonalization. Had we chosen to replace this potential before applying the diagonalization procedure would have in fact led to the same contribution to the Band Hamiltonian. Actually, it is straightforward to check that this early replacement would have led to consider the potential $\frac{1}{2}\sum_{\alpha }V\left( \mathbf{R}^{\left(
\alpha \right) }-\mathbf{R}^{\left( \beta \right) }\right) $ in the Hamiltonian (and no more the potential $v$). This expression mixing the particles at an early stage would have altered the diagonalization process. Actually, the diagonalization matrix wouldn’t have been the simple tensor product of individual particles diagonalization matrices, but would have mixed the various spaces at first order in $\hbar $. The result would be, at the order $\hbar ^{2}$, in crossed contributions in the Berry phase for $%
\mathbf{K}^{\left( \alpha \right) }$ (not for $\mathbf{R}^{\left( \alpha
\right) }$ as can directly been checked). Namely, the Berry phase for $%
\mathbf{K}^{\left( \alpha \right) }$ would acquire a supplementary contribution proportional to $\nabla _{R_{m}^{\left( \alpha \right) }}\nabla
_{R_{l}^{\left( \beta \right) }}\sum_{\beta \neq \alpha }V\left( \mathbf{R}%
^{\left( \alpha \right) }-\mathbf{R}^{\left( \beta \right) }\right) \left(
\mathcal{A}_{0}^{R_{m}^{\left( \beta \right) }}\right) _{N^{\left( \beta
\right) }M^{\left( \beta \right) }}\hat{\delta}_{N^{\left( \beta \right)
}M^{\left( \beta \right) }}$. Given our generalized results for the Diagonalized Hamiltonian, this term will not contribute to the Band Hamiltonian nor to the dynamical operators at order $\hbar ^{2}$. This is the reason why we choose to treat the electrostatic potential in a less rigorous, but symmetric with respect to the vector potential, way.
### $P$ particles Hamiltonian
We can now gather all our results to obtain the final form for the Hamiltonian
$$\begin{aligned}
H_{N} &=&\sum_{\alpha }\hat{\varepsilon}_{0N^{\left( \alpha \right)
}}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha \right)
}\right) +\frac{1}{2}\sum_{\alpha }V\left( \mathbf{r}^{\left( \alpha \right)
}-\mathbf{r}^{\left( \beta \right) }\right) -\frac{1}{2}\sum_{\alpha }\left(
\mathbf{\mu }\left( \mathbf{x}^{\left( \alpha \right) }\right) \mathbf{.B}%
^{\left( \alpha \right) }\left( \mathbf{x}^{\left( \alpha \right) }\right) +%
\mathbf{B}^{\left( \alpha \right) }\left( \mathbf{x}^{\left( \alpha \right)
}\right) \mathbf{.\mu }\left( \mathbf{x}\right) \right) \\
&&-\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right) \mathbf{.}\nabla _{R_{k}}%
\mathbf{B}^{\left( \alpha \right) }\left( \mathbf{x}^{\left( \alpha \right)
}\right) +\mathbf{B}^{\left( \alpha \right) }\mathbf{.\tilde{\mu}.B}^{\left(
\alpha \right) } \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\left[ \sum_{M^{\left( \alpha \right)
}}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right) _{N^{\left(
\alpha \right) }M^{\left( \alpha \right) }}\left( \mathcal{A}%
_{0}^{R_{m}^{\alpha }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha
\right) }}\right] \nabla _{R_{m}^{\left( \alpha \right) }}\nabla
_{R_{l}^{\left( \alpha \right) }}\sum_{\beta \neq \alpha }V\left( \mathbf{r}%
^{\left( \alpha \right) }-\mathbf{r}^{\left( \beta \right) }\right) \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\frac{\left[ \left[ \varepsilon
_{0}^{\left( \alpha \right) }\left( \mathbf{x}\right) ,\mathcal{A}%
_{0}^{R_{m}^{\left( \alpha \right) }}\right] _{N^{\left( \alpha \right)
}M^{\left( \alpha \right) }}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha
\right) }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha \right)
}}\nabla _{R_{m}^{\left( \alpha \right) }}\nabla _{R_{l}^{\left( \alpha
\right) }}\right] \sum_{\beta \neq \alpha }V\left( \mathbf{r}^{\left( \alpha
\right) }-\mathbf{r}^{\left( \beta \right) }\right) }{\hat{\varepsilon}%
_{0M^{\left( \alpha \right) }}^{\left( \alpha \right) }-\hat{\varepsilon}%
_{0N^{\left( \alpha \right) }}} \\
+ &&\frac{1}{2}\int dy^{3}dx^{3}dt\hat{\jmath}^{i}\mathbf{\left( \mathbf{x}%
,t\right) }\hat{G}\left( \mathbf{x-y}\right) \hat{\jmath}^{j}\mathbf{\left(
\mathbf{y},t\right) }\end{aligned}$$
Application : Hamiltonian at the lowest order
---------------------------------------------
We have seen in the previous section that, due to the complexity to the Maxwell equation, the Full Hamiltonian to the order $\hbar ^{2}$ cannot be computed exactly but has in fact to be computed as a power series in the current. The Formula we gave included first order corrections for the Green function. Here, to give an application of our method we will neglect these first order corrections and consider the Hamiltonian at the lowest order in current, that is quadratic in current. We will see that it will result in a kind of generalized Breit Hamiltonian, including magnetization-magnetization interaction. To do so, we divide our work in two steps. First computing the Hamiltonian without electromagnetic interaction and second including the electromagnetic corrections.
### Coulombian interaction
If we neglect, the magnetic interactions, three simplifications arise. First, we can cancel all contributions to the magnetic field. Second, in the first order Berry phases, $\mathcal{A}_{0}^{R_{l}^{\left( \alpha \right)
\beta }}$ is null, except if $\alpha =\beta $. As before, we will note $%
\mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}$ the zeroth order Berry phase, keeping in mind there are no crossed terms. Third, the crystal momentum dynamical variables are unchanged through the transformation. Thus $%
\mathbf{k}^{\left( \beta \right) }=\mathbf{K}^{\left( \beta \right) }$. The Hamiltonian reduces thus to : $$\begin{aligned}
H_{N} &=&\sum_{\alpha }\hat{\varepsilon}_{0N^{\left( \alpha \right)
}}^{\left( \alpha \right) }\left( \mathbf{k}^{\left( \alpha \right) }\right)
+\frac{1}{2}\sum_{\alpha }V\left( \mathbf{r}^{\left( \alpha \right) }-%
\mathbf{r}^{\left( \beta \right) }\right) \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\left[ \sum_{M^{\left( \alpha \right)
}}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right) _{N^{\left(
\alpha \right) }M^{\left( \alpha \right) }}\left( \mathcal{A}%
_{0}^{R_{m}^{\alpha }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha
\right) }}\right] \nabla _{R_{m}^{\left( \alpha \right) }}\nabla
_{R_{l}^{\left( \alpha \right) }}\sum_{\beta \neq \alpha }V\left( \mathbf{r}%
^{\left( \alpha \right) }-\mathbf{r}^{\left( \beta \right) }\right) \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\frac{\left[ \left[ \varepsilon
_{0}^{\left( \alpha \right) }\left( \mathbf{x}\right) ,\mathcal{A}%
_{0}^{R_{m}^{\left( \alpha \right) }}\right] _{N^{\left( \alpha \right)
}M^{\left( \alpha \right) }}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha
\right) }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha \right)
}}\nabla _{R_{m}^{\left( \alpha \right) }}\nabla _{R_{l}^{\left( \alpha
\right) }}\right] \sum_{\beta \neq \alpha }V\left( \mathbf{r}^{\left( \alpha
\right) }-\mathbf{r}^{\left( \beta \right) }\right) }{\hat{\varepsilon}%
_{0M^{\left( \alpha \right) }}^{\left( \alpha \right) }-\hat{\varepsilon}%
_{0N^{\left( \alpha \right) }}}\end{aligned}$$ where the dynamical variables in that set up are given by : $$\begin{aligned}
\mathbf{r}_{N}^{\left( \alpha \right) } &=&\mathbf{R}^{\left( \alpha \right)
}+\emph{A}_{N^{\left( \alpha \right) }}^{\mathbf{R}\left( ^{\alpha }\right) }
\\
&\equiv &\mathbf{R}^{\left( \alpha \right) }+\mathcal{A}_{NN}^{\mathbf{R}%
^{\left( \alpha \right) }} \\
\mathbf{k}_{N}^{\left( \alpha \right) } &=&\mathbf{K}^{\left( \alpha \right)
}\end{aligned}$$ with $$\begin{aligned}
\emph{A}_{N^{\left( \alpha \right) }}^{\mathbf{R}\left( ^{\alpha }\right) }
&=&\left( \mathcal{A}^{\mathbf{R}}\right) _{N^{\left( \alpha \right)
}N^{\left( \alpha \right) }}=\left( \hbar \mathcal{A}_{0}^{\mathbf{R}%
}\right) _{N^{\left( \alpha \right) }N^{\left( \alpha \right) }}+\frac{\hbar
^{2}}{8}\left( \mathcal{A}_{0}^{R_{l}}\right) _{N^{\left( \alpha \right)
}P}\nabla _{R_{l}}\left( \mathcal{A}_{0}^{\mathbf{R}}\right) _{PN^{\left(
\alpha \right) }} \\
&&+\frac{\hbar ^{2}}{2}\left( \left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{N^{\left( \alpha \right) }P^{\left( \alpha \right) }}\left( \mathcal{A}%
_{0}^{R_{l}}\right) _{P^{\left( \alpha \right) }N^{\left( \alpha \right)
}}+\left( \mathcal{A}_{0}^{R_{l}}\right) _{N^{\left( \alpha \right)
}P^{\left( \alpha \right) }}\left( \mathcal{A}_{0}^{\mathbf{R}}\right)
_{P^{\left( \alpha \right) }N^{\left( \alpha \right) }}\right) \hat{\delta}%
_{P^{\left( \alpha \right) }N^{\left( \alpha \right) }}\frac{\nabla
_{R_{l}}V\left( \mathbf{R}\right) }{\hat{\varepsilon}_{0P^{\left( \alpha
\right) }}-\hat{\varepsilon}_{0N^{\left( \alpha \right) }}} \\
&&+H.C.\end{aligned}$$
### Electromagnetic interaction at the lowest order and magnetization-magnetization interaction
We now turn to the electromagnetic part of the interaction in our Hamiltonian for $P$ particles. It is found by isolating, at the second order in the current the interaction terms particles-electromagnetic field plus the internal Field Hamiltonian.
In the non relativistic limit which is of interest for us, it takes the form : $$\begin{aligned}
&&H_{\text{magnetic}}=-\sum_{\alpha }e\left( \frac{\partial \left( \hat{%
\varepsilon}_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left(
\mathbf{k}^{\left( \alpha \right) }\right) +D^{\left( \alpha \right)
}\right) +\sum_{\beta \neq \alpha }V\left( \mathbf{r}^{\left( \alpha \right)
}-\mathbf{r}^{\left( \beta \right) }\right) }{\partial k_{j}^{\left( \alpha
\right) }}\right) _{\mid \mathbf{A}=0}A^{j}\left( \mathbf{r}^{\left( \alpha
\right) }\right) \\
&&+\frac{1}{2}e^{2}\left( \frac{\partial ^{2}\left( \hat{\varepsilon}%
_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left( \mathbf{k}%
^{\left( \alpha \right) }\right) +D^{\left( \alpha \right) }\right)
+\sum_{\beta \neq \alpha }V\left( \mathbf{r}^{\left( \alpha \right) }-%
\mathbf{r}^{\left( \beta \right) }\right) }{\partial k_{j}^{\left( \alpha
\right) }\partial k_{k}^{\left( \alpha \right) }}\right) _{\mid \mathbf{A}%
=0}A^{j}\left( \mathbf{r}^{\left( \alpha \right) }\right) A^{k}\left(
\mathbf{r}^{\left( \alpha \right) }\right) \\
- &&\frac{1}{2}\sum_{\alpha }\left( \mathbf{\mu }\left( \mathbf{x}^{\left(
\alpha \right) }\right) \mathbf{.B}^{\left( \alpha \right) }\left( \mathbf{x}%
^{\left( \alpha \right) }\right) +\mathbf{B}^{\left( \alpha \right) }\left(
\mathbf{x}^{\left( \alpha \right) }\right) \mathbf{.\mu }\left( \mathbf{x}%
\right) \right) -\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right) \mathbf{.}%
\nabla _{R_{k}}\mathbf{B}^{\left( \alpha \right) }\left( \mathbf{x}^{\left(
\alpha \right) }\right) +\mathbf{B}^{\left( \alpha \right) }\mathbf{.\tilde{%
\mu}.B}^{\left( \alpha \right) } \\
&&+\frac{1}{2}\int dy^{3}dx^{3}dt\hat{\jmath}^{i}\mathbf{\left( \mathbf{x}%
,t\right) }\hat{G}\left( \mathbf{x-y}\right) \hat{\jmath}^{j}\mathbf{\left(
\mathbf{y},t\right) }\end{aligned}$$ Moreover, in first approximation, $\hat{G}\left( \mathbf{x-y}\right) $ can be approximated by $G_{0}\left( \mathbf{x-y}\right) $ so that $$\begin{aligned}
\frac{1}{2}\int dy^{3}dx^{3}dt\hat{\jmath}^{i}\mathbf{\left( \mathbf{x}%
,t\right) }\hat{G}\left( \mathbf{x-y}\right) \hat{\jmath}^{j}\mathbf{\left(
\mathbf{y},t\right) } &\simeq &\frac{1}{2}\int dy^{3}dx^{3}\hat{\jmath}_{i}%
\mathbf{\left( \mathbf{x},t\right) }G_{0}^{ij}\left( \mathbf{x-y}\right)
\hat{\jmath}_{j}\mathbf{\left( \mathbf{y},t\right) } \\
&=&\int d^{3}\mathbf{x}d^{3}\mathbf{y}\hat{\jmath}_{i}\mathbf{\left( \mathbf{%
x},t\right) }\frac{\delta ^{ij}+\frac{\left( \mathbf{x-y}\right) _{i}\left(
\mathbf{x-y}\right) _{j}}{\left\vert \mathbf{x-y}\right\vert ^{2}}}{%
2\left\vert \mathbf{x-y}\right\vert }\hat{\jmath}_{j}\mathbf{\left( \mathbf{y%
},t\right) }\end{aligned}$$ As seen before, the current decomposes as : $$\hat{\jmath}_{i}\mathbf{\left( \mathbf{y},t\right) =\jmath }_{1i}\mathbf{%
\left( \mathbf{y},t\right) }+\mathcal{M}_{l}^{ef}\mathbf{\left( \mathbf{y}%
,t\right) }\varepsilon ^{lmi}\nabla _{m}^{\left( \mathbf{y}\right) }$$ (we have used the fact that in the definition of the magnetization, the gradient of a delta acts as minus a derivative) where the magnetization $%
\mathcal{M}_{i}^{ef}$ and the current $\mathbf{\jmath }_{1}$ is given by : $$\begin{aligned}
\mathcal{M}_{i}^{ef}\left( \mathbf{y}\right) &=&\sum_{\alpha }\mathcal{M}%
_{i}^{ef\left( \alpha \right) }\left( \mathbf{x}^{\left( \alpha \right)
}\right) \delta \left( \mathbf{y-r}^{\left( \alpha \right) }\right) \\
&=&\sum_{\alpha }\left[ \mu _{i}\left( \mathbf{x}^{\left( \alpha \right)
}\right) +\hat{\mu}_{i}^{k}\left( \mathbf{x}^{\left( \alpha \right) }\right)
\nabla _{R_{k}}\right] \delta \left( \mathbf{y-r}^{\left( \alpha \right)
}\right)\end{aligned}$$ with $\mathcal{M}_{i}^{ef\left( \alpha \right) }$ is the particle $\alpha $ individual particle magnetization, and $$\begin{aligned}
\mathbf{\jmath }_{1i}\mathbf{\left( \mathbf{y},t\right) } &=&e\sum_{\alpha }%
\frac{\partial \left( \hat{\varepsilon}_{0N^{\left( \alpha \right)
}}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha \right)
}\right) +D^{\left( \alpha \right) }\right) }{\partial k_{j}^{\left( \alpha
\right) }}\left[ \left( \delta _{i}^{j}-\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{j}^{\left( \alpha
\right) }}+\mathbf{\emph{A}}_{j,i}\right) \right] _{\mid \mathbf{A}=0}\delta
\left( \mathbf{y-r}^{\left( \alpha \right) }\right) \\
&&-\sum_{\alpha }\frac{\partial v\left( \mathbf{r}^{\left( \alpha \right)
}\right) }{\partial r_{j}^{\left( \alpha \right) }}\left[ \frac{\partial
\tilde{\emph{A}}^{R_{j}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left(
\alpha \right) }}\left( \delta _{i}^{n}-\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{n}^{\left( \alpha
\right) }}+\mathbf{\emph{A}}_{n,i\mid \mathbf{A}=0}\right) +\frac{\partial
\tilde{\emph{A}}^{R_{j}^{\left( \alpha \right) }}}{\partial A^{i}\left(
\mathbf{r}^{\left( \alpha \right) }\right) }\right] _{\mid \mathbf{A}%
=0}\delta \left( \mathbf{y-r}^{\left( \alpha \right) }\right)\end{aligned}$$ ultimately, as seen in the previous section $v\left( \mathbf{r}^{\left(
\alpha \right) }\right) $ can be replaced by $\sum_{\beta }V\left( \mathbf{r}%
^{\left( \alpha \right) }-\mathbf{r}^{\left( \beta \right) }\right) $.
These formula allow to derive directly the potential and the magnetic field in our approximation : $$\begin{aligned}
\mathbf{A}\left( \mathbf{x}\right) &=&\sum_{\alpha }\frac{\jmath _{1}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) +}\frac{\left[ \jmath _{1}%
\mathbf{\left( \mathbf{r}^{\left( \alpha \right) }\right) .}\left( \mathbf{%
x-r}^{\left( \alpha \right) }\right) \right] \left( \mathbf{x-r}^{\left(
\alpha \right) }\right) }{\left| \mathbf{x-r}^{\left( \alpha \right)
}\right| ^{2}}}{2\left| \mathbf{x-r}^{\left( \alpha \right) }\right| }+\frac{%
\mathcal{M}_{m}^{ef}\left( \mathbf{r}^{\left( \alpha \right) }\right) \times
\left( \mathbf{x-r}^{\left( \alpha \right) }\right) }{\left| \mathbf{x-r}%
^{\left( \alpha \right) }\right| ^{2}} \\
\mathbf{B}\left( \mathbf{x}\right) &=&\sum_{\alpha }\frac{\mathbf{\jmath }%
_{1}\times \left( \mathbf{x-r}^{\left( \alpha \right) }\right) }{\left|
\mathbf{x-r}^{\left( \alpha \right) }\right| ^{3}}+\mathbf{\nabla \times }%
\frac{\mathcal{M}_{m}^{ef}\left( \mathbf{r}^{\left( \alpha \right) }\right)
\times \left( \mathbf{x-r}^{\left( \alpha \right) }\right) }{\left| \mathbf{%
x-r}^{\left( \alpha \right) }\right| ^{3}}\end{aligned}$$ We can now compute the various terms appearing in $H_{\text{magnetic}}$. The first term, involving the magnetization is given by :
$$\begin{aligned}
&&-\frac{1}{2}\sum_{\alpha }\left( \mathbf{\mu }\left( \mathbf{x}^{\left(
\alpha \right) }\right) \mathbf{.B}\left( \mathbf{x}^{\left( \alpha \right)
}\right) +\mathbf{B}\left( \mathbf{x}^{\left( \alpha \right) }\right)
\mathbf{.\mu }\left( \mathbf{x}^{\left( \alpha \right) }\right) \right) -%
\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}^{\left( \alpha \right) }\right)
\mathbf{.}\nabla _{R_{k}}\mathbf{B}\left( \mathbf{x}^{\left( \alpha \right)
}\right) +\mathbf{B.\tilde{\mu}\left( \mathbf{x}^{\left( \alpha \right)
}\right) .B}^{\left( \alpha \right) } \\
&=&-\frac{1}{2}\sum_{\alpha ,\beta ,\alpha \neq \beta }\left( \mu _{l}\left(
\mathbf{r}^{\left( \alpha \right) }\right) \varepsilon ^{lmi}\nabla
_{m}G_{0}^{ij}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right) \jmath _{1j}\mathbf{\left( \mathbf{r}^{\left( \beta
\right) }\right) +H.C.}\right. \\
&&\left. \mathbf{-}\hat{\mu}_{l}^{k}\left( \mathbf{r}^{\left( \alpha \right)
}\right) \varepsilon ^{lmi}\nabla _{k}\nabla _{m}G_{0}^{ij}\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) \jmath
_{1j}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right) }\right) \\
&&+\sum_{\alpha ,\beta ,\alpha \neq \beta }\tilde{\mu}_{lp}\mathbf{\left(
\mathbf{r}^{\left( \alpha \right) }\right) }\varepsilon ^{lmi}\nabla
_{m}G_{0}^{ij}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right) \jmath _{1j}\mathbf{\left( \mathbf{r}^{\left( \beta
\right) }\right) }\varepsilon ^{pmi}\nabla _{m}G_{0}^{ij}\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \gamma \right) }\right) \jmath
_{1j}\mathbf{\left( \mathbf{r}^{\left( \gamma \right) }\right) } \\
&&+\sum_{\alpha ,\beta ,\alpha \neq \beta }\mu _{l}\left( \mathbf{r}^{\left(
\alpha \right) }\right) \varepsilon ^{lri}\varepsilon ^{jmp}\nabla
_{r}\nabla _{p}G_{0}^{ij}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r%
}^{\left( \beta \right) }\right) \mu _{m}\left( \mathbf{r}^{\left( \beta
\right) }\right)\end{aligned}$$
The gradient, in all these expressions and in the sequel, is understood as acting on the first variable (here $\mathbf{r}^{\left( \alpha \right) }$).
The second relevant term is the energy of the internal magnetic field : $$\begin{aligned}
&&\frac{1}{2}\int dy^{3}dx^{3}\hat{\jmath}_{i}\mathbf{\left( \mathbf{x}%
,t\right) }G_{0}^{ij}\left( \mathbf{x-y}\right) \hat{\jmath}_{j}\mathbf{%
\left( \mathbf{y},t\right) } \\
&=&\frac{1}{2}\sum_{\alpha ,\beta ,\alpha \neq \beta }\jmath _{1i}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) }G_{0}^{ij}\left( \mathbf{r%
}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) \jmath
_{1j}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right) } \\
&&\mathbf{+}\frac{1}{2}\sum_{\alpha ,\beta ,\alpha \neq \beta }\left( \mu
_{l}\left( \mathbf{r}^{\left( \alpha \right) }\right) +\hat{\mu}%
_{l}^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right) \nabla _{k}\right)
\varepsilon ^{lmi}\nabla _{m}G_{0}^{ij}\left( \mathbf{r}^{\left( \alpha
\right) }\mathbf{-r}^{\left( \beta \right) }\right) \jmath _{1j}\mathbf{%
\left( \mathbf{r}^{\left( \beta \right) }\right) +H.C.} \\
&&-\frac{1}{2}\sum_{\alpha ,\beta ,\alpha \neq \beta }\mu _{l}\left( \mathbf{%
r}^{\left( \alpha \right) }\right) \varepsilon ^{lri}\varepsilon
^{jmp}\nabla _{r}\nabla _{p}G_{0}^{ij}\left( \mathbf{x-y}\right) \mu
_{m}\left( \mathbf{r}^{\left( \beta \right) }\right)\end{aligned}$$ We gather these two contributions to get : $$\begin{aligned}
&&-\frac{1}{2}\sum_{\alpha }\left( \mathbf{\mu }\left( \mathbf{x}^{\left(
\alpha \right) }\right) \mathbf{.B}^{\left( \alpha \right) }\left( \mathbf{x}%
^{\left( \alpha \right) }\right) +\mathbf{B}^{\left( \alpha \right) }\left(
\mathbf{x}^{\left( \alpha \right) }\right) \mathbf{.\mu }\left( \mathbf{x}%
\right) \right) -\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right) \mathbf{.}%
\nabla _{R_{k}}\mathbf{B}^{\left( \alpha \right) }\left( \mathbf{x}^{\left(
\alpha \right) }\right) +\mathbf{B}^{\left( \alpha \right) }\mathbf{.\tilde{%
\mu}.B}^{\left( \alpha \right) } \\
&&+\frac{1}{2}\int dy^{3}dx^{3}dt\hat{\jmath}^{i}\mathbf{\left( \mathbf{x}%
,t\right) }\hat{G}\left( \mathbf{x-y}\right) \hat{\jmath}^{j}\mathbf{\left(
\mathbf{y},t\right) } \\
&=&\frac{1}{2}\sum_{\alpha ,\beta ,\alpha \neq \beta }\jmath _{1i}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) }G_{0}^{ij}\left( \mathbf{r%
}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) \jmath
_{1j}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right) } \\
&&+\frac{1}{2}\sum_{\alpha ,\beta ,\alpha \neq \beta }\mu _{l}\left( \mathbf{%
r}^{\left( \alpha \right) }\right) \varepsilon ^{lri}\varepsilon
^{jmp}\nabla _{r}\nabla _{p}G_{0}^{ij}\left( \mathbf{x-y}\right) \mu
_{m}\left( \mathbf{r}^{\left( \beta \right) }\right) \\
&&+\sum_{\alpha ,\beta ,\alpha \neq \beta }\tilde{\mu}_{lp}\mathbf{\left(
\mathbf{r}^{\left( \alpha \right) }\right) }\varepsilon ^{lmi}\nabla
_{m}G_{0}^{ij}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right) \jmath _{1j}\mathbf{\left( \mathbf{r}^{\left( \beta
\right) }\right) }\varepsilon ^{pmi}\nabla _{m}G_{0}^{ij}\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \gamma \right) }\right) \jmath
_{1j}\mathbf{\left( \mathbf{r}^{\left( \gamma \right) }\right) }\end{aligned}$$ The last term of interest for us, can be developer in the following way : $$\begin{aligned}
&&-\sum_{\alpha }e\left( \frac{\partial \left( \hat{\varepsilon}_{0N^{\left(
\alpha \right) }}^{\left( \alpha \right) }\left( \mathbf{k}^{\left( \alpha
\right) }\right) +D^{\left( \alpha \right) }\right) +\sum_{\beta \neq \alpha
}V\left( \mathbf{r}^{\left( \alpha \right) }-\mathbf{r}^{\left( \beta
\right) }\right) }{\partial k_{j}^{\left( \alpha \right) }}\right) _{\mid
\mathbf{A}=0}A^{j}\left( \mathbf{r}^{\left( \alpha \right) }\right) \\
&&+\frac{1}{2}e^{2}\left( \frac{\partial ^{2}\left( \hat{\varepsilon}%
_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left( \mathbf{k}%
^{\left( \alpha \right) }\right) +D^{\left( \alpha \right) }\right)
+\sum_{\beta \neq \alpha }V\left( \mathbf{r}^{\left( \alpha \right) }-%
\mathbf{r}^{\left( \beta \right) }\right) }{\partial k_{j}^{\left( \alpha
\right) }\partial k_{k}^{\left( \alpha \right) }}\right) _{\mid \mathbf{A}%
=0}A^{j}\left( \mathbf{r}^{\left( \alpha \right) }\right) A^{k}\left(
\mathbf{r}^{\left( \alpha \right) }\right) \\
&=&-\sum_{\alpha }\mathbf{\jmath }_{1}\mathbf{\left( \mathbf{r}^{\left(
\alpha \right) }\right) A}\left( \mathbf{r}^{\left( \alpha \right) }\right) +%
\frac{1}{2}e^{2}\left( \frac{\partial ^{2}\left( \hat{\varepsilon}%
_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left( \mathbf{k}%
^{\left( \alpha \right) }\right) +D^{\left( \alpha \right) }\right) }{%
\partial k_{j}^{\left( \alpha \right) }\partial k_{k}^{\left( \alpha \right)
}}\right) _{\mid \mathbf{A}=0}A^{j}\left( \mathbf{r}^{\left( \alpha \right)
}\right) A^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right) \\
&=&-\sum_{\alpha ,\beta ,\alpha \neq \beta }\jmath _{1i}\mathbf{\left(
\mathbf{r}^{\left( \alpha \right) }\right) }G_{0}^{ij}\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) \jmath
_{1j}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right) } \\
&&\mathbf{-}\frac{1}{2}\left[ \sum_{\alpha ,\beta ,\alpha \neq \beta }\left(
\mu _{l}\left( \mathbf{r}^{\left( \alpha \right) }\right) +\hat{\mu}%
_{l}^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right) \nabla _{k}\right)
\varepsilon ^{lmi}\nabla _{m}G_{0}^{ij}\left( \mathbf{r}^{\left( \alpha
\right) }\mathbf{-r}^{\left( \beta \right) }\right) \jmath _{1j}\mathbf{%
\left( \mathbf{r}^{\left( \beta \right) }\right) +H.C.}\right] \\
&&+\frac{1}{2}e^{2}\left( \frac{\partial ^{2}\left( \hat{\varepsilon}%
_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left( \mathbf{k}%
^{\left( \alpha \right) }\right) +D^{\left( \alpha \right) }\right) }{%
\partial k_{j}^{\left( \alpha \right) }\partial k_{k}^{\left( \alpha \right)
}}\right) _{\mid \mathbf{A}=0}A^{j}\left( \mathbf{r}^{\left( \alpha \right)
}\right) A^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right)\end{aligned}$$ The first equality is implied by construction of $\mathbf{\jmath }_{1}%
\mathbf{\left( \mathbf{r}^{\left( \alpha \right) }\right) }$ since this last quantity was precisely defined as $\sum_{\alpha }e\left( \frac{\partial
\left( \hat{\varepsilon}_{0N^{\left( \alpha \right) }}^{\left( \alpha
\right) }\left( \mathbf{k}^{\left( \alpha \right) }\right) +D^{\left( \alpha
\right) }\right) +\sum_{\beta \neq \alpha }V\left( \mathbf{r}^{\left( \alpha
\right) }-\mathbf{r}^{\left( \beta \right) }\right) }{\partial k_{j}^{\left(
\alpha \right) }}\right) _{\mid \mathbf{A}=0}$ .
We can now use the following facts : $$\begin{aligned}
G_{0}^{ij}\left( \mathbf{x-y}\right) &=&\frac{\delta ^{ij}+\frac{\left(
\mathbf{x-y}\right) _{i}\left( \mathbf{x-y}\right) _{j}}{\left| \mathbf{x-y}%
\right| ^{2}}}{2\left| \mathbf{x-y}\right| } \\
\varepsilon ^{lmi}\nabla _{m}G_{0}^{ij}\left( \mathbf{x-y}\right)
&=&\varepsilon ^{lji}\frac{\left( \mathbf{x-y}\right) _{i}}{\left| \mathbf{%
x-y}\right| ^{3}} \\
\varepsilon ^{lri}\varepsilon ^{jmp}\nabla _{r}\nabla _{p}G_{0}^{ij}\left(
\mathbf{x-y}\right) &=&\frac{\delta ^{ij}-3\frac{\left( \mathbf{x-y}\right)
_{i}\left( \mathbf{x-y}\right) _{j}}{\left| \mathbf{x-y}\right| ^{2}}-\frac{%
8\pi }{3}\delta ^{ij}\delta \left( \mathbf{x-y}\right) }{\left| \mathbf{x-y}%
\right| ^{3}}\end{aligned}$$ Reintroducing the effective individual magnetization operator as : $$\mathcal{M}^{ef\left( \alpha \right) }\left( \mathbf{r}^{\left( \alpha
\right) }\right) =\mathbf{\mu }\left( \mathbf{r}^{\left( \alpha \right)
}\right) +\mathbf{\mu }^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right)
\nabla _{k}$$ one can rewrite ultimately at the order $\hbar ^{2}$ :
$$\begin{aligned}
H_{\text{magnetic}} &=&-\sum_{\alpha \neq \beta }\frac{1}{2}\frac{\mathbf{%
\jmath }_{1}\mathbf{\left( \mathbf{r}^{\left( \alpha \right) }\right)
.\jmath }_{1}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right) }+%
\frac{\jmath _{1}\mathbf{\left( \mathbf{r}^{\left( \alpha \right) }\right) .}%
\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right) \jmath _{1}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right)
}.\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta
\right) }\right) }{\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}%
^{\left( \beta \right) }\right| ^{2}}}{2\left| \mathbf{r}^{\left( \alpha
\right) }\mathbf{-r}^{\left( \beta \right) }\right| } \\
&&\mathbf{+}\frac{1}{2}\sum_{\alpha \neq \beta }\left[ \mathcal{M}^{ef\left(
\alpha \right) }.\left( \frac{\left( \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right) }{\left| \mathbf{r}^{\left(
\alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right| ^{3}}\times
\mathbf{\jmath }_{1}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right)
}\right) \mathbf{+H.C.}\right] \\
&&+\frac{1}{2}\sum_{\alpha \neq \beta }\frac{\mathcal{M}^{ef\left( \alpha
\right) }.\mathcal{M}^{ef\left( \beta \right) }-3\frac{\mathcal{M}%
_{i}^{ef\left( \alpha \right) }.\left( \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right) \mathcal{M}^{ef\left( \beta
\right) }.\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right) }{\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r%
}^{\left( \beta \right) }\right| ^{2}}-\frac{8\pi }{3}\mathcal{M}^{ef\left(
\alpha \right) }.\mathcal{M}^{ef\left( \beta \right) }\delta \left( \mathbf{r%
}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) }{%
\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right| ^{3}} \\
&&\mathbf{+}\sum_{\alpha \neq \beta ,\alpha \neq \gamma ,\beta \neq \gamma
}\left( \jmath _{1i}\mathbf{\left( \mathbf{r}^{\left( \gamma \right)
}\right) \times \frac{\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}%
^{\left( \beta \right) }\right) }{\left| \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right| ^{3}}}\right) \mathbf{.}\tilde{\mu%
}\mathbf{\left( \mathbf{r}^{\left( \alpha \right) }\right) .}\left( \frac{%
\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right) }{\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right| ^{3}}\mathbf{\times }\jmath _{1j}\mathbf{\left(
\mathbf{r}^{\left( \beta \right) }\right) }\right) \\
&&+\sum_{\alpha \neq \beta }\frac{1}{2}\left( \frac{\jmath _{1}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) -}\frac{\jmath _{1}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) .}\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) \left(
\mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right) }{\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right| ^{2}}}{2\left| \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right| }+\mathcal{M}^{ef\left( \alpha
\right) }\times \frac{\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}%
^{\left( \beta \right) }\right) }{\left| \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right| ^{3}}\right) _{j} \\
&&\times \frac{\partial ^{2}\left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{k}^{\left( \alpha \right)
}\right) +D^{\left( \alpha \right) }\right) }{\partial k_{j}^{\left( \alpha
\right) }\partial k_{k}^{\left( \alpha \right) }} \\
&&\times \left( \frac{\jmath _{1}\mathbf{\left( \mathbf{r}^{\left( \alpha
\right) }\right) -}\frac{\jmath _{1}\mathbf{\left( \mathbf{r}^{\left( \alpha
\right) }\right) .}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}%
^{\left( \beta \right) }\right) \left( \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right) }{\left| \mathbf{r}^{\left(
\alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right| ^{2}}}{2\left|
\mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right| }+\mathcal{M}^{ef\left( \alpha \right) }\times \frac{\left( \mathbf{%
r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) }{%
\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right| ^{3}}\right) _{k}\end{aligned}$$
Note here that the double scalar product involving the 2 tensor $\tilde{\mu}%
\mathbf{\left( \mathbf{r}^{\left( \alpha \right) }\right) }$ is performed on each of its indices separately.
The interpretation of the various terms can done by comparison to the very similar Breit Hamiltonian for the Dirac electron. Given the current is decomposed in two parts due to the velocity of the particles and their magnetization, the magnetic part of the Hamiltonian is mainly a current-current interaction. The first term is the usual current-current interaction involving $\mathbf{\jmath }_{1}\mathbf{\left( \mathbf{r}^{\left(
\alpha \right) }\right) }$ only, whereas the second term mixing $\mathbf{%
\jmath }_{1}\mathbf{\left( \mathbf{r}^{\left( \alpha \right) }\right) }$ and the magnetization of other particles is a magnetization-orbit coupling between different particles. This interaction is formally similar to the spin-orbit interaction term, except that here we are here in a non relativistic context, and that the magnetization arise as a band phenomenon. The third term is for the magnetization-magnetization interaction term which is formally similar to the spin-spin interaction term. The two last terms are new with respect to the usual Breit Hamiltonian. The fourth one is again a magnetization orbit coupling but of higher order, since it involves triples of particles. The last term is a correction to the energy due to the development to the second order of the free energy. It is a shift of the individual particles energy operator due to the field created by other particles. It should be present in the Breit Hamiltonian but is in fact neglected while considering the quantum field derivation of this last one since it is of order higher than $\frac{1}{m^{2}}$ for the Dirac case ( [Landau1]{}).
### Full Hamiltonian
Since our development was in the second order in the currents we can simply gather the magnetic part and the electric part of the Hamiltonian to obtain ultimately the full $P$ particles Hamiltonian : $$\begin{aligned}
H_{N} &=&\sum_{\alpha }\hat{\varepsilon}_{0N^{\left( \alpha \right)
}}^{\left( \alpha \right) }\left( \mathbf{k}^{\left( \alpha \right) }\right)
+\frac{1}{2}\sum_{\alpha \neq \beta }V\left( \mathbf{r}^{\left( \alpha
\right) }-\mathbf{r}^{\left( \beta \right) }\right) \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\left[ \sum_{M^{\left( \alpha \right)
}}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right) _{N^{\left(
\alpha \right) }M^{\left( \alpha \right) }}\left( \mathcal{A}%
_{0}^{R_{m}^{\alpha }}\right) _{M^{\left( \alpha \right) }N^{\left( \alpha
\right) }}\right] \nabla _{R_{m}^{\left( \alpha \right) }}\nabla
_{R_{l}^{\left( \alpha \right) }}\sum_{\beta \neq \alpha }V\left( \mathbf{r}%
^{\left( \alpha \right) }-\mathbf{r}^{\left( \beta \right) }\right) \\
&&+\frac{\hbar ^{2}}{4}\sum_{\alpha }\frac{\left[ \left[ \hat{\varepsilon}%
_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left( \mathbf{k}%
^{\left( \alpha \right) }\right) ,\mathcal{A}_{0}^{R_{m}^{\left( \alpha
\right) }}\right] _{N^{\left( \alpha \right) }M^{\left( \alpha \right)
}}\left( \mathcal{A}_{0}^{R_{l}^{\left( \alpha \right) }}\right) _{M^{\left(
\alpha \right) }N^{\left( \alpha \right) }}\nabla _{R_{m}^{\left( \alpha
\right) }}\nabla _{R_{l}^{\left( \alpha \right) }}\right] \sum_{\beta \neq
\alpha }V\left( \mathbf{r}^{\left( \alpha \right) }-\mathbf{r}^{\left( \beta
\right) }\right) }{\hat{\varepsilon}_{0M^{\left( \alpha \right) }}^{\left(
\alpha \right) }-\hat{\varepsilon}_{0N^{\left( \alpha \right) }}} \\
&&-\sum_{\alpha \neq \beta }\frac{1}{2}\frac{\mathbf{\jmath }_{1}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) .\jmath }_{1}\mathbf{%
\left( \mathbf{r}^{\left( \beta \right) }\right) }+\frac{\jmath _{1}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) .}\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) \jmath
_{1}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right) }.\left(
\mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right) }{\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right| ^{2}}}{2\left| \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right| } \\
&&\mathbf{+}\frac{1}{2}\sum_{\alpha \neq \beta }\left[ \left( \mathcal{M}%
^{ef\left( \alpha \right) }\times \frac{\left( \mathbf{r}^{\left( \alpha
\right) }\mathbf{-r}^{\left( \beta \right) }\right) }{\left| \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right| ^{3}}%
\right) .\mathbf{\jmath }_{1}\mathbf{\left( \mathbf{r}^{\left( \beta \right)
}\right) +H.C.}\right] \\
&&+\frac{1}{2}\sum_{\alpha \neq \beta }\frac{\mathcal{M}^{ef\left( \alpha
\right) }.\mathcal{M}^{ef\left( \beta \right) }-3\frac{\mathcal{M}%
_{i}^{ef\left( \alpha \right) }.\left( \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right) \mathcal{M}^{ef\left( \beta
\right) }.\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right) }{\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r%
}^{\left( \beta \right) }\right| ^{2}}-\frac{8\pi }{3}\mathcal{M}^{ef\left(
\alpha \right) }.\mathcal{M}^{ef\left( \beta \right) }\delta \left( \mathbf{r%
}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) }{%
\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right| ^{3}} \\
&&\mathbf{+}\sum_{\alpha \neq \beta }\jmath _{1i}\mathbf{\left( \mathbf{r}%
^{\left( \alpha \right) }\right) .}\left( \mathbf{\frac{\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) }{\left|
\mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right| ^{3}}\times }\tilde{\mu}\mathbf{\left( \mathbf{r}^{\left( \alpha
\right) }\right) \times }\frac{\left( \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right) }{\left| \mathbf{r}^{\left(
\alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right| ^{3}}\right)
.\jmath _{1j}\mathbf{\left( \mathbf{r}^{\left( \beta \right) }\right) } \\
&&+\sum_{\alpha \neq \beta }\frac{1}{2}\left( \frac{\jmath _{1}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) -}\frac{\jmath _{1}\mathbf{%
\left( \mathbf{r}^{\left( \alpha \right) }\right) .}\left( \mathbf{r}%
^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right) \left(
\mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right) }{\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right| ^{2}}}{2\left| \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right| }+\mathcal{M}^{ef\left( \alpha
\right) }\times \nabla _{k}\frac{\left( \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right) }{\left| \mathbf{r}^{\left(
\alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right| ^{3}}\right) _{j}
\\
&&\times \frac{\partial ^{2}\left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{k}^{\left( \alpha \right)
}\right) +D^{\left( \alpha \right) }\right) }{\partial k_{j}^{\left( \alpha
\right) }\partial k_{k}^{\left( \alpha \right) }} \\
&&\times \left( \frac{\jmath _{1}\mathbf{\left( \mathbf{r}^{\left( \alpha
\right) }\right) -}\frac{\jmath _{1}\mathbf{\left( \mathbf{r}^{\left( \alpha
\right) }\right) .}\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}%
^{\left( \beta \right) }\right) \left( \mathbf{r}^{\left( \alpha \right) }%
\mathbf{-r}^{\left( \beta \right) }\right) }{\left| \mathbf{r}^{\left(
\alpha \right) }\mathbf{-r}^{\left( \beta \right) }\right| ^{2}}}{2\left|
\mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right| }+\mathcal{M}^{ef\left( \alpha \right) }\times \nabla _{k}\frac{%
\left( \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left( \beta \right)
}\right) }{\left| \mathbf{r}^{\left( \alpha \right) }\mathbf{-r}^{\left(
\beta \right) }\right| ^{3}}\right) _{i}\end{aligned}$$
with : $$\begin{aligned}
\mathbf{\jmath }_{1i}\mathbf{\left( \mathbf{r}^{\left( \alpha \right)
}\right) } &=&e\sum_{\alpha }\frac{\partial \left( \hat{\varepsilon}%
_{0N^{\left( \alpha \right) }}^{\left( \alpha \right) }\left( \mathbf{k}%
^{\left( \alpha \right) }\right) +D^{\left( \alpha \right) }\right) }{%
\partial k_{j}^{\left( \alpha \right) }}\left[ \left( \delta _{i}^{j}-%
\mathbf{\emph{A}^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}%
_{j}^{\left( \alpha \right) }}+\mathbf{\emph{A}}_{j,i}\right) \right] _{\mid
\mathbf{A}=0} \\
&&-\sum_{\alpha }\frac{\partial \sum_{\beta }V\left( \mathbf{r}^{\left(
\alpha \right) }-\mathbf{r}^{\left( \beta \right) }\right) }{\partial
r_{j}^{\left( \alpha \right) }}\left[ \frac{\partial \tilde{\emph{A}}
^{R_{j}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left( \alpha \right) }%
}\left( \delta _{i}^{n}-\mathbf{\emph{A}^{R_{i}^{\left( \alpha \right) }}}%
\nabla _{\mathbf{r}_{n}^{\left( \alpha \right) }}+\mathbf{\emph{A}}_{n,i\mid
\mathbf{A}=0}\right) +\frac{\partial \tilde{\emph{A}}^{R_{j}^{\left( \alpha
\right) }}}{\partial A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }%
\right] _{\mid \mathbf{A}=0} \\
\mathcal{M}^{ef\left( \alpha \right) }\left( \mathbf{r}^{\left( \alpha
\right) }\right) &=&\mathbf{\mu }\left( \mathbf{r}^{\left( \alpha \right)
}\right) +\mathbf{\mu }^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right)
\nabla _{k}\end{aligned}$$ Recall at that point that we have performed a development in currents (or field) to the lowest order. As such in all our expression the electromagnetic field is set to $0$. As a consequence, the variables $%
\mathbf{k}^{\left( \alpha \right) }$, $\mathbf{r}^{\left( \alpha \right) }$ are computed with $\mathbf{A}=0$.
The dynamical variables are thus given by : $$\begin{aligned}
\mathbf{r}_{N}^{\left( \alpha \right) } &=&\mathbf{R}^{\left( \alpha \right)
}+\emph{A}_{N^{\left( \alpha \right) }}^{\mathbf{R}\left( ^{\alpha }\right) }
\\
\mathbf{k}_{N}^{\left( \alpha \right) } &=&\mathbf{K}^{\left( \alpha \right)
}\end{aligned}$$ Apart from the magnetic part of the Hamiltonian, already discussed, we have included the usual electrostatic part, as well as the Darwin term, which is analogous to the eponymous term in the Breit Hamiltonian. Note however that this term in our context is not only a contact term (i.e. of Dirac delta type) since its form depends ultimately on the form of the Berry phases that depend on the structure of the system at stake.
Let us also remark ultimately that the expansion performed, is different from the one chosen in [@Landau1] to derive the Breit Hamiltonian. Actually, the derivation of this last one through the context of quantum field involves a power expansion in $\frac{1}{m}$, which is in fact performed at the second order.
Conclusion
==========
The diagonalization of the Hamiltonian for a Bloch electron in a magnetic field is an old problem is solid state physics initiated principally by Blount [@Blount], who developed a general procedure for the removal of the interband matrix elements based on a asymptotic series expansion in the fields strength. In this way, the actual effective one-band Hamiltonian was obtained to the second order in the magnetic field. In this paper,we came back, in a way, to this old problem, but by trying instead to derive an in-band Hamiltonian as a series expansion in the Planck constant. A first attempt in this direction was done when we provided a procedure at order $%
\hbar $ for an arbitrary matrix valued Hamiltonian [@PIERRESEMIDIAG]. This method resulted in an effective diagonal Hamiltonian in terms of gauge-covariant but noncanonical, actually noncommutative, coordinates. It has also revealed that a generalized Peierls substitution taking into account a Berry phase term must be considered for the semiclassical treatment of electrons in a magnetic field [@PIERREEBS]. In particular, the Bohr-Sommerfeld quantization condition when reformulated in terms of the generalized Peierls substitution leads to a modification of the semiclassical quantization rules as well as to a generalization of the cross-sectional area derived by Roth [@ROTH]. Soon later, the semiclassical diagonalization was extended to any order, when we developed another method of diagonalization where $\hbar $ was considered as a running parameter. This method allowed us to solve formally the problem in terms of a differential equation for the diagonal Hamiltonian with respect to Planck constant $\hbar $, which could, in principle, be solved recursively by a series expansion in $\hbar $ [@SERIESPIERRE]. As an example, the energy spectrum of a Bloch electron in an electric field was derived to all order in $\hbar .$ However, in the presence of a magnetic field, the differential approach beyond the semiclassical turned out to be very complicated.
In this paper, we solved the Hamiltonian diagonalization for a Bloch electron in an electromagnetic field to the second order in $\hbar $ by applying a radically different procedure, which was recently developed and provides a general diagonal expression for any kind of matrix valued Hamiltonian [@DIAGOEXACT] (this diagonal expression is also solution of the differential equation of [@SERIESPIERRE]). The main advantage of our method with respect to Blount’s procedure is obviously that our result is valid even in the presence of strong external electromagnetic fields. But a second important advantage with an expansion in $\hbar $ is that it allows us to consider particles in interaction. Indeed, although the removal of the interband matrix elements to the second order in $\hbar $ for a Bloch electron is important in itself, the principal objective of this paper was the determination of the effective Hamiltonian of interacting Bloch electrons living in different energy bands. As we have shown, even if the electrostatic interaction dominates the magnetic one, effects like magnetic moment-moment interaction mediated by the magnetic part of the full electromagnetic interaction requires a computation to second order in $\hbar
$. Comparison with other methods is difficult because to our knowledge other results for interacting Bloch electrons do not exist. Nevertheless because of the strong analogy with the Dirac equation it is possible to try a comparison with the Breit Hamiltonian for the Dirac electrons [@BREIT]. As for Breit, we found that the electronic current is made of two contributions: one comes from the velocity and the other is a magnetic moment current similar to the spin current for Dirac [@PIERREDIRAC]. It is this last one which is responsible for the magnetic moment-moment interaction similar to the spin-spin interaction for Dirac. Another important interaction revealed by our approach is the moment-orbit coupling by analogy with the spin-orbit one. From the results of this paper, our goal in the future is to consider the physical relevance of interactions bringing into play magnetic moments.
Appendix
========
Since the right hand side of the Maxwell equation involves $-\int d^{3}y%
\left[ \delta ^{ij}\delta ^{3}\left( \mathbf{x-y}\right) +\frac{\partial ^{2}%
}{\partial x^{i}\partial x^{j}}\frac{1}{4\pi \left\vert \mathbf{x-y}%
\right\vert }\right] \frac{\delta }{\delta A^{j}\left( \mathbf{y,t}\right) }%
H_{mat}$ the operator $\delta O\left( \mathbf{x}\right) $ acts on $\mathbf{A}%
\left( \mathbf{x,}t\right) $ as : $$\delta O\left( \mathbf{x}\right) \mathbf{A}\left( \mathbf{x,}t\right)
=-\sum_{\alpha }\int d^{3}y\left[ \delta ^{ir}\delta ^{3}\left( \mathbf{x-y}%
\right) +\frac{\partial ^{2}}{\partial x^{i}\partial x^{r}}\frac{1}{4\pi
\left\vert \mathbf{x-y}\right\vert }\right] \delta O_{1}\left( \mathbf{y}%
\right) \mathbf{A}\left( \mathbf{y,}t\right)$$ where the operator $\delta O_{1}\left( \mathbf{x}\right) $ is obtained by isolating in $-\frac{\delta }{\delta A^{j}\left( \mathbf{y,t}\right) }%
H_{mat} $ the linear term in the electromagnetic potential. Collecting the relevant terms yields directly : $$\begin{aligned}
&&\left( \delta O_{1}\left( \mathbf{x}\right) \mathbf{A}\left( \mathbf{x,}%
t\right) \right) _{i} \\
&=&-\sum_{\alpha }\frac{\partial }{\partial \pi _{j}^{\left( \alpha \right)
}\left( \mathbf{x,t}\right) }\left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha
\right) }\right) +D^{\left( \alpha \right) }\right) \\
&&\times \left[ \left( \delta _{k}^{j}-\tilde{\emph{A}}\mathbf{%
^{R_{k}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{j}^{\left( \alpha
\right) }}\right) \right] \left[ \frac{\partial \tilde{\emph{A}}%
^{R_{l}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left( \alpha \right) }%
}\left( \delta _{i}^{n}-\mathbf{\emph{A}^{R_{i}^{\left( \alpha \right) }}}%
\nabla _{\mathbf{r}_{n}^{\left( \alpha \right) }}\right) \right] \nabla
_{l}A^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right) \\
&&+e\sum_{\alpha }\frac{\partial }{\partial \pi _{j}^{\left( \alpha \right)
}\left( \mathbf{x,t}\right) }\left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha
\right) }\right) +D^{\left( \alpha \right) }\right) \times \\
&&\left( -\nabla _{\mathbf{r}_{j}^{\left( \alpha \right) }}\left( \frac{%
\partial \tilde{\emph{\ A}}^{R_{n}^{\left( \alpha \right) }}}{\partial
A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }\right) +e\nabla _{%
\mathbf{r}_{m}^{\left( \alpha \right) }}\left( \frac{\partial \tilde{\emph{A}%
}^{R_{n}^{\left( \alpha \right) }}}{\partial \pi _{i}^{\left( \alpha \right)
}}-\frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{%
\partial \pi _{u}^{\left( \alpha \right) }}\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{u}^{\left( \alpha
\right) }}\right) +\mathbf{\frac{d\mathbf{\emph{A}}_{u,r}}{dA^{i}\left(
\mathbf{r}^{\left( \alpha \right) }\right) }}\nabla _{\mathbf{r}_{u}^{\left(
\alpha \right) }}\nabla _{\mathbf{r}_{n}^{\left( \alpha \right) }}\right)
A^{n}\left( \mathbf{r}^{\left( \alpha \right) }\right) \\
&&-e\sum_{\alpha }\frac{\partial v\left( \mathbf{r}^{\left( \alpha \right)
}\right) }{\partial r_{j}^{\left( \alpha \right) }}\frac{\partial \tilde{%
\emph{\ A}}^{R_{j}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left(
\alpha \right) }}\times \\
&&\left( \nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}\left( \frac{%
\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{\partial
A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }\right) -e\nabla _{%
\mathbf{r}_{m}^{\left( \alpha \right) }}\left( \frac{\partial \tilde{\emph{A}%
}^{R_{n}^{\left( \alpha \right) }}}{\partial \pi _{i}^{\left( \alpha \right)
}}-\frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{%
\partial \pi _{u}^{\left( \alpha \right) }}\mathbf{\emph{A}^{R_{i}^{\left(
\alpha \right) }}}\nabla _{\mathbf{r}_{u}^{\left( \alpha \right) }}\right) +%
\mathbf{\frac{d\mathbf{\emph{A}}_{u,r}}{dA^{i}\left( \mathbf{r}^{\left(
\alpha \right) }\right) }}\nabla _{\mathbf{r}_{u}^{\left( \alpha \right)
}}\nabla _{\mathbf{r}_{n}^{\left( \alpha \right) }}\right) A^{n}\left(
\mathbf{r}^{\left( \alpha \right) }\right) \\
&&+e\sum_{\alpha }\frac{\partial v\left( \mathbf{r}^{\left( \alpha \right)
}\right) }{\partial r_{j}^{\left( \alpha \right) }}\left( \delta _{k}^{j}-%
\tilde{\emph{A}}\mathbf{^{R_{k}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}%
_{j}^{\left( \alpha \right) }}\right) \left[ \nabla _{l}\left( -e\frac{%
\partial \tilde{\emph{A}}^{R_{l}^{\left( \alpha \right) }}}{\partial \pi
_{n}^{\left( \alpha \right) }}\left( \delta _{i}^{n}-\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{n}^{\left( \alpha
\right) }}\right) \right) \right] A^{k}\left( \mathbf{r}^{\left( \alpha
\right) }\right) \\
&&-\frac{1}{2}\sum_{\alpha }\mathbf{B}^{\left( \alpha \right) }\left(
\mathbf{x}^{\left( \alpha \right) }\right) .\left[ \frac{d\pi _{j}^{\left(
\alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }%
\frac{\partial }{\partial \pi _{j}^{\left( \alpha \right) }\left( \mathbf{x,t%
}\right) }+\frac{dr_{j}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) }\frac{\partial }{\partial r_{j}^{\left(
\alpha \right) }}\right] \left( \mathbf{\mu }\left( \mathbf{x}^{\left(
\alpha \right) }\right) +\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right)
\mathbf{.}\nabla _{R_{k}}\mathbf{.B}^{\left( \alpha \right) }\left( \mathbf{x%
}^{\left( \alpha \right) }\right) \right) \\
&&+\frac{1}{2}e\sum_{\alpha }\left[ \left( \mathbf{\mu }\left( \mathbf{x}%
^{\left( \alpha \right) }\right) \times \mathbf{\nabla }\right) _{p}.\left(
\nabla _{l}A^{p}\left( \mathbf{r}^{\left( \alpha \right) }\right) \left(
\frac{\partial \tilde{\emph{A}}^{R_{l}^{\left( \alpha \right) }}}{\partial
\pi _{n}^{\left( \alpha \right) }}\left( \delta _{i}^{n}-\tilde{\emph{A}}%
\mathbf{^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{n}^{\left(
\alpha \right) }}\right) \right) \right) \delta \left( \mathbf{x-r}^{\left(
\alpha \right) }\right) \right] \\
&&-\frac{1}{2}\left[ \mathbf{B}\left( \mathbf{x}^{\left( \alpha \right)
}\right) \mathbf{.\tilde{\mu}}\left( \mathbf{x}^{\left( \alpha \right)
}\right) \times \mathbf{\nabla }\delta \left( \mathbf{x-R}^{\left( \alpha
\right) }\right) -\mathbf{\nabla }\delta \left( \mathbf{x-R}^{\left( \alpha
\right) }\right) \times \mathbf{\tilde{\mu}}\left( \mathbf{x}^{\left( \alpha
\right) }\right) \mathbf{.B}\left( \mathbf{x}^{\left( \alpha \right)
}\right) \right] _{i} \\
&&-\left( \frac{\delta }{\delta \mathbf{A}\left( \mathbf{x}\right) }\frac{%
\delta }{\delta \mathbf{A}_{j}\left( \mathbf{x}\right) }H_{mat}\right)
_{\mid \mathbf{A}=0}\mathbf{A}_{j}\left( \mathbf{x}\right) +\mathbf{B}%
^{\left( \alpha \right) }\mathbf{.\tilde{\mu}.B}^{\left( \alpha \right) }\end{aligned}$$
The Fourier transform of this operator is straightforwardly obtained as being equal to : $$\begin{aligned}
&&\delta O_{1}\left( \mathbf{p}\right) \\
&=&-\sum_{\alpha }\frac{\partial }{\partial \pi _{j}^{\left( \alpha \right)
}\left( \mathbf{x,t}\right) }\left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha
\right) }\right) +D^{\left( \alpha \right) }\right) \\
&&\times \left[ \left( \delta _{k}^{j}-\tilde{\emph{A}}\mathbf{%
^{R_{k}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{j}^{\left( \alpha
\right) }}\right) \right] \left[ \frac{\partial \tilde{\emph{A}}%
^{R_{l}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left( \alpha \right) }%
}\left( \delta _{i}^{n}-\tilde{\emph{A}}\mathbf{^{R_{i}^{\left( \alpha
\right) }}}\nabla _{\mathbf{r}_{n}^{\left( \alpha \right) }}\right) \right]
p_{l}A^{k}\left( \mathbf{r}^{\left( \alpha \right) }\right) \\
&&+e\sum_{\alpha }\frac{\partial }{\partial \pi _{j}^{\left( \alpha \right)
}\left( \mathbf{x,t}\right) }\left( \hat{\varepsilon}_{0N^{\left( \alpha
\right) }}^{\left( \alpha \right) }\left( \mathbf{\pi }^{\left( \alpha
\right) }\right) +D^{\left( \alpha \right) }\right) \times \\
&&\left( -\nabla _{\mathbf{r}_{j}^{\left( \alpha \right) }}\left( \frac{%
\partial \tilde{\emph{\ A}}^{R_{n}^{\left( \alpha \right) }}}{\partial
A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }\right) +e\nabla _{%
\mathbf{r}_{m}^{\left( \alpha \right) }}\left( \frac{\partial \tilde{\emph{A}%
}^{R_{n}^{\left( \alpha \right) }}}{\partial \pi _{i}^{\left( \alpha \right)
}}-\frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{%
\partial \pi _{u}^{\left( \alpha \right) }}\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{u}^{\left( \alpha
\right) }}\right) +\mathbf{\frac{d\mathbf{\emph{A}}_{u,r}}{dA^{i}\left(
\mathbf{r}^{\left( \alpha \right) }\right) }}\nabla _{\mathbf{r}_{u}^{\left(
\alpha \right) }}\nabla _{\mathbf{r}_{n}^{\left( \alpha \right) }}\right)
A^{n}\left( \mathbf{r}^{\left( \alpha \right) }\right) \\
&&-e\sum_{\alpha }\frac{\partial v\left( \mathbf{r}^{\left( \alpha \right)
}\right) }{\partial r_{j}^{\left( \alpha \right) }}\frac{\partial \tilde{%
\emph{\ A}}^{R_{j}^{\left( \alpha \right) }}}{\partial \pi _{n}^{\left(
\alpha \right) }}\times \\
&&\left( \nabla _{\mathbf{r}_{m}^{\left( \alpha \right) }}\left( \frac{%
\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{\partial
A^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }\right) -e\nabla _{%
\mathbf{r}_{m}^{\left( \alpha \right) }}\left( \frac{\partial \tilde{\emph{A}%
}^{R_{n}^{\left( \alpha \right) }}}{\partial \pi _{i}^{\left( \alpha \right)
}}-\frac{\partial \tilde{\emph{A}}^{R_{n}^{\left( \alpha \right) }}}{%
\partial \pi _{u}^{\left( \alpha \right) }}\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{u}^{\left( \alpha
\right) }}\right) +\mathbf{\frac{d\mathbf{\emph{A}}_{u,r}}{dA^{i}\left(
\mathbf{r}^{\left( \alpha \right) }\right) }}\nabla _{\mathbf{r}_{u}^{\left(
\alpha \right) }}\nabla _{\mathbf{r}_{n}^{\left( \alpha \right) }}\right)
A^{n}\left( \mathbf{r}^{\left( \alpha \right) }\right) \\
&&+e\sum_{\alpha }\frac{\partial v\left( \mathbf{r}^{\left( \alpha \right)
}\right) }{\partial r_{j}^{\left( \alpha \right) }}\left( \delta _{k}^{j}-%
\tilde{\emph{A}}\mathbf{^{R_{k}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}%
_{j}^{\left( \alpha \right) }}\right) \left[ \nabla _{l}\left( -e\frac{%
\partial \tilde{\emph{A}}^{R_{l}^{\left( \alpha \right) }}}{\partial \pi
_{n}^{\left( \alpha \right) }}\left( \delta _{i}^{n}-\tilde{\emph{A}}\mathbf{%
^{R_{i}^{\left( \alpha \right) }}}\nabla _{\mathbf{r}_{n}^{\left( \alpha
\right) }}\right) \right) \right] A^{k}\left( \mathbf{r}^{\left( \alpha
\right) }\right) \\
&&-\frac{1}{2}\sum_{\alpha }\mathbf{B}^{\left( \alpha \right) }\left(
\mathbf{x}^{\left( \alpha \right) }\right) \left[ \frac{d\pi _{j}^{\left(
\alpha \right) }}{dA^{i}\left( \mathbf{r}^{\left( \alpha \right) }\right) }%
\frac{\partial }{\partial \pi _{j}^{\left( \alpha \right) }\left( \mathbf{x,t%
}\right) }+\frac{dr_{j}^{\left( \alpha \right) }}{dA^{i}\left( \mathbf{r}%
^{\left( \alpha \right) }\right) }\frac{\partial }{\partial r_{j}^{\left(
\alpha \right) }}\right] \left( \mathbf{\mu }\left( \mathbf{x}^{\left(
\alpha \right) }\right) +\mathbf{\hat{\mu}}^{k}\left( \mathbf{x}\right)
\mathbf{.}\nabla _{R_{k}}\mathbf{.B}^{\left( \alpha \right) }\left( \mathbf{x%
}^{\left( \alpha \right) }\right) \right) \\
&&+\mathbf{B}^{\left( \alpha \right) }\mathbf{.\tilde{\mu}.B}^{\left( \alpha
\right) }\end{aligned}$$
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|
---
abstract: 'If $S=\{v_1,\ldots, v_k\}$ is an ordered subset of vertices of a connected graph $G$ and $e$ is an edge of $G$, then the vector $r_G(e|S) = (d_G(v_1,e), \ldots, d_G(v_k,e))$ is the edge metric $S$-representation of $e$. If the vertices of $G$ have pairwise different edge metric $S$-representations, then $S$ is an edge metric generator for $G$. The cardinality of a smallest edge metric generator is the edge metric dimension ${\rm edim}(G)$ of $G$. A general sharp upper bound on the edge metric dimension of hierarchical products $G(U)\sqcap H$ is proved. Exact formula is derived for the case when $|U| = 1$. An integer linear programming model for computing the edge metric dimension is proposed. Several examples are provided which demonstrate how these two methods can be applied to obtain the edge metric dimensions of some applicable graphs.'
author:
- 'Sandi Klavžar $^{a,b,c}$'
- 'Mostafa Tavakoli $^{d,}$[^1]'
title: Edge metric dimensions via hierarchical product and integer linear programming
---
$^a$ Faculty of Mathematics and Physics, University of Ljubljana, Slovenia\
[[email protected]]{}
$^b$ Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia\
$^c$ Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia\
$^d$ Department of Applied Mathematics, Faculty of Mathematical Sciences,\
Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran\
[m$\[email protected]]{}
[**Key words:**]{} metric dimension; edge metric dimension; hierarchical product; integer linear programming; molecular graph
[**AMS Subj. Class:**]{} 05C12; 05C76
Introduction {#sec:intro}
============
Graphs considered in this paper are connected, finite, and simple. If $G$ is a graph and $u, v\in V(G)$, then $d_G(u,v)$ denotes the shortest-path distance between $u$ and $v$. If $S=\{v_1,\ldots,v_k\}$ is an ordered subset of $V(G)$, then the [*metric $S$-representation*]{} of a vertex $u\in V(G)$ is the vector $r_G(u|S) = (d_G(v_1,u), \ldots, d_G(v_k,u))$. The set $S$ [*distinguishes*]{} vertices $u$ and $v$ if $r_G(u|S)\neq r_G(v|S)$ and $S$ is a [*metric generator*]{} for $G$ if each pair of vertices of $G$ is distinguished by $S$. A metric generator of smallest cardinality is called a [*metric basis*]{} for $G$, its order being the [*metric dimension*]{} ${\rm dim}(G)$ of $G$.
The sources for the metric dimension are papers [@Harary; @Slater]. Afterwards the concept was studied in depth, classical references include [@bailey-2011; @caceres-2007; @Chartrand-2000], papers dealing with applications of the metric dimension in modeling of real world problems include [@johnson-1993; @khuller-1996], while for some of the recent developments we refer to [@vetrik-2020; @zhang-2020]. Several variations of the concept were also studied such as the local metric dimension [@okamoto-2010], independent resolving sets [@Chartrand-2003], strong resolving sets [@Oellermann-Peters-Fransen], and $k$-metric generators [@Estrada-Moreno]. Distinguishing edges instead of vertices seems an utmost natural variation, hence it comes as a surprise that the edge metric dimension was introduced only recently in [@kelenc-2018] as follows.
Let $G$ be a graph. If $u\in V(G)$ and $xy\in E(G)$, then the distance $d_G(u,xy)$ between $u$ and $xy$ is $\min\{d_G(u,x),d_G(u,y)\}$. If $S=\{v_1,\ldots,v_k\}$ is an ordered subset of $V(G)$, then the [*edge metric $S$-representation*]{} of an edge $e\in E(G)$ is the vector $$r_G(e|S) = (d_G(v_1,e), \ldots, d_G(v_k,e))\,.$$ $S$ is an [*edge metric generator*]{} for $G$ if the edges of $G$ have pairwise different edge metric $S$-representations. A smallest edge metric generator is an [*edge metric basis*]{} for $G$, its cardinality is the [*edge metric dimension*]{} ${{\rm edim}}(G)$ of $G$.
When someone thinks of a smart city, an intelligent transportation system (ITS) may quickly come to mind. Self-driving cars will probably soon play a crucial role in an ITS. Clearly, a self-driving car needs to determine its position on the city’s streets uniquely, hence each street needs a code which uniquely determines its location. If we represent the city with a graph $G$, where the edges of $G$ correspond to streets, then an edge metric generator of $G$ provides unique codes for the streets.
The seminal paper [@kelenc-2018] on the edge metric dimension brings a wealth of results, including a proof that the problem of finding the edge metric dimension of a graph is NP-hard and some approximation results for the invariant. It is also shown that ${\rm dim}(G)$ and ${{\rm edim}}(G)$ are in general incomparable, but it seems that in most cases ${\rm dim}(G) \le {{\rm edim}}(G)$ holds. In a subsequent paper [@zubrilina-2018] several problems from [@kelenc-2018] are answered, in particular, a classification of the graphs $G$ of order $n$ for which ${{\rm edim}}(G) = n-1$ holds is given. These graphs were also investigated in [@zhu-2019] where a polynomial algorithm is developed for their recognition. Papers [@adawiyah-2019; @filipovic-2019; @yang-2019; @zhang-2020b] determine the edge metric dimension for some families of graphs. Finally, in [@peterin-2019] the edge metric dimension of the join of graphs, the lexicographic product of graphs, and the corona product of graphs is reported.
In the next section we study the edge metric dimension of hierarchical products $G(U)\sqcap H$ of graphs. We prove a general sharp upper bound on ${{\rm edim}}(G(U)\sqcap H)$ and an exact result for the case when $|U| = 1$. Earlier known results on the corona product of graphs can be deduced from these results. In Section \[sec:ILP\] we propose an integer linear programming model for computing the edge metric dimension. In the final section several examples are provided that demonstrate how the methods proposed in the previous two sections can be applied to obtain the edge metric dimension of some interesting graphs, notably from mathematical chemistry.
To conclude the introduction we extend (edge) metric generators to vertex and edge subsets as follows. If $X\subseteq V(G)$, then $S\subseteq V(G)$ is a [*metric generator for $X$*]{} if the vertices from $X$ have pairwise different metric $S$-representations. A smallest metric generator for $X$ is a [*metric basis*]{} for $X$, its cardinality being the [*metric dimension*]{} ${\rm dim}_G(X)$ [*for $X$*]{}. In this notation, ${\rm dim}_G(V(G)) = {\rm dim}(G)$. Similarly, if $F\subseteq E(G)$, then $S\subseteq V(G)$ is an [*edge metric generator for $F$*]{} if the edges from $F$ have pairwise different edge metric $S$-representations. A smallest edge metric generator for $F$ is an [*edge metric basis*]{} for $F$, its cardinality is the [*edge metric dimension*]{} ${{\rm edim}}_G(F)$ [*for $F$*]{}. So ${{\rm edim}}_G(E(G)) = {{\rm edim}}(G)$.
Hierarchical products {#sec:hierarchical-products}
=====================
In this section we consider the edge metric dimension of the hierarchical product of graphs and mention in passing that the metric dimension and the fractional metric dimension of these products were studied in [@feng-2013], and the local metric dimension in [@klavzar-2020].
If $G$ and $H$ are graphs and $U\subseteq V(G)$, then the [*hierarchical product*]{} $G(U)\sqcap H$ of $G$ and $H$ (with respect to $U$) has the vertex set $V(G)\times V(H)$ and the edge set $$\{(g,h)(g',h'):\ gg'\in E(G), h=h'\} \cup \{(g,h)(g',h'):\ g=g'\in U, hh'\in E(H)\}\,.$$ Note that $G(U)\sqcap H$ contains $n(G)$ subgraphs isomorphic to $G$, they are called [*$G$-layers*]{}. Similarly, $G(U)\sqcap H$ contains $|U|$ subgraphs isomorphic to $H$, these are [*$H$-layers*]{}. The operation $\sqcap$ (for two and also more factors) was in the seminal paper [@barriere-2009] named the [*generalized hierarchical product*]{}, here we follow the reasonable suggestion from [@anderson-2017] to simplify the naming to the hierarchical product.
If $U\subseteq V(G)$ and $u,v\in V(G)$, then we say that a $u,v$-walk $W$ is a [*$u,v$-walk through $U$*]{} if $W$ is an $u,v$-walk in $G$ that contains some vertex of $U$, where the latter vertex could be one of $u$ and $v$. With $d_{G(U)}(u,v)$ we denote the length of a shortest $u,v$-walk through $U$. With this notation we can state the following fundamental observation from [@barriere-2009].
\[prp:distance\] If $G$ is a graph with $U\subseteq V(G)$ and $H$ is a graph, then $$d_{G(U)\sqcap H}((g,h),(g',h'))=\begin{cases}
d_{G(U)}(g,g')+d_H(h,h'); & h\neq h',\\
d_G(g,g'); & h=h'.
\end{cases}$$
To state our results, we need some more preparation. If $v$ is a vertex of a graph $G$ and $k\in {\mathbb N}_0$, then let $E_G(v,k)$ be the set of edges of $G$ that are at distance $k$ from $v$, that is, $$E_G(v,k) = \{e\in E(G):\ d_G(v,e) = k\}\,.$$ If $F\subseteq E(G)$ and $|F|\ge 2$, then we say that $X\subseteq V(G)$ is an [*equidistant discriminator for*]{} $F$, if $X$ is an edge metric generator for $F$. In the case when $|F|\le 1$, we define $\emptyset$ to be the only equidistant discriminator for $F$. We analogously define equidistant discriminators for vertex subsets of $G$. With this terminology we set $${{\rm edim}}(G(U)) = \min\{ \mid\! \bigcup_{u\in U\atop k\ge 0} S_G(u,k)\! \mid\ :\ S_G(u,k)\ {\rm equidistant\ discriminator\ for}\ E_G(v,k) \}\,.$$ That is, ${{\rm edim}}(G(U))$ is the cardinality of a smallest set of vertices which distinguish all pairs of edges that are equidistant from some vertex from $U$. In addition, we set $${{\rm edim}}^+(G(U)) = \min \left\{ \mid \left(\bigcup_{u\in U, k\ge 0} S_G(u,k)\right) \cup S_G(U) \mid \right\}\,,$$ where the minimum is taken over all equidistant discriminators $S_G(u,k)$ for $E_G(u,k)$ and over all equidistant discriminators $S_G(U)$ for $U$. After this preparation we can state the following bound.
\[t1\] If $G$ and $H$ are graphs and $U\subseteq V(G)$ with $|U|>1$, then $${{\rm edim}}(G(U)\sqcap H)\leq n(H) ({{\rm edim}}^+(G(U))+1)\,.$$
[[**Proof. **]{}]{}Note that the assumption $|U|>1$ implies that also $n(G) > 1$. To simplify the notation, set $X = G(U)\sqcap H$ for the rest of the proof. Let $S_G(u,k)$, $u\in U$, $k\ge 0$, be equidistant discriminators for $E_G(u,k)$, and $S_G(U)$ be an equidistant discriminator for $U$ which together realize ${{\rm edim}}^+(G(U))$. Set $S^T(G) = \bigcup_{u\in U, k\ge 0} S_G(u,k)$ and let $S=(S^T(G)\cup S_G(U))\times V(H)$. Select further a vertex $w\in U$ and set $S'=\{(w,y):\ y\in V(H)\}$. We claim that $S\cup S'$ is an edge metric generator for $X$. For this sake let $e$ and $f$ be arbitrary, different edges of $X$, and consider the following cases.
[**Case 1**]{}: $e$ and $f$ are both in $H$-layers.\
Suppose first that $e$ and $f$ are in the same $H$-layer, say $e = (g,h)(g,h')$ and $e = (g,h'')(g,h''')$. If $\{h,h'\}\cap \{h'',h'''\} = \emptyset$, that is, if $e$ and $f$ are not adjacent, then consider an arbitrary vertex $v\in S^T(G)$ and note that the vertex $(v,h)\in S$ distinguishes $e$ and $f$. In the second subcase suppose that $e$ and $f$ are adjacent, say $h = h''$. Now the vertex $(v,h')\in S$ distinguishes $e$ and $f$.
Suppose second that $e$ and $f$ are in different $H$-layers, say $e = (g,h)(g,h')$ and $e = (g',h'')(g',h''')$, where $g\ne g'$. Select a vertex $v\in S_G(U)$ which distinguishes $g$ and $g'$, that is, $d_G(v,g)\neq d_G(v,g')$. In the first subcase suppose that $\{h,h'\}\cap \{h'',h'''\} \ne \emptyset$, say $h=h''$. It this case the vertex $(v,h)\in S$ distinguishes $e$ and $f$. Suppose next that $\{h,h'\}\cap \{h'',h'''\} = \emptyset$. Then $d_X((v,h),e)=d_X((v,h), (g,h))$ and we may without loss of generality assume that $d_X((v,h),f)=d_X((v,h), (g',h''))$. If $d_X((v,h),e) \ne d_X((v,h),f)$, then $(v,h)$ distinguishes $e$ and $f$. Suppose next that $d_X((v,h),e) = d_X((v,h),f)$. Then $d_{G(U)}(v,g)=d_{G(U)}(v,g')+d_H(h,h'')$. Thus $d_{G(U)}(v,g)+d_H(h,h'')>d_{G(U)}(v,g)=d_{G(U)}(v,g')+d_H(h,h'')>d_G(v,g')+d_H(h'',h'')$ and so $d_X((v,h''),e) > d_X((v,h''),f)$. Therefore, the vertex $(v,h'')\in S$ distinguishes $e$ and $f$.
[**Case 2**]{}: $e$ and $f$ are are both in $G$-layers.\
Let $e=(g,h)(g',h)$ and $f=(g'',h')(g''',h')$. First, we check the case $h=h'$. In this case, if there exists a vertex $v\in U$ such that $d(v,gg')=d(v,g''g''')$, then there exists a vertex $u\in S^T(G)$ for which $d_G(u,gg')\neq d_G(u,g''g''')$ holds. Consequently, for the vertex $(u,h)\in S$ we have $d_X((u,h),e) \neq d_X((u,h),f)$. Otherwise, the vertex $(w,h)\in S'$ distinguishes $e$ and $f$.\
Now we investigate the case $h\neq h'$. In this case, if $d_X((u,h),e)= d_X((u,h),f)$ for each $(u,h)\in S$, then, again by a similar argument as applied in Case 1, there exists $(u,h')$ in $S$ that $d_X((u,h'),e)\neq d_X((u,h'),f)$.
[**Case 3**]{}: $e$ is in a $G$-layer and $f$ is in a $H$-layer.\
Let $e=(g,h)(g',h)$ and $f=(g'',h')(g'',h'')$. The case $h\notin \{h',h''\}$ can be proved by a similar technique as used in Case 1 and so we will only check the case $h=h'$. If $d_X((u,h),e)= d_X((u,h),f)$ for each $(u,h)\in S$, then $\min\{d_G(g,u),d_G(g',u)\}=d_G(u,g'')$ for each $u\in S^T(G)\cup S_G(U)$. Therefore, $d_X((u,h''),f)=d_X((u,h''), (g'',h''))=d_G(u,g'')<d_X((u,h''),e)$. So we have detected the vertex $(u,h'')\in S$ such that $d_X((u,h''),e) \neq d_X((u,h''),f)$.
We conclude that every pair of edges from $X$ is distinguished by a vertex of $S\cup S'$, and consequently ${{\rm edim}}(X) \leq n(H) ({{\rm edim}}^+(G(U))+1)$. [$\square$ ]{}
Consider the cases in which $|U|=1$ or $U\cap (S^T(G)\cup S_G(U))\neq\emptyset$, where $S^T(G)$ and $S_G(U)$ are defined as in the proof of Theorem \[t1\]. If there exists $z\in U\cap (S^T(G)\cup S_G(U))$, then replacing the vertex set $S'$ in the proof of Theorem \[t1\] with the set$\{(z,y):\ y\in V(H)\}$ we can prove along with the lines of the proof that $S$ is an edge metric generator for $G(U)\sqcap H$. In the case when $|U|=1$, Cases 1 and 3 from the proof are valid for $G(U)\sqcap H$, and we do not need the vertices of $S'$ in Case 2. Summarizing this discussion we have the following fact. If $|U|=1$ or $U\cap (S^T(G)\cup S_G(U))\neq\emptyset$, then $$\label{eq:1}
{{\rm edim}}(G(U)\sqcap H) \leq n(H)\,{{\rm edim}}^+(G(U))\,.$$
Consider $P_{11}(U)\sqcap P_2$, where $V(P_{11}) = \{v_1,\ldots, v_{11})$ and $U=\{v_{2k-1}:\ k\in [6]\}$, see Fig. \[fig3\].
![$P_{11}(U)\sqcap P_2$, where $U=\{v_1,v_3,v_5,v_7,v_9,v_{11}\}$.[]{data-label="fig3"}](fig3.eps)
Then we can select $S_{P_{11}}(U) = \{v_1\}$ and $S^T(P_{11})=\{v_1\}$, and so ${{\rm edim}}^+(P_{11}(U))=1$. From we then infer that $S=(S_{P_{11}}(U)\cup S^T(G))\times V_H=\{v_1\}\times V_H$ (black vertices in the figure) is an edge metric generator for $P_{11}(U) \sqcap P_2$, hence ${\rm edim}(P_{11}(U) \sqcap P_2) \leq 2$. From [@kelenc-2018 Remark 1] we know that ${\rm edim}(G)=1$ if and only if $G$ is a path, therefore we conclude that ${\rm edim}(P_{11}(U) \sqcap P_2)= 2$. This demonstrates that is sharp.
We now focus on hierarchical products $G(U)\sqcap H$, where $|U| = 1$. If $U = \{u\}$, then we simplify the notation $G(\{u\})$ to $G(u)$. If $G$ is a path and $u$ its end vertex, then we say that $G(u)$ is a [*rooted path*]{}.
\[t2\] If $X = G(u)\sqcap H$, where $G(u)$ is not a rooted path, and $n(H)\ge 2$, then $${\rm edim}(X) = n(H) \cdot {{\rm edim}}(G(u))\,.$$
[[**Proof. **]{}]{}From we know that ${\rm edim}(X) \le n(H) \cdot {{\rm edim}}^+(G(u))$. Since $|U|=1$, the equidistant discriminator for $U$ is the empty set, that is, $S_G(U) = \emptyset$, and consequently $ {{\rm edim}}(X)\leq n(H)\cdot {{\rm edim}}(G(u))$.
Let $S^T(G) = \bigcup_{u\in U, k\ge 0} S_G(u,k)$, where $S_G(u,k)$ are equidistant discriminators for $E_G(u,k)$ that realize ${{\rm edim}}(G(U))$. Then we know that $S=S^T(G)\times V(H)$ is an edge metric generator for $X$. We wish to show that $|S| = {{\rm edim}}(X)$ and assume by way of contradiction that there is an edge metric generator $S'$ for $X$ such that $|S'|<|S|$. By the pigeonhole principle there exists a $G$-layer of $X$, denote it with $G_h$ (here $h$ is the vertex of $H$ to which the $G$-layer corresponds), such that $|S'\cap V(G_h)| < |S\cap V(G_h)|$. Let $S'_h = S' \cap V(G_h)$ and note that $S'_h$ is not an equidistant discriminator for $E(G_h)$ and $|S'_h|<|S^T(G)|$. Hence there exist $k\ge 0$ and edges $e, f \in E(G_h)\cap \{(g,h)(g',h):\ gg'\in E_G(u,k)\}$ such that $d_X(x,e) = d_X(x,f)$ holds for each vertex $x \in S'_h$. Since $d_X((u,h),e)=d_{X}((u,h),f)$, it follows that the equality $d_X(v,e) = d_X(v,f)$ holds also for each $v\in S'\setminus V(G_h)$. But this means that $S'$ is not an edge metric generator for $X$, a contradiction. [$\square$ ]{}
Note that the rooted paths $G(u)$ are the only graphs for which $S^T(G) = \emptyset$ and consequently ${{\rm edim}}(G(u)) = 0$. This is the reason that the rooted paths are excluded in Theorem \[t2\].
To conclude the section we consider the corona product of graphs. Recall that the [*corona product*]{} $G\odot H$ of graphs $G$ and $H$ is obtained from the disjoint union of $G$ and $n(G)$ copies of $H$, by joining by an edge every vertex from the $i^{\rm th}$ copy of $H$ with the $i^{\rm th}$ vertex of $G$. (See [@Corona] for more information on this product.) The key observation is that $$G\odot H = (H+v)(v) \sqcap G\,,$$ where $H+v$ denotes the join of $H$ and the one vertex graph with the vertex $v$. More precisely, $H+v = H+K_1$, where $V(K_1) = \{v\}$. Then Theorem \[t2\] implies that $${{\rm edim}}(G\odot H) = {{\rm edim}}((H+v)(v) \sqcap G) = n(G)\cdot {{\rm edim}}((H+v)(v))\,.$$ From here it is not difficult to deduce [@zubrilina-2018 Theorem 4.1] which determines the edge metric dimension for the join of $K_1$ and an arbitrary graph, and [@peterin-2019 Theorem 6] that determines the edge metric dimension of corona products of nontrivial graphs. We can reformulate and combine these two results as follows. Let $\mathcal{F}$ be the family of graphs consisting of all graphs $G$ such that that $d_G(v,e)\leq 1$ holds for each $e\in E(G)$ and each $v\in V(G)$.
\[t3\] If $G$ is a connected graph, and $H$ is a graph with more than one vertex, then $${\rm edim}(G\odot H)=
\begin{cases}
n(H); & n(G) = 1\ \text{and}\ H\in \mathcal{F}, \\
n(G)(n(H)-1); & \text{otherwise}.
\end{cases}$$
Integer linear programming model {#sec:ILP}
================================
An integer linear programming model, ILPM for short, for finding the metric dimension and a metric basis for a graph has been presented in [@Chartrand-2000]. Following this approach we introduce an ILPM for finding the edge metric basis for a given graph as follows.
Let $G$ be a graph with $V(G) = \{v_1,\ldots,v_n\}$ and $E(G) = \{e_1,\ldots,e_m\}$. Let $D_G=[d_{ij}]$ be an $m\times n$ matrix, where $d_{ij}=d_G(e_i,v_j)$ for $i\in [m]$ and $j\in [n]$. For $x_i \in \{0,1\}$, $i\in [n]$, define the function $F$ by $$F(x_1,\ldots, x_n)=x_1+\cdots+x_n\,,$$ and minimize $F$ subject to the constraints $$|d_{i1}-d_{j1}|x_1+|d_{i2}-d_{j2}|x_2 + \cdots + |d_{in}-d_{jn}|x_n>0,\ 1 \le i < j \le m\,.$$ Then note that if $x'_1, \ldots , x'_n$ is a set of values for which $F$ attains its minimum, then $W = \{v_i:\ x'_i =1\}$ is an edge metric basis for $G$.
For example, consider $K_3$ with the vertex set $\{v_1,v_2,v_3\}$ and edges $e_1=v_1v_2$, $e_2=v_2v_3$, and $e_3=v_1v_3$. Then $D_{K_3}=\begin{pmatrix}
0 & 0& 1\\
1 & 0 & 0\\
0 & 1 & 0
\end{pmatrix}$. Thus, minimize $F(x_1,x_2,x_3)=x_1+x_2+x_3$ subject to the constraints $x_1+x_3 > 0$, $x_2 + x_3 > 0$, $x_1+x_2 > 0$, $x_1, x_2, x_3 \in \{0, 1\}$. Then $F$ attains its minimum for $x_1=0$, $x_2=1$, and $x_3=1$, hence $W=\{v_2,v_3\}$ is an edge metric basis for $K_3$.
Applications {#sec:applications}
============
In this section we demonstrate how the results from the previous sections can be applied to compute the edge metric dimension of interesting graphs.
Let $G_1,\ldots,G_k$ be rooted graphs with respective root vertices $r_1,\ldots,r_k$. The [*bridge-cycle graph*]{} $BC(G_1,\ldots,G_k; r_1,\ldots,r_k)$ is the graph obtained from the disjoint union of $G_1,\ldots,G_k$ by joining the vertices $r_i$ and $r_{i+1}$ for all $i\in [r-1]$ and adding the edge $r_1r_k$, see Fig. \[fig7\].
![The bridge-cycle graph $BC(G_1,\ldots,G_k; r_1,\ldots,r_k)$.[]{data-label="fig7"}](fig7.eps)
If $G_1=\cdots=G_k=G$ and $r=r_1$, where $G(r)$ is not a rooted path, then we infer that $BC(G_1,\ldots,G_k; r_1,\ldots,r_k)\cong G(r)\sqcap C_k$. Theorem \[t2\] then implies that $${\rm edim}(BC(G_1,\ldots,G_k; r_1,\ldots,r_k))={\rm edim}(G(r)\sqcap C_k)=k\cdot {\rm edim}(G(r))\,.$$
The examples from the rest of this section come from chemical graph theory. Consider first the (molecular) graph of truncated cube, it is denoted by $\Gamma$ and drawn in Fig. \[fig4\].
![$W(U)\sqcap P_2=\Gamma$ where $U=\{g_1,g_4,g_7,g_{10}\}$.[]{data-label="fig4"}](fig4.eps)
As the figure shows, $\Gamma$ is isomorphic to the hierarchical product $W(U)\sqcap P_2$ (see the figure for $W$), where $U=\{g_1,g_4,g_7,g_{10}\}$. Then, by the proof of Theorem \[t1\], $S=\{g_1,g_6\}\times V(H) = \{(g_1,h_1),(g_1,h_2),(g_6,h_1),(g_6,h_2)\}$ because $S^T(W) = S_W(U) = \{g_1,g_6\}$. Thus ${\rm edim}^+(W(U))=2$, and so by , ${\rm edim}(\Gamma)={\rm edim}(W(U)\sqcap P_2)\leq 4$. On the other hand, the exact value of ${\rm edim}(\Gamma)$ computed by the ILPM from Section \[sec:ILP\] is equal to $3$. The black vertices from Fig. \[fig4\] form an edge metric bases of $\Gamma$ found by the ILPM.
Continuing with examples from chemical graph theory, recall that a [*fullerene*]{} is a plane, $3$-connected, cubic graph with only pentagonal and hexagonal faces. The literature on fullerenes is huge, see for instance [@fullerene1] for more informations about their electronic and structural properties and the recent survey [@andova-2016]. More generally, the term fullerene is also used for such graphs where other lengths of faces are present, cf. [@shi-2019; @zhao-2018]. For instance, the graph $(BN)_{16}$ from Fig. \[fig2\] is an example of a $(4,6)$-fullerene.
![ (a) $K_{1,3}(U)\sqcap P_2$ where $\{u_1,u_2\}$ (b) $W(U)\sqcap P_2$ where $\{u_1,\ldots,u_4\}$ (c) $W'(U)\sqcap P_2$ where $\{u_1,\ldots,u_8\}$.[]{data-label="fig2"}](fig2.eps)
By Fig. \[fig2\] and Theorem \[t1\], we have ${\rm edim}(K_{1,3}(U)\sqcap P_2)\leq 4$, ${\rm edim}(W(U)\sqcap P_2)\leq 4$, and ${\rm edim}((BN)_{16})={\rm edim}(W'(U)\sqcap P_2)\leq 4$. On the other hand, using the ILPM from the previous section we get ${\rm edim}(K_{1,3}(U)\sqcap P_2)= 2$, ${\rm edim}(W(U)\sqcap P_2)= 2$, and ${\rm edim}((BN)_{16})= 3$. The black vertices show form the edge metric bases found by the ILPM.
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[^1]: Corresponding author
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abstract: 'We present interferometric observations of three ultraluminous infrared galaxies (ULIRGs; UGC 5101, Mrk 273, and IRAS 17208$-$0014) in the 3-mm wavelength range, using the Nobeyama Millimeter Array. Both the HCN (J=1–0) and HCO$^{+}$ (J=1–0) molecular lines were observed simultaneously. HCN emission was clearly detected at the nuclear positions of these ULIRGs, and HCO$^{+}$ emission was detected at the nuclear positions of UGC 5101 and IRAS 17208$-$0014. The HCN to HCO$^{+}$ brightness-temperature ratios toward the nuclei of the three ULIRGs were derived and compared with those of lower luminosity galaxies known to be dominated by active galactic nuclei (AGNs) or starbursts. In UGC 5101 and Mrk 273, where there is evidence for obscured AGNs from previous observations at other wavelengths, we found high HCN/HCO$^{+}$ ratios ($>$1.8) that are in the range found for AGN-dominated galaxies. In IRAS 17208$-$0014, where the presence of a powerful obscured AGN has been unclear, the ratio ($\sim$1.7) is in between the observed values for starburst- and AGN-dominated galaxies. The high HCN/HCO$^{+}$ brightness-temperature ratios in UGC 5101 and Mrk 273 could be the consequence of an HCN abundance enhancement, which is expected from chemical effects of the central X-ray emitting AGN on the surrounding dense molecular gas. Our proposed millimeter interferometric method based on HCN/HCO$^{+}$ ratios may be an effective tool for unveiling elusive buried AGNs at the cores of ULIRGs, especially because of the negligible dust extinction at these wavelengths.'
author:
- Masatoshi Imanishi
- Kouichiro Nakanishi
- Kotaro Kohno
title: 'Millimeter Interferometric Investigations of the Energy Sources of Three Ultraluminous Infrared Galaxies, UGC 5101, Mrk 273, and IRAS 17208$-$0014, based on HCN to HCO$^{+}$ Ratios'
---
Introduction
============
Ultraluminous infrared galaxies (ULIRGs), discovered during the [*IRAS*]{} all-sky survey, radiate the bulk of their very large luminosities ($L> 10^{12}L_{\odot}$) as dust emission in the infrared [@sam96]. Hence, powerful energy sources, such as starbursts or active galactic nuclei (AGNs), must be present hidden behind dust. Since distant ULIRGs dominate the cosmic infrared background emission, the sum of the dust-obscured activity in the universe [@bla99], establishing the energy sources of ULIRGs is closely related to understanding the connections between obscured AGNs and starbursts. Distant ULIRGs are too faint to investigate in detail with existing facilities and so the study of nearby ULIRGs continues to be an important means of understanding the distant ULIRG population.
It is now generally accepted that the majority of nearby ULIRGs are energetically dominated by compact ($<$500 pc), highly dust-obscured nuclear cores, rather than by extended ($>$kpc), weakly obscured (A$_{\rm V}$ $<$ 15 mag) starburst activity in host galaxies [@soi00]. Therefore, a primary goal in the study of nearby ULIRGs is to establish whether the obscured compact cores are very compact starbursts or AGNs. Optical spectroscopy is an often used way to distinguish an AGN from a starburst. In the presence of an AGN obscured by dust in a torus geometry, clouds that are photo-ionized by the radiation from the AGN develop along the torus axis above a torus scale height (the so-called narrow line regions: NLRs) [@ant93]. The NLRs produce optical emission lines whose flux ratios differ from the clouds in starburst galaxies; galaxies that contain an AGN obscured by torus-shaped dust are classified optically as Seyfert 2s [@vei87]. Hence, this type of obscured AGN can be identified by using optical spectroscopy. However, the nuclear regions of ULIRGs contain sufficiently highly concentrated molecular gas and dust that putative AGNs may be obscured by dust in virtually all directions [@sam96]. Such [*buried*]{} AGNs [^1] without significant NLRs are classified optically as “non-Seyferts”, and so very difficult to find [@mai03]; however, they are predicted to be abundant in the universe [@fab02]. In fact, the initial evidence of powerful buried AGNs has been detected in several optical non-Seyfert ULIRGs [@dud97; @soi00; @imd00; @tra01; @idm01; @imm03; @ima03; @idm06], so that it is essential to understand the contribution from buried AGNs to the total energy output of the ULIRG population.
One of the most effective ways to distinguish a buried AGN from a very compact starbust as the primary energy source in the core of a ULIRG is to investigate whether there is a strong X-ray-emitting source. A luminous buried AGN should exhibit intrinsically strong X-ray emission at E $>$ 2 keV from the vicinity of the accretion disk around the central supermassive black hole [@elv94], whereas in a starburst, the intrinsic X-ray emission above $E\sim2$ keV, relative to bolometric luminosity, is significantly weaker [@ran03]. Therefore, the detection of intrinsically strong, but highly absorbed, X-ray emission can provide strong evidence for a powerful buried AGN. Although moderately X-ray absorbed, Compton thin (N$_{\rm H}$ $<$ 10$^{24}$ cm$^{-2}$) AGNs are easy to find with currently available X-ray data at E = 2–10 keV [@fra03; @pta03; @imt04], it is very difficult to unveil and quantify the luminosities of Compton thick (N$_{\rm H}$ $>$ 10$^{24}$ cm$^{-2}$) buried AGNs using such data. Given that most of the AGNs detected so far in optical Seyfert-2 ULIRGs [@fra00; @iwa01; @bra03; @iwa05] and non-Seyfert galaxies with L$_{\rm IR}$ $\lessapprox$ 10$^{12}$L$_{\odot}$ [@don96; @vig99; @del02] suffer from Compton-thick absorption, it is very likely that the majority of the putative buried AGNs in optical non-Seyfert ULIRGs are also Compton thick. High-sensitivity observations at E $>$ 10 keV using a next-generation X-ray satellite are required to properly evaluate the energetic importance of buried AGNs in a large ULIRG sample.
Another approach to investigating whether a ULIRG core has an intrinsically strong X-ray emitting source is to search for signatures of the effects of the luminous X-ray source on the surrounding interstellar medium. Around a buried AGN, X-ray dissociation regions (XDRs; Maloney et al. 1996) should develop, rather than the photo-dissociation regions (PDRs) usually found in starburst regions. XDRs and PDRs can be distinguished if they show different flux ratios of emission lines. In particular, emission lines in the far-infrared to millimeter wavelength range are potentially a powerful tool, because low dust extinction at these wavelengths makes it possible to uncover the signatures of XDRs around powerful buried AGNs that are usually deeply embedded in dust. @koh01 [@koh03] and @koh05 proposed that the HCN (J=1–0) ($\lambda$=3.38 mm), HCO$^{+}$ (J=1–0) ($\lambda$=3.36 mm), and CO(J=1–0) ($\lambda$=2.60 mm) lines are useful for this purpose, and argued that AGN-dominated galaxies (XDRs) show systematically larger HCN/HCO$^{+}$ and HCN/CO ratios in brightness temperature than starburst-dominated galaxies (PDRs). @ima04 pointed out that the HCN/HCO$^{+}$ ratio is particularly effective for investigating the nature of heavily obscured ULIRG cores for the following reasons: (1) since both molecular lines probe high-density molecular gas ($n_{\rm H_2}$ $>$10$^{4}$ cm$^{-3}$), their relative ratios are not strongly contaminated by more extended lower density gas, whose relative fraction can vary among different galaxies, and (2) the wavelengths of the two lines are sufficiently close that they are observable simultaneously with wide-band receiving systems, reducing the net observing time and systematic errors. Since HCN and HCO$^{+}$ are both moderately strong lines, this method is applicable to a larger number of galaxies than that proposed by @use04, which uses various faint molecular lines for the purpose of distinguishing between XDRs and PDRs.
Interferometric maps are preferable to single-dish large-aperture observations for investigating the cores of ULIRGs in the millimeter wavelength range, because they reduce the contamination from extended star-formation emission in the host galaxy. We applied this millimeter interferometric method to the infrared luminous galaxy NGC 4418 (L$_{\rm IR}$ $\sim$ 10$^{11}$L$_{\odot}$), which likely contains a powerful buried AGN [@dud97; @spo01], and were successful in finding signatures of a buried AGN from the HCN/HCO$^{+}$ ratio [@ima04], demonstrating the potential of this method. In this paper, we report the application of this interferometric HCN/HCO$^{+}$ method to three ULIRGs: UGC 5101, Mrk 273, and IRAS 17208$-$0014. Throughout this paper, we adopt $H_{0}$ $=$ 75 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm M}$ = 0.3, and $\Omega_{\rm
\Lambda}$ = 0.7.
Targets
=======
The target ULIRGs were selected based on the following criteria: 1) Strong HCN emission is detected with single-dish radio telescopes, so that the study of HCN and HCO$^{+}$ emission-line ratios at the nuclei is feasible with the Nobeyama Millimeter Array (NMA), our observing facility. (2) They are nearby ($z <$ 0.06), so that redshifted HCN and HCO$^{+}$ emission lines are observable with the NMA receiving systems. The three ULIRGs UGC 5101, Mrk 273, and IRAS 17208$-$0014 were selected as the first targets. Table 1 summarizes the detailed information about these sources. An angular scale of 1$''$ corresponds to a physical size of 0.7–0.8 kpc for the three ULIRGs.
UGC 5101 (z = 0.040) is included in the original ULIRG sample [@san88a] and is a well-studied ULIRG. It is classified optically as a LINER [@vei95] or as a Seyfert 2 in a slightly different classification scheme by a different group [@gon99]. @idm01 and @imm03 performed infrared 3–4 $\mu$m spectroscopy and detected polycyclic aromatic hydrocarbon (PAH) emission at 3.3 $\mu$m, a starburst indicator, as well as signatures of a powerful AGN deeply buried at the core. They inferred that the infrared luminosity of UGC 5101 is dominated by the buried AGN, rather than the detected starburst activity. Radio VLBI observations [@smi98; @lon03] and a near-infrared high-spatial-resolution image [@sco00] showed compact, high surface-brightness emission, which was interpreted as an AGN by these authors. Similar compact emission was detected in the infrared at 12.5 $\mu$m [@soi00]. The presence of a powerful obscured AGN was confirmed by subsequent 2–10 keV X-ray observations [@ima03]. @arm04 detected weak high-excitation forbidden emission lines originating in the NLRs in a mid-infrared 5–40 $\mu$m high-resolution spectrum. Judging from the detection of NLR-originating emission lines in the optical [@gon99] and mid-infrared [@arm04], there may be a small angular range that is transparent to the AGNs’ ionizing photons (mostly UV and soft X-rays). HCN (J=1–0) data based on a single-dish telescope are available [@gao04] and indicate that if a significant fraction of the detected HCN emission comes from the nucleus of UGC 5101, then an investigation of the HCN/HCO$^{+}$ ratio at the core should be feasible using the NMA with a reasonable integration time.
Mrk 273 (z = 0.038) is also a well-studied ULIRG [@san88a]. It is classified optically as a Seyfert 2 [@vei99]. Its Seyfert-2 classification and the detection of strong NLR-originating high-excitation forbidden emission lines in the mid-infrared 5–11 $\mu$m spectrum [@gen98] suggest that the dust distribution around the putative AGN is torus-shaped and that a significant amount of ionizing radiation is escaping the AGN. 2–10 keV X-ray data have revealed the presence of a moderately absorbed Compton-thin AGN [@iwa99; @xia02; @bal05]. The presence of an obscured AGN in Mrk 273 is also suggested from (1) a smaller equivalent width of 3.3 $\mu$m PAH emission than the typical value for starburst galaxies [@imd00], (2) deep 9.7 $\mu$m silicate dust absorption and weak PAH emission features in the 8–13 $\mu$m spectrum [@dud99; @soi02], and (3) compact high surface brightness mid-infrared emission at 12.5 $\mu$m [@soi00]. The HCN flux from Mrk 273 observed with a single-dish telescope is as strong as that from UGC 5101 [@gao04], so that investigations of nuclear HCN and HCO$^{+}$ emission are also feasible for Mrk 273.
IRAS 17208$-$0014 (z = 0.042) is classified optically as an HII-region [@vei95]. Unlike other ULIRGs, the mid-infrared 8–23 $\mu$m emission of IRAS 17208$-$0014 is spatially extended, with no prominent core [@soi00]. Its radio emission as detected with the VLA is relatively compact [@mar89; @con96], but its surface brightness is not high enough to indicate strongly the presence of an AGN [@dia99]. Its 8–13 $\mu$m spectrum is typical of starburst galaxies, with prominent PAH emission features [@dud99] Its 3–4 $\mu$m spectrum also displays a clear 3.3 $\mu$m PAH emission feature, but the detection of a moderately strong 3.1 $\mu$m ice absorption feature suggests an energy source more centrally-concentrated than the surrounding dust, such as a buried AGN [@ris06; @ima06]. However, in IRAS 17208$-$0014, the AGN signature is much weaker than UGC 5101 and Mrk 273. The detected HCN flux using a single-dish telescope is the largest among the three ULIRGs [@gao04].
Observations and Data Reduction
===============================
Interferometric observations of HCN (J=1–0) ($\lambda_{\rm rest}$ = 3.3848 mm or $\nu_{\rm rest}$ = 88.632 GHz) and HCO$^{+}$ (J=1–0) ($\lambda_{\rm rest}$ = 3.3637 mm or $\nu_{\rm rest}$ = 89.188 GHz) were performed with the NMA and RAINBOW Interferometer at Nobeyama Radio Observatory (NRO) between December 2004 and March 2005. Table 2 summarizes the detailed observing log. The NMA consists of six 10-m antennas, and observations were performed using the D (the longest baseline was 82 m), C (163 m), and AB (351 m) configurations. The RAINBOW interferometer is a seven-element combined array that includes six 10-m antennas (NMA) and the NRO 45-m telescope. The RAINBOW observations were performed when the NMA array was in the AB configuration, and the longest baseline was 410 m. The sensitivity of RAINBOW is about twice that of the NMA array in the 3-mm wavelength range, because the inclusion of the NRO 45-m telescope increases the total aperture size by a factor of four.
The backend was the Ultra-Wide-Band Correlator (UWBC) [@oku00], which was configured to cover 1024 MHz with 128 channels at 8-MHz resolution. The central frequency was set to 85.49, 85.67, and 85.24 GHz for UGC 5101, Mrk 273, and IRAS 17208$-$0014, respectively, so that both the redshifted HCN and HCO$^{+}$ lines were observable simultaneously. A bandwidth of 1024 MHz corresponds to $\sim$3500 km s$^{-1}$ at $\nu$ $\sim$ 86 GHz. The field of view at this frequency is $\sim$26$''$ (full width at half maximum; FWHM) and $\sim$77$''$ (FWHM) for RAINBOW and NMA, respectively. Since the Hanning window function was applied to reduce side lobes in the spectra, the effective resolution was widened to 16 MHz or 55 km s$^{-1}$ at $\nu$ $\sim$ 86 GHz. To calibrate the passband across the 128 channels, the bright quasars 3C 84 (UGC 5101), 0355+508 (Mrk 273), and 3C 454.3 and 3C279 (IRAS 17208$-$0014) were observed every observing day. Quasars 0954+658, 1418+546, and 1741$-$038 were used to calibrate temporal variation in the visibility amplitude and phase for UGC 5101, Mrk 273, and IRAS 17208$-$0014, respectively.
The UVPROC-II package developed at NRO [@tsu97] and the AIPS package of the National Radio Astronomy Observatory were used for standard data reduction. Corrections for the antenna baselines, bandpass properties, and the time variation of the visibility amplitude and phase were applied to all of the data. A fraction of the data with large phase scatter due to bad radio seeing was removed from our analysis. After clipping a small fraction of data of unusually high amplitude, the data were Fourier-transformed using a natural [*UV*]{} weighting. The flux calibration was made using the observations of the appropriate quasars, 1749+096 (UGC 5101), 1741$-$038 and 0355+508 (Mrk 273), or 1741$-$038 (IRAS 17208$-$0014), whose flux levels relative to Uranus or Neptune had been measured at least every month between December 2004 and March 2005. A conventional CLEAN method was applied to deconvolve the synthesized beam pattern. The total net on-source integration times were $\sim$29 hours (UGC 5101), $\sim$8 hours (Mrk 273), and $\sim$18 hours (IRAS 17208$-$0014). The synthesized beam sizes are 5$\farcs$8 $\times$ 4$\farcs$5 (position angle is 45$^{\circ}$ west of north; PA = $-$45$^{\circ}$) for UGC 5101, 1$\farcs$9 $\times$ 1$\farcs$5 (PA = $-$66$^{\circ}$) for Mrk 273, and 3$\farcs$1 $\times$ 2$\farcs$4 (PA = $-$1$^{\circ}$) for IRAS 17208$-$0014. Absolute positional uncertainties of NMA/RAINBOW maps were estimated to be much less than 1 arcsec.
Results
=======
HCN (J=1–0) or HCO$^{+}$ (J=1–0) data are available for several ULIRGs [@sol92; @cur00; @gao04], but observations have generally been performed with single-dish telescopes. @rad91 showed an interferometric HCN map of the ultraluminous infrared galaxy Arp 220. With this exception, no detailed morphological information for HCN emission exists for ULIRGs. Therefore, this work allows us to reveal for the first time the spatial distribution of HCN (J=1–0) in the ULIRG population other than Arp 220. Ours are also the first published interferometric maps of ULIRGs for HCO$^{+}$ (J=1–0) emission.
Figure 1 shows the contours of the continuum emission from UGC 5101, Mrk 273, and IRAS 17208$-$0014. Signals away from the HCN and HCO$^{+}$ emission lines were combined to produce the continuum maps. Strong continuum emission is found in all ULIRGs and the continuum peaks are spatially coincident with the nuclear positions of these ULIRGs. Table 3 summarizes the estimated continuum flux levels. The detection significance is $>6\sigma$ (UGC 5101), $>4\sigma$ (Mrk 273), and $>10\sigma$ (IRAS 17208$-$0014). The spatial extents of the continuum emission agree with the beam sizes within $\lesssim$50%.
After the continuum emission was subtracted, we investigated the spatial distribution of the HCN and HCO$^{+}$ emission. Figure 2 shows the integrated intensity maps of the HCN emission for UGC 5101, Mrk 273, and IRAS 17208$-$0014. Figure 3 displays the HCO$^{+}$ integrated intensity maps for UGC 5101 and IRAS 17208$-$0014. HCO$^{+}$ emission is not clearly detected from the nucleus of Mrk 273 in our interferometric map ($<3\sigma$). With this exception, the HCN and HCO$^{+}$ emission lines are clearly ($>10\sigma$) detected. The detected emission lines are coincident with the nuclear positions of the ULIRGs and has spatial extents consistent with the beam sizes within $\lesssim$50%. For UGC 5101 and IRAS 17208$-$0014, the signal-to-noise ratios around the HCN and HCO$^{+}$ lines are good enough to present channel maps, which are shown in Figures 4 and 5, respectively.
Figure 6 shows continuum-subtracted nuclear spectra around the HCN and HCO$^{+}$ emission lines. The presence of HCO$^{+}$ emission from Mrk 273 is unclear, but HCN emission from all the ULIRGs, and HCO$^{+}$ emission from UGC 5101 and IRAS 17208$-$0014 are clearly recognizable. Figure 7 presents the Gaussian fits to these detected emission lines. The central velocity and line width of the Gaussian were determined independently for the HCN and HCO$^{+}$ lines. The HCN emission from UGC 5101 suggests the presence of double peaks, which may be related to the nuclear rotating molecular disk probed by the CO (J=1–0) line [@gen98]. Therefore, a double Gaussian fit was also attempted for the HCN line of UGC 5101. For the HCN and HCO$^{+}$ lines from IRAS 17208$-$0014, we added a constant continuum to subtly adjust the residual continuum level in the vicinity of the HCN and HCO$^{+}$ lines. Table 4 summarizes the Gaussian fitting results.
The local standard of rest (LSR) peak velocities of HCN emission from Mrk 273 (11310 km s$^{-1}$) and IRAS 17208$-$0014 (12860 km s$^{-1}$) agree, within 25 km s$^{-1}$, with those of CO peak velocities (11324 km s$^{-1}$ for Mrk 273 and 12837 km s$^{-1}$ for IRAS 17208$-$0014; Downes & Solomon 1998). HCN and CO [@dow98] line profiles are also similar for the nuclei of Mrk 273 and IRAS 17208$-$0014. For the double Gaussian fit of HCN emission from UGC 5101, the velocity difference of the two components is $\sim$400 km $^{-1}$, which is roughly comparable to that of CO ($\sim$480 km s$^{-1}$; Genzel et al. 1998).
The integrated fluxes of HCN and HCO$^{+}$ at the peak position, estimated from the Gaussian fits, are summarized in Table 5. For the HCN emission from UGC 5101, the single and double Gaussian fits give fluxes that are consistent to within $<$5%. Therefore, we adopt the values based on single Gaussian fits for all sources. The derived HCN fluxes correspond to $\sim$60% (UGC 5101), $\sim$30% [^2] (Mrk 273), and $\sim$55% (IRAS 17208$-$0014) of the HCN flux measurements from single-dish radio telescopes [@gao04].
The HCN/HCO$^{+}$ ratios in brightness temperature ($\propto$ $\lambda^{2}$ $\times$ flux density) are also summarized in Table 5. For the purpose of detecting the effects of putative AGNs on the surrounding interstellar medium at ULIRG cores, we regard our HCN/HCO$^{+}$ ratios as a more appropriate tool than the ratios previously derived from single-dish telescope observations [@ngu92; @sol92], because (1) we can reduce the contamination from the extended flux of the host galaxy, and (2) in our NMA data, the HCN and HCO$^{+}$ lines were observed simultaneously with the same receiver and same correlator unit, so that possible systematic errors should be minimal. Although the [*absolute*]{} flux calibration uncertainty of the NMA and RAINBOW data might be as large as $\sim$20%, this uncertainty does not affect the HCN/HCO$^{+}$ ratios. The uncertainty in the ratios is dominated by statistical noise and fitting errors (see Figure 7).
Discussion
==========
Comparison of HCN/HCO$^{+}$ Ratios with Other Galaxies
------------------------------------------------------
Figure 8 shows the HCN/HCO$^{+}$ and HCN/CO ratios in brightness temperature for UGC 5101, Mrk 273, and IRAS 17208$-$0014, together with previously obtained data for nearby starburst and Seyfert galaxies [@koh01; @koh05], and the AGN-powered infrared luminous galaxy NGC 4418 [@ima04]. The ratios for the three main nuclei of the infrared luminous merging galaxy Arp 299 (Imanishi et al. 2006, in preparation) are also plotted. @koh01 and @koh05 found empirically that AGN-dominated galaxies tend to have higher HCN/HCO$^{+}$ and HCN/CO ratios than starburst-dominated galaxies: AGN-dominated galaxies should be distributed on the upper right-hand side, whereas starburst-dominated galaxies should occupy the lower left region in Figure 8.
The HCN/CO ratio in [*brightness temperature*]{} (abscissa) may increase if the HCN [*abundance*]{} relative to CO is enhanced by X-ray radiation from an AGN [@lep96]. However, another possible explanation for the enhanced HCN/CO brightness-temperature ratio is an increase in molecular gas density. Since the dipole moment of HCN ($\mu$ $\sim$ 3 debye; Millar et al. 1997) is much larger than that of CO ($\mu$ $\sim$ 0.1 debye; Millar et al. 1997), HCN emission traces a much higher density molecular gas ($n_{\rm H_2}$ $>$ 10$^{4}$ cm$^{-3}$) than is traced by CO ($n_{\rm H_2}$ $\sim$ 10$^{2-3}$ cm$^{-3}$). The HCN/CO brightness-temperature ratio can increase if a galaxy contains a larger fraction of high-density molecular gas (e.g., Solomon et al. 1992; Helfer & Blitz 1993; Kohno et al. 1999; Curran et al. 2000). In fact, ULIRGs are expected to have a larger fraction of high-density molecular gas than less infrared-luminous galaxies do [@sam96], so that HCN/CO brightness-temperature ratios can be high in ULIRGs, even without the HCN abundance enhancement caused by powerful AGNs. Furthermore, the HCN and CO data were taken on different observing runs, with different beam sizes, under different weather conditions, all of which could introduce additional systematic uncertainties. Therefore, the interpretation of HCN/CO brightness-temperature ratios in ULIRGs is not simple.
Such uncertainties are much reduced if we use the HCN/HCO$^{+}$ ratios. Both HCN and HCO$^{+}$ ($\mu$ $\sim$4 debye; Botschwina et al. 1993) probe high-density molecular gas ($n_{\rm H_2}$ $>$ 10$^{4}$ cm$^{-3}$), so that the HCN/HCO$^{+}$ brightness-temperature ratio is insensitive to the different fraction of dense molecular gas, relative to diffuse one, in individual galaxies. Furthermore, since both the HCN and HCO$^{+}$ emission lines were observed simultaneously with NMA/RAINBOW for all the sources plotted in Figure 8, systematic errors are not a serious concern. For this reason, we use the HCN/HCO$^{+}$ ratios (ordinate) in our discussion of the possible presence of powerful AGNs.
Our new millimeter interferometric data for UGC 5101 and Mrk 273 reveal that the HCN/HCO$^{+}$ brightness-temperature ratios are high and in the range expected for AGN-dominated galaxies (HCN/HCO$^{+}$ $\gtrsim$ 2; Kohno 2005). This is consistent with the fact that previous data have found the signatures of powerful AGNs in UGC 5101 and Mrk 273 ($\S$2).
The interpretation of our data for IRAS 17208$-$0014 is rather unclear. No strong evidence for a luminous buried AGN in IRAS 17208$-$0014 has been found at other wavelengths ($\S$2). The HCN/HCO$^{+}$ brightness-temperature ratio for IRAS 17208$-$0014 ($\sim$1.7) is located in between the expected values for AGN- and starburst-dominated galaxies. A powerful AGN possibly hidden in IRAS 17208$-$0014 may slightly increase the HCN/HCO$^{+}$ ratio, but the presence of an AGN is not clear from our data.
Interpretations of HCN/HCO$^{+}$ Ratios
---------------------------------------
In ULIRGs, the profiles of HCN (J=1–0) and CO (J=1–0) lines are generally similar (Solomon et al. 1992; Gao & Solomon 2004; This paper), despite the difference of their optical depths. The similar line profiles are naturally explained by the widely accepted mist model [@sol87], where a molecular gas cloud consists of a large number of small clumps, with a filling factor of $<$1 at any fixed velocity. In this model, even if each clump is optically thick for a molecular line, we can probe clumps at the background side of the molecular gas cloud.
If each clump is optically thin for a molecular line, its flux will increase linearly with increasing abundance. However, each clump is thought to be optically thick for CO (J=1–0) lines [@sol87; @sco87]. It is not clear whether each clump is optically thick or thin for HCN (J=1–0) and HCO$^{+}$ (J=1–0) lines, but we can roughly estimate their optical depths ($\tau$), based on the formula of $$\begin{aligned}
\tau & \propto & A \times \lambda^{3} \times abundance, \end{aligned}$$ where [*A*]{} is the Einstein coefficient and is proportional to $\mu^{2}$, where $\mu$ is the value of dipole moment [@sco87]. If we adopt the HCN and HCO$^{+}$ abundance, relative to CO, are 10$^{-4}$ $\sim$ 5$\times$10$^{-3}$ [@bla87], the optical depths of both HCN (J=1–0) and HCO$^{+}$ (J=1–0) lines are 0.2–2 times as large as that of CO (J=1–0) line. Thus, this simple estimate suggests that each clump can be optically thick for HCN (J=1–0) and HCO$^{+}$ (J=1–0) lines. Nguyen-Q-Rieu (1992) also showed that each clump can have optical depth slightly larger than unity for HCN (J=1–0) line, if the volume filling factor of the clump is as small as f $\lesssim$ 0.1 [@sol87; @kod05].
For an optically thick case, although the relation between flux and abundance is not so simple, their positive correlation is still expected for the following main reason. Assuming that each clump has a decreasing radial density profile ($\propto$ r$^{-1.5}$; Gierence et al. 1992), line emission comes mainly from the layer where an optical depth reaches unity [@gie92]. When a molecular abundance increases (decreases), the $\tau$ = 1 layer for that molecular line moves outward (inward). Consequently, the surface area of the layer, or its area filling factor, becomes larger (smaller), increasing (decreasing) the flux of that molecular line. Unless the optical depth is extremely large, this effect can be significant and flux will increase with increasing abundance, if not completely linearly, as was demonstrated for the optically thick CO (J=1–0) line [@ver95; @wil95; @sak96; @ari96]. Hence, qualitatively, the high HCN/HCO$^{+}$ ratio in the brightness temperature can be explained by an enhanced HCN/HCO$^{+}$ abundance ratio in dense molecular gas from which the bulk of the observed HCN and HCO$^{+}$ fluxes originate.
Theoretical models allow the calculation of the abundance of molecules, including HCN and HCO$^{+}$, as a function of distance from a central UV illuminating source or X-ray emitting AGN deeply buried in molecular gas and dust [@lep96; @mei05]. @lep96 found that the abundance of HCN can be enhanced relative to HCO$^{+}$ if a strong X-ray source is present inside molecular gas. @mei05 calculated the effects of a central X-ray-illuminating AGN on the surrounding molecular gas. They found that the HCN/HCO$^{+}$ abundance ratio in uniformly-distributed dense molecular gas (10$^{5.5}$ cm$^{-3}$), as probed by HCN and HCO$^{+}$, around a luminous X-ray-emitting source can be substantially higher than the ratio in dense molecular gas illuminated only by a central UV source (their Fig.10, model 4). Thus, we have some theoretical basis for the enhanced HCN abundance, relative to HCO$^{+}$, in the presence of an X-ray-emitting AGN.
However, further sophisticated model calculations are definitely required to [*quantitatively*]{} interpret our observational results. Models available to date [@lep96; @mei05] have only calculated the HCN/HCO$^{+}$ abundance ratio and showed that the abundance ratio is a function of distance from the central illuminating source. To convert from the HCN/HCO$^{+}$ abundance ratio to brightness-temperature ratio, we have to know the region from where the bulk of the observed HCN and HCO$^{+}$ fluxes come, which has not been explicitly mentioned in any available model calculations. Furthermore, the models assume uniform dust/gas distribution for simplicity. If the distribution is clumpy, which is more realistic [@sol87], the abundance ratio can change substantially.
In order for our millimeter interferometric HCN/HCO$^{+}$ method to help to detect buried AGN signatures, high-density molecular gas must be concentrated near the core region where putative AGNs reside and can affect the gas. If high-density molecular gas is distributed in between double nuclei, for example as in NGC 6240 [@nak05], then our method may no longer work. Furthermore, shocks may enhance HCO$^{+}$ emission [@raw00; @raw04]. If shocks driven by outflows or superwind from starbursts coexist with powerful buried AGNs, and if there is a large amount of high-density molecular gas that is affected by the shocks, then the HCN/HCO$^{+}$ brightness-temperature ratio may decrease compared to that of a pure buried AGN. Thus, some fraction of the galaxies containing powerful AGNs can show low HCN/HCO$^{+}$ ratios and may be missed when searching based on the HCN/HCO$^{+}$ ratio.
Despite these remained ambiguities, this millimeter interferometric method can empirically be an effective means of revealing the presence of deeply buried AGNs at ULIRG cores, because of the negligible dust extinction in this wavelength range. If this method is established from a larger number of sources, future observations with high spatial resolution and high sensitivity using ALMA are potentially a powerful probe for optically and even 2–10 keV X-ray-elusive buried AGNs in ULIRG cores.
Summary
=======
We performed millimeter interferometric observations of the HCN (J=1–0) and HCO$^{+}$ (J=1–0) lines of the three ultraluminous infrared galaxies, UGC 5101, Mrk 273, and IRAS 17208$-$0014, using NMA/RAINBOW. These are the first interferometric maps of a ULIRG sample for which the HCN (J = 1–0) and HCO$^{+}$ (J = 1–0) molecular lines were observed simultaneously. We extracted the HCN to HCO$^{+}$ brightness-temperature ratios at the cores of these ULIRGs, and compared the ratios with those previously estimated for less-luminous, AGN-dominated and starburst-dominated galaxies. We then looked for possible signatures of the influence of AGNs, strong X-ray emitters, on the surrounding interstellar medium at the nuclei. We found that:
1. Continuum and HCN emission were clearly detected in all ULIRGs. HCO$^{+}$ emission was found in UGC 5101 and IRAS 17208$-$0014. The spatial distribution of the emission, when detected, is coincident with the nuclear positions of these ULIRGs.
2. UGC 5101 and Mrk 273 showed high HCN/HCO$^{+}$ ratios in brightness temperature, as found in AGN-dominated galaxies. Previous observations had suggested luminous obscured AGNs in UGC 5101 and Mrk 273, and so our millimeter interferometric data may be revealing the signatures of the effects of a luminous X-ray-emitting AGN on the surrounding interstellar medium at the nucleus.
3. The HCN/HCO$^{+}$ ratio for IRAS 17208$-$0014 was in between the regions occupied by AGN- and starburst-dominated galaxies. There was no clear evidence for a hidden AGN in IRAS 17208$-$0014 based on previously obtained data at other wavelengths. The presence of a luminous buried AGN in IRAS 17208$-$0014 remains uncertain based on currently available millimeter interferometric data.
4. Although the sample size is still small, this millimeter interferometric HCN/HCO$^{+}$ method succeeded in confirming AGN signatures in the two ULIRGs (UGC 5101 and Mrk 273) that were previously known to possess luminous obscured AGNs. Since dust extinction is negligible at this wavelength range, we argue that our method is potentially an important tool for unveiling powerful, but deeply buried, AGNs in optical “non-Seyfert” ULIRGs in the ALMA era.
We are grateful to NRO staff for their support during our observing runs and to the anonymous referee for his/her valuable comments. We thank S. Ishizuki, J. Koda, N. Z. Scoville, T. Sawada, and S. Takano for useful discussions about the properties of molecular gas. We also thank S. K. Okumura for providing us detailed information on the NMA. M.I. was supported by a Grant-in-Aid for Scientific Research (16740117). NRO is a branch of the National Astronomical Observatory, National Institutes of Natural Sciences, Japan. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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[cccccccc]{} UGC 5101 & 0.040 & 0.25 & 1.03 & 11.54 & 20.23 & 45.5 (12.0) & cool\
Mrk 273 & 0.038 & 0.24 & 2.28 & 21.74 & 21.38 & 45.7 (12.1) & cool\
IRAS 17208$-$0014 & 0.042 & 0.20 & 1.66 & 31.14 & 34.90 & 45.9 (12.3) & cool\
[lcl]{} UGC 5101 & D & 2004 Dec 22, 23, 24\
& RAINBOW & 2005 Jan 20\
& C & 2005 Mar 9, 10\
Mrk 273 & RAINBOW & 2005 Jan 22 and 31\
IRAS 17208$-$0014 & AB & 2005 Jan 26–28, Feb 7\
& C & 2005 Mar 6, 7, 16–19, 26\
[ccc]{} UGC 5101 & 6 & 85.5\
Mrk 273 & 6 & 85.6\
IRAS 17208$-$0014 & 9 & 85.3\
[ccccc]{} UGC 5101 & 11830 & 11990 & 700 & 330\
& 11585+11985 & 11990 & 270+375 & 330\
Mrk 273 & 11310 & & 450 &\
IRAS 17208$-$0014 & 12860 & 12820 & 530 & 470\
[cccccc]{} UGC 5101 & 6.6 & 2.7 & 2.5 & 50 & a\
& 3.6+2.7 & & $>$2,$\sim$1 & &\
Mrk 273 & 4.8 & $<$2.7 & $>$1.8 & 78 & b\
IRAS 17208$-$0014 & 14 & 8.1 & 1.7 & 132 & b\
![ Continuum maps of UGC 5101, Mrk 273, and IRAS 17208$-$0014. The crosses show the coordinates of the ULIRG nuclei. The coordinates in J2000 are (9$^{h}$35$^{m}$51.61$^{s}$, $+$61$^{\circ}$21$'$11$\farcs$4) for UGC 5101, (13$^{h}$44$^{m}$42.12$^{s}$, $+$55$^{\circ}$53$'$13$\farcs$5) for Mrk 273, and (17$^{h}$23$^{m}$21.95$^{s}$, $-$00$^{\circ}$17$'$00$\farcs$7) for IRAS 17208$-$0014. The nuclear coordinates of UGC 5101 are adopted from VLA radio data [@con91], while those of Mrk 273 and IRAS 17208$-$0014 are based on the CO peaks in millimeter interferometric maps [@dow98]. The contours are $-$3, 3, 4.5, and 6 mJy beam$^{-1}$ for UGC 5101, $-$4.5, $-$3, 3, 4.5, and 6 mJy beam$^{-1}$ for Mrk 273, and $-$4, 4, 6, and 8 mJy beam$^{-1}$ for IRAS 17208$-$0014. ](f1a.eps "fig:") ![ Continuum maps of UGC 5101, Mrk 273, and IRAS 17208$-$0014. The crosses show the coordinates of the ULIRG nuclei. The coordinates in J2000 are (9$^{h}$35$^{m}$51.61$^{s}$, $+$61$^{\circ}$21$'$11$\farcs$4) for UGC 5101, (13$^{h}$44$^{m}$42.12$^{s}$, $+$55$^{\circ}$53$'$13$\farcs$5) for Mrk 273, and (17$^{h}$23$^{m}$21.95$^{s}$, $-$00$^{\circ}$17$'$00$\farcs$7) for IRAS 17208$-$0014. The nuclear coordinates of UGC 5101 are adopted from VLA radio data [@con91], while those of Mrk 273 and IRAS 17208$-$0014 are based on the CO peaks in millimeter interferometric maps [@dow98]. The contours are $-$3, 3, 4.5, and 6 mJy beam$^{-1}$ for UGC 5101, $-$4.5, $-$3, 3, 4.5, and 6 mJy beam$^{-1}$ for Mrk 273, and $-$4, 4, 6, and 8 mJy beam$^{-1}$ for IRAS 17208$-$0014. ](f1b.eps "fig:") ![ Continuum maps of UGC 5101, Mrk 273, and IRAS 17208$-$0014. The crosses show the coordinates of the ULIRG nuclei. The coordinates in J2000 are (9$^{h}$35$^{m}$51.61$^{s}$, $+$61$^{\circ}$21$'$11$\farcs$4) for UGC 5101, (13$^{h}$44$^{m}$42.12$^{s}$, $+$55$^{\circ}$53$'$13$\farcs$5) for Mrk 273, and (17$^{h}$23$^{m}$21.95$^{s}$, $-$00$^{\circ}$17$'$00$\farcs$7) for IRAS 17208$-$0014. The nuclear coordinates of UGC 5101 are adopted from VLA radio data [@con91], while those of Mrk 273 and IRAS 17208$-$0014 are based on the CO peaks in millimeter interferometric maps [@dow98]. The contours are $-$3, 3, 4.5, and 6 mJy beam$^{-1}$ for UGC 5101, $-$4.5, $-$3, 3, 4.5, and 6 mJy beam$^{-1}$ for Mrk 273, and $-$4, 4, 6, and 8 mJy beam$^{-1}$ for IRAS 17208$-$0014. ](f1c.eps "fig:")
  
 
![ Channel maps of HCN and HCO$^{+}$ emission for UGC 5101. [*Left*]{}: HCN emission. The contours are $-$6, 6, and 10 mJy beam$^{-1}$. The r.m.s. noise level is $\sim$2.5 mJy beam$^{-1}$. [*Right*]{}: HCO$^{+}$ emission. The contours are $-$6, 6, and 10 mJy beam$^{-1}$. The r.m.s. noise level is $\sim$2.5 mJy beam$^{-1}$. ](f4a.eps "fig:") ![ Channel maps of HCN and HCO$^{+}$ emission for UGC 5101. [*Left*]{}: HCN emission. The contours are $-$6, 6, and 10 mJy beam$^{-1}$. The r.m.s. noise level is $\sim$2.5 mJy beam$^{-1}$. [*Right*]{}: HCO$^{+}$ emission. The contours are $-$6, 6, and 10 mJy beam$^{-1}$. The r.m.s. noise level is $\sim$2.5 mJy beam$^{-1}$. ](f4b.eps "fig:")
![ Channel maps of HCN and HCO$^{+}$ emission for IRAS 17208$-$0014. [*Left*]{}: HCN emission. The contours are $-$9, 9, 15, 21, and 27 mJy beam$^{-1}$. The r.m.s. noise level is $\sim$3 mJy beam$^{-1}$. [*Right*]{}: HCO$^{+}$ emission. The contours are $-$9, 9, 12, and 15 mJy beam$^{-1}$. The r.m.s. noise level is $\sim$3 mJy beam$^{-1}$. ](f5a.eps "fig:") ![ Channel maps of HCN and HCO$^{+}$ emission for IRAS 17208$-$0014. [*Left*]{}: HCN emission. The contours are $-$9, 9, 15, 21, and 27 mJy beam$^{-1}$. The r.m.s. noise level is $\sim$3 mJy beam$^{-1}$. [*Right*]{}: HCO$^{+}$ emission. The contours are $-$9, 9, 12, and 15 mJy beam$^{-1}$. The r.m.s. noise level is $\sim$3 mJy beam$^{-1}$. ](f5b.eps "fig:")
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[^1]: We use the term “obscured AGN” if the AGN is obscured along our line of sight, and “buried AGN” if the AGN is obscured along virtually all sightlines.
[^2]: This value seems low, but we have no clear answer. The flux difference between the single-dish and interferometric observations may be due to discrepancies in the absolute flux scale between the two observations, or the presence of extended HCN emission unrecovered by our interferometric observations (less likely), or possibly a substantial overestimate of the HCN flux by Gao & Solomon (2004), coming from relatively large scatters in their single-dish data (their Figure 2).
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abstract: 'We consider evolution of a double quantum dot (DQD) two-electron spin qubit which is continuously weakly measured with a linear charge detector (quantum point contact). Since the interaction between the spins of two electrons depends on their charge state, the charge measurement affects the state of two spins, and induces non-trivial spin dynamics. We consider the regimes of strong and weak coupling to the detector, and investigate the measurement-induced spin dynamics both analytically and numerically. We observe emergence of the negative-result evolution and the system stabilization due to an analog of quantum Zeno effect. Moreover, unitary evolution between the triplet and a singlet state is induced by the negative-result measurement. We demonstrate that these effects exist for both strong and weak coupling between the detector and the DQD system.'
address: 'Ames Laboratory, Iowa State University, Ames, Iowa 50014, U.S.A.'
author:
- 'Rusko Ruskov, Viatcheslav V. Dobrovitski, and Bruce N. Harmon'
title: 'Manipulation of double-dot spin qubit by continuous noisy measurement'
---
Introduction
============
Quantum dots (QDs) recently have attracted much attention as very suitable candidates for studying evolution of a single quantum system. Moreover, QDs are very promising candidates for quantum information processing. An electron spin in a quantum dot is a good representation for a qubit, being a natural two-state quantum system. Manipulations of the single QDs and double-QD (DQD) systems can be achieved using external magnetic fields and gate voltages [@HansonReview], which implement injection of an electron (or pair of electrons for DQD) in a given spin state, or some unitary rotation in a two-spin space. However, it is not easy to achieve the full set of transformations in the Hilbert space of two coupled spins [@QDman1; @QDman2; @QDman3; @QDman4]: some transformations require a strong gradient of magnetic fields on a nanometer scale, and advanced techniques are needed for performing spin rotations rapidly and reliably.
Thus, it is interesting to explore other possible ways of manipulating electron spins in DQD structures. In particular, it is natural to study whether measurement of the charge state of the DQD may help drive the desired evolution of the two-spin system. The weak continuous measurement [@Dalibard; @Carmichael; @WisemanMilburn; @Kor-99-01], which monitors the system in question, and therefore affects its evolution, may provide an additional and useful tool for controlling the electron spins in DQD systems. At some level of description of a probed quantum system plus detector, it is postulated that the fundamental measurement process consists of direct particle detections (e.g., photons, electrons, etc.) or absence of any detection. The absence of a detection for specific time interval constitutes a negative result. In both cases (“detection” or “no detection”) the quantum state of the system evolves according to the provided information and the evolution can be described using quantum Bayesian inference.[@Kor-99-01] E.g., for a two-level atom which did not decay by time $t$ the evolution of the atom’s density matrix takes the form: $$\begin{aligned}
&&\rho_{ee}( t) = \frac{\rho_{ee}(0)\, e^{-\Gamma_{sp}t}}{\cal N} ,\
\rho_{gg}( t) = \frac{\rho_{gg}(0)}{\cal N} \ \
\label{atom-Bayes1}\\
&& |\rho_{eg}( t)| = |\rho_{eg}(0)|\,
\sqrt{\frac{\rho_{ee}(t)\,\rho_{gg}(t)}{\rho_{ee}(0)\,\rho_{gg}(0)}} , \qquad
\label{atom-Bayes2}\end{aligned}$$ where $e^{-\Gamma_{sp}t}$ is the conditional probability not to decay from the excited state $|e\rangle$ (ground state $|g\rangle$ is supposed stable), $\Gamma_{sp}$ is the spontaneous decay rate, and ${\cal N}\equiv
\rho_{gg}(0) + \rho_{ee}(0)\, e^{-\Gamma_{sp}t}$. The classical Bayesian update of the diagonal elements, Eq.(\[atom-Bayes1\]), reflects an information-related evolution. Also, Eq.(\[atom-Bayes2\]) shows that the coherence ratio $|\rho_{eg}|/\sqrt{\rho_{ee}\,\rho_{gg}}$ is conserved; in particular, a pure state remains pure.
The above evolution has been proposed and discussed mainly in the context of quantum optics. [@Epstein-Dicke]${}^{,}$[@Dalibard; @Carmichael; @WisemanMilburn] For solid state systems, it has been realized recently, using an artificial two-level system: a superconducting “phase” qubit, measured via tunneling of its superconducting phase.[@Katz-Sci] The detection of tunneling was considered as an instantaneous process[@Katz-Sci; @Pryadko] in the highly non-linear detector (dc-SQUID) that implies detector relaxation rates much faster than the qubit “decay rate”.
![Schematic of a DQD spin system measured by a QPC; the QPC current $I(t)$ is affected by the charge state of the DQD. In quantum Zeno regime the tunneling between dots is suppressed while the spin qubit undergoes continuous negative-result evolution.[]{data-label="schematic"}](fig1.eps){width="2.8in"}
In this paper we consider another realization of a solid state qubit and a linear detector measuring continuously. Nevertheless, it can behave similarly to the above examples in certain measurement regimes. We will consider a semiconductor double quantum dot (DQD) spin system (Fig.\[schematic\]) where two electrons can be in either of the dots and the relevant states are spin-charge states.[@LossDiVi; @Kouwenhoven-Nat1; @Marcus-Sci; @HansonReview] At relatively small time scales (from ns to $\mu$s) the electron tunneling between the dots preserves the system’s total spin that allows spin-selective evolution and leads to the so called spin-to-charge conversion.[@HansonReview] This allows the measurement of the DQD spin system by a charge sensitive detector such as a quantum point contact (QPC)[@Kouw-prl94; @Marcus-Sci] or single electron transistor (SET).[@Rimberg-nat]
For our system (Fig.\[schematic\]) the spin qubit[@Marcus-Sci] is formed by the singlet spin state $|S(1,1)\rangle$ and the triplet state $|T_0(1,1)\rangle$ that has a zero spin projection on an external magnetic field (here, e.g., $(1,1)$ denotes a charge state when one electron is in the left dot and the other is in the right dot). Spin conservation allows transition between the singlet $|S(1,1)\rangle$ and the localized singlet, $|S(0,2)\rangle$, while a transition from the triplet state is spin-blocked. Thus, the state $|S(0,2)\rangle$ can be used for preparation and subsequent measurement of the spin qubit. For a weak measurement the system-detector coupling $\Gamma$ is finite and defines the time scale of measurement evolution (in particular, $1/\Gamma$ is of the order of the time needed to distinguish the two different charge states by the QPC). The spin system evolution will be an interplay of the non-unitary dynamics due to measurement and the dynamics associated with the system’s internal Hamiltonian.
The purpose of this work is to develop the theory of quantum evolution of a DQD two-electron spin system under the continuous measurement by a linear charge detector such as a QPC that is based on the Bayesian approach.[@Kor-99-01; @Kor-osc; @Ruskov-neg] Initially motivated by experiment[@Marcus-Sci] we pose the question: “How the measurement by a QPC will affect the spin qubit state and is it possible to use continuous measurement for manipulation of the quantum state ?” The system state is continuously updated according to a given measurement record due to quantum back action.[@Kor-99-01] On typical system’s times when many electrons pass through the QPC, the quantum evolution is conditioned on the fluctuating detector current $I(t)$.
We identify regimes when the system-detector coupling becomes large relative to a typical system frequency, that leads to suppression of coherent tunneling transitions. The associated stabilization of the states in continuous measurement (see, e.g., Ref.[@Kor-osc; @RusKor-ent; @Ruskov-Zeno]) is an analog of the well known quantum Zeno effect.[@Khalfin; @MisraSudarshan; @Peres] Since the system-detector coupling is finite, the Zeno stabilization does not last forever. The stabilized states switch to each other with a rate $\Gamma_{sw} \ll \Gamma$. The switching happens on a time scale $1/\Gamma$. In fact, on this time scale the switching transition becomes irreversible. For an ensemble averaged description of the state evolution we will show that the irreversibility corresponds to an exponential decay (compare with Ref.[@Pryadko]) with the decay rate $\Gamma_{sw}$.
Correspondingly, in the case of a spin qubit we show via numerical simulation of the measurement process that the switching rate $\Gamma_{sw}$ plays the role of the spontaneous decay rate $\Gamma_{sp}$ for the two-level atom described above. Despite the quantum back action noise reflecting the noisy output, the spin qubit state will evolve according to Eqs.(\[atom-Bayes1\]), (\[atom-Bayes2\]) in a spin-charge basis that corresponds to the Zeno stabilized states. In particular, in our analysis the negative-result evolution of the qubit subsystem is not postulated but emerges from continuous noisy measurement evolution and Hamiltonian evolution at a microscopic level.
The paper is organized as follows. In Sec.II we review/derive the Bayesian equations for weak continuous measurement of a DQD two-electron spin system. In Sec. III we investigate the measurement collapse scenarios for different coupling regimes and give qualitative and quantitative explanations of the obtained results. In Sec. IV we estimate the detector-system coupling and other important parameters that could be relevant for an experiment to confirm our results. We also comment on the role of various sources of decoherence.
DQD two electron system and measurement device
==============================================
Hamiltonian and time scales {#Hamiltonian_time_scales}
---------------------------
The system of two electrons confined in the DQD, and the quantum point contact as a charge sensing detector are shown in Fig.\[schematic\]. External gate voltages form the potential profile of the DQD and are well controlled in the experiment.[@HansonReview] In particular, changing the energy difference between the dots, $\delta$, allows continuous tuning of the charge configuration between $(1,1)$ and $(0,2)$: when the electrons are in the right dot $(0,2)$ the ground state is the spin singlet $|S(0,2)\rangle$. The interdot tunneling coupling $\Omega$ allows coherent transitions between the singlet spin-charge states $|S(1,1)\rangle$ and $|S(0,2)\rangle$ which are close in energy (for $\delta=0$, the states $|S(1,1)\rangle$ and $|S(0,2)\rangle$ are in resonance). Transitions between the triplet states $|T(1,1)\rangle$ and $|T(0,2)\rangle$ are suppressed due to large energy mismatch: the highly energetic triplet states $|T_{\pm,0}(0,2)\rangle$ are well separated in energy from the singlet $|S(0,2)\rangle$[@Taylor] ($\Delta E_{ST} \sim 400 \mu eV$) due to tight confinement and on site exchange interaction. In the $(1,1)$ configuration the spin qubit subspace is formed by the singlet $|S(1,1)\rangle$ and the triplet state $|T_{0}(1,1)\rangle$ with zero spin projection on an external magnetic field $B$; the applied field removes the triplet degeneracy by splitting off the $|T_{\pm}(1,1)\rangle$ states (splitting $\Delta_{\pm} \approx 2.5 \mu\mbox{eV}$ for the GaAs DQD described in Ref.[@Marcus-Sci]). Thus, the DQD spin system is considered below as a three state quantum system (qutrit).
Introducing the short notations for the relevant three states: $|1\rangle \equiv |S(0,2)\rangle$, $|2\rangle \equiv |S(1,1)\rangle$, $|3\rangle \equiv |T_0(1,1)\rangle$, we write the system hamiltonian[@JouravlevNazarov; @Taylor]: $$\begin{aligned}
{\cal H}_{DQD}=
-\delta\ |1\rangle \langle 1|
+ \frac{\Omega}{2}\ (|1\rangle \langle 2| + |2\rangle \langle 1|)
&&\nonumber\\
+ \varepsilon _S\ |2\rangle \langle 2| +
\varepsilon _T\ |3\rangle \langle 3|
\label{H-2spin} .\end{aligned}$$ The presence of the $|T_{\pm,0}(0,2)\rangle$ states induces small exchange energies $\varepsilon _S$, $\varepsilon _T$ which can be taken into account perturbatively;[@Taylor] in what follows we will neglect these energies that are of the order of $\Omega^2/\Delta E_{ST} \ll \Omega$. Other possible terms in the Hamiltonian ${\cal H}_{DQD}$ (that are present for a general qutrit system[@CavesMilburn]) are suppressed due to conservation of spin.
In the total Hamiltonian of system plus detector $${\cal H} = {\cal H}_{DQD} + {\cal H}_{det} + {\cal H}_{int} ,$$ the (low transparency) QPC detector [@Gurvitz] is described with $$\begin{aligned}
{\cal H}_{det}=\sum_l E_l c_l^\dagger c_l + \sum_u E_u c_u^\dagger c_u
+\sum_{l,u} (T c_u^\dagger c_l + \mbox{H.c.}) ; &&
\label{detector}\end{aligned}$$ here the operator $c^\dagger_l$ ($c^\dagger_u$) creates an electron in the lower (upper) lead of the detector (Fig.\[schematic\]) and the tunneling $T$ between leads is assumed energy independent.
The system–detector interaction can be written in analogy with the qubit case; for a general qutrit measured by a hypothetical detector that can distinguish all three states one formally writes the interaction Hamiltonian as proportional to the two \[$SU(3)$\] generators, that are diagonal in the spin-charge basis: $$\begin{aligned}
{\cal H}_{int}=\sum_{l,r} \left[ A\, (|1\rangle \langle 1|
- |2\rangle \langle 2|) \otimes c_u^\dagger c_l \right. &&
\nonumber\\
\left. + B\, (|1\rangle \langle 1|
+ |2\rangle \langle 2| - 2\, |3\rangle \langle 3|) \otimes
c_u^\dagger c_l + \mbox{H.c.} \right]
\label{int-2spin} .\end{aligned}$$ Introducing detector transmission probability $T_k$ for each system state $|k\rangle$, the average currents can be expressed as $I_k =2\pi T_k \rho_l \rho_u e^2 V/\hbar,\ k=1,2,3$, where $V$ is the detector bias voltage, and $\rho_{l}$ and $\rho_{u}$ are the densities of states in the lower and upper lead. In the case of QPC, which cannot distinguish between states $|2\rangle \equiv |S(1,1)\rangle$ and $|3\rangle \equiv |T_0(1,1)\rangle$ the tunneling amplitudes $A$ and $B$ are not independent: $T_2=|T - A + B|^2=T_3=|T - 2 B|^2$, since the corresponding average currents are equal, $I_2=I_3\equiv I_{(1,1)}$. For the state $|1\rangle \equiv |S(0,2)\rangle$ the average current $I_{(0,2)}\neq I_{(1,1)}$ and it can be distinguished from the $(1,1)$ states.
In what follows we consider a situation when the internal detector dynamics given by Eqs.(\[detector\]), (\[int-2spin\]) is much faster than the two-spin dynamics due to ${\cal H}_{DQD}$. Also, it is assumed that the voltage applied to detector is large so that a typical detector decoherence time is much smaller than the typical electron tunneling times; thus, $\hbar/eV \ll e/I_{(0,2)},e/I_{(1,1)}\ll 1/\Omega$. The first inequality implies that coherences between different electron passages in the QPC can be neglected. Thus, the QPC detector will behave essentially classically on the typical time scale of the DQD two-spin dynamics.
Continuous quantum evolution according to the result
----------------------------------------------------
The quantum state evolution of an open quantum system[@Carmichael; @Davies] has an analog in a classical state estimation procedure;[@Holevo] it takes into account the actual measurement record that is imperfectly correlated with the system state. In a situation when the QPC is a weakly responding detector, $|\Delta I|\equiv |I_{(0,2)} - I_{(1,1)}| \ll I_{(0,2)}, I_{(1,1)}$, when every tunneling electron brings a little information about the system state, it is reasonable to condition the system evolution on the quasicontinuous noisy detector current $I(t)$.[@Kor-99-01] Typical measurement time (that reflects accumulation of a signal-to-noise ratio of order one) can be introduced[@Shnirman]${}^{,}$[@Kor-99-01], $\tau_{meas} = 2 S_0/(\Delta I)^2 \sim 1/\Gamma$, where $S_0\simeq 2e I_{(1,1)}$ is the low-frequency spectral density of the detector shot noise. \[We have neglected small differences between shot noises.\] The finite measurement time implies that given the noisy current $I(t)$, the system state is updated gradually. Below we sketch the derivation of the measurement evolution equations for a qutrit related to the “informational” Bayesian approach.[@Kor-99-01; @Kor-ent]
For the measurement evolution alone, the most straightforward way is to consider the elementary act of scattering of an incoming electron, $|i\rangle$, off the QPC tunnel barrier that depends on the charge state of the system, $|k\rangle$. The scattered state is expressed as a linear transform of the initial state: $|f\rangle=\hat{S}_k |i\rangle$ where the scattering matrix is (see, e.g., Ref.) $$\hat{S}_k=\left(
\begin{array}{cc}
r_k & t_k^* \\
t_k & -r_k^* \\
\end{array}
\right) ,$$ and $r_k$, $t_k$ are the reflection and transmission amplitudes. Following Jordan and Korotkov,[@JordanKorotkov] every tunneling electron can be mapped to an ancilla qubit, whose basis states are the scattering states: $|R\rangle$ for a reflected electron, and $|T\rangle$ for a transmitted electron. Scattering will entangle the states of the ancilla qubit and the qutrit. Then projective measurement on the ancilla will lead to POVM (positive operator-valued measure) measurement operators[@NielsenChuang] of the form: $\hat{M}_R=\mbox{diag}\{r_1,r_2,r_3\}$ and $\hat{M}_T=\mbox{diag}\{t_1,t_2,t_3\}$ that satisfy the completeness condition $\hat{M}_R^\dagger\,\hat{M}_R + \hat{M}_T^\dagger\,\hat{M}_T=1$. The diagonal form of these operators in the qutrit basis (the spin-charge basis) follows from the diagonal form of the interaction Hamiltonian ${\cal H}_{int}$, Eq.(\[int-2spin\]). For collecting electrons in the upper lead, the count of an electron updates the qutrit density matrix according to a POVM formula $\hat{\rho}^{'}=\hat{M}_T \hat{\rho} \hat{M}_T^\dagger/{\cal P}_T$ where ${\cal P}_T = \mbox{Tr}[\hat{M}_T^\dagger\,\hat{M}_T\hat{\rho}]=
\sum_k \rho_{kk} T_k$ is the total probability to find an electron in the upper lead and $T_k=|t_k|^2$ are the transmission probabilities introduced in the previous section. If an electron was not counted a similar update takes place using the measurement operator $\hat{M}_R$ and the reflection probabilities, $1-T_k=|r_k|^2$. In the basis where $\hat{M}_{R,T}$ are diagonal these evolutions take the form of Bayesian updates.[@JordanKorotkov]
The evolution rate of the system density matrix will be related to the rate of tunneling through the QPC. Introducing an average number $A\,\delta t$ of (independent) tunneling attempts per time $\delta t$, the average current[@Landauer] (given the system state $|k\rangle$) is then $I_k=e A T_k$. For times of the order of individual tunneling times, $\delta t \sim e/I_k$, we can consider a negative-result evolution of the qutrit density matrix. For the evolution of the diagonal elements in case of no tunneling we write: $$\rho_{kk}(\delta t) = \rho_{kk}(0)\,(1-T_k)^{A\,\delta t}/{Norm} .
\label{neg-rez-elementary}$$ ($Norm$ is a proper normalization.) For a low transparency QPC, $T_k \ll 1$, the number of attempts is large, $A\,\delta t \sim 1/T_k \gg 1$ so that $(1-T_k)^{A\,\delta t}\approx e^{-T_k A\,\delta t}=e^{-\frac{I_k}{e}\delta t}$. Thus, we obtain a negative-result evolution similar to Eq.(\[atom-Bayes1\]) where the spontaneous decay rate $\Gamma_{sp}$ is replaced by the tunneling rates $\frac{I_k}{e}$, each for every $k=1,2,3$. The evolution of the non-diagonal elements can be shown to be of the form similar to Eq.(\[atom-Bayes2\]).
The above evolution can be hardly observed, since for typical system times, $1/\Omega,1/\Gamma \gg e/I_k$, many electrons pass through QPC. For $N$ independent attempts one can derive the conditional probability for $m$ successful tunnelings[@JordanKorotkov] (per time $\tau$) given the system is in state $|k\rangle$ $${\cal P}(m,N|k) = C_N^m T_k^m (1-T_k)^{N-m}
\label{independent-probab} ,$$ so that the Bayesian update of the system (qutrit) density matrix can be shown to be of the form: $\rho_{kk}^{'} = \rho_{kk} {\cal P}(m,N|k) /Norm$ and $\rho_{kl}^{'} = \rho_{kl} \sqrt{\rho_{kk}^{'}\rho_{ll}^{'}/\rho_{kk}\rho_{ll}}$.
We mention that the update related to Eq.(\[independent-probab\]), is just a simple composition of $N$ elementary updates, each corresponding to tunneling $(1)$ or no tunneling $(0)$ of a single electron: e.g., for a result $\tilde{Q}=(0,1,\ldots,0)$ the measurement operator $\hat{M}_{\tilde{Q}}$ is just a multiplication of $N$ diagonal operators, $\hat{M}_{\tilde{Q}}=\hat{M}_{R_1} \hat{M}_{T_2} \ldots \hat{M}_{R_N}$. Also note that Eq.(\[independent-probab\]) sums up over $C_N^m$ identical possibilities since QPC cannot distinguish different sequences $\tilde{Q}_1,\tilde{Q}_2,\ldots$ with the same total charge tunneled to the right lead. By representing the (random) number $m$ of tunneling electrons per time interval $\tau$ through the QPC current, $m\equiv \tau \bar{I}(t,\tau)=\int_t^{t+\tau} I(t') dt'$, and using De Moivre-Laplace limit theorem ($N\gg 1$, $T_k$ fixed) we can replace ${\cal P}(m,N|k)$ by a Gaussian distribution for the measurement result $\bar{I}(t,\tau)$, given the state $|k\rangle$: $P_k(\bar{I})=\sqrt{\tau/\pi S_0} \exp{[-(\bar{I}(t,\tau) - I_k)^2 \tau/S_0]}$, where $I_k\simeq e M T_k/\tau$ is the average current, and $S_0$ is the shot noise spectral density for a low transparency and weakly responding QPC. The update of the system density matrix can be written again as a Bayesian (informational) evolution:[@Kor-99-01] $$\rho_{kl}(t+\tau) =
\rho_{kl}(t) \frac{\sqrt{P_k(\bar{I})}\,\sqrt{P_l(\bar{I})}}{P(\bar{I})}
\label{quantBayes},$$ with the total probability of a particular result $\bar{I}$ given by $P(\bar{I})=\sum_k \rho_{kk}(t) P_k(\bar{I})$.
Differentiating Eq.(\[quantBayes\]) over $\tau$ (at $\tau \to 0$) one obtains a stochastic equation for the qutrit state evolution (in Stratonovich form[@Oksendal]). Taking into consideration the (internal) Hamiltonian evolution and possible dephasing, one can write: $$\begin{aligned}
&&\dot{\rho}_{kl} =
\frac{\rho_{kl}}{S_0} \sum_j \rho_{jj}\,
\left\{ (I_k - I_j) \left(I(t) - \frac{I_k + I_j}{2}\right)\right.
\nonumber \\
&&\left. \qquad\ {} + (I_l - I_j) \left(I(t) - \frac{I_l + I_j}{2}\right) \right\}
\nonumber \\
&&\qquad\ {} -\frac{\imat}{\hbar} [{\cal H}_{DQD}, \rho]_{kl}
-\gamma_{kl}\, \rho_{kl} .
\label{Bayes1}\end{aligned}$$ Here $I(t)$ is the formal limit (at $\tau \to 0$) of the observed detector signal $\bar{I}(t,\tau)$. For numerical simulations of a measurement one complements[@Kor-99-01] Eq.(\[Bayes1\]) by $$I(t) = \sum_k \rho_{kk}(t) I_k + \xi(t)
\label{current-model}$$ that is consistent with the statistics of $P(\bar{I})$; here $\xi(t)$ is a white noise with a spectral density $S_{\xi}=S_0$. Eq.(\[Bayes1\]) is of the same form as an analogous evolution equation for a system of $N$ qubits;[@Kor-ent] here the summation is over the three qutrit states. The dephasing rates, $\gamma_{kl}$, are related to the detector non-ideality;[@Kor-99-01; @back-action] experimentally[@Buks] and theoretically[@Kor-99-01; @back-action; @AverinSukhorukov] a QPC is close to an ideal detector. An SET is usually highly non-ideal[@Kor-99-01], however it may reach ideality close to 1 in the co-tunneling or Cooper pair tunneling regime.[@cotun] It is worthwhile to note that the measurement evolution and the Hamiltonian evolution enter in Eq.(\[Bayes1\]) independently; they just reflect the measurement (POVM) and unitary postulates applied to the system at a coarse grained time $t \gg e/I_k \gg \hbar/eV$ where the evolution is noisy and quasicontinuous. In Sec. III we will show via numerical solutions of Eq.(\[Bayes1\]) that the continuous measurement evolution and the Hamiltonian evolution interplay non-trivially so that new effective (negative-result) measurement and Hamiltonian evolutions of the system arise at a larger time scale.
Ensemble averaged evolution of the system
-----------------------------------------
A total ignorance of a particular measurement result $\bar{I}(t)$ corresponds to a situation when the QPC detector is considered just as a part of a (Markovian) environment surrounding the system. Correspondingly, the density matrix available to such an observer (denoted as $\langle\rho_{kl}\rangle$) will be quite different from that described by Eq.(\[Bayes1\]). The density matrix $\langle\rho(t)\rangle$ can be related to $\rho(t)$, Eq.(\[Bayes1\]) by a formal procedure of an ensemble averaging over possible results $\bar{I}(t)$ at every time moment $t$, similar to the classical probabilities.[@Holevo] The averaging can be performed (for a sufficiently small time interval $\tau$), e.g., by using the total probability of a particular result $\bar{I}$, given by $P(\bar{I})$ and then simply adding the Hamiltonian evolution. The result will be a standard master equation[@Leggett] $$\langle\dot{\rho}_{kl}\rangle = -\Gamma_{kl} \, \langle\rho_{kl}\rangle
-\frac{\imat}{\hbar} [{\cal H}_{DQD}, \langle\rho\rangle]_{kl}
\label{Gen-master}$$ with ensemble-averaged dephasing rates $$\Gamma_{kl} = (I_k - I_l)^2/4 S_0 + \gamma_{kl} .
\label{decohrates}$$ For a quantum limited (ideal) detector, $\gamma_{kl}=0$, the dephasing rates produced due to averaging are just $(I_k - I_l)^2/4 S_0$ which is the minimum allowed by quantum mechanics.[@Kor-99-01; @Kor-osc; @back-action; @Ruskov-Bell]
The individual dephasings, $\gamma_{kl}\neq 0$, may be a consequence of partial ignorance of the measurement result[@Kor-99-01] and are parameterized by the detector ideality (efficiency) $\eta$ ($0\leq \eta \leq 1$): $\gamma_{kl} = (\eta^{-1} -1)\,(I_k - I_l)^2/4 S_0$, i.e., $\Gamma_{kl}=(I_k - I_l)^2/4 S_0\eta$. Other sources of decoherence of the DQD system will be discussed in Sec. IV.
Emergence of negative-result evolution in the qubit subspace
============================================================
When the unitary evolution of the three-state system is taken into account one has to explore the full Bayesian Eq.(\[Bayes1\]), which, as a rule, does not provide simple solutions. In what follows, using Eq.(\[Bayes1\]), we will perform numerical simulation of the measurement process for various regimes of the system-detector dynamics. The non-trivial interplay of the quantum dynamics can be seen if one compares the effects of measurement evolution alone vs. total evolution. The measurement alone tends to collapse the system (qutrit) in either the state $|1\rangle$ or to the qubit subspace $\{|2\rangle,|3\rangle\}$ leaving states $|2\rangle$ and $|3\rangle$ unresolved. However, adding the continuous coherent mixing of $|1\rangle$ and $|2\rangle$ by ${\cal H}_{DQD}$ leads to effective resolution of the states $|2\rangle$ and $|3\rangle$, so that a continuous collapse happens either to the state $|3\rangle$ or to the remaining spin-singlet subspace.
If the continuous collapse happens to the singlet subspace, at small coupling $\Gamma$ the system will perform quantum oscillations, that are weakly perturbed by the measurement. When the coupling becomes large the picture is qualitatively different: one reaches the regime of Zeno stabilization of the system’s singlet states. The latter is characterized by long time intervals when the QPC current is either $I_{(1,1)}$ or $I_{(0,2)}$, interupted by rare switching between them; the qutrit state is correspondingly $|2\rangle \equiv |S(1,1)\rangle$ or $|1\rangle \equiv |S(0,2)\rangle$. If the system was initially into the qubit subspace, the continuous collapse takes the form of a slow negative-result evolution of the qubit state until it eventually switches to $|S(0,2)\rangle$. The negative-result evolution is a Bayesian evolution conditioned by the information that the “qubit did not switch”. This evolution emerges in the Zeno regime as a solution of the underlying Bayesian stochastic evolution, Eq.(\[Bayes1\]). It is important to note that the key feature for establishing a negative-result evolution is the irreversibility of the switching event (compare with Ref.). Within the stochastic measurement evolution, the irreversibility is consistent with an important property of Eq.(\[Bayes1\]) that is seen in numerical simulations: given a measurement record $I(t)$, two evolutions that start from different initial states, $\rho^{(1)}(0)$ or $\rho^{(2)}(0)$, will become undistinguishable[@Kor-99-01; @Ruskov-Zeno] at a time scale of the order or greater than $1/\Gamma$. In other words the system “forgets” its initial state, $\rho(0)$ so that further evolution is dominated by the result, $I(t)$ itself. The regime of current stabilization, (that is essentially classical[@Ruskov-Bell]) then would correspond to irreversibility of the switching event.
Collapse scenarios
------------------
For a numerical simulation of the measurement process we use Eq.(\[Bayes1\]), supplemented with the relation for the current signal, Eq.(\[current-model\]), where we incorporated the current level degeneracy for the states $|2\rangle$ and $|3\rangle$ ($I_2=I_3=I_{(1,1)}$). In order to understand the collapse scenarios it is instructive to look at the stochastic equation for the relevant density matrix elements transformed from Eq.(\[Bayes1\]) to their Itô form[@Oksendal] (below we use $\hbar=1$). $$\begin{aligned}
&& \dot{\rho}_{11} =
-\Omega\, \mbox{Im}\rho_{12} + \rho_{11}\,(1-\rho_{11})\,\frac{\Delta I}{2S_0}\,\xi(t), \ \
\label{r11-Ito}\\
&& \dot{\rho}_{33} =
- \rho_{11} \rho_{33}\,\frac{2 \Delta I}{S_0}\,\xi(t),
\label{r33}\end{aligned}$$ $\qquad\qquad\qquad\dot{\rho}_{22}=-\dot{\rho}_{11}-\dot{\rho}_{33}$, and $$\begin{aligned}
&& \dot{\rho}_{12} =
i\, \delta\, \rho_{12} + i\, \frac{\Omega}{2}\,(\rho_{11}-\rho_{22}) -\Gamma_{12}\, \rho_{12}
\nonumber\\
&& \qquad\ \ {} - (2 \rho_{11} - 1)\,\rho_{12}\,\frac{\Delta I}{S_0}\,\xi(t),
\label{r12-Ito}\\
&& \dot{\rho}_{13} =
i\, \delta\, \rho_{13} - i\, \frac{\Omega}{2}\,\rho_{23} -\Gamma_{13}\, \rho_{13}
\nonumber\\
&& \qquad\ \ {} - (2 \rho_{11} - 1)\,\rho_{13}\,\frac{\Delta I}{S_0}\,\xi(t),
\label{r13-Ito}\\
&& \dot{\rho}_{23} =
- i\, \frac{\Omega}{2}\,\rho_{13} - \rho_{11}\,\rho_{23}\,\frac{2 \Delta I}{S_0}\,\xi(t).
\label{r23-Ito}\end{aligned}$$ Here $\Delta I \equiv I_{(0,2)}-I_{(1,1)}$ is the current difference between the charge subspaces that can be distinguished by the QPC. The system-detector coupling explicitly enters in Eqs.(\[r12-Ito\]),(\[r13-Ito\]) as $\Gamma_{12}=\Gamma_{13}\equiv (\Delta I)^2/4 S_0 + \gamma_{12}=\Gamma=(\Delta I)^2/4 S_0 \eta$
In the absence of the Hamiltonian, ${\cal H}_{DQD}=0$, the equation for $\rho_{11}$ becomes a pure noise and decouples; it can be heuristically considered as a (position dependent) Wiener process for the restricted variable $0\leq\rho_{11}\leq 1$ with a diffusion coefficient, $\sim \rho_{11} (1-\rho_{11})$, approaching a minimum (zero) at the endpoints, $\rho_{11}=0$ or $\rho_{11}=1$. Just as in the single qubit case[@KorJordan-undo] it suggests that the endpoints are the two possible attractors for a given realization of the measurement. In the qutrit case, the measurement also preserves the ratio $\rho_{22}/\rho_{33}\equiv\alpha$ and the measurement evolution happens on a ray $\rho_{33}=(1-\rho_{11})/(1+\alpha)$ in the physical triangle of the plane $(\rho_{11},\rho_{33})$ \[$0\leq\rho_{kk}\leq1, k=1,2,3$\]. Switching on the Hamiltonian mixes the density matrix components leaving $\rho_{33}$ unaffected; geometrically this is represented as a horizontal moving in the plane $(\rho_{11},\rho_{33})$ that changes $\alpha$ \[causing the system to change from one ray to another\]. From this point of view it is easy to imagine how the new attractors become $\rho_{33}=0$ (a horizontal ray) or the point $\rho_{33}=1$. Finally, it is clear from Eq.(\[r33\]) that if the system is in either of the subspaces, $\{|1\rangle,|2\rangle\}$ or $|3\rangle$, it will remain there: neither Hamiltonian nor measurement evolution mixes the subspaces, which is true for any DQD parameters, $\delta$, $\Omega$. The decoupling of subspaces is a consequence of the spin blockade and indistinguishability of the states $|2\rangle$ and $|3\rangle$ by measurement.
The decoupling implies that the system will continuously collapse to one of the subspaces under a weak continuous measurement as seen in numerical simulations, Fig.\[weak\]. Since the corresponding ensemble averaged equation for $\rho_{33}(t)$ is simply $d \langle\rho_{33}(t)\rangle /dt =0$ (ensemble averaging implies just nullifying of the noise in Itô form of the equations[@Oksendal]), then the ensemble averaged density matrix element is conserved: $\langle\rho_{33}(t)\rangle = \rho_{33}(0)$. Thus, it must be that the fraction of members of the ensemble that collapses to $\rho_{33}(t)=1$ (after a sufficient measurement time $t$) is just $\rho_{33}(0)$, which means that the probability of collapse to either of the subspaces is $\rho_{33}(0)$ and $1-\rho_{33}(0)$ respectively (i.e., the usual probability rules apply).
System evolution at small coupling. Quantum oscillations and spin blockade {#small_coupling}
--------------------------------------------------------------------------
In order to characterize the emergence of a negative result evolution from continuous noisy measurement we will first consider the case when the system may establish quantum oscillations weakly perturbed by the measurement, i.e., when negative-result evolution is not yet established. In particular, this happens for small system-detector coupling, $\Gamma \ll \Omega$ while the quantum dots detuning $\delta$ is also small. For simplicity, we first consider the case of an ideal detector, $\eta=1$.
![(Color online) Weak coupling regime, $\Gamma\ll\Omega$, at zero detuning. (a) A realization of $\rho_{33}(t)$ evolution (mixed initial state) showing collapse to the spin-singlet subspace $\{|1\rangle,|2\rangle\}$ (dashed line); weakly perturbed coherent oscillations of $\rho_{11}(t)$ (thick line) into the subspace. (b) Qutrit purification due to measurement with an ideal detector, $\eta=1$.[]{data-label="weak"}](fig2ab.eps){width="3.2in"}
An example of the three-state system evolution is shown on Fig.\[weak\]a. In this particular realization of the measurement process, while starting from a mixed initial state ($\rho_{11}(0)=0$, $\rho_{22}(0)=\rho_{33}(0)=1/2$, $\rho_{ij}(0)=0, i\neq j$) $\rho_{33}(t) \to 0$ (dashed line on Fig.\[weak\]a), i.e., the system is continuously collapsed to the spin-singlet subspace, $\{|1\rangle\equiv|S(0,2)\rangle,|2\rangle\equiv|S(1,1)\rangle\}$ after a transition time of the order of $1/\Gamma$. On the same time scale, weakly perturbed quantum oscillations are established in this subspace \[solid line on Fig.\[weak\]a) for $\rho_{11}(t)$\].
The oscillation scenario in case of collapse to the singlet subspace is easily understood. Once $\rho_{33}(t)$ gets to zero, Eqs.(\[r11-Ito\]),(\[r12-Ito\]) just become of the same form as that describing quantum oscillations for a single qubit.[@Kor-osc] For zero detunning, $\mbox{Re}\rho_{12}$ decouples ($\mbox{Re}\rho_{12}(t)\rightarrow 0$), while $\rho_{11}(t)-\rho_{22}(t)=\cos[ \Omega t +\varphi(t)]$ and $2 \mbox{Im}\rho_{12}(t)=\sin[ \Omega t +\varphi(t)]$ oscillate with a unity amplitude (for $\eta=1$), with phase $\varphi(t)$ that slowly diffuses in time.[@Kor-osc] Correspondingly, the oscillating scenario can be distinguished by the average current $\langle I(t) \rangle_t = (I_{(0,2)} + I_{(1,1)})/2$ or by the current spectral density $S_I(\omega) = 2\int_{-\infty}^{\infty} K_I(\tau)\exp(i\omega \tau) d\tau$ (with $K_I(\tau)=\langle I(t) I(t+\tau)\rangle_t - \langle I(t) \rangle_t^2$ being the current correlation function). Since the evolution of $\rho_{33}(t)$ is transitional in character (see Fig.\[weak\]), it will not affect the long-time average $K_I(\tau)$. Then using, e.g., the methods of Ref., it can be shown that the detector power spectrum will have the same form as in the one-qubit case[@Kor-osc] for any DQD parameters, $\delta$, $\Omega$. In particular, in the weak coupling regime (for $\delta=0$) the spectrum $S_I(\omega)$ exhibits a Lorenzian peak at the Rabi frequency $\Omega$ with a signal-to-noise ratio of $4\eta$, and a width $\Gamma = (I_{(0,2)} - I_{(1,1)})^2/4S_0\eta$.
On (Fig.\[weak\]b) the qutrit purity is plotted. \[We have defined it as $\mbox{Pur}(t) = (3\mbox{Tr}[\hat{\rho}(t)^2] - 1)/2$, so that $\mbox{Pur}=1$ for a pure state and $\mbox{Pur}=0$ for a totally mixed state.\] It is seen that the qutrit is eventually reaching a pure state \[even though with a random phase, $\varphi(t)$\] for a time of the order of $1/\Gamma$. The purification of the qutrit state is yet another demonstration of non-trivial interplay of dynamics. Indeed, Hamiltonian evolution alone conserves purity, while a measurement alone leaves states $|2\rangle$ and $|3\rangle$ unresolved. The purification of the state is due to an effective resolution of the states $|2\rangle$ and $|3\rangle$ and the fact that no information is lost with an ideal measurement.
For a measurement with a non-ideal detector, $\eta < 1$, simulations show (if collapse happened to the spin-singlet subspace) that the amplitude of quantum oscillations is less than $1$ and fluctuates in time. Correspondingly, the average purity $\langle \mbox{Pur}(t)\rangle_t$ saturates at some lower value. Using Eqs.(\[r11-Ito\])-(\[r23-Ito\]) we have derived in the weak coupling regime (compare with Ref.[@QinRusKor]) $\langle Pur(t)\rangle_t \simeq 1 +(3/4)[1/2\eta-\sqrt{(1+1/2\eta)^2-2}]$ so that the qutrit state remains mixed; for small $\eta$ the qutrit purity approaches $1/4+3/2\eta$. If collapse happens to $\rho_{33}(t) \to 1$, the state will purify even for measurement with a non-ideal detector. In this scenario the average current is $I_{(1,1)}$ and the power spectrum is flat, $S_I(\omega)=S_0$ \[the detector signal is just $I(t) = I_{(1,1)} + \xi(t)$\].
Large coupling. Zeno stabilization and emergence of negative-result evolution
-----------------------------------------------------------------------------
We now turn to the case of relatively large system detector coupling, $\Gamma \gg \Omega$; the DQD detuning energy, $\delta$, can be either small or large. Numerical simulations confirm that collapse scenarios remain the same in the large coupling regime consistent with our argumentation in Sec. III A. However, in the strong coupling case the evolution qualitatively changes; the typical collapse time to either of the subspaces, {$|1\rangle,|2\rangle$} or $|3\rangle$, becomes much longer than $1/\Gamma$. Also, instead of quantum oscillations within the spin-singlet subspace $\{|1\rangle\equiv|S(0,2)\rangle$, $|2\rangle\equiv|S(1,1)\rangle\}$, associated with $\Omega$, we have relatively long stabilization of the system state (Fig. \[strong-Zeno\]) in one of the two states since the measurement is trying to localize and “freeze” the system in a definite charge state. This is a manifestation of the quantum Zeno effect:[@Khalfin; @MisraSudarshan] in the case of continuous measurement the detector is always coupled to the system, so the approach to quantum Zeno regime corresponds to the limit of stronger and stronger coupling. For a finite coupling the long stabilization periods will be interrupted by rare switching events between the subspaces (the switching transition time is of the order of $1/\Gamma$; see below).
![(Color online) Strong coupling regime, $\Gamma\gg\Omega$. Zeno stabilization in the qubit subspace, $\rho_{11}=0$, for a long time until it switches to $\rho_{11}=1$ (thick dashed line). The noisy qubit evolution of a mixed initial state shown for $\rho_{22}-\rho_{33}$ and $|\rho_{23}|$ vs. postulated negative-result evolution given by Eqs.(\[SW-rate\])-(\[Bayes-upd2\]) (long dashed and short dashed lines). []{data-label="strong-Zeno"}](fig3.eps){width="3.2in"}
Anticipating the scenario of a negative-result evolution in the qubit subspace outlined in the beginning of Sec. III, we turn to the calculation of the switching rate between the two singlet states, which will set the time scale of non-unitary dynamics. From an ensemble averaged point of view the qubit “decay” from $|S(1,1)\rangle$ to $|S(0,2)\rangle$ (triplet state $|T_0(1,1)\rangle$ is spin blocked) is described as a smooth decrease of $\langle\rho_{11}(t)\rangle$ and therefore the switching rate would be most easily obtained if we consider the master equation (\[Gen-master\]) that follows from averaging of measurement evolution.
Due to current level degeneracy, $I_2=I_3=I_{(1,1)}$, the dephasing rate $\Gamma_{23}$, Eq.(\[decohrates\]), vanishes just because the detector cannot distinguish between the corresponding states (no information can be obtained and therefore no information can be lost). Introducing $\Gamma_{12}=\Gamma_{13}=\Gamma \equiv (\Delta I)^2/4 S_0 \eta$ the master equation reads $$\begin{aligned}
\lefteqn{ \langle\dot{\rho}_{kl}\rangle =
-\frac{\imat}{\hbar} [{\cal H}_{DQD}, \langle\rho\rangle]_{kl} }
&& \label{Master} \\
&& \qquad {} - \Gamma\ \left(
\begin{array}{ccc}
0 & \langle\rho_{12}\rangle & \langle\rho_{13}\rangle \\
\langle\rho_{21}\rangle & 0 & 0 \\
\langle\rho_{31}\rangle & 0 & 0 \\
\end{array}
\right)_{kl} \nonumber\end{aligned}$$ In order to calculate the switching rate from the state $|1\rangle =|S(0,2)\rangle$ we start with the initial condition $\rho_{11}=1$. Using Eq.(\[Master\]) one derives the small time evolution: $$\langle\rho_{11}(t)\rangle = 1 - \frac{\Omega^2 t^2}{4} + \cdots
\label{small-t} .$$ Notice, that there is no linear term in the expansion, Eq.(\[small-t\]), which tells us that the decay is not exponential at a small time scale. This fact makes quantum Zeno effect physics possible. The $t^2$-coefficient turns out to be $-(\Omega)^2/4 = -\mbox{Tr}[\hat{\rho}^2 {\cal H}^2_{DQD}] +
\mbox{Tr}[\hat{\rho}{\cal H}_{DQD}\hat{\rho}{\cal H}_{DQD}]$, i.e. it is determined by coherent (Hamiltonian) evolution alone, consistent with the discussion in Refs. .
It is instructive to find an approximate solution for $\langle\rho_{11}(t)\rangle$ considering in (\[Master\]) terms proportional to $\Omega$ as a small perturbation ($\Omega \ll \Gamma$, for strong coupling). To first order in the perturbation $\Omega$ we obtain for $\mbox{Im}\langle\rho_{12}(t)\rangle$: $$\begin{aligned}
\lefteqn{\mbox{Im}\langle\rho_{12}(t)\rangle =e^{-\Gamma t}\,\left[\mbox{Re}\rho_{12}(0)\,\sin{\delta t} +
\mbox{Im}\rho_{12}(0)\,\cos{\delta t} \right] }
&& \nonumber\\
&& \ {}+ \frac{\Omega}{\Gamma^2+\delta^2}\,\left\{\frac{\Gamma}{2}
+ e^{-\Gamma t}\,\left[ \delta\,\sin{\delta t}- \Gamma\cos{\delta t}\right] \right\}.
\label{pert-omeg}\end{aligned}$$ Using the relation, $\langle\rho_{11}(t)\rangle=\rho_{11}(0) - \Omega \int_0^t \mbox{Im}\langle\rho_{12}(t')\rangle dt'$, we can obtain an approximate solution for $\langle\rho_{11}(t)\rangle$ and its small $t$ expansion coincides with first few terms of the expansion, Eq.(\[small-t\]). However on a time scale $t^* \gtrsim 1/\Gamma$, as seen in Eq.(\[pert-omeg\]), the exponential terms drop out and one reaches an expansion that has a linear term: $\langle\rho_{11}(t^*)\rangle \simeq 1 - \frac{\Omega^2\Gamma}{2(\Gamma^2+\delta^2)}\,t^*$. Similarly, one can show that if one starts from an initial state in the {$|2\rangle$, $|3\rangle$}-subspace then on a coarse grained time scale $\langle\rho_{22}(t^*)\rangle \simeq
\rho_{22}(0)\,\left[1 - \frac{\Omega^2\Gamma}{2(\Gamma^2+\delta^2)}\,t^*\right]$; $\langle\rho_{33}(t)\rangle$ is just conserved. By solving numerically the master equation (\[Master\]) for $\langle\rho_{11}\rangle$, $\langle\rho_{22}\rangle$, $\mbox{Re}\langle\rho_{12}\rangle$, $\mbox{Im}\langle\rho_{12}\rangle$ one confirms that the decay of the subspaces is indeed exponential for large times and the switching rate between the subspaces is $$\Gamma_{sw} =\frac{\Omega^2\Gamma}{2(\Gamma^2+\delta^2)}
\label{SW-rate} .$$ Note that the strong coupling limit when $\Gamma \gg \Omega$ ($\delta$ is arbitrary) implies that $\Gamma_{sw} \ll \Gamma$, i.e. the subspace life time is much longer than both $1/\Gamma$ and $1/\Omega$ implying quantum Zeno stabilization.[@EnsAverageView] The exponential decay at times $t^* \gtrsim 1/\Gamma$ is a sign of the irreversibility of the measurement[@Peres-book] that appears as a switching event.
Having calculated the switching rate between the spin-charge states we can write (postulate) an ansatz for the time evolution of $\rho(t)$ according to a given result. Starting from the qubit subspace, for times $t^* \gtrsim 1/\Gamma$ one will be able to discriminate between the two current values ($I_{(1,1)}$ or $I_{(0,2)}$) and thus to distinguish whether the system has decayed (switched) to the third state or not. The conditional probability for the state $|2\rangle = |S(1,1)\rangle$ not to decay by time $t$ is given by $P_2(t)=\exp{(-\Gamma_{sw}t)}$, while analogous probability for $|3\rangle = |T_0(1,1)\rangle$ is $P_3(t)=1$ (due to spin-blockade). Using the quantum Bayes rule, similar to the two-level atom, Eqs.(\[atom-Bayes1\]),(\[atom-Bayes2\]) \[see also Eq.(\[quantBayes\])\], one can write the effective negative-result evolution of the spin qubit subsystem given that it did not decay by time $t$: $$\begin{aligned}
&&\rho_{22}( t) = \frac{\rho_{22}(0)\, P_2(t)}{P_{tot}(t)} ,\
\rho_{33}( t) = \frac{\rho_{33}(0)}{P_{tot}(t)} \ \
\label{Bayes-upd1}\\
&& \rho_{23}( t) = \rho_{23}(0)\,
\sqrt{\frac{\rho_{22}(t)\,\rho_{33}(t)}{\rho_{22}(0)\,\rho_{33}(0)}}
\ e^{ -i \phi(t,\delta,\Omega,\Gamma) } , \qquad
\label{Bayes-upd2}\end{aligned}$$ where the total probability not to decay is given by $P_{tot}(t)=\rho_{22}(0)\, P_2(t) + \rho_{33}(0)\, P_3(t)$, and $\phi(t,\delta,\Omega,\Gamma)$ is an accumulated phase (see below).
Consistency of the informational (Bayesian) approach requires that the negative-result ansatz for the $\rho(t)$-evolution be reproduced by the underlying evolution, Eq.(\[Bayes1\]), according to the noisy record $I(t)$. In Fig.\[strong-Zeno\] we show the negative-result evolution for $\rho_{22}(t) - \rho_{33}(t)$ and $|\rho_{23}(t)|$ defined by Eqs.(\[SW-rate\]),(\[Bayes-upd1\]),(\[Bayes-upd2\]), versus the density matrix evolution generated through simulation of the noisy measurement process via Eqs.(\[Bayes1\]),(\[current-model\]). The (noisy) evolution in the qubit subspace is quite regular, and the state eventually approaches $|3\rangle \equiv |T_0(1,1)\rangle$ (if the system did not switch). One can see that the two evolutions well agree in the strong coupling regime; the agreement is already established at $\Gamma \gtrsim 5 \Omega$. The agreement reveals a non-trivial property of the Bayesian stochastic evolution Eq.(\[Bayes1\]). It means that the postulated negative-result evolution given by Eqs.(\[Bayes-upd1\]), (\[Bayes-upd2\]), can be actually derived from Eq.(\[Bayes1\]) in the Zeno regime, as an interplay of measurement evolution and Hamiltonian evolution at an underlying microscopic level.
The time scale of the negative-result evolution is set by the switching rate $\Gamma_{sw}$, Eq.(\[SW-rate\]), in which the detector rate $\Gamma=(\Delta I)^2/4 S_0 \eta \equiv (\Delta I)^2/4 S_0 + \gamma_{12}$ is the total rate, including the measurement rate and the additional rate due to detector non-ideality. Via numerical simulations we have confirmed the dependence of $\Gamma_{sw}$ on $\eta$. We note that while the rate $\gamma_{12}$ would lead to dephasing in the spin-singlet subspace {$|1\rangle,|2\rangle$} \[see Eq.(\[r12-Ito\])\], it affects the spin qubit coherently[@non-ideality-neg] in the sense that Eq.(\[Bayes-upd2\]) preserves the coherence ratio $|\rho_{23}|/\sqrt{\rho_{22}\,\rho_{33}}$. The reason is the indistinguishability of the states $|S(1,1)\rangle$ and $|T_0(1,1)\rangle$ by the measurement. Particularly, a pure state remains pure, which we have confirmed numerically. (A similar conclusion was drawn in Ref. from a completely different viewpoint.) The mixed state will generally purify (see the discussion in Ref.). The final qubit (and qutrit) purification happens on the same time scale $\Gamma_{sw}^{-1} \gg \Gamma^{-1}$ as the collapse to the spin-singlet or spin-triplet subspaces.
The update for the non-diagonal element $\rho_{23}$ in (\[Bayes-upd2\]) reflects not only the conservation of coherence, but includes an accumulated phase $\phi(t,\delta,\Omega,\Gamma)$, which remains undefined by the negative-result ansatz. Stochastic numerical simulations by Eqs.(\[Bayes1\]),(\[current-model\]) show that the phase is linear in time even for small detuning, $|\delta| \lesssim \Omega$: $\phi(t) = \varepsilon^{\mbox{\small eff}}_{23}(\delta,\Omega,\Gamma)\,t$ and vanishes for $\delta=0$ (while noisy, it stabilizes just for times $\gtrsim 1/\Gamma$). The coefficient $\varepsilon^{\mbox{\small eff}}_{23}$ can be interpreted as an energy splitting between the spin qubit states $|2\rangle$ and $|3\rangle$ induced by the negative-result measurement in the presence of the localized singlet state, $|1\rangle$. For small detuning, $\delta \sim \Omega$, the energy splitting is small, $|\varepsilon^{\mbox{\small eff}}_{23}|\ll |\varepsilon^{pert}_{23}|\simeq \frac{\Omega^2}{4|\delta|}$; it has the same sign as the exchange splitting, $\varepsilon^{pert}_{23}$, that would be induced perturbatively.
The effective energy splitting can be derived if one compares ensemble averaging of the negative-result evolution given by Eqs.(\[Bayes-upd1\]),(\[Bayes-upd2\]) with the ensemble averaged evolution of the “original” stochastic equations (\[Bayes1\]). Since the negative-result evolution is the Bayesian evolution at a coarse grained time scale $t^* \gtrsim 1/\Gamma$, its ensemble averaging must coincide with the evolution given by the master equation (\[Master\]) considered at times $\gtrsim 1/\Gamma$. Averaging of the negative-result evolution is straightforward using the probability not to decay, $P_{tot}(t)$. Starting from the qubit subspace it gives, e.g., for the non-diagonal matrix element: $$\langle\rho_{23}(t)\rangle = \rho_{23}(0)\, e^{-\Gamma_{sw}\,t/2}\,
e^{-i\,\varepsilon^{\mbox{\footnotesize eff}}_{23}\, t} .
\label{r23-ens-large-t}$$ On the other hand Eq.(\[Master\]) gives for the ensemble averaged evolution of $\langle\rho_{23}(t)\rangle$, $\langle\rho_{13}(t)\rangle$: $$\begin{aligned}
&&\langle{\dot{\rho}}_{23}(t)\rangle = - i\,\frac{\Omega}{2}\langle\rho_{13}(t)\rangle
\label{r23-ens}\\
&& \langle{\dot{\rho}}_{13}(t)\rangle = i\, \delta\,
\langle\rho_{13}(t)\rangle - i\,\frac{\Omega}{2}\,
\langle\rho_{23}(t)\rangle - \Gamma\, \langle\rho_{13}(t)\rangle
\label{r13-ens} . \qquad\\end{aligned}$$ It can be solved exactly, and for the initial values $\rho_{13}(0)=0$, $\rho_{23}(0)$, we obtain: $$\langle\rho_{23}(t)\rangle = \rho_{23}(0)\, e^{-\tilde{\Gamma} t/2}\,
\left( \cosh{\tilde{\Omega} t} +
\frac{2\tilde{\Gamma} }{\tilde{\Omega} } \,
\sinh{\tilde{\Omega} t} \right)
\label{complex-sol}$$ where $\tilde{\Gamma} \equiv \Gamma - i\,\delta$, $\tilde{\Omega} \equiv \sqrt{\tilde{\Gamma}^2-\Omega^2}/2$. Taking the strong coupling limit $\Gamma \gg \Omega$ ($\delta$ is arbitrary), at times $t^* \gtrsim 1/\Gamma$ when some contributions to Eq.(\[complex-sol\]) are exponentially suppressed \[similar to Eq.(\[pert-omeg\])\], we reproduce the evolution for $\langle\rho_{23}(t)\rangle$, Eq.(\[r23-ens-large-t\]), with an energy splitting: $$\varepsilon^{\mbox{\small eff}}_{23}(\delta,\Omega,\Gamma)
= \frac{\Omega^2 \delta}{4 (\Gamma^2 + \delta^2)}
= \Gamma_{sw} \frac{\delta}{2 \Gamma} ;
\label{eff-split}$$ it approaches $\varepsilon^{pert}_{23} \simeq \frac{\Omega^2}{4\delta}$ for large $\delta$. Eq.(\[eff-split\]) is confirmed with a very good accuracy by the results obtained through direct simulation of the stochastic Bayesian evolution equations (\[Bayes1\]). Eqs.(\[Bayes-upd1\]),(\[Bayes-upd2\]),(\[eff-split\]) suggest that the spin-charge states under continuous strong measurement can be re-interpreted as the “new” energy states, if the quantum evolution is monitored at times $t^* \gtrsim 1/\Gamma$.
Zeno stabilization at small coupling and large detuning
-------------------------------------------------------
Interestingly, Zeno stabilization can take place even at small coupling, if the detuning is sufficiently large, $\Gamma \ll \Omega \ll \delta$. Qualitatively, this can be understood by kinematical reasons as illustrated on the spin-singlet Bloch sphere, Fig.\[kinematic-Zeno\]. In the stochastic Bayesian equations (\[Bayes1\]), for large detuning $\delta \gg \Omega$, the Hamiltonian evolution is a fast rotation with $\Omega_{sys}=\sqrt{\delta^2 + \Omega^2}$ around an axis close to the $z$-axis. On the other hand, the measurement alone tries to localize the state to either of the poles (the singlet states) acting along the meridians. The angular velocity along the meridians is $\Omega \ll \Omega_{sys}$; also the oscillation amplitude is $\sim \frac{\Omega}{\delta}\ll 1$, so that the Hamiltonian evolution is effectively suppressed due to averaging. Thus, even though the coupling is small, $\Gamma \ll \Omega$, it might be large with respect to an effective evolution rate along the meridians which quantifies a quantum Zeno effect. Numerical simulation of noisy measurement evolution shows that the DQD two-electron system either collapses to the spin-singlet subspace (similar to that shown in Fig.\[strong-Zeno\]), or ends up at the triplet state. In the first scenario, the system is stabilized for a relatively long time $\Gamma_{sw}^{-1} \gg \Gamma^{-1}$ in one of the singlet states while performing rare switchings between them.
![(Color online) Bloch sphere in the spin-singlet subspace illustrates the Zeno stabilization in case of weak coupling and large detuning. The projection of the Hamilton evolution (rotation around an axis close to $z$-axis) along the meridians (measurement evolution) is effectively small. []{data-label="kinematic-Zeno"}](fig5.eps){width="1.8in"}
The switching rate $\Gamma_{sw}$ is given by the same formula, Eq.(\[SW-rate\]), since the $\Omega$-terms in the master equation are still small perturbations with respect to the remaining terms as in Eq.(\[pert-omeg\]). Solution of the master equation (\[Master\]) confirms that at a time scale $t^* \geq 1/\Gamma$ the decay from the subspaces is exponential as was in the case of a strong coupling, with the same switching rate. Correspondingly, numerical simulation of noisy measurement evolution of the DQD two-electron system, as shown on Fig.\[large-detun\]a,b, is in a good agreement with the negative-result evolution described by Eqs.(\[Bayes-upd1\]),(\[Bayes-upd2\]). Generally, the agreement is established already at $\delta \gtrsim 3 \Omega$. Here, the initial qubit state was chosen to be a pure state and it remains pure as well as the purity of the total qutrit state (not shown). The negative-result evolution in the spin qubit subspace takes place as long as the system did not switch to the third state. In Fig.\[large-detun\] a realization of the measurement process is shown when the system did not switch at all. Indeed, in a situation when only a $\sigma_z$-evolution is present in addition to the negative-result evolution, Eqs.(\[Bayes-upd1\]),(\[Bayes-upd2\]), the probability not to decay is given by $P_{tot}(t)=\rho_{33}(0)+\rho_{22}(0) e^{-\Gamma_{sw} t}$. So, it approaches $\rho_{33}(0)$, remaining finite for large times.
![(Color online) Evolution in the regime of small coupling and large detuning. (a) A realization when the qubit did not switch: $\rho_{11}(t)$ remains zero always. Evolution of $\rho_{22}-\rho_{33}$ shows collapse to the state $|3\rangle\equiv|T_0(1,1)\rangle$. (b) Oscillation of coherences, $\mbox{Re}\rho_{23}$ and $\mbox{Im}\rho_{23}$ (thin dashed and solid lines), and their envelop $|\rho_{23}|$ (thick solid line) vs. negative-result evolution of $|\rho_{23}|$ (dashed line) according to Eqs.(\[SW-rate\])-(\[Bayes-upd2\]). []{data-label="large-detun"}](fig4ab.eps){width="3.2in"}
The induced energy splitting leads to a $\sigma_z$ evolution for $\mbox{Re}\rho_{23}$, $\mbox{Im}\rho_{23}$, whose amplitude envelope is provided by the negative-result evolution, Fig.\[large-detun\]b. For large detuning and small coupling the energy splitting is mainly due to perturbative influence of the third state: $\varepsilon^{\mbox{\small eff}}_{23} \simeq \varepsilon^{pert}_{23} = \Omega^2/4\delta$ as it follows from Eq.(\[eff-split\]).
Available experimental parameters
=================================
[*1. Coupling strength and switching rate.*]{} The possibility to detect single electronic charges in quantum dots (QD) via quantum point contact or single electron transistor was recently demonstrated in experiments of different groups. [@Marcus-Sci; @Kouw-prl94; @Rimberg-nat] The detector coupling, $\Gamma = (\Delta I)^2/4 S_0\eta$, essentially determines the measurement time $\tau_{meas} \sim 1/\Gamma$ to reach a signal-to-noise ratio of order one, where $\Delta I$ is the current difference signal in the detector due to the presence (or absence) of an extra electron charge $e$ in the QD. For the charge sensitivity of an rf-SET $\delta q = 2.4\times 10^{-5} e/\sqrt{Hz}$, provided in Ref., one estimates $\Gamma^{SET}_{exp} \approx 10^7\, s^{-1}$. Recent experiment of Rimberg group[@Rimberg-new2] demonstrated a charge sensitivity of an rf-SET of $\delta q = 2.4\times 10^{-6} e/\sqrt{Hz}$. Due to quadratic dependence on sensitivity, $\Gamma \propto (\delta q)^{-2}$, this amounts to a two orders of magnitude improvement,[@improvement] $\Gamma^{SET}_{exp}\simeq 10^{9}\ s^{-1}$. For an rf-QPC,[@rf-QPC] we estimated $\Gamma^{QPC}_{exp} \simeq 10^{6}\ s^{-1}$. The high detector ideality of a QPC also may be lost in the rf-regime.[@Gambetta]
The above estimates show that, in principle, both a weak coupling as well as a strong coupling regime of a negative-result evolution are experimentally reachable. For a typical tunneling,[@Taylor] $t_c \approx 10\ \mu eV$ the characteristic frequency $\Omega = 2\, t_c/\hbar \approx 3\times 10^{10}\ s^{-1}$ is large. Given the range of DQD detuning, $|\delta| \approx 0 - 300\ \mu eV$, one can reach a quantum Zeno stabilization if a relatively large detuning is taken. Smaller values of tunneling, $t_c \approx 1\ \mu eV$ or even $t_c \approx 0.1\ \mu eV$, are also reachable[@Kouwenhoven-Sci] so that a strong coupling regime may be realized. Typical values of switching rate $\Gamma_{sw}$ for the presented parameters are in the range $2\times 10^{5} - 5\times 10^{7} s^{-1}$.
[*2. Decoherence due to charge fluctuations.*]{} The decoherence in the DQD two-electron system has various sources. The coupling to uncontrollable detector degrees of freedom leads to detector non-ideality, $\eta$, partially discussed in Section II. Another mechanism of decoherence is due to systems’s coupling to background charge fluctuations[@CoishLoss; @HuDasSarma; @Taylor; @RomitoGefen] that will lead to a non-zero dephasing $\gamma_{23}$ in the spin-qubit subspace. It was argued[@CoishLoss; @HuDasSarma; @Taylor] that such dephasing will be well suppressed in the far-detuned regime $\delta \gg t_c$, where it may be $\gamma_{23}\approx 10^3 - 10^5\ s^{-1}$. However, close to the charge degeneracy, $\delta \lesssim t_c$, the dephasing may strongly increase[@HuDasSarma; @RomitoGefen] to $\gamma_{23}\approx 10^6 - 10^7\ s^{-1}$ due to higher sensitivity to fluctuations of the DQD parameters. In the latter case it will be difficult to see the effect of Zeno stabilization since $\gamma_{23}$ becomes comparable to the switching rate $\Gamma_{sw}$.
[*3. Decoherence due to phonons.*]{} Yet another decoherence mechanism (also related to charge degrees of freedom) is due to coupling to a phonon environment.[@LeggettRevModPhys; @Brandes] Physically, the relevant process is the double-dot inelastic tunneling[@Marcus-Sci; @Kouwenhoven-Sci] from state $|2\rangle = |S(1,1)\rangle$ to state $|1\rangle = |S(0,2)\rangle$ quantified by the inelastic rate $\Gamma_{in}(\delta)$. Relaxation process associated with the inelastic tunneling may lead to a contribution to the switching rate $\Gamma_{sw}$, assuming a weak environment, $\Gamma_{in}\ll \Omega,\Gamma$. Associated contribution to the dephasing $\gamma_{12}$ (also of the order of $\Gamma_{in}$) is expected not to destroy the negative-result evolution in this case.
Estimations of the inelastic rate[@Taylor] give the range of $0.01 - 100 neV$ (corresponding to approximately $10^{4} - 10^{8}\ s^{-1}$) depending on the energy splitting $\Delta E$ between the relevant states. Due to generic cubic dependence on the splitting, $\Gamma_{in} \propto (\Delta E)^3$, one can hope to find a range of not too large detuning $\delta$ so that $\Gamma_{in}$ is in the range of $10^{4} - 10^{5}\ s^{-1}$. Thus, eventually one may expect to manage the inequality $\Gamma_{in} \ll \Gamma_{sw}$, so that the Zeno stabilization will effectively “fight” against decoherence. We note that various models of boson environment may affect the predicted negative-result evolution in different ways that deserve a separate investigation.
[*4. Implications of the QD nuclei.*]{} The contact hyperfine interaction of the electron spins with the surrounding nuclei spins in the DQD leads to entanglement with the uncontrolable spin-bath degrees of freedom;[@Dobrovitski; @Dobrovitski1; @Zurek; @Glazman] in quasi-classical language it can be described as an effect of “inhomogeneous broadening” quantified by the random nuclear Overhauser field. [@nuclear-quasistatic; @Taylor; @Dobr1; @Dobrovitski-rev] The dephasing caused by such effects in the spin qubit have been experimentally measured[@Marcus-Sci; @HansonReview] (dephasing time is of the order of $\sim 10^{-8}\ s$); such strong dephasing may conceal any interesting quantum evolution. For a non-interacting bath it was shown that this type of decoherence can be completely eliminated by using various spin-echo techniques.[@Witzel] Realistically, for a weakly interacting bath, application of such techniques would reduce the decoherence by a few orders of magnitude.[@Witzel; @Zhang] For a GaAs DQD spin qubit system a “true” decoherence time of the order of $\sim 10^{-6}\ s$ was measured experimentally using simple spin-echo.[@Marcus-Sci; @HansonReview]
This suggests that spin-echo techniques could be of use also to reveal the negative-result evolution we are discussing in this paper. We leave for a future project the investigation of various possibilities for application of spin-echo techniques in conjunction with a negative-result measurement evolution in the context of manipulation and/or preparation of the state of a DQD spin qubit.
Conclusion
==========
We have shown that a negative-result evolution of a spin qubit can effectively emerge out of noisy measurement of a DQD spin system by a linear detector such as a quantum point contact. The evolution emerges as an interplay of measurement (non-unitary) dynamics and Hamiltonian dynamics of the three-level system, when quantum Zeno stabilization of the spin qubit subspace takes place.
Besides implications to the theory of quantum measurements, our results may be of practical use for manipulation of a DQD spin qubit (for papers discussing quantum measurements as an important resource see, e.g., Refs.). Recent experiments on a single phase qubit provide an interesting manipulation of its state via negative-result measurement.[@Katz-Sci; @Katz-Nature] These advances support the hope that experimental implementation of negative-result evolution is also possible in a DQD spin qubit system.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
The authors would like to thank A.N. Korotkov, L.P. Pryadko, and A.J. Rimberg for fruitful discussions and remarks. This work at Ames Laboratory was supported by the Department of Energy — Basic Energy Sciences under contract No. DE-AC02-07CH11358.
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|
---
abstract: |
We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When $B$ is non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When $B$ is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong long-range effect.
In the non-gauge invariant case, we exhibit a strong Aharonov-Bohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limiting absorption principle.
We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential.
author:
- Sylvain Golénia and Sergiu Moroianu
title: Spectral analysis of magnetic Laplacians on conformally cusp manifolds
---
[2000 Mathematics Subject Classification: 35P20, 46N50, 47A10, 47A40, 81Q10.]{}
Keywords: Magnetic fields, long range effect, cusp pseudodifferential operators, pure point spectrum, Mourre estimate, limiting absorption principle.
Introduction
============
Let $X$ be a smooth manifold of dimension $n$, diffeomorphic outside a compact set to a cylinder $(1,\infty)\times M$, where $M$ is a possibly disconnected closed manifold. On $X$ we consider asymptotically conformally cylindrical metrics, i.e., perturbations of the metric given near the border $\{\infty\}\times M$ by: $$\begin{aligned}
\label{mc}
g_p=y^{-2p}(dy^2+h),&&y\to\infty\end{aligned}$$ where $h$ is a metric on $M$ and $p>0$. If $p=1$ and $h$ is flat, the ends are *cusps*, i.e., complete hyperbolic of finite volume. For $p>1$ one gets the (incomplete) metric horns.
The refined properties of the essential spectrum of the Laplace-Beltrami operator $\Delta_p:=d^*d$ have been studied by Froese and Hislop [@FH] in the complete case. For the unperturbed metric , they get $$\begin{aligned}
\label{e:Ise}
\sigma_{\rm ess}(\Delta_p)=[\kappa(p), \infty), \mbox{ where }
\left\{\begin{array}{ll}
\kappa(p)=0, &\mbox{ for } p \pp 1 \\
\kappa(1)=\left(\frac{n-1}{2}\right)^2. &
\end{array}\right.\end{aligned}$$ The singular continuous part of the spectrum is empty and the eigenvalues distinct from $\kappa(p)$ are of finite multiplicity and may accumulate only at $\kappa(p)$. Froese and Hislop actually show a limiting absorption principle, a stronger result, see also [@DHS; @FHP; @FHP1; @H] for the continuation of their ideas. Their approach relies on a positive commutator technique introduced by E. Mourre in [@mou], see also [@ABG] and references therein. See for instance [@Guillope; @Kumura] for different methods.
Consider more generally a conformal perturbation of the metric . Let $\rho\in{\mathcal{C}}^\infty(X,{\mathbb{R}})$ be such that $\inf_{y\in X} (\rho(y))\pg -1$. Consider the same problem as above for the metric $$\label{e:Ipertu}
\tilde g_p=(1+\rho) g_p, \mbox{ for large } y.$$ To measure the size of the perturbation, we compare it to the lengths of geodesics. Let $L\in{\mathcal{C}}^\infty(X)$ be defined by $$\begin{aligned}
\label{e:IL}
L\geq 1,\quad L(y)=\left\{\begin{array}{ll}
\frac{y^{1-p}}{1-p}&\mbox{ for } p \pp 1 \\
\ln(y) &\mbox{ for } p = 1
\end{array}\right., \mbox{ for } y \mbox { big enough}.\end{aligned}$$
In [@FH], one essentially asks that $$L^2\rho, L^2d\rho \mbox{ and } L^2\Delta_g\rho \mbox{ are in } L^\infty(X).$$ to obtain the absence of singular continuous spectrum and local finiteness of the point spectrum. On one hand, one knows from the perturbation of a Laplacian by a short-range potential $V$ that only the speed of the decay of $V$ is important to conserve these properties. On the other hand, in [@GG] and in a general setting, one shows that only the fact that $\rho$ tends to $0$ is enough to ensure the stability of the essential spectrum. Therefore, it is natural to ask whether the decay of the metric (without decay conditions on the derivatives) is enough to ensure the conservation of these properties. In this paper, we consider that $\rho=\rho_{\rm sr }+\rho_{\rm lr }$ decomposes in short-range and long-range components. We ask the long-range component to be radial. We also assume that there exists $\varepsilon\pg 0$ such that $$\label{e:Iconf}\begin{split}
L^{1+\varepsilon}\rho_{\rm sr } \text{ and }
d\rho_{\rm sr }, \Delta_g\rho_{\rm sr } \in L^\infty(X),\\
L^{\varepsilon}\rho_{\rm lr }\text{, }
L^{1+\varepsilon}d\rho_{\rm lr } \text{ and } \Delta_g \rho_{\rm sr }
\in L^\infty(X).
\end{split}$$ Going from $2$ to $1+\varepsilon$ is not a significant improvement as it relies on the use of an optimal version of the Mourre theory instead of the original theory, see [@ABG] and references therein. Nevertheless, the fact that the derivatives are asked only to be bounded and no longer to decay is a real improvement due to our method. We prove this result in Theorem \[t:mourre0\]. In the Mourre theory, one introduces a conjugate operator to study a given operator. The conjugate operator introduced in [@FH] is too rough to handle very singular perturbations. In our paper, we introduce a conjugate operator local in energy to avoid the problem. We believe that our approach could be implemented easily in the manifold settings from [@bouclet; @DHS; @FH; @FHP; @FHP1; @H; @KT] to improve results on perturbations of the metric.
A well-known dynamical consequence of the absence of singular continuous spectrum and of the local finiteness of the point spectrum is that for an interval ${\mathcal{J}}$ that contains no eigenvalue of the Laplacian, for all $\cchi\in{\mathcal{C}}^\infty_c(X)$ and $\phi\in L^2(X)$, the norm $\|\cchi e^{it\Delta_p} E_{\mathcal{J}}(\Delta_p) \phi\|$ tends to $0$ as $t$ tends to $\pm \infty$. In other words, if you let evolve long enough a particle which is located at scattering energy, it eventually becomes located very far on the exits of the manifold. Add now a magnetic field $B$ with compact support and look how strongly it can interact with the particle. Classically there is no interaction as $B$ and the particle are located far from each other. One looks for a quantum effect.
The Euclidean intuition tells us that is no essential difference between the free Laplacian and the magnetic Laplacian $\Delta_A$, where $A$ is a magnetic potential arising from a magnetic field $B$ with compact support. They still share the spectral properties of absence of singular continuous spectrum and local finiteness of the point spectrum, although a long-range effect does occur and destroys the asymptotic completeness of the couple $(\Delta, \Delta_A)$; one needs to modify the wave operators to compare the two operators, see [@LT]. However, we point out in this paper that the situation is dramatically different in particular on hyperbolic manifolds of finite volume, even if the magnetic field is very small in size and with compact support. We now go into definitions and describe our results.
A magnetic field $B$ is a smooth real exact $2$-form on $X$. There exists a real $1$-form $A$, called vector potential, satisfying $dA=B$. Set ${d_A}:=d+iA\wedge: {{\mathcal{C}}^{\infty}_c}(X)\to{{\mathcal{C}}^{\infty}_c}(X,T^*X)$. The magnetic Laplacian on ${{\mathcal{C}}^{\infty}_c}(X)$ is given by $\Delta_A:={d_A}^*{d_A}$. When the manifold is complete, $\Delta_A$ is known to be essentially self-adjoint, see [@shubin]. Given two vector potentials $A$ and $A'$ such that $A-A'$ is exact, the two magnetic Laplacians $\Delta_A$ and $\Delta_{A'}$ are unitarily equivalent, by gauge invariance. Hence when $H^1_{{\rm dR}}(X)=0$, the spectral properties of the magnetic Laplacian do not depend on the choice of the vector potential, so we may write $\Delta_B$ instead of $\Delta_A$.
The aim of this paper is the study of the spectrum of magnetic Laplacians on a manifold $X$ with the metric , which includes the particular case . In this introduction we restrict the discussion to the complete case, i.e. $p\leq 1$. We focus first on the case of gauge invariance, i.e., $H^1_{{\rm dR}}(X)=0$, and we simplify the presentation assuming that the boundary is connected. We classify magnetic fields.
\[d:intro\] Let $X$ be the interior of a compact manifold with boundary ${\overline{X}}$. Suppose that $H^1_{{\rm dR}}(X)=0$ and that $M=\partial {\overline{X}}$ is connected. Let $B$ be a magnetic field on $X$ which extends smoothly to a $2$-form on ${\overline{X}}$. We say that $B$ is *trapping* if
- either $B$ does not vanish identically on $M$, or
- $B$ vanishes on $M$ but defines a non-integral cohomology class $[2\pi B]$ inside the relative cohomology group $H^2_{{\rm dR}}(X,M)$.
Otherwise, we say that $B$ is *non-trapping*. This terminology is motivated by the spectral consequences a) and c) of Theorem \[t:Ithmag\]. The definition can be generalized to the case where $M$ is disconnected (Definition \[magtrap\]). The condition of $B$ being trapping can be expressed in terms of any vector potential $A$ (see Definition \[def7\] and Lemma \[l:compsupp\]). When $H^1_{{\rm dR}}(X)\neq 0$, the trapping condition makes sense only for vector potentials, see Section \[s:mfc\] and Theorem \[t:IAB\].
Let us fix some notation. Given two Hilbert spaces ${\rond{H}}$ and ${\rond{K}}$, we denote by ${\mathcal{B}}({\rond{H}}, {\rond{K}})$ and ${\mathcal{K}}({\rond{H}}, {\rond{K}})$ the bounded and compact operators acting from ${\rond{H}}$ to ${\rond{K}}$, respectively. Given $s\geq 0$, let ${\rond{L}}_{s}$ be the domain of $L^s$ equipped with the graph norm. We set ${\rond{L}}_{-s}:={\rond{L}}_{s}^*$ where the adjoint space is defined so that ${\rond{L}}_{s}\subset L^2(X, g_p) \subset {\rond{L}}_{s}^*$, using the Riesz lemma. Given a subset $I$ of ${\mathbb{R}}$, let $I_{\pm}$ be the set of complex numbers $x\pm iy$, where $x\in I$ and $y>0$. For simplicity, in this introduction we state our result only for the unperturbed metric .
\[t:Ithmag\] Let $1\geq p>0$, $g_p$ the metric given by . Suppose that $H_1(X,{\mathbb{Z}})=0$ and that $M$ is connected. Let $B$ be a magnetic field which extends smoothly to ${\overline{X}}$. If $B$ is trapping then:
- The spectrum of $\Delta_B$ is purely discrete.
- The asymptotic of its eigenvalues is given by $$\label{e:Ithmag}
N_{B,p}(\lambda) \approx \begin{cases}
C_1\lambda^{n/2}&
\text{for $1/n< p$,}\\
C_2\lambda^{n/2}\log \lambda &\text{for $p=1/n$,}\\
C_3\lambda^{1/2p}&\text{for $0<p<1/n$}
\end{cases}$$ in the limit $\lambda\to\infty$, where $C_3$ is given in Theorem \[t:thmag\], and $$\label{e:Ic}
C_1=\frac{{\operatorname{Vol}}(X,g_p){\operatorname{Vol}}(S^{n-1})}{n(2\pi)^n},\quad
C_2=\frac{{\operatorname{Vol}}(M,h){\operatorname{Vol}}(S^{n-1})}{2(2\pi)^n}.$$
If $B$ is non-trapping with compact support in $X$ then
- The essential spectrum of $\Delta_B$ is $[\kappa(p), \infty)$.
- The singular continuous spectrum of $\Delta_B$ is empty.
- The eigenvalues of $\Delta_B$ are of finite multiplicity and can accumulate only in $\{\kappa(p)\}$.
- Let ${\mathcal{J}}$ a compact interval such that ${\mathcal{J}}\cap \big(\{\kappa(p)\}\cup
\sigma_{\rm pp}(H)\big)=\emptyset$. Then, for all $s\in ]1/2, 3/2[$ and all $A$ such that $dA=B$, there is $c$ such that $$\|(\Delta_A-z_1)^{-1} - (\Delta_A-z_2)^{-1} \|_{{\mathcal{B}}({\rond{L}}_s, {\rond{L}}_{-s})}
\leq c \|z_1-z_2\|^{s-1/2},$$ for all $z_1, z_2 \in {\mathcal{J}}_{\pm}$.
The statements a) and b) follow from general results from [@wlom]. This part relies on the Melrose calculus of cusp pseudodifferential operators (see e.g., [@meni96c]) and is proved in Theorem \[t:thmag\] for the perturbed metric . We start from the basic observation that for smooth vector potentials, the magnetic Laplacian belongs to the cusp calculus with positive weights. For this part, we can treat the metric which is quasi-isometric (but not necessarily asymptotically equivalent) to . Moreover, the finite multiplicity of the point spectrum (which is possibly not locally finite) in e) follows from Appendix \[fm\] for this class of metrics.
The statement $c)$ follows directly from the analysis of the free case in Section \[s:free\]. The perturbation of the metric is considered in Proposition \[p:thema\] and relies on general results on stability of the essential spectrum shown in [@GG]. The points d), f) and e) rely on the use of an optimal version of Mourre theory, see [@ABG]. They are developed in Theorem \[t:mourre0\] for perturbations satisfying . Scattering theory under short-range perturbation of a potential and of a magnetic field is also considered.
The condition of being trapping (resp. non-trapping) is discussed in Section \[s:inv\] and is equivalent to having empty (resp. non-empty) essential spectrum in the complete case. The terminology arises from the dynamical consequences of this theorem and should not be confused with the classical terminology. Indeed, when $B$ is trapping, the spectrum of $\Delta_B$ is purely discrete and for all non-zero $\phi$ in $L^2(X)$, there is $\cchi\in{\mathcal{C}}^\infty_c(X)$ such that $1/T \int_{0}^T \|\cchi e^{it \Delta_B} \phi\|^2 dt$ tends to a non-zero constant as $T$ tends to $\pm \infty$. On the other hand, taking ${\mathcal{J}}$ as in f), for all $\cchi\in{\mathcal{C}}^\infty_c(X)$ one gets that $\cchi e^{it \Delta_B}E_{\mathcal{J}}(\Delta_A) \phi$ tends to zero, when $B$ is non-trapping and with compact support.
If $H^1_{{\rm dR}}(M)\neq 0$ (take $M=S^1$ for instance), there exist some trapping magnetic fields with *compact support*. We construct an explicit example in Proposition \[p:contreex\]. We are able to construct some examples in dimension $2$ and higher than $4$ but there are topological obstructions in dimension $3$, see section \[s:inv\]. As pointed out above regarding the Euclidean case, the fact that a magnetic field with compact support can turn off the essential spectrum and even a situation of limiting absorption principle is somehow unexpected and should be understood as a strong long-range effect.
We discuss other interesting phenomena in Section \[s:coupling\]. Consider $M=S^1$ and take a trapping magnetic field $B$ with compact support and a coupling constant $g\in{\mathbb{R}}$. Now remark that $\Delta_{g B}$ is non-trapping if and only if $g$ belongs to the discrete group $c_B{\mathbb{Z}}$, for a certain $c_B\neq 0$. When $g\notin c_B{\mathbb{Z}}$ and $p\geq 1/n$, the spectrum of $\Delta_{g B}$ is discrete and the eigenvalue asymptotics do not depend either on $B$ or on $g$. It would be very interesting to know whether the asymptotics of embedded eigenvalues, or more likely of resonances, remain the same when $g\in c_B{\mathbb{Z}}$, (see [@tanya] for the case $g=0$). It would be also interesting to study the inverse spectral problem and ask if the magnetic field could be recovered from the knowledge of the whole spectrum, since the first term in the asymptotics of eigenvalues does not feel it.
Assume now that gauge invariance does not hold, i.e., $H^1_{{\rm dR}}(X)\neq 0$. In quantum mechanics, it is known that the choice of a vector potential has a physical meaning. This is known as the Aharonov-Bohm effect [@AB]. Two choices of magnetic potential may lead to in-equivalent magnetic Laplacians. In ${\mathbb{R}}^2$ with a bounded obstacle, this phenomenon can be seen through a difference of wave phase arising from two non-homotopic paths that circumvent the obstacle. Some long-range effect appears, for instance in the scattering matrix like in [@R; @RY; @RY2], in an inverse-scattering problem [@N; @W] or in the semi-classical regime [@BR]. See also [@helffer] for the influence of the obstacle on the bottom of the spectrum. In all of these cases, the essential spectrum remains the same.
In section \[s:AB\], we discuss the Aharonov-Bohm effect in our setting. In light of Theorem \[t:Ithmag\], one expects a drastic effect. We show that the choice of a vector potential can indeed have a significant spectral consequence. For one choice of vector potential, the essential spectrum could be empty and for another choice it could be a half-line. This phenomenon is generic for hyperbolic surfaces of finite volume, and also appears for hyperbolic $3$-manifolds. We focus the presentation on magnetic fields $B$ with compact support. We say that a smooth vector potential $A$ (i.e., a smooth $1$-form on ${\overline{X}}$) is trapping if $\Delta_{A}$ has compact resolvent, and non-trapping otherwise. By Theorems \[t:thmag\] and \[t:mourre0\], for $p\leq 1$, this is equivalent to Definition \[def7\]. It also follows that when the metric is of type , $A$ is trapping if and only if a)–b) of Theorem \[t:Ithmag\] hold for $\Delta_{A}$, while $A$ is non-trapping if and only if $\Delta_{A}$ satisfies c)–f) of Theorem \[t:Ithmag\].
\[t:IAB\] Let $X$ be a complete oriented hyperbolic surface of finite volume and $B$ a smooth magnetic field on the compactification ${\overline{X}}$.
- If $X$ has at least $2$ cusps, then for all $B$ there exists both trapping and non-trapping vector potentials $A$ such that $B=dA$.
- If $X$ has precisely $1$ cusp, choose $B=dA=dA'$ where $A,A'$ are smooth vector potentials for $B$ on ${\overline{X}}$. Then $$\begin{aligned}
\text{$A$ is trapping $\Longleftrightarrow A'$
is trapping $\Longleftrightarrow \int_X B\in 2\pi {\mathbb{Z}}$.}\end{aligned}$$
This follows from Corollary \[81\]. More general statements are valid also in dimension $3$, see Section \[hyp\].
This implies on one hand that for a choice of $A$, as one has the points c), d) and e), a particle located at a scattering energy escapes from any compact set; on the other hand taking a trapping choice, the particle will behave like an eigenfunction and will remain bounded. It is interesting that the dimension $3$ is exceptional in the Euclidean case [@Y] and that we are able to construct examples of such a behavior in any dimension.
In the first appendix, we discuss the key notion of $C^1$ regularity for the Mourre theory and make it suitable to the manifold context and for our choice of conjugate operator. As pointed in [@GG0], this is a key hypothesis in the Mourre theory in order to apply the Virial theorem and deduce the local finiteness of the embedded eigenvalues. In the second appendix, we recall that (cusp) elliptic, not necessarily fully elliptic, cusp operators have $L^2$ eigenvalues of finite multiplicity. Finally, in the third appendix, we give a criteria of stability of the essential spectrum, by cutting a part of the space, encompassing incomplete manifolds. Some of the results concerning the essential spectrum and the asymptotic of eigenvalues were already present in the unpublished preprint [@GMo].
Acknowledgements. {#acknowledgements. .unnumbered}
-----------------
We acknowledge useful discussions with Barbu Berceanu, Dan Burghelea, Jan Dereziński, Vladimir Georgescu, Bernard Helffer, Andreas Knauf, François Nicoleau, Marius Măntoiu and Radu Purice. We are also grateful to the referee for helpful remarks.
The authors were partially supported from the contract MERG 006375, funded by the European Commission. The second author was partially supported by the contracts 2-CEx06-11-18/2006 and CNCSIS-GR202/19.09.2006 (Romania).
Cusp geometry {#lcm}
=============
Definitions {#s:gene}
-----------
This section follows closely [@wlom], see also [@meni96c]. Let ${\overline{X}}$ be a smooth $n$-dimensional compact manifold with closed boundary $M$, and $x:{\overline{X}}\to[0,\infty)$ a boundary-defining function. A *cusp metric* on ${\overline{X}}$ is a complete Riemannian metric $g_0$ on $X:={\overline{X}}\setminus M$ which in local coordinates near the boundary takes the form $$\label{cume}
g_0=a_{00}(x,y)\frac{dx^2}{x^4}+\sum_{j=1}^{n-1} a_{0j}(x,y)
\frac{dx}{x^2}dy_j+\sum_{i,j=1}^{n-1}a_{ij}(x,y)dy_idy_j$$ such that the matrix $(a_{\alpha\beta})$ is smooth and non-degenerate down to $x=0$. For example, if $a_{00}=1$, $a_{0j}=0$ and $a_{ij}$ is independent of $x$, we get a product metric near $M$. If we set $y=1/x$, a cusp metric is nothing but a quasi-isometric deformation of a cylindrical metric, with an asymptotic expansion for the coefficients in powers of $y^{-1}$. We will focus on the *conformally cusp* metric $$\begin{aligned}
\label{pme}
g_p:=x^{2p}g_0,\end{aligned}$$ where $p>0$. Note that is a particular case of such metric.
Let ${\mathcal{I}}\subset{{\mathcal{C}}^{\infty}}({\overline{X}})$ be the principal ideal generated by the function $x$. Recall [@meni96c] that a *cusp vector field* is a smooth vector field $V$ on ${\overline{X}}$ such that $dx(V)\in{\mathcal{I}}^2$. The space of cusp vector fields forms a Lie subalgebra ${{}^c{\mathcal{V}}}$ of the Lie algebra ${\mathcal{V}}$ of smooth vector fields on ${\overline{X}}$. In fact, there exists a natural vector bundle ${{}^cT{\overline{X}}}$ over ${\overline{X}}$ whose space of smooth sections is ${{}^c{\mathcal{V}}}$, and a natural map ${{}^cT{\overline{X}}}\to TX$ which induces the inclusion ${{}^c{\mathcal{V}}}\hookrightarrow {\mathcal{V}}$. Let $E,F\to{\overline{X}}$ be smooth vector bundles. The space of cusp differential operators ${\mathrm{Diff}_c}({\overline{X}},E,F)$ is the space of those differential operators which in local trivializations can be written as composition of cusp vector fields and smooth bundle morphisms down to $x=0$.
The *normal operator* of $P\in{\mathrm{Diff}_c}({\overline{X}},E,F)$ is the family of operators defined by $${\mathbb{R}}\ni\xi\mapsto{\mathcal{N}}(P)(\xi):=
\left(e^{i\xi/x}Pe^{-i\xi/x}\right)_{|x=0}\in{\mathrm{Diff}}(M,E_{|M},F_{|M}).$$
\[ex1\] ${\mathcal{N}}(x^2{\partial_x})(\xi)=i\xi$.
Note that $\ker {\mathcal{N}}={\mathcal{I}}\cdot{\mathrm{Diff}_c}$, which we denote again by ${\mathcal{I}}$. The normal operator map is linear and multiplicative. It is also invariant under the conjugation by powers of $x$. Namely, if $P\in{\mathrm{Diff}_c}$ and $s\in{\mathbb{C}}$ then $x^s Px^{-s}\in{\mathrm{Diff}_c}$ and ${\mathcal{N}}(x^s Px^{-s})={\mathcal{N}}(P)$. Concerning taking the (formal) adjoint, one needs to specify the volume form on the boundary.
\[lema6\] Let $P\in{\mathrm{Diff}_c}({\overline{X}},E,F)$ be a cusp operator and $P^*$ its adjoint with respect to $g_0$. Then ${\mathcal{N}}(P^*)(\xi)$ is the adjoint of ${\mathcal{N}}(P)(\xi)$ with respect to the metric on $E_{|M},F_{|M}$ induced by restriction, for the volume form ${a_{0}}^{1/2} {\operatorname{vol}}_{h_0}$, where the metric $h_0$ on $M$ is defined from rewriting $g_0$ as in .
The principal symbol of a cusp operator on $X$ extends as a map on the cusp cotangent bundle down to $x=0$. This implies that a cusp operator of positive order cannot be elliptic at $x=0$ in the usual sense. A cusp operator is called cusp-elliptic if its principal symbol is invertible on ${{}^cT^*{\overline{X}}}\setminus\{0\}$ down to $x=0$.
A cusp operator is called *fully elliptic* if it is cusp-elliptic and if its normal operator is invertible for all values of $\xi\in{\mathbb{R}}$.
An operator $H\in x^{-l}{\mathrm{Diff}_c}^k({\overline{X}},E,F)$ is called a cusp differential operator of type $(k,l)$.
Fix a product decomposition of $X$ near $M$, compatible with the boundary-defining function $x$. This gives a splitting of the cusp cotangent bundle on ${\overline{X}}$ in a neighborhood of $M$: $$\label{decom}
{{}^cT^*{\overline{X}}}\simeq T^*M\oplus\langle x^{-2}dx\rangle.$$
\[prop3\] The de Rham differential $d:{{\mathcal{C}}^{\infty}}(X)\to {{\mathcal{C}}^{\infty}}(X,T^*X)$ restricts to a cusp differential operator $d:{{\mathcal{C}}^{\infty}}({\overline{X}})\to{{\mathcal{C}}^{\infty}}({\overline{X}},{{}^cT^*{\overline{X}}})$. Its normal operator in the decomposition is $${\mathcal{N}}(d)(\xi)=
\begin{bmatrix}d^M\\i\xi\end{bmatrix}$$ where $d^M$ is the partial de Rham differential in the $M$ factor of the product decomposition.
Let $\omega\in{{\mathcal{C}}^{\infty}}({\overline{X}})$ and decompose $d\omega$ according to : $$d\omega=d^M\omega + {\partial_x}(\omega) dx=d^M\omega+x^2{\partial_x}(\omega)\frac{dx}{x^2}.$$ Since $d^M$ commutes with $x$, it follows from the definition that ${\mathcal{N}}(d^M)=d^M$. The result follows using Example \[ex1\].
Relative de Rham cohomology {#rdR}
---------------------------
Recall [@BT] that the cohomology of ${\overline{X}}$ and the relative cohomology groups of $({\overline{X}},M)$ (with real coefficients) can be computed using smooth differential forms as follows let $\Lambda^*({\overline{X}})$ denote the space of forms smooth on ${\overline{X}}$ down to the boundary. Let $\Lambda^*({\overline{X}},M)$ denote the subspace of those forms whose pull-back to $M$ vanishes. These spaces form complexes for the de Rham differential (because $d$ commutes with pull-back to $M$) and their quotient is the de Rham complex of $M$: $$0\to \Lambda^*({\overline{X}},M)\hookrightarrow
\Lambda^*({\overline{X}})\to\Lambda^*(M)\to 0.$$ The induced long exact sequence in cohomology is just the long exact sequence of the pair $({\overline{X}},M)$.
Cusp de Rham cohomology
-----------------------
Notice that $d$ preserves the space of cusp differential forms. Indeed, since $d$ is a derivation and using Lemma \[prop3\], it suffices to check this property for a set of local generators of ${{\mathcal{C}}^{\infty}}({\overline{X}},{{}^cT^*{\overline{X}}})$. Choose local coordinates $(y_j)$ on $M$ and take as generators $x^{-2}dx$ and $dy_j$, which are closed.
Let ${{}^c\!H}^*(X)$ denote the cohomology of the complex of cusp differential forms $({{\mathcal{C}}^{\infty}}({\overline{X}},\Lambda^*({{}^cT{\overline{X}}})), d)$ with respect to the de Rham differential.
\[dec\] ${{}^c\!H}^k(X)=H^k(X)\oplus H^{k-1}(M)^2$.
The short exact sequence of de Rham complexes $$0\to \Lambda^*(T{\overline{X}})\hookrightarrow
\Lambda^*({{}^cT{\overline{X}}})\to \Lambda^{*-1}(TM)^2\to 0,$$ where the second map is given by $$\label{restfr}
\Lambda^k({{}^cT{\overline{X}}})\ni\omega\mapsto \left((x^2{\partial_x}\lrcorner \omega)_{x=0},
(\partial_x(x^2{\partial_x}\lrcorner \omega))_{x=0}\right),$$ gives rise to a long exact sequence in cohomology. Now the composition $$\Lambda^*(T{\overline{X}})\hookrightarrow \Lambda^*({{}^cT{\overline{X}}})\hookrightarrow \Lambda^*(TX)$$ is a quasi-isomorphism, since de Rham cohomology can be computed either with smooth forms on ${\overline{X}}$, or with smooth forms on $X$. Thus in cohomology the map induced from $\Lambda^*(T{\overline{X}})\hookrightarrow
\Lambda^*({{}^cT{\overline{X}}})$ is injective.
The magnetic Laplacian {#sectionmagn}
======================
The magnetic Laplacian on a Riemannian manifold
-----------------------------------------------
A *magnetic field* $B$ on the Riemannian manifold $(X,g)$ is an exact real-valued $2$-form. A *vector potential* $A$ associated to $B$ is a $1$-form such that $dA=B$. We form the *magnetic Laplacian* acting on ${{\mathcal{C}}^{\infty}}(X)$: $$\Delta_A:={d_A}^*{d_A}.$$ This formula makes sense for complex-valued $1$-forms $A$. Note that when $A$ is real, ${d_A}$ is a metric connection on the trivial bundle $\underline{{\mathbb{C}}}$ with the canonical metric, and $\Delta_A$ is the connection Laplacian.
If we alter $A$ by adding to it a real exact form, say $A'=A+df$, the resulting magnetic Laplacian satisfies $$\Delta_{A'}=e^{-if}\Delta_Ae^{if}$$ so it is unitarily equivalent to $\Delta_A$ in $L^2(X,g)$. Therefore if $H^1_{\mathrm{dR}}(X)=0$ (for instance if $\pi_1(X)$ is finite; see [@BT]) then $\Delta_A$ depends, up to unitary equivalence, only on the magnetic field $B$. This property is called *gauge invariance*. For a more refined analysis of gauge invariance, see [@gruber].
One usually encounters gauge invariance as a consequence of $1$-connectedness (i.e., $\pi_1=0$). But in dimensions at least $4$, every finitely presented group (in particular, every finite group) can be realized as $\pi_1$ of a compact manifold. Thus the hypothesis $\pi_1=0$ is unnecessarily restrictive, it is enough to assume that its abelianisation is finite.
While the properties of $\Delta_A$ in ${\mathbb{R}}^n$ with the flat metric are quite well understood, the (absence of) essential spectrum of magnetic Laplacians on other manifolds has not been much studied so far. One exception is the case of bounded geometry, studied in [@KS]. However our manifolds are not of bounded geometry because the injectivity radius tends to $0$ at infinity.
Magnetic fields and cohomology {#s:mfc}
------------------------------
Recall that $A$ is a (smooth) *cusp $1$-form* (not to be confused with the notion of cusp form from automorphic form theory) on ${\overline{X}}$ if $A\in{{\mathcal{C}}^{\infty}}(X,T^*X)$ is a real-valued $1$-form satisfying near $\partial X$ $$\label{aft}
A=\varphi(x)\frac{dx}{x^2}+\theta(x)$$ where $\varphi\in{{\mathcal{C}}^{\infty}}({\overline{X}})$ and $\theta\in{{\mathcal{C}}^{\infty}}([0,\varepsilon)\times M,
\Lambda^1(M))$, or equivalently $A$ is a smooth section in ${{}^cT^*{\overline{X}}}$ over ${\overline{X}}$.
\[p:cuspduB\] Let $B$ be a cusp $2$-form. Suppose that $B$ is exact on $X$, and its image by the map is exact on $M$. Then there exists a smooth cusp $1$-form $A$ on ${\overline{X}}$ such that $dA=B$.
Note that $B$ is exact as a form on $X$, so $dB=0$ on $X$. By continuity, $dB=0$ on ${\overline{X}}$ (in the sense of cusp forms) so $B$ defines a cusp cohomology $2$-class. By hypothesis, this class maps to $0$ by restriction to $X$. Now the pull-back of $B$ to the level surfaces $\{x=\varepsilon\}$ is closed; by continuity, the image of $B$ through the map is closed on $M$. Assuming that this image is exact, it follows from Proposition \[dec\] that $B$ is exact as a cusp form.
By Lemmata \[lema6\] and \[prop3\], $\Delta_A$ is a cusp differential operator of order $(2p,2)$.
\[def7\] Let $A$ be a (complex-valued) cusp vector potential. Given a connected component $M_0$ of $M$, we say $A$ is a *trapping* vector potential on $M_0$ if
- either the restriction $\varphi_0:=\varphi(0)$ is not constant on $M_0$,
- or $\theta_0:=\theta(0)$ is not closed on $M_0$,
- or the cohomology class $[{\theta_0}_{|M_0}]\in H^1_{\mathrm{dR}}(M_0)$ does not belong to the image of $$2\pi H^1(M_0;{\mathbb{Z}})\to H^1(M_0,{\mathbb{C}})\simeq H^1_{\mathrm{dR}}(M_0)\otimes{\mathbb{C}}.$$
and *non-trapping on $M_0$* otherwise.
We say that $A$ is *trapping* if it is trapping on each connected component of $M$. The vector potential is said to be *non-trapping* if it is non-trapping on at least one connected component of $M$. If $A$ is non-trapping on all connected component of $M$, we say that it is *maximal non-trapping*.
The trapping notion can be expressed solely in terms of the magnetic field $B
=dA$ when $H^1(X)=0$, see Lemma \[l:compsupp\].
We comment briefly the terminology. When $A$ is constant in $x$ near $M$, the multiplicity of the absolutely continuous part of the spectrum of $\Delta_A$ will be given by the number of connected component of $M$ on which $A$ is non-trapping. Hence, taking $A$ maximal non-trapping maximizes the multiplicity of this part of the spectrum.
We refer to [@BT] for an exposition of cohomology with integer coefficients. The trapping property is determined only by the asymptotic behavior of $A$. More precisely, if $A'$ is also of the form with $\varphi(0)=0$ and $\theta(0)=0$ then $A$ is a (non-)trapping vector potential if and only if $A+A'$ is.
The term “trapping" is motivated by dynamical consequences of Theorems \[p:pmag\] and \[t:mourre0\] and has nothing to do with the classical trapping condition. This terminology is also supported by the examples given in Section \[s:nonstab\].
For a trapping vector potential, $x^{2p}\Delta_A$ is a fully-elliptic cusp operator. In turn, this implies that $\Delta_A$ has empty essential spectrum so from a dynamical point of view, a particle can not diffuse, in other words it is trapped in the interior of $X$. Indeed, given a state $\phi\in L^2(X)$, there exists $\cchi$ (the characteristic function of a compact subset of $X$) such that $1/T\int_0^T \|\cchi e^{it\Delta_A} \phi\|^2 dt$ tends to a positive constant as $T$ goes to infinity.
On the other hand, if $A$ is a non-trapping vector potential, then $\Delta_A$ is not Fredholm between the appropriate cusp Sobolev spaces. If the metric is an exact cusp metric and complete, we show that $\Delta_A$ has nonempty essential spectrum also as an unbounded operator in $L^2$, given by $[\kappa(p), \infty)$ by Proposition \[p:thema\]. We go even further and under some condition of decay of $\varphi$ and $\theta$ at infinity, we show that there is no singular continuous spectrum for the magnetic Laplacian and that the eigenvalues of ${\mathbb{R}}\setminus \{\kappa(p)\}$ are of finite multiplicity and can accumulate only in $\{\kappa(p)\}$. Therefore given a state $\phi$ which is not an eigenvalue of $\Delta_A$, one obtains that for all $\cchi$, $\cchi e^{it\Delta_A}\phi$ tends to $0$ as $t\to\infty$.
When $M$ is connected, the class of non-trapping vector potentials is a group under addition but that of trapping vector potential is not. When $M$ is disconnected, none of these classes is closed under addition. Directly from the definition, we get however:
\[addtrap\] Let $A$ be a maximal non-trapping vector potential and let $A'$ be a $1$-form smooth up to the boundary. Then $A'$ is trapping if and only if $A+A'$ is.
Let $B$ be a smooth magnetic field on ${\overline{X}}$ (i.e., a $2$-form) whose pull-back to $M$ vanishes. Since $B$ is exact, it is also closed, thus it defines a relative de Rham class as in Subsection \[rdR\]. If this class vanishes, we claim that there exists a vector potential $A$ for $B$ which is maximal non-trapping. Indeed, let $A\in \Lambda^1({\overline{X}},M)$ be any (relative) primitive of $B$. Then $A$ is clearly a cusp form, the singular term $\phi(0)$ vanishes, and the pull-back of $A$ to each boundary component vanishes by definition, in particular it defines the null $1$-cohomology class. From Remark \[addtrap\] we get
\[cohcl\] Let $B$ be a cusp magnetic field. Let $B'$ be a smooth magnetic field on ${\overline{X}}$ which vanishes on the boundary and which defines the zero relative cohomology class in $H^2({\overline{X}},M)$. Then $B$ admits (non-)trapping vector potentials if and only if $B+B'$ does.
Note that when $H^1({\overline{X}})\neq 0$, a given magnetic field may admit both trapping and non-trapping vector potentials. See Theorem \[t:IAB\] and Sections \[s:AB\] and \[hyp\].
The trapping case
=================
The absence of essential spectrum
---------------------------------
In this section, given a smooth cusp $1$-form, we discuss the link between its behavior at infinity and its trapping properties.
\[p:pmag\] Let $p>0$, $g_p$ a metric on $X$ given by near $\partial X$ and $A$ a smooth cusp $1$-form given by . Then $\Delta_A$ is a weighted cusp differential operator of order $(2p,2)$. If $A$ is trapping then $\Delta_A$ is essentially self-adjoint on ${\mathcal{C}}_c^\infty(X)$, it has purely discrete spectrum and its domain is $x^{2p}H^2(X,g_p)$.
If $p\leq 1$ then $g_p$ is complete so $\Delta_A$ is essentially self-adjoint [@shubin]. This fact remains true for a trapping $A$ in the incomplete case, i.e. $p>1$.
Using Lemma \[prop3\], we get $${\mathcal{N}}({d_A})(\xi)=
\begin{bmatrix} d^M+i\theta_0\\
i(\xi+\varphi_0)\end{bmatrix}.$$ Suppose that $x^{2p}\Delta_A$ is not fully elliptic, so there exists $\xi\in{\mathbb{R}}$ and $0\neq u\in\ker({\mathcal{N}}(x^{2p}\Delta_A)(\xi))$. By elliptic regularity, $u$ is smooth. We replace $M$ by one of its connected components on which $u$ does not vanish identically, so we can suppose that $M$ is connected. Using Lemma \[lema6\], by integration by parts with respect to the volume form ${a_0}^{1/2}dh_0$ on $M$ and the metric $h_0$ on $\Lambda^1(M)$, we see that $u\in\ker({\mathcal{N}}(\Delta_A)(\xi))$ implies $u\in\ker({\mathcal{N}}({d_A})(\xi))$. Then $$\begin{aligned}
\label{xvp}
(\xi+\varphi_0)u= 0\mbox{ and }(d^M+i\theta_0)u= 0, \end{aligned}$$ so $u$ is a global parallel section in the trivial bundle ${\underline{{\mathbb{C}}}}$ over $M$, with respect to the connection $d^M+i\theta_0$. This implies $$\label{zudg}
0=(d^M)^2 u=d^M(-iu\theta_0)
=-i(d^M u)\wedge\theta_0-iud^M\theta_0=-iud^M\theta_0.$$ By uniqueness of solutions of ordinary differential equations, $u$ is never $0$, so $d^M\theta_0=0$. Furthermore, from , we see that $\varphi_0$ equals the constant function $-\xi$. It remains to prove the assertion about the cohomology class $[\theta_0]$.
Let ${\tilde{M}}$ be the universal cover of $M$. Denote by ${\tilde{u}}$, ${\tilde{\theta}_0}$ the lifts of $u,\theta_0$ to ${\tilde{M}}$. The equation $(d^M+i\theta_0)u=0$ lifts to $$\label{ecu}
(d^{{\tilde{M}}}+i{\tilde{\theta}_0}){\tilde{u}}=0.$$ The $1$-form ${\tilde{\theta}_0}$ is closed on the simply connected manifold ${\tilde{M}}$, hence it is exact (by the universal coefficients formula, $H^1({\tilde{M}},{\mathbb{C}})=H_1({\tilde{M}},{\mathbb{C}})=H_1({\tilde{M}};{\mathbb{Z}})\otimes{\mathbb{C}}$, and $H_1({\tilde{M}};{\mathbb{Z}})$ vanishes as it is the abelianisation of $\pi_1({\tilde{M}})$). Let $v\in{{\mathcal{C}}^{\infty}}({\tilde{M}})$ be a primitive of ${\tilde{\theta}_0}$, i.e., $d^{{\tilde{M}}}v={\tilde{\theta}_0}$. Then, from , ${\tilde{u}}=Ce^{-iv}$ for some constant $C\neq 0$.
The fundamental group $\pi_1(M)$ acts to the right on ${\tilde{M}}$ via deck transformations. The condition that ${\tilde{u}}$ be the lift of $u$ from $M$ is the invariance under the action of $\pi_1(M)$, in other words $${\tilde{u}}(y)={\tilde{u}}(y[\gamma])$$ for all closed loops $\gamma$ in $M$. This is obviously equivalent to $$\begin{aligned}
v(y[\gamma])-v(y)\in 2\pi{\mathbb{Z}},&&\forall y\in \tilde{M}.\end{aligned}$$ Let $\tilde{\gamma}$ be the lift of $\gamma$ starting in $y$. Then $$\begin{aligned}
v(y[\gamma])-v(y)=&\int_{\tilde{\gamma}}d^{{\tilde{M}}}v
=\int_{\tilde{\gamma}}{\tilde{\theta}_0}=\int_\gamma \theta_0.\end{aligned}$$ Thus the solution ${\tilde{u}}$ is $\pi_1(M)$-invariant if and only if the cocycle $\theta_0$ evaluates to an integer multiple of $2\pi$ on each closed loop $\gamma$. These loops span $H_1(M;{\mathbb{Z}})$, so $[\theta_0]$ lives in the image of $H^1(M;{\mathbb{Z}})$ inside $H^1(M,{\mathbb{C}})={\mathrm{Hom}}(H_1(M;{\mathbb{Z}}),{\mathbb{C}})$. Therefore the solution $u$ must be identically $0$ unless $\varphi_0$ is constant, $\theta_0$ is closed and $[\theta_0]\in 2\pi H^1(M;{\mathbb{Z}})$.
Conversely, if $\varphi_0$ is constant, $\theta_0$ is closed and $[\theta_0]\in 2\pi H^1(M;{\mathbb{Z}})$ then ${\tilde{u}}=e^{-iv}$ as above is $\pi_1(M)$-invariant, so it is the lift of some $u\in{{\mathcal{C}}^{\infty}}(M)$ which belongs to $\ker({\mathcal{N}}(x^{2p}\Delta_A)(\xi))$ for $-\xi$ equal to the constant value of $\varphi_0$.
The conclusion of the theorem is now a consequence of general properties of the cusp calculus [@wlom Theorem 17]. Namely, since $\Delta_A$ is fully elliptic, there exists an inverse in $x^{2p}\Psi_c^{-2}(X)$ (a micro-localized version of ${\mathrm{Diff}}_c(X)$) modulo compact operators. If $p>0$, this pseudo-inverse is itself compact. The operators in the cusp calculus act by closure on a scale of Sobolev spaces. It follows easily that for $p\geq 0$, a symmetric fully elliptic cusp operator in $x^{2p}\Psi_c^{2}(X)$ is essentially self-adjoint, with domain $x^{2p}H^2_c(X)$. Thus $\Delta_A$ is self adjoint with compact inverse modulo compact operators, which shows that the spectrum is purely discrete.
The cusp calculus [@meni96c] is a particular instance of Melrose’s program of micro-localizing boundary fibration structures. It is a special case of the fibered-cusp calculus [@mame99] and can be obtained using the groupoid techniques of [@lani].
Eigenvalue asymptotics for trapping magnetic Laplacians
-------------------------------------------------------
If $A$ is a trapping vector potential, the associated magnetic Laplacian has purely discrete spectrum. In this case we can give the first term in the eigenvalue growth law.
\[t:thmag\] Let $p>0$, $g_p$ a metric on $X$ given by near $\partial X$ and $A\in{{\mathcal{C}}^{\infty}}(X,T^*X)$ a complex-valued trapping vector potential in the sense of Definition \[def7\]. Then the eigenvalue counting function of $\Delta_A$ satisfies $$\label{e:thmag}
N_{A,p}(\lambda) \approx \begin{cases}
C_1\lambda^{n/2}&
\text{for $1/n< p<\infty$,}\\
C_2\lambda^{n/2}\log \lambda &\text{for $p=1/n$,}\\
C_3\lambda^{1/2p}&\text{for $0<p<1/n$}
\end{cases}$$ in the limit $\lambda\to\infty$, where $$\label{c12}\begin{split}
C_1=&\frac{{\operatorname{Vol}}(X,g_p){\operatorname{Vol}}(S^{n-1})}{n(2\pi)^n},\\
C_2=&\frac{{\operatorname{Vol}}(M,{a_0}^{1/2}h_0){\operatorname{Vol}}(S^{n-1})}{2(2\pi)^n}.
\end{split}$$ If moreover we assume that $g_0$ is an exact cusp metric, then $$C_3=\frac{\Gamma\left(\tfrac{1-p}{2p}\right) }
{2\sqrt{\pi}\Gamma\left(\tfrac{1}{2p}\right)}
\zeta\left(\Delta_A^{h_0}, \frac{1}{p}-1\right),$$ where $\Delta_A^{h_0}$ is the magnetic Laplacian on $M$ with potential $A_{|M}$ with respect to the metric $h_0$ on $M$ defined in Section \[lcm\].
We stress that the constants $C_1$ and $C_2$ do not depend on the choice of $A$ or $B$, but only on the metric. This fact provides some very interesting coupling constant effect, see Section \[s:coupling\].
Note also that the hypotheses of the theorem are independent of the choice of the vector potential $A$ inside the class of cusp $1$-forms. Indeed, assume that $A'=A+dw$ for some $w\in{{\mathcal{C}}^{\infty}}(X)$ is again a cusp $1$-form. Then $dw$ must be itself a cusp form, so $$dw=x^2{\partial_x}w \frac{dx}{x^2}+d^M w\in{{\mathcal{C}}^{\infty}}({\overline{X}},{{}^cT^*{\overline{X}}}).$$ Write this as $dw=\varphi'\frac{dx}{x^2}+\theta'_x$. For each $x>0$ the form $\theta'_x$ is exact. By the Hodge decomposition theorem, the space of exact forms on $M$ is closed, so the limit $\theta'_0$ is also exact. Now $dw$ is an exact cusp form, in particular it is closed. This implies that $x^2{\partial_x}\theta'=d^M \varphi'$. Setting $x=0$ we deduce $d^M(\varphi')_{x=0}=0$, or equivalently $\varphi'_{|x=0}$ is constant. Hence the conditions from Theorem \[t:thmag\] on the vector potential are satisfied simultaneously by $A$ and $A'$.
From Theorem \[p:pmag\], the operator $x^{2p}\Delta_A$ is fully elliptic when $A$ is trapping. The result follows directly from [@wlom Theorem 17]. Let us explain the idea: the complex powers $\Delta_A^{-s}$ belong to the cusp calculus, and are of trace class for sufficiently large real part of $s$. The trace of the complex powers is holomorphic for such $s$ and extends meromorphically to ${\mathbb{C}}$ with two families of simple poles, coming from the principal symbol and from the boundary. The leading pole governs eigenvalue asymptotics, by the Delange theorem. In case the leading pole is double (by the superposition of the two families of poles), we get the logarithmic growth law.
The explicit computation of the constants (given by [@wlom Prop. 14 and Lemma 16]) is straightforward.
Analysis of the free case for non-trapping potentials {#s:free}
=====================================================
We have shown in Theorems \[p:pmag\] and \[t:thmag\] that the essential spectrum of the magnetic Laplacian $\Delta_A$ is empty when $A$ is trapping (see Definition \[def7\]), and we have computed the asymptotics of the eigenvalues. We now consider the case of a non-trapping vector potential $A$. One can guess that in this last case, the essential spectrum is not empty when the metric is complete. In this section we concentrate on the unperturbed metric with the model non-trapping potential . We take advantage of the decomposition in low- and high-energy functions from Section \[s:decomp\]. The computation of the essential spectrum is based on Proposition \[p:free’\] and on the diagonalization of the magnetic Laplacian performed in Section \[s:diago\]. In Section \[s:conjop\], we construct a local conjugate operator and state the Mourre estimate (Theorem \[t:mourre\_0\]).
In Section \[s:srlr\] we introduce the classes of perturbation under which we later give a limiting absorption principle. The perturbation theory is developed in Section \[s:ntp\]. We refer to Proposition \[p:thema\] for the question of the essential spectrum and to Theorem \[t:mourre0\] for its refined analysis under short/long range perturbations.
We localize the computation on the end $X':=(0,\varepsilon) \times M\subset X$. We assume that on $X':=(0,\varepsilon)\times M\subset X$ we have $$\begin{aligned}
g_p=&x^{2p}\left(\frac{dx^2}{x^4}+h_0\right),\label{gp'}\\
A_{\rm f}=&Cdx/x^2+\theta_0,\label{acon}\end{aligned}$$ where $C$ is a constant, $\theta_0$ is closed and independent of $x$, and the cohomology class $[\theta_0]\in H^1(M)$ is an integer multiple of $2\pi$. By a change of gauge, one may assume that $C=0$. Indeed, it is enough to subtract from $A_{\rm f}$ the exact $1$-form $d(-C/x)$.
The high and low energy functions decomposition {#s:decomp}
-----------------------------------------------
Set ${{d_{{\tilde{\theta}}}}}:=d^M+i\theta_0\wedge$. We now decompose the $L^2$ space as follows: $$\label{e:decomp}
L^2(X')={\rond{H}}_{\rm l} \oplus {\rond{H}}_{\rm h},$$ where ${\rond{H}}_{\rm l}:={\rond{K}}\otimes \ker({{d_{{\tilde{\theta}}}}})$ with ${\rond{K}}:=L^2\left((0,\varepsilon), x^{np-2} dx\right)$, and where ${\rond{H}}_{\rm h}={\rond{K}}\hat{\otimes} \ker({{d_{{\tilde{\theta}}}}})^\perp$. We do not emphasize the dependence on $\varepsilon$ for these spaces as the properties we are studying are independent of $\varepsilon$. The subscripts ${\rm l},{\rm h}$ stand for low and high energy, respectively. The importance of the *low energy functions* space ${\rond{H}}_{\rm l}$ is underlined by the following:
\[p:free’\] The magnetic Laplacian $\Delta_{A_{\rm f}}$ stabilizes the decomposition of $L^2(X')$. Let $\Delta_{A_{\rm f}}^l$ and $\Delta_{A_{\rm f}}^h$ be the Friedrichs extensions of the restrictions of $\Delta_{A_{\rm f}}$ to smooth compactly supported functions in ${\rond{H}}_{\rm l}$, respectively in ${\rond{H}}_{\rm h}$. Then $\Delta_{A_{\rm f}}^h$ has compact resolvent, while $$\Delta^l_{A_{\rm f}}=\big(D^*D+c_0^2x^{2-2p}\big)\otimes 1,$$ where $c_0:=((2-n)p-1)/2$, and $D:=x^{2-p}{\partial_x}-c_0x^{1-p}$ acts in ${{\rond{K}}_{k,\varepsilon}}$.
By Proposition \[p:cut\], the essential spectrum is prescribed by the space of low energy functions. This kind of decomposition can be found in various places in the literature, see for instance [@lott] for applications to finite-volume negatively-curved manifolds.
\[r:free’\] The dimension of the kernel of ${{d_{{\tilde{\theta}}}}}$ equals the number of connected components of $M$ on which $A_f$ non-trapping. Indeed, take such a connected component $M_1$ and let $v$ be a primitive of $\theta_0$ on its universal cover. On different sheets of the cover, $v$ changes by $2\pi{\mathbb{Z}}$ so $e^{iv}$ is a well-defined function on $M_1$ which spans $\ker({{d_{{\tilde{\theta}}}}})$. This decomposition is also valid in the trapping case, only that the low energy functions space is then $0$. Using Proposition \[p:cut\], we obtain the emptiness of the essential spectrum for the unperturbed metric and along the way a special case of Theorem \[p:pmag\] (we do not recover the full result in this way, since the metric can *not* be reached perturbatively from the metric ).
We decompose the space of $1$-forms as the direct sum . Recall that $\delta_M$ is the adjoint of $d^M$ with respect to $h_0$. We compute $$\label{e:diam}\begin{split}
d_{A_{\rm f}}=&
\begin{bmatrix}
d^M+i\theta_0\wedge\\
x^2{\partial_x}\end{bmatrix}\\
d_{A_{\rm f}}^*=& x^{-np}\begin{bmatrix}\delta_M-i\theta_0\lrcorner& -x^2{\partial_x}\end{bmatrix}x^{(n-2)p}\\
\Delta_{A_{\rm f}}=& x^{-2p}({{d_{{\tilde{\theta}}}}}^*{{d_{{\tilde{\theta}}}}}-(x^2{\partial_x})^2-(n-2)px (x^2{\partial_x})).
\end{split}$$ On the Riemannian manifold $(M,h)$, ${{d_{{\tilde{\theta}}}}}^*{{d_{{\tilde{\theta}}}}}$ is non-negative with discrete spectrum. Since $-(x^2{\partial_x})^2-(n-2)px (x^2{\partial_x})=(x^2{\partial_x})^*(x^2{\partial_x})$ is non-negative, one has that $\Delta_{A_{\rm f}}^h\geq \varepsilon^{-2p} \lambda_1$, where $\lambda_1$ is the first non-zero eigenvalue of ${{d_{{\tilde{\theta}}}}}^*{{d_{{\tilde{\theta}}}}}$. By Proposition \[p:cut\], the essential spectrum is independent of $\varepsilon$. By letting $\varepsilon \rightarrow 0$ we see that it is empty; thus $\Delta_{A_{\rm f}}^h$ has compact resolvent. The assertion on $\Delta_{A_{\rm f}}^l$ is a straightforward computation.
Diagonalization of the free magnetic Laplacian {#s:diago}
----------------------------------------------
In order to analyze the spectral properties of $\Delta_{A_{\rm f}}$ on $(X,g_p)$, where ${A_{\rm f}}$ is given by with $C=0$ and $g_p$ by , we go into some “Euclidean variables”. We concentrate on the complete case, i.e. $p\leq 1$. We start with and work on ${\rond{K}}$. The first unitary transformation is $$\begin{aligned}
L^2\left( x^{np-2} dx\right)\to L^2\left( x^{p-2} dx\right)&&
\phi\mapsto x^{(n-1)p/2}\phi.\end{aligned}$$ Then we proceed with the change of variables $z:=L(x)$, where $L$ is given by . Therefore, ${\rond{K}}$ is unitarily sent into $L^2\left((c,\infty),
dz\right)$ for a certain $c$. We indicate operators and spaces obtained in the new variable with a subscript $0$.
Thanks to this transformation, we are pursuing our analysis on the manifold $X_0=X$ endowed with the Riemannian metric $$\begin{aligned}
\label{e:gp''}
dr^2+h,&&r\to\infty,\end{aligned}$$ on the end $X_0'=[1/2,\infty)\times M$. The subscript $r$ stands for radial. The magnetic Laplacian is unitarily sent into an elliptic operator of order $2$ denoted by $\Delta_0$. On ${\mathcal{C}}^\infty_c(X_0')$, it acts by $$\label{e:H_0}
\Delta_0:= Q_p\otimes {d_{\tilde\theta}}^*d_{\tilde\theta}+(-\partial^2_r +
V_p)\otimes 1,$$ on the completed tensor product $L^2([1/2,\infty), dr)\hat \otimes L^2(M, h)$, where $$\begin{aligned}
V_p(r)=\begin{cases} ((n-1)/2)^2\\c_0r^{-2}\end{cases}
&& \mbox{and } Q_p(r)=\begin{cases}
e^{2r}\\
((1-p)r)^{2p/(1-p)}
\end{cases} &&\mbox{for } \begin{cases}
p=1\\
p<1.
\end{cases}\end{aligned}$$ Recall that $c_0$ is defined in Proposition \[p:free’\] and $d_{\tilde\theta}=d_M+i\theta_0\wedge$.
We denote also by $L_0$ the operator of multiplication corresponding to $L$, given by , in the new variable $r$. It is bounded from below by a positive constant, equals $1$ on the compact part and on the trapping ends, and equals $r\mapsto r$ on the non-trapping ends.
Let ${\rond{H}}^s_0:={\mathcal{D}}((1+\Delta_0)^{s/2})$ for $s>0$. By identifying ${\rond{H}}_0$ with ${\rond{H}}^*_0$ by the Riesz isomorphism, by duality, we define ${\rond{H}}^s_0$ for $s<0$ with ${\rond{H}}^{-s}_0$. We need the next well-known fact.
\[l:cchi\] For every $\gamma\in{\mathcal{C}}^\infty_c(X)$, we have that $\gamma:{\rond{H}}^s\subset{\rond{H}}^s$ for all $s\in{\mathbb{R}}$
A computation gives that there is $c$ such that $\|(\Delta_0+i)\gamma\varphi\|\leq c \|(\Delta_0+i)\varphi\|$, for all $\varphi\in{\mathcal{C}}^\infty_c(X_0)$. Since ${\mathcal{C}}^\infty_c(X_0)$ is a core for $\Delta_0$, we get the result for $s=2$. By induction we get it for $s\in 2{\mathbb{N}}$. Duality and interpolation give it for $s\in{\mathbb{R}}$.
The local conjugate operator {#s:conjop}
----------------------------
In this section, we construct some conjugate operators in order to establish a Mourre estimate. By mimicking the case of the Laplacian, see [@FH], one may use the following localization of the generator of dilations. Let $\xi\in{\mathcal{C}}^\infty([1/2,\infty))$ such that the support of $\xi$ is contained in $[2,\infty)$ and that $\xi(r)=r$ for $r\geq 3$ and let $\tilde \cchi\in{\mathcal{C}}^\infty([1/2,\infty))$ with support in $[1,\infty)$, which equals $1$ on $[2,\infty)$. By abuse of notation, we denote $\tilde\cchi\otimes 1\in{\mathcal{C}}^\infty(X_0)$ with the same symbol. Let $\cchi:=1-\tilde\cchi$. On ${\mathcal{C}}^\infty_c(X_0)$ we set: $$\label{e:S0}
S_\infty:= \left( -i (\xi \partial_r +\partial_r \xi)\otimes P_0\right) \tilde\cchi,$$ where $P_0$ is the orthogonal projection onto $\ker({{d_{{\tilde{\theta}}}}})$. The presence of $P_0$ comes from the decomposition in low and high energies.
\[r:sa\] By considering a $C_0$-group associated to a vector field on ${\mathbb{R}}$ like in [@ABG Section 4.2], one shows that $-i(\xi \partial_r +\partial_r \xi)$ is essentially self-adjoint on ${\mathcal{C}}^\infty_c({\mathbb{R}})$. This $C_0$-group acts trivially away from the support of $\tilde{\cchi}$ and then it is easy to construct another $C_0$-group $G_0$ which acts like the first on $L^2({\mathrm{supp}}(\tilde{\cchi}))\otimes\ker({{d_{{\tilde{\theta}}}}})$, and trivially on the rest of $L^2(X_0)$. Let $\overline{S_\infty}$ be the generator of $G_0$. Since $G_0(t)$ leaves invariant ${\mathcal{C}}^\infty_c(X_0)$ the Nelson lemma implies that $\overline{S_\infty}$ is the closure of $S_\infty$, in other words $\overline{S_\infty}$ is essentially self-adjoint on ${\mathcal{C}}^\infty_c(X_0)$. This kind of approach has been used in [@bouclet] for instance.
Therefore, we denote below also by $S_\infty$ the self-adjoint closure of the operator defined by on ${\mathcal{C}}^\infty_c(X_0)$.
However this operator is *not* suitable for very singular perturbations like that of the metric considered in this paper. To solve this problem, one should consider a conjugate operator more “local in energy”. Concerning the Mourre estimate, as it is local in energy for the Laplacian, one needs only a conjugate operator which fits well on this level of energy. Considering singular perturbation theory, the presence of differentials in is a serious obstruction; the idea is to replace the conjugate operator with a multiplication operator in the analysis of perturbations, therefore reducing the rôle of derivatives within it. The approach has been used for Dirac operators for instance in [@GM] to treat very singular perturbations. The case of Schrödinger operators is summarized in [@ABG Theorem 7.6.8]. We set: $$\label{e:SR}
S_R:=\tilde\cchi \left(\big(\Phi_R(-i\partial_r) \xi +\xi
\Phi_R(-i\partial_r)\big)\otimes P_0\right) \tilde\cchi,$$ where $\Phi_R(x):=\Phi(x/R)$ for some $\Phi\in{\mathcal{C}}^\infty_c({\mathbb{R}})$ satisfying $\Phi(x)=x$ for all $x\in[-1,1]$. The operator $\Phi_R(-i\partial_r)$ is defined on ${\mathbb{R}}$ by ${\rond{F}}^{-1}\Phi_R{\rond{F}}$, where ${\rond{F}}$ is the unitary Fourier transform. Let us also denote by $S_R$ the closure of this operator.
Unlike , $S_R$ does not stabilize ${\mathcal{C}}^\infty_c(X_0)$ because $\Phi_R(-i\partial_r)$ acts like a convolution with a function with non-compact support. This subspace is sent into $\tilde\cchi {\rond{S}}({\mathbb{R}})$, where ${\rond{S}}({\mathbb{R}})$ denotes the Schwartz space. To motivate the subscript $R$, note that $S_R$ tends strongly in the resolvent sense to $S$, as $R$ goes to infinity. We give some properties of $S_R$. The point (\[p2\]) is essential to be able to replace $S_R$ by $L_0$ in the theory of perturbations. The point (\[p3\]) is convenient to be able to express a limiting absorption principle in terms of $L_0$, which is very explicit. Of course these two points are false for $S_\infty$ and this explains why we can go further in the perturbation theory compared to the standard approach.
\[l:sa\] Let $S_R$ denote the closure of the unbounded operator .
1. For all $R\in[1,\infty]$, the operator $S_R$ is essentially self-adjoint on ${\mathcal{C}}^\infty_c(X)$.
2. For $R$ finite, $L_0^{-2}S_R^2: {\mathcal{C}}^\infty_c(X_0)
\rightarrow {\mathcal{D}}(\Delta_0)$ extends to a bounded operator in ${\mathcal{D}}(\Delta_0)$. \[p2\]
3. For $R$ finite, ${\mathcal{D}}(L_0^s)\subset {\mathcal{D}}(|S_R|^s)$ for all $s\in[0,2]$. \[p3\]
The case $R=\infty$ is discussed in Remark \[r:sa\], so assume that $R$ is finite. We compare $S_R$ with $L_0$, defined in Section \[s:diago\], which is essentially self-adjoint on ${\mathcal{C}}^\infty_c(X)$. Noting that it stabilizes the decomposition , we write also by $L_{0,\rm l}$ its restriction to ${\rond{H}}_{\rm l}$, which is simply the multiplication by $r$. On ${\mathcal{C}}^\infty_c(X_0)$, $$S_R= \tilde\cchi\big(2\Phi_R(-i\partial_r)\xi L_{0,\rm l}^{-1}+
[\xi, \Phi_R(-i\partial_r)] L_{0,\rm l}^{-1} \big)\otimes P_0 \tilde\cchi L_0$$ Noting that $\xi L_{0,\rm l}^{-1}$ is bounded and that $\xi'\in
L^\infty$ and using Lemma \[l:pseudo\], we get $\|S_R \varphi\|\leq a
\|L_0 \varphi\|$, for all $\varphi\in{\mathcal{C}}^\infty_c(X_0)$.
On the other hand, $[S_R, L_0]$ is equal to the bounded operator $$\begin{aligned}
[S_R, L_0]=& \tilde\cchi\big([\Phi_R(-i\partial_r), L_{0,\rm l}]\xi
L_{0,\rm l}^{-1}\big) \otimes P_0 \tilde\cchi L_0\\ &
+ L_0 \tilde\cchi\big( L_{0,\rm l}^{-1} \xi [\Phi_R(-i\partial_r), L_{0,\rm l}] \big) \otimes P_0\tilde\cchi. \end{aligned}$$ This gives $|\langle S_R \varphi, L_0\varphi \rangle -\langle L_0\varphi,
S_R\varphi \rangle| \leq b \|L_0^{1/2}\varphi\|^2$, for all $\varphi\in{\mathcal{C}}^\infty_c(X_0)$. Finally, one uses [@RS2 Theorem X.37] to conclude that $S_R$ is essentially self-adjoint.
On ${\mathcal{C}}^\infty_c(X_0)$, we have $$\label{e:LS}
L^{-2}_0S_R^2= (2\cchi L^{-1}_{0,\rm l}\xi
\Phi_R(-i\partial_r)\cchi\otimes P_0 +
\cchi L^{-1}_{0,\rm l} [\Phi_R(-i\partial_r), \xi]\cchi\otimes P_0)^2.$$ All these terms are bounded in $L^2(X_0)$ by Lemma \[l:pseudo\] and by density. We now compute $\Delta_0 L^{-2}_0S_R^2$. By Lemma \[l:cchi\], it is enough to show that $\Phi_R(-i\partial_r)$ and $[\Phi_R(-i\partial_r), \xi]$ stabilize the domain $\Delta$ in $L^2({\mathbb{R}})$. The first one commutes with $\Delta$. For the second one, we compute on ${\mathcal{C}}^\infty_c({\mathbb{R}})$. Since $[\Phi_R(-i\partial_r), \xi]$ is bounded in $L^2({\mathbb{R}})$, it is enough to show that the commutator $[\Delta,[\Phi_R(-i\partial_r), \xi]]$ is also bounded in $L^2({\mathbb{R}})$. By Jacobi’s identity, it is equal to $[\Phi_R(-i\partial_r), [\Delta, \xi]] = [\Phi_R(-i\partial_r), 2
\xi'\partial_r + \xi'']= 2\Phi_R(-i\partial_r)\partial_r \xi'-2
\Phi_R(-i\partial_r) \xi'' + 2 \xi'\Phi_R(-i\partial_r) +\xi''$. This is a bounded operator in $L^2({\mathbb{R}})$ and we get point (2).
We now note that is bounded in $L^2(X_0)$. Then, since $S_R^2L_0^{-2}$ is also bounded, we get $\|S_R^2\varphi\|^2 \leq c
\|L^2_0\varphi\|$ for all $\varphi \in {\mathcal{C}}^\infty(X_0)$. Taking a Cauchy sequence, we deduce ${\mathcal{D}}(L^2_0)\subset {\mathcal{D}}(S_R^2)$. An argument of interpolation gives point (3).
The aim of this section is the following Mourre estimate.
\[t:mourre\_0\] Let $R\in [1,\infty]$. Then $e^{itS_R} {\rond{H}}^2_0\subset {\rond{H}}^2_0$ and $\Delta_0\in {\mathcal{C}}^{2}(S_R, {\rond{H}}^2_0, {\rond{H}}_0)$. Given an interval ${\mathcal{J}}$ inside $\sigma_{\rm ess}(\Delta_0)$, there exist $\varepsilon_R> 0$ and a compact operator $K_R$ such that $$E_{\mathcal{J}}(\Delta_0) [\Delta_0,iS_R] E_{\mathcal{J}}(\Delta_0) \geq (4\inf({\mathcal{J}})
- \varepsilon_R) E_{\mathcal{J}}(\Delta_0)+ K_R$$ holds in the sense of forms, and such that $\varepsilon_R$ tends to $0$ as $R$ goes to infinity.
The regularity assumptions follow from Lemmata \[l:regu0\] and \[l:regu\]. The left hand side of is the commutator $[\Delta_0,
iS_R]$ in the sense of forms. It extends to a bounded operator in ${\mathcal{B}}({\rond{H}}^2_0, {\rond{H}}^{-2}_0)$ since ${\mathcal{C}}^\infty_c(X_0)$ is a core for $\Delta_0$. We can then apply the spectral measure and obtain the inequality using Lemma \[l:mourre\_0\].
Compared to the method from [@FH Lemma 2.3] (for the case $R=\infty$), we have a relatively more direct proof based on Lemma \[l:mourre\_0\]. However this has no real impact on applications of the theory.
We now go in a series of lemmata to prove this theorem. Given a commutator $[A,B]$, we denote its closure by $[A,B]_0$.
\[l:commu\] For $R=\infty$, the commutators $[\Delta_0,iS_\infty]_0$ and $[[\Delta_0,iS_\infty],iS_\infty]_0$ belong to ${\mathcal{B}}({\rond{H}}^2_0, {\rond{H}}_0)$. For $R$ finite, $[\Delta_0,iS_R]_0$ and $[[\Delta_0,iS_\infty],iS_R]_0$ belong to ${\mathcal{B}}({\rond{H}}_0)$. Moreover, if $p=1$, all higher commutators extend to bounded operators in ${\mathcal{B}}({\rond{H}}^2_0,
{\rond{H}}_0)$ for $R=\infty$ and in ${\mathcal{B}}({\rond{H}}_0)$ for $R<\infty$.
Let $\varphi\in{\mathcal{C}}^{\infty}_c(X_0')$ such that $\varphi=\varphi_{\mathbb{R}}\otimes \varphi_M$ where $\varphi_M\in{\mathcal{C}}^\infty(M)$ and $\varphi_{\mathbb{R}}\in{\mathcal{C}}^\infty_c([1/2,
\infty))$. Note that $P_0\varphi_M\in{\mathcal{C}}^\infty(M)$ by the Hodge decomposition, since $d_A^2=0$. Applying the brackets to $\varphi$, by a straightforward computation, we get $$\label{e:commu}
[\Delta_0,iS_\infty]=-(4\xi'\partial^2_r+4\xi''\partial_r+\xi'''-2V_p')\tilde
\cchi\otimes P_0.$$ By linearity and density, we get $\|[\Delta_0,iS_\infty]\varphi\|\leq
C\|(\Delta_0+i)\varphi\|$ for all $\varphi\in{\mathcal{C}}^\infty_c(X_0')$. Take now $\varphi\in{\mathcal{C}}^\infty_c(X_0)$, considering the support of the commutator, we get $$\begin{aligned}
\|[\Delta_0,iS_\infty]\varphi\|&=& \|[\Delta_0,iS_\infty]\widetilde\Xi\varphi\|
\\
&\leq& C\|(\Delta_0+i) \varphi\| + \|[\Delta_0, \widetilde\Xi] \varphi\|
\leq C'\|(\Delta_0+i) \varphi\|.\end{aligned}$$ where $\widetilde\Xi\in{\mathcal{C}}^{\infty}(X)$ with support in $X_0'$ such that $\widetilde\Xi|_{[1,\infty)\times M}=1$. Therefore, since ${\mathcal{C}}^\infty_c(X_0)$ is a core for $\Delta_0$, we conclude $[\Delta_0,iS_\infty]_0\in{\mathcal{B}}({\rond{H}}^2_0, {\rond{H}}_0)$.
In the same way we compute $$\begin{aligned}
\nonumber
[[\Delta_0,iS_\infty], iS_\infty]&=& -\big((16(\xi')^2-8\xi\xi'')\partial^2_r + (24
\xi'\xi''-8\xi\xi''')\partial_r \\\label{e:commu2}
&&+4 (\xi'')^2+4\xi'''-2 \xi'''' - 4V_p''\big)\tilde\cchi\otimes P_0.\end{aligned}$$ and get $[[\Delta_0,iS_\infty],iS_\infty]_0\in{\mathcal{B}}({\rond{H}}^2_0, {\rond{H}}_0)$. For $p=1$, the boundedness of higher commutators follows easily by induction ($V_p=0$ in this case).
We compute next the commutators of $\Delta_0$ with $S_R$. As above we compute for $\varphi=\varphi_r\otimes\varphi_M$. For brevity, we write $\Phi_R$ instead of $\Phi_R(-i\partial_r)$.
As $\Phi_R$ is not a local operator, we first note that the commutator $[\Delta_0, S_R]$ could be taken in the operator sense. Indeed, $\tilde\cchi$ sends $\varphi_r$ to ${\mathcal{C}}^\infty_c({\mathbb{R}})$ (note that $[1/2,\infty)$ is injected in a canonical way into ${\mathbb{R}}$), then $\Phi_R \xi +\xi\Phi_R$ sends to the Schwartz space ${\rond{S}}({\mathbb{R}})$ and finally $\tilde\cchi$ sends to $\tilde\cchi{\rond{S}}({\mathbb{R}})$ which belongs to ${\mathcal{D}}(\Delta_0)$.
We compute $[\partial_r^2, \tilde\cchi \big(\Phi_R \xi +\xi
\Phi_R\big) \tilde\cchi]\otimes P_0$. Against $\varphi_r\otimes \varphi_M$, we have: $$\begin{aligned}
\nonumber
[\partial_r^2, \tilde\cchi \Phi_R \xi +\xi
\Phi_R \tilde\cchi]&=& \Xi [\partial_r^2, \tilde\cchi \Phi_R \xi +\xi
\Phi_R \tilde\cchi] \Xi = \Xi [\partial_r^2, \Phi_R r + r\Phi_R ] \Xi
+\Psi_{\mathrm{comp}}\\ \label{e:commu5}
&=& 4\Xi \partial_r\Phi_R \Xi +\Psi_{\mathrm{comp}}= 4\tilde\cchi \partial_r\Phi_R
\tilde\cchi +\Psi_{\mathrm{comp}},\end{aligned}$$ where $\Psi_{\mathrm{comp}}$ denotes a pseudo-differential operator with compact support such that its support in position is in the interior of $X_0'$. For $p<1$, the potential part $V_p$ arises. We treat its first commutator: $$\begin{aligned}
\nonumber
[V_p, \tilde\cchi \Phi_R \xi +\xi \Phi_R\tilde\cchi]
&=&\tilde\Xi[V_p, 2\Phi_R r -i\Phi'_R]\tilde\Xi +\Psi_{\mathrm{comp}}\\ \label{e:commu6}
&=&\tilde\cchi\big(2[V_p,\Phi_R]\xi-i[V_p, \Phi_R'] \big)\tilde\cchi+\Psi_{\mathrm{comp}},\end{aligned}$$ where $\Phi_R'=\Phi_R'(-i\partial_r)$. Applying Lemma \[l:pseudo\], we get that $[V_p, \Phi_R]\xi$ and $[V_p, \Phi_R']$ are bounded in $L^2({\mathbb{R}})$ also. Therefore, using like above $\tilde{\Xi}$, we get $\|[\Delta_0,iS_R]\varphi\|\leq C\|\varphi\|$ for all $\varphi\in{\mathcal{C}}^\infty_c(X_0)$. This implies that $[\Delta_0,iS_R]_0\in{\mathcal{B}}({\rond{H}}_0)$.
For higher commutators, the $n$-th commutator with $\tilde\cchi
\Phi_R \xi +\xi \Phi_R \tilde\cchi$ is given by $2^n\tilde\cchi
\partial_r^n\Phi_R \tilde\cchi +\Psi_{\mathrm{comp}}$. Note that $\partial_r^n\Phi_R$ is a compactly supported function of $\partial_r$, so the contribution of this term is always bounded.
Consider now the second commutator of $V_p$. As above, since we work up to $\tilde\Xi$, $\tilde{\cchi}$ and $\Psi_{\mathrm{comp}}$, it is enough to show that the next commutator defined on ${\rond{S}}({\mathbb{R}})$ extend to bounded operators in $L^2({\mathbb{R}})$. We treat only the most singular part of the second commutator: $$\begin{aligned}
\nonumber
[[V_p, \Phi_R]r, \Phi'_R r]&=& [[V_p, \Phi_R], \Phi_R r]r + [V_p, \Phi_R][r,\Phi_R]r
\\ \nonumber
&\hspace*{-4cm }=& \hspace*{-2cm } [[V_p, \Phi_R], \Phi_R ]r^2 + \Phi_R [[V_p, \Phi_R], r]r -i [V_p, \Phi_R]\Phi_R'r
\\ \nonumber
&\hspace*{-4cm }=& \hspace*{-2cm } [[V_p, \Phi_R], \Phi_R ]r^2 - \Phi_R [[r, \Phi_R], V_p]r -i [V_p, \Phi_R][\Phi_R',r] -i [V_p, \Phi_R]r \Phi_R'
\\ \nonumber
&\hspace*{-4cm }=& \hspace*{-2cm } [[V_p, \Phi_R], \Phi_R ]r^2 + i \Phi_R [\Phi_R', V_p]r + [V_p, \Phi_R] \Phi_R'' -i [V_p, \Phi_R]r \Phi_R'\end{aligned}$$ These terms extend to bounded operators by Lemma \[l:pseudo\].
The following lemma is the key-stone for the Mourre estimate.
\[l:mourre\_0\] For all $R\in[1,\infty]$, there exists $K\in{\mathcal{K}}({\rond{H}}^2_0, {\rond{H}}^{-2}_0)$ and $N_R\in{\mathcal{B}}({\rond{H}}^2_0, {\rond{H}}^{-2}_0)$ such that $$\begin{aligned}
\nonumber
\langle \Delta_0\varphi, iS_R\varphi \rangle+ \langle iS_R
\varphi, \Delta_0\varphi \rangle &=&
\\ \label{e:mourre_0}
&& \hspace*{-4cm}4\langle \varphi,
(\Delta_0-\inf(\sigma_{\rm ess}(\Delta_0)) \varphi\rangle + \langle \varphi, N_R
\varphi\rangle +\langle \varphi, K\varphi\rangle, \end{aligned}$$ for all $\varphi\in{\mathcal{C}}^\infty_c(X_0)$ and such that $\|N_R\|_{{\mathcal{B}}({\rond{H}}^2_0,
{\rond{H}}^{-2}_0)}$ tends to $0$ as $R$ goes to infinity.
First note that the essential spectrum of $\Delta_0$ is $[V_p(\infty), \infty)$, by Proposition \[p:thema\]. We now act in three steps. Let $\Xi$ be like in the proof of Lemma \[l:commu\] and let $\varphi\in{\mathcal{C}}^\infty_c(X_0)$. Since $\tilde\cchi$ and $\Xi$ have disjoint supports, one has for all $R$ that $\langle \varphi,
[\Delta_0,iS_R] \varphi\rangle= \langle\widetilde\Xi \varphi, [\Delta_0,iS_R]
\widetilde\Xi\varphi\rangle$.
For the first step, we start with $R=\infty$. By and since $\xi'=1$ on $[2,\infty)$, the Rellich-Kondrakov lemma gives $$\label{e:Mstep1}
\langle \widetilde\Xi \varphi, [\Delta_0,iS_\infty] \widetilde\Xi\varphi\rangle
=\langle \widetilde\Xi \varphi, 4(\Delta_0 -V_p(\infty))(1\otimes
P_0)\widetilde\Xi \varphi \rangle+ \langle \varphi, K_1 \varphi \rangle$$ for a certain $K_1\in{\mathcal{K}}({\rond{H}}^2_0,{\rond{H}}_0^{-2})$. Indeed, $1-\xi', V_p-V_p(\infty), \xi'',\xi''', V_p'$ belong to ${\mathcal{K}}({\rond{H}}^1_0,{\rond{H}}_0)$ since they tend to $0$ at infinity.
We consider now $R$ finite. We add and . We have $$\label{e:Mstep1'}
\langle \widetilde\Xi \varphi, [\Delta_0,iS_R] \widetilde\Xi\varphi\rangle
=\langle \widetilde\Xi \varphi, 4(\Delta_0 -V_p(\infty) - T_R)(1\otimes
P_0)\widetilde\Xi \varphi \rangle
+ \langle \varphi, K_2 \varphi \rangle$$ for a certain $K_2=K_2(R)\in{\mathcal{K}}({\rond{H}}^2_0,{\rond{H}}_0^{-2})$ and with $T_R=
\partial_r(\partial_r-\Phi_R(\partial_r))$. The compactness of $K_2$ follows by noticing that $L_0^{-1}\in {\mathcal{K}}({\rond{H}}^2_0,{\rond{H}}_0)$ and that $L_0[V_p, i S_R]_0\in {\mathcal{B}}({\rond{H}}_0,{\rond{H}}_0)$, by Lemma \[l:pseudo\]. We control the size of $T_R$ by showing that $\|\widetilde\Xi T_R (1\otimes P_0)\widetilde\Xi \|_{{\mathcal{B}}({\rond{H}}^2_0,
{\rond{H}}^{-2}_0)}$ tends to $0$ as $R$ goes to infinity. By Lemma \[l:cchi\], one has that $\widetilde\Xi$ stabilizes ${\rond{H}}^{\pm
2}_0$, therefore $-\partial_r^2 \widetilde\Xi$ belongs to ${\mathcal{B}}({\rond{H}}^2_0, L^2({\mathbb{R}}))$. It remains to note that $(-\partial_r^2+i)^{-2}T_R$ tends to $0$ in norm by functional calculus, as $R$ goes to infinity.
The second step is to control the high energy functions part. Consider the Friedrichs extension of $\Delta_0$ on ${\rond{H}}_{0,h}:=L^2([1/2,
\infty))\otimes P_0^\perp L^2(M)$. We have: $$\label{e:Mstep2}
\langle \widetilde\Xi \varphi, \Delta_0 P_0^\perp\widetilde\Xi\varphi\rangle=
\langle (\Delta_0 P_0^\perp+i)^{-1}(\Delta_0 P_0^\perp+i)
\widetilde\Xi \varphi, \Delta_0 P_0^\perp\widetilde\Xi\varphi\rangle
=\langle \varphi, K_1 \varphi\rangle$$ where $K_1\in{\mathcal{K}}({\rond{H}}_0^2, {\rond{H}}_0^{-2})$. Indeed, note first that $\Delta_0 P_0^\perp\tilde\cchi \in
{\mathcal{B}}({\rond{H}}_0^2,{\rond{H}}_{0,h})$ and that $(\Delta_0 P_0^\perp+i)^{-1}
\in{\mathcal{K}}({\rond{H}}_{0,h})$, since $Q_p$ in goes to infinity. Therefore the left hand side belongs ${\mathcal{K}}({\rond{H}}_0^2,{\rond{H}}_0)$ and the right hand side belongs to ${\mathcal{B}}({\rond{H}}^2_0,{\rond{H}}_0)$.
The third step is to come back on the whole manifold. It is enough to note that $[\Delta_0,\tilde\cchi]\in{\mathcal{K}}({\rond{H}}^1_0,
{\rond{H}}_0)$ and to add , with .
We now turn to the regularity assumptions. Lemma \[l:C1\] plays a central rôle.
\[l:regu0\] For $R\in [1, \infty]$, one has $\Delta_0\in{\mathcal{C}}^1(S_R)$ and $e^{itS_R} {\mathcal{D}}(\Delta_0) \subset{\mathcal{D}}(\Delta_0)$.
We start by showing that $\Delta_0\in{\mathcal{C}}^1(S_R)$. We check the hypothesis of Lemma \[l:C1\]. Let $\cchi_n(r):=\cchi(r/n)$ and ${\rond{D}}={\mathcal{C}}^\infty_c(X_0)$. Remark that ${\mathrm{supp}}(\cchi_n')\subset[n,2n]$ and that $\xi \cchi_n^{(k)}$ tends strongly to $0$ on $L^2({\mathbb{R}}^+)$, for any $k\geq 1$. By the uniform boundedness principle, this implies that $\sup_n\|\cchi_n\|_{{\mathcal{D}}(H)}$ is finite.
Remark \[r:sa\] and Lemma \[l:sa\] give that ${\rond{D}}$ is a core for $S_R$. Assumption (1) is obvious, assumption (2) holds since $(1-\cchi_n)$ has support in $[2n,\infty)$ and assumption (3) follows from the fact that $H$ is elliptic, so the resolvent of $\Delta_0$ sends ${\rond{D}}$ into ${\mathcal{C}}^\infty(X_0)$. The point follows from Lemma \[l:mourre\_0\]. We now show that is true. Let $\phi\in{\mathcal{C}}^\infty(X_0)\cap{\mathcal{D}}(\Delta_0)$. We have $[\Delta_0,\cchi_n]\phi=[\Delta_0,\cchi_n]\tilde\cchi\phi=2
\cchi_n'\partial_r\tilde\cchi\phi + \cchi_n''\tilde\cchi\phi$. We have $iS_\infty[\Delta_0,\cchi_n]\phi=
2\xi\cchi_n'\partial^2_r P_0\tilde \cchi\phi +(4\cchi\xi_n''
+\xi'\cchi_n')\partial_r P_0 \tilde\cchi\phi+
(2\xi\cchi_n'''+\xi'\cchi_n'')P_0 \tilde\cchi\phi$ and, for a finite $R$, we get $iS_R[\Delta_0,\cchi_n]\phi=
\tilde\cchi(2\Phi_R(\partial_r)\xi + [\xi, \Phi_R(\partial_r)])
(2\cchi_n'\partial_r P_0\tilde\cchi\phi + \cchi_n'' P_0\tilde\cchi\phi)$. Both terms are tending to $0$ because of the previous remark, Lemma \[l:cchi\] and the fact that $[\xi, \Phi_R(\partial_r)]$ is bounded by Lemma \[l:pseudo\]. From that, we can apply the lemma and obtain $H\in{\mathcal{C}}^1(S_R)$.
By Lemma \[l:commu\], we have that $[\Delta_0, iS_R]_0\in{\mathcal{B}}({\rond{H}}_0^2,{\rond{H}}_0)$ and [@GG0 Lemma 2] gives that $e^{itA}{\rond{H}}_0^2\subset{\rond{H}}_0^2$.
The invariance of the domain under the group $e^{itS_R}$ implies that $e^{itS_R}{\rond{H}}^s_0\subset{\rond{H}}^s_0$ for $s\in [-2, 2]$ by duality and interpolation. This allows one to define the class ${\mathcal{C}}^k(S_R,
{\rond{H}}^{s}_0,{\rond{H}}^{-s}_0)$ for $s\in [-2,2]$, for instance; we recall that a self-adjoint operator $H$ is in this class if $t\mapsto e^{itS_R}He^{-itS_R}$ is strongly $C^k$ from ${\rond{H}}^{s}_0$ to ${\rond{H}}^{-s}_0$.
\[l:regu\] Let $R\in [1, \infty]$. Then $\Delta_0$ belongs to ${\mathcal{C}}^2(S_R, {\rond{H}}^2_0,{\rond{H}}_0)$ for $p\leq 1$.
From Lemma \[l:commu\], the commutators with $S_R$ extend to bounded operators from ${\rond{H}}^2_0$ to ${\rond{H}}_0$.
We finally give an estimation of commutator that we have used above.
\[l:pseudo\] Let $f\in{\mathcal{C}}^{0}({\mathbb{R}})$ with polynomial growth, $\Phi_j\in{\mathcal{C}}^\infty_c({\mathbb{R}})$ and $g\in{\mathcal{C}}^k({\mathbb{R}})$ with bounded derivatives. Let $k\geq 1$. Assume that $\sup_{t\in{\mathbb{R}}, |s-t|\leq 1}|f(t)g^{(l)}(s)|<\infty$, for all $1\leq l\leq k$. Then the operator $f[\Phi_1(-i\partial_r), [\Phi_2(-i\partial_r)\ldots
[\Phi_k(-i\partial_r), g]\ldots]$, defined on ${\mathcal{C}}^\infty_c({\mathbb{R}})$, extends also to a bounded operator.
Take $k=1$. We denote with a hat the unital Fourier transform. We get $$(f[\Phi(-i\partial_r), g]\varphi)(t)= \frac{1}{2\pi}\int
\widehat{\Phi}(s-t)f(t)(g(s)-g(t))\varphi(s) ds,$$ for $\varphi\in{\mathcal{C}}^\infty_c({\mathbb{R}})$. In order to show that the $L^2$ norm of the left hand side is uniformly bounded by $\|f\|_2$, we separate the integral in $|s-t|$ lower and bigger than one. We start with the first part. Recalling $t^k\widehat\Phi(t)=(-1)^k\widehat{\Phi^{(k)}}(t)$, $\Phi$ is replaced by $\Phi'$ when we divide by $|s-t|$ the term with $g$. Now since $\widehat{\Phi'}\in L^1$, $\sup_{t\in{\mathbb{R}}, |s-t|\leq
1}|f(t) (g(s)-g(t))/ (s-t)|$ is finite, and the convolution by a $L^1$ function is bounded in $L^2$, we control this part of the integral. We turn to the part $|s-t|\geq 1$. Let $R\in{\mathbb{N}}$ such that $|f(t)|\leq C(1+ |t|^R)$. We let appear the Fourier transform of $\Phi^{(R+1)}$. Now to conclude, note that $\sup_{s,t\in{\mathbb{R}}}|(g(s)-g(t))/ (s-t)|$ is finite and that $t\mapsto|f(t)/(s-t)^R|$ is in $L^\infty$ uniformly in $s$. For higher $k$, one repeats the same decomposition and let appear the terms in the $l$-th derivative of $g$ by regrouping terms.
A short-range and long-range class of perturbations {#s:srlr}
---------------------------------------------------
In the early versions of Mourre theory, one asked $[[H,S_R], S_R]$ to be $H-$bounded to obtain refined results of the resolvent like the limiting absorption principle and the Hölder regularity of the resolvent. In this section, we check the optimal class of regularity ${\mathcal{C}}^{1,1}(S_R)$, for $R$ finite. This is a weak version of the two-commutators hypothesis. We refer to [@ABG] for definition and properties. This is the optimal class of operators which give a limiting absorption principle for $H$ in some optimal Besov spaces associated to the conjugate operator $S_R$.
The operator $\Delta_0$ belongs to ${\mathcal{C}}^2(S_R)$ by Lemma \[l:regu\] and therefore also to ${\mathcal{C}}^{1,1}(S_R)$. We now consider perturbations of $\Delta_0$ which are also in ${\mathcal{C}}^{1,1}(S_R)$. We define two classes.
Consider a symmetric differential operator $T:
{\mathcal{D}}(\Delta_0)\rightarrow {\mathcal{D}}(\Delta_0)^*$. Take $\theta_{\rm
sr}\in{\mathcal{C}}_c^\infty((0,\infty))$ not identically $0$; $V$ is said to be *short-range* if $$\label{e:sr}
\int_1^\infty \Big\|\theta_{\rm sr}\Big(\frac{L_0}{r}\Big) T \Big\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_0),
{\mathcal{D}}(\Delta_0)^*)} dr <\infty.$$ and to be *long-range* if $$\begin{aligned}
\int_1^\infty & \Big\|[T, L_0]\theta_{\rm lr}\Big(\frac{L_0}{r}\Big)
\Big\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_0), {\mathcal{D}}(\Delta_0)^*)}
+ \Big\|\tilde\Xi[T, P_0]L_0\theta_{\rm
lr}\Big(\frac{L_0}{r}\Big)\tilde\Xi\Big\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_0),
{\mathcal{D}}(\Delta_0)^*)}\nonumber
\\
&+ \Big\|\tilde\Xi[T, \partial_r]P_0 L_0\theta_{\rm
lr}\Big(\frac{L_0}{r}\Big)\tilde\Xi \Big\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_0),
{\mathcal{D}}(\Delta_0)^*)} \frac{dr}{r} <\infty.\label{e:lr}\end{aligned}$$ where $\theta_{\rm lr}$ is the characteristic function of $[1,\infty)$ in ${\mathbb{R}}$ and where $\widetilde\Xi\in{\mathcal{C}}^{\infty}(X)$ with support in $X_0'$ such that $\widetilde\Xi|_{[1,\infty)\times M}=1$.
The first condition is evidently satisfied if there is $\varepsilon
\pg 0$ such that $$\label{e:sr'}
\|L^{1+\varepsilon}_0 T\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_0),
{\mathcal{D}}(\Delta_0)^*)} \pp \infty$$ and the second one if $$\begin{aligned}
\nonumber
\|L^{\varepsilon}_0 [T,L_0]\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_0), {\mathcal{D}}(\Delta_0)^*)} &+
\|L^{1+\varepsilon}_0
\widetilde\Xi[T,P_0]\widetilde\Xi\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_0),
{\mathcal{D}}(\Delta_0)^*)}
\\ \label{e:lr'}
&+ \|L^{1+\varepsilon}_0
\widetilde\Xi[T,\partial_r]P_0\widetilde\Xi\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_0),
{\mathcal{D}}(\Delta_0)^*)}\pp \infty \end{aligned}$$ The condition with $P_0$ essentially tells that the non-radial part of $T$ is a short-range perturbation. This is why we will ask the long-range perturbation to be radial. To show that the first class is in ${\mathcal{C}}^{1,1}(S_R)$ for $R$ finite, one use [@ABG Theorem 7.5.8]. The hypotheses are satisfied thanks to Lemmata \[l:sa\] and \[l:L\]. Concerning the second class, one shows that $[T,
S_R]\in{\mathcal{C}}^{0,1}(S_R)$ by using [@ABG Proposition 7.5.7] (see the proof of [@ABG Proposition 7.6.8] for instance).
We go back to the $x$ coordinate. For ${\rond{G}}={\mathcal{D}}(\Delta^{1/2})$, the short and long-range perturbation of the electric and magnetic perturbations are given by:
\[l:magne\] Let $V\in L^\infty(X)$ and $\tilde A\in L^\infty(X, T^*X)$. If $\|L^{1+\varepsilon}\tilde A\|_\infty\pp \infty$ (respectively $\|L^{1+\varepsilon}V\|_\infty<\infty$) then the perturbation $\big(
d_{A_{\rm f}}^* (i\tilde A\wedge) + (i\tilde A\wedge)^* d_{A_{\rm
f}})$ (resp. $V$) is short-range in ${\mathcal{B}}({\rond{G}}, {\rond{G}}^*)$. If $\tilde A$ is radial, $\|L^{\varepsilon}\tilde A\|\pp \infty$ and $\|L^{1+\varepsilon} x^{2-p}\partial_x \tilde A\|\pp \infty$, where these norms are in ${\mathcal{B}}(L^2(X, g), L^2(X, \Lambda^1, g))$ (respectively $V$ radial, and $\|L^{1+\varepsilon} x^{2-p}\partial_x
V\|_\infty\pp \infty$) then the same perturbation is long-range in ${\mathcal{B}}({\rond{G}}, {\rond{G}}^*)$.
We deal with the magnetic perturbation. Start with the short-range. We have $\langle L^{1+\varepsilon}(i\tilde A\wedge)^*
d_{A_{\rm f}}) f\rangle= \langle d_A f, L^{1+\varepsilon}(i\tilde
A\wedge) f \rangle $ and on the other hand, we have $\langle L^{1+\varepsilon} d_{A_{\rm f}}^* (i\tilde A\wedge) f,
f\rangle$ $=\langle [L^{1+\varepsilon}, d_{A_{\rm f}}^*]L^{-\varepsilon
} L^{\varepsilon}
(i\tilde A\wedge) f, f\rangle + \langle L^{1+\varepsilon}(i\tilde
A\wedge) f, d_A f\rangle$. This is bounded by $\|f\| +\|d_A f\|^2$ uniformly in $f\in{\mathcal{C}}^\infty_c(X)$.
We deal now with the long-range perturbation by checking . The condition with $L^\varepsilon$ is treated as above. In the variable of the free metric $g_p$ , $\partial_r$ is given by $\partial_L:=x^{2-p}\partial_x-(n-1)p/2\,
x^{1-p}$. We extend $\partial_L$ on $1$-forms by setting $\partial_L=x^{2-p}\partial_x-(n+1)p/2\,
x^{1-p}$. Note it is symmetric on $1$-forms with compact support on the cusp. First, we have on smooth functions with compact support on the cusp that: $$\label{e:comm1}
[d_{A_{\rm f}}, \partial_L]P_0= (2(1-p)x^{1-p}d_{A_{\rm f}} + c(1-p)x^{2(1-p)}x^{p-2}dx\wedge)P_0.$$ Here, we used $d_{A_{\rm f}}P_0=(dx\wedge \partial_x\cdot)P_0$. Note that $x^{p-2}dx\wedge$ is a bounded operator from function to $1$-forms. In the following, we drop $P_0$ and $\Xi$ to lighten the notation. For $\varphi\in{\mathcal{C}}^\infty_c(X)$, we have $$\begin{aligned}
\langle L^{1+\varepsilon} \varphi, [
d_{A_{\rm f}}^* (i\tilde A\wedge) + (i\tilde A\wedge)^* d_{A_{\rm
f}}] \varphi\rangle &=& \langle [\partial_L, d_{A_{\rm f}}]
L^{1+\varepsilon} \varphi, \tilde A\wedge \varphi\rangle + \\
&&\hspace*{-6cm}
+\langle \tilde A\wedge L^{1+\varepsilon} \varphi, [d_{A_{\rm f}},
\partial_L]\varphi\rangle + \langle d_{A_{\rm f}}
L^{1+\varepsilon}\varphi, [\tilde A\wedge, \partial_L]\rangle+
\langle[\partial_L, \tilde A\wedge] L^{1+\varepsilon}\varphi,
d_{A_{\rm f}}\varphi\rangle, \end{aligned}$$ Once $d_{A_{\rm f}}$ commuted with $L^{1+\varepsilon}$, the two last terms are controlled by the assumption on $\|[\partial_L, \tilde A\wedge]
L^{1+\varepsilon}\|$. The two first ones are $0$ for $p=1$ using . When $p\pp 1$, note that $x^{1-p}L^{1+\varepsilon }=c
L^{\varepsilon}$ and control the term using $L^\varepsilon \tilde A$ bounded.
We now describe the perturbation of the metric following the two classes. We keep the notation from Theorem \[t:mourre0\]. We introduce the canonical unitary transformation due to the change of measure. Set $\rho$ to be $\rho_{\rm sr}$, $\rho_{\rm lr}$ or $\rho_{\rm t}$. Let $U$ be the operator of multiplication by $(1+\rho)^{-n/4}$ in $L^2(X,
g)$ and $V$ the operator of multiplication by $(1+\rho)^{(2-n)/4}$ in $L^2(X,T^*X, g)$. The operator $U$ is a unitary operator from $ L^2(X,
g)$ onto $L^2(X, \tilde g)$ and $V$ a unitary operator from $L^2(X,
T^*X, g)$ onto $L^2(X, T^*X, \tilde g)$.
\[l:uni\] Let $\widetilde\Delta_{A_{\rm f}}$ be the magnetic Laplacian of vector potential $A_{\rm f}$ acting in $L^2(X, \tilde g)$. Let $W_0:= U^{-1}\widetilde\Delta_{A_{\rm f}} U$. Then,
1. On ${\mathcal{C}}_c^{\infty}(X)$, $W_0$ acts by $U d_{A_{\rm f}}^* V^2
d_{A_{\rm f}} U$. In $L^2(X,g)$, it is essentially self-adjoint and its domain is ${\mathcal{D}}(\Delta_{A_{\rm f}})$.
2. One has $(W_0- \Delta_{A_{\rm f}} +i)^{-1}- (\Delta_{A_{\rm
f}}+i)^{-1}$ is compact.
3. For $\rho_{\rm sr}$ and $\rho_{\rm t}$, $W_0$ is a short-range perturbation of the magnetic Laplacian $\Delta_{A_{\rm f}}$ in the space ${\mathcal{B}}({\mathcal{D}}(\Delta_{A_{\rm f}}), {\mathcal{D}}(\Delta_{A_{\rm f}})^*)$.
4. For $\rho_{\rm lr}$, $W_0$ is a long-range perturbation of $\Delta_{A_{\rm f}}$ in ${\mathcal{B}}({\mathcal{D}}(\Delta_{A_{\rm f}}),
{\mathcal{D}}(\Delta_{A_{\rm f}})^*)$.
We write $\widetilde\Delta_{A_{\rm f}}$ with the help of the operator $d_{A_{\rm f}}$. Since the manifold is complete, $\widetilde\Delta_{A_{\rm f}}$ is essentially self-adjoint on ${\mathcal{C}}^{\infty}_c(X)$. In particular it corresponds to $\tilde
d_{A_{\rm f}}^*\tilde d_{A_{\rm f}}$, the Friedrichs extension. This is equal to $U^2 d_{A_{\rm f}}^* V^2 d_{A_{\rm f}}$ in $L^2(X,
\tilde g)$. Now remark that $(1+\rho)^\alpha$ stabilizes ${\mathcal{D}}(\Delta_{A_{\rm f}})$, ${\mathcal{D}}(d_{A_{\rm f}})$ and ${\mathcal{D}}(d_{A_{\rm
f}}^*)$, for all $\alpha\in{\mathbb{R}}$ to obtain the first point.
We now compare the two operators in $L^2(X, g)$. We compute on ${\mathcal{C}}^\infty_c(X)$. $$\label{e:D}\begin{split}
D:=W_0 - \Delta_{A_{\rm f}} &= U^{-1}
U^2 d_{A_{\rm f}}^* V^2 d_{A_{\rm f}} U - d_{A_{\rm f}}^* d_{A_{\rm f}}
\\
&=
U^{-1}\widetilde\Delta_{A_{\rm f}}(U-1) +
U d_{A_{\rm f}}^* (V^2-1) d_{A_{\rm f}}+
(U-1)\Delta_{A_{\rm f}}.
\end{split}$$ We focus on point $(3)$. The two first terms need a justification. We start with the first term. $$L^{1+\varepsilon} U^{-1} \widetilde\Delta_{A_{\rm f}}(U-1)=
U^{-1} \big(\widetilde\Delta_{A_{\rm f}} +
[L^{1+\varepsilon}, \widetilde\Delta_{A_{\rm f}}]L^{-1-\varepsilon}\big)
L^{1+\varepsilon}(U-1).$$ Using Lemma \[l:L\] and the invariance of the domain under $(1+\rho)^\alpha$, we obtain that $(\widetilde\Delta_{A_{\rm f}} +
[L^{1+\varepsilon}, \widetilde\Delta_{A_{\rm f}}]L^{-\varepsilon})^*$ is bounded from ${\mathcal{D}}(\Delta_{A_{\rm f}})$ to $L^2(X, g)$. Again using properties of $(1+\rho)^\alpha$, for all $\varphi\in{\mathcal{C}}^\infty_c(X)$ we get $$\begin{aligned}
\lefteqn{|\langle \varphi, L^{1+ \varepsilon}
U^{-1}\widetilde\Delta_{A_{\rm f}}(U-1) \varphi \rangle |}\\&&&&=
|\langle (\widetilde\Delta_{A_{\rm f}} +
[L^{1+\varepsilon}, \widetilde\Delta_{A_{\rm f}}]L^{-1-\varepsilon})^*
U^{-1}\varphi, L^{1+\varepsilon }(U-1)\varphi \rangle |
\leq c \|(\Delta_{A_{\rm f}}+i) \varphi\|^2.\end{aligned}$$ For the second term, we have $L^{1+ \varepsilon }\big(U
d_{A_{\rm f}}^* (V^2-1) d_{A_{\rm f}}\big)=U \big( d_{A_{\rm f}}^* +[L^{1+
\varepsilon}, d_{A_{\rm f}}^*] L^{-1 - \varepsilon }\big)$ $L^{1+
\varepsilon } (V^2-1) d_{A_{\rm f}}$. By Lemma \[l:L\] and the invariance of ${\mathcal{D}}(d_{A_{\rm f}})$ by $\rho^\alpha$, we obtain $$\begin{aligned}
\lefteqn{|\langle \varphi, L^{1+ \varepsilon }\big(U d_{A_{\rm f}}^* (V^2-1)
d_{A_{\rm f}}\big)
\varphi\rangle|}\\&&&&=
|\langle
\big( d_{A_{\rm f}} + L^{-1 - \varepsilon } [d_{A_{\rm f}}, L^{1+
\varepsilon}]\big)U \varphi, L^{1+
\varepsilon } (V^2-1) d_{A_{\rm f}}\varphi\rangle|\leq c \|(\Delta_{A_{\rm f}}+i) \varphi\|^2, \end{aligned}$$ for all $\varphi\in{\mathcal{C}}^\infty_c(X)$. To finish, use the fact that ${\mathcal{C}}^\infty_c(X)$ is a core for $\Delta_{A_{\rm f}}$.
We now deal with $(4)$ by checking . The real point to check is that $\|L^{1+\varepsilon }\tilde\Xi[D,
\partial_L]P_0\tilde\Xi\|_{{\mathcal{B}}({\mathcal{D}}(\Delta), {\mathcal{D}}(\Delta)^*)}$ is finite. We take $\partial_L$ like in the proof of Lemma \[l:magne\]. First, $$\label{e:comm2}
[d^*d, \partial_L]P_0= c(1-p)x^{3(1-p)}P_0.$$ We start with the easy part of $D$. We drop $\tilde\Xi$ and $P_0$ for clarity. We have: $$\begin{aligned}
\langle L^{1+\varepsilon }\varphi, [(U-1)d^*d, \partial_L] \varphi \rangle=
\langle L^{1+\varepsilon}\varphi, [U, \partial_L] d^*d \varphi
\rangle+ \langle L^{1+\varepsilon}(U-1)\varphi, [ d^*d, \partial_L]
\varphi \rangle. \end{aligned}$$ This is bounded by $\|(\Delta +i)\varphi\|^2$. Indeed, the first term follows since $L^{1+\varepsilon}[U, \partial_L]$ is bounded in $L^2(X,g)$. The second one is $0$ for $p=1$ and equals otherwise to $c
\langle \varphi, L^{\varepsilon -2}(U-1) \varphi\rangle$ by . Turn now to: $$\begin{aligned}
\lefteqn{\langle L^{1+\varepsilon}\varphi, [Ud^*(V^2-1)d,
\partial_L]\varphi\rangle}\\&
=&\langle dU L^{1+\varepsilon}\varphi, [V^2, \partial_L]d\varphi\rangle
+\langle [U, \partial_L] L^{1+\varepsilon}\varphi, d^*(V^2-1)d\varphi\rangle
\\&&
+\langle (V^2-1)dU L^{1+\varepsilon}\varphi, [d, \partial_L]
\varphi\rangle + \langle [d, \partial_L] U L^{1+\varepsilon}\varphi,
(V^2-1)d\varphi \rangle. \end{aligned}$$ The first is controlled by commuting $L^{1+\varepsilon }$ with $d$ like above and by using that $L^{1+\varepsilon }[V^2, \partial_L]$ is bounded in $L^2(X, \Lambda^1, g)$. For the second one, $[U, \partial_L] L^{1+\varepsilon}$ is bounded in $L^2(X, \Lambda^1, g)$. Turn the two last ones and use , for $p=1$ this is $0$. Focus on the very last one for example. Now commute $L^{1+\varepsilon}$ with $d$ like above. The most singular term being $\langle d U \varphi,
x^{1-p}L^{1+\varepsilon}(V^2-1) d\varphi\rangle$. Now remember that $x^{1-p}L^{1+\varepsilon }=c L^{\varepsilon}$ and use the fact that $L^\varepsilon (V^2-1)$ is bounded to control it. To conclude, repeat the same arguments for $[U^{-1}\tilde\Delta(U-1), \partial_L]$.
We turn to point (2), $W_0$ and $\Delta_{A_{\rm f}}$ have the same domain. We take the proof of Lemma \[l:pertuA\] replacing ${\mathcal{G}}$ with this domain. We then obtain a rigorous version of . Therefore, it remains to check that $W_0-\Delta_{A_{\rm f}}\in{\mathcal{K}}({\mathcal{D}}(\Delta_{A_{\rm f}}),
{\mathcal{D}}(\Delta_{A_{\rm f}})^*)$. This comes directly using Rellich-Kondrakov Lemma and .
Finally, we gather various technicalities concerning the operator $L$.
\[l:L\] We have that $dL$ is with support in $(0, \varepsilon)\times M_{\rm nt}$ and $dL=
f(x)dx$ where $f:(0, \varepsilon )\rightarrow {\mathbb{R}}$ such that $f$ is $0$ in a neighborhood of $\varepsilon$ and such that $f(x)= -x^{p-2}$ for $x$ small enough. Moreover:
1. The operator $L^{-\varepsilon }d(L^{1+\varepsilon })\wedge$ belongs to ${\mathcal{B}}\big(L^2(X, g), L^2(X,T^*X, g)\big)$ and the commutator $L^{-\varepsilon }[\Delta_{A_{\rm f}}, L^{1+
\varepsilon}]$ with initial domain ${\mathcal{C}}^\infty_c(X)$ extends to a bounded operator in ${\mathcal{B}}\big( {\mathcal{D}}(\Delta_{A_{\rm f}}), L^2(X, g)\big)$.
2. $e^{itL}{\mathcal{D}}(\Delta_{A_{\rm f}})\subset {\mathcal{D}}(\Delta_{A_{\rm f}})$ and $\|e^{itL}\|_{{\mathcal{B}}({\mathcal{D}}(\Delta_{A_{\rm f}}))}\leq c
(1+t^2)$.
3. $L^{-1-\varepsilon }{\mathcal{D}}(\Delta_{A_{\rm f}})\subset
{\mathcal{D}}(\Delta_{A_{\rm f}})$.
With the diagonalization of Section \[s:diago\], the operator $\Delta_{A_{\rm f}}$ is given by . The operator $L$ corresponds to the operator $L_0$ of multiplication by $r\otimes 1_{M_{\rm nt}}$ on $(c, \infty)$ in this variable and by $1$ on the rest of the manifold. Hence, points (1) and (3) are easily obtained. Moreover $e^{itL_0}/(1+t^2)$ and its first and second derivative belong to $L^2(X_0)$, uniformly in $t$, from which (2) follows.
The non-trapping case for perturbed metrics {#s:ntp}
===========================================
The essential spectrum
----------------------
In this section, we compute the essential spectrum of a magnetic Laplacian given by a non-trapping vector potential. Unlike the trapping case, it is non-empty in the complete case. To show this, we apply perturbation techniques to the results of the previous section.
We restrict ourselves to conformal perturbations of *exact* cusp metrics. In a fixed product decomposition of $X$ near $M$ we rewrite as $$\label{cume2}
g_0=a \left(\frac{dx}{x^2}+\alpha(x)\right)^2+h(x)$$ where $a\in{{\mathcal{C}}^{\infty}}({\overline{X}})$, $\alpha\in{{\mathcal{C}}^{\infty}}([0,\varepsilon)\times M,\Lambda^1(M))$ and $h\in {{\mathcal{C}}^{\infty}}([0,\varepsilon)\times M, S^2TM)$. By [@wlom Lemma 6], the function $a_0:=a(0)$, the metric $h_0:=h(0)$ and the class (modulo exact forms) of the $1$-form $\alpha_0:=\alpha(0)$, defined on $M$, are independent of the chosen product decomposition and of the boundary-defining function $x$ inside the fixed cusp structure.
\[defex\] The metric $g_0$ is called *exact* if $a_0=1$ and $\alpha_0$ is an exact $1$-form.
If $\alpha_0=df$ is exact, then by replacing $x$ with the boundary-defining function $x'=x/(1+xf)$ inside the same cusp structure, we can as well assume that $\alpha_0=0$ (see [@wlom]). It follows that $g_0$ is quasi-isometric to a cylindrical metric near infinity.
\[p:thema\] Let $(X,\tilde g_p)$ be a Riemannian manifold with a conformal *exact* cusp metric $\tilde g_p:=(1+\rho)g_p$, where $g_0$ is exact, $g_p=x^{2p}g_0$ and $$\rho\in L^\infty(X; {\mathbb{R}}), \quad \inf_{x\in X}\rho(x)\pg -1, \quad
\rho(x)\rightarrow 0,
\mbox{ as } x\rightarrow 0.$$ Let $A$ be a non-trapping vector potential given by . Then
- For $0<p\leq 1$, the Friedrichs extension of $\Delta_A$ has essential spectrum $ {\sigma_{\mathrm{ess}}}(\Delta_A)= [\kappa(p), \infty)$, where $\kappa(p)=0$ for $p<1$ and $\kappa(1)=(n-1)/2$. Moreover, if $\rho$ is smooth, the $\Delta_A$ is essential self-adjoint on ${\mathcal{C}}^\infty_c(X)$.
- If $p>1$ and $\tilde g_p:=g_p$ is the unperturbed metric given in , then every self-adjoint extension of $\Delta_A$ has empty essential spectrum.
Note that for $p>1$, the unperturbed metric $g_p$ given in is essentially of *metric horn* type [@lepe].
Using the Weyl theorem, Lemma \[l:pertuA\] and by changing the gauge, we can suppose without loss of generality that $A$ is of the form with $C=0$.
We start with the complete case. The essential self-adjointness follows from [@shubin]. In the exact case, $g_p$ is quasi-isometric to the metric . Using [@GG Theorem 9.4] (see Theorem 9.5 for the case of the Laplacian), to compute the essential spectrum we may replace $h(x)$ in by the metric $h_0:=h(0)$ on $M$, extended to a symmetric $2$-tensor constant in $x$ near $M$, and we may set $\rho=0$. By Proposition \[p:cut\], computing ${\sigma_{\mathrm{ess}}}(\Delta_A)$ is the same as computing ${\sigma_{\mathrm{ess}}}(\Delta^l_A)$ on $X'$ of Proposition \[p:free’\]. By the results of Section \[s:diago\], this is given on $L^2({\mathbb{R}}^+)$ by ${\sigma_{\mathrm{ess}}}(-\Delta+V_p)
=[V_p(\infty), \infty)$.
Let now $p> 1$. The metric is no longer complete and $(X,g_p)$ is not proper; one can not apply [@GG Theorem 9.4]. By Lemma \[l:indices\] and by the Krein formula, all self-adjoint extensions have the same essential spectrum. So it is enough to consider the Friedrichs extension of $\Delta_A$. We now use Propositions \[p:free’\] and \[p:cut\]. The operator $D^*D$ is non-negative, so the spectrum of $\Delta_A^0$ is contained in $[\varepsilon^{2-2p}c_0,\infty)$. By Proposition \[p:cut\], the essential spectrum does not depend on the choice of $\varepsilon$. Now we remark that $p>1$ implies $c_0\neq 0$. Indeed, the equality would imply that $1/p\in{\mathbb{Z}}$, which is impossible. Thus by letting $\varepsilon \to 0$ we conclude that the essential spectrum is empty.
We have used above a general lemma about compact perturbations of magnetic Laplacians:
\[l:pertuA\] Let $A$ and $A'$ in $L^\infty(X, T^*X)$ be two magnetic fields on a smooth Riemannian manifold $(X,g)$ (possibly incomplete) with a measurable metric $g$. Suppose that $A-A'$ belongs to $L^\infty_0(X,
T^*X)$. Let $\Delta_A={d_A}^*d_A$ and $\Delta_{A'}={d_{A'}}^*d_{A'}$. Then $(\Delta_A+i)^{-1}- (\Delta_{A'}
+i)^{-1}$ is compact.
Here, $L^\infty_0(X, T^*X)$ denotes the space of those forms of $L^\infty(X,T^*X)$ which are norm limit of compactly supported forms. Note that this lemma holds without any modification for a $C^1$ manifold equipped with a (RM) structure, see [@GG section 9.3]. Unlike the result on the stability of the essential spectrum of the (magnetic) Laplacian from [@GG Theorem 9.5], we do not ask for the completeness of the manifold. A magnetic perturbation is much less singular than a perturbation of the metric. Note that the form domain of $\Delta_A$ and of $\Delta_A$ is given by ${\rond{G}}:={\mathcal{D}}(d)$, because $A$ and $A'$ are in $L^\infty(X,
T^*X)$. We write with a tilde the extension of the magnetic Laplacians to ${\mathcal{B}}({\rond{G}}, {\rond{G}}^*)$. We aim to give a rigorous meaning to $$\label{33}
(\Delta_A+i)^{-1}-(\Delta_{A'}+i)^{-1}=(\Delta_A+i)^{-1}(\Delta_{A'}-\Delta_A) (\Delta_{A'}+i)^{-1}.$$ We have $(\Delta_A+i)^{-1*}{\rond{H}}\subset{\rond{G}}$. This allows one to deduce that ($\Delta_A+i)^{-1}$ extends to a unique continuous operator ${\rond{G}}^*\rightarrow{\rond{H}}$. We denote it for the moment by $R$. From $R(\Delta_A+i)u=u$ for $u\in {\mathcal{D}}(\Delta_A)$ we get, by density of ${\mathcal{D}}(\Delta_A)$ in ${\rond{G}}$ and continuity, $R(\widetilde{\Delta_A}+i)u=u$ for $u\in{\rond{G}}$, in particular $$(\Delta_{A'}+i)^{-1}= R(\widetilde{\Delta_A}+i)(\Delta_{A'}+i)^{-1}.$$ Clearly, $$(\Delta_A+i)^{-1}= (\Delta_A+i)^{-1}(\Delta_{A'}+i)(\Delta_{A'}+i)^{-1}=
R(\widetilde{\Delta_{A'}}+i)(\Delta_{A'}+i)^{-1}.$$ We subtract the last two relations to get $$(\Delta_A+i)^{-1}-(\Delta_{A'}+i)^{-1}=R(\widetilde{\Delta_{A'}}
-\widetilde{\Delta_A})(\Delta_{A'}+i)^{-1}$$ Since $R$ is uniquely determined as the extension of $(\Delta_A+i)^{-1}$ to a continuous map ${\rond{G}}^*\rightarrow{\rond{H}}$, one may keep the notation $(\Delta_A+i)^{-1}$ for it. With this convention, the rigorous version of (\[33\]) that we shall use is: $$\label{34}
(\Delta_A+i)^{-1}-(\Delta_{A'}+i)^{-1}=(\Delta_A+i)^{-1}(\widetilde
\Delta_{A'}-\widetilde \Delta_A)(\Delta_{A'}+i)^{-1}.$$ Since $A''=i( A-A')\in L^\infty_0(X, T^*X)$, the Rellich-Kondrakov lemma gives that $A''\wedge$ belongs to ${\mathcal{K}}({\rond{G}}, L^2(X, T^*X))$. Therefore, $\widetilde\Delta_{A'}-\widetilde\Delta_{A} = (A''\wedge)^* d_{A} -
d_{A}^*A''\wedge\in {\mathcal{K}}({\rond{G}},{\rond{G}}^*)$. This gives the announced compactness.
The spectral and scattering theory
----------------------------------
In this section, we refine the study of the essential spectrum given in Proposition \[p:thema\] for non-trapping vector potential. As the essential spectrum arises only in the complete case, we will suppose that $p\leq 1$.
We give below our main result in the study of the nature of the essential spectrum and in scattering theory under short-range perturbation. It is a consequence of the Mourre theory [@mou] with an improvement for the regularity of the boundary value of the resolvent, see [@ggm] and references therein.
We treat some conformal perturbation of the metric $\eqref{gp'}$. To our knowledge, this is the weakest hypothesis of perturbation of a metric obtained so far using Mourre theory. Compared to previous approaches, we use a conjugate operator which is local in energy and therefore can be compared directly to a multiplication operator. We believe that this procedure can be implemented to all known Mourre estimates on manifolds to improve the results obtained by perturbation of the metric.
We fix $A_{\rm f}$ a non-trapping vector potential of the form . By a change of gauge, one can suppose that $A_{\rm f}=\theta_0$ which is constant in a neighborhood of $M$. Let $M_{\rm t}$ (resp. $M_{\rm nt}$) be the union of the connected components of $M$ on which $A_{\rm f}$ is trapping (resp. non-trapping). Let $L$ be the operator of multiplication by a smooth function $L\geq 1$ which measures the length of a geodesic going to infinity in the directions where $\theta_0$ is non-trapping: $$\begin{aligned}
\label{e:L}
L(x)=\begin{cases}
-\ln(x)& \text{for $p=1$},\\
-\frac{x^{p-1}}{p-1}& \text{for $p< 1$}
\end{cases}\end{aligned}$$ on $(0, \varepsilon/4)\times M_{\rm nt}$ for small $x$, and $L=1$ on the trapping part $\big((0, \varepsilon /2)\times M_{\rm nt} \big)^c$. Given $s\geq 0$, let ${\rond{L}}_{s}$ be the domain of $L^s$ equipped with the graph norm. We set ${\rond{L}}_{-s}:={\rond{L}}_{s}^*$. Using the Riesz theorem, we obtain the scale of spaces ${\rond{L}}_{s}\subset L^2(X, \tilde g) \subset {\rond{L}}_{s}^*$, with dense embeddings and where $\tilde g$ is defined in the theorem below. Given a subset $I$ of ${\mathbb{R}}$, let $I_{\pm}$ be the set of complex number $x\pm iy$, where $x\in I$ and $y>0$. The thresholds $\{\kappa(p)\}$ are given in Proposition \[p:thema\].
For shorthand, perturbations of *short-range* type (resp.*trapping* type) are denoted with the subscript ${\rm sr}$ (resp. ${\rm t}$); they are supported in $(0, \varepsilon)\times
M_{\rm nt}$ (resp. in $((0, \varepsilon/2)\times M_{\rm nt})^c$). We stress that the class of “trapping type” perturbations is also of short-range nature, in the sense described in Section \[s:srlr\], even if no decay is required. This is a rather amusing phenomenon, linked to the fact that no essential spectrum arises from the trapping cusps. The subscript $\rm lr$ denotes *long-range* type perturbations, also with support in $(0, \varepsilon)\times M_{\rm nt}$. We ask such perturbations to be *radial*, i.e. independent of the variables in $M$. In other words, a perturbation $W_{\rm lr}$ satisfies $W_{\rm lr}(x,m)= W_{\rm
lr}(x,m')$ for all $m,m'\in M$.
\[t:mourre0\] Fix $\varepsilon >0$. Let $X$ be endowed with the metric $\tilde g =
(1+\rho_{\rm sr}+\rho_{\rm lr}+\rho_{\rm t}) g$, where $g=g_p$ is given in for some $0<p\leq 1$; $\rho_{\rm sr}$, $\rho_{\rm lr}$ and $\rho_{\rm t}$ belong to ${\mathcal{C}}^\infty(X)$, such that $$\begin{aligned}
\inf_{x\in X}(\rho_{\rm sr}(x)+ \rho_{\rm lr}(x) + \rho_{\rm
t}(x))\pg -1,&& \rho_{\rm t}(x) = o(1) \text{ as }
x\rightarrow 0\end{aligned}$$ and such that $$\begin{aligned}
L^{1+\varepsilon}\rho_{\rm sr},
d_{A_{\rm f}}\rho_{\rm sr},
\Delta_{A_{\rm f}}\rho_{\rm sr},
L^{\varepsilon}\rho_{\rm lr},
L^{1+\varepsilon}d_{A_{\rm f}}\rho_{\rm lr},
\Delta_{A_{\rm f}}\rho_{\rm lr},
d_{A_{\rm f}}\rho_{\rm t},
\Delta_{A_{\rm f}}\rho_{\rm t} \text{ belong to $L^\infty$.}\end{aligned}$$ In ${\rond{H}}= L^2(X, \tilde g)$, let $\widetilde\Delta_A$ be the magnetic Laplacian with a non trapping potential $A= A_{\rm f} + A_{\rm lr } + A_{\rm sr}+ A_{\rm t}$, where $A_{\rm f}$ is as in , $A_{\rm sr}$, $A_{\rm lr}$ and $A_{\rm t}$ are in ${\mathcal{C}}^\infty(X, T^*X)$ such that: $$\begin{aligned}
\|L^{1+\varepsilon}A_{\rm sr}\|_\infty,\,
\|L^{\varepsilon}A_{\rm lr}\|_\infty,\,
\|L^{1+\varepsilon}{\mathcal{L}}_{x^{(2-p)}\partial_{x}} A_{\rm lr}\|_\infty\pp
\infty \mbox{ and } A_{\rm t}= o(1), \end{aligned}$$ where ${\mathcal{L}}$ denotes the Lie derivative. Let $V= V_{\rm loc}+ V_{\rm sr}+ V_{\rm lr} + V_{\rm t}$ and $V_{\rm
lr}$ be some potentials, where $V_{\rm loc}$ is measurable with compact support and $\widetilde\Delta_{A_{\rm f}}$-compact and $V_{\rm sr}$, $V_{\rm lr}$ and $V_{\rm t}$ are in $L^\infty(X)$ such that: $$\|L^{1+\varepsilon}V_{\rm sr}\|_\infty,\,
\|L^{1+\varepsilon}d_{A_{\rm f}}V_{\rm lr}\|_\infty<\infty
\mbox{ and } V_{\rm t}=o(1).$$ Consider the magnetic Schrödinger operators $H_0=
\widetilde\Delta_{A_{\rm f}+A_{\rm lr}}+V_{\rm lr}$ and $H=
\widetilde\Delta_{A} + V$. Then
1. $H$ has no singular continuous spectrum.
2. The eigenvalues of ${\mathbb{R}}\setminus \{\kappa(p)\}$ have finite multiplicity and no accumulation points outside $\{\kappa(p)\}$.
3. Let ${\mathcal{J}}$ a compact interval such that ${\mathcal{J}}\cap \big(\{\kappa(p)\}\cup
\sigma_{\rm pp}(H)\big)=\emptyset$. Then, for all $s\in (1/2, 3/2)$, there exists $c$ such that $$\|(H-z_1)^{-1} - (H-z_2)^{-1} \|_{{\mathcal{B}}({\rond{L}}_s, {\rond{L}}_{-s})} \leq c
\|z_1-z_2\|^{s-1/2},$$ for all $z_1, z_2 \in {\mathcal{J}}_{\pm}$.
4. Let ${\mathcal{J}}={\mathbb{R}}\setminus \{\kappa(p)\}$ and let $E_0$ and $E$ be the continuous spectral component of $H_0$ and $H$, respectively. Then, the wave operators defined as the strong limit $$\Omega_{\pm}={\mathrm{s-}\!\lim}_{t\rightarrow \pm \infty} e^{itH}e^{-itH_0}E_0({\mathcal{J}})$$ exist and are complete, i.e. $\Omega_{\pm} {\rond{H}}= E({\mathcal{J}}){\rond{H}}$.
Any smooth $1$-form $A$ on ${\overline{X}}$ is a short-range perturbation of a free vector potential $A_{\rm f}$ as in .
If one is interested only in the free metric $g_p$, the conclusions of the theorem hold for $V_{\rm loc}$ in the wider class of $\widetilde\Delta_{A_{\rm f}}$-form compact perturbations, using [@ABG Theorem 7.5.4]. Similar results should hold for a smooth metric on ${\overline{X}}$ and Dirichlet boundary conditions, as in [@bouclet].
We start with $\Delta_{A_{\rm f}}$ in $L^2(X, g)$. In section \[s:conjop\], we transform it unitarily into $\Delta_0$ given by . For $R$ finite, we construct a conjugate operator $S_R$ to $\Delta_0$ given by . Theorem \[t:mourre\_0\] gives a Mourre estimate for $\Delta_0$ and the regularity of $\Delta_0$ compared to the self-adjoint operator $S_R$. We go back by unitary transform into $L^2(X,
g)$. Since the dependence on $R$ is no longer important, we denote simply by $S$ the image of the conjugate operator $S_R$. Therefore, we have $\Delta_{A_{\rm f}}\in {\mathcal{C}}^{2}(S, {\mathcal{D}}(\Delta_{A_{\rm f}}), L^2(X,
g))$ and given ${\mathcal{J}}$ an open interval included in $\sigma_{\rm
ess}(H_0)$, there is $c>0$ and a compact operator $K$ such that the inequality $$\label{e:T}
E_{\mathcal{J}}(T) [T,iS] E_{\mathcal{J}}(T) \geq c E_{\mathcal{J}}(T)+ K$$ holds in the sense of forms in $L^2(X,g)$, for $T=\Delta_{A_{\rm f}}$.
Let $W_0$ be the unitary conjugate of $\widetilde\Delta_{A_{\rm f}}$ acting in $L^2(X,g)$. By Lemma \[l:uni\], $W_0\in{\mathcal{C}}^{1,1}(S,{\mathcal{D}}(\Delta_{A_{\rm f}}), {\mathcal{D}}(\Delta_{A_{\rm f}})^*)$ is a sum of short and long-range perturbation as described in Section \[s:srlr\]. In particular, we get $W_0\in{\mathcal{C}}^{1}_{\rm u}(S,{\mathcal{D}}(\Delta_{A_{\rm f}}),
{\mathcal{D}}(\Delta_{A_{\rm f}})^*)$. By the point (2) of Lemma \[l:uni\] and [@ABG Theorem 7.2.9] the inequality holds for $T$ (up to changing $c$ and $K$).
We now go into $L^2(X, \tilde g)$ using $U$ defined before Lemma \[l:uni\]. We write the conjugate operator obtained in this way by $\tilde S$. Therefore, $\tilde\Delta_{A_{\rm f}}$ belongs to ${\mathcal{C}}^{1,1}(\tilde
S, {\mathcal{D}}(\Delta_{\tilde A_{\rm f}}), {\mathcal{D}}(\Delta_{\tilde A_{\rm f}})^*)$ and given ${\mathcal{J}}$ an open interval included in $\sigma_{\rm ess}(H_0)$, there is $c>0$ and a compact operator $K$ such that $$\label{e:tildeT}
E_{\mathcal{J}}(\tilde T) [\tilde T,i\tilde S] E_{\mathcal{J}}(\tilde T) \geq c E_{\mathcal{J}}(\tilde T)+ K$$ holds in the sense of forms in $L^2(X, \tilde g)$ for $\tilde T=\tilde
\Delta_{A_{\rm f}}$. We now add the perturbation given by $A_{\rm sr}, A_{\rm lr}, A_{\rm t}, V_{\rm sr}, V_{\rm lr},$ and $V_{\rm t}$. Note that $H$ has the same domain as $H_0$ and that $(H+i)^{-1}- (H_0+i)^{-1}$ is compact by Rellich-Kondrakov lemma and Lemma \[l:pertuA\]. By Lemma \[l:magne\], we obtain $H\in{\mathcal{C}}^{1,1}(S,{\mathcal{D}}(H),{\mathcal{D}}(H)^*)$. As above, the inequality \[e:tildeT\] is true for $\tilde T=H$.
We now deduce the different claims of the theorem. The first comes from [@ABG Theorem 7.5.2]. The second ones is a consequence of the Virial theorem. For the third point first note that ${\rond{L}}_s\subset{\mathcal{D}}(|A|^s)$ for $s\in [0,2]$ by Lemma \[l:sa\] and use [@ggm] for instance (see references therein). Finally, the last point follows from [@ABG Theorem 7.6.11].
The non-stability of the essential spectrum and of the situation of limiting absorption principle {#s:nonstab}
=================================================================================================
In ${\mathbb{R}}^n$, with the flat metric, it is well-known that only the behavior of the magnetic field at infinity plays a rôle in the computation of the essential spectrum. Moreover, [@KS Theorem 4.1] states that the non-emptiness of the essential spectrum is preserved by the addition of a bounded magnetic field, even if it can become purely punctual. Concerning a compactly support magnetic field, the essential spectrum remains the same, see [@MS]. However, it is well-known that one obtains a long-range effect from it, in other words it acts on particles which have support away from it. In the case of ${\mathbb{R}}^n$ with a hole of some kind, this phenomena are of special physical interests and are related to the Aharonov-Bohm effect, see section \[s:AB\] and references therein.
In contrast with the Euclidean setting, Theorem \[t:thmag\] indicates that in general the essential spectrum may vanish under *compactly supported* perturbations of the magnetic field. In the next sections, we discuss this effect both with and without the hypothesis of gauge invariance, and we investigate the coupling constant effect.
The case $H^1(X)= 0$. {#s:inv}
---------------------
In this section, we assume gauge invariance. We first characterize trapping condition in terms of the magnetic field, see Definition \[def7\] for the case of a magnetic potential. We recall that if $H^1(X)= 0$, given a magnetic field $B$ the spectral properties of the magnetic Laplacian $\Delta_A$ will not depend on the choice of vector potential $A$ such that $dA=B$. Indeed, given $A,
A'$ such that $dA=dA'$, the operators $\Delta_A$ and $\Delta_{A'}$ are unitarily equivalent by a gauge transformation. Therefore, we denote the magnetic Laplacian by $\Delta_B$ and express the condition of being (non-)trapping in function of $B$.
Let $p>0$ and let $X$ be the interior a compact manifold $\overline{X}$ endowed with the metric $g_p$ given by . For simplicity, assume that $B$ is a smooth $2$-form on ${\overline{X}}$ such that its restriction to $X$ is exact. Then there exists $A\in
{{\mathcal{C}}^{\infty}}({\overline{X}},T^*{\overline{X}})$ such that $B=dA$ (since the cohomology of the de Rham complex on ${\overline{X}}$ equals the singular cohomology of ${\overline{X}}$, hence that of $X$). Let $$\begin{aligned}
M=\sqcup_{\alpha\in{\mathcal{A}}}M_\alpha\end{aligned}$$ be the decomposition of the boundary $M$ into its connected components. Set $${\mathcal{A}}_0:=\{\alpha\in{\mathcal{A}}; H^1(M_\alpha;{\mathbb{R}})=0\}.$$ For some ${\mathcal{B}}\subset {\mathcal{A}}$ set $M_{\mathcal{B}}=\sqcup_{\beta\in{\mathcal{B}}}M_\beta$ and consider the long exact cohomology sequence of the pair $({\overline{X}},M_{\mathcal{B}})$ with real coefficients: $$H^1({\overline{X}};{\mathbb{R}})\longrightarrow H^1(M_{\mathcal{B}};{\mathbb{R}})\stackrel{\partial}{\longrightarrow}
H^2({\overline{X}},M_{\mathcal{B}};{\mathbb{R}})\stackrel{i}{\longrightarrow} H^2({\overline{X}};{\mathbb{R}})$$ Since we assume that $H^1(X;{\mathbb{R}})=0$ it follows that the connecting map $\partial$ is injective. If $B$ vanishes under pull-back to $M_{\mathcal{B}}$ then (since it is exact on $X$) it defines a class in $H^2({\overline{X}},M_{\mathcal{B}};{\mathbb{R}})$ which vanishes under the map $i$, so it belongs to the image of the injection $\partial$. We denote by $[B]_\beta$ the component of $[B]$ inside $\partial H^1(M_\beta)\subset H^2({\overline{X}},M_{\mathcal{B}};{\mathbb{R}})$.
\[magtrap\] Assume $H^1(X)=0$. Let $B$ be a smooth exact $2$-form on ${\overline{X}}$. Denote by ${\mathcal{B}}$ the set of those $\alpha\in{\mathcal{A}}$ such that $B$ vanishes identically on $M_\alpha$. The field $B$ is called *trapping* if for each $\beta\in{\mathcal{B}}$, the component $[B]_\beta \in\partial H^1(M_\beta)\subset H^2({\overline{X}},M_{\mathcal{B}};{\mathbb{R}})$ is not integral, i.e., it does not live in the image of the map of multiplication by $2\pi$ $$H^2({\overline{X}},M_{\mathcal{B}};{\mathbb{Z}})\stackrel{2\pi\cdot}{\longrightarrow}H^2({\overline{X}},M_{\mathcal{B}};{\mathbb{R}}),$$ and *non-trapping* otherwise.
This definition is consistent with Definition \[d:intro\] when $M$ is connected. Note that if $B$ is trapping then ${\mathcal{B}}$ must contain the index set ${\mathcal{A}}_0$ defined above.
In order to apply Theorem \[t:mourre0\] and Theorem \[t:thmag\] we use the following lemma:
\[l:compsupp\]
1. \[compsupp1\] Let $A$ be a smooth vector potential on ${\overline{X}}$ such that $dA=0$ in a neighbourhood of $M=\partial X$. Then there exists a smooth vector potential $A'$, *constant in $x$* in a neighborhood of $M$, such that $A=A'$ on $M$ and $d(A-A')=0$.
2. \[compsupp2\] Assume that $H^1({\overline{X}},{\mathbb{R}})$ vanishes. Let $B$ be a trapping magnetic field on ${\overline{X}}$. Then every vector potential for $B$ will be trapping.
3. \[compsupp3\] Assume moreover that $H_1(X;{\mathbb{Z}})=0$. Let $B$ be non-trapping such that $\imath^*_M B=0$. Then every vector potential for $B$ will be non-trapping.
Recall that $\pi_1(X)=0\Longrightarrow H_1(X;{\mathbb{Z}})=0
\Longrightarrow H^1_{{\rm dR}}(X)=0$.
\[compsupp1\]) Let us show that one can choose $A$ to be *constant in $x$* near the boundary, in the sense that near $M$ it is the pull-back of a form from $M$ under the projection $\pi:[0,\varepsilon)\times M\to M$ for $\varepsilon$ small enough. Indeed, $A-\pi^* \imath_M^*A$ is closed on the cylinder $[0,\varepsilon)\times M$ and vanishes when pulled-back to $M$. Now $M$ is a deformation-retract of the above cylinder, so the map of restriction to $M$ induces an isomorphism in cohomology and thus the cohomology class $[A-\pi^* \imath_M^*A]\in H^1([0,\varepsilon)\times M)$ must be zero. Let $f\in{{\mathcal{C}}^{\infty}}({\overline{X}},{\mathbb{R}})$ be a primitive of this form for $x\leq \varepsilon/2$, then $A-df$ is the desired constant representative.
\[compsupp2\]) Consider the commutative diagram $$\label{cohoz}
\begin{CD}
H^{1}(M_{\mathcal{B}};{\mathbb{R}})@>{\partial}>>H^{2}({\overline{X}},M_{\mathcal{B}};{\mathbb{R}})@>>>H^2({\overline{X}},{\mathbb{R}})\\
@AA{2\pi\cdot}A@AA{2\pi\cdot}A@AA{2\pi\cdot}A\\
H^{1}(M_{\mathcal{B}};{\mathbb{Z}})@>{\partial}>>H^{2}({\overline{X}},M_{\mathcal{B}};{\mathbb{Z}})@>>>H^2({\overline{X}},{\mathbb{Z}})
\end{CD}$$ where the horizontal maps come form the long exact sequence of the pair $({\overline{X}},M)$ and the vertical maps are multiplication by $2\pi$. Let $B$ be trapping and choose a vector potential $A$ for $B$, smooth on ${\overline{X}}$. We claim that $A$ is trapping. Indeed, $A$ is not closed on the components $M_\alpha$, $\alpha\in{\mathcal{A}}\setminus{\mathcal{B}}$, while it is closed on $M_{\mathcal{B}}$. We note that $\partial[A_{|M_{\mathcal{B}}}]=[dA]=[B]$, so $\partial [A_{|M_\beta}]=[B]_\beta$. Assume that for some $\beta\in{\mathcal{B}}$, the class $[A_{|M_\beta}]$ were integral. Then using the first square from diagram , it would follow that $[B]_\beta$ was also integral, contradiction.
\[compsupp3\]) If $B$ is non-trapping, there exists $\beta\in{\mathcal{B}}$ and $b\in H^2({\overline{X}},M_{\mathcal{B}},{\mathbb{Z}})$ with $[B]_\beta=2\pi b$. The image of $[B]_\beta\in\partial H^1(M_\beta,{\mathbb{R}})$ in $H^2({\overline{X}},{\mathbb{R}})$ is zero, thus $b$ maps to a torsion element in $H^2({\overline{X}},{\mathbb{Z}})$. From $H_1(X,{\mathbb{Z}})=0$ we see using the universal coefficients theorem $$0\to {\mathrm{Ext}}(H_1({\overline{X}},{\mathbb{Z}}),{\mathbb{Z}})\to H^2({\overline{X}};{\mathbb{Z}})\to {\mathrm{Hom}}(H_2({\overline{X}},{\mathbb{Z}}),{\mathbb{Z}})\to 0$$ that $H^2({\overline{X}};{\mathbb{Z}})$ is torsion-free. Thus $b$ comes from some $a\in H^1(M_{\mathcal{B}},{\mathbb{Z}})$. By commutativity we have $\partial(2\pi a)=[B]_\beta$. Since $\partial$ is an injection, it follows that every vector potential smooth on ${\overline{X}}$ for $B$ will define an integral cohomology class on $M_\beta$, thus will be non-trapping.
A spectacular example is a compactly supported magnetic field which induces very strong long-range effects. If $H^1({\overline{X}};{\mathbb{Z}})=0$ and $B$ be is an exact $2$-form with compact support in $X$, then $B$ is maximal non-trapping (see Definition \[def7\]) if and only if its class in $H^2({\overline{X}},M;{\mathbb{R}})$ is integral. We will stress more on this aspect in the following section.
We can also construct compactly supported magnetic fields for which the consequences of Theorem \[t:thmag\] hold true. We summarize this fact in the next proposition and give an explicit construction in the proof.
\[p:contreex\] Let $X$ be the interior of a compact manifold ${\overline{X}}$ with boundary $M=\partial {\overline{X}}$, endowed with a conformally cusp metric $g_p$. Assume that $H^1(X)=0$ and $H^1(M_j)\neq 0$ for every connected component $M_j$ of the boundary. Then there exists a non-zero smooth magnetic field $B$ with *compact support* such that the essential spectrum of $\Delta_B$ is empty and such that for $p\geq 1/n$ the growth law of the eigenvalues does not depend on $B$ and is given by . Such fields $B$ are generic inside compactly supported magnetic fields.
We construct $A$ like in (\[aft\]) satisfying the hypotheses of Theorem \[t:thmag\]. We take $\varphi_0$ to be constant. Let $\psi\in{{\mathcal{C}}^{\infty}}([0, \varepsilon))$ be a cut-off function such that $\psi(x)=0$ for $x\in[3\varepsilon/4,\varepsilon)$ and $\psi(x)=1$ for $x\in[0, \varepsilon/2)$. Since $H^1(M_j)\neq 0$, there exists a closed $1$-form $\beta_j$ on $M$ which is not exact. Up to multiplying $\beta_j$ by a real constant, we can assume that the cohomology class $[\beta_j]\in H^1_{\mathrm{dR}}(M_j)$ does not belong to the image of $2\pi H^1(M_j;{\mathbb{Z}})\to H^1(M_j;{\mathbb{R}})\simeq
H^1_{\mathrm{dR}}(M_j)$. Let $\beta$ denote the form on $M$ which equals $\beta_j$ on $M_j$. Choose $A$ to be $\psi(x)\beta$ for $\varepsilon>x>0$ and extend it by $0$ to $X$. The magnetic field $B=dA=\psi'(x)dx\wedge\beta$ has compact support in $X$.
By Theorem \[t:thmag\], $\Delta_A$ has purely discrete spectrum with the Weyl asymptotic law eigenvalues independent of $B$.
The relative cohomology class $[B]$ lives in the direct sum $\oplus_j \partial H^1(M_j,{\mathbb{R}})\subset H^2({\overline{X}},M,{\mathbb{R}})$. The field $B$ is non-trapping if at least one of its components in this decomposition lives in the image of $H^1(M_j,{\mathbb{Z}})$. Since we assume all $H^1(M_j,{\mathbb{R}})$ to be nonzero, the space of non-trapping magnetic fields is a finite union of subspaces of codimension at least $1$.
We now show that the cohomological hypothesis about $X$ and $M$ can be satisfied in all dimensions greater than or equal to $2$, and different from $3$.
In dimension $2$, take $X={\mathbb{R}}^2$ endowed with the metric . Consider, for instance, the metric $r^{-2p}(dr^2+d\sigma^2)$ given in polar coordinates. Here $M$ is the circle at infinity and $x=1/r$ for large $r$. Thus $b_1(X)=0$ while $b_1(M)\neq 0$. The product of this manifold with a closed, connected, simply connected manifold $Y$ of dimension $k$ yields an example in dimension $2+k$ with the same properties. Indeed, by the Künneth formula, the first cohomology group of ${\mathbb{R}}^2\times Y$ vanishes, while $H^1(S^1\times Y)\simeq
H^1(S^1)={\mathbb{Z}}$. Clearly $k$ cannot be $1$ since the only closed manifold in dimension $1$ is the circle. Thus the dimension $3$ is actually exceptional.
For orientable $X$ of dimension $3$, the assumptions $H^1(X)=0$ and $H^1(M)\neq 0$ cannot be simultaneously fulfilled. Indeed, we have the following long exact sequence (valid actually regardless of the dimension of $X$) $$\label{les3}
H^1({\overline{X}})\stackrel{i_M}{\longrightarrow} H^1(M)\stackrel{\delta}{\longrightarrow} H^2({\overline{X}},M).$$ If $\dim(X)=3$, the spaces $H^1({\overline{X}})$ and $H^2({\overline{X}},M)$ are isomorphic by Poincaré duality, hence $H^1({\overline{X}})=0$ implies $H^2({\overline{X}},M)=0$ and so (by exactness) $H^1(M)=0$. It should be possible to build a non-orientable example in dimension $3$ such that one could apply Proposition \[p:contreex\] but we were not able to construct one.
We finally give an example not covered by Lemma \[l:compsupp\] but such that the conclusion of Theorem \[t:thmag\] holds. We considered so far magnetic fields on $\overline X$ with vector potential smooth on ${\overline{X}}$. One may consider also more singular magnetic fields arising from Proposition \[p:cuspduB\].
### Example
Let $X$ be any conformally cusp manifold, without any cohomological assumptions. Suppose that $B=df\wedge dx/x^2$ where $f$ is a function on $X$ smooth down to the boundary $M$ of $X$. Assume that $f$ is not constant on any connected component of $M$. Then the essential spectrum of the magnetic operator (which is well-defined by $B$ if $H^1(X)=0$) is empty. This follows from the fact that $A:=fdx/x^2$ is trapping.
Note that the pull-back to the border of the above magnetic field is zero.
The coupling constant effect {#s:coupling}
----------------------------
In flat Euclidean space it is shown in [@hn], under some technical hypotheses, that the spectrum has a limit as the coupling constant tends to infinity. In contrast, in the next example, we exhibit the creation of essential spectrum for periodic values of the coupling constant. We will focus on the properties of $\Delta_{gB}$ for some coupling constant $g\in{\mathbb{R}}$. In order to be able to exploit the two sides of this work we concentrate here on the metric . We assume that $M$ is *connected*, $H^1(X,{\mathbb{Z}})=0$ and $H^1(M)\neq 0$.
Given $B$ a magnetic potential with compact support, $g B$ is non-trapping if and only its class in $H^2({\overline{X}},M;{\mathbb{R}})$ is integral. Let $G_B$ be the discrete subgroup of those $g\in{\mathbb{R}}$ such that $g B$ is non-trapping. As this subgroup is possibly $\{0\}$, we start with some exact form $B$ which represents a nonzero cohomology class in $H^2({\overline{X}},M;{\mathbb{Z}})$; then by exactness of the relative cohomology long sequence, $[B]$ lives in the image of the injection $H^1(M;R)\stackrel{\partial}{\longrightarrow}H^2({\overline{X}},M;{\mathbb{R}})$. With these restrictions, $G_B$ is a non-zero discrete subgroup of ${\mathbb{Q}}$. Now we apply Theorem \[t:thmag\] for the trapping case and Theorem \[t:mourre0\] for the non-trapping case. We obtain that
1. For $g\in G_B$, the essential spectrum of $\Delta_{g B}$ is given by $[\kappa(p), \infty)$, where $\kappa(p)$ is defined in Proposition \[p:thema\]. The spectrum of $\Delta_{g B}$ has no singular continuous part and the eigenvalues of ${\mathbb{R}}\setminus \{\kappa(p)\}$ are of finite multiplicity and can accumulate only in $\kappa(p)$.
2. For $g\notin G_B$, the spectrum $\Delta_{g B}$ is discrete and if $p\geq 1/n$, the asymptotic of the eigenvalues depend neither on $g$ nor on $B$.
We now describe the long-range effect regarding the coupling constant. Take a state $\phi\in
L^2(X)$ such that $\phi$ is not an eigenvalue of the free Laplacian $\Delta_0$ and is located in a energy higher than $\kappa(p)$. Since the Fourier transform of an absolutely continuous measure (comparing to the Lebesgue measure) tends to $0$ at infinity, we obtain, for each $g\in G_{g B}$ and for each $\cchi$ operator of multiplication by the characteristic function of compact support that $\cchi e^{it\Delta_{g B}}\phi\rightarrow 0$ as $t\rightarrow \infty$. If one considers $\cchi$ being 1 above the support of the magnetic field, then after some time the norm of $\phi$ above this zone is arbitrary small. Classically, the particle stops interacting with the magnetic field. Let us denote by $\phi'$ the particle at this moment. Now, if we switch on the interaction with intensity as small as one desires one gets $g\notin G_{g B}$ and then the spectrum of $\Delta_{g B}$ is discrete. Therefore there exists $\cchi'$ operator of multiplication by the characteristic function of compact support such that $1/T
\int_{0}^{T} \|\cchi' e^{it\Delta_{g B}}\phi'\|^2 dt$ tends to a positive constant, as $T\rightarrow \infty$. The particle is caught by the magnetic field even thought they are far from being able to interact classically.
In other words, switching on the interaction of the magnetic field with compact support has destroyed the situation of limiting absorption principle. This is a strong long-range effect.
For the sake of utmost concreteness, take $X={\mathbb{R}}^2$ endowed with the metric $r^{-2p}(dr^2+d\theta^2)$ in polar coordinates, for $r$ big enough and $1\geq p>0$. The border $M$ is $S^1$ and $H^2(\overline{X}, M;{\mathbb{R}})\simeq {\mathbb{Z}}$. Then for every closed $2$-form $B$ with compact support and non-zero integral, the group $G_B$ defined above is non-zero.
The case $H^1(X)\neq 0$, the Aharonov-Bohm effect {#s:AB}
-------------------------------------------------
Gauge invariance does not hold in this case, so one expects some sort of Aharonov-Bohm effect [@AB]. Indeed, given two vectors potential arising from the same magnetic field, the associated magnetic Laplacians $\Delta_A$ and $\Delta_{A'}$ might be unitarily in-equivalent. The vector potential acquires therefore a certain physical meaning in this case.
In flat ${\mathbb{R}}^n$ with holes, some long-range effect appears, for instance in the scattering matrix like in [@R; @RY; @RY2; @Y], in an inverse-scattering problem [@N; @W] or in the semi-classical regime [@BR]. See also [@helffer] for the influence of the obstacle on the bottom of the spectrum.
In all the above cases the essential spectrum remains the same. In light of Proposition \[p:contreex\], one can expect a much stronger effect in our context. We now give some examples of magnetic fields with compact support such that there exists a non-trapping vector potential $A$, constant in $x$ in a neighborhood of $M$, and a trapping vector potential $A'$, such that $dA=dA'=B$. To ease the presentation, we stick to the metric . For $A$, one applies Theorem \[t:mourre0\] and obtain that the essential spectrum of $\Delta_{A}$ is given by $[\kappa(p), \infty)$, that $\Delta_{A}$ has no singular continuous part and the eigenvalues of ${\mathbb{R}}\setminus \{\kappa(p)\}$ have finite multiplicity and can accumulate only to $\{\kappa(p)\}$. For $A'$, one applies Theorem \[t:thmag\] to get the discreteness of the spectrum of $\Delta_{A'}$ and to obtain that the asymptotic of eigenvalues depends neither on $A$ nor on $B$ for $n\geq 1/p$. In the next section, we describe how generic this situation is for hyperbolic manifolds of dimension $2$ and $3$.
The easy step is to construct the non-trapping vector potential $A$ constant in a neighborhood of $M$, more precisely we construct $A$ to be *maximal* non-trapping, see Definition \[def7\]. Indeed, prescribe a closed $1$-form $\theta$ on $M$ defining integral cohomology $1$-classes on each component of the boundary, and extend it smoothly to $X$, constant in $x$ in a neighborhood on $M$, like in the proof of Proposition \[p:contreex\]. The magnetic field $B:=dA$ has then compact support and one may apply Theorem \[t:mourre0\] for $\Delta_A$. We now construct $A'$ by adding to $A$ a closed form $\alpha$, smooth on $\overline{X}$. Since $\alpha$ is closed, $A'$ and $A$ define the same magnetic field. In light of Remark \[addtrap\], $A'$ is trapping if and only if $\alpha$ is. We can then apply Theorem \[t:thmag\] to $\Delta_{A'}$. It remains to show that closed, trapping $\alpha$ do exist. We start with a concrete example.
Consider the manifold ${\overline{X}}=(S^1)^{n-1}\times [0,1]$ with a metric $g_p$ as in near the two boundary components. Let $\theta_i\in {\mathbb{R}}$ be variables on the torus $(S^1)^{n-1}$, so $e^{i\theta_i}\in S^1$. Take the vector potential $A$ to be $0$, it is (maximal) non-trapping. Choose now $\alpha=A'$ to be the closed form $\mu d\theta_1$ for some $\mu\in{\mathbb{R}}$. It is constant in a neighborhood of $(S^1)^{n-1}$. The class $[i_M^*(A')]$ is an integer multiple of $2\pi$ if and only if $\mu\in{\mathbb{Z}}$. In other words, $A'$ is non-trapping if and only if $\mu\in{\mathbb{R}}\setminus{\mathbb{Z}}$. Note that here the magnetic field $B$ vanishes.
In order to show the existence of such $\alpha$ in a more general setting, we assume that the first Betti number of each connected component of the boundary is non-zero. It is enough to find some closed $\alpha$, smooth on ${\overline{X}}$, which on each boundary component represents a non-zero cohomology class. Then, up to a multiplication by a constant, $\alpha$ will be trapping. When $X$ is orientable and $\dim(X)$ is $2$ or $3$, one proceeds as follows.
\[p23\] Let ${\overline{X}}$ be a compact manifold with non-empty boundary $M$. Assume that one of the following hypotheses holds:
1. $\dim(X)=2$ and $M$ is disconnected;
2. $\dim(X)=2$ and $X$ is non-orientable;
3. $\dim(X)=3$, $X$ is orientable and none of the connected components of $M$ are spheres.
Then there exists a closed smooth form $\alpha\in\Lambda^1({\overline{X}})$, constant in $x$ near the boundary, such that for all connected components $M_j$ of $M$, the class $[\alpha_{|M_j}]\in H^1(M_j;{\mathbb{R}})$ is non-zero.
Any cohomology class on ${\overline{X}}$ admits a smooth representative $\alpha$ up to the boundary. Moreover since $\alpha$ is closed, one can choose $\alpha$ to be constant in $x$ near the boundary using Lemma \[l:compsupp\]. Thus, in cohomological terms, the proposition is equivalent to finding a class $[\alpha]\in H^1({\overline{X}})$ whose pull-back to each connected component of $M$ is non-zero, i.e., $H^1(M_j)\ni i_{M_j}[\alpha]\neq 0$.
Consider first the case $\dim(X)=2$. If $X$ is non-orientable, $H^2({\overline{X}},M)=0$ so $\delta$ is the zero map. If $X$ is oriented, the boundary components are all oriented circles, so $H^1(M_j)\simeq {\mathbb{R}}$; the compactly supported cohomology $H^2({\overline{X}},M)$ is isomorphic to ${\mathbb{R}}$ via the integration map, and the boundary map $\delta:H^1(M)\to H^2({\overline{X}},M)$ restricted to $H^1(M_j)$ is just the identity map of ${\mathbb{R}}$ under these identifications. Thus the kernel of $\delta$ is made of $\nu$-tuples (where $\nu$ is the number of boundary components) $(a_1,\ldots,a_\nu)$ of real numbers, with the constraint $\sum a_j=0$ in the orientable case. By exactness, this space is also the image of the restriction map $H^1({\overline{X}})\to H^1(M)$. Clearly there exist such tuples with non-zero entries, provided $\nu\geq 2$ in the orientable case. Thus the conclusion follows for $\dim(X)=2$.
Assume now that $\dim(X)=3$. Then the maps $i_M$ and $\delta$ from the relative long exact sequence are dual to each other under the intersection pairing on $M$, respectively on $H^1({\overline{X}})\times H^2({\overline{X}},M)$: $$\int_M i_M(\alpha)\wedge\beta=\int_X \alpha\wedge\delta \beta.$$ These bilinear pairings are non-degenerate by Poincaré duality; in particular since the pairing on $H^1(M)$ is skew-symmetric, it defines a symplectic form. It follows easily that the subspace $$L:=i_M(H^1(X))\subset H^1(M)$$ is a Lagrangian subspace (i.e., it is a maximal isotropic subspace for the symplectic form). Now the symplectic vector space $H^1(M)$ splits into the direct sum of symplectic vector spaces $H^1(M_j)$, By hypothesis, the genus $g_j$ of the oriented surface $M_j$ is at least $1$ so $H^1(M_j)$ is non-zero for all $j$. It is clear that the projection of $L$ on each $H^1(M_j)$ must be non-zero, otherwise $L$ would not be maximal. Hence, there exists an element of $L=i_M(H^1(X))$ which restrict to non-zero classes in each $H^1(M_j)$, as desired.
\[r:int\] If one is interested in some coupling constant effect, it is interesting to choose the closed form $\alpha$ so that every $[\alpha_{|M_j}]$ are non-zero integral classes. In dimension $2$, this amounts to choosing non-zero integers with zero sum. In dimension $3$, as $L=i_M(H^1(X))=i_M(H^1(X,{\mathbb{Z}}))\otimes {\mathbb{R}}$ is spanned by integer classes, we can find an integer class in $L$ with non-zero projection on all $H^1(M_j)$. For real $g$, the vector potential $g\alpha$ is therefore non-trapping precisely for $g$ in a discrete subgroup $g_0{\mathbb{Z}}$ for some $g_0\in{\mathbb{Q}}$.
Application to hyperbolic manifolds {#hyp}
===================================
We now examine in more detail how this Aharonov-Bohm effect arises in the context of hyperbolic manifolds of finite volume in dimension $2$ and $3$.
These are conformally cusp manifold with $p=1$, with unperturbed metric of the form and such that every component $M_j$ of the boundary is a circle when $\dim(X)=2$, respectively a flat torus when $\dim(X)=3$: indeed, outside a compact set, the metric takes the form $g=dt^2+e^{-2t}h$, where $t\in[0,\infty)$, and this is of the form after the change of variables $x:=e^{-t}$. We denote by ${\overline{X}}$ the compactification of $X$ by requiring that $x:=e^{-t}$ be a boundary-defining function for “infinity”. These manifolds and their boundary components always have non-zero first Betti number.
For complete hyperbolic *surfaces* with cusps, every smooth $2$-form $B$ on ${\overline{X}}$ must be exact because $H^2({\overline{X}};{\mathbb{R}})$ is always zero for a non-closed surface. Call $A$ a smooth primitive of $B$. If $B$ vanishes at $M$ then $A$ is necessarily closed over $M$. In terms of cohomology classes, we have $[B]=\delta_{\mathbb{R}}[A_{|M}]$ where $\delta_{\mathbb{R}}$ is the connecting morphism from the sequence with real coefficients. Notice that $A$ can be chosen to define an integer class on $M$ if and only if $[B]$ is integer. Indeed, $H^2({\overline{X}};{\mathbb{Z}})$ is also $0$ for a compact surface with non-empty boundary, so if $[B]$ is integer, it must lie in the image of $\delta_{\mathbb{Z}}$ for the sequence with integer coefficients. Conversely, if $[A]$ is integer, i.e., $[A]=2\pi[A_{\mathbb{Z}}]$ (see diagram with $M$ in the place of $M_{\mathcal{B}}$, where the horizontal maps are now *surjective*) then $[B]=2\pi\delta_{\mathbb{Z}}[A_{\mathbb{Z}}]$ is also integer. We summarize these remarks in the following
\[81\] Let $B$ be a smooth $2$-form on the compactification of a complete hyperbolic surface $X$ with cusps, and denote by $[B]\in H^2({\overline{X}},M,{\mathbb{R}})$ its relative cohomology class.
- If either $X$ is non-orientable, or $X$ is orientable with at least two cusps, then $B$ admits both trapping and non-trapping vector potentials.
- If $X$ is orientable with precisely one cusp, then $B$ admits only non-trapping vector potentials if $[B]$ is integral, while if $[B]$ is not integral then $B$ admits only trapping vector potentials.
First note that $B$ is closed since it is of maximal degree; it is exact since $H^2({\overline{X}})=0$ for every surface with boundary; moreover its pull-back to the $1$-dimensional boundary also vanishes, so $B$ defines a relative de Rham class. By Corollary \[cohcl\], the existence of trapping and non-trapping vector potentials depends only on this class.
If $X$ is orientable and has precisely $1$ cusp, then the map $\delta:H^1(M)\to H^2({\overline{X}},M)$ is an isomorphism both for real and for integer coefficients. Thus $[B]$ is integer if and only if $[A_{|M}]$ is integer. Since the boundary is connected, $A$ is trapping if and only if the cohomology class of its restriction to the boundary is non-integer.
If $X$ is oriented and has at least two cusps, identify $H^2({\overline{X}},M)$ and each $H^1(M_j)$ with ${\mathbb{Z}}$, so that the boundary map restricted to $H^1(M_j)$ is the identity. We can write $[B]$ first as a sum $\sum\alpha_j$ of non-integer numbers, then also as a sum where at least one term is integer. Let $A$ be a $1$-form on ${\overline{X}}$ which restricts to closed forms of cohomology class $\alpha_j$ on $M_j=S^1$. Then $B-dA$ represents the $0$ class in $H^2({\overline{X}},M)$, so after adding to $A$ a form vanishing at the boundary, we can assume that $B=dA$. Now when all $\alpha_j$ are non-integers, $A$ is trapping, while in the other case it is non-trapping as claimed.
If $X$ is non-orientable, the class $[B]$ vanishes. It is enough to find trapping and non-trapping vector potentials for the zero magnetic field, which is done as in the orientable case.
When $\dim(X)=3$, we have:
Let $X$ be an orientable complete hyperbolic $3$-manifold of finite volume. Then every magnetic field $B$ smooth on the compactification ${\overline{X}}$ admits trapping vector potentials.
Assume that the pull-back of $B$ to the boundary $M$ vanishes. If $X$ has precisely one cusp, then there exists a rational (i.e., containing integer classes) infinite cyclic subgroup $G\subset H^2({\overline{X}},M,{\mathbb{R}})$ so that $B$ admits a non-trapping vector potential if and only if $[B]\in G$. In general, one of the following alternative statements holds:
1. Either every magnetic field smooth on ${\overline{X}}$ and vanishing at $M$ admits a non-trapping vector potential, or
2. Generically, magnetic fields smooth on ${\overline{X}}$ and vanishing at $M$ do not admit non-trapping vector potentials.
There exists moreover $q\in {\mathbb{Z}}^*$ such that if $[B]$ is integer, then $qB$ admits non-trapping vector potentials.
For the existence of trapping vector potentials we use the closed form $\alpha$ from Proposition \[p23\]. Let $A$ be any vector potential for $B$. It suffices to note that for $u\in{\mathbb{R}}$, the form $A+u\alpha$ is another vector potential for $B$, which is trapping on each connected component of $M$ except possibly for some discrete values of $u$.
Let $h$ denote the number of cusps of $X$. Both the Lagrangian subspace $L\subset H^1(M)$ and the image space $\partial H^1(M)\subset H^2({\overline{X}},M)$ have dimension $h$. By hypothesis, the cohomology class of $B$ on ${\overline{X}}$ is $0$ so by exactness of , the relative cohomology class $[B]$ lives in $\partial H^1(M)$.
Assume first that $X$ has precisely one cusp. Let $A$ be a vector potential for $B$ (smooth on ${\overline{X}})$. We can change $A$ by adding to it any class in the line $L$ without changing $[B]$. Notice that $H^1(M)={\mathbb{Z}}^2$. The line $L$ has an integer generator (given by the image of $H^1({\overline{X}},{\mathbb{Z}})\to H^1(M,{\mathbb{Z}})$). Without loss of generality, we can assume that $L$ is not the horizontal axis in ${\mathbb{Z}}^2$. It follows that the translates of all integer points in ${\mathbb{Z}}^2$ in directions parallel to $L$ form a discrete subgroup of ${\mathbb{Q}}$. Thus $B$ admits non-trapping vector potentials if and only if the cohomology class $[B]$ inside the $1$-dimensional image $\partial H^1(M)$ lives inside a certain infinite cyclic discrete subgroup. In particular, if $B$ is irrational (i.e., no positive integer multiple of $B$ is an integral class) then $B$ does not admit non-trapping vector potentials.
In the general case, assume first that there exists a boundary component $M_j$ so that $L$ projects surjectively onto $H^1(M_j,{\mathbb{R}})$. Let $A$ be a vector potential for an arbitrary magnetic field $B$ which vanishes at $M$. Let $[A'_j]\in H^1(M_j,{\mathbb{R}}) $ be such that $[A]_{H^1(M_j)}+[A'_j]$ is integer. Let $[A']\in L$ be an element whose component in $H^1(M_j)$ is $[A'_j]$. Choose a representative $A'$ and extend it to a smooth $1$-form on ${\overline{X}}$, constant in $x$ near the boundary. Then $A+A'$ is a non-trapping vector potential. From the definition of $L$, the form $dA'$ defines the zero class in relative cohomology, so from Corollary \[cohcl\] we get the assertion on $B$.
If the assumption on $L$ is not fulfilled, we claim that $$\hat{L}_j:=L\cap \oplus_{i\neq j}H^1(M_i)$$ has dimension $h-1$ for all $j$. Indeed, this dimension cannot be $h$ (since the projection of $L$ on $H^1(M_j)$ is not zero) and it cannot be $h-2$ (since the projection is not surjective). It follows easily that $\hat{L}_j$ is a Lagrangian subspace of $\oplus_{i\neq j}H^1(M_i)$. Let $v$ be a vector in $L\setminus \hat{L}_j$. The component $\hat{v}_j$ of $v$ in $\oplus_{i\neq j}H^1(M_i)$ is clearly orthogonal (with respect to the symplectic form) to $\hat{L}_j$, so by maximality it must belong to $\hat{L}_j$. Thus we may subtract this component to obtain, for each $j$, a non-zero element of $L\cap
H^1(M_j)$. These elements may be taken integral since $L$ has integer generators. Since $L=\ker\partial$, it follows that the image of $\partial$ is the direct sum of the images $\partial(H^1(M_j))$. As in the case of only one cusp, we see that $B$ has a non-trapping potential if and only if at least one of the components of $[B]$ in this decomposition belong to a certain cyclic subgroup containing integer classes.
Set $q$ to be the least common denominator of the generators of these subgroups for all $j$. If $[B]$ is integer, it follows that every vector potential for $qB$ must be maximal non-trapping.
The $C^1$ condition in the Mourre theory {#s:c1}
========================================
In this appendix, we give a general criterion of its own interest to check the, somehow abstract, hypothesis of regularity ${\mathcal{C}}^1$ which is a key notion in the Virial theorem within Mourre’s theory, see [@ABG] and [@GG0]. Let $A$ and $H$ be two self-adjoint operators in a Hilbert space ${\rond{H}}$. The commutator $[H,iA]$ is defined in the sense of forms on ${\mathcal{D}}(A)\cap{\mathcal{D}}(H)$. Suppose that the commutator $[H,iA]$ extends to ${\mathcal{B}}({\mathcal{D}}(H), {\mathcal{D}}(H)^*)$ and denote by $[H,iA]_0$ the extension. Suppose also that the following Mourre estimate holds true on an open interval ${\mathcal{I}}$, i.e. there is a constant $c>0$ and a compact operator $K$ such that $$\label{e:mourreK}
E_{\mathcal{I}}(H)[H,iA]_0 E_{\mathcal{I}}(H)\geq c E_{\mathcal{I}}(H) +K,$$ where $E_{\mathcal{I}}(H)$ denotes the spectral measure of $H$ above ${\mathcal{I}}$. Take now $\lambda\in{\mathcal{I}}$ which is not an eigenvalue of $H$. Set ${\mathcal{I}}_n:=(\lambda-1/n, \lambda+1/n)$. Then $E_{{\mathcal{I}}_n}(H)$ tends strongly to $0$ as $n\to\infty$, so $E_{{\mathcal{I}}_n}(H)KE_{{\mathcal{I}}_n}(H)$ tends in norm to $0$. Hence for $n$ big enough and for some $0<c'\leq c$, one gets the strict Mourre estimate $$\label{e:mourrestrict}
E_{{\mathcal{I}}_n}(H)[H,iA]_0 E_{{\mathcal{I}}_n}(H)\geq c' E_{{\mathcal{I}}_n}(H).$$
By supposing that $H\in{\mathcal{C}}^1(A)$ (see below) or that $e^{itA}{\mathcal{D}}(H)\subset{\mathcal{D}}(H)$, the Virial theorem holds true, i.e. $\langle f, [H,iA]_0 f\rangle=0$ for every eigenvector $f$ of $H$. Note that $f$ has no reason to lie in ${\mathcal{D}}(A)$ and that the expansion of the commutator $[H-\lambda, iA]$ over $f$ is formal.
The Virial theorem is crucial to the study of embedded eigenvalues of $H$. Assuming , it implies the local finiteness of the point spectrum of $H$ over ${\mathcal{I}}$, i.e., that the sum of the multiplicities of the eigenvalues of $H$ inside ${\mathcal{I}}$ is finite. To see this, apply to a infinite sequence $(f_n)_{n\in{\mathbb{N}}}$ of orthonormal eigenvectors of $H$. Then, since $\langle f_n, K f_n \rangle$ tends to $0$ as $n$ goes to infinity, one obtains a contradiction with the positivity of $c$. Assuming , the Virial theorem implies directly that $H$ has no eigenvalue in ${\mathcal{I}}_n$.
We stress that the hypothesis $[H,iA]_0\in{\mathcal{B}}({\mathcal{D}}(H), {\mathcal{D}}(H)^*)$ does not imply the Virial theorem. A counterexample is given in [@GG0]. If one adds some conditions on the second order commutator of $H$ and $A$, e.g. like in [@CFKS], one deduces from a limiting absorption principle and therefore the absence of eigenvalues in ${\mathcal{I}}_n$. In turn, assuming , we deduce that the set of eigenvalues of $H$ in ${\mathcal{I}}$ is closed. It is not known whether the multiplicity of the point spectrum must be locally finite when the Virial theorem does not hold.
Checking the Virial theorem, or a sufficient condition for it like the $C^1$ condition, is sometimes omitted in the Mourre analysis in a manifold context. To our knowledge, no result exists actually to show directly the $C^1$ regularity in a manifold context. On a class of exponentially growing manifolds, Bouclet [@bouclet] circumvents the problem by showing a stronger fact, i.e., the invariance of the domain. This method does not seem to work for our local conjugate operator, see Section \[s:conjop\]. Besides giving an abstract criterion for the ${\mathcal{C}}^1$ condition, we will explain under which additional condition we can recover the invariance of the domain from it.
Given $z\in\rho(H)$, we denote by $R(z)=(H-z)^{-1}$. For $k\in{\mathbb{N}}$, we recall that $H\in{\mathcal{C}}^k(A)$ if for one $z\notin\sigma(H)$ (then for all $z\notin\sigma(H)$) the map $t\mapsto e^{-itA}R(z)e^{itA}$ is $C^k$ in the strong topology. We recall a result following from Lemma 6.2.9 and Theorem 6.2.10 of [@ABG].
\[th:abg\] Let $A$ and $H$ be two self-adjoint operators in the Hilbert space ${\rond{H}}$. The following points are equivalent:
1. $H\in{\mathcal{C}}^1(A)$.
2. For one (then for all) $z\notin \sigma(H)$, there is a finite $c$ such that $$\begin{aligned}
\label{e:C1a}
|\langle A f, R(z) f\rangle - \langle R(\overline{z}) f, Af\rangle| \leq c
\|f\|^2, \mbox{ for all $f\in{\mathcal{D}}(A)$}. \end{aligned}$$
3. 1. There is a finite $c$ such that for all $f\in {\mathcal{D}}(A)\cap{\mathcal{D}}(H)$: $$\label{e:C1b}
|\langle Af, H f\rangle- \langle H f, Af\rangle|\leq
c(\|H f\|^2+\|f\|^2).$$
2. For some (then for all) $z\notin \sigma(H)$, the set $\{f\in{\mathcal{D}}(A) \mid R(z)f\in{\mathcal{D}}(A)$ and $R(\overline{z})f\in{\mathcal{D}}(A)
\}$ is a core for $A$.
Note that in practice, condition (3.a) is usually easy to check and follows from the construction of the conjugate operator. The condition (3.b) could be more delicate. This is addressed in the next lemma, inspired by [@bouclet].
\[l:C1\] Let ${\rond{D}}$ be a subspace of ${\rond{H}}$ such that ${\rond{D}}\subset{\mathcal{D}}(H)\cap{\mathcal{D}}(A)$, ${\rond{D}}$ is a core for $A$ and $H{\rond{D}}\subset{\rond{D}}$. Let $(\cchi_n)_{n\in
{\mathbb{N}}}$ be a family of bounded operators such that
1. $\cchi_n{\rond{D}}\subset {\rond{D}}$, $\cchi_n$ tends strongly to $1$ as $n\to\infty$, and $\sup_n \|\cchi_n\|_{{\mathcal{D}}(H)}<\infty$.
2. $A\cchi_n f \rightarrow Af$, for all $f\in{\rond{D}}$, as $n\to\infty$,
3. There is $z\notin \sigma(H)$, such that $\cchi_n R(z) {\rond{D}}\subset
{\rond{D}}\mbox{ and } \cchi_n R(\overline{z}) {\rond{D}}\subset{\rond{D}}$.
Suppose also that for all $f\in{\rond{D}}$ $$\label{e:comm}
\lim_{n\rightarrow \infty} A [H, \cchi_n] R(z)f= 0
\mbox{ and }
\lim_{n\rightarrow \infty} A [H, \cchi_n] R(\overline{z})f=
0.$$ Finally, suppose that there is a finite $c$ such that $$\label{e:C1}
\hspace*{1cm} |\langle Af, H f\rangle- \langle H f, Af\rangle|\leq
c(\|H f\|^2+\|f\|^2),\quad \forall f\in{\rond{D}}.$$ Then one has $H\in{\mathcal{C}}^1(A)$.
Note that is well defined by expanding the commutator $[H,\cchi_n]$ and by using (3) and $H{\rond{D}}\subset {\rond{D}}$. By polarization and by applying to $\cchi_n
R(\overline{z})f$ and to $\cchi_n R(z)f$, with $f\in{\rond{D}}$, we see that there exists $c<\infty$ such that $$\begin{aligned}
\nonumber
\lefteqn{ |\langle A\cchi_n R(\overline{z})f, H \cchi_n R({z})f\rangle-
\langle H \cchi_n R(\overline{z}) f, A\cchi_n R({z}) f\rangle|}\\&&
\leq c\|(H+i)\cchi_n R(\overline{z})f\|\cdot \|(H+i)\cchi_n R(z)f\|, \label{e:A7}\end{aligned}$$ for all $f\in{\rond{D}}$. By condition (1), the right-hand side is bounded by $C\|f\|^2$ for some $C$. We expand the left hand side of by commuting $H$ with $\cchi_n$: $$\begin{aligned}
& | \langle \cchi_n R(\overline{z})f, A\cchi_n f\rangle- \langle
A\cchi_n f, \cchi_n R({z}) f\rangle\\
& + \langle \cchi_n R(\overline{z})f, A[H,\cchi_n] R({z})f\rangle-
\langle A[H, \cchi_n] R(\overline{z}) f, \cchi_n R({z}) f\rangle|. \end{aligned}$$ Using , the second line vanishes as $n$ goes to infinity. Taking in account the assumptions (1) and (2), we deduce: $$|\langle R(\overline{z})f, A f\rangle- \langle A f, R({z}) f\rangle |
\leq C\|f\|^2, \quad \forall f\in{\rond{D}}.$$ Finally, since ${\rond{D}}$ is a core for $A$, we obtain . We conclude that $H\in{\mathcal{C}}^1(A)$.
The hypotheses of the lemma are easily satisfied in a manifold context with $H$ being the Laplacian and $A$ its conjugate operator, constructed as a localization on the ends of the generator of dilatation like for instance in [@FH]. Let ${\rond{D}}={{\mathcal{C}}^{\infty}_c}(X)$ and $\cchi_n$ a family of operators of multiplication by smooth cut-off functions with compact support. The fact that $A$ is self-adjoint comes usually by some consideration of $C_0$-group associated to some vector fields and using the Nelson Lemma and the invariance of ${\rond{D}}$ under the $C_0$-group give that ${\rond{D}}$ is a core for $A$, see Remark \[r:sa\]. The hypothesis (3) follows then by elliptic regularity. The only point to really check is . At this point one needs to choose more carefully the family $\cchi_n$. In Lemma \[l:regu0\], we show that the hypotheses of Lemma \[l:C1\] hold for the standard conjugate operator and for our local conjugate operator.
The invariance of the domain is desirable in order to deal, in a more convenient way, directly with operators and no longer with resolvents. On a manifold the $C_0$-group $e^{itA}$ is not explicit and it could be delicate to deal with the domain of $H$ directly. However, one may obtain this invariance of the domain using [@GG0] and a $C^1(A)$ condition. We recall:
If $H\in {\mathcal{C}}^1(A)$ and $[H, iA]:{\mathcal{D}}(H)\rightarrow
{\rond{H}}$ then $e^{itA}{\mathcal{D}}(H)\subset{\mathcal{D}}(H)$, for all $t\in{\mathbb{R}}$.
In light of this lemma, one understands better the importance of having some $C^1(A)$ criteria. On one hand, one can easily check the invariance of the domain. On the other hand, if the commutator belongs only to ${\mathcal{B}}({\mathcal{D}}(H), {\mathcal{D}}(H)^*)$ and not ${\mathcal{B}}({\mathcal{D}}(H), {\rond{H}})$, one may turn to another version of the Mourre Theory like in [@ABG] when $H$ has a spectral gap or like in [@GJ; @S] in the other case.
Finite multiplicity of $L^2$ eigenvalues {#fm}
========================================
In order to classify all maximal symmetric extension of a given cusp-elliptic operator $H$ (see section \[s:gene\] for definitions), one computes the defect indices, i.e., $\dim\ker (H^*\pm i)$. If they are equal and finite, one concludes that all maximal symmetric extensions of $H$ are self-adjoint. By the Krein formulae, one hence obtains that the difference of the resolvent of two maximal extensions is finite rank. This implies by Weyl’s theorem that the essential spectrum is the same for all self-adjoint extensions. Moreover, by Birman’s theorem, the wave operators associated to a pair of such extensions exist and are complete. On the other hand, note that if the defect indices are not finite, one may have maximal symmetric extensions which are not self-adjoint.
It is also interesting to control the multiplicity of eigenvalues embedded in the essential spectrum.
In the next lemma we assume that $X$ is a conformally cusp manifold with respect to the metric . We fix a vector bundle $E$ over ${\overline{X}}$ (for instance the bundle of cups differential forms, although in this paper we only use the case where $E$ is the trivial bundle ${\mathbb{C}}$) endowed with a smooth metric up to $\partial {\overline{X}}=M$.
\[l:indices\] Let $\Delta$, acting on ${{\mathcal{C}}^{\infty}}_c(X, E)$, be a cusp-elliptic differential operator in $x^{-2p}{\mathrm{Diff}}^k(X,E)$ for some $p,k>0$. Then the dimension of any $L^2$-eigenspace of $\Delta^*$ is finite.
Remember that if the operator $\Delta$ is bounded from below, then the defect indices are the same. This lemma guarantees that they are also finite. This point is not obvious when the manifold is not complete even if $\Delta$ is a Laplacian. Of course, this result is based on ideas that can be traced back to [@melaps] and which are today quite standard. This lemma generalizes a result of [@GMo].
We start by noticing that $\Delta$ can be regarded as an unbounded operator in a larger $L^2$ space. Namely, let $L^2_{\varepsilon}$ be the completion of ${{\mathcal{C}}^{\infty}_c}(X,E)$ with respect to the volume form $e^{-\frac{2\varepsilon}{x}}dg_p$ for some $\varepsilon>0$. Clearly then $L^2_{\varepsilon}$ contains $L^2$. A distributional solution of $\Delta-\lambda$ in $L^2$ is evidently also a distributional solution of $\Delta-\lambda$ in $L^2_{\varepsilon}$. Thus the conclusion will follow by showing that $\Delta$ has in $L^2_{\varepsilon}$ a unique closed extension with purely discrete spectrum. The strategy for this is by now clear. First we conjugate $\Delta$ through the isometry $$\begin{aligned}
L^2_{\varepsilon}\to L^2,&&\phi\mapsto e^{-\frac{\varepsilon}{x}}\phi.\end{aligned}$$ We get an unbounded operator $e^{-\frac{\varepsilon}{x}}
\Delta e^{\frac{\varepsilon}{x}}$ in $L^2$, which is unitarily equivalent to $\Delta$ (acting in $L^2_{\varepsilon}$). Essentially from the definition, see Section \[s:gene\], $${\mathcal{N}}(x^{2p}e^{-\frac{\varepsilon}{x}}\Delta
e^{\frac{\varepsilon}{x}})(\xi)= {\mathcal{N}}(x^{2p}\Delta)(\xi+i\varepsilon).$$ The normal operator is a polynomial in $\xi$, in particular it is entire. Then by analytic Fredholm theory [@melaps Prop. 5.3], the set of complex values of $\xi$ for which ${\mathcal{N}}(x^{2p}\Delta)(\xi)$ is not invertible, is discrete. Thus there exists $\varepsilon \pg 0$ such that ${\mathcal{N}}(x^{2p}\Delta)(\xi+i\varepsilon)$ is invertible for all $\xi\in{\mathbb{R}}$. For such $\varepsilon$ the operator $\Delta$ in $L^2_\varepsilon$ is unitarily equivalent to a fully elliptic cusp operator of order $(k,2p)$ in $L^2$. It is then a general fact about the cusp algebra [@wlom Theorem 17] that such an operator has a unique closed extension and admits a compact inverse modulo compact operators in $L^2_\varepsilon$. In particular, its eigenvalues have finite multiplicity. As noted above, the eigenspaces of $\Delta$ in $L^2$ are contained in the eigenspaces of $\Delta$ in $L^2_\varepsilon$ for the same eigenvalue.
As a corollary, the magnetic Laplacians for the metric and for vector potentials which are smooth cusp $1$-forms, have finite multiplicity eigenvalues.
Stability of the essential spectrum {#s:stab}
===================================
It is well-known that the essential spectrum of an elliptic differential operator on a complete manifold can be computed by cutting out a compact part and studying the Dirichlet extension of the remaining operator on the non-compact part (see e.g., [@donelili]). This result is obvious using Zhislin sequences, but the approach from loc. cit. fails in the non-complete case. For completeness, we give below a proof which has the advantage to hold in a wider context and for a wider class of operator, pseudodifferential operators for instance.
We start with a general lemma. We recall that a *Weyl sequence* for a couple $(H,\lambda)$ with $H$ a self-adjoint operator and $\lambda\in{\mathbb{R}}$, is a sequence $\varphi_n\in{\mathcal{D}}(H)$ such that $\|\varphi_n\|=1$, $\varphi_n\rightharpoonup 0$ (weakly) and such that $(H-\lambda)\varphi_n\rightarrow 0$, as $n$ goes to infinity. It is well-known that $\lambda\in\sigma_{\rm ess}(H)$ if and only if there is Weyl sequence for $(H,\lambda)$.
\[l:cut\] Let $H$ be a self-adjoint operator in a Hilbert space ${\rond{H}}$. Let $\varphi_n$ be a Weyl sequence for the couple $(H,\lambda)$. Suppose that there is a closed operator $\Phi$ in ${\rond{H}}$ such that:
1. $\Phi {\mathcal{D}}(H)\subset{\mathcal{D}}(H)$,
2. $\Phi(H+i)^{-1}$ is compact,
3. $[H,\Phi]$ is a compact operator from ${\mathcal{D}}(H)$ to ${\rond{H}}$.
Then there is $\widetilde{\varphi}_n\in{\mathcal{D}}(H)$ such that $(1-\Phi)\widetilde{\varphi}_n$ is a Weyl sequence for $(H,\lambda)$.
First we note (2) implies that $\Phi\varphi_n$ goes to $0$. Indeed, we have $\Phi\varphi_n=\Phi(H+i)^{-1}\big((H-\lambda)\varphi_n
+(i+\lambda) \varphi_n \big)$ and the bracket goes weakly to $0$. Similarly, using (3) we get that $[H,\Phi]\varphi_n\rightarrow 0$. Therefore we obtain $\|(1-\Phi)\varphi_n\|\geq 1/2$ for $n$ large enough. We set $\widetilde\varphi_n:=\varphi_n/\|(1-\Phi)\varphi_n\|$. Note that $(1-\Phi)\widetilde\varphi_n\rightharpoonup 0$. Finally, $(H-\lambda)(1-\Phi)\widetilde\varphi_n\rightarrow 0$ since $[H,\Phi]\varphi_n\rightarrow 0$.
This shows that the essential spectrum is given by a “non-compact” part of the space. We now focus on Friedrichs extension. Given a dense subspace ${\rond{D}}$ of a Hilbert space ${\rond{H}}$ and a positive symmetric operator on ${\rond{D}}$. Let ${\rond{H}}_1$ be the completion of ${\rond{D}}$ under the norm given by ${\rond{Q}}(\varphi)^2=\langle H\varphi,
\varphi\rangle+\|\varphi\|^2$. The domain of the Friedrichs extension of $H$, is given by ${\mathcal{D}}(H_{\rond{F}})=\{f\in {\rond{H}}_1 \mid {\rond{D}}\ni g\mapsto \langle
Hg,f\rangle+\langle g,f\rangle $ extends to a norm continuous function on ${\rond{H}}\}$. For each $f\in{\mathcal{D}}(H_{\rond{F}})$, there is a unique $u_f$ such that $\langle
Hg,f\rangle + \langle g,f\rangle = \langle g,u_f\rangle$, by Riesz theorem. The Friedrichs extension of $H$ is defined by setting $H_{\rond{F}}f:=u_f-f$. It is a self-adjoint extension of $H$, see [@RS2].
Let $(X,g)$ be a smooth Riemannian with distance $d$. We fix $K$ a smooth compact sub-manifold of $X$ of same dimension. We endow it with the induced Riemannian metric. We set $X'=X\setminus K$. In the following, we embed $L^2(X')$ in $L^2(X)$. We will need the next definition within the proof.
We say that $\Phi$ is a (smooth) *cut-off function* for $K$ if $\Phi\in{\mathcal{C}}_c^\infty(X)$ and $\Phi|_K=1$. We say that it is an $\varepsilon$-cut-off is ${\mathrm{supp}}(\Phi)\subset
B(K,\varepsilon)$.
We are now able to give a result of stability of the essential spectrum.
\[p:cut\] Let $d$ be a differential form of order $1$ on $C^\infty_c(X)\rightarrow C^\infty(X,\Lambda^1)$ with injective symbol away from the $0$ section of the cotangent bundle. We denote by $d_X$ and $d_{X'}$ the closure of $d$ in $L^2(X)$ and $L^2(X')$, respectively. Consider $\Delta_X:=d_X^*d_X$ and $\Delta_{X'}=d_{X'}^*d_{X'}$, the Friedrichs extensions of the operator $d^*d$, acting on ${\mathcal{C}}_c^\infty(X)$ and ${\mathcal{C}}_c^\infty(X')$, respectively. One has $\sigma_{\rm{ess}}(\Delta_X)=
\sigma_{\rm{ess}}(\Delta_{X'})$.
Let $f\in L^2(B(K, \varepsilon)^c)$ and let $\Phi$ be a $\varepsilon$-cut-off for $K$. We first show that $f\in{\mathcal{D}}(d_X)$ if and only if $f\in{\mathcal{D}}(d_{X'})$. Suppose that $f\in{\mathcal{D}}(d_X)$, then for all $\eta>0$, there is $\varphi\in {\mathcal{C}}_c^\infty(X)$ such that $\|f-\varphi_n\|+ \|df-d\varphi_n\|<\eta$. Because of the support of $f$, one obtain that $\|\Phi\varphi\|<\eta\|\Phi\|_\infty$ and that $\|[d,\Phi]\varphi\|<\eta\|[d,\Phi]\|_\infty$. Therefore $(1-\Phi)\varphi_n\in {\mathcal{C}}_c^\infty(X')$ and is Cauchy in ${\mathcal{D}}(d_{X'})$, endow with the graph norm. By uniqueness of the limit, one obtains that $f\in{\mathcal{D}}(d_{X'})$ and that $d_X f=d_{X'}f$. The opposite implication is obvious. Using again the $\varepsilon$-cut-off, one shows that $g\in{\mathcal{D}}(d^*_X)$ if and only if $g\in{\mathcal{D}}(d^*_{X'})$ and that $d^*_{X}g= d^*_{X'}g$ for $g\in L^2(\Lambda^1 (B(K, \varepsilon)^c)$. Finally, we obtain that $f\in{\mathcal{D}}(\Delta_X)$ if and only if $f\in{\mathcal{D}}(\Delta_{X'})$ and that $\Delta_X f=\Delta_{X'} f$, for $f\in L^2(B(K,
\varepsilon)^c)$.
From the definition of the Friedrichs extension and the injectivity of the symbol of $d$, the domain of $\Delta_X$, $\Delta_{X'}$ is contained in $H^1_0(X)\cap H^2_{\mathrm{loc}}(X)$, respectively in $H^1_0(X')\cap H^2_{\mathrm{loc}}(X')$. By taking the same $\Phi$ as above and using the Rellich-Kondrakov lemma, the hypotheses of Lemma \[l:cut\] are satisfied. Finally, we apply it to $\Delta_X$ and $\Delta_{X'}$ and since the Weyl sequence is with support away from $K$, the first part of the proof gives us the double inclusion of the essential spectra.
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Sylvain Golénia,
Mathematisches Institut der Universität Erlangen-Nürnberg Bismarckstr. 1 1/2, 91054 Erlangen, Germany
E-mail: [email protected]
Sergiu Moroianu,
Institutul de Matematică al Academiei Române, P.O. Box 1-764, RO-014700 Bucharest, Romania
and
Şcoala Normală Superioară Bucharest, Calea Griviţei 21, Bucharest, Romania
E-mail: [email protected]
|
---
abstract: 'FIRBACK is a one of the deepest surveys performed at 170 $\mu m$ with ISOPHOT onboard ISO, and is aimed at the study of cosmic far infrared background sources. About 300 galaxies are detected in an area of four square degrees, and source counts present a strong slope of 2.2 on an integral “logN-logS” plot, which cannot be due to cosmological evolution if no K-correction is present. The resolved sources account for less than 10% of the Cosmic Infrared Background at 170 $\mu m$. In order to understand the nature of the sources contributing to the CIB, and to explain deep source counts at other wavelengths, we have developed a phenomenological model, which constrains in a simple way the luminosity function evolution with redshift, and fits all the existing deep source counts from the mid-infrared to the submillimetre range.'
author:
- 'H. Dole'
- 'R. Gispert'
- 'G. Lagache'
- 'J-L. Puget'
- 'H. Aussel'
- 'F.R. Bouchet'
- 'P. Ciliegi'
- 'D.L. Clements'
- 'C.J. Cesarsky'
- 'F.X. Désert'
- 'D. Elbaz'
- 'A. Franceschini'
- 'B. Guiderdoni'
- 'M. Harwit'
- 'R. Laureijs'
- 'D. Lemke'
- 'R. McMahon'
- 'A.F.M. Moorwood'
- 'S. Oliver'
- 'W.T. Reach'
- 'M. Rowan-Robinson'
- 'M. Stickel'
title: FIRBACK Source Counts and Cosmological Implications
---
Introduction
============
The Cosmic Infrared Background (CIB), due to the accumulation of galaxy emission at all redshifts along the line of sight in an instrument beam, is a powerful tool for studying galaxy evolution. FIRBACK ([@puget99a], [@dole99]), one of the deepest surveys performed at $170\,\mu m$, is aimed at the study of the CIB, in two complementary ways:
- study the resolved sources (this paper)
- study the background fluctuations ([@lagache99a], [@lagache99b])
Throughout this paper we use a cosmology with $h=0.65$, $\Omega=1$ and $\Lambda=0$.
The FIRBACK Survey
==================
FIRBACK, is a survey of 4 square degrees in 3 high galactic latitude fields, chosen to have as low an HI column-density as possible, typically $N_H \simeq 10^{20} cm^{-2}$, and if possible multiwavelength coverage. Observations were carried with ESA’s Infrared Space Observatory (ISO, [@kessler96]) with the ISOPHOT photometer [@lemke96] in raster mode (AOT P22) with the `C200` camera and `C_160` broadband filter centered at $\lambda = 170\, \mu m$. A detailed description of the reduction, data processing, and calibration will be discussed in [@lagache2000], whereas the analysis of the complete survey will be discussed in [@dole2000].
Source Counts at $170\,\mu m$
=============================
Preliminary FIRBACK integral source counts at $170\,\mu m$ (Fig. \[firback\_count\]), not corrected for incompleteness, show a strong slope of 2.2 between 120 and 500 mJy. This strong slope is not explained by a K-correction or cosmological evolution alone: both must be present; the K-correction is the ratio, at a given wavelength, of the emitted flux over the redshifted flux.\
Non (or low) evolution scenarii, or extrapolation of IRAS counts, are not able to reproduce the observed counts. For illustration, we plot in Fig. \[firback\_count\] the non-evolution model from [@franceschini98] and the evolution model A from [@guiderdoni98]. On the other hand, evolutionary models from [@franceschini98] and from [@guiderdoni98] (model E: evolution + ULIRGs) give a better agreement. At this wavelength, FIRBACK sources account for only 3% of the background.
![FIRBACK integral source counts at $170\,\mu m$ not corrected for incompleteness. Models from [@guiderdoni98]: A (dot) with evolution, E (solid) with evolution + ULIRGs. Models from [@franceschini98]: without evolution (dot-dash), with evolution (dash).[]{data-label="firback_count"}](figure1.eps){width=".85\textwidth"}
Modelling the Evolution of Galaxies
===================================
Method
------
Our philosophy is to make a phenomenological model that explains all the observed deep source counts and reproduces the CIB in the mid-IR to submillimetre range. One way to do this is to constrain the evolution with redshift of the luminosity function (LF) in the infrared, given:
- templates of galaxy spectra
- the energy density available at each redshift
For the first point, we used template galaxy spectra based on IRAS colors [@maffei94] modified to account for recent ISO observations, in particular the absorption feature near $10\,\mu m$ at high luminosity; PAH features are present in the mid-infrared [@desert90], even if their strength seems larger than the observations: this is not a problem because the right amount of energy is present in each peak, and we convolve the spectrum with the filter spectral response (Fig. \[template\_spectra\]). For the second point, we used the inversion of the CIB spectrum by [@gispert2000] and [@puget99b], which gives with good accuracy the energy density available at redshifts between 1 and 3. This energy density is the integral of the luminosity function at each redshift, but there is no unique solution for the LF shape.
![Template spectra used for simulating source counts; from bottom to top: $L =
10^{9},\, 10^{10},\, 10^{11},\, 10^{12},\, 10^{13}\, L_{\odot}$.[]{data-label="template_spectra"}](figure2.eps){width=".85\textwidth"}
Evolving Luminosity Function
----------------------------
The question is: how to evolve the LF with redshift ?\
Fig. \[lf\_z0\] represents the LF of [@sanders96] at a redshift z=0, renormalized to $h=0.65$ and having the same integral as [@soifer91]. Our constraint on the LF redshift variation is the energy density available at each redshift [@gispert2000]. Usually, authors apply a pure density evolution to the LF as a function of redshift (i.e. a vertical shift), or a pure luminosity evolution (a horizontal shift). This does not work for FIRBACK source counts: There is no alternative other than adding the evolution to one part of the LF only. This part is constrained by IR and submm observations: this is the bright end of the LF. Fig. \[lf\_z0\] represents our decomposition of the local LF into two parts:
- left part: “normal” galaxies
- right part: ULIRG’s, centered on a luminosity $L_{ULIRG} \simeq 2.0 \times 10^{11}
L_{\odot}$, where $L_{ULIRG}$ is the free parameter; we get the same value as [@tan99]
In our model, “without evolution” means that the local LF (Fig. \[lf\_z0\]a) is taken, and is the same at every redshift. In the evolutionary scenario, only the ULIRG part moves, in such a way that the integral of the LF equals the constraint given by the CIB inversion at each redshift. The maximum is reached at $z \simeq 2.5$, and Fig. \[lf\_z0\]b represents the LF at this redshift. We neglect at this stage the evolution of “normal” galaxies, which do not play a crucial role in the mid-infrared to submillimetre wavelength range.
![a: Luminosity Function at z=0 (solid line); normal galaxy (dot-dash); ULIRG (dash-dash). b: Luminosity Function at z=2.5 (solid line); normal galaxy (dot-dash) ULIRG (dash-dash) and local LF (dots).[]{data-label="lf_z0"}](figure3.eps){width="95.00000%"}
Model of Source Counts at $170\,\mu m$
--------------------------------------
The model at $170\,\mu m$, together with our data, is presented in Fig. \[create\_counts170\]. The brightest point of the observed counts is compatible with all our models, in particular the non-evolutionary scenario, which is expected for local sources. We also show the effect of the K-correction, which steepens the integral source count slope. Our evolutionary scenario fits the data within the error bars. Most of the background is expected to be resolved into sources once we are able to detect sources at the mJy level. This wavelength region, in which evolutionary effects are particularly important and where there are good prospects for detecting higher redshift sources because of the K-correction, is nowadays probably the best-suited range for probing galaxy evolution from space in the far-infrared.
![Observed Counts and Models at $170\,\mu m$, with (solid line) and without (dashed line) evolution. The model with evolution and without K-correction is the dot-dash line.[]{data-label="create_counts170"}](figure4.eps){width=".85\textwidth"}
Models of Source Counts at other wavelengths
--------------------------------------------
Data at $15\,\mu m$ [@elbaz99], $90\,\mu m$ [@efstathiou99] and $850\,\mu m$ [@barger99], with our models are presented in Fig. \[create\_counts15\], Fig. \[create\_counts90\] and Fig. \[create\_counts850\] respectively.\
At $15\,\mu m$, both the slopes and the little “waves” in the counts appear in the models with the combined effects of the K-correction and the evolution. At $90\,\mu m$, the K-correction does not emphasize the differences between the scenarii of evolution or non-evolution. At $850\,\mu m$, the “non smooth” appearance of our model is due to the discretization of the LF.
![Models at $15\,\mu m$ with (solid line) and without (dashed line) evolution, and observed counts from Elbaz et al. [@elbaz99] (thin lines).[]{data-label="create_counts15"}](figure5.eps){width=".85\textwidth"}
![Models at $90\,\mu m$ with (solid line) and without (dashed line) evolution, and observed counts from Efstathiou et al. [@efstathiou99] (squares) and IRAS counts [@efstathiou99] (solid area)[]{data-label="create_counts90"}](figure6.eps){width=".85\textwidth"}
![Models at $850\,\mu m$ with (solid line) and without (dashed line) evolution, and observed counts from Barger et al. [@barger99] (diamonds).[]{data-label="create_counts850"}](figure7.eps){width=".85\textwidth"}
Discussion
==========
The bright end luminosity function evolution model
--------------------------------------------------
Our model of the evolving LF scenario fits most of the existing deep survey data from space (mid and far infrared) and ground (submillimetre). It is also compatible with the observational estimate of the LF in [@lilly99]. The strong observational constrain of multiwavelength source counts is thus explainable by a simple evolutionary law of the LF: the Bright End Luminosity Function Evolution (BELFE) model. One simple model is in agreement with all up-to-date observables and reproduces the background.
Redshift and Nature of the sources
----------------------------------
Another crucial observational test for our model is the predicted vs observed redshift distribution. Although the statistics are poor, we have some evidence that most of the ISOCAM sources lie at redshift between 0 and 1.4 with a median at 0.8 [@aussel99], and that most of the SCUBA sources lie at redshifts greater than 2 [@barger99]. Our predicted redshift distributions at these two wavelengths are in agreement with the existing observations.\
What about FIRBACK $170\,\mu m$ sources ? Our predicted redshift distribution shows that most of the sources lie at redshifts below 1.5, with a median comparable to that of the ISOCAM sources. This means that we are sensitive both to local sources and sources beyond redshift $z
=1$. All the FIRBACK sources with known redshifts (less than 10) [@dennefeld99], [@scott2000] are in this range. Two submillimetre sources are at $z > 1$, and a few visible sources are at $z \simeq 0.2$.\
It is difficult to address the question differences between PHOT and CAM sources, because their redshift distributions (both expected and observed) are similar. About half of the $170\,\mu m$ sources have $15\,\mu m$ counterparts.
Conclusion
==========
We have presented results from the FIRBACK survey, one of the largest ISO programs, dealing with resolved sources of the CIB: our source counts at $170\,\mu m$ show strong evolution. This evolution is explained by a simple law involving the redshift of the luminosity function, which differs from pure density or luminosity evolution: the “Bright End Luminosity Function Evolution” model. The model fits all the existing source counts at 15, 90, 170 and 850 $\mu m$, and also predicts a redshift distribution in agreement with the (sometimes sparse) observations. This powerful tool, based on observational constraints on the CIB spectrum inversion and the local Luminosity Function, and on assumed template galaxy spectra, not only agrees with existing data but also is able to make useful predictions on source counts or CIB fluctuations. These predictions may be useful for planning the utilization of major telescopes of the future, such as SIRTF, Planck, FIRST and ALMA. All the FIRBACK materials are available at: `http://wwwfirback.ias.fr`.
[7]{}
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|
---
abstract: 'We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time discretization of geodesic calculus presented in [@RuWi12b], which just requires an approximation of the squared Riemannian distance that is typically easy to compute. First we obtain first order discrete covariant derivatives via a Schild’s ladder type discretization of parallel transport. Second order discrete covariant derivatives are then computed as nested first order discrete covariant derivatives. These finally give rise to an approximation of the curvature tensor. First and second order consistency are proven for the approximations of the covariant derivative and the curvature tensor. The findings are experimentally validated on two-dimensional surfaces embedded in ${\mathbb{R}}^3$. Furthermore, as a proof of concept the method is applied to the shape space of triangular meshes, and discrete sectional curvature indicatrices are computed on low-dimensional vector bundles.'
author:
- Alexander Effland
- Behrend Heeren
- Martin Rumpf
- Benedikt Wirth
bibliography:
- 'bibtex/all.bib'
- 'bibtex/own.bib'
title: |
Consistent Curvature Approximation\
on Riemannian Shape Spaces
---
Introduction {#sec:intro}
============
Over the last two decades there has been a growing interest in modeling general shape spaces as Riemannian manifolds. Applications range from medical imaging and computer vision to geometry processing in computer graphics. Besides the computation of a rigorous distance between shapes defined as the length of a geodesic curve, other tools from Riemannian geometry proved to be practically useful as well. The geometric logarithm offers a representation of large scale shape variability in a linear tangent space, the exponential map is a tool for shape extrapolation, and parallel transport allows to transfer edited details along animation sequences.
The Riemann curvature tensor represents a higher order characterization of the local geometry of the manifold. In this paper, we discuss how to numerically approximate the curvature tensor as well as the sectional curvature, a generalization of the Gauß curvature, on Riemannian manifolds. The method is based on a variational time discretization of the Riemannian path energy which has been developed in [@RuWi12b] and has previously been used to deduce a corresponding discretization of exponential map, logarithm, and parallel transport. It requires a local approximation ${\mathcal{W}}$ of the squared Riemannian distance ${\mathrm{dist}}$ such that ${\mathcal{W}}{[{y}, \tilde{y}]}={\mathrm{dist}}^2({y},\tilde{y})+O({\mathrm{dist}}^3({y},\tilde{y}))$ for ${y}$ and $\tilde {y}$ being points on the manifold. In the case of shape spaces such approximations are usually substantially easier to compute than the underlying Riemannian metric and the squared distance itself. Our approximation of the Riemann curvature tensor is based on replacing the second order covariant derivatives in its definition by corresponding *covariant difference quotients*. We prove the consistency of these difference quotients as well as the resulting approximation of the Riemann curvature tensor. Furthermore, we experimentally validate the presented approach on embedded surfaces and show its applicability in a concrete example of a high-dimensional shape space. More precisely, we consider the space of triangular surfaces along with a physical deformation model. In the computer graphics community this type of model is known as *Discrete Shells* and was first introduced by Grinspun [@GrHiDeSc03]. This approach turned out to be very effective as well as efficient and has been used and modified by various authors. In particular, it has also been investigated in the context of Riemannian shape space theory in [@HeRuWa12; @HeRuSc14].
The paper is organized as follows: In Section \[sec:related\] we discuss some examples of shape spaces and different approaches for the computation of curvature with special emphasis on shape spaces. Section \[sec:review\] reviews the discrete geodesic calculus from [@RuWi12b] which is the starting point of our computational scheme for the Riemann curvature tensor. Since the computation of the discrete curvature tensor is based on discrete covariant derivatives, we first investigate first and second order approximations of the covariant derivative in Section \[sec:cov\]. Then, in Section \[sec:curvatureTensor\] we prove consistency estimates for a corresponding first and second order approximation of the curvature tensor. Finally, Section \[sec:ShellSpace\] discusses the application of our approach to the study of curvature on the space of discrete shells as a proof of concept.
Related work {#sec:related}
============
Over the last two decades Riemannian calculus on shape spaces has attracted a lot of attention. In particular, it allows to transfer many important concepts from classical geometry to these usually high or even infinite-dimensional spaces.
#### Some examples of shape spaces.
Prominent examples with a full-fledged geometric theory are spaces of planar curves with a curvature-based metric [@MiMu07], an elastic metric [@SrJaJo06], Sobolev-type metrics [@ChKePo05; @MiMu07; @SuYeMe07; @MiMu07; @Br16; @BaBrHa17] or a Riemannian distance computed based on the square root velocity function on the curves [@SrKlJo11; @Br16]. Spaces of surfaces have also been considered as Riemannian manifolds. Sobolev metrics on the space of surfaces have been presented by Bauer [@BaHaMi11]. In [@BaHaMi12] they extended the Riemannian approach for the space of curves from [@MiMu07] introducing a suitable metric on tangent spaces to the space of embeddings or immersions. In the context of geometry processing Kilian [@KiMiPo07] studied geodesics in the space of triangulated surfaces, where the metric is derived from the in-plane membrane distortion. Since then, a variety of other Riemannian metrics have been investigated on the space of surfaces [@LiShDi10; @KuKlDi10; @BaBr11]. In [@HeRuWa12; @HeRuSc14] a metric was proposed that measures membrane distortion as well as bending. In image processing the large deformation diffeomorphic metric mapping (LDDMM) framework is based on the theory of diffeomorphic flows: Dupuis [@DuGrMi98] showed that the associated flow is actually a flow of diffeomorphisms. In [@HaZaNi09], Hart exploited the optimal control perspective on the LDDMM model with the motion field as the underlying control. Vialard [@ViRiRu12a; @ViRiRu12] studied methods from optimal control theory to accurately estimate the initial momentum of the flow and to relate it to the Hamiltonian formulation of geodesics. Lorenzi and Pennec [@LoPe13] applied the framework to compute geodesics and parallel transport using Lie group methods. The metamorphosis model [@MiYo01; @TrYo05] generalizes the flow of diffeomorphism approach by allowing additional intensity variations along the flow of diffeomorphically deformed images.
In this paper we aim at the numerical computation of the Riemann curvature tensor and sectional curvatures on Riemannian manifolds. Our focus is on high-dimensional shape manifolds and not on low-dimensional embedded surfaces even though we show numerical experiments for a two-dimensional embedded surface to confirm our theoretical findings. Nevertheless, let us briefly review some of the numerous approaches for the approximation of curvature on two-dimensional discrete surfaces.
#### Numerical approximation of curvature on discrete 2D surfaces.
One of the earliest approaches to numerically estimate the curvature tensor at the vertices of a polyhedral approximation of a surface was proposed by Taubin [@Ta95a]. In detail, this ansatz exclusively works for embedded two-dimensional manifolds in ${\mathbb{R}}^3$ and relies on the observation that the directional curvature function corresponds to a quadratic form, which can be estimated with linear complexity and from which the curvature tensor can be retrieved. Starting from the triangulation of a two-dimensional manifold embedded in ${\mathbb{R}}^3$, Meyer [@MeDeScBa] defined discrete operators representing, for instance, the mean and the Gaussian curvature by spatial averaging of suitably rescaled geometric quantities. Using a mixed finite element/finite volume discretization as well as a Voronoi decomposition principal curvatures can be robustly estimated. Cohen-Steiner and Morvan [@CoMo03] proposed an integral approximation of the curvature tensor on smooth or polyhedral surfaces using normal cycles and proved its linear convergence if the polyhedral triangulation is Delaunay. Hildebrandt [@HiPoWa06] showed that if a sequence of polyhedral surfaces isometrically embedded in ${\mathbb{R}}^3$ converges to a differentiable manifold ${\mathcal{M}}$ with respect to the Hausdorff distance, then the convergence of the normal field is equivalent to the convergence of the metric tensor, which itself is equivalent to the convergence of the surface area. In addition, under suitable assumptions they established the convergence of geodesic curves and the mean curvature functionals on discrete surfaces to their limit counterparts on ${\mathcal{M}}$. In [@KaSiNo07], Kalogerakis approximated the second fundamental form of a possibly noisy surface using M-estimation, which amounts to an efficient data fitting approach using the method of iteratively reweighted least squares. Here, the surface is either represented by polygonal meshes or by point clouds. Hildebrandt and Polthier [@HiPo11] introduced generalized shape operators for smooth and polyhedral surfaces as linear operators on Sobolev spaces and provided error estimates to approximate the generalized shape operator on smooth surfaces by that on polyhedral surfaces, from which several geometric quantities can be recovered. Starting from a weak formulation of the Ricci curvature, Fritz [@Fr13] employed a surface finite element method to approximate the Ricci curvature on isometrically embedded hypersurfaces ${\mathcal{M}}$ and proved that the rate of convergence is $\frac{2}{3}$ and $\frac{1}{3}$ with respect to the $L^2({\mathcal{M}})$- and $H^1({\mathcal{M}})$-norm, respectively.
#### Explicit computation of curvature on shape spaces.
Michor and Mumford [@MiMu04] studied the Riemann curvature tensor on the space of smooth planar curves and gave an explicit formula for the sectional curvature. They could show that for large, smooth curves all sectional curvatures turn out to be nonnegative, whereas for curves with high frequency perturbation sectional curvatures are nonpositive. Younes [@YoMiSh08] investigated a metric on the space of plane curves derived as the limit case of a scale invariant metric of Sobolev order 1 from [@MiMu07] which allows the explicit computation of geodesics and sectional curvature. Micheli [@MiMiMu12] showed how to compute sectional curvature on the space of landmarks with a metric induced by the flow of diffeomorphisms. They were able to evaluate the sectional curvature for pairs of tangent directions to special geodesics along which only two landmarks actually move. A formula for the derivatives of the inverse of the metric is at the core of this approach. In [@MiMiMu13] the formula for the sectional curvature on the landmark space was generalized to special infinite-dimensional weak Riemannian manifolds. A brief overview over the geometry of shape spaces with a particular emphasis on sectional curvature in different shape spaces can be found in the contribution by Mumford [@Mu12].
A brief review of continuous and discrete geodesic calculus {#sec:review}
===========================================================
To keep this paper self-contained, we first collect all required ingredients of the discrete geodesic calculus introduced in [@RuWi12b] and state our overall assumptions. Let us remark that some results reviewed here hold also under weaker assumptions. For a comprehensive discussion we refer to [@RuWi12b]. Let ${\mathbf{V}}$ be a separable, reflexive Banach space that is compactly embedded in a Banach space ${\mathbf{Y}}$. Let ${\mathcal{M}}$ be the weak closure of an open path-connected subset of ${\mathbf{V}}$, potentially with a smooth boundary. We consider $({\mathcal{M}}, g)$ as a Riemannian manifold with metric $g : {\mathcal{M}}\times {\mathbf{V}}\times{\mathbf{V}}\to{\mathbb{R}}$, which is uniformly bounded and ${\mathbf{V}}$-coercive in the sense $c^\ast \|v\|^2_{\mathbf{V}}\leq g_{y}(v,v) \leq C^\ast \|v\|^2_{\mathbf{V}}$. Furthermore, $g$ is $C^3({\mathbf{Y}}\times{\mathbf{V}}\times{\mathbf{V}};{\mathbb{R}})$-smooth. Here, the tangent space $T_{y}{\mathcal{M}}$ is identified with ${\mathbf{V}}$.
#### Geodesic paths.
A geodesic path ${\mathbf{y}}=({\mathbf{y}}(t))_{t\in [0,1]}$ on ${\mathcal{M}}$ is defined as a local minimizer of the path energy $$\begin{aligned}
\label{eq:contenergy}
{\mathcal{E}}[{\mathbf{y}}] = \int_0^1 g_{{\mathbf{y}}(t)}(\dot{\mathbf{y}}(t),\dot{\mathbf{y}}(t)) {\,\mathrm{d}}t\end{aligned}$$ for fixed end positions ${\mathbf{y}}(0)={y}_A$ and ${\mathbf{y}}(1) = {y}_B$ with ${y}_A,{y}_B\in{\mathcal{M}}$. Then the Riemannian distance ${\mathrm{dist}}({y}_A,{y}_B)$ is given as the square root of the minimal path energy. Given $v\in {\mathbf{V}}$, one may ask for the end point ${\mathbf{y}}(1)$ of a geodesic $({\mathbf{y}}(t))_{t\in[0,1]}$ with ${\mathbf{y}}(0) = y$ and $\dot {\mathbf{y}}(0) = v$. This map is called the exponential map with $\exp_y(v):= {\mathbf{y}}(1)$.
The discrete geodesic calculus is based on a local, usually easily computable approximation of the squared Riemannian distance ${\mathrm{dist}}^2$ by a smooth functional ${\mathcal{W}}: {\mathcal{M}}\times{\mathcal{M}}\to {\mathbb{R}}$ with an extension of ${\mathcal{W}}$ to ${\mathbf{Y}}\times{\mathbf{Y}}$ by $\infty$. ${\mathcal{W}}$ is assumed to be weakly lower semi-continuous and locally consistent with the squared Riemannian distance in the sense $$\begin{aligned}
{\mathcal{W}}{[{y}, \tilde{y}]}={\mathrm{dist}}^2({y},\tilde{y})+O({\mathrm{dist}}^3({y},\tilde{y}))\,. \label{eq:consW}\end{aligned}$$ The consistency implies ${\mathcal{W}}{[{y}, {y}]} = 0\,,\quad {\mathcal{W}}_{,2}{[{y}, {y}]}(v) = 0\,,\quad {\mathcal{W}}_{,22}{[{y}, {y}]}(v,w) = 2 g_{y}(v,w)$ for any $v,w\in{\mathbf{V}}$. Here, ${\mathcal{W}}_{,j}$ denotes the derivative with respect to the $j$th component, and in analogy ${\mathcal{W}}_{,ij}$ the second order derivative in the $i$th and $j$th component. Furthermore, we also observe [@RuWi12b Lemma 4.6] that ${\mathcal{W}}_{,1}{[{y}, {y}]}(v) = 0$ and $$\begin{aligned}
\label{eq:secondDerivIdentities}
{\mathcal{W}}_{,11}{[{y}, {y}]}(v,w) = -{\mathcal{W}}_{,12}{[{y}, {y}]}(v,w) = -{\mathcal{W}}_{,21}{[{y}, {y}]}(v,w) = {\mathcal{W}}_{,22}{[{y}, {y}]}(v,w)\,.\end{aligned}$$ Differentiating once again we achieve for any $u,v,w\in{\mathbf{V}}$ that $$\begin{aligned}
\label{eq:thirdDerivSymm}
\begin{split}
&{\mathcal{W}}_{,221}{[{y}, {y}]}(u,v,w)+{\mathcal{W}}_{,222}{[{y}, {y}]}(u,v,w)={\mathcal{W}}_{,111}{[{y}, {y}]}(u,v,w)+{\mathcal{W}}_{,112}{[{y}, {y}]}(u,v,w)\\
=&-{\mathcal{W}}_{,121}{[{y}, {y}]}(u,v,w)-{\mathcal{W}}_{,122}{[{y}, {y}]}(u,v,w)=-{\mathcal{W}}_{,211}{[{y}, {y}]}(u,v,w)-{\mathcal{W}}_{,212}{[{y}, {y}]}(u,v,w)\,.
\end{split}\end{aligned}$$ With the function ${\mathcal{W}}$ at hand one can define the discrete path energy $$\begin{aligned}
{{\mathbf{E}}}[({y}_0,\ldots,{y}_K)] =K\sum_{k=1}^K {\mathcal{W}}{[{y}_{k-1}, {y}_k]} \label{eq:discretePathEnergy}\end{aligned}$$ on a discrete $K$-path $({y}_0,\ldots,{y}_K)\in{\mathcal{M}}^{K+1}$. A *discrete geodesic* of order $K$ (or $K$-geodesic) is then defined as a local minimizer of ${\mathbf{E}}[({y}_0,\ldots,{y}_K)]$ for fixed end points ${y}_0 = {y}_A$ and ${y}_K = {y}_B$. The Euler–Lagrange conditions of a discrete geodesic are $$\begin{aligned}
\label{eq:EL}
{\mathcal{W}}_{,2}[{y}_{k-1},{y}_{k}] + {\mathcal{W}}_{,1}[{y}_{k},{y}_{k+1}] =0\end{aligned}$$ for $k=1,\ldots, K-1$. It is shown in [@RuWi12b Theorems 4.1 & 4.2] that under the above assumptions continuous geodesics exists and are locally unique (for ${\mathrm{dist}}({y}_A,{y}_B)$ sufficiently small). Furthermore, [@RuWi12b Theorems 4.3 & 4.7] ensure the existence of discrete geodesics and their local uniqueness. Finally, using a suitable extension via piecewise geodesic paths it is shown in [@RuWi12b Theorem 4.8] that the discrete path energy $\Gamma$–converges to the continuous path energy for $K\to \infty$ and that due to equi-coerciveness discrete (extended) minimizers converge to a continuous minimizer. If $({y}_0, {y}_1, \ldots, y_K)$ is a discrete $K$-geodesic connecting ${y}_0 = {y}_A$ and ${y}_K = {y}_B$, the displacement $K ({y}_1-{y}_0) \in {\mathbf{V}}$ is considered as the discrete logarithm of ${y}_B$ at ${y}_A$. Vice versa, if ${y}_0$ and ${y}_1 ={y}_0+v$ are given for $v\in{\mathbf{V}}$ one can iteratively solve for ${y}_{k+1}$ given ${y}_k$ and ${y}_{k-1}$ and compute a discrete exponential map which is consistent with the discrete logarithm by construction. For the analysis of convergence to the corresponding continuous counterparts we refer the reader to [@RuWi12b Theorems 5.1 & 5.10].
#### Covariant derivative.
The differentiation of vector fields on manifolds leads to the notion of the covariant derivative. Before we proceed, let us fix the notation and introduce some abbreviations. In the following, let ${y}\in {\mathcal{M}}$ be an arbitrary but fixed point and $v\in {\mathbf{V}}$ an arbitrary but fixed tangent vector. For some $\epsilon > 0$, let ${\mathbf{y}}= ({\mathbf{y}}(t))_{t\in (-\epsilon,\epsilon)}$ be a path with ${\mathbf{y}}(0) = {y}$ and $\dot {\mathbf{y}}(0) = v$. Furthermore, we consider a vector field $w$ along the path, $t \mapsto w({\mathbf{y}}(t))$ for all $|t| < \epsilon$, and the corresponding covariant derivative $t \mapsto {{\frac{D}{{\,\mathrm{d}}t}}}w({\mathbf{y}}(t))$, which is again a vector field along the path for all $|t| < \epsilon$. The covariant derivative ${{\frac{D}{{\,\mathrm{d}}t}}}w$ of $w$ along the path ${\mathbf{y}}$ is uniquely defined – due to the coercivity of the metric – by $$\begin{aligned}
\label{eq:defCovDeriv}
g_{{\mathbf{y}}(t)}\left({{\frac{D}{{\,\mathrm{d}}t}}}(w \circ {\mathbf{y}})(t) , z\right) = g_{{\mathbf{y}}(t)}\left(\tfrac{\mathrm{d}}{\mathrm{d} t}(w \circ {\mathbf{y}})(t), z\right)+ g_{{\mathbf{y}}(t)}(\Gamma_{{\mathbf{y}}(t)}(w\circ {\mathbf{y}}(t),\dot {\mathbf{y}}(t)),z)
\end{aligned}$$ for all $z\in {\mathbf{V}}$ and all $t\in (-\epsilon,\epsilon)$, where the dot expresses differentiation with respect to $t$. Here, the Christoffel operator $\Gamma: {\mathcal{M}}\times {\mathbf{V}}\times {\mathbf{V}}\to {\mathbf{V}};\; ({y},v,w) \mapsto \Gamma_{y}(v,w)$ is defined by $$\begin{aligned}
2 g_{y}(\Gamma_{{y}}(v,w),z) = \left(D_{y}g_{y}\right)(w)(v,z) - \left(D_{y}g_{y}\right)(z)(v,w) + \left(D_{y}g_{y}\right)(v)(w,z)\end{aligned}$$ for all tangent vectors $z \in {\mathbf{V}}$. Note that ${{\frac{D}{{\,\mathrm{d}}t}}}(w\circ {\mathbf{y}})(0)$ does depend on the point ${y}={\mathbf{y}}(0)$ and the direction $v \in {\mathbf{V}}$ but not on the specific choice of the path, for which reason we will also use the short notation
$
{{\frac{D}{{\,\mathrm{d}}t}}}w({y})
\quad\text{instead of}\quad
{{\frac{D}{{\,\mathrm{d}}t}}}(w\circ{\mathbf{y}})(0)
$
whenever the tangent vector $v$ is clear from the context. A vector field $w$ is parallel along a curve ${\mathbf{y}}$ if $ {{\frac{D}{{\,\mathrm{d}}t}}}(w \circ {\mathbf{y}})(t)=0$ for all $t$. For given $w({\mathbf{y}}(0))$ a parallel vector field along ${\mathbf{y}}$ can be generated solving the differential equation $\frac{\,\mathrm{d}}{{\,\mathrm{d}}t}(w\circ {\mathbf{y}})(t) = - \Gamma_{{\mathbf{y}}(t)}(w\circ {\mathbf{y}}(t),\dot {\mathbf{y}}(t))$ with initial data $w({\mathbf{y}}(0))$. The associated parallel transport map ${{\mathrm{P}}}_{{\mathbf{y}}(\tau \leftarrow 0)} w({\mathbf{y}}(0)) = w({\mathbf{y}}(\tau))$, which maps initial data $w({\mathbf{y}}(0)) \in {\mathbf{V}}$ to output data $w({\mathbf{y}}(\tau))\in {\mathbf{V}}$ for $\tau\in {\mathbb{R}}$ and $w$ parallel along ${\mathbf{y}}$, is a linear isomorphism from ${\mathbf{V}}$ to ${\mathbf{V}}$. Here, we use the notation ${\mathbf{y}}(\tau \leftarrow 0)$ for the curve segment $s \mapsto {\mathbf{y}}(s\tau)$ for $s \in [0,1]$. The covariant derivative at ${y}= {\mathbf{y}}(0)$ can be written as the limit of difference quotients, $$\begin{aligned}
{{\frac{D}{{\,\mathrm{d}}t}}}(w\circ {\mathbf{y}})(0) = \lim_{\tau \to 0} \frac{ {{\mathrm{P}}}_{{\mathbf{y}}(\tau \leftarrow 0)}^{-1} \tau w({\mathbf{y}}(\tau)) - \tau w({\mathbf{y}}(0))}{\tau^2}\,,\end{aligned}$$ where ${{\mathrm{P}}}_{{\mathbf{y}}(\tau \leftarrow 0)}^{-1} = {{\mathrm{P}}}_{{\mathbf{y}}(0 \leftarrow \tau)}$ (the above notation is chosen for consistency with the discrete approximation to be introduced further below).
A discrete parallel transport can be defined via an iterative construction of (discrete) geodesic parallelograms called Schild’s ladder. However, for the definition of discrete curvature it suffices to introduce only a single step of this scheme. For ${y}\in \mathring{{\mathcal{M}}}$ and sufficiently small vectors $w,\, v\in {\mathbf{V}}$ such a single step corresponds to the discrete transport of $w$ along the line segment from ${y}$ to ${y}+v$.
The single step of Schild’s ladder is achieved by constructing a discrete geodesic parallelogram ${y},{y}+v,z,{y}+w$ (where $({y},{y}+v)$, $({y},{y}+w)$ and $({y}+v,z)$ are viewed as discrete $1$-geodesics) and taking $z-({y}+v)\in{\mathbf{V}}$ as the result of the discrete transport. In the following we explain this procedure in detail. First, we compute a discrete $2$-geodesic $({y}+w, {c}, {y}+v)$, where existence and uniqueness of ${c}$ follows from [@RuWi12b Theorems 4.3 & 4.7] for $\|v\|_{\mathbf{V}}$ and $\|w\|_{\mathbf{V}}$ sufficiently small. Then, we determine $z \in {\mathcal{M}}$ such that $({y},{c}, z)$ is a discrete $2$-geodesic. This time, existence and uniqueness is ensured by [@RuWi12b Lemma 5.6] for $\|v\|_{\mathbf{V}},\,\|w\|_{\mathbf{V}}$ sufficiently small. Note that the constructed $2$-geodesics form the diagonals of the parallelogram with $c$ being the center point. Finally, the discrete transport of $w$ along the line segment from $y$ to $y+v$ is given by $$\begin{aligned}
\label{eq:Ptransport}
{{\mathbf{P}}}_{{y}+v,{y}} w \coloneqq z - ({y}+v)\,.\end{aligned}$$ The system of Euler–Lagrange equations associated with reads $$\begin{aligned}
\label{eq:ELeqTransp}
\begin{split}
{\mathcal{W}}_{,2}[{y}+w,{c}] + {\mathcal{W}}_{,1}[{c},{y}+v] &=0,\\
{\mathcal{W}}_{,2}[{y},{c}] + {\mathcal{W}}_{,1}[{c},z] &=0
\end{split}\end{aligned}$$ for given ${y},\,v,\, w$ and unknown ${c}$ and $z$. Iterating this scheme along a polygonal curve on $\mathring{{\mathcal{M}}}$ one can define a consistent discrete transport of a vector along that polygonal curve. Let us remark that we actually need to introduce a scaling of the vector field $w$ and $v$ by $\tau$ before being transported and the corresponding rescaling afterwards to ensure consistency ([@RuWi12b Theorem 5.11]). The transport map is invertible for small $w$ and $v$. Note that in general ${{\mathbf{P}}}_{{y}+v,{y}}^{-1}$ does not coincide with ${{\mathbf{P}}}_{{y},{y}+v}$ because ${\mathcal{W}}$ is not assumed to be symmetric. With this discrete parallel transport at hand we can finally define a one-sided covariant difference quotient of a general vector field $w$ along the (polygonal) path ${\mathbf{y}}(t)={y}+tv$ by $$\begin{aligned}
\label{eq:CovPlus}
{{\frac{\mathbf{{D}}^{{\tau}}}{{\,\mathrm{d}}t}}} w({y})\coloneqq\frac{ {{\mathbf{P}}}_{{y}+\tau v,{y}}^{-1} \tau w({y}+\tau v) - \tau w({y})}{\tau^2}\end{aligned}$$ (again we suppress the vector $v\in{\mathbf{V}}$ in the notation since it will always be clear from the context). Depending on the sign of $\tau$ this is a forward ($\tau >0$) or a backward ($\tau <0$) difference quotient. Computing is based on solving the following system of Euler–Lagrange equations associated with the *inverse* discrete parallel transport ${{\mathbf{P}}}_{{y}+\tau v,{y}}^{-1}(\tau w) = z -y$ for given ${y},\,v,\, w$ and unknowns ${c},z$: $$\begin{aligned}
\label{eq:ELeqInverseTransp}
\begin{split}
{\mathcal{W}}_{,2}[z,{c}] + {\mathcal{W}}_{,1}[{c},{y}+\tau v] &=0,\\
{\mathcal{W}}_{,2}[{y},{c}] + {\mathcal{W}}_{,1}[{c},{y}+\tau (v+w)] &=0.
\end{split}\end{aligned}$$
Consistency of discrete covariant derivatives {#sec:cov}
=============================================
Instead of the one-sided covariant difference quotient defined in one can also consider a central covariant difference quotient of a vector field $w$ along the (polygonal) path ${\mathbf{y}}(t)={y}+tv$ by $$\begin{aligned}
\label{eq:CovCentral}
{{\frac{\mathbf{{D}}^{{\pm\tau}}}{{\,\mathrm{d}}t}}} w({y}) = \frac{ {{\mathbf{P}}}_{{y}+\tau v,{y}}^{-1} \tau w({y}+\tau v) + {{\mathbf{P}}}_{{y}-\tau v,{y}}^{-1} \left(- \tau w({y}-\tau v) \right)}{2\tau^2}\, .\end{aligned}$$ Above, ${{\mathbf{P}}}_{{y}-\tau v,{y}}^{-1} \left(- \tau w({y}-\tau v)\right)$ is the appropriately reflected construction of ${{\mathbf{P}}}_{{y}+\tau v,{y}}^{-1} \tau w({y}+\tau v)$. Note that it does in general not equal $- {{\mathbf{P}}}_{{y}-\tau v,{y}}^{-1} \left( \tau w({y}-\tau v)\right)$. Theorem \[lemma:consistency\] below states the consistency of both the one-sided covariant difference quotient and the central covariant difference quotient . The associated estimates will then be used to prove the consistency of different approximations of the continuous Riemann curvature tensor. In fact, a proof for the first order consistency of the one-sided covariant difference quotient can already be found in [@RuWi12b Theorem 5.13] as a corollary of the convergence of the discrete parallel transport. Here, we use a different approach based on the implicit function theorem, which is substantially simpler and also allows to prove the second order consistency of the central covariant difference quotient with little extra effort.
\[lemma:consistency\] Let and the assumptions from the beginning of Section\[sec:review\] hold true, and let $m\in{\mathbb{N}}$, ${y}\in \mathring{\mathcal{M}}$, $w\in C^{m-1}({\mathcal{M}};{\mathbf{V}})$, and ${\mathcal{W}}\in C^m({\mathcal{M}}\times{\mathcal{M}};{\mathbb{R}})$ (which by automatically implies ${g}\in C^{m-2}({\mathcal{M}}\times{\mathbf{V}}\times{\mathbf{V}};{\mathbb{R}})$). We consider the covariant derivative of $w$ at ${y}$ in a fixed given direction $v\in {\mathbf{V}}$. If $m\geq4$ we have $$\begin{aligned}
\left\|{{\frac{\mathbf{{D}}^{{\tau}}}{{\,\mathrm{d}}t}}} w ({y}) - {{\frac{D}{{\,\mathrm{d}}t}}}w ({y}) \right\|_{\mathbf{V}}\leq C \tau\end{aligned}$$ for $\tau$ sufficiently small. Furthermore, if $m\geq5$, then we have $$\begin{aligned}
\left\| {{\frac{\mathbf{{D}}^{{\pm \tau}}}{{\,\mathrm{d}}t}}} w ({y}) - {{\frac{D}{{\,\mathrm{d}}t}}}w ({y}) \right\|_{\mathbf{V}}\leq C \tau^2\end{aligned}$$ for ${y}\in \mathring{\mathcal{M}}$ and $\tau$ sufficiently small. In both estimates the constant $C>0$ is independent of $\tau$ and within sufficiently small balls $B\subset\mathring{\mathcal{M}}$ also independent of ${y}\in B$.
Without loss of generality we may assume ${y}=0$, and we consider $\tau \mapsto {\mathbf{w}}(\tau)= w\circ{\mathbf{y}}(\tau)$ to be the vector field along ${\mathbf{y}}(\tau) = 0 + \tau v$. By and the definition of ${{\mathbf{P}}}_{\tau v,0}^{-1} \tau w(\tau v)$ we have $\left( {{\frac{\mathbf{{D}}^{{\tau}}}{{\,\mathrm{d}}t}}} w \right)(0) = \tau^{-2} ( {z}(\tau)-\tau {\mathbf{w}}(0))$, where ${z}(\tau)$ is the fourth corner of the discrete geodesic parallelogram $0,\tau v,\tau v+\tau {\mathbf{w}}(\tau),{z}(\tau)$ with center ${c}(\tau)$ and where ${z}$ and ${c}$ satisfy .
Introducing $F: {\mathbb{R}}\times {\mathbf{V}}\times {\mathbf{V}}\to {\mathbf{V}}'\times{\mathbf{V}}'$ with $$\begin{aligned}
\label{eq:F}
F(\tau, {z}, {c})(r) =
\begin{pmatrix}
{\mathcal{W}}_{,2}{[{z}, {c}]}(r) + {\mathcal{W}}_{,1}{[{c}, \tau v]}(r) \\
{\mathcal{W}}_{,2}{[0, {c}]}(r) + {\mathcal{W}}_{,1}[{c},\tau v+ \tau {\mathbf{w}}(\tau)](r)
\end{pmatrix}\end{aligned}$$ for $r\in {\mathbf{V}}$, the condition on ${z}(\tau)$ and ${c}(\tau)$ can be rewritten as $F(\tau, {z}(\tau), {c}(\tau))(r) = 0$ for all $r\in{\mathbf{V}}$. In what follows, we skip the tangent vector $r$ to which $F$ is applied, remembering that in all derivatives of ${\mathcal{W}}$ the first differentiation is in direction $r$. Now the implicit function theorem tells us that there exists a neighbourhood $U$ of the origin in ${\mathbb{R}}\times {\mathbf{V}}\times {\mathbf{V}}$ such that the set $\{(\tau, {z}, {c})\in U\,|\,F(\tau, {z}, {c}) =0\}$ is the graph of a map $\tau \mapsto ({z}, {c})(\tau)$, $$F(\tau, {z}(\tau), {c}(\tau)) = 0\,.$$ Furthermore, we can differentiate with respect to $\tau$ and obtain $$\begin{aligned}
\label{eq:dTauFequalsZero}
0= \frac{\mathrm{d}}{\mathrm{d} \tau} F(\tau, {z}(\tau), {c}(\tau)) = \partial_\tau F(\tau, {z}(\tau), {c}(\tau)) + \partial_{({z},{c})} F(\tau, {z}(\tau), {c}(\tau))
\begin{pmatrix} \dot {z}(\tau)\\ \dot {c}(\tau) \end{pmatrix}\,,\end{aligned}$$ where the dot expresses differentiation with respect to $\tau$. Inserting into we obtain the block operator equation $$\begin{aligned}
\label{eq:IFTderivative}
\begin{pmatrix}
{\mathcal{W}}_{,21}{[{z}, {c}]} & {\mathcal{W}}_{,22}{[{z}, {c}]} + {\mathcal{W}}_{,11}{[{c}, \tau v]} \\
0 & {\mathcal{W}}_{,22}{[0, {c}]} + {\mathcal{W}}_{,11}{[{c}, \tau v+ \tau {\mathbf{w}}]}
\end{pmatrix}
\begin{pmatrix} \dot {z}\\ \dot {c}\end{pmatrix}
= -
\begin{pmatrix}
{\mathcal{W}}_{,12}{[{c}, \tau v]}(v) \\
{\mathcal{W}}_{,12}{[{c}, \tau v+ \tau {\mathbf{w}}]} (v+ {\mathbf{w}}+ \tau \dot {\mathbf{w}})
\end{pmatrix}
\,,\end{aligned}$$ where we abbreviated ${c}= {c}(\tau)$, ${z}= {z}(\tau)$ and ${\mathbf{w}}= {\mathbf{w}}(\tau)$. Obviously, we have $({z}(0), {c}(0))=(0,0)$. Using $2g = {\mathcal{W}}_{,11}{[0, 0]} = {\mathcal{W}}_{,22}{[0, 0]} = - {\mathcal{W}}_{,12}{[0, 0]}$ for $g=g_0$ we can evaluate this operator equation as $$\begin{aligned}
\label{eq:IFTderivativeZero}
\begin{pmatrix}
-2g & 4g \\
0 & 4g
\end{pmatrix}
\begin{pmatrix} \dot {z}(0)\\ \dot {c}(0) \end{pmatrix}
=
\begin{pmatrix}
2g \,v \\
2g \,v + 2g \,{\mathbf{w}}(0)
\end{pmatrix}
\,,\end{aligned}$$ which leads to $\dot {c}(0)=\tfrac12 (v+{\mathbf{w}}(0))$ and $\dot {z}(0)= {\mathbf{w}}(0)$. Next, we differentiate once more in $\tau$ and obtain $$\begin{aligned}
\label{eq:IFTsecondDerivative}
&\begin{pmatrix}
{\mathcal{W}}_{,21}{[{z}, {c}]} & {\mathcal{W}}_{,22}{[{z}, {c}]} + {\mathcal{W}}_{,11}{[{c}, \tau v]} \\
0 & {\mathcal{W}}_{,22}{[0, {c}]} + {\mathcal{W}}_{,11}{[{c}, \tau v+ \tau {\mathbf{w}}]}
\end{pmatrix}
\begin{pmatrix} \ddot {z}\\ \ddot {c}\end{pmatrix}
=
\begin{pmatrix}
R_1(\tau) \\
R_2(\tau)
\end{pmatrix}\, ,\end{aligned}$$ where $R_1$ and $R_2$ are functionals mapping $\tau$ into ${\mathbf{V}}'$ and act on $r \in {\mathbf{V}}$ with $$\begin{aligned}
R_1(\tau)(r) =& - {\mathcal{W}}_{,211}{[z, c]}(r,\dot {z},\dot {z}) - {\mathcal{W}}_{,212}{[z, c]}(r,\dot {z},\dot {c})- {\mathcal{W}}_{,221}{[z, c]}(r,\dot {c},\dot {z}) - {\mathcal{W}}_{,222}{[z, c]}(r,\dot {c},\dot {c})\\
& - {\mathcal{W}}_{,111}{[c, \tau v]}(r,\dot {c},\dot {c}) - {\mathcal{W}}_{,112}{[c, \tau v]}(r,\dot {c},v) - {\mathcal{W}}_{,121}{[c, \tau v]}(r,v,\dot {c}) - {\mathcal{W}}_{,122}{[c, \tau v]}(r,v,v)\\
=& - {\mathcal{W}}_{,211}{[z, c]}(r,\dot {z},\dot {z}) - 2{\mathcal{W}}_{,212}{[z, c]}(r,\dot {z},\dot {c}) - {\mathcal{W}}_{,222}{[z, c]}(r,\dot {c},\dot {c})\\
& - {\mathcal{W}}_{,111}{[c, \tau v]}(r,\dot {c},\dot {c}) - 2{\mathcal{W}}_{,112}{[c, \tau v]}(r,\dot {c},v) - {\mathcal{W}}_{,122}{[c, \tau v]}(r,v,v)\,, \\
R_2(\tau)(r) =& - {\mathcal{W}}_{,222}{[0, c]}(r,\dot {c},\dot {c}) - {\mathcal{W}}_{,111}{[c, \tau (v+{\mathbf{w}})]}(r,\dot {c},\dot {c}) - {\mathcal{W}}_{,112}{[c, \tau (v+{\mathbf{w}})]}(r,\dot {c},v+{\mathbf{w}}+\tau \dot {\mathbf{w}})\\
& - {\mathcal{W}}_{,121}{[c, \tau (v+{\mathbf{w}})]}(r,v+{\mathbf{w}}+\tau \dot {\mathbf{w}},\dot {c}) - {\mathcal{W}}_{,122}{[c, \tau (v+{\mathbf{w}})]}(r,v+{\mathbf{w}}+\tau \dot {\mathbf{w}},v+{\mathbf{w}}+\tau \dot {\mathbf{w}})\\
& - {\mathcal{W}}_{,12}{[c, \tau (v+{\mathbf{w}})]} (r, 2 \dot {\mathbf{w}}+ \tau \ddot {\mathbf{w}}) \\
=& - {\mathcal{W}}_{,222}{[0, c]}(r,\dot {c},\dot {c}) - {\mathcal{W}}_{,111}{[c, \tau (v+{\mathbf{w}})]}(r,\dot {c},\dot {c}) - 2{\mathcal{W}}_{,112}{[c, \tau (v+{\mathbf{w}})]}(r,\dot {c},v+{\mathbf{w}}+\tau \dot {\mathbf{w}})\\
& - {\mathcal{W}}_{,122}{[c, \tau (v+{\mathbf{w}})]}(r,v+{\mathbf{w}}+\tau \dot {\mathbf{w}},v+{\mathbf{w}}+\tau \dot {\mathbf{w}})- {\mathcal{W}}_{,12}{[c, \tau (v+{\mathbf{w}})]} (r, 2 \dot {\mathbf{w}}+ \tau \ddot {\mathbf{w}}) \, .\end{aligned}$$ Evaluating at $\tau=0$ gives $$\begin{aligned}
\label{eq:2nd}
&\begin{pmatrix}
-2g &4g \\
0 & 4g
\end{pmatrix}
\begin{pmatrix} \ddot {z}(0)\\ \ddot {c}(0) \end{pmatrix}
=
\begin{pmatrix}
R_1(0) \\
R_2(0)
\end{pmatrix}\end{aligned}$$ with $$\begin{aligned}
R_1(0)(r) =& -{\mathcal{W}}_{,211}{[0, 0]}(r,{\mathbf{w}}(0),{\mathbf{w}}(0)) - {\mathcal{W}}_{,212}{[0, 0]}(r,{\mathbf{w}}(0),v+{\mathbf{w}}(0)) \\
& -\tfrac14 {\mathcal{W}}_{,222}{[0, 0]}(r,v+{\mathbf{w}}(0),v+{\mathbf{w}}(0)) -\tfrac14 {\mathcal{W}}_{,111}{[0, 0]}(r,v+{\mathbf{w}}(0),v+{\mathbf{w}}(0))\\
& -{\mathcal{W}}_{,112}{[0, 0]}(r,v+{\mathbf{w}}(0),v) -{\mathcal{W}}_{,122}{[0, 0]}(r,v,v)\,, \\
R_2(0)(r) =& -\tfrac14 {\mathcal{W}}_{,222}{[0, 0]}(r,v+{\mathbf{w}}(0),v+{\mathbf{w}}(0)) -\tfrac14 {\mathcal{W}}_{,111}{[0, 0]}(r,v+{\mathbf{w}}(0),v+{\mathbf{w}}(0)) \\
& -{\mathcal{W}}_{,112}{[0, 0]}(r,v+{\mathbf{w}}(0),v+{\mathbf{w}}(0)) -{\mathcal{W}}_{,122}{[0, 0]}(r,v+{\mathbf{w}}(0),v+{\mathbf{w}}(0)) - {\mathcal{W}}_{,12}{[0, 0]} (r, 2\dot {\mathbf{w}}(0)) \,,\end{aligned}$$ where we used $\dot {c}(0)=\tfrac12 (v+{\mathbf{w}}(0))$ and $\dot {z}(0) = {\mathbf{w}}(0)$ from above. Solving yields $g(\ddot c(0),r) = \tfrac{R_2(0)(r)}4$ and $$\begin{aligned}
g(\ddot z(0),r) = &\frac12 R_2(0)(r) - \frac12 R_1(0)(r) \notag\\
=& \tfrac12\big(
[-{\mathcal{W}}_{,121}-{\mathcal{W}}_{,122}]{[0, 0]}(r,v+{\mathbf{w}}(0),v+{\mathbf{w}}(0))
+{\mathcal{W}}_{,212}{[0, 0]}(r,{\mathbf{w}}(0),v+{\mathbf{w}}(0))\notag\\
&+{\mathcal{W}}_{,112}{[0, 0]}(r,v+{\mathbf{w}}(0),v)
+{\mathcal{W}}_{,122}{[0, 0]}(r,v,v)
+{\mathcal{W}}_{,211}{[0, 0]}(r,{\mathbf{w}}(0),{\mathbf{w}}(0))
\big)\notag\\
&- {\mathcal{W}}_{,12}{[0, 0]} (r, \dot {\mathbf{w}}(0)) \notag\\
=&\tfrac12\big(
[-{\mathcal{W}}_{,121}-{\mathcal{W}}_{,122}+{\mathcal{W}}_{,212}+{\mathcal{W}}_{,211}]{[0, 0]}(r,{\mathbf{w}}(0),{\mathbf{w}}(0))\notag\\
&+[-{\mathcal{W}}_{,121}-{\mathcal{W}}_{,122}+{\mathcal{W}}_{,112}+{\mathcal{W}}_{,122}]{[0, 0]}(r,v,v)\notag\\
&+[-{\mathcal{W}}_{,121}-{\mathcal{W}}_{,112}-2{\mathcal{W}}_{,122}+{\mathcal{W}}_{,212}+{\mathcal{W}}_{,112}]{[0, 0]}(r,{\mathbf{w}}(0),v)
\big)- {\mathcal{W}}_{,12}{[0, 0]} (r, \dot {\mathbf{w}}(0)) \notag\\
=&-\tfrac12[{\mathcal{W}}_{,211}+{\mathcal{W}}_{,122}]{[0, 0]}(r,v,{\mathbf{w}}(0)) - {\mathcal{W}}_{,12}{[0, 0]} (r, \dot {\mathbf{w}}(0))\,.\label{eq:metricZConsistency}\end{aligned}$$ Note that implies $(-{\mathcal{W}}_{,121}-{\mathcal{W}}_{,122}+{\mathcal{W}}_{,212}+{\mathcal{W}}_{,211}){[0, 0]} = 0$ and $(-{\mathcal{W}}_{,121}-{\mathcal{W}}_{,122}+{\mathcal{W}}_{,212}){[0, 0]} = -{\mathcal{W}}_{,211}{[0, 0]}$ which was used in the last step.
From $g(v,w) = \tfrac12 {\mathcal{W}}_{,11}{[0, 0]}(v,w)$ we deduce $$D_{y}g(r)(v,w) = \tfrac12 \left({\mathcal{W}}_{,111}{[0, 0]}(v,w,r) + {\mathcal{W}}_{,112}{[0, 0]}(v,w,r) \right)\,.$$ Hence, using the definition of the covariant derivative in , we obtain at ${\mathbf{y}}(0) ={y}= 0$ $$\begin{aligned}
&g\left({{\frac{D}{{\,\mathrm{d}}t}}}w({y}),r\right) =
g(\dot {\mathbf{w}}(0),r) + g(\Gamma_0({\mathbf{w}}(0),v),r)\\
=& - \frac12 {\mathcal{W}}_{,12}{[0, 0]} (r, \dot {\mathbf{w}}(0)) + \frac12 \left(D_{y}g({\mathbf{w}}(0))(r,v) +D_{y}g(v)(r,{\mathbf{w}}(0))-D_{y}g(r)(v,{\mathbf{w}}(0))\right)\\
=& \frac14 \big( {\mathcal{W}}_{,111}{[0, 0]}(r,v,{\mathbf{w}}(0)) + {\mathcal{W}}_{,112}{[0, 0]}(r,v,{\mathbf{w}}(0)) + {\mathcal{W}}_{,111}{[0, 0]}(r,{\mathbf{w}}(0),v) \\
&+ {\mathcal{W}}_{,112}{[0, 0]}(r,{\mathbf{w}}(0),v) - {\mathcal{W}}_{,111}{[0, 0]}(v,{\mathbf{w}}(0),r) - {\mathcal{W}}_{,112}{[0, 0]}(v,{\mathbf{w}}(0),r) \big) - \frac12 {\mathcal{W}}_{,12}{[0, 0]} (r, \dot {\mathbf{w}}(0))\\
=& \frac14 \big( {\mathcal{W}}_{,111} + {\mathcal{W}}_{,112} + {\mathcal{W}}_{,121} - {\mathcal{W}}_{,211}\big){[0, 0]}(r,v,{\mathbf{w}}(0)) - \frac12 {\mathcal{W}}_{,12}{[0, 0]} (r, \dot {\mathbf{w}}(0)) \\
=& -\frac14 \big( {\mathcal{W}}_{,122} + {\mathcal{W}}_{,211}\big){[0, 0]}(r,v,{\mathbf{w}}(0)) - \frac12 {\mathcal{W}}_{,12}{[0, 0]} (r, \dot {\mathbf{w}}(0))\,,\end{aligned}$$ where implied $({\mathcal{W}}_{,111} + {\mathcal{W}}_{,112}){[0, 0]} = -({\mathcal{W}}_{,121} + {\mathcal{W}}_{,122}){[0, 0]}$ in the last step. Then, taking into account we obtain $$\begin{aligned}
g\left( \frac12\ddot z(0),r\right) = g\left({{\frac{D}{{\,\mathrm{d}}t}}}w(0),r\right)\, .\end{aligned}$$ Next, we consider the Taylor expansion of $\tau \mapsto {z}(\tau)$ at $\tau =0$, $$\begin{aligned}
{z}(\tau) =& {z}(0) + \dot z(0) \tau + \frac12 \ddot {z}(0) \tau^2 + \frac16 \dddot {z}(0) \tau^3 + R(\tau)\\
=& 0 + w(0) \tau + {{\frac{D}{{\,\mathrm{d}}t}}}w(0) \,\tau^2 + \frac16 \dddot {z}(0) \tau^3 + R(\tau)\end{aligned}$$ with the remainder term $R(\tau)=o(\tau^3)$ if ${\mathcal{W}}\in C^4({\mathcal{M}}\times{\mathcal{M}};{\mathbb{R}})$ and $R(\tau)=O(\tau^4)$ if ${\mathcal{W}}\in C^5({\mathcal{M}}\times{\mathcal{M}};{\mathbb{R}})$. This implies $$\begin{aligned}
{{\frac{D}{{\,\mathrm{d}}t}}}w(0) =& \frac1{\tau^2} \left( {z}(\tau)-\tau w(0)\right) + O(\tau)
={{\frac{\mathbf{{D}}^{{\tau}}}{{\,\mathrm{d}}t}}} w(0) + O(\tau)\,,\end{aligned}$$ which establishes first order consistency of the one-sided covariant difference quotient. Similarly, for the central covariant difference quotient we obtain $$\begin{aligned}
{{\frac{\mathbf{{D}}^{{\pm \tau}}}{{\,\mathrm{d}}t}}} w(0)
&=\frac12\left( \frac1{\tau^2} \left( {z}(\tau)-\tau w(0)\right) + \frac1{\tau^2} \left( {z}(-\tau)+\tau w(0)\right)\right)
= \frac{{z}(\tau) + {z}(-\tau)}{2\tau^2} \\
&= \frac1{2\tau^2}\Big(w(0) \tau + {{\frac{D}{{\,\mathrm{d}}t}}}w(0)\, \tau^2 +\frac{1}{6}\dddot {z}(0) \tau^3 + O(\tau^4)
- w(0) \tau +{{\frac{D}{{\,\mathrm{d}}t}}}w(0)\, \tau^2 -\frac{1}{6}\dddot {z}(0) \tau^3 + O(\tau^4)\Big) \\
&= {{\frac{D}{{\,\mathrm{d}}t}}}w (0) + O(\tau^2)\,.\end{aligned}$$ The differentiability requirements in the statement of the theorem directly follow by inspection of .
Discrete Approximation of the Riemann Curvature Tensor {#sec:curvatureTensor}
======================================================
For ${y}\in {\mathcal{M}}$ and a pair of tangent vectors $v,\,w \in {\mathbf{V}}$ the Riemannian curvature tensor is a linear mapping ${\mathrm{R}}_{y}(v,w) :{\mathbf{V}}\to {\mathbf{V}}$ with $$\label{eqn:RiemannCurvature}
\left({\mathrm{R}}_{y}(v,w)z\right)= \left({{\frac{D}{{\,\mathrm{d}}t}}}{{\frac{D}{{\,\mathrm{d}}s}}}z - {{\frac{D}{{\,\mathrm{d}}s}}}{{\frac{D}{{\,\mathrm{d}}t}}}z \right)({y})\,,$$ where $z$ is an extension of the tangent vector $z$ by a smooth vector field (in our context, for instance, by a vector field constant on ${\mathcal{M}}$) and ${\mathbf{y}}(t,s)$ a smooth mapping ${\mathbf{y}}: [-\epsilon, \epsilon]^2 \to {\mathcal{M}}$ with ${\mathbf{y}}(0,0)={y}$, $\partial_t {\mathbf{y}}(0,0) = v$ and $\partial_s {\mathbf{y}}(0,0) = w$ for some $\epsilon >0$. Examples of such mappings are ${\mathbf{y}}(t,s) =\exp_{y}(sw+tv)$ on general manifolds. In the case ${\mathcal{M}}\subset {\mathbf{V}}$ we simply choose ${\mathbf{y}}(t,s)={y}+sw+tv$. To define a discrete counterpart we consider nested covariant difference quotients with a smaller step size in the inner difference quotient. In detail, using the first order one-sided covariant difference quotient we define $$\label{eq:RTensor}
\left({\mathbf{R}}^\tau_{y}(v,w)z\right)= \left({{\frac{\mathbf{{D}}^{{\tau}}}{{\,\mathrm{d}}t}}} {{\frac{\mathbf{D}^{{\tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{\mathbf{D}^{{\tau}}}{{\,\mathrm{d}}s}}} {{\frac{\mathbf{{D}}^{{\tau^\beta}}}{{\,\mathrm{d}}t}}} z \right)({y})$$ with $\beta\geq2$. Using the central covariant difference quotient we obtain the corresponding discrete Riemannian curvature tensor $$\begin{aligned}
\left({\mathbf{R}}^{\pm\tau}_{y}(v,w)z\right)= \left({{\frac{\mathbf{{D}}^{{\pm\tau}}}{{\,\mathrm{d}}t}}} {{\frac{\mathbf{D}^{{\pm\tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{\mathbf{D}^{{\pm\tau}}}{{\,\mathrm{d}}s}}} {{\frac{\mathbf{{D}}^{{\pm\tau^\beta}}}{{\,\mathrm{d}}t}}} z \right)({y})\end{aligned}$$ with $\beta \geq \frac32$. The consistency results for the different discrete Riemannian curvature tensors are given in the following theorem.
\[thm:consRiemCurvTensor\] Let and the assumptions from the beginning of Section\[sec:review\] hold, and let $m\in{\mathbb{N}}$ as well as ${\mathcal{W}}\in C^{m}({\mathcal{M}}\times{\mathcal{M}};{\mathbb{R}})$ and ${g}\in C^{m}({\mathcal{M}}\times{\mathbf{V}}\times{\mathbf{V}};{\mathbb{R}})$. For ${y}\in \mathring{\mathcal{M}}$, $v,w,z \in {\mathbf{V}}$ and $\tau$ sufficiently small we have $$\begin{aligned}
\left({\mathbf{R}}^{\tau}_{y}(v,w)z\right) = \left({\mathrm{R}}_{y}(v,w)z\right) + O(\tau)\end{aligned}$$ with respect to the ${\mathbf{V}}$ norm for $m\geq4$ and $\beta\geq2$, and we have $$\begin{aligned}
\left({\mathbf{R}}^{\pm\tau}_{y}(v,w)z\right) = \left({\mathrm{R}}_{y}(v,w)z\right) + O(\tau^2)\end{aligned}$$ with respect to the ${\mathbf{V}}$ norm for $m\geq5$ and $\beta \geq \frac32$.
We first prove the estimate for the discrete Riemannian curvature tensor based on one-sided covariant difference quotients. It follows from the stability of the midpoint and the endpoint in $2$-geodesics [@RuWi12b Lemmas 5.5 & 5.8] that $$\begin{aligned}
\| {{\mathbf{P}}}_{{y}+\tau v,{y}}^{-1} w \|_{\mathbf{V}}\leq C \| w \|_{\mathbf{V}}\end{aligned}$$ for some constant $C>0$ and all $\tau$ and $\| w \|_{\mathbf{V}}$ sufficiently small. In fact, from the local differentiability of discrete geodesics with respect to their end points [@RuWi12b Theorem 4.7] and the local differentiability of the discrete exponential map with respect to base point and direction [@RuWi12b Lemma 5.6 & Remark 5.7] the discrete parallel transport is differentiable. Thus, the difference between the first term of and of can be estimated as $$\begin{aligned}
&\left\|\left({{\frac{\mathbf{{D}}^{{\tau}}}{{\,\mathrm{d}}t}}} {{\frac{\mathbf{D}^{{\tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{D}{{\,\mathrm{d}}t}}}{{\frac{D}{{\,\mathrm{d}}s}}}z \right)({y}) \right\|_{{\mathbf{V}}}
\leq \left\|{{\frac{\mathbf{{D}}^{{\tau}}}{{\,\mathrm{d}}t}}} \left( {{\frac{\mathbf{D}^{{\tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{D}{{\,\mathrm{d}}s}}}\right)({y})\right\|_{{\mathbf{V}}} + \left\|\left({{\frac{\mathbf{{D}}^{{\tau}}}{{\,\mathrm{d}}t}}} {{\frac{D}{{\,\mathrm{d}}s}}}z - {{\frac{D}{{\,\mathrm{d}}t}}}{{\frac{D}{{\,\mathrm{d}}s}}}z \right)({y})\right\|_{{\mathbf{V}}} \\
& \leq \frac1{\tau^2} \left\| \left( {{\mathbf{P}}}_{{y}+\tau v,{y}}^{-1} \tau \left( {{\frac{\mathbf{D}^{{\tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{D}{{\,\mathrm{d}}s}}}z \right) ({y}+\tau v) \right)
- \tau \left( {{\frac{\mathbf{D}^{{\tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{D}{{\,\mathrm{d}}s}}}z \right) ({y}) \right\|_{{\mathbf{V}}}+ O(\tau) \\
&\leq \frac1{\tau^2} (1+C) \tau \max_{\sigma \in \{0,1\}}\left\| \left( {{\frac{\mathbf{D}^{{\tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{D}{{\,\mathrm{d}}s}}}z \right) ({y}+\tau \sigma v) \right\|_{{\mathbf{V}}} + O(\tau)
= \frac{1+C}{\tau} O(\tau^\beta) + O(\tau) = O(\max\{\tau,\tau^{\beta-1}\})\,.\end{aligned}$$ The second term $\left({{\frac{\mathbf{D}^{{\tau}}}{{\,\mathrm{d}}s}}} {{\frac{\mathbf{{D}}^{{\tau^\beta}}}{{\,\mathrm{d}}t}}} z - {{\frac{D}{{\,\mathrm{d}}s}}}{{\frac{D}{{\,\mathrm{d}}t}}}z \right)({y})$ can be treated in analogy and thus the claim follows for the choice $\beta\geq2$. Let us remark that the above application of Theorem \[lemma:consistency\] requires ${\mathcal{W}}\in C^4({\mathcal{M}}\times{\mathcal{M}},{\mathbb{R}})$ and ${{\frac{D}{{\,\mathrm{d}}s}}}z \in C^3({\mathcal{M}};{\mathbf{V}})$, which is fulfilled if $g\in C^4({\mathcal{M}}\times{\mathbf{V}}\times {\mathbf{V}})$. In the case of the discrete Riemannian curvature tensor based on central covariant difference quotients we proceed along the same lines and obtain $$\begin{aligned}
\left\|\left({{\frac{\mathbf{{D}}^{{\pm \tau}}}{{\,\mathrm{d}}t}}} {{\frac{\mathbf{D}^{{\pm \tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{D}{{\,\mathrm{d}}t}}}{{\frac{D}{{\,\mathrm{d}}s}}}z \right)({y}) \right\|_{{\mathbf{V}}}
\leq& \left\|{{\frac{\mathbf{{D}}^{{\pm \tau}}}{{\,\mathrm{d}}t}}} \left( {{\frac{\mathbf{D}^{{\pm \tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{D}{{\,\mathrm{d}}s}}}\right)({y})\right\|_{{\mathbf{V}}} + \left\|\left({{\frac{\mathbf{{D}}^{{\pm \tau}}}{{\,\mathrm{d}}t}}} {{\frac{D}{{\,\mathrm{d}}s}}}z - {{\frac{D}{{\,\mathrm{d}}t}}}{{\frac{D}{{\,\mathrm{d}}s}}}z \right)({y})\right\|_{{\mathbf{V}}} \\
\leq& \frac{2C \tau}{\tau^2} \max_{\sigma \in \{-1,1\}} \left\| \left({{\frac{\mathbf{D}^{{\pm \tau^\beta}}}{{\,\mathrm{d}}s}}} z - {{\frac{D}{{\,\mathrm{d}}s}}}z \right) ({y}+\tau \sigma v)\right\|_{{\mathbf{V}}} + O(\tau^2)\\
= & \frac{2C}{2\tau} O(\tau^{2\beta}) + O(\tau^2) = O(\max\{\tau^{2\beta-1},\tau^2\})\end{aligned}$$ which for $\beta\geq\frac32$ implies the second order consistency of the discrete Riemannian curvature tensor. This time, the application of Theorem \[lemma:consistency\] requires ${\mathcal{W}}\in C^5({\mathcal{M}}\times{\mathcal{M}},{\mathbb{R}})$ and ${{\frac{D}{{\,\mathrm{d}}s}}}z \in C^4({\mathcal{M}};{\mathbf{V}})$, which is fulfilled if $g\in C^5({\mathcal{M}}\times{\mathbf{V}}\times {\mathbf{V}})$.
Finally, for ${y}\in\mathring{\mathcal{M}}$ and a pair of tangent vectors $v,\,w \in {\mathbf{V}}$ the sectional curvature $$\begin{aligned}
\kappa_{y}(v,w) := \frac{ {g}_{y}\left( v, {\mathrm{R}}_{y}(v,w)w\right) } { {g}_{y}(v,v){g}_{y}(w,w) - {g}_{y}^2(v,w)}\end{aligned}$$ is the Gaussian curvature of the surface $\{ \exp_{y}(sv + tw) : s,t\in (-\epsilon, \epsilon) \}$ at ${y}$. Having the notion of a consistent discrete Riemann curvature tensor, we define a corresponding discrete sectional curvature as $$\begin{aligned}
\mathbf{\kappa}_{y}^{\tau}(v,w) := \frac{ {g}_{y}\left( v, {\mathbf{R}}^\tau_{y}(v,w)w\right) } { {g}_{y}(v,v){g}_{y}(w,w) - {g}_{y}^2(v,w)}\end{aligned}$$ and likewise $\mathbf{\kappa}^{\pm\tau}(v,w)$ using ${\mathbf{R}}^{\pm\tau}$. Under the assumptions of Theorem \[thm:consRiemCurvTensor\] we then have $$\begin{aligned}
\left| \kappa_{y}- \mathbf{\kappa}_{y}^{\tau} \right| (v,w) &= \frac{ \left|{g}_{y}\left( v, \Big[{\mathrm{R}}_{y}(v,w) - {\mathbf{R}}^\tau_{y}(v,w)\Big] w\right) \right|} { \left|{g}_{y}(v,v){g}_{y}(w,w) - {g}_{y}^2(v,w) \right|} \leq C \left\| v\right\|_{\mathbf{V}}\, \left\| \Big[{\mathrm{R}}_{y}(v,w) - {\mathbf{R}}^\tau_{y}(v,w)\Big] w \right\|_{\mathbf{V}}= O(\tau)\, , \end{aligned}$$ where a second order consistency of $\mathbf{\kappa}_{y}^{\pm\tau}$ follows analogously. Altogether, we have proven the following:
\[thm:consSecCurv\] Let ${y}\in\mathring{\mathcal{M}}$ and $v,\,w \in {\mathbf{V}}$. Under the assumptions of Theorem \[thm:consRiemCurvTensor\] we have $\left| \kappa_{y}- \mathbf{\kappa}_{y}^{\tau} \right| (v,w) = O(\tau)$ and $|\kappa_{y}- \mathbf{\kappa}_{y}^{\pm\tau}|(v,w) = O(\tau^2)\, .$
Figures \[fig:torus\] and \[fig:torusConvergence\] illustrate the numerical computation and convergence of the sectional curvature on a two-dimensional embedded surface.
![Left: Gaussian curvature on two tori (each with center line radius of $\sqrt2$) numerically approximated via $\kappa^{\pm\tau}(v,w)$ for $\tau=10^{-2}$ and $\beta=\frac32$. The tori were parameterized by toroidal and poloidal angles, $v=(1,0)$, $w=(0,1)$, and ${\mathcal{W}}$ is taken as the squared Euclidean distance in the embedding space ${\mathbb{R}}^3$. The black arrows on the left indicate the position and tangent vectors for which Figure \[fig:torusConvergence\] shows a convergence study of the approximated sectional curvature. Right: Corresponding approximation error $(\kappa^{\pm\tau}-\kappa)(v,w)$. []{data-label="fig:torus"}](images/WillmoreTorusCurvature "fig:"){width="\unitlength"}![Left: Gaussian curvature on two tori (each with center line radius of $\sqrt2$) numerically approximated via $\kappa^{\pm\tau}(v,w)$ for $\tau=10^{-2}$ and $\beta=\frac32$. The tori were parameterized by toroidal and poloidal angles, $v=(1,0)$, $w=(0,1)$, and ${\mathcal{W}}$ is taken as the squared Euclidean distance in the embedding space ${\mathbb{R}}^3$. The black arrows on the left indicate the position and tangent vectors for which Figure \[fig:torusConvergence\] shows a convergence study of the approximated sectional curvature. Right: Corresponding approximation error $(\kappa^{\pm\tau}-\kappa)(v,w)$. []{data-label="fig:torus"}](images/ThinTorusCurvature "fig:"){width="\unitlength"}![Left: Gaussian curvature on two tori (each with center line radius of $\sqrt2$) numerically approximated via $\kappa^{\pm\tau}(v,w)$ for $\tau=10^{-2}$ and $\beta=\frac32$. The tori were parameterized by toroidal and poloidal angles, $v=(1,0)$, $w=(0,1)$, and ${\mathcal{W}}$ is taken as the squared Euclidean distance in the embedding space ${\mathbb{R}}^3$. The black arrows on the left indicate the position and tangent vectors for which Figure \[fig:torusConvergence\] shows a convergence study of the approximated sectional curvature. Right: Corresponding approximation error $(\kappa^{\pm\tau}-\kappa)(v,w)$. []{data-label="fig:torus"}](images/WillmoreTorusCurvatureError "fig:"){width="\unitlength"}![Left: Gaussian curvature on two tori (each with center line radius of $\sqrt2$) numerically approximated via $\kappa^{\pm\tau}(v,w)$ for $\tau=10^{-2}$ and $\beta=\frac32$. The tori were parameterized by toroidal and poloidal angles, $v=(1,0)$, $w=(0,1)$, and ${\mathcal{W}}$ is taken as the squared Euclidean distance in the embedding space ${\mathbb{R}}^3$. The black arrows on the left indicate the position and tangent vectors for which Figure \[fig:torusConvergence\] shows a convergence study of the approximated sectional curvature. Right: Corresponding approximation error $(\kappa^{\pm\tau}-\kappa)(v,w)$. []{data-label="fig:torus"}](images/ThinTorusCurvatureError "fig:"){width="\unitlength"} \
-4 ![Left: Gaussian curvature on two tori (each with center line radius of $\sqrt2$) numerically approximated via $\kappa^{\pm\tau}(v,w)$ for $\tau=10^{-2}$ and $\beta=\frac32$. The tori were parameterized by toroidal and poloidal angles, $v=(1,0)$, $w=(0,1)$, and ${\mathcal{W}}$ is taken as the squared Euclidean distance in the embedding space ${\mathbb{R}}^3$. The black arrows on the left indicate the position and tangent vectors for which Figure \[fig:torusConvergence\] shows a convergence study of the approximated sectional curvature. Right: Corresponding approximation error $(\kappa^{\pm\tau}-\kappa)(v,w)$. []{data-label="fig:torus"}](images/colorbar "fig:"){width="\unitlength"} 3$-5\cdot10^{-5}$ ![Left: Gaussian curvature on two tori (each with center line radius of $\sqrt2$) numerically approximated via $\kappa^{\pm\tau}(v,w)$ for $\tau=10^{-2}$ and $\beta=\frac32$. The tori were parameterized by toroidal and poloidal angles, $v=(1,0)$, $w=(0,1)$, and ${\mathcal{W}}$ is taken as the squared Euclidean distance in the embedding space ${\mathbb{R}}^3$. The black arrows on the left indicate the position and tangent vectors for which Figure \[fig:torusConvergence\] shows a convergence study of the approximated sectional curvature. Right: Corresponding approximation error $(\kappa^{\pm\tau}-\kappa)(v,w)$. []{data-label="fig:torus"}](images/colorbar "fig:"){width="\unitlength"} $7\cdot10^{-5}$
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Discrete shell space {#sec:ShellSpace}
====================
In this section, we consider the manifold ${\mathcal{M}}$ of *discrete shells*, triangle meshes with fixed mesh connectivity whose deformations are modelled based on a physical shell model. The concept of discrete shells has been introduced by Grinspun et al. [@GrHiDeSc03] as a tool for efficient and reliable physical simulation of shell deformations. Fröhlich and Botsch [@FrBo11] studied the effective discretization of membrane and bending distortion. In [@HeRuWa12; @HeRuSc14] this approach was investigated in the context of shape space theory with a metric representing a physical dissipation due to membrane and bending distortion. In fact, a suitable Riemannian metric on ${\mathcal{M}}$ is induced by the Hessian of an appropriate elastic energy functional. In the following, we briefly summarize the underlying physical model and discuss its discretization, for details we refer the reader to [@HeRuWa12; @HeRuSc14].
Before we introduce a discrete surface model based on triangle meshes, let us briefly investigate the case of smooth surfaces ${{y}}\subset {\mathbb{R}}^3$ representing the middle layer of a thin sheet of some material. For a homogeneous, isotropic and elastic material, the deformation energy ${\mathcal{W}}_{{y}}[\phi]$ of an elastic deformation $\phi: {{y}}\to {\mathbb{R}}^3$ can be captured by membrane and bending contributions, reflected by the change of first (metric) and second (curvature) fundamental forms, respectively. In the following, we distinguish between the undeformed reference surface ${{y}}$ and the deformed surface ${{y}}^\phi = \phi({{y}})$. For parametric surfaces ${{y}}$ we denote by $g$ and $h$ the matrix representation of the first and second form, respectively, in a parameter domain $\omega \subset {\mathbb{R}}^2$. To this end, the membrane distortion induced by $\phi$ can be described pointwise by a (reduced) Cauchy-Green strain tensor $A[\phi] = g^{-1}g^\phi \in {\mathbb{R}}^{2,2}$, and the bending distortion is captured pointwise by the relative shape operator $B[\phi] = g^{-1}(h - h^\phi) \in {\mathbb{R}}^{2,2}$. The total elastic energy is then given by $$\begin{aligned}
\label{eq:smoothShellEnergy}
{\mathcal{W}}_{{y}}[\phi]= \int_\omega W_{{\mbox{{\tiny mem}}}}(A[\phi])\, \sqrt{\det g} {\,\mathrm{d}}\xi +\delta^2 \int_\omega W_{{\mbox{{\tiny bend}}}}(B[\phi]) \,\sqrt{\det g} {\,\mathrm{d}}\xi ,\end{aligned}$$ where $\delta >0$ represents the physical thickness of the thin elastic material. For $W_{{\mbox{{\tiny mem}}}}$, we make use of a classical hyperelastic energy density as in equation (8) in [@HeRuWa12]. Additionally, we set $W_{{\mbox{{\tiny bend}}}}(B[\phi]) = ({\mathrm{tr}}B[\phi])^2$, which measures the squared error of the mean curvature. It has been shown in [@HeRuSc14] that the Hessian of ${\mathcal{W}}_{{y}}$ induces a Riemannian metric on the space of surfaces modulo rigid body motions.
Now we consider an approximation of the smooth surface by a triangle mesh – which we shall also denote by ${{y}}$. Let $\mathcal{V} = \{1, \ldots, n\}$, $\mathcal{T} \subset \mathcal{V}\!\times\!\mathcal{V}\!\times\!\mathcal{V}$ and $\mathcal{E} \subset \mathcal{V}\!\times\!\mathcal{V}$ be the set of vertices, triangles and edges of ${{y}}$, respectively. A (discrete) deformation of ${{y}}$ can be identified with a mapping $\phi:\mathcal{V}\to{\mathbb{R}}^3$ which is interpolated piecewise linearly over triangles. Obviously, $D\phi$ is constant on triangles $t \in \mathcal{T}$ which leads to an elementwise constant, discrete first fundamental form $g = g_t$. In particular, we can make use of the same energy density $W_{{\mbox{{\tiny mem}}}}$ as above now measuring changes in the discrete first fundamental form. For the discretization of the bending term, which requires second derivatives of surface positions in the smooth case, we consider the widely used *discrete shell* energy [@GrHiDeSc03], where bending is quantified by changes of dihedral angles $\theta_e$ living on edges $e \in \mathcal{E}$. The discrete version of the elastic deformation energy reads $$\begin{aligned}
\label{eq:discreteShellsEnergy}
{{{\mathcal{W}}}}_{{y}}[\phi] = \sum_{t \in \mathcal{T}} a_t \, W_{{\mbox{{\tiny mem}}}}(g^{-1}_t g^\phi_t ) + \delta^2 \sum_{e \in \mathcal{E}} \frac{(\theta_e - \theta^\phi_e)^2}{d_e}l_e^2\, ,\end{aligned}$$ where $a_t$ is the volume of triangle $t$, $l_e$ is the length of edge $e$ and $d_e = \tfrac13(a_t + a_{t'})$ if $e = t \cap t'$. In fact, it can be shown that the simplified bending term in can be derived formally from $W_{{\mbox{{\tiny bend}}}}(B[\phi])$ by an appropriate linearization [@He16]. For a more rigorous discussion of the discretization of the bending term we refer to [@CoMo03] and [@GlOl19]. Again, it has been shown in [@HeRuSc14] that the Hessian of induces a Riemannian metric on the space of triangle meshes modulo rigid body motions.
Now we define the shape space of discrete shells ${\mathcal{M}}$ as the set of all triangle meshes with the same connectivity, for two discrete shells ${{y}},\tilde{{y}}\in{\mathcal{M}}$ there is a 1-to-1 correspondence between all vertices, faces and edges. In particular, a discrete shell ${{y}}$ is uniquely described by its vector of vertex positions living in ${\mathbb{R}}^{3n}$. To this end, the functional $({y},\tilde {y}) \mapsto {\mathcal{W}}{[{y}, \tilde {y}]}$ required in the discrete Riemannian calculus is simply defined as ${\mathcal{W}}{[{{y}}, \tilde {{y}}]} := {{{\mathcal{W}}}}_{{y}}[\phi]$ with ${{{\mathcal{W}}}}_{{y}}$ given by and $\phi$ is the *unique* discrete deformation such that $\tilde{{y}}= \phi({{y}})$.
In the following, we will first consider a toy example (a deformation of a triangulated sphere) to investigate the convergence behaviour of the discrete sectional curvature in the space of discrete shells when increasing the temporal and/or spatial resolution as well as the dependence on the (squared) thickness $\mu := \delta^{2}$ of the elastic material, which we shall also denote as *bending weight*. Finally, we consider a family of almost isometric deformations of a more complex cactus shape and compute sectional curvature patterns.
#### Sectional curvature near spherical discrete shell.
Let ${{y}}\in{\mathbb{R}}^{3n}$ be the vector of vertex positions of a triangluation of the unit sphere $S^2$ with vertices lying on $S^2$. Furthermore, we consider tangent vectors $v$ and $w$ corresponding to a stretching by a factor $\tfrac32$ in the first and second component of the vertex positions, respectively. Figure \[fig:secCurv\_DiscreteShells\_sphere\_convergence\] displays the decay of the relative error $\left| \frac{\kappa - \kappa^\tau}{\kappa}\right|(v,w)$ and $\left| \frac{\kappa - \kappa^{\pm \tau}}{\kappa}\right|(v,w)$ as a function of the time step size $\tau$ for different spatial resolutions and a bending weight $\mu = 10^{-3}$. Here, the value of the discrete sectional curvature for the smallest time step $\tau = \tau_{\min}$ was used as ground truth. As predicted by our analytical results, we observe a quadratic convergence when using the central difference quotient and a convergence which is experimentally still better than linear but not yet quadratic when using the one-sided difference quotient. Figure \[fig:secCurv\_DiscreteShells\_sphere\_convergenceInH\_differentMus\] shows the experimental convergence of $\kappa^{\pm\tau}$ for fixed $\tau = 10^{-2}$ and $\mu = 10^{-3}$ for decreasing grid size $h$ of the underlying triangular mesh as well as the dependence on the thickness parameter $\mu$ in the discrete shell model for fixed $\tau$ and $h$. In fact, one observes second order convergence with respect to a uniform refinement of triangles as well as a decay of the sectional curvature for decreasing values of $\mu$.
plot coordinates [ ( 0.5, 0.580974 ) ( 0.25, 0.225961 ) ( 0.125, 0.0819605 ) ( 0.0625, 0.0291723 ) ( 0.03125, 0.0105529 ) ( 0.015625, 0.00406301 ) ]{}; plot coordinates [ ( 0.5, 0.64949 ) ( 0.25, 0.250004 ) ( 0.125, 0.0905021 ) ( 0.0625, 0.0324547 ) ( 0.03125, 0.0120694 ) ( 0.015625, 0.00498563 ) ]{}; plot coordinates [ ( 0.5, 0.66911 ) ( 0.25, 0.256239 ) ( 0.125, 0.0921003 ) ( 0.0625, 0.0324804 ) ( 0.03125, 0.0115667 ) ( 0.015625, 0.00429072 ) ]{}; plot coordinates [ ( 0.5, 0.674391 ) ( 0.25, 0.258243 ) ( 0.125, 0.092983 ) ( 0.0625, 0.0329849 ) ( 0.03125, 0.0119458 ) ( 0.015625, 0.00464054 ) ]{}; plot coordinates [ ( 0.5, 0.675157 ) ( 0.25, 0.257523 ) ( 0.125, 0.0917174 ) ( 0.0625, 0.031528 ) ( 0.03125, 0.0104235 ) ( 0.015625, 0.00309561 ) ]{}; plot coordinates [ (0.4, 0.2) (0.04, 0.002) (0.4, 0.002) (0.4, 0.2) ]{};
plot coordinates [ ( 0.5, 0.30027 ) ( 0.125, 0.00513938 ) ( 0.0625, 0.00188461 ) ( 0.03125, 0.000533602 ) ( 0.015625, 0.000131361 ) ]{}; plot coordinates [ ( 0.5, 0.325127 ) ( 0.125, 0.00672624 ) ( 0.0625, 0.00234951 ) ( 0.03125, 0.000670019 ) ( 0.015625, 0.000180093 ) ]{}; plot coordinates [ ( 0.5, 0.33226 ) ( 0.125, 0.00719349 ) ( 0.0625, 0.00248957 ) ( 0.03125, 0.000714752 ) ( 0.015625, 0.000199649 ) ]{}; plot coordinates [ ( 0.5, 0.334127 ) ( 0.125, 0.00730559 ) ( 0.0625, 0.00251646 ) ( 0.03125, 0.000718043 ) ( 0.015625, 0.000194521 ) ]{}; plot coordinates [ ( 0.5, 0.334636 ) ( 0.125, 0.0073022 ) ( 0.0625, 0.00249226 ) ( 0.03125, 0.000633795 ) ( 0.015625, 0.000163492 ) ]{}; plot coordinates [ (0.4, 0.02) (0.04, 0.0002) (0.4, 0.0002) (0.4, 0.02) ]{};
plot coordinates [ ( 1.0, 0.173603 ) ( 0.5, 0.042268 ) ( 0.25, 0.00997576 ) ( 0.125, 0.00194468 ) ]{}; plot coordinates [ (0.8, 0.032) (0.2, 0.002) (0.8, 0.002) (0.8, 0.032) ]{};
plot coordinates [ (0.1, 0.023806) (0.05, 0.0180375) (0.025, 0.0125942) (0.0125, 0.00855959) (0.00625, 0.00602473) (0.003125, 0.00458566) (0.0015625, 0.00381388) (0.00078125, 0.00341265) (0.000390625, 0.00320767) (0.000195313, 0.00310399) ]{};
#### Sectional curvature for a triangulated cactus shape.
As a more complex geometric model, we consider a triangulated cactus having $n = 5261$ vertices and a vector of vertex positions ${{y}}$. We investigate different pairs of displacement vectors as tangent vector arguments for the computation of sectional curvatures. In Figure \[fig:secCurv\_DiscreteShells\_cactus\_dissipation\], we consider subspaces spanned by the eigenmodes $v_i$ of the Hessian of the elastic energy and compute the sectional curvature for all pairs from the set of eigenmodes of the lowest $i = 1, \ldots, 8$ eigenvalues. All sectional curvatures $\kappa_{ij} := \kappa_{{y}}(v_i, v_j)$ turn out to be negative. Furthermore, one observes varying values of sectional curvature reflecting the specific type of coupling in the underlying pair of eigenmodes. In particular, $|\kappa_{ij}|$ turns out to be relatively large if $i,j \in \{1,2,7\}$, since all of the corresponding modes represent a strong displacement of the right arm. The same is observed with respect to the left arm if $i,j \in \{5,6\}$. Furthermore, all two-dimensional subspaces involving the vibration mode $i=4$ are comparably flat. In this sense, the second order representation of the shell space manifold reflects the physics of the underlying shell model.
file \[meta=index 2\][images/shells/secCurv\_fullCactus\_dissipationModes\_0p01mu\_tau1\_negative\_8x8.dat]{};
(4,2.5) (0,1.4) (0.15,1.5)[$1$]{} (1,1.4) (1.15,1.5)[$2$]{} (2,1.4) (2.15,1.5)[$3$]{} (3,1.4) (3.15,1.5)[$4$]{} (0,-0.2) (0.15,-0.1)[$5$]{} (1,-0.2) (1.15,-0.1)[$6$]{} (2,-0.2) (2.15,-0.1)[$7$]{} (3,-0.2) (3.15,-0.1)[$8$]{}
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In contrast, in Figure \[fig:secCurv\_DiscreteShells\_cactus\_deform\] we consider manually designed deformations $\phi_{1},\ldots,\phi_{16}$ of ${{y}}$. These deformations are generated as follows: The foot of the cactus (the *fixed handle region*) is fixed while certain *free handle regions* (e.g. the top of the cactus, or tip of the arms) have been deformed smoothly in space. In Figure \[fig:secCurv\_DiscreteShells\_cactus\_deform\] (top row) we show deformed shapes ${{y}}_i := \phi_i({{y}})$ for $i = 1, \ldots, 16$. Finally, we compute the sectional curvatures of the subspaces spanned by pairs of the discrete logarithms $v_{i}$ of ${{y}}_i$ at ${{y}}$ for $i\in \{1,\ldots,16\}$ (computed for $K=8$) and show the corresponding confusion matrix with entries $\kappa_{ij} := \kappa_{{{y}}}^{\pm\tau}(v_i, v_j)$ for $i,j \in \{1,\ldots,16\}$. The colour coding of the confusion matrix is as in Figure \[fig:secCurv\_DiscreteShells\_cactus\_dissipation\]. Note that we display $|\kappa_{ij}|$ since $\kappa_{ij} < 0$ for almost all choices of $i \neq j$. Additionally, we take into account substantially different resolutions and tesselations of the triangulated cactus model (grey, orange and green). However, the resulting pattern of the confusion matrices for the different triangulations are still similar, although the scaling of the sectional curvatures varies. This appears to be related to the fact that the discrete elastic energy ${\mathcal{W}}_{{y}}(\phi_i)$ already differs significantly when comparing the grey, green and orange model, respectively, as shown in the histogram in Figure \[fig:secCurv\_DiscreteShells\_cactus\_deform\]. Compared to Figure \[fig:secCurv\_DiscreteShells\_cactus\_dissipation\], for this set of tangent vectors (based on manual generation) a clearer separation of certain subspaces can be identified in the confusion matrix. For example, the subset $\{v_1, v_2, v_3\}$ represents deformations which are solely supported on the *left* arm, whereas the subset $\{v_4, \ldots, v_{10}\}$ represents deformations which are mainly supported on the *right* arm.
(15,.5) (-0.25,-0.) (0.75,-0.) (1.75,-0.) (2.75,-0.) (3.75,-0.) (4.75,-0.) (5.75,-0.) (6.7,-0.) (7.7,-0.) (8.7,-0.) (9.75,-0.) (10.75,-0.) (11.5,-0.) (12.75,-0.) (13.6,-0.) (14.6,-0.)
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file \[meta=index 2\][images/shells/secCurv\_fullCactus\_0p01mu\_tau0\_fixedBdry\_FINAL\_negative\_reordered.dat]{};
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file[images/shells/secCurv\_coarseCactus\_0p01mu\_tau0\_fixedBdry\_FINAL\_negative\_reordered.dat]{};
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file[images/shells/secCurv\_fullCactus\_20pcFlippedEdges\_0p01mu\_tau0\_fixedBdry\_FINAL\_negative\_reordered.dat]{};
#### Positive sectional curvature and nonuniqueness of geodesics.
It is well-known that estimates of the injectivity radius of the exponential map on a manifold depend on sectional curvature bounds and in particular that positive sectional curvature may be associated with nonuniqueness of geodesics. In Figure\[fig:positiveCurvature\] we provide a corresponding example in the space of discrete shells: We consider a flat rectangle with two bumps pointing in opposite directions, where the height of the bumps is indicated by the parameters $\zeta$ and $\eta$. This example was first given in [@HeRuWa12 Fig.5] to show nonuniqueness of shortest geodesics. Indeed, the extremal grey shells in Figure\[fig:positiveCurvature\] correspond to $(\zeta>0,\eta<0)$ and $(\zeta<0,\eta>0)$, respectively. Thus, to get from one shell to the other, both bumps need to be flipped through. If the bumps are flipped through simultaneously, this induces a strong distortion energy since then both bumps compete for in-plane area. Therefore, it is energetically more favourable to flip both bumps through one after another, and indeed there are two shortest (discrete) geodesics connecting both shells (green and orange) that only differ in the order of the flips. Intuitively this means that somewhere in between these two geodesics there is a region in shell space that is avoided by geodesics since (thinking in terms of a chart) there the Riemannian metric becomes large. At that point in the chart where the Riemannian metric is locally maximal the sectional curvature should be positive. This intuition is confirmed by computing the sectional curvature at the central flat rectangle in the directions of $\zeta$ and $\eta$, which turns out to almost reach the value $1000$. As for the previous example, we confirmed (but do not show here) robustness of this behaviour with respect to triangulation modifications and convergence in $\tau$ and in the triangulation width.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Alexander Effland acknowledges support from the European Research Council under the Horizon 2020 program, ERC starting grant HOMOVIS (No. 640156). Behrend Heeren and Martin Rumpf acknowledge support of the Collaborative Research Center 1060 funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) and the Hausdorff Center for Mathematics, funded by the DFG under Germany’s Excellence Strategy - GZ 2047/1, Project-ID 390685813. Benedikt Wirth was supported by the Alfried Krupp Prize for Young University Teachers awarded by the Alfried Krupp von Bohlen und Halbach-Stiftung and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) via Germany’s Excellence Strategy through the Cluster of Excellence “Mathematics Münster: Dynamics – Geometry – Structure” (EXC 2044) at the University of Münster.
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---
abstract: 'It is widely believed that the perovskite Sr$_2$RuO$_4$ is an unconventional superconductor with broken time reversal symmetry. It has been predicted that superconductors with broken time reversal symmetry should have spontaneously generated supercurrents at edges and domain walls. We have done careful imaging of the magnetic fields above Sr$_2$RuO$_4$ single crystals using scanning Hall bar and SQUID microscopies, and see no evidence for such spontaneously generated supercurrents. We use the results from our magnetic imaging to place upper limits on the spontaneously generated supercurrents at edges and domain walls as a function of domain size. For a single domain, this upper limit is below the predicted signal by two orders of magnitude. We speculate on the causes and implications of the lack of large spontaneous supercurrents in this very interesting superconducting system.'
author:
- 'J.R. Kirtley,$^{1,2,3}$ C. Kallin,$^4$ C.W. Hicks,$^1$ E.-A. Kim,$^{5,6}$ Y. Liu,$^{7}$ K.A. Moler,$^{1,5}$ Y. Maeno,$^8$ and K.D. Nelson,$^7$'
title: 'Upper limit on spontaneous supercurrents in Sr$_2$RuO$_4$'
---
Introduction
============
The perovskite superconductor Sr$_2$RuO$_4$[@maeno1994] (T$_c$=1.5K) is believed for a number of reasons to have unconventional pairing symmetry.[@mackenzie2003] Muon spin resonance experiments are consistent with the generation of large but sparse internal magnetic fields when Sr$_2$RuO$_4$ becomes superconducting, indicating a superconducting state with broken time reversal symmetry.[@luke1998] This finding is supported by the observation of the onset at the superconducting transition temperature of a Kerr effect rotation of light polarization upon reflection,[@xia2006] consistent with large ($\sim$50-100$\mu$m) domains with broken chiral symmetry. The first phase sensitive Josephson tunneling measurements suggested a static $\pi$-phase shift between opposite faces of a Sr$_2$RuO$_4$ single crystal,[@nelson2004] a result obtained only when the sample was prepared by controlled slow cooling, implying that the sample could be made to possess a small even number of domain walls separating the opposite faces. On the other hand, more recent Josephsen tunneling measurements point toward small ($\sim$1$\mu$m) dynamic order parameter domains.[@kidwingara2006] Magnetic imaging of the $ab$ face of a single crystal of Sr$_2$RuO$_4$ using a micron sized SQUID shows vortex coalescence on a scale of $\sim$10 $\mu$m that may be related to a domain structure in the superconducting order parameter.[@dolocan2005] These observations, as well as others, have been interpreted in terms of a superconducting order parameter with spin triplet,[@nmr] chiral $p_x \pm ip_y$ Cooper pairing symmetry. However, Hall bar microscopy measurements[@bjornsson2005] did not observe the magnetic fields expected at the surface or edges of a superconductor with broken chiral symmetry. Here, we report on new scanning SQUID microscope measurements, and we also further analyze some of the Hall bar measurements, showing that, if they exist, the local fields at the surfaces and edges of Sr$_2$RuO$_4$ single crystals are much smaller than those expected at the surfaces and edges of a chiral p-wave superconductor.
Although $p_x \pm i p_y$ pairing symmetry is fully gapped, specific heat, [@nishizaki2000] nuclear relaxation rate, [@ishida2000] and thermal conductivity measurements [@tanatar2001a; @tanatar2001b; @izawa2001] all show a power law temperature dependence, suggesting the presence of line nodes. Among other suggestions, one possibility is that the $\gamma$ band, the band with the primary contribution to superconductivity in Sr$_2$RuO$_4$, has nodeless $p_x \pm i p_y$ pairing symmetry, but induces superconductivity with a line of nodes in the other ($\alpha$ and $\beta$) bands. [@zhitomirsky2001] Anisotropy in the gap function [@miyake1999] has been supported experimentally by specific heat [@deguchi2004a; @deguchi2004b] and ultrasound attenuation [@lupien2001] measurements.
The issue of the broken time reversal symmetry in the superconducting state of Sr$_2$RuO$_4$, aside from intrinsic interest, has taken on new urgency with several proposals for error tolerant quantum logic elements taking advantage of this property. [@zagoskin2003a; @zagoskin2003b; @dassarma2006]
There are useful analogies between a chiral $p_x \pm ip_y$ superconductor and a ferromagnet.[@kidwingara2006; @rice2006] A single domain ferromagnet has a uniform magnetization which is equivalent to the field produced by a current sheet circulating around the surface, in the appropriate geometry, while a single domain $p_x + ip_y$ superconductor carries an intrinsic angular momentum of $\hbar$ per Cooper pair,[@kita1997] which one would expect to lead to an actual surface current sheet, confined within a healing length proportional to the coherence length of the surface.[@stone2004] However, the field generated by this current must be screened inside the superconductor by a diamagnetic shielding current flowing within the penetration depth of the surface, so that $B=0$ inside the superconductor. The net result is a spontaneous magnetization within the healing length plus the penetration depth of the sample edges, which is greatly reduced from that expected from the simple ferromagnetic analogy, but which is still substantial and, using parameters appropriate to Sr$_2$RuO$_4$, predicted to give rise to local fields as large as 1mT under certain assumptions.[@matsumoto1999; @kwon2001] The superconductor can also support domains in which regions of $p_x + ip_y$ coexist with regions of $p_x - ip_y$ order. Although the net magnetization vanishes at the boundaries between such domains, the local fields, which extend over the penetration depth on either side of the wall, can be as large as 2mT.[@matsumoto1999; @kwon2001] Other than direct phase sensitive measurements, the detection of such fields would be one of most direct confirmations of a superconducting order parameter with time reversal symmetry breaking since the spontaneous boundary and domain wall supercurrents are expected by symmetry[@sigrist1991] and would have no other obvious explanation. It is therefore appropriate to attempt to image the magnetic fields arising from these spontaneous supercurrents using scanning magnetic microscopy.
Magnetic imaging
================
We have performed scanning magnetic imaging of the $ab$ and $ac$ faces of single crystals of Sr$_2$RuO$_4$. The magnetic images reported here were made at Stanford with a dilution refrigerator based Hall bar/SQUID microscope [@bjornsson2001] with a base temperature below 100mK, and at IBM with a $^3$He based scanning SQUID microscope with a base temperature below 300mK. Our SQUID sensors had square pickup loops 8 $\mu$m on a side; the Hall bars had roughly square effective areas 0.5 $\mu$m on a side. The Hall bar measurements were made in a residual field of about 2.5$\mu$T; the SQUID measurements were made in a residual field of 75nT, compensated for fields perpendicular to the scanning direction to less than 10nT using a small Helmholtz coil. The SQUID measurements were made after cooling the samples through the superconducting transition temperature at a rate of about 1mK/sec. Some of the Hall bar data discussed in this paper has been reported previously. [@bjornsson2005] However, here we make a more quantitative comparison of this data with theory.
The Sr$_2$RuO$_4$ single crystals used in our experiments were grown using a floating zone method.[@mao2000] The samples used for the IBM SQUID measurements were mounted in epoxy and polished so that either the $ab$ or $ac$ face was part of a smooth plane, allowing scanning across the edges of the crystal.[@nelson2004] Some of the samples used for SQUID microscopy were the same as for phase sensitive experiments on the pairing symmetry of Sr$_2$RuO$_4$, [@nelson2004] and had layers of SiO and Au$_{0.5}$In$_{0.5}$ (T$_c$= 0.4-0.5K) deposited on some of the crystal faces perpendicular to the scanned face. These additional layers should have had no effect on the magnetic imaging experiments reported here. The critical temperature of the crystals were measured to be $>$ 1.4K using scanning and bulk susceptometry measurements.
![SQUID microscope image of the $ab$ face of a Sr$_2$RuO$_4$ single crystal, cooled in a field $B_z<$10 nT and imaged at T=0.27K with an 8 $\mu$m square pickup loop. [**a**]{} Pseudocolor image with full-scale variation of 0.2 $\Phi_0$ ($\Phi_0 = h/2e$) in magnetic flux through the SQUID pickup loop. The dashed line in [**a**]{} shows the outlines of the crystal. [**b**]{} Same image as [**a**]{}, but with the pseudocolor scale expanded to 0.02 $\Phi_0$. The dashed line in [**b**]{} shows the line traced by the cross-section in [**c**]{}. The dashed rectangle in [**b**]{} shows the area of the image expanded in [**d**]{}. [**e**]{} is a histogram of pixel values for the data displayed in [**d**]{}. []{data-label="fig:abimage"}](ABIMAGE.PS){width="3.5in"}
Figure \[fig:abimage\] shows a SQUID microscope image of the $ab$ face of a Sr$_2$RuO$_4$ single crystal. The largest feature evident in this image (Fig. \[fig:abimage\]a) is an isolated Abrikosov vortex. When the pseudocolor scale is expanded to $\Delta \Phi_s = 0.02 \Phi_0$ (Fig. \[fig:abimage\]b) magnetic features become apparent in the epoxy and along the edges of the crystal. We believe these features are not due to the superconductivity of the Sr$_2$RuO$_4$ because they are unchanged from cooldown to cooldown in different fields. Figure \[fig:multcrs\] compares images from 3 different cooldowns of the same crystal, in nominal ambient plus compensating fields perpendicular to the scanning plane of zero (Fig. \[fig:multcrs\][**a**]{}), 10nT (Fig. \[fig:multcrs\][**b**]{}), and 15nT (Fig. \[fig:multcrs\][**c**]{}). The number and positions of the Abrikosov vortices in the top (ab) face of the crystal, and an interlayer vortex emerging from the left (ac face) edge (Fig. \[fig:multcrs\][**b**]{}) of the crystal change from cooldown to cooldown, but the sharp features at the edge of the sample are remarkably reproducible. These edge features may be the result of the polishing process, such as topographical or magnetic features from particles trapped in the epoxy. Note that features very similar to the edge features are apparent in the epoxy far from the sample edge. Above the sample itself the flux image is relatively smooth, with a broad background (Fig. \[fig:abimage\]c). We believe that this broad background is the result of magnetic flux coupled into the SQUID through sections outside of the pickup loop. A clear demonstration of this effect appears in Ref. . On top of the broad background, two steps in the cross-section (Fig. \[fig:abimage\]c) correspond to the edges of the crystal. We believe that these steps are due to small supercurrents circulating around the entire sample due to uncompensated residual fields (see Figure \[fig:srocrs\]). Figure \[fig:abimage\]d shows a magnified image of a section of the crystal (indicated by the box in Fig. \[fig:abimage\]b), with no magnetic features larger than a few m$\Phi_0$ over an area of several hundred microns on a side.
![Comparison of SQUID microscope images of an $ab$ face of a Sr$_2$RuO$_4$ crystal after three different cooldowns in slightly different magnetic fields.[]{data-label="fig:multcrs"}](MULTCRS.PS){width="3.5in"}
![SQUID microscope image of the $ac$ face of a Sr$_2$RuO$_4$ single crystal cooled in nominally zero field and imaged at T=0.27K with an 8 $\mu$m square pickup loop. [**a**]{} Pseudocolor image $\Delta \Phi_s = 0.8 \Phi_0$. The dashed line in [**a**]{} shows the positions of the outer edges of the crystal. [**b**]{} Same image as [**a**]{}, but with $\Delta \Phi_s = 0.08 \Phi_0$ A few interlayer vortices with both positive and negative signs are visible near the lower-left edge of the crystal. The dashed line in [**b**]{} is along the $a$-axis and shows the data traced by the cross-section in [**c**]{}. The arrows in [**c**]{} indicate the edges of the crystal. The dashed square in [**b**]{} shows the area of the image expanded in [**d**]{}. The diagonal stripes visible in [**d**]{} are due to 60 Hz noise. [**e**]{} is a histogram of pixel values for the data displayed in [**d**]{}. []{data-label="fig:acimage"}](ACIMAGE.PS){width="3.5in"}
Similar results were obtained when SQUID microscope images were taken of the $ac$ face of a Sr$_2$RuO$_4$ single crystal (Figure \[fig:acimage\]). In this case there were a number of interlayer vortices with flux both emerging from and entering into the crystal surface near the left edge of the crystal (Fig. \[fig:acimage\]b). Just as for the $ab$ face, there were sharp magnetic features along the edges of the crystal and in the epoxy which did not appear to be correlated with the superconductivity of the Sr$_2$RuO$_4$, as well as broad magnetic backgrounds, but sharp magnetic features were absent from large areas of the crystal face.
![[**a**]{} Scanning Hall bar image of $ab$ face of Sr$_2$RuO$_4$ single crystal, cooled in $\sim 2.5 \mu$T and imaged at a temperature below 100mK using a Hall bar with a sensor area 0.5$\mu$m on a side. In this image the mean of each scan line was subtracted from the raw data to remove slow drift in the sensor Hall voltage. [**b**]{} Same area as [**a**]{} but with an expanded pseudocolor scale. The dashed line in [**b**]{} shows the line traced by the data cross-section in [**c**]{}. The dashed square in [**b**]{} shows the area for which a histogram of pixel inductance values are displayed in [**d**]{}. []{data-label="fig:hallimag"}](HALLIMAG.PS){width="3.5in"}
The samples used in the Hall bar measurements were cleaved. 1$\mu$m diameter, $\sim$1 $\mu$m deep holes were milled on a 20 $\mu$m grid on the upper surface using a focussed ion beam, to create artificial edges.[@bjornsson2005] Figure \[fig:hallimag\] shows a scanning Hall bar image of the $ab$ face, with a regular array of 1 micron holes at a pitch of 20 microns, of a Sr$_2$RuO$_4$ single crystal. There are a few Abrikosov vortices apparent in this image, but the area away from these vortices is featureless. In particular, no features were observed in connection with the edges or interiors of the 1 micron holes. Since the 1 $\mu$m deep holes did not even act as effective pinning centers for the vortices, they may not have served as significant singularities to create edge currents. The outer edges of the crystal were not scanned in the Hall bar measurements.
Modeling
========
Matsumoto and Sigrist [@matsumoto1999] (MS) have solved the Bogoliubov-De Gennes equations using a quasi-classical approximation for the cases of an edge between a semi-infinite, ideal $p_x +i p_y$ superconductor and vacuum, and a domain boundary between a $p_x + i p_y$ superconductor and a $p_x - i p_y$ superconductor. Their solutions are fully self-consistent so that they include the effect of screening currents. They predict substantial supercurrents and consequent magnetic fields spontaneously generated at edges and domain boundaries. For example, the peak magnetic fields in these calculations correspond to 1 mT for edges, and 2 mT for domain walls using values for the coherence length ($\xi_0$=66 nm) and penetration depth ($\lambda_L$= 190 nm) suitable for Sr$_2$RuO$_4$. However, some modeling is required to compare our experimental results with the MS predictions because we measure the magnetic fields above the surface, rather than inside the sample.
The simplest approach to this problem is to assume that the magnetic fields at the surface of the sample are the same as those in the bulk. This neglects field spreading and any change in superconducting shielding due to the finite sample geometry. However, in our case the size of the magnetic sensor and its spacing from the sample are large relative to the coherence length and penetration depth, so that the field averaging from these effects are larger than the additional effects of field spreading and changes in superconducting shielding. The field averaging effects from finite sensor size and height can be shown rigorously to be larger than field spreading, for example, in the similar problem of vortex fields spreading from the surface of a superconductor and imaged with a SQUID microscope.[@chang1992; @kirtley1999] In the remainder of this section, we will neglect changes in the currents near the surface due to the finite sample geometry. We will show below that the effect of finite sample geometry only leads to suppression of the expected signal by 30% compared to what is expected from the edge currents of an infinite sample. The finite sample geometry effects for edge currents are expected to be similar to those for domain walls and are also discussed in the following section on surface screening effects.
It is well known [@roth1988] that if the normal component of the magnetic field $B_z(x,y,z)$ is known at all points of a surface $z=0$ the magnetic field in free space at a height $z$ above that surface is given by $$\tilde{B}_z(k_x,k_y,z)=\tilde{B}_z(k_x,k_y,z=0)e^{-kz},
\label{eq:wikswo_trick}$$ where $\tilde{B}_z(k_y,k_y,z)$ is the 2-dimensional Fourier transform of $B_z(x,y,z)$ and $k=\sqrt{k_x^2+k_y^2}$. To model the magnetic signals in our experimental SQUID and Hall bar microscope geometries, we assume a particular domain structure with the magnetic fields, $B_z$, at each edge and domain boundary, at the surface $z=0$, taken to be those predicted by Matsumoto and Sigrist [@matsumoto1999] for an infinite sample. We then propagate the fields to a height $z$ using Eq. \[eq:wikswo\_trick\], integrate over an area appropriate for the SQUID or Hall bar sensor to obtain a magnetic flux, and divide by the area of the sensor for the case of the Hall bar to get an average magnetic field. We will refer to this model as the “extended-Matsumoto-Sigrist” model to distinquish it both from the prediction made by Matsumoto-Sigrist for an ideal (infinite) geometry and from the more accurate model which includes additional screening effects due to the finite geometry, as discussed in the next section. The original Matsumoto-Sigrist results are scaled in field by $B_c=\Phi_0/2\sqrt{2}\pi\xi_0\lambda_L$, where $\Phi_0=h/2e$ is the superconducting flux quantum, $\xi_0$ is the coherence length and $\lambda_L$ is the London penetration depth. For the modeling presented here we take $\xi_0 = 66 nm$ and $\lambda_L = 190 nm$.[@mackenzie2003]
![ Cross-section through the image of the $ab$ face of Sr$_2$CuO$_4$ displayed in Figure \[fig:abimage\] (solid line). The short-dashed line is the prediction for a superconducting disk in a uniform residual field of 3 nT. The long-dashed line (with a peak at $\Phi_s/\Phi_0$ = 1.1) is the prediction for a single domain $p_x+ip_y$ superconductor of the extended-Matsumoto-Sigrist model as described in the text, assuming a square pickup loop 8$\mu$m on a side, at a height of 3$\mu$m above the sample. Here the superconductor is positioned to the left of 0$\mu$m, with epoxy to the right. []{data-label="fig:srocrs"}](srocrs.ps){width="3.5in"}
Figure \[fig:srocrs\] compares the results of this calculation (long-dashed line) with the experimental cross-section of the image shown in Figure \[fig:abimage\] (solid line). Also shown for comparision is the predicted cross-section for an ideal superconducting disk in a uniform residual field[@jackson] of 3 nT. The small steps in flux at the edges of the crystals in Figures \[fig:abimage\] and \[fig:acimage\] can be attributed to shielding of a very small residual background field. These steps are much smaller than the peaks predicted by the extended-Matsumoto-Sigrist model for a single domain.
![ Predicted magnetic fluxes through an 8 $\mu$m square pickup loop, 3 $\mu$m above the sample surface, for a 64 $\mu$m square $p_x \pm p_y$ superconductor with various domain sizes, using the predictions for the edge and domain wall currents of Matsumoto and Sigrist as described in the text. The dashed lines in the figure show the positions of the cross-sections displayed in Fig. \[fig:fluxncrs\]. []{data-label="fig:fluxndom"}](fluxndoms.ps){width="3.5in"}
![ Cross-sections through the modeling images of Fig. \[fig:fluxndom\] for various domain sizes. []{data-label="fig:fluxncrs"}](FLUXNCRS.PS){width="3.5in"}
Figure \[fig:fluxndom\] shows the results from the modeling outlined above for a series of domain sizes using parameters appropriate for our SQUID measurements. In these calculations it was assumed that the domains were square, and extended infinitely far in the negative $z$ direction (perpendicular to the crystal face). Fig. \[fig:fluxncrs\] shows cross-sections through the modeling results as indicated by the dashed lines in Fig. \[fig:fluxndom\]. As expected, the magnetic fields above the edges and domain boundaries are averaged over a length set by both the height of the sensor above the sample surface and its size. This leads to a rapid decrease in the predicted signal when the domains become smaller than a critical length. (In this modeling the magnetic signal for a domain size of 4 microns vanishes everywhere except at the sample corners because, due to the symmetry of the domains with respect to the sensor, there are exactly as many positive as negative contributions to the flux through the 8 $\mu$m diameter pickup loop. For this reason, we show the predicted flux for 32/6=5.3 rather than 4 micron domains.) The calculated peak values for the SQUID flux signal for edges and domain boundaries are plotted in Fig. \[fig:peakvd\]a as a function of domain size. The lower dashed line in Fig. \[fig:peakvd\]a is an estimate of the noise in the SQUID images above the interior of the crystals, taken to be the rms noise of the flux distribution shown in Fig. \[fig:acimage\]e (2.5m$\Phi_0$). The upper dashed line is the rms value of the flux distribution above the sample edges in Fig. \[fig:abimage\]a (8.5m$\Phi_0$). Comparable modeling results using parameters appropriate for our Hall bar measurements are shown in Fig. \[fig:peakvd\]b. In this case the dashed line represents the rms noise value of the field distribution in Fig. \[fig:hallimag\]d (3.5$\mu$T). We do not display an experimental limit on the possible edge currents set by the Hall bar experiments because of uncertainties associated with the hole geometry and surface damage induced by the focussed ion beam in these experiments.[@bjornsson2005]
In order to place limits on the possible field magnitude and domain sizes consistent with our results, we assume that the magnitude of the spontaneous supercurrents can vary, but that the spatial distribution of spontaneous supercurrents is as calculated by Matsumoto and Sigrist. With this assumption we can scale the results, for example, in Figs. \[fig:peakvd\]a,b vertically by assuming a scaling field $B_s$ different from $B_c=\Phi_0/2\sqrt{2}\pi\xi_0\lambda_L$. In order for the spontaneous supercurrents to be unobservable in our experiments, the scaling factor and domain size must be in the region below and to the left of the lines in Figs. \[fig:peakvd\]c,d. Either the spontaneous currents are substantially smaller than calculated from the extended-Matsumoto-Sigrist model, or the domains are small. For example, for the SQUID measurements, the magnitude of the supercurrents at the edge must be a factor of 100 smaller than those prediced by MS if the domains are 10 or more microns in size.
![ [**a**]{} Plots of the predicted peak flux signals for an 8$\mu$m square SQUID pickup loop, 3 $\mu$m above the sample surface, for a 64 $\mu$m square $p_x \pm ip_y$ superconductor with various domain sizes, using the predictions of Matsumoto and Sigrist [@matsumoto1999] for the spontaneously generated edge and domain supercurrents. The dashed lines represent the estimated SQUID noise in the measurements within the sample (lower line) and at the sample edges (upper line). [**b**]{} Plots of the predicted peak fields for a square Hall bar 0.5 $\mu$m on a side, 1.2$\mu$m above the sample surface, with the corresponding Hall bar noise floor. [**c**]{} Upper limits on the size of the scaling fields $B_s$, normalized by $B_c= \Phi_0/2 \sqrt{2} \pi \xi_0 \lambda_L$, as a function of domain size, given by our failure to observe spontaneously generated supercurrents at edges and domain walls in the SQUID measurements. In this figure the extended-Matsumoto-Sigrist predictions are represented by $B_s/B_c$=1. [**d**]{} Upper limits on $B_s/B_c$ as a function of domain size set by the Hall bar measurements. []{data-label="fig:peakvd"}](PEAKVD.PS){width="3.5in"}
Surface screening effects
=========================
In our modeling we have neglected the fact that the magnetic fields at the sample surface will be somewhat reduced from their bulk values. In principle, one can calculate the surface fields by self-consistently solving the Bogliubov-de Gennes equations in the appropriate geometry. Here, we simply estimate the errors involved in neglecting surface screening effects using a London approach. Following Ref. , the superconductor is assumed to fill the half-space $z<0$. If the change in the penetration depth close to the surface is neglected, the magnetic field ${\bf B}$ inside the superconductor can be decomposed as ${\bf B=B_0+B_1}$, where ${\bf B_0}$ is the particular solution given by Matsumoto and Sigrist [@matsumoto1999] of the inhomogeneous London’s equation for a domain wall and ${\bf B_1}$ is a general homogenous solution chosen to satisfy the matching conditions at $z=0$. The London’s equation for the particular solution can be written as [@bluhm2007] $$k(K+k)\Phi_K = K \hat{e_z} \cdot {\bf \tilde{B}}_0 ({\bf k},0)+i{\bf k} \cdot \left ({\bf \tilde{B}}_0(\bf{k},0)-4\pi
{\bf \tilde{M}}({\bf k},0) \right ),
\label{eq:phik}$$ where $k=\sqrt{k_x^2+k_y^2}$, $K=\sqrt{k^2+1/\lambda^2}$, the magnetic field ${\bf B}$ above the superconductor is given by ${\bf B} = - \nabla \Phi_K$, ${\bf \tilde{B}_0}$ and ${\bf \tilde{M}}$ are the 2-dimensional Fourier transforms in $x$ and $y$ of the inhomogeneous solution to London’s equation and the volume magnetization respectively, and ${\bf k} = k_x {\bf \hat{e}}_x + k_y {\bf \hat{e}}_y$. However, if the domain walls are assumed to be parallel to the $z$ axis, both ${\bf B}_0$ and ${\bf M}$ have only $z$ components, and Eq. (\[eq:phik\]) reduces to $$\Phi_K = \frac{K}{k(k+K)} \tilde{B}_{0z}({\bf k},0),
\label{eq:phik2}$$ where $\tilde{B}_{0z}$ is the $z$-component of ${\bf \tilde{B}}_{0}$. Then $${\bf \tilde{B}}({\bf k},z) = \frac{(i {\bf k}+k \hat{e}_z) K}{k(k+K)} \tilde{B}_{0z}({\bf k},0)e^{-kz}.
\label{eq:b}$$ In our case we are only interested in the $z$-component of the field outside of the superconductor, which takes the particularly simple form $$\tilde{B}_z({\bf k},z)=\frac{K}{k+K} \tilde{B}_{0z}({\bf k},0)e^{-kz}.
\label{eq:bz}$$
![ Comparison of the predicted magnetic field sensed by a 0.5 $\mu$m square Hall bar, 1.2$\mu$m above a single domain in a $p_x \pm ip_y$ superconductor, using the spontaneous domain currents predicted by Matsumoto and Sigrist,[@matsumoto1999] with (dashed line) and without (solid line) surface screening effects as described in the text. The solid line corresponds to the extended-Matsumoto-Sigrist model. []{data-label="fig:hallcrcm"}](HALLCRCM.PS){width="3.5in"}
The modeling in the previous section, which neglects surface shielding, is equivalent to Eq. \[eq:bz\] in the limit $\lambda \rightarrow 0$. Fig. \[fig:hallcrcm\] shows the effects of surface screening on the fields predicted for a single domain boundary for parameters appropriate for our Hall bar measurements. Even in this case the effects of screening are relatively small, because the penetration depth is smaller than the measuring height and size of the Hall bar. Surface screening effects would be even smaller (a few percent) for the case of SQUID imaging, because of the larger size of the sensor. The geometry for considering the effects of superconducting shielding on the edge fields is more complex than for the case of the domain boundary, as one needs to consider a superconductor bounded by both $z$ and at least one of $x$ or $y$. However, again the edge and surface effects will be confined on the scale of the penetration depth which is much smaller than the distance to the probe or the probe size. Therefore, we do not believe that the simple model presented above will be more than a factor of two different from a full calculation.
Discussion
==========
If the superconductivity of Sr$_2$RuO$_4$ breaks time-reversal symmetry, it should spontaneously generate supercurrents at domain boundaries and sample edges. The fact that no magnetic fields due to such supercurrents were observed using scanning magnetic microscopy places significant limits on the size of these currents and the size of the domains, as shown in Fig. \[fig:peakvd\]. In particular, from the combined Hall bar and SQUID measurements, we conclude that if the spontaneous supercurrents at a domain wall are of the size expected from the calculations of Matsumoto and Sigrist[@matsumoto1999] and the modelling done here, one can set a conservative upper limit on the domain size of 1.5 microns for both interior and edge domains. Alternatively, if the domains intersecting the ab face are 10 microns or more in size, we conclude that the spontaneous supercurrents at edges are a factor of 100 smaller than expected from the calculations of Matsumoto and Sigrist combined with our modelling.
Calculations of the self-consistent screening currents employed in our modelling have assumed an ideal $p_x\pm ip_y$ superconducting gap symmetry.[@matsumoto1999] However, for Sr$_2$RuO$_4$, the gap in the $ab$ plane is believed to be anisotropic.[@deguchi2004a; @deguchi2004b] In addition, three different bands contribute to the Fermi surface in Sr$_2$RuO$_4$.[@mackenzie2003] These properties are likely to impact on the magnitude of the self-consistent screening currents, although, a priori, it is not clear whether the magnitude would be increased or decreased from the values calculated by MS. On the other hand, muon spin resonance observed internal fields which are roughly consistent with the predicted values.[@luke1998] If these observed fields are due to internal domain walls, it suggests that the surface currents must be reduced by two or more orders of magnitude from their bulk values if the domains are larger than 10 microns. It is difficult to imagine what could so strongly reduce the surface fields at the ab surface due to domain boundaries. The surface screening effects are small; the surfaces are cleaved; and roughness even to the depth of a hundred angstroms or so will not substantially reduce the fields detected at the Hall probe or SQUID. This suggests that domains intersecting the ab plane are either so sparse as to not have been scanned or are smaller than a few microns. Another possibility is that the domains intersecting the ab surface are shallow, with a depth along the c-axis noticeably less than the penetration depth. In this case, the spontaneous currents and fields could be too weak and spread out in the ab layers to be detected. However, we note that either small domains or domains shallower than the optical skin depth would also interfere with observations of the Kerr effect rotation.
Sufficient roughness of the ac or bc faces can be expected to have a more noticeable effect on the edge or boundary currents. The samples used for the SQUID measurements on these faces were polished and AFM imaging on typical samples show them to be smooth to 5 nm (rms).[@karlthesis] MS assumed specular scattering from the edge in which case one component of the order parameter is suppressed while the other component is slightly enhanced. For diffuse scattering from a rough edge, both components will be suppressed and this will reduce the surface currents and resulting magnetic fields. Although self-consistent calculations have not been carried out for this case, the effect of surface roughness on the two component order parameter has been studied,[@nagato98] and one finds that the two components heal over quite different length scales. Using Ginzburg-Landau and London theory to estimate the resulting change in the surface magnetization, one finds that, even for completely diffuse scattering from a rough surface, the reduction in surface magnetization is less than 30%.[@kallin]
Domain walls cost energy because they disrupt the superconducting order. Unlike a ferromagnet, there is no balancing of this energy due to dipolar forces because spontaneous screening currents ensure that the magnetic field, or local magnetization, is zero inside the superconductor. Therefore, in principle, a single domain $p_x + ip_y$ superconductor is possible. However, domains will naturally form as the sample is cooled through $T_c$ and as extended objects, these domains are susceptible to pinning by defects and impurities in the sample. Therefore, one expects domains to be present although their density may be controlled by sample purity and slow cooling in a field. Muon spin resonance experiments were interpreted as evidence for dilute domains,[@luke1998] Kerr effect measurements suggested domain sizes in the range of 50 to 100 microns,[@xia2006] while the first phase sensitive Josephson tunneling measurements are consistent with no domains (or a small even number of domain walls between opposing faces of the crystal).[@nelson2004] On the other hand, more recent Josephson tunneling measurements were interpreted as evidence for dynamic domains of $\sim$1 micron on average,[@kidwingara2006] although one would extract larger domain sizes if finite domains perpendicular to the c-axis were included in the modelling. All of these measurements, except for muon spin resonance, would see reduced signals if the domain size along the c-axis becomes small and this would affect the measurements reported here as well. Unless the fields at domain walls are reduced by more than an order of magnitude in size from the predicted values, the Hall bar measurements suggest domain sizes of either less than 1.5 microns in size over the ab face or large enough that no domain wall fell in the 100$\times$100 $\mu$m scan area.
Earlier work has reported that large domains can be flipped by fields of the order of a mT or larger[@xia2006] and that small surface domains are influenced by fields $< 0.1\mu$T.[@kidwingara2006] While the data presented here was taken on samples cooled in fields less than 2.5$\mu$T, Hall data taken on samples cooled in up to a mT was very similar to that shown here except for the presence of more trapped vortices.[@bjornsson2005] In principle, very fast domain wall motion could result in zero time-averaged edge current and zero time-averaged domain wall current. However, previous experiments[@nelson2004; @kidwingara2006; @xia2006] suggest that the domain wall motion would be slow on our experimental time scale, which is 10 secconds per line scan for the scanning SQUID microscope data shown here. Therefore, it is unlikely that dynamic behavior of the domains prevents the observation of the signal in this experiment.
In conclusion, scanning magnetic microscopy measurements place quite severe limits on the size of edge currents and/or on domain sizes in Sr$_2$RuO$_4$. The different experimental results taken as evidence for $p_x + ip_y$ pairing come to quite different conclusions about domain sizes. Since there are now detailed predictions for the field profile in the vicinity of domain walls in the bulk, muon spin resonance could now, in principle, provide detailed information about the validity of these predictions as well as quantitative information about the density of domains in the bulk. In addition, either slow muons[@sigrist] or beta-NMR[@salman2006] could be used to probe the surface region and to look for fields due to spontaneous edge currents as well as domains near the surface. Scanning magnetic microscopy is still one of the most direct probes of domains intersecting the surface and of edge currents and further improvements in sensitivity may either confirm or rule out their existence.
Acknowledgements
================
We would like to thank H. Bluhm, J. Berlinsky, S.B. Chung, M. Franz, H. Hilgenkamp, T.-L. Ho, M. Matsumoto, M. Sigrist and M. Stone for many useful discussions and H. Bluhm for also sharing the results of his calculations before publication. We would also like to thank P. Bj[ö]{}rnsson for sharing his Hall bar data with us. This work was supported by DOE at Stanford under grant DE-AC02-76SF00515. Work at Penn State was supported by DOE under grant DE-FG02-04ER46159. JRK was supported by the Center for Probing the Nanoscale, an NSF NSEC, NSF Grant No. PHY-0425897, and by the Dutch NWO Foundation. In addition, CK acknowledges the hospitality of the Stanford Institute for Theoretical Physics and the Kavli Institute for Theoretical Physics during this collaboration and the support of the National Science Foundation under Grant No. PHY05-51164.
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abstract: 'We use supernovae measurements, calibrated by the local determination of the Hubble constant $H_0$ by SH0ES, to interpolate the distance-redshift relation using Gaussian process regression. We then predict, independent of the cosmological model, the distances that are measured with strong lensing time delays. We find excellent agreement between these predictions and the measurements. The agreement holds when we consider only the redshift dependence of the distance-redshift relation, independent of the value of $H_0$. Our results disfavor the possibility that lens mass modeling contributes a 10% bias or uncertainty in the strong lensing analysis, as suggested recently in the literature. In general our analysis strengthens the case that residual systematic errors in both measurements are below the level of the current discrepancy with the CMB determination of $H_0$, and supports the possibility of new physical phenomena on cosmological scales. With additional data our methodology can provide more stringent tests of unaccounted for systematics in the determinations of the distance-redshift relation in the late universe.'
author:
- Shivam Pandey
- Marco Raveri
- Bhuvnesh Jain
bibliography:
- 'biblio.bib'
title: 'A model independent comparison of supernova and strong lensing cosmography: implications for the Hubble constant tension'
---
Direct measurements of the Hubble constant, $H_0$, and its inference from the Cosmic Microwave Background (CMB) are showing a discrepancy of increasing statistical significance within the $\Lambda$CDM cosmological model (see [@Verde:2019ivm] for a recent review).
Measurements of $H_0$ based on supernovae, calibrated with cepheid variables [@Riess:2019cxk], and the results of CMB observations from the [*Planck*]{} satellite [@Aghanim:2018eyx] differ at a confidence level equivalent to $4.4$ standard deviations, ruling out the possibility that this discrepancy is due to a statistical fluctuation.
The challenging aspect of this tension is to understand if it is related to residual unaccounted systematic effects in the data or to a breakdown of the standard cosmological model that could result in the discovery of new physical phenomena on cosmological scales.
In this context, distance measurements from strong lensing time delays, as reported recently in [@Wong:2019kwg], are crucial as they do not share any source of systematic uncertainty with the other two probes. These use the variability of quasars lensed into multiple images by elliptical galaxies to estimate the time delay along the different lines of sight. By fitting for a model for the lens mass distribution, they constrain a combination of lens and source distances as detailed below. Sources and lenses are separated by cosmological distances so that the measured combination depends on both the amplitude and shape of the distance-redshift relation.
In this letter we carry out a consistency test of the distance-redshift relation measured by the supernovae in the Pantheon compilation (SN) [@Scolnic:2017caz] and the available time delay and lens distance measurements from H0LiCOW [@Suyu:2016qxx; @Wong:2019kwg]. Our tests span all redshifts that are probed and is independent of the underlying cosmological model. We carry out tests with the SN distances calibrated with local measurements of the Hubble constant by SH0ES [@Riess:2019cxk] as well as tests that are free of the $H_0$ calibration. We leverage the large number of measured SN distances to reliably interpolate between them, using Gaussian Process (GP) regression, and then predict the time delay and lens distances that should be observed. Since the two data sets that we consider do not share the same sources of systematic uncertainties this provides an end to end test that is sensitive to residual problems with either dataset. In addition to testing for a bias in the distance-redshift relation, we also test for under-estimated uncertainties by comparing the scatter in the predicted and measured distances.
Our tests compare SN and strong lensing measurements and are thus not sensitive to the possibility that both experiments have residual systematic bias affecting both of them in the same way. Although we cannot exclude this possibility, systematic biases that preserve the agreement, across redshift, of both probes seem unlikely. We also test the consistency of both, SN and H0LiCOW with Baryon Acoustic Oscillation (BAO) measurements in a model independent way.
These tests are complementary to other consistency tests between SN and H0LiCOW that have recently been discussed in literature [@Taubenberger:2019qna; @Arendse:2019hev; @Liao:2019qoc] and to studies probing the sources of systematic effects in the data sets considered here [@Rigault:2014kaa; @Rigault:2018ffm; @Jones:2018vbn; @Schneider:2013wga; @Xu:2015dra; @Unruh:2016adf; @Sonnenfeld:2017dca; @Kochanek:2019ruu; @Camarena:2019moy].
{width="\textwidth"}
Data and methodology {#data-and-methodology .unnumbered}
====================
We start by considering the six strong lensing time delay measurements, of which five were analyzed blindly, from the H0LiCOW collaboration in [@Wong:2019kwg; @Suyu:2009by; @Suyu:2013kha; @Wong:2016dpo; @Birrer:2018vtm; @Rusu:2019xrq; @Chen:2019ejq]. The difference of excess time delays between two lensed images (and angular positions $\mathbf{\theta}_i$ and $\mathbf{\theta}_j$) of a source (at angular position $\mathbf{\beta}$) is given by: $$\begin{aligned}
\label{Eq:TimeDelay}
\Delta t_{ij} = \frac{D_{\Delta t}}{c} \Bigg[ \frac{(\mathbf{\theta}_i - \mathbf{\beta})^2}{2} - \psi (\mathbf{\theta}_i) - \frac{(\mathbf{\theta}_j - \mathbf{\beta})^2}{2} + \psi (\mathbf{\theta}_j) \Bigg] \,,\end{aligned}$$ where, $\psi(\mathbf{\theta})$ is the lens potential. By measuring time delays and modelling the gravitational potential of the source and the lens one can infer the time delay distance: $$\begin{aligned}
\label{Eq:TimeDelayDistance}
D_{\Delta t} \equiv (1+z_{l})\frac{D_{l}D_{s}}{D_{{l}{s}}} \,,\end{aligned}$$ where, hereafter, subscripts ${l}$ and ${s}$ indicates quantities referring to the lens and source respectively and $D$ is the angular diameter distance of different objects. As we can see, the measured time delay distance is sensitive to the expansion history of the universe through its dependence on the angular diameter distance at two different redshifts. For the measured systems the spread in redshift is large, with the lens redshift ranging in $z\in [0.3,0.7]$ and the source redshifts ranging in $z\in [0.6,1.7]$ thus making the time delay distance measurements sensitive to the shape of the distance-redshift relation.
In addition to the time delay distance measurements, lens kinematic data can be used to estimate the angular diameter distance of the lens, $D_{l}$, by comparing the dynamical mass with the lensing mass (which depends on distances) [@Paraficz:2009xj; @Chen:2019ejq; @Jee:2014uxa; @Birrer:2015fsm; @Jee:2019hah].
The measurements of $D_{\Delta t}$ and $D_{l}$ are in general correlated since they use different aspects of the same data [@Birrer:2015fsm; @Birrer:2018vtm]. Since the H0LiCOW collaboration has not yet publicly released the full posterior of the two distance measurements for all but one of its observations, we separately consider the marginalized constraints on $D_{\Delta t}$ and $D_{l}$.
We observe that, if we consider logarithmic distances, $\log_{10}D_{\Delta t}$ and $\log_{10}D_{l}$, then the posterior distribution of the H0LiCOW measurements becomes practically indistinguishable from Gaussian. The reason why logarithmic distances are likely to be Gaussian distributed, or very close to that, is that they are computed as the difference of well measured quantities rather than their ratios. We discuss further details on the treatment of the strong lensing measurements in Appendix \[App:SLGaussianTest\].
This allows us to convert the constraints on $D_{\Delta t}$ and $D_{l}$ from [@Wong:2019kwg] into constraints on $\log_{10}D_{\Delta t}$ and $\log_{10}D_{l}$, properly accounting for the Jacobian of the transformation, and to consider the latter to be Gaussian distributed. We check that cosmological results obtained by fitting the logarithmic measurements reproduce the ones reported in [@Wong:2019kwg], as detailed in Appendix \[App:SLGaussianTest\].
We then consider the Pantheon SN compilation [@Scolnic:2017caz] that provides accurate measurements of relative distances across the redshift range $z\in [0.01,2.26]$ with $1048$ SN measurements. We use the measurement of the Hubble constant of $H_0=74.03\pm 1.42$ from the SH0ES project [@Riess:2019cxk]. Cepheid variables are used to calibrate the absolute magnitude of the SN so that the measured distance modulus can be used to directly estimate luminosity distances. Further details on the calibration of the SN distance modulus can be found in Appendix \[App:SN\_cal\]. While the SH0ES analysis is the most mature and precise of the local measurements of $H_0$, analyses with alternative approaches are underway. A recent analysis using the Tip of the Red Giant Branch method by the Carnegie-Chicago Hubble Program [@Freedman:2019jwv] yields a lower value of $H_0$ with somewhat larger uncertainty than SH0ES (but there is some debate about their analysis, see [@Yuan:2019npk]). We test this alternative result also in Appendix \[App:SL\_Bias\]. Note that the SN sample cannot be calibrated using the $H_0$ determination from CMB measurements in a model independent way. Not surprisingly, when using the standard cosmological model determination of the sound horizon scale, [@Macaulay:2018fxi; @Alam:2016hwk; @Aubourg:2014yra] find $H_0$ consistent with the CMB value. Finally we note that bias corrections of SN luminosities assume the $\Lambda$CDM model and in principle this breaks model independence, but as discussed in [@Kessler:2016uwi] it is a very small effect.
Given that the redshift range spanned by SN observations is populated by a large number of measurements we can interpolate between them to obtain an estimate of distances as a function of redshift. This is achieved by Gaussian process regression of the measured distance modulus. The Gaussian process kernel and kernel parameters are chosen so that the Gaussian process inference is as close as possible to the binned Pantheon SN sample. We find that this procedure is flexible enough to fully capture all the features that are present in the binned SN sample, starting from the full sample, thus effectively obtaining a version of the binned SN sample that is continuous in redshift. We also test the GP covariance matrix, and compare it to the result from polynomial interpolation which tends to significantly underestimate the error bars at intermediate redshifts. Further details on the implementation of the Gaussian process regression are discussed in Appendix \[App:SN\_GP\].
{width="\textwidth"}
With the Gaussian process for the SN we can now predict, with Eq. , the time delay and lens distances that H0LiCOW should have observed, independently of the cosmological model. We note that the predicuted distances from SN are correlated; we take this into account when computing the statistical significance of the reported results.
Results {#results .unnumbered}
=======
In Fig. \[Fig:DistanceComparison\] we show the results of the SN prediction for the time delay and lens distance measurements. In Panel a) we can see that the time delay distance measurement predicted by SN agrees with the direct measurements from H0LiCOW. The uncertainties in the two methods are comparable. In Panel b) the lens distance as directly measured by H0LiCOW and predicted by SN are compared. As in the previous case these are largely in agreement. The error bars of the SN predicted distances are significantly smaller than the H0LiCOW ones because this quantity is directly measured by a large number of SN at intermediate redshifts.
To precisely quantify the agreement between the H0LiCOW measurements and the SN ones, we exploit the fact that both logarithmic distances are close to Gaussian distributed. The difference between the two, weighted by the inverse sum of the two covariances is then chi squared distributed with number of degrees of freedom equal to the number of data points. This results in a probability of agreement of $85.7\%$ for the time delay distances and $63.6\%$ for the lens distance. Both results indicate very good agreement.
Further, in Fig. \[Fig:DistanceComparison\], there are no outlier measurements. Since the two distributions are very close, possible non-Gaussianities in the tails of the distributions are not expected to change the results significantly. This also rules out the possibility of a systematic uncertainty in the strong lensing sample at a level larger than the reported error bars.
Beyond the above tests, it is crucial to test whether the SN and strong lensing measurements agree on the amplitude and “shape” (the redshift dependence) of the distance-redshift relation. The amplitude test relates to possible discrepancies in the determination of the Hubble constant, while being fully independent of the expansion history. The shape test would tell us if there is agreement between the two measurements regardless of the overall calibration that measures the Hubble constant.
The amplitude of the distance-redshift relation can be tested by looking at the average residual logarithmic distance. The H0LiCOW data measure the average amplitude to be $\langle \log_{10}D_{\Delta t} \rangle = 3.514 \pm 0.012$ while the SN predict the average amplitude to be $\langle \log_{10}D_{\Delta t}^{\rm SN} \rangle = 3.517 \pm 0.017$. These two determinations are in full agreement at $89\%$ confidence level and provide a test at about $5\%$ precision. For lens distances we similarly have $\langle \log_{10}D_{\Delta t} \rangle = 3.026 \pm 0.044$ and $\langle \log_{10}D_{\Delta t}^{\rm SN} \rangle = 3.0416 \pm 0.0088$ which are again in full agreement at about $10\%$ precision.
To test the redshift dependence of the measurements we can consider distance ratios, or differences in logarithmic distances. Since the Hubble constant enters as an overall distance multiplier, its value cancels in the ratio. We then take both data sets and consider the ratio with respect to the system with the lowest lens redshift. Although arbitrary, this choice does not influence the outcome of the test. All possible choices would just be different linear combinations of the same data thus leaving the statistical significance unchanged.
We show in Fig. \[Fig:DistanceRatioComparison\] the comparison of distance ratios, as predicted by SN measurements and as estimated by H0LiCOW. As in Fig. \[Fig:DistanceComparison\], the ratios also show agreement between the two measurements. And since logarithmic distance differences are also close to Gaussian distributed, we can easily compute the statistical significance of their agreement. The time delay distance ratios are in agreement at the $78.2\%$ level and the lens distance ratios are in agreement at the $54.5\%$ confidence level. We note that the H0LiCOW measurements are incompatible with no redshift dependence and hence detect the shape of the distance-redshift relation at $34\sigma$ in Panel a) and $2.6\sigma$ in Panel b).
We can further compare these distance ratios with predictions from the BAO measurements. The BAO measurements are sensitive to the ratio of angular diameter distance of the galaxies and the photo-baryon sound horizon scale at the epoch of decoupling. Therefore, we can take the ratio of BAO measurements at two different redshifts and remove any sensitivity to the overall calibration with the sound horizon. In this analysis we consider three BAO measurements from BOSS DR12 data [@Alam:2016hwk] and one measurement from the eBOSS DR14 LRG data [@Bautista:2017wwp]. We use the lowest redshift DR12 measurement as our reference BAO measurement in the distance ratio test. The comparison between the BAO and SN predicted distance ratios can be performed as before and results are in agreement at the 95.4% confidence level. To qualitatively compare the results to the H0LiCOW measurements we shift the overall amplitude to match the H0LiCOW reference redshift. We compare the distance ratios of all three probes in Fig. \[Fig:DistanceRatioComparison\]. All measurements show remarkable consistency in this test, which is independent of $H_0$ and of the cosmological model.
Moreover, if we use the SN Gaussian process to calibrate the BAO angular diameter distances, as discussed in [@Aylor:2018drw], we can directly measure the sound horizon scale independently of the cosmological model. Individually the four calibrated BAO measurements that we consider are in good agreement on the sound horizon determination, giving: $137.39 \pm 3.91$ Mpc, $136.05 \pm 3.83$ Mpc and $137.48 \pm 3.92$ Mpc for the three SDSS DR12 observations in increasing order of their effective redshift; $133.77 \pm 6.24$ Mpc for the SDSS DR14 LRG observation. Jointly the four measurements result in a sound horizon measurement of $135.92 \pm 3.26$ Mpc, which is in $3.3 \sigma$ tension with the [*Planck*]{} results of $147.09 \pm 0.26 $ Mpc [@Aghanim:2018eyx]. This effectively accounts for a large portion of the $4.4 \sigma$ Hubble constant tension. Since the sound horizon is constant after recombination, this part of the Hubble constant tension cannot be resolved by introducing new physics between the redshift of the BAO measurements and recombination.
Summary and discussion {#summary-and-discussion .unnumbered}
======================
We have presented a new way of testing the consistency of distance measurements from supernovae and strong lensing time delays that is independent of the cosmological model. Our method exploits the power of SN observations across a large range of redshifts to directly predict the outcome of strong lensing measurements. We use Gaussian process regression to interpolate across redshift and show that these provide robust results. Tests of the covariance matrix show that it is more reliable than polynomial interpolation which tends to underestimate the error bars at intermediate redshifts.
We devise three types of tests sensitive to the distance-redshift relation: one directly tests distances (and therefore biases in $H_0$), one tests their calibration and another tests distance ratios. While the first test is sensitive to both calibration and redshift dependent systematic effects, the other two single out and test these two aspects independently.
We find that all tests report excellent agreement between the Pantheon supernovae, calibrated with the SH0ES distance ladder, and H0LiCOW time delay measurements. Given the model independence of our tests we conclude that, at present sensitivity, there is no indication for the presence of unaccounted systematic effects in either data set. In particular, if the distance-redshift relation inferred from SN is correct, there is no evidence of a residual bias due to mass modeling uncertainties in the strong lensing data. It is possible that both measurements have a bias of the same sign, magnitude and redshift dependence – this unlikely scenario would evade our tests.
The possibility that uncertainties in mass modeling, in particular the mass sheet degeneracy, have biased the strong lensing determination of $H_0$ has been discussed in the literature [@Schneider:2013wga; @Xu:2015dra; @Unruh:2016adf; @Sonnenfeld:2017dca; @Kochanek:2019ruu]. Recently [@Kochanek:2019ruu] has suggested that this leads to at least a 10% level of uncertainty on $H_0$ in present and near future measurements. We explicitly check the impact an unknown residual systematic would have on both $D_{\Delta t}$ and $D_{{l}}$. We create fake strong lensing observations and test for several types of unaccounted for errors:
- Biases in the distance-redshift relation that directly impact $H_0$ or the amplitude of the distance-redshift relation. An 8% bias that would fully reconcile the tension with [*Planck*]{} is disfavored at nearly 3$\sigma$.
- Redshift dependent biases. We test the ratio of distances as well as a bias that scales as $(1+z)$, motivated by uncertainties in lens mass modeling. These are also disfavored at over 2$\sigma$.
- Underestimated errors. We artificially increase the covariance in the lensing measurements to obtain 10% uncertainty on $H_0$. We then find that the probability of disagreement with SN inferred distances drops to 0.05%, i.e. over 3$\sigma$ evidence against inflated errors.
Unknown measurement systematic effects and/or incorrect lens modelling are likely to affect different systems differently, so they would show up as an unexpected redshift dependence of the measurements. On the other hand if they affected all lenses the same way, they would affect the amplitude. These problems are not guaranteed to show up as a discrepancy in the determination of some cosmological parameter, so our tests provide an independent check. We discuss the detailed results of these tests in Appendix \[App:SL\_Bias\].
Strong lensing and SN determination of distances almost certainly do not share the same systematic uncertainties. The consistency we have found between the two sets of measurements implies that the discrepancy with the CMB is more likely due to new physical phenomena, or a potential systematic error in the CMB analysis (which is considered unlikely as it has been tested extensively [@Addison:2015wyg; @Aghanim:2016sns; @Hou:2017smn; @Aylor:2017haa; @Aghanim:2019ame]). With reduced uncertainties, as expected by increases in the number of SN [@Abbott:2018wog; @Scolnic:2019apa; @Hlozek:2019vjs; @Shajib:2019toy] and new lens systems [@Huber:2019ljb; @Yildirim:2019vlv], we expect our methodology to provide more stringent tests for systematic uncertainties. In the near term joint analysis of the full $D_{\Delta t}$ and $D_{{l}}$ posterior will also strengthen our tests. With a factor of two reduction in statistical uncertainties, we could rule out a 5 percent bias at the 3$\sigma$ level, independent of the cosmological model – this would certainly strengthen the case for new physics.
We find that measurements of BAO distances are also in agreement with predictions from SN. Once calibrated with SN, the BAO measurements are in tension with the CMB over the determination of the sound horizon. We find that this accounts for a large part of the statistical significance of the Hubble constant tension. Since the sound horizon is constant after recombination, this tension is largely independent of the expansion history between the redshift of the measured BAOs, of about $z\sim 0.7$, and recombination. The structure of the CMB peaks measures the angular size of the sound horizon very precisely. Thus an explanation of the Hubble constant tension that would significantly change the inference of the CMB parameters must rely on new physical phenomena before recombination (see [@Knox:2019rjx] and references therein).
We implicitly assumed in our analysis that the universe is spatially flat. It is possible to perform the same tests assuming a curved universe. Since this will introduce an extra free parameter our tests would show stronger consistency. We defer an analysis of time delay distances and their implications for curvature to a future study (see [@Collett:2019hrr; @Arendse:2019hev] for related analyses).
We thank Eric Baxter, Gary Bernstein, Simon Birrer, Dillon Brout, Neal Dalal, Wayne Hu, Mike Jarvis and Sherry Suyu for helpful discussions and comments on the paper. SP is supported in part by the U.S. National Science Foundation award AST-1440226. MR and BJ are supported in part by NASA ATP Grant No. NNH17ZDA001N, and by funds provided by the Center for Particle Cosmology. BJ is supported in part by the US Department of Energy Grant No. DE-SC0007901. Computing resources were provided by the University of Chicago Research Computing Center through the Kavli Institute for Cosmological Physics at the University of Chicago.
The Supernovae calibration with the Hubble constant {#App:SN_cal}
===================================================
In this appendix we discuss the details of the SN distance modulus calibration with measurements of the Hubble constant. In doing so we mostly follow [@MRinprep]. The only difference with [@MRinprep] is that, in this work, we add directly the calibration to the SN data points and add its contribution to the SN covariance instead of using an additional parameter to describe the calibration. We have verified explicitly that these two approaches are equivalent.
The Pantheon collaboration provides measurements of the SN magnitude relative to a fiducial and arbitrary absolute magnitude, $m-M_{\rm fid}$ with its covariance. The Pantheon likelihood is then usually marginalized over the value of the fiducial magnitude so that the SN catalog is only using relative SN distances to constrain cosmological parameters.
The SN distance modulus, $m-M$ is related to luminosity distances, $D_L$, by $$\begin{aligned}
\label{Eq:DistanceModulus}
m-M = 5 \log_{10} \frac{D_L(z)}{10\,{\rm pc}} \,.\end{aligned}$$ To use this distance modulus to measure distances we then clearly need to fix the difference between the fiducial and absolute SN magnitude.
Calibrating the absolute magnitude of the very low redshift SN is precisely the way in which the Hubble constant in [@Riess:2019cxk] is measured. We can then use that determination of the Hubble constant to fix the Pantheon absolute magnitude.
The Pantheon fiducial absolute magnitude, $M_{\rm fid}$, is chosen to correspond to $H_0=70$ [@Kessler:2009yy], quoted here and throughout in units of kms$^{-1}$Mpc$^{-1}$.
The difference between the fiducial and measured absolute magnitude is then given by: $$\begin{aligned}
\label{Eq:SNCalibration}
M-M_{\rm fid} = 5 \log_{10}\frac{H_0}{H_0^{\rm fid}} \,.\end{aligned}$$ The covariance of the calibrated distance modulus has to take into account the uncertainty of the determination of the SN absolute magnitude so that: $$\begin{aligned}
\label{Eq:SNCalibratedCovariance}
{\rm cov}(m-M)_{ij} = {\rm cov}(m-M_{\rm fid})_{ij} +\left( \frac{5}{\ln 10} \frac{\sigma_{H_0}}{H_0}\right)^2 \,,\end{aligned}$$ where the indexes $i,j$ run over all the SN in the catalog.
The fully calibrated SN compilation is then sensitive to both the amplitude and the shape of the distance-redshift relation.
We check our implementation by fitting the resulting measurements with the flat $\Lambda$CDM model in which the luminosity distance, as a function of redshift, is given by: $$\begin{aligned}
\label{Eq:LCDMLuminosityDistance}
D_L(z) = \frac{1+z}{H_0} \int_0^z\frac{dz'}{\sqrt{\Omega_m (1+z')^3 + 1-\Omega_m}} \,,\end{aligned}$$ where $\Omega_m$ denotes the total matter density in units of critical density today.
In Fig. \[Fig:SNParameterTest\] we show the joint posterior of $H_0$ and $\Omega_m$ as obtained by fitting both the full and binned calibrated Pantheon SN catalog. The figure also shows the results reported in [@Riess:2019cxk] and [@Scolnic:2017caz] for the two flat $\Lambda$CDM parameters respectively, showing that our procedure is fully consistent.
![ \[Fig:SNParameterTest\] The marginalized posterior of the $\Lambda$CDM parameters from the calibrated full and binned Pantheon SN sample. The filled contour corresponds to the 68% C.L. region while the continuous contour shows the 95% C.L. region. The two dashed lines show the constraints on the two cosmological parameters separately, as obtained in [@Riess:2019cxk] for $H_0$ and [@Scolnic:2017caz] for $\Omega_m$ respectively. We do not show them in the joint contour panel for visual clarity.](figure_AppA_1){width="\columnwidth"}
We further notice that, in order to directly infer distances from the SN distance modulus, we have to apply a correction that takes into account the difference between redshifts in the heliocentric, $z_{\rm h}$, and CMB, $z_{\rm CMB}$, reference frames. The luminosity distance at a given redshift in the CMB frame is then given by: $$\begin{aligned}
5\log_{10}\frac{D_L(z_{\rm CMB})}{10\,{\rm pc}} = m-M -5\log_{10}\frac{(1+z_{\rm h})}{(1+z_{\rm CMB})} \,.\end{aligned}$$ We observe that this correction is overall very small.
Distribution of measured time delay distances {#App:SLGaussianTest}
=============================================
In this appendix we discuss the Gaussianity of the measured time delay and lens distance measurements, detailing the analysis of the public posterior distribution of the SDSS 1206+433 system by the H0LiCOW collaboration. The estimated distances are obtained as ratios of well measured quantities that are close to Gaussian distributed, but their ratios are not Gaussian distributed. Logarithmic distances, on the other hand, are sums and differences of well measured quantities and should have a distribution that is closer to Gaussian.
![ \[Fig:SLGaussianPosterior\] The joint posterior of logarithmic time delay and lens distances for the SDSS 1206+4332 strong lensing system. The filled contour corresponds to the 68% C.L. region while the continuous contour shows the 95% C.L. region. The black dashed lines show the posterior of the two separate logarithmic distances as inferred from Eq. . ](figure_AppB_1){width="\columnwidth"}
In Fig. \[Fig:SLGaussianPosterior\] we show the posterior of logarithmic time delay and lens distances as obtained form the publicly available posterior samples of the SDSS 1206+433 system, together with its Gaussian approximation. The full posterior is very close to the Gaussian approximation. The 95% C.L. region shows slight deviations from Gaussianity that are hard to tell apart from sampling noise.
We can then proceed and derive the Gaussian approximation for the two separate distance measurements under the logarithmic distribution assumption. The H0LiCOW collaboration reports the mean distance together with the $84\%$ and $16\%$ confidence level constraints, in Table 2 of [@Wong:2019kwg]. Given two constraints, $v_1$ and $v_2$, at two different confidence levels, $p_1$ and $p_2$, for a random variable $X$, assuming that $\log X$ is Gaussian distributed, we can compute the mean, $\langle \log X \rangle$ and variance, ${\rm var}(\log X)$, of $\log X$ as: $$\begin{aligned}
\label{Eq:SLLogGaussianization}
\langle \log X \rangle =& \frac{-{\rm Erf}^{-1}(2 p_2-1)|\log v_1 -\log v_2|}{|{\rm Erf}^{-1}(2 p_2-1) -{\rm Erf}^{-1}(2 p_1-1)|} \,, \nonumber \\
{\rm var}(\log X) =& \frac{|\log v_1 -\log v_2|}{\sqrt{2}|{\rm Erf}^{-1}(2 p_1-1) -{\rm Erf}^{-1}(2 p_2-1)|} \,,\end{aligned}$$ where ${\rm Erf}^{-1}$ denotes the inverse error function. This transformation takes into account the Jacobian transformation of the two variables.
$z_{{l}}$ $D_{\Delta t}$ (Mpc) $\log_{10} D_{\Delta t}$ $\log_{10} D^{\rm SN}_{\Delta t}$ $D_{{l}}$ (Mpc) $\log_{10} D_{{l}}$ $\log_{10} D^{\rm SN}_{{l}}$
--------------------------------------------- ----------- ---------------------- -------------------------- ----------------------------------- ---------------------- --------------------- ------------------------------
B1608+656 [@Suyu:2009by] 0.630 $5156^{+296}_{-236}$ $3.714\pm 0.022$ $3.689 \pm 0.025$ $1228^{+177}_{-151}$ $3.090\pm 0.058$ $3.123 \pm 0.010$
RXJ1131-1231 [@Suyu:2013kha; @Chen:2019ejq] 0.295 $2096^{+98}_{-83}$ $3.323\pm 0.019$ $3.339 \pm 0.015$ $804^{+141}_{-112}$ $2.908\pm 0.068$ $2.929 \pm 0.009$
HE 0435-1223 [@Wong:2016dpo; @Chen:2019ejq] 0.454 $2707^{+183}_{-168}$ $3.433\pm 0.028$ $3.427 \pm 0.018$ $-$ $-$ $-$
SDSS 1206+4332 [@Birrer:2018vtm] 0.745 $5769^{+589}_{-471}$ $3.764\pm 0.040$ $3.758 \pm 0.032$ $1805^{+555}_{-398}$ $3.261\pm 0.113$ $3.154 \pm 0.010$
WFI2033-472 [@Rusu:2019xrq] 0.657 $4784^{+399}_{-248}$ $3.686\pm 0.029$ $3.672 \pm 0.028$ $-$ $-$ $-$
PG 1115+080 [@Chen:2019ejq] 0.311 $1470^{+137}_{-127}$ $3.167\pm 0.039$ $3.216 \pm 0.014$ $697^{+186}_{-144}$ $2.844\pm 0.102$ $2.958 \pm 0.010$
We report in Table \[Tab:SLGaussianResults\] the results of computing the logarithmic time delay distances from the measurements in [@Wong:2019kwg]. We observe that these results are in good agreement with the log-normal likelihoods reported for three of the lenses in [@Suyu:2009by; @Suyu:2013kha; @Wong:2016dpo]. Moreover, we show the logarithmic distance posterior derived from the results in Table \[Tab:SLGaussianResults\] for the SDSS 1206+4332 strong lensing system. This is indistinguishable, for all practical purposes, from the Gaussian approximation and the full posterior.
![ \[Fig:SLSNLCDMComparison\] The joint marginalized posterior distribution of the two flat $\Lambda$CDM parameters, $\Omega_m$ and $H_0$, as obtained from the calibrated Pantheon SN sample, H0LiCOW time delays, and [*Planck*]{} CMB observations. The filled contour corresponds to the 68% C.L. region while the lines shows the 95% C.L. region. ](figure_AppB_2){width="\columnwidth"}
As a further check we fit to the measurements (both individuallhy and jointly) the flat $\Lambda$CDM model finding good agreement with the results in [@Wong:2019kwg]. Since we do not have access to the joint time delay and lens distance posterior we cannot combine the two measurements. We therefore show in Fig. \[Fig:SLSNLCDMComparison\] the joint posterior on $\Omega_m$ and $H_0$ for the time delay and lens distance measurements. The time delay result measures $H_0=73.1\pm 1.8$, in good agreement with [@Wong:2019kwg]. As a reference we show the results from the fully calibrated Pantheon sample, highlighting their agreement with H0LiCOW within the $\Lambda$CDM model, and the results from [*Planck*]{} CMB measurements.
Gaussian process regression on the Supernova data {#App:SN_GP}
==================================================
In this appendix we describe the details of the SN Gaussian process (GP) regression [@Rasmussen:2005:GPM:1162254] that allows for the computation of the predicted time delay distances.
The goal of our GP construction is to build, starting from the full SN catalog, a continuous version of the SN sample that is in quantitative agreement with the binned data that the Pantheon collaboration provides.
The first step in the construction of the GP is to define the time scale that is used. The binned SN data points are not equally spaced in redshift [@Scolnic:2017caz; @Brout:2018jch]. In order to ensure that the GP recovers the relative weights of the binned data points as solely determined by the data covariance, and that it does not introduce artefacts that depend on the relative distance between the points, we find a unique mapping between a uniform redshift spacing and the redshift values of the binned Pantheon data. This effectively defines a new time coordinate in which the binned SN sample is equispaced and this coordinate is then used to define the GP.
In order to reliably compare the estimated time delay distances, we need to make sure that the GP has enough freedom to describe all variations in the measured distance modulus from the Pantheon data.
The fluctuations in the SN binned data points are small compared to overall variations of the distance-redshift relation. To help the GP in recovering them we subtract from the data the best fit $\Lambda$CDM distance modulus that is added back to the GP predictions. Notice that this is just a convenient choice to work with data points that are scattered around zero.
The degree of flexibility of the Gaussian process depends on the choice of the kernel function describing relative correlation between different points with different time coordinates. We consider three different kernels to interpolate the full SN catalog: the exponential squared kernel; the Matern kernels of order $\nu = 9/2$; the rational quadratic kernel. The first two kernels have two free parameters describing the strength and the characteristic time scale of the correlation. The last kernel, in addition to these two parameters, has a parameter, the decay exponent, that describes the relative weighting of the large and small scale variations.
To all these kernels we add a constant offset that we need to accurately reproduce the correlation between the SN measurements induced by the shared amplitude calibration, as in Eq. .
The kernel parameters are usually selected by maximizing the marginal likelihood (i.e. the evidence) of the observed data predicted using the GP. This method is, however, not guaranteed to converge to a unique working solution. We find that, in our case, this does not provide a good solution for the GP.
To select the best kernel parameters we then minimize the Kullback-Leibler (KL) divergence [@kullback1951information] between the GP prediction and the binned Pantheon data.
The KL divergence represents the information difference in going from the GP to the full binned data and, in this case, serves as a measure of how much the two distributions differ. The KL divergence, $D(P_{\rm d}||P_{\rm gp})$, for the two Gaussian distributions, $P_{\rm gp}$ and $P_{\rm d}$, is given by: $$\begin{aligned}
\label{Eq:KLOptimization}
D(P_{\rm d}||P_{\rm gp}) \equiv \frac{1}{2 \ln 2} \Bigg[ & (y_d - y_{gp})^T \Sigma_d^{-1} (y_d - y_{gp}) \nonumber \\
& -\ln \frac{{\rm det} \ \Sigma_{gp} }{{\rm det} \ \Sigma_d} - d + {\rm tr}(\Sigma_{gp} \Sigma_d^{-1}) \Bigg] \,, \nonumber \\\end{aligned}$$ where $y_d$ and $y_{gp}$ are the distance modulus residuals from the binned SN data and predictions from the GP at a particular set of kernel parameters, $\Sigma_d$ is the covariance of the binned data, $\Sigma_{gp}$ is the Gaussian process predicted covariance conditioned on the full Pantheon data and $d$ is the number of the binned data points.
We find that this optimization problem, for all the kernels that we consider, has always a unique and well defined solution.
This also allows us to compare the performances of different kernels in matching the binned SN data. We find that using the rational quadratic kernel leads to the lowest KL divergence value, by a marginal amount, and hence we use this as our fiducial choice of kernel to produce the main results of this paper.
![ \[Fig:AppBinnedGPComparison\] Comparison of the binned Pantheon data and the predictions from the GP on the full SN sample, with the best available kernel and kernel parameters. The continuous line represents the predicted GP mean while the color band represents the GP predicted uncertainty. In this figure the best fit $\Lambda$CDM model has been subtracted to highlight the features in the binned SN sample. ](figure_AppC_1){width="\columnwidth"}
In Fig. \[Fig:AppBinnedGPComparison\] we compare the results obtained from the GP regression of the full SN sample that we discussed, for the best matching kernel, to the binned Pantheon measurements. As we can see this results in excellent agreement between the two on the reconstructed distance modulus, and its variance, that preserves all the features of the binned data points. We highlight that our goal is to preserve all features in the distance measurements that are present in the binned sample rather than smoothing it further.
![\[Fig:AppBinnedGPCorrelationComparison\] The correlation matrix of the binned calibrated Pantheon distance modulus compared to the correlation matrix predicted by the GP regression of the full SN sample. ](figure_AppC_2){width="\columnwidth"}
We also test that the Gaussian process is recovering the correlation between different data points as in the binned SN sample. These correlations are induced by the common shared calibration of the SN and joint correction of systematic effects, as discussed in [@MRinprep]. In Fig. \[Fig:AppBinnedGPCorrelationComparison\] we compare the binned Pantheon correlation matrix with the GP predicted correlation. As we can see these correlations are not negligible, as discussed in [@MRinprep], and are faithfully reproduced by the GP regression of the full SN sample.
![ \[Fig:AppGPLCDMParamPosterior\] The comparison of the posterior of the $\Lambda$CDM parameters as obtained by fitting the full Pantheon data set and by fitting the GP prediction evaluated at the redshifts of the binned SN sample. The filled contour corresponds to the 68% C.L. region while the continuous contour shows the 95% C.L. region. This ensures that the GP is a faithful representation of the SN data and is not introducing any subtle changes in the predictions across redshift. ](figure_AppC_3){width="\columnwidth"}
To further test the fidelity of the GP regression we want to ensure that the kernel smoothing is not introducing subtle inconsistencies in either the mean and the covariance of the predicted measurements. These include over smoothing the data that would result in a wrongly overestimated data constraining power. To do so we take the GP predicted mean and covariance at the redshifts of the binned Pantheon data set and we fit them with the $\Lambda$CDM model as if they were the true data points. In Fig. \[Fig:AppGPLCDMParamPosterior\] we show the excellent agreement between the full Pantheon posterior and the posterior obtained fitting the GP predictions. Furthermore we have checked that, if we were to take more data points, the results will still perfectly agree with the full data set, meaning that the GP is not introducing spurious data constraining power.
To further ensure the stability of our results to the specific choice of the GP kernel we run the end to end analysis for all the three kernels that we consider. In Fig. \[Fig:AppDifferentKernelTest\] we report the results, as in Fig. \[Fig:DistanceComparison\] and Fig. \[Fig:DistanceRatioComparison\], clearly showing quantitative agreement between different kernels.
{width="\textwidth"}
![ \[Fig:AppPolynomianTest\] Comparison of the binned Pantheon data and the polynomial fit of degree 5 and 15. The continuous line represents the mean of the polynomial fit while the color band represents the predicted uncertainty. Comparing this to Fig. \[Fig:AppBinnedGPComparison\] we see that using polynomial fit results in wrong error predictions across redshift. This highlights the need for a more sophisticated GP regression technique to recover unbiased estimate of covariance between the points. In this figure the best fit $\Lambda$CDM model has been subtracted to highlight the features in the binned SN sample. ](figure_AppC_5){width="\columnwidth"}
Finally we can compare our predictions to the case of fitting the binned distance modulus with a polynomial in log-redshift. We consider a 5-degree and 15-degree polynomial and show the results of the fit in Fig. \[Fig:AppPolynomianTest\]. Qualitatively the mean behavior reproduces the results of the GP for the two different degrees, with the higher order polynomial preserving more details of the data. The covariance of the polynomial fit is, however, largely misestimating the covariance of the true data points. In particular the low degree polynomial case is underestimating the covariance so that using this technique would clearly give overly constrained results. On the other hand the high order polynomial is overestimating the covariance at the end points while underestimating it toward the center of the redshift distribution. This is because, outside of the radius of convergence of the polynomial expansion, polynomials grow unbounded. This is then responsible for over constrained results inside the convergence radius and under constrained results outside of it. We have also compared our results with some of the polynomial models considered in [@Arendse:2019hev], by changing the interpolation time scales and we reach similar conclusions.
All these issues are not present in our GP construction that overall gives a more faithful representation of the data.
Tests with biased fake H0LiCOW Data {#App:SL_Bias}
===================================
In this section we detail the tests on fake H0LiCOW data created by biasing the measured $D_{\Delta t}$ and $D_{{l}}$ by a specified percentage. We change both the mean and covariance of the measurements to keep the signal to noise ratio of the measurements constant. We then analyze the consistency of these fake data sets with SN and [*Planck*]{} data in both a model independent way and within the $\Lambda $CDM model. We show the comparison of these shifted data with the original data and SN predictions in Fig. \[Fig:BaisedDistanceComparison\]. We find that a +8% shift in $D_{\Delta t}$ and +15% shift in $D_{{l}}$ predictions would resolve the $H_0$ tension between [*Planck*]{} and H0LiCOW, but would result in a 2.85$\sigma$ tension between SN and H0LiCOW on the determination of $H_0$ (the tension increases to over 3$\sigma$ for a 10% shift in $D_{\Delta t}$. ) We can also do this comparison in a model independent way by comparing the amplitude of the shifted H0LiCOW data points with the SN predictions. This comparison results in a tension of 1.45$\sigma$ over the amplitude of $\log_{10} D_{\Delta t}$ and 0.6$\sigma$ in $\log_{10} D_{{l}}$ with respect to SN measurements.
A negative bias of -8% for time delay distances and -15% for lens distances would decrease the agreement of H0LiCOW with both [*Planck*]{} and SN. In particular, considering time delay distances, the $H_0$ determination would be in 5.8$\sigma$ tension with [*Planck*]{} and 2.2$\sigma$ tension with SN. The model independent amplitude test would report a 1.8$\sigma$ tension between SN and H0LiCOW on both time delay and lens distance measurements.
Notice that these tests would leave the model independent shape test invariant since we are considering a multiplicative bias that is the same for all H0LiCOW measurements.
We also test a redshift dependent bias parameterized by $0.5(z_{\rm lens}-z_{\rm mean})$ where $z_{\rm mean}$ is the mean redshift of the lenses. This results in a mean bias amplitude of 8% in $D_{\Delta t}$ and 15% bias in $D_{{l}}$, consistent with the constant bias case. Unknown measurement systematics or incorrect lens modelling are likely to affect different systems differently, so they could show up as an unexpected redshift dependence. These problems are not guaranteed to show up as a discrepancy in the determination of cosmological parameters, so our tests provide an independent check. We get a 2.7$\sigma$ tension between these fake measurements and SN predictions for the time delay distance test and a 2.5$\sigma$ tension for time delay distance ratio test. The values for the lens distance tests are given in Table \[Tab:SLBiasComp\].
Finally, we test the possibility that the random error is underestimated by inflating the covariance of the measurements to give a 10% uncertainty on the resulting $H_0$. This could represent a scenario where lens mass modeling adds to the scatter. We inflate the error bar on $D_{\Delta t}$ measurements by a factor four (we do not consider $D_{l}$ since these already result in a $10\%$ determination of $H_0$). In this case, comparing SN predictions to the fake data gives a probability of disagreement of $0.05\%$ for the full test and $0.2\%$ for the shape test. Both these results therefore signal a statistically significant excess confirmation.
We have also checked the alternative value of $H_0$ obtained using the TRGB [@Freedman:2019jwv] calibration of local SN. Since their value of $H_0$ is lower, we get slightly higher tension with strong lensing though not at high significance given the larger error bars in the determination of $H_0$. The global test of time delay and lens distance is in agreement at the $58.8\%$ and $45.7\%$ confidence level respectively. The amplitude test gives slightly higher tension with a probability of agreement of $25.1\%$ and $37.7\%$ respectively for time delay and lens distances. All of these results are summarized in the Table \[Tab:SLBiasComp\].
{width="\textwidth"}
[ l c c c c c c c ]{} &
-----------------
$H_0$
test ($\sigma$)
-----------------
&
---------------------
full $D_{\Delta t}$
test ($\sigma$)
---------------------
&
--------------------------
$D_{\Delta t}$ Amplitude
test ($\sigma$)
--------------------------
&
----------------------
$D_{\Delta t}$ Shape
test ($\sigma$)
----------------------
&
-----------------
full $D_{{l}}$
test ($\sigma$)
-----------------
&
---------------------
$D_{{l}}$ Amplitude
test ($\sigma$)
---------------------
&
-----------------
$D_{{l}}$ Shape
test ($\sigma$)
-----------------
\
Original & 0.4 & 0.2 & 0.1 & 0.3 & 0.5 & 0.3 & 0.6\
TRGB Calibration & 1.0 & 0.5 & 1.1 & 0.3 & 0.7 & 0.8 & 0.6\
Constant Bias & 2.9 & 0.6 & 1.5 & 0.3 & 0.5 & 1.0 & 0.6\
Redshift-dependent Bias & $0.3$ & $2.6$ & $0.2$ & $2.5$ & $1.1$ & $0.4$ & $1.3$\
Increased Uncertainty & 0.1 & 0.0006 & 0.2 & 0.002 & $-$ & $-$ & $-$\
|
---
abstract: 'We investigate strong lensing by non-singular finite isothermal ellipsoids taking into account the influence of the matter along the line of sight and in the close lens vicinity. We compare three descriptions of light propagation: the full approach taking into account all matter inhomogeneities along the rays, the single plane approach, where we take into account the influence of the strong lens neighbours but neglect the foreground and background objects, and the single lens approach. In each case we simulate many strong lensing configurations placing a point source at the same redshift but in different locations inside the region surrounded by caustics. We further analyze configurations of four or five images. For every simulated strong lensing configuration we attempt to fit a simplified lens model using a single isothermal ellipsoid or a single isothermal ellipsoid with external shear. The single lens fits to configurations obtained in the full approach are rejected in majority of cases with 95% significance. For configurations obtained in the single plane approach the rejection rate is substantially lower. Also the inclusion of external shear in simplified modeling improves the chances of obtaining acceptable fits, but the problem is not solved completely. The quantitative estimates of the rates of rejection of simplified models depend on the required accuracy of the models, and we present few illustrative examples, which show that both matter close to the lens and matter along the rays do have important influence on lens modeling. We also estimate the typical value of the external shear and compare the fitted parameters of the simplified models with the parameters of the lenses used in the simulations.'
author:
- |
M. Jaroszynski$^{1}$[^1] and Z. Kostrzewa-Rutkowska$^{1}$\
$^{1}$University of Warsaw Observatory, Al. Ujazdowskie 4, 00-478 Warsaw, Poland
date: 'Accepted 2012 April 26; Received 2012 April 25; in original form 2011 November 10'
title: 'Background, foreground and nearby matter influence on strong gravitational lenses'
---
\[firstpage\]
gravitational lensing: strong and weak - large-scale structure of the Universe
Introduction
============
Strong gravitational lensing of distant sources has low probability and is usually attributed to a single concentration of matter close to the line of sight, while the influence of anything else is neglected (see e.g. [@K06] for a review of strong lensing). In some cases simple, single component models of lenses are not satisfactory. Even the first lensed quasar observed, QSO 0957+561 A,B [@b32; @b33] requires a combination of a galaxy and a galaxy cluster to model its basic properties [@Y81].
The influence of another object along the line of sight on the main lens properties has been investigated by @KA88. @KKS97 argue that adding an external shear to an elliptic lens model greatly improves the fits of four image configurations, but they conclude that the shear is probably related to the main lens. @BK96 investigates the influence of the density fluctuations along the line of sight on the accuracy of Hubble constant determination based on time delays measured in some of the multi-image QSO systems. @ChKK03 check the influence of the substructure not related to the main lens on the observed flux ratios in lensed images. @WBO04 study the probability of giant arcs using the results of a large cosmological simulation and multi-layer approach to backward ray tracing. They find that the matter on the way slightly increases the probability of arcs. @WBO05 address specifically the question of secondary lenses on the line of sight. According to this study the role of secondary lenses is a strong function of source redshift: it can be neglected in 95% of cases for a source at $z_\mathrm{S}=1$, but is important in 38% of cases at $z_\mathrm{S}=3.8$. Using single plane approach introduces systematic errors to the estimated lens parameters. In a recent paper @AN11 investigate the influence of the secondary lenses in the line of sight on the accuracy of estimating the dark energy density.
It is probably impossible to quantify the influence of the matter along the line of sight and of the strong lens close (but possibly unobservable) neighbours in a single case of real observed multiple image configuration. In this paper we are going to obtain a statistical measure of such an influence performing simulations of many multiple image configurations using a realistic model of light propagation in an inhomogeneous Universe model. To find the role of matter not belonging to the strong lens, we follow the rays including or not the lens neighbours and inhomogeneities along the rays. Again, the direct comparison between individual image configurations obtained as a result of different descriptions of light propagation seems useless, since one should compare positions and flux ratios of images of a point source [*at the same position*]{} and it is not clear how to compare source positions between the different models of light propagation. If image configuration is considered, it is the source position relative to the caustic structure which counts and since the caustic structures are different the problem persists.
Instead of making comparisons on a one to one basis, we compare the properties of all four or five image configurations between the models of light propagation. If the influence of matter outside the strong lens were unimportant, each lens - image configuration would be similar to some configuration obtained with a single lens model. To check whether this is the case we attempt to fit such simplified model to all image configurations considered. The success rate of the fitting procedure applied to configurations obtained with a different model of light propagation statistically measures the similarity of this model to the single lens model.
In Sec. 2 we describe our models of light propagation. Sec. 3 presents tools used to compare different models and the results of such comparison. Discussion and conclusions follow in Sec. 4.
Model of the light propagation
==============================
Deflection angles
-----------------
The results of the Millennium Simulation [@s05] give the matter distribution (the positions and velocities of $\sim 10^{10}$ simulation particles) in the 64 epochs corresponding to the redshifts $0 \le z_i \le127$. In our calculations we assume, that the propagation of a ray between redshifts $z_{i_1} \equiv (z_{i-1}+z_i)/2$ and $z_{i_2} \equiv (z_{i+1}+z_i)/2$ is affected by matter distribution given for the epoch $z_i$. We are interested in sources at high redshifts ($z \ge 1$), so there are always many such matter layers on the ray path, and each of them can be considered thin. Thus the multilayer approach [@b1] with layers corresponding to the Millennium epochs is natural. The position of the ray in the $N$-th layer in the angular coordinates is given as $$\bm{\beta}_N = \bm{\beta}_1
- \sum_{i=1}^{N-1}~\frac{d_{iN}}{d_{N}}~\bm{\alpha}_i(\bm{\beta}_i)$$ where $d_{iN}$ is the angular diameter distance as measured by an observer at epoch $i$ to the source at epoch $N$, $d_N$ - the angular diameter distance to the same source measured by a present ($z=0$) observer, and $\bm{ \alpha}_i(\bm{ \beta}_i)$ is the deflection angle in the $i$-th layer at the position $\bm{\beta}_i$. In a flat cosmological model the angular diameter distances in the lens equation can be replaced by comoving distances. Below we use comoving distances $D(z)$, denoting $D_i \equiv D(z_i)$ for short. Since the comoving distances are additive in a flat model, one has $d_{iN}/d_N=(D_N-D_i)/D_N$. In the calculations we apply more efficient recurrent formula of @b34, equivalent to the above equation.
To calculate the deflection angle in a given layer we need a description of its matter distribution. This is done in two steps. The averaged matter density is defined on a coarse grid of $256^3$ cells. The gravitationally bound haloes have been described by @b11 and @b4. Both kinds of data are accessible from the Virgo - Millennium Database [@b23]. @JK10 use the same information on matter distribution, but their calculation of lensing effect is based on different approach.
Since only the positions, virial masses, virial radii, and virial velocities of the haloes can be obtained from the Database, the detailed matter distributions of each halo must be supplemented. We use non singular isothermal ellipsoids (NIE) as models of individual lenses. The averaged matter density includes also the matter belonging to haloes, so the halo mass models must be compensated by a shallow negative density distribution (see below).
In constructing the matter layers we follow @b10, randomly rotating and shifting simulation cubes corresponding to different epochs, which eliminates the consequences of periodic boundary conditions used in their calculations. The comoving thickness of the $i$-th layer $D_{i_2}-D_{i_1}$ is of the order of $10^2~\mathrm{Mpc}$, always smaller than the simulation cube size $500/h~\mathrm{Mpc}$. The 2D layers are not periodic in general. @b17 define specific projection directions to obtain periodic 2D density distributions and use 2D Fourier transforms in calculations of the deflection angles. Our 3D grid ($256^3$) is small enough to employ spectral methods. Using Poisson equation we calculate 3D gravitational acceleration ${\bm g}$ on the grid. The deflection of a ray passing the $i$-th layer at position $\bm{\beta}_i$ due to the averaged (or [*background*]{}) matter distribution is given as: $$\bm{\alpha}_i^\mathrm{bcg}(\bm{\beta}_i) = \frac{2}{(1+z_i)c^2}~
\int_{D_{i_1}}^{D_{i_2}}~{\bm g}_\perp(D,\bm{\beta}_i)~dD$$ where ${\bm g}_\perp$ is the component of gravitational acceleration perpendicular to the ray, and its value at any location is obtained by the interpolation on the 3D grid. $(D,\bm{\beta})$ are used as coordinates. Since our simulated maps of the sky cover typically regions of a few minutes of arc in size, the position components $(D\beta_x,D\beta_y)$ may be treated as Cartesian coordinates in the plane perpendicular to the propagation direction, and the comoving distance $D$ serves as radial coordinate. Because the integration should be performed over the proper distance, instead of comoving one, the result is divided by the factor $1+z_i$.
For each individual halo we apply the model of non-singular isothermal ellipsoid as described by @K94. The deflections in 2D real notation are given by @K06: $$\alpha_x=
\frac{\alpha_0}{q^\prime}~\mathrm{arctan}
\left(\frac{q^\prime~x}{\omega+r_0}\right)~~
\alpha_y
=\frac{\alpha_0}{q^\prime}~\mathrm{artanh}
\left(\frac{q^\prime~y}{\omega+q^2r_0}\right)$$ where: $$\omega(x,y,q,r_0)=\sqrt{q^2(x^2+r_0^2)+y^2}~~~~q^\prime=\sqrt{1-q^2}$$ The ray crosses the lens plane at $(x,y)$, the lens centre is placed at the origin of the coordinate system, the major axis along $x$. The axis ratio is given by $q$, $r_0$ is the core radius, and $\alpha_0$ is the deflection angle parameter (see below).
The convergence $\kappa$ (which is proportional to the surface mass density) has the required (NIE) form: $$\kappa(x,y,q,r_0)=\frac{1}{2}(\alpha_{x,x}+\alpha_{y,y})
=\frac{1}{2}~\frac{\alpha_0}{\omega(x,y,q,r_0)}$$ Far from the centre the 2D matter distribution becomes elliptical, as required.
The NIE matter distribution is infinite with divergent total mass. To obtain a finite mass lens we follow @K06 combining two NIE distributions with different core radii $r_1 \ll r_2$. Inside $r_1$ the surface density is approximately constant, isothermal in between, and sharply falling outside $r_2$: $$\kappa(x,y,q)=\kappa(x,y,q,r_1)-\kappa(x,y,q,r_2)
\rightarrow \frac{1}{4}~\frac{\alpha_0(r_2^2-r_1^2)}{\omega^3(x,y,q,0)}$$ which gives a non-singular, finite mass lens with the deflection angle: $$\bm{\alpha}(x,y,q)=\bm{\alpha}(x,y,q,r_1)-\bm{\alpha}(x,y,q,r_2)
\rightarrow \alpha_0(r_2-r_1)\frac{{\bm r}}{r^2}$$ For large radii $r \gg r_2$ the lens acts as a spherical finite mass contained within $r_2$. We identify the mass and radius of the lens with the virial mass and radius of the halo, so the characteristic deflection angle is given as: $$\alpha_0=
\frac{4GM_\mathrm{vir}}{c^2 (r_2-r_1)} \approx
\frac{4GM_\mathrm{vir}}{c^2 r_\mathrm{vir}}
~~~~~~~~r_2=r_\mathrm{vir}$$
To compensate the lens we use a circular disk of radius $r_\mathrm{lim}\gg r_\mathrm{vir}$ with constant surface density and with the deflection angle: $$\bm{\alpha}_\mathrm{cmp}({\bm r})
=\frac{4GM_\mathrm{vir}}{c^2}\times\left\{
\begin{array}{ll}
{\bm r}/r_\mathrm{lim}^2 & \mbox{if $r\le r_\mathrm{lim}$}\\
{\bm r}/r^2 & \mbox{if $r> r_\mathrm{lim}$}
\end{array}
\right.$$ where $$M_\mathrm{vir} \equiv \frac{4}{3}\pi\rho_0r_\mathrm{lim}^3$$ defines the size of the disk. $\rho_0$ is the average matter density in the Universe at the lens redshift. In our approach we always use $r_1=0.01r_2$, so we avoid singularity at the lens centre, but we do not assign any physical meaning to this parameter.
Finally we check numerically the dependence of the deflection angle on various parameters. We use the formula for a compensated lens: $$\bm{\alpha}({\bm r})=\bm{\alpha}(x,y,q,r_1)
-\bm{\alpha}(x,y,q,r_2)-\bm{\alpha}_\mathrm{cmp}({\bm r})$$
![The deflection angle of the compensated lens. For illustration we use: $r_1=1$, $r_2=10^2r_1$, and $r_{lim}=10^2r_2$. The plots show the radial deflection component $\alpha_r$ as function of radius for fixed polar angle $\phi=90^\circ$, $60^\circ$, $30^\circ$, and $0^\circ$ (top - down). Dashed lines are used for $q=0.3$, and solid lines for $q=0.7$.[]{data-label="alfa-r"}](fig1.eps){width="84mm"}
We illustrate the dependence of deflection angle on position in Fig. \[alfa-r\], using arbitrary units and a case with $r_1 \ll r_2 \ll r_\mathrm{lim}$. The compensation is not exact, but at $r\approx r_\mathrm{lim}$ the deflection angle drops by two orders of magnitude.
A compensated lens has only a finite range. Its radius depends on the properties of each individual halo, but investigating the deflection by haloes at any position in a given layer we take into account only the finite number of lenses: $$\bm{\alpha}^\mathrm{haloes}_i(\bm{\beta})
=\sum_j~\bm{\alpha}(\bm{\beta}-\bm{\beta}_j)
~~\mathrm{where:}~~D_i|\bm{\beta}-\bm{\beta}_j| \le r^\mathrm{lim}_j$$ The expression $\bm{\alpha}(\bm{\beta}-\bm{\beta}_j)$ stands for the deflection at the position $\bm{\beta}$ caused by the lens at position $\bm{\beta}_j$ within its limiting range.
The total deflection in the $i$-th layer $$\bm{\alpha}_i(\bm{\beta}_i) =
\bm{\alpha}^\mathrm{bcg}_i(\bm{\beta}_i)
+\bm{\alpha}^\mathrm{haloes}_i(\bm{\beta}_i)$$ can be calculated for a ray passing at any location $\bm{\beta}_i$.
Maps of the sky
---------------
We calculate deflection angles for all layers of interest storing them on the $2048^2$ grids with $0.1~\mathrm{arcsec}$ resolution. Using a set of stored deflection angles for all layers and interpolating on the grids one can apply Eq.(1), mapping a region of $\approx 3^\prime \times3^\prime$ in the sky. We investigate light propagation from sources at $z_S\approx 2$ taking into account the influence of several thousands haloes in the field of view and some in adjacent regions, if they are within their $r_\mathrm{lim}$ ranges. We construct eight separate maps pointing in random directions.
When investigating multiple image properties we need zoomed maps of smaller parts of the sky. For better resolution we use finer grids with deflections interpolated from previous calculations with the help of bi-cubic spline, so the interpolated deflection derivatives are continuous. The backward ray shooting on a finer grid is repeated.
Treating each of the individual haloes in the field of view as an isolated lens of known velocity dispersion and using SIS model we calculate its Einstein radius $r_E$ for a source at given redshift $z_S$ in a homogeneous Universe model. In each simulated [*map*]{} of the sky we find 10 lenses with the largest Einstein radii. We expect that these [*dominating*]{} lenses have the highest probability of producing multiple images.
In the single SIS lens case the second image is possible only for a source within the Einstein ring. The brighter of the two images lies within $2~r_E$ from the lens centre and the dimmer within $1~r_E$, so we expect that in more complicated problem, including lens ellipticity and the influence of other lenses the multiple images lie within few Einstein radii from each other.
We check for the presence of other haloes inside the circle of the radius $3~r_E$ surrounding a dominating lens. If they are present we enlarge the region of interest including $3~r_E$ zones around all companions. Finally we repeat backward ray shooting inside a square on the sky including the region of interest. The fine grids giving the deflection angles in consecutive layers encompass still larger areas, so one can follow rays deflected off the main region. The result of the ray shooting is a vector array: $$\bm{\beta}_N^{kl}=\bm{\beta}_N(\bm{\beta}_1^{kl})$$ where $\bm{\beta}_N^{kl}$ gives the positions in the source plane of rays apparently coming from the directions $\bm{\beta}_1^{kl}$ on the observer’s sky. Superscripts $k$, $l$ enumerate the rays. The $\bm{\beta}_N(\bm{\beta}_1)$ relation is given in Eq.(1).
Three propagation models
------------------------
{width="176mm"}
To estimate the influence of the foreground and background matter in the beam and also of the lens neighbours belonging to the same redshift layer we consider three scenarios for the backward ray shooting. Very similar method is employed by @b26. In the [*full*]{} approach ([*Case 1*]{}) we include deflection of rays by matter in all layers between the source and observer, as described above. In the [*one layer*]{} approach ([*Case 2*]{}) we include deflection by all matter in the layer of dominating lens only, neglecting the influence of other layers. In the [*single lens*]{} approach ([*Case 3*]{}) only the deflection by the dominant lens is taken into account.
Image finding
-------------
To find critical lines in the sky we solve the standard equation [@SEF92]: $$\det~\mathsf{A}=0~~~~~~~~~~~~
\mathsf{A}\equiv\left|\left|\frac{\partial\bm{\beta}_N}
{\partial\bm{\beta}_1}\right|\right|$$ where $\mathsf{A}$ is the deformation matrix. The equation is solved approximately on the grid and then refined by iterations. Finding of caustic lines in the source plane is then straightforward.
Usually the critical lines and caustic structures obtained in three [*Cases*]{} of backward ray shooting are similar and the influence of matter outside the dominating lens is not apparent. There are however several cases showing the influence of smaller mass haloes on the critical lines and (to a lesser extent) caustics as shown in the upper row of Fig. \[panel\]. An example of a strong caustic perturbation is shown in the lower row of Fig. \[panel\].
We start image finding using approximate methods on the grid. We replace the point source by an extended surface brightness profile with the Gaussian shape and the characteristic radius of few pixels. The related surface luminosity in the sky is given by: $$I^\mathrm{obs}(\bm{\beta}_1)=I^\mathrm{src}(\bm{\beta}_N(\bm{\beta}_1))$$ The local maxima of the observed surface luminosity are the positions of the images of the source centre. We find brightness maxima on the grid and use them as approximate solutions to the lens equation. The improved positions are obtained by iterations.
Using the above method we find multiple images of point sources placed behind lenses. For each lens we try few hundreds to several thousands source positions (depending on the size of the caustics region) distributing them evenly. Each position considered represents the same surface area in the source plane. Finding a source at any location has the same probability if they are distributed randomly. (We neglect here the amplification bias caused by the strong lensing.) The procedure of image finding for many source positions is repeated for each of the ten strongest lenses belonging to each of the eight simulated maps of the small fragments of the sky obtained with the three different models of light propagation.
Fitting the simulated strong lenses with simplified models
==========================================================
In the case of pure elliptical, non-singular lens one would expect (in general) 1, 3, or 5 images. Our numerical method using at the beginning a small extended source and having finite resolution may miss some of the images. (See the Discussion). In [*Cases 1,2*]{} the external influence may also change the number of images. Since configurations with more images probably better constrain the lens, we concentrate on cases with five or four images.
To estimate the influence of the galaxies in the lens vicinity and the matter along the line of sight on the properties of the strong lensing, we attempt to model all our simulated cases of multiple imaging using a single lens model in a uniform Universe. The single lens we are using in modeling is a non-singular finite isothermal ellipsoid used also in the simulations for each of the haloes. When tracing the rays in simulations we interpolate all deflection angles from earlier computed arrays. Single lens modeling uses analytically calculated deflection angles and their derivatives, so it serves also as a test of ray tracing simulations in [*Case 3*]{}.
We also try a more sophisticated lens model using the same non-singular finite isothermal ellipsoids with external shear. In this way the tidal influence of masses close to the rays is at least partially represented.
Simulations give the lens position, image positions and image amplifications corresponding to their energy fluxes. These parameters would also be observed in reality. The source position $\bm{\beta}_\mathrm{S}$ is unknown but it must be the same for all the images, and we use it as a model parameter. The source luminosity is also unknown, but the observed fluxes must be proportional to the lens amplifications in corresponding positions. Thus the model has to reproduce amplification ratios and not their absolute values. The intrinsic lens parameters (axis ratio $q$, characteristic deflection angle $\alpha_0$, the virial radius $r_\mathrm{vir}$, and the lens position angle in the sky $\phi$) are not so easy to measure and we treat them as unknown model parameters. Also the shear components $\gamma_1$, $\gamma_2$ belong to the class of intrinsic parameters. Below we use the value of shear defined as $$\gamma \equiv \sqrt{\gamma_1^2+\gamma_2^2}$$
We treat the lens position, image positions and image fluxes ratios given by the simulations as observed quantities, each with its own uncertainty. We attempt to fit every simulated strong lensing case with our simplified model. We are going to reproduce the observed lens position, image positions and image flux ratios (or differences in image stellar magnitudes) looking for a minimum of $\chi^2$ taking all three constraints into account: $$\chi^2=\chi^2_\mathrm{L}+\chi^2_\mathrm{I}+\chi^2_\mathrm{m}$$ The first term, controlling the lens position, used in the model has the obvious form: $$\chi^2_\mathrm{L}=
\frac{(\bm{\beta}_\mathrm{L}-\bm{\beta}_\mathrm{L}^{~0})^2}
{\sigma_\mathrm{L}^2}$$ where the subscript $\mathrm{L}$ stands for “lens” and $\sigma_\mathrm{L}$ is the assumed accuracy of the lens position.
Using the simplified model we can calculate the source positions $\bm{\beta}_\mathrm{S}^{~i}$ related to observed image positions $\bm{\beta}_\mathrm{I}^{~i0}$. The deformation matrix $\mathsf{A}_{(i)}$ and magnification matrix $\mathsf{M}_{(i)}$ can be calculated at each image position: $$\bm{\beta}_\mathrm{S}^i
\equiv\bm{\beta}_\mathrm{S}(\bm{\beta}_\mathrm{I}^{~i0})
~~~~~~~~
\mathsf{A}_{(i)}\equiv\left|\left|\frac{\partial\bm{\beta}_\mathrm{S}^{~i}}
{\partial\bm{\beta}_\mathrm{I}^{~i0}}\right|\right|
~~~~~~\mathsf{M}_{(i)}\equiv\mathsf{A}_{(i)}^{-1}$$ The mismatch between $\bm{\beta}_\mathrm{S}^{i}$ and $\bm{\beta}_\mathrm{S}$ implies magnified mismatch between modeled and observed image positions, which gives for $\chi^2_\mathrm{I}$ ([@K06]): $$\chi^2_\mathrm{I}=\sum_i~
\frac{\left|\mathsf{M}_{(i)}\mathbf{\cdot}
~\left(\bm{\beta}_\mathrm{S}^i-\bm{\beta}_\mathrm{S}\right)\right|^2}
{\sigma_\mathrm{I}^2}$$ The fitting statistic for flux ratios is given as: $$\chi^2_\mathrm{m}=\sum_i
\frac{\left(m^i-m^{0i}-\left<m^i-m^{0i}\right>\right)^2}
{\sigma_\mathrm{m}^2}$$ In all cases the quantities with extra superscript “$0$” are taken from simulations and mimic the observed values while the quantities without this superscript are given by the model.
We use the following characteristic values for the accuracy parameters: $\sigma_\mathrm{I}=0.01~\mathrm{arcsec}$, $\sigma_\mathrm{L}=0.1~\mathrm{arcsec}$, and $\sigma_\mathrm{m}=0.1$ or $10$ (in stellar magnitudes) This choice takes into account the fact that image positions are easier to measure than the lens position. With the small value of the flux measurement error ($\sigma_\mathrm{m}=0.1$) we use the fluxes as important factor in modeling. The other value makes the fluxes unimportant in the model fitting. Using it one can check to which extent models well reproducing image positions are also good in reproducing ratios of their fluxes.
We start fitting process using a model with all parameter values, also the intrinsic lens parameters, taken from the simulation. This is not a general approach, but it is fast in finding solutions close to the starting point in parameter space (if they are present). It is also our goal to obtain the “observable” parameters as close to simulated values as possible, but the intrinsic lens parameters are not constrained a’priori and have no preferred values.
Our approach to model fitting is not general, since we are probing only the small region of parameter space and there is a chance that we miss some acceptable models of the lens. On the other hand, if the influence of the environment were weak, the resulting lens model should be close to the original lens used in the simulations. Failure to find a model which is similar to the original can be interpreted as the result of the strong influence of the matter in the rays vicinity.
The number of the degrees of freedom ($DOF$) depends on the number of images ($N_\mathrm{im}$) and on the number of intrinsic model parameters, which is $N_\mathrm{par}=4$ for an elliptic lens ($q$, $\alpha_0$, $r_\mathrm{vir}$, $\phi$) and $N_\mathrm{par}=6$ if the external shear ($\gamma_1$, $\gamma_2$) is taken into account. The lens position is both observed and modeled, so it has no impact. The source position requires 2 parameters, and the source internal luminosity (which we do not use explicitly in modeling) - another one. When modeling the flux ratios with $\sigma_\mathrm{m}=0.1$ we effectively use the magnification information from the simulations, but for $\sigma_\mathrm{m}=10$ it is practically lost. In calculations we use: $DOF=3(N_\mathrm{im}-1)-N_\mathrm{par}$ (if $\sigma_\mathrm{m}=0.1$) or $DOF=2(N_\mathrm{im}-1)-N_\mathrm{par}$ (if $\sigma_\mathrm{m}=10$). (That implies that modeling of four-image configurations including shear and neglecting flux ratios is impossible, since $DOF=0$ in this case). Using tables of $\chi^2$ distribution we reject fits if they give $$\chi^2 > \chi_{0.95}^2(DOF)$$ where $\chi_{0.95}^2$ corresponds to 95% significance for a given $DOF$.
The influence of the matter distribution outside the lens on the lensing process can be characterized by the difficulty in obtaining an acceptable single lens model in such more complicated situations. To measure the rate of success in fitting procedure we assign the same weight to each investigated source position since they are distributed evenly in the source plane. Thus the more massive lenses have greater influence on the results as compared with less massive. This is the simplest consequence of the observational selection. More subtle effects, like magnification bias, influence of image configuration etc are too difficult to model within our approach.
Since we perform our simulations in eight different, randomly chosen regions of the synthetic Universe, we can estimate the role of the cosmic variance in our simulations. Calculating the rate of success for each of the regions separately and finding the dispersion of the results, we get an error estimate.
[@ccccc@]{} Shear & no & no & yes & yes\
$\sigma_\mathrm{m}$ & 0.1 & 10. & 0.1 & 10.\
& $0.08\pm0.13$ & $0.11\pm0.13$ & $0.43\pm0.17$ & $0.68\pm0.22$\
[*Case 2*]{} & $0.38\pm0.21$ & $0.41\pm0.18$ & $0.59\pm0.22$ & $0.78\pm0.19$\
[*Case 3*]{} & $.9998\pm .0002$ &$.9998\pm.0001$& $.9997\pm.0002$ & $.9998\pm.0001$\
\[accept1\]
In Table. 1 we show the chances that a simulated strong lensing case has an acceptable simplified model. In [*Case 3*]{} acceptable models are possible to obtain in almost 100% of simulated lens-image configurations, which shows that our fully numerical (used in obtaining lens - image configurations) and semi-analytical (used in fitting models) descriptions of the same physical approach are in complete agreement.
The results depend on the adopted values of position accuracy, (see below) but the difficulty in modeling realistic configurations using a single elliptical lens without external shear is apparent, even if the reproduction of correct flux ratios is not required. The inclusion of external shear substantially improves the chances of obtaining acceptable solutions with simplified lens models, but still there is no guarantee of success, especially if one requires the reproduction of the observed flux ratios.
The present day astrometry gives better accuracy of image positions than assumed in majority of our simulations. On the other hand the micro-lensing may change the measured fluxes of individual images in a random way, which diminishes their role as model constraints. In Table. 2 we show the results of our approach with increased required accuracy of positions of the lens, the images, or both, for models with shear, reproducing flux ratios.
[@ccccc@]{} $\sigma_\mathrm{I}$ & 0.01 & 0.01 & 0.0025 & 0.0025\
$\sigma_\mathrm{L}$ & 0.1 & 0.025 & 0.1 & 0.025\
& $0.43\pm0.17$ & $0.39\pm0.18$ & $0.18\pm0.10$ & $0.16\pm0.10$\
[*Case 2*]{} & $0.59\pm0.22$ & $0.47\pm0.18$ & $0.34\pm0.20$ & $0.28\pm0.17$\
[*Case 3*]{} & $.9997\pm .0002$ & $.9997\pm .0002$ & $.9997\pm .0001$ & $.998\pm .001$\
\[accept3\]
In the following Fig. \[gamma\] we show the probability $P(>\gamma)$ that an acceptable fit uses external shear value greater than $\gamma$. The plots refer to fits taking into account image positions but not their flux ratios.
![The integral probability distribution of the external shear value $\gamma$ for acceptable models. The results for models neglecting flux ratios ($\sigma_\mathrm{m}=10$) in [*Case 1*]{} are shown as solid lines (blue), and in [*Case 2*]{} - as dashed lines (red). For comparison the results for models taking into account flux ratios ($\sigma_\mathrm{m}=0.1$) in [*Case 1*]{} are shown as dotted lines (black). The thick lines correspond to lower accuracy ($\sigma_\mathrm{I}=0.01$, $\sigma_\mathrm{L}=0.1$) and the thin lines - to models of higher accuracy ($\sigma_\mathrm{I}=0.0025$, $\sigma_\mathrm{L}=0.025$).[]{data-label="gamma"}](fig3.eps){width="84mm"}
The influence of lens neighbours and matter in the lens foreground/background is also illustrated by the changes in the intrinsic lens parameters when comparing the fitted model with the original lens used in the simulation. In reality one has only one set of lens parameters obtained from modeling and nothing to compare with, nevertheless our plots show the likely systematic errors resulting from the model approximations.
![The integral probability distribution showing the errors in the fitted values of characteristics deflection angle $\alpha_0$. Conventions follow Fig. \[gamma\]. $\alpha_0$ is the deflection angle of the lens used in simulations. $\alpha_0^\mathrm{fit}$ is its fitted value.[]{data-label="alpha"}](fig4.eps){width="84mm"}
![The integral probability distribution showing the errors in the fitted values of the lens virial mass. Line/colour conventions follow Fig. \[gamma\]. $M_\mathrm{vir}$ is the virial mass of the lens used in simulations. $M_\mathrm{vir}^\mathrm{fit}$ is its fitted value.[]{data-label="mass"}](fig5.eps){width="84mm"}
In Fig. \[alpha\] we show (on a logarithmic graph) the integral distribution of the ratios of the fitted values of the deflection angle parameter to its original values used for the same lenses in simulations. Similarly Fig. \[mass\] shows the distribution of the ratios between fitted and original virial masses.
Discussion and conclusions
==========================
In simulations we replace each halo taken from the Millennium Simulation by a specific kind of matter distribution with the same mass and size, acting as non-singular isothermal ellipsoid with finite mass. We believe that this particular choice of lens model has little impact on our results, because our aim is to investigate the influence of the environment on the lens action, not the action itself. In particular a different choice of the relative size of the lens core (we always use $r_1/r_2=0.01$) may change the properties of the radial caustic of each lens, but has no influence on the action of matter outside it.
We consider lenses with non-singular matter distribution so the number of images should be odd in all cases. In our calculations (which start from an approximate solution of lens equation) the solutions with four images are rare, 40 times less frequent than the five-image cases. The brightest images in our four-image configurations are on average brighter by $\approx 1.5^\mathrm{mag}$ as compared to the brightest images in five-image cases. Also the flux ratio of the brightest image to the next one is substantially higher (by $\approx 0.6^\mathrm{mag}$) in the four-image configurations. This suggests that our four-image solutions correspond to source positions close to the caustics, which produce close image pairs unresolvable by our approximate method of image finding. Since four-image configurations are rare, they have little impact on the statistics of the acceptability of fits.
The matter belonging to gravitationally bound haloes makes only a part of the whole matter distribution. The distribution of the unbound component we are using is known with limited resolution of $\approx 2~h^{-1}\mathrm{Mpc}$ in 3D. Since the influence of low density regions averages along the lines of sight on distances of several thousands megaparsecs, the averaging inside cells of few megaparsecs across probably has little meaning.
Our simulations cover eight small ($3^\prime\times 3^\prime$), randomly chosen regions in the sky. In each of the regions we choose ten lenses, with largest Einstein ring for a source at $z \approx 2$ according to approximate, [*SIS*]{} lens formula. These are the strongest lenses in the region, giving the largest image separations and having the largest area closed by caustics in the source plane. That does not mean that every lensed source must be related to one of the lenses investigated, but for its random position in the region of interest this would happen in a vast majority of cases.
Our numerical experiment shows that matter close to the line of sight and matter in the close vicinity of the lens do have an important influence on its lensing properties. Our statement is of statistical nature: we have shown than in large part of image configurations obtained as a result of simulations taking into account the presence of matter concentrations near the lens it is impossible to obtain acceptable models neglecting their influence. The problem is more severe in case when simulations take into account not only lens close neighbours but also matter inhomogeneities along the beam. It shows that both close neighbours and matter along the line of sight are important. Our results are in general agreement with @WBO04 [@WBO05; @AN11], but we address different aspect of the problem.
The quantitative characteristics of the effect depends on the required accuracy of the simplified modeling. It certainly depends on the source redshift as well, but we have not pursued this aspect of the problem, limiting ourselves to sources at $z_\mathrm{s}\approx2$.
Some examples of the expected rate of success, when using simplified lens model to fit realistic image configurations, are shown in Tables. 1 and 2. We assume that every position of the source has the same probability. The selection of a sample of multiple image systems may also depend on the lens amplification, image configuration and other secondary issues, which are difficult to simulate and are neglected.
Our results in Tables. 1 and 2 take into account the fifth image, if present. In real cases the fifth images are not observed or very faint and difficult to use in modeling. We have performed some extra calculations neglecting the fifth images altogether. Of course it is always easier to model a less complicated configuration of images, so the rate of success is higher in this case. For instance the first column of Table. 1 would read: 0.16, 0.48, 0.9996 (instead of 0.08, 0.38, 0.9998) and the third column: 0.64, 0.68, 0.9995 (instead of 0.43, 0.59, 0.9997). The changes are substantial, but do not qualitatively affect our conclusion.
Our calculations give typical values of the external shear, which is used by the acceptable simplified models. When modeling configurations resulting from the full approach (taking into account all matter inhomogeneities along the rays) we obtain median shear value $\gamma\approx0.09$ (compare Fig. \[gamma\]). For the single plane approach (which neglects lens foreground and background objects) we get $\gamma\approx0.03$. Both numbers weakly depend on the required accuracy of modeling and do not change when the fifth image is neglected. Comparison of these two numbers and the inspection of Table. 1 suggest that shear produced by the matter along the beam and the matter in the lens vicinity cannot be neglected in majority of cases. It also shows that contribution to the shear by background and foreground objects is significant.
We have also compared the fitted values of lens parameters with the parameters of the lens used in the simulation. If the flux ratios are not modeled, the fitted parameters may substantially differ from their original values. For instance the characteristic bending angle of the lens $\alpha_0$ may differ by a factor greater than $\approx 1.2$ in 50% of cases (compare Fig. \[alpha\]). This would imply similar errors in estimates of mass distribution in the central part of the lens. For the virial mass ($M_\mathrm{vir}\propto\alpha_0 r_\mathrm{vir}$) the discrepancies may exceed factor $\approx1.5$ in 50% of cases (compare Fig. \[mass\]). The models reproducing also the flux ratios give much better accuracy of the intrinsic lens parameters fitting (dotted lines in Fig. \[alpha\], \[mass\]).
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to the Anonymous Referee, whose critical remarks greatly improved the paper. The Millennium Simulation databases used in this paper and the web application providing on-line access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory. We are grateful to Volker Springel for providing us with the smoothed Millennium density distribution in the early stage of this project. This work has been supported in part by the Polish National Science Centre grant N N203 581540.
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\[lastpage\]
[^1]: E-mail: [email protected] (MJ); [email protected] (ZKR)
|
---
abstract: 'We propose a simple phenomenological model predicting, through Lorentz symmetry breaking, a CPT-violating asymmetry between particle and antiparticle states in neutrino oscillations involving sterile neutrinos. Such a model is able to explain the apparently observed anomalous excess of low-energy $\nu_e$-like events, reported by the MiniBooNE collaboration, as well as the non-observation of the corresponding anomalous excess of $\ov{\nu}_e$-like events. The present model leads to very specific physical predictions in the neutrino oscillations scenario, and account for the observed anomalies in terms of only one CPT-Lorentz violation parameter of the order of the Grand-Unification energy scale.'
author:
- 'S. Esposito'
- 'G. Salesi'
title: 'CPT-violating neutrino oscillations'
---
In recent years, compelling evidence for the phenomenon of neutrino oscillations [@NuOsc] has been firmly established by atmospheric [@BE12], solar [@BE345], reactor [@BE678] and long-baseline accelerator [@BE910] neutrino experiments. With a confidence level of more than $7 \sigma$, indeed, we now know that solar $\nu_e$’s convert to $\nu_\mu$ or $\nu_\tau$, while atmospheric $\nu_\mu$’s disappear (most likely converting to $\nu_\tau$) at a level of more than $15 \sigma$. Moreover, the KamLAND experiment has found that reactor $\ov{\nu}_e$’s disappear over a distance of about 180 km, also observing a distortion of their energy spectrum; while accelerator $\nu_\mu$’s disappear over distances of 250 km (K2K experiment) and 735 km (MINOS experiment), again with a distortion in the energy spectrum. All these results clearly imply that different flavored neutrinos are massive and mix according to what originally proposed several decades ago [@PMNS]. Further possible evidence comes from the LSND experiment [@LSND], reporting positive signals for $\ov{\nu}_\mu \rightarrow \ov{\nu}_e$ transitions, but such a result has not been confirmed by any other experiment. Very recently, the MiniBooNE collaboration [@MINIBOONE] has reported the absence of signal in the 475-3000 MeV energy range due to short-baseline ${\nu}_\mu \rightarrow {\nu}_e$ oscillations with a squared mass difference $\Delta m^2$ compatible with the indication of $\ov{\nu}_\mu \rightarrow \ov{\nu}_e$ oscillations found in the LSND experiment. Analogous missing events have been apparently observed in searches for $\ov{\nu}_\mu \rightarrow \ov{\nu}_e$ oscillations, although the MiniBooNE analysis of antineutrino data is not as precise as that of neutrino data because of a much lower statistics. Interestingly enough, the MiniBooNE collaboration has also reported an anomalous excess of low-energy $\nu_e$-like events, while a corresponding anomalous excess of antineutrino events has not been observed.
A possible explanation of the observed low-energy $\nu_e$ anomaly has been proposed [@GL08] in terms of very short-baseline disappearance of $\nu_e$’s due to oscillations into sterile neutrinos generated by a large squared mass difference (20 eV$^2\lesssim\Delta{m}^{2}\lesssim 330$ eV$^{2}$). This hypothesis is motivated by the anomalous ratio between measured and predicted $^{71}$Ge production rates observed in the Gallium radioactive source experiments GALLEX [@gallex] and SAGE [@sage], and has been carefully confronted with the lack of any evidence of very short-baseline $\ov{\nu}_e$ disappearance in existing reactor experiments [@GL09]. The thorough analysis on the experimental data reported in [@GL09] apparently implies a possible CPT violation in the neutrino sector. Indeed, in very simple terms, the apparently observed $\nu_e$/$\ov{\nu}_e$ dissymmetry would point toward an asymmetry between the survival probability $P(\nu_e \rightarrow \nu_e)$ of electron neutrinos and that $P(\ov{\nu}_e \rightarrow \ov{\nu}_e)$ of electron antineutrinos against transitions into a sterile state, which is a clear signal of a CPT violation [@NuOsc]. At 99.71% confidence level, quantity $A_{ee} \equiv P(\nu_e \rightarrow \nu_e) - P(\ov{\nu}_e \rightarrow \ov{\nu}_e)$ is negative, with a best-fit value of [@GL09] \[2\] A\_[ee]{} = - 0.165 . The possibility of a CPT violation in the neutrino sector is not a completely novel topic [@SME] [@KostMewes], and experimental tests of such a violation have also been performed [@MINOSCPT] [@LSNDCPT], with negative evidence. However, the result (\[2\]) is very specific, and it is of some relevance to take it into due consideration because of its physical consequences. As a matter of fact, a CPT violation would signal, through the CPT theorem, a violation of the Lorentz symmetry, an issue that has been considered extensively in recent times [@LV]. Let us also recall that neither Lorentz symmetry violation nor CPT violation automatically implies a non-vanishing asymmetry $A_{ee} \neq 0$, as also remarked in [@KostMewes] (while the contrary is always true, as abovesaid [@NuOsc]). In the following we shall put forward a very simple CPT and Lorentz violating theoretical model for neutrino oscillations keeping our phenomenological discussion as general as possible.
For the sake of simplicity, we consider only the mixing (with an angle $\theta$) between an active electron neutrino state $\nu_e$ and a sterile (under weak interaction) state $\nu_s$, with masses $m_1$ and $m_2$, respectively. Standard simple calculations [@NuOsc] show that the survival probability for a beam of $\nu_s$ with momentum $p$ traveling for a time $t$ from the source can be generally cast in the following form: \[3\] P(\_e \_e) = 1 - \^2 2 \^2 t . In the usually considered ultrarelativistic case ($p \gg m$, distance $x \simeq t$), the energies $E_1$, $E_2$ of the two states approximate to \[4\] E\_[1,2]{} p + , so that we recover the standard formula \[5\] P(\_e \_e) 1 - \^2 2 \^2 x with $\Delta m^2 \equiv m_2^2 - m_1^2$. The same formula (\[5\]) applies also for antineutrinos, $P(\nu_e \rightarrow \nu_e) = P(\ov{\nu}_e \rightarrow \ov{\nu}_e)$, since the change of sign in $E_1$, $E_2$ has no effect on the survival probability in (\[3\]) or (\[5\]).
At variance with the standard approach, let us assume that the dispersion relation between neutrino (and, possibly, also for other particles) energy and momentum is not that leading to Eqs. (\[4\]), $E^2 = p^2 + m^2$, but is modified by effects beyond the Standard Model. Such a possibility, indeed, has been extensively investigated by means of quite different approaches, sometimes extending, sometimes abandoning the formal and conceptual framework of Einstein’s Special Relativity. Moreover, some authors suspect that the Lorentz symmetry breaking may play a role in extreme astrophysical phenomena as, e.g., the observation of ultra-high energy cosmic rays with energies [@UHECR] beyond the Greisen-Zatsepin-Kuzmin [@GZK] cut-off, and of gamma rays bursts with energies beyond 20 TeV originated in distant galactic sources [@Markarian]. Lorentz-breaking observable effects appear [@LVN] in Grand-Unification Theories, in M-Theory and String theories, in (Loop) Quantum Gravity, in foam-like quantum spacetimes; in spacetimes endowed with a nontrivial topology or with a discrete structure at the Planck length, or with a (canonical or noncanonical) noncommutative geometry; in so-called “effective field theories” and “extensions” of the Standard Model including Lorentz violating dimension-5 operators; in theories with a variable speed of light or variable physical constants. An interesting theoretical approach to Lorentz symmetry violation is found in DSR [@GAC; @NCG; @Deformed] working in $k$-deformed Lie-algebra noncommutative ($k$-Minkowski) spacetimes, in which both the Planck scale and the speed of light act as characteristic scales of a 6-parameter group of spacetime 4-rotations with deformed but preserved Lorentz symmetries. In any of these theories, the most important consequence of a Lorentz violation is the modification of the ordinary energy-momentum dispersion relation, at energy scales usually assumed to be of the order of the Planck energy, by means of additional terms which vanish in the low-momentum limit. Such terms may produce nice physical effects, explaining for example baryogenesis [@EPL], and several constraints (in the neutrino sector) on them have been obtained [@MPLA] from existing data coming from laboratory experiments. Consequences of a nonstandard dispersion law on neutrino oscillations has been analyzed in [@ChenYang] and [@Coleman]; observable effects of relativistic non-covariance on the flavor eigenstates have been investigated in [@Various]. In many of the theories above mentioned, the modified dispersion relation takes the general form \[6\] E\^2 = p\^2 + m\^2 - E p\^2 , where $\lambda$ is a Lorentz violation parameter (with the dimensions of an inverse energy) satisfying the relation $\lambda E \ll 1$. In such approximation, deducing $E$ from Eq. (\[6\]) we obtain (neglecting $\lambda^2$-order terms) \[7\] E = - , where, as usual, the upper (lower) sign refers to particle (antiparticle) states. In the ultrarelativistic limit achieved in the neutrino oscillation experiments, we thus have \[8\] E ( p + ) - . As for the mass, the $\lambda$ parameter is, in general, particle-dependent and, as shown in [@EPL] on very general grounds, the requiring of CPT invariance would lead to a change of sign for $\lambda$ when we pass from particle to antiparticle states. For our purpose, instead, we take a [*different*]{} view, and assume that CPT does not hold anymore and $\lambda$ can have the same value for particles and antiparticles. Consequently, the energy difference between $\nu_e$ and $\nu_s$ in Eq. (\[3\]) is given by \[9\] E\_2 - E\_1 - , while, for $\ov{\nu}_e$ and $\ov{\nu}_s$, \[10\] \_2 - \_1 - - - . For simplicity we have introduced the parameters \[11\]
[rcl]{} M\^2(p) & & m\^2 - p\^3\
&=& m\^2 ,\
& &\
\^2(p) & & m\^2 + p\^3\
&=& m\^2 ,
that can be interpreted as “effective”, momentum-dependent, squared mass differences. Note that, since $p \gg m$, the “correction” terms in Eqs. (\[11\]), that is $\displaystyle \left( \frac{p^2}{\Delta m^2} \right)p\,\Delta\lambda$, may [*not*]{} in principle be vanishingly small, depending on how much ultrarelativistic is the regime considered. Since $\ov{\Delta M}^2(p) \neq \Delta {M}^2(p)$, the survival probabilities for neutrino and antineutrino states $$\begin{aligned}
P(\nu_e \rightarrow \nu_e) & \simeq & 1 - \sin^2 2 \theta \, \sin^2 \frac{\Delta M^2(p)}{4 p} \, x \, \label{12} \\
P(\ov{\nu}_e \rightarrow \ov{\nu}_e) & \simeq & 1 - \sin^2 2 \theta \, \sin^2 \frac{\ov{\Delta M}^2(p)}{4 p} \, x \, \label{13}\end{aligned}$$ result to be [*not*]{} identical. Let us remark that the effect introduced here phenomenologically, although driven by a violation of the Lorentz symmetry, is substantially different from analogous effects introduced earlier by other authors [@Coleman], with different phenomenological predictions (and in absence of a CPT violation), as a simple comparison shows.
Very roughly, in order to explain the apparently observed asymmetry (\[2\]), the model proposed may be implemented as follows. For simplicity, let as assume that for ordinary $\nu_e$’s we have $\lambda_e \approx 0$, while for the sterile neutrino the $\lambda$ parameter is significatively different from zero: $\Delta \lambda \simeq - \lambda_s$. If the non-observation of an anomalous excess of low-energy antineutrino events, reported by the MiniBooNE collaboration, will be confirmed, it means that, in the given energy range, $P(\ov{\nu}_e \rightarrow \ov{\nu}_e) \simeq 1$, that is $\ov{\Delta M}^2(p) \simeq 0$ or, from (\[11\]), \[14\] \_s . Because of Eq.\[11\] this relation automatically implies $\Delta {M}^2 \simeq 2 \Delta {m}^2$, so that the survival probability for neutrinos, $P(\nu_e \rightarrow \nu_e)$, is significatively different from 1, thus explaining the observed anomalous excess of low-energy $\nu_e$-like events. Note that such a scenario is favored for “low-energy” events since, otherwise, the formula (\[14\]) would require an exceedingly small $\lambda_s$.
Remarkably, if Eq.(\[14\]) holds true, $P(\nu_e \rightarrow \nu_e)$ has a form and an effective energy dependence similar to the standard one in (\[5\]), but with $\Delta m^2$ replaced by $2 \Delta m^2$. This allows a very rapid estimate of the order of magnitude for the $\lambda_s$ parameter. By using [@GL09] $\Delta m^2 \sim 100$ eV$^2$, $p \sim 1 $ GeV, from Eq. (\[14\]) we obtain \[15\] \_s\^[-1]{} 10\^[16]{} , which is a completely reasonable estimate once assumed that the Lorentz violation is induced by effects taking place, while approaching the Planck energy scale, around the Grand-Unification scale.
Summing up, we have proposed a very simple phenomenological model which predicts a CPT-violating asymmetry between particle and antiparticle states in neutrino oscillations involving a sterile neutrino. If the recent analysis in [@GL09] is further confirmed by novel experimental results, our model would imply that the standard energy-momentum dispersion relation for this sterile neutrino is modified according to Eq.(\[6\]), with a CPT/Lorentz violation parameter $\lambda$ of the order of the Grand-Unification energy scale. This would directly affect the survival probabilities for neutrino and antineutrino oscillations (\[12\]), (\[13\]), with a possible extreme suppression of antineutrino transitions with respect to neutrino oscillations which, instead, would take place at a sizeable level.
Future neutrino oscillation experiments will allow to compare the present model with the standard scenario of flavor (or other) oscillations. Indeed, the dependence in our model of $P(\nu_e \rightarrow \nu_e)$ and $P(\ov{\nu}_e \rightarrow \ov{\nu}_e)$ on the energy of the neutrino beam is [*different*]{} from the standard case \[compare Eqs. (\[12\]), (\[13\]) with Eq. (\[5\])\], and this may be easily tested in experiments with different baselines and energy ranges. We then expect that further interesting results may be achieved in a near future, along the lines depicted in the present paper.
[**Acknowledgements**]{}
The authors are indebted with M. Laveder for his suggestions to work about the topic of the present paper and for very useful discussions.
[99]{}
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|
---
abstract: 'We present an algorithm for computing the Lyndon factorization of a string that is given in grammar compressed form, namely, a Straight Line Program (SLP). The algorithm runs in $O(n^4 + mn^3h)$ time and $O(n^2)$ space, where $m$ is the size of the Lyndon factorization, $n$ is the size of the SLP, and $h$ is the height of the derivation tree of the SLP. Since the length of the decompressed string can be exponentially large w.r.t. $n, m$ and $h$, our result is the first polynomial time solution when the string is given as SLP.'
author:
- Tomohiro I
- Yuto Nakashima
- Shunsuke Inenaga
- Hideo Bannai
- |
\
Masayuki Takeda
bibliography:
- 'ref.bib'
title: Efficient Lyndon factorization of grammar compressed text
---
Conclusions and open problem
============================
Lyndon words and Lyndon factorization are important concepts of combinatorics on words, with various applications. Given a string in terms of an SLP of size $n$, we showed how to compute the Lyndon factorization of the string in $O(n^4 + mn^3h)$ time using $O(n^2)$ space, where $m$ is the size of the Lyndon factorization and $h$ is the height of the SLP. Since the decompressed string length $N$ can be exponential w.r.t. $n, m$ and $h$, our algorithm can be useful for highly compressive strings.
An interesting open problem is to compute the Lyndon factorization from a given LZ78 encoding [@LZ78]. Each LZ78 factor is a concatenation of the longest previous factor and a single character. Hence, it can be seen as a special class of SLPs, and this property would lead us to a much simpler and/or more efficient solution to the problem. Noting the number $s$ of the LZ78 factors is $\Omega(\sqrt{N})$, a question is whether we can solve this problem in $o(s^2) + O(m)$ time.
|
---
abstract: 'Single atom catalysts (SACs) based on 2D materials are identified to be efficient in many catalytic reaction. In this work, the catalytic performance of Pd/Pt embedded planar carbon nitride (CN) for CO oxidation, has been investigated via spin$-$polarized density functional theory calculations. We find that Pd/Pt can be firmly anchored in the porous CN monolayer due to the strong hybridization between Pd/Pt$-$$d$ orbitals and adjacent N$-$2$p$ orbitals. The resulting high adsorption energy and large diffusion barrier of Pd/Pt, ensuring the remarkable stability of the catalyst Pd/Pt@CN during CO oxidation reaction. The three distinct CO reaction mechanisms, namely, Eley$-$Rideal (ER), Langmuir$-$Hinshelwood (LH), and Tri$-$molecular Eley$-$Rideal (TER), are taken into consideration comparatively. Intriguingly, the oxidation reaction on Pd@CN prefers to proceed through the less common TER mechanism, where two CO molecules and one O$_{\rm 2}$ molecule need to cross a small reaction barrier of 0.48 eV, and finally dissociate into two CO$_{\rm 2}$ molecules. However, the LH mechanism is the most relevant one on Pt@CN with a rate$-$limiting reaction barrier of 0.68 eV. Moreover, the origin of SAC’s reactivity enhancement is the electronic “acceptance$-$donation” interaction caused by orbital hybridization between Pd/Pt and preadsorbed O$_{\rm 2}$/CO. Our findings are expected to widen the catalytic application of carbon based 2D materials.'
address: 'Department of Physics, Faculty of Science, Ningbo University, Ningbo-315211, P.R. China'
author:
- 'Yong-Chao Rao'
- 'Xiang-Mei Duan$^{\ast}$'
bibliography:
- 'CN\_SAC\_for\_CO\_oxidation.bib'
title: 'Pd/Pt embedded CN monolayer as efficient catalysts for CO oxidation'
---
Introduction
============
To control the emission of environmental$-$harmful pollutants and remove CO contaminants from H$_{\rm 2}$ gas fuel in fuel cells, the low$-$temperature CO oxidation reaction has received sustained attention.[@liu2012recent] Typical and conventional catalysts for CO oxidation mainly consisting of noble metals, especially for Pd[@nakai2006mechanism; @gong2004systematic], Pt[@eichler2002co; @alavi1998co] and Au,[@liu2002catalytic; @kimble2004reactivity] have been abroadly investigated for decades. However, the large$-$scale applications are restrained by the high cost and limited storage of these noble metals in earth.[@judai2004low; @zhang2001density] Gradually accepted methods, downsizing the catalysts to atomic scale, for extreme case, SACs, can not only enhance the reactivity of catalysts, but also reduce the cost of fundamental scientific research and industrial applications.[@qiao2011single; @moses2013co; @thomas2005single] Meanwhile, it is an indisputable fact that high stability and no cluster formation of individual metal atom on the supporting material are the key prerequisites for catalyst to maintain its chemical activity.[@uzun2010site] Therefore research on finding an appropriate single-atom anchoring platform remains a major challenge.
Benefiting from the large surface area, high thermal stability, easily fabrication, two$-$dimensional (2D) materials, such as graphene[@krishnan2017new; @li2010co] or defective graphene,[@esrafili2019exploring; @zhang2015single],even Graphene assembled nanotube,[@Lu2017CO] Graphdiyne,[@Xu2018First] Boron sheet, [@zhu2019single] MoS$_{\rm 2}$,[@du2015mos; @Ma2015CO] h$-$BN[@Mao2014A; @gao2013co] and graphic carbon nitride (CN,[@Li2016Graphitic] C$_{\rm 3}$N$_{\rm 4}$[@gao2016single; @Zheng2017Molecule] and C$_{\rm 2}$N[@yu2018c; @li2018cu; @Ma20163]), have been considered as prominent supports to host single atoms in various catalytic reactions (CO oxidation, O$_{\rm 2}$ reduction/evolution, H$_{\rm 2}$ evolution, N$_{\rm 2}$ reduction, $etc$). In particular, a porous 2D framework containing nitrogen can provide rich electron pairs to capture metal ions in the ligand. More importantly, the uniform nitrogen coordinators provides unique identification of active sites. Among these nitrogen$-$doped 2D carbon networks, CN has triggered intensive research in He separation,[@rao2019theoretical] H$_{\rm 2}$ storage[@Chen2018A] and O$_{\rm 2}$ evolution reaction.[@li2018cu] Although intrinsic CN is inert in electrocatalytic processes, it is still attractive to study whether the transition metals$-$embedded CN system can exhibit promising catalytic performance with high stability and superior reactivity in the CO oxidation reaction.
In this work, we have systematically investigated the catalytic performance of two representative species, Pd and Pt, anchored CN systems in CO oxidation process by employing first$-$principles calculations. The results show that TER mechanism is the most preferable one for CO oxidation on Pd@CN with a rate$-$determining barrier of 0.46 eV, while Pt@CN prefers the LH mechanism with a barrier of 0.68 eV. Moreover, the activation nature of O$_{\rm 2}$ and CO are interpreted through electron “acceptance$-$donation” interaction between gas molecules and transition metals. We predict that the Pd/Pt embedded CN system is an efficient, low cost and stable SACs with potential application in CO oxidation.
Computational details
=====================
The spin$-$polarized first$-$principles calculations were conducted in Vienna [*ab initio*]{} simulation package (VASP).[@Kresse1996Efficiency] The Projected Augmented Wave (PAW) in conjunction with Generalized Gradient Approximation (GGA) in the framework of Perdew$-$Burke$-$Ernzerhof (PBE) form with van der Waals (vdW) of DFT$-$D2 scheme was used to describe the electron$-$exchange correction effect,[@Perdew1998Generalized; @Grimme2010Semiempirical] and a cutoff energy of 500 eV was adopted. The geometries were optimized until the energy and force were converged to 10$^{\rm -4}$ eV and 10$^{\rm -2}$ eV/[Å]{}. To avoid interaction between two adjacent images, the vacuum space was set to be 20 Å. Also, The Monkhorst$-$Pack meshes of $3\times3\times1$ were used in sampling the Brillouin zone. The on$-$site coulomb interaction (U) of 4 eV and exchange interaction (J) of 1 eV were applied to describe partially filled $d$$-$orbitals by considering coulomb and exchange corrections.[@Zhou2011Magnetism] The reaction paths were investigated by the climbing nudged elastic band (CI$-$NEB) method.[@Henkelman2000A] To evaluate the thermodynamic stability of Pd and Pt anchored CN systems, we carried out the first$-$principles molecular dynamics (MD) simulations at 300 K with a time step of 1 fs. Bader charge analysis[@Henkelman2006A] was used to obtain the amount of transferred electrons.
Results and discussion
======================
Structure and Stability of Pd/Pt embedded CN
--------------------------------------------
The optimized lattice parameter, corresponding C$-$N and C$-$C bond length of CN monolayer are 7.12, 1.34, and 1.51 Å, respectively, coinciding with reported results.[@Li2016Graphitic; @rao2019theoretical; @Chen2018A] We used $2\times2$ CN supercell containing 24 carbon and 24 nitrogen atoms, creating a hole surrounding by six $sp{^{\rm 2}}$ bonded nitrogen atoms, which are beneficial for Pd/Pt anchoring. Taking Pd@CN as an example, it is indeed as expected that the Pd bonds to two N edge atoms with bond length of 2.20 [Å]{} and keeps its planar geometric structure undistorted after structural optimization \[shown in Fig. 1(a) and (b)\]. Moreover, the energy band structures of for pure CN, Pd@CN and Pt@CN are presented in Fig. S1 (a) (b) (c), respectively. For Pd and Pt decorated CN, there are two bands dominated by N 2$p$ orbitals \[verified by Fig. 1(e) and (f)\] crossing the Fermi level, which results in a semiconductor to metal transition. The enhancement in conductivity makes Pd/Pt@CN promising for the electrocatalystic reaction of CO oxidation. It is well known that transition metal (TM) atoms tend to form cluster on most of the substrate. The preference of clustering can be avoid when the electronic interaction between TM atoms and the substrate is sufficiently strong, and the mobility of TM atoms is low. We first check the possibility of Pd and Pt aggregation on CN sheet. The binding energy ($E_{\rm b}$) of Pd/Pt on the CN monolayer is defined by $E_{\rm b}$ = $E_{\rm {Pd/Pt@CN}}$ $-$ $E_{\rm {Pd/Pt}}$ $-$ $E_{\rm CN}$, where $E_{\rm {Pd/Pt@CN}}$ is the total energy of Pd/Pt@CN, $E_{\rm {Pd/Pt}}$ and $E_{\rm CN}$ are the energies of spin$-$polarized Pd/Pt atoms and pure CN sheet, respectively. The considerably large $E_{\rm b}$ of $-$3.35 eV and $-$3.66 eV for Pd and Pt confirm that the metal atoms robustly bind with the adjacent N atoms. Besides, the relatively larger ratios, $E_{\rm b}$/$E_{\rm c}$ (cohesive energy, 0.85 for Pd, and 0.62 for Pt, respectively), further confirm that the two single TM atoms are not inclined to form cluster on CN.[@kittel2005introduction] The CI$-$NEB and MD methods were used to examine the dynamic and thermal stabilities of Pd/Pt@CN. As depicted in Fig. S2 (c), the large diffusion barriers, 2.88 eV and 2.45 eV for Pd and Pt, furthermore demonstrate that it is hard for single Pd/Pt to move on CN. MD simulations at room temperature \[showed in Fig. S2 (d) and (e)\] confirm that Pd/Pt@CN can keep its structure intact during CO oxidation reaction. The electronic structure analyses show that Pd and Pt donate charge mainly to the edged $sp{^{\rm 2}}$$-$bound N atoms, as shown in Fig. 1(c) and (d). There are strongly overlapped peaks between Pd$-$4$d$/Pt$-$5$d$ and N$-$2$p$ orbitals in the projected DOS profiles \[see Fig. 1(e) and (f)\]. Especially, a peak of localized Pt$-$5$d$ orbital is situated near the Fermi level, resulting in an increase in the chemical activity of Pt@CN toward O$_{\rm 2}$ and CO.
{width="9cm"}
O$_{\rm 2}$ and CO adsorption on Pd/Pt@CN
-----------------------------------------
For SACs, the efficiency strongly depends on its ability to bind the gas molecules over catalytic active centers. So the preliminary investigation of O${\rm _2}$ and CO adsorption over active sites, here Pd and Pt, is meaningful and essential. The adsorption energy ($E_{\rm a}$) of a single molecule (O$_{\rm 2}$, CO and CO$_{\rm 2}$) and O$_{\rm 2}$/CO+CO co$-$adsorption on the substrate were obtained by $E_{\rm a}$ = $E_{\rm {molecule+Pd/Pt@CN}}$ $-$ $E_{\rm molecule}$ $-$ $E_{\rm {Pd/Pt@CN}}$, where $E_{\rm {molecule+Pd/Pt@CN}}$, $E_{\rm molecule}$, $E_{\rm {Pd/Pt@CN}}$ presents the total energies of hybrid structures and isolated parts, respectively. Since the similarity of molecules adsorption on both Pd/CN and Pt/CN, the activation mechanisms of O$_{\rm 2}$ and CO on Pd@CN are taken as examples to discuss and we do not illustrate more details about gas molecules adsorption on Pt@CN (See Fig. S3) in the following part.
![The side views of optimized structures and corresponding charge density difference plots with an iso$-$surface value of 0.002 [*e*]{}/bohr${^{3}}$, as well as spin$-$polarized PDOS for (a) (b) (c) O$_{\rm 2}$ and (d) (e) (f) CO adsorption on Pd@CN, respectively. The blue and yellow iso-surfaces represent the electron depletion and accumulation, and the Fermi level is set zero. ](figure2.pdf){width="9cm"}
As shown in Fig. 2 (a), the O${\rm _2}$ molecule prefers to lie and parallel to the plane of CN sheet by side$-$on configuration, with the O$-$O bond length of 1.36 Å, yielding a moderate adsorption energy of $-$0.69 eV. For comparison, the end$-$on configuration was taken into account, while the O${\rm _2}$ molecule will recline into the side$-$on one during the structural optimization. An general consequence of O${\rm _2}$ adsorption on the Pd@CN is that the distance of O$-$O bond is stretched from 1.23 [Å]{} (in gas phase) to 1.36 Å, indicating that the adsorbed O${\rm _2}$ is activated by the Pd@CN, which is of much significance for CO oxidation. Meanwhile, the Pd atom, originally locating in the undistorted plane of CN \[shown in Fig. 1(b)\], is also somehow pulled out from the sheet with two Pd$-$O distance of 2.03 [Å]{} \[see Fig. 2(a)\]. Moreover, As depicted in Fig. 2(b), the charge accumulation region exhibits a feature of donuts surrounding two O atoms, and charge depletion is mainly around Pd, confirming that the charge transfer from Pd to O${\rm _2}$, which leads to the elongation of O$-$O bond. Bader analysis indicates that the O${\rm _2}$ molecule gains 0.48 $e$ from the substrate. For the two systems Pd@CN and Pt@CN (Fig. S3), the more electron transfer from the substrate to O${\rm _2}$, the larger elongation of O$-$O bond.
Since the reactivity of Pd@CN system originates from the activated O${\rm _2}$ adsorption state at the interface, so it is of significance to explore bonding characteristic between O${\rm _2}$ and Pd. As shown in Fig. 2(c), the O${\rm _2}$ molecule initially donates 1$\pi$ orbital electrons to the Pd$-$4$d$ orbital, and then the occupied Pd$-$4$d$ orbital interacts with the 2$\pi$$^{\rm \ast}$ orbital of O${\rm _2}$ molecule, causing a reverse charge transfer from Pd to O${\rm _2}$ molecule, and partially occupying the 2$\pi$$^{\rm \ast}$ orbital of O${\rm _2}$ (a spin$-$down peak occurs near the Fermi level). The accessibility of moderate electron donation and back$-$donation between adsorbed molecule and active site is highly desirable for the reactivity in most SACs, especially for the diatomic catalytic reactions, such as CO oxidation[@du2015mos] and N${\rm _2}$ reduction.[@zhu2019single; @ling2018metal]
The most stable CO adsorption configuration as well as corresponding charge density difference plot, PDOS are presented in Fig. 2(d) (e) (f), respectively. The CO prefers an end$-$on configuration with the formation of a Pd$-$C bond (1.89 Å). The adsorption energy ($E{\rm _a}$ = $-$0.97 eV) is moderate, which may avoid CO poisoning and surely benefits for the reaction. Similar to the adsorption of O${\rm _2}$, the Pd atom is also pulled out of the CN plane upon CO adsorption. The bond distance of adsorbed CO is slightly elongated by 0.02 [Å]{} as compared with that of free CO (1.14 [Å]{}), owing to the less electron transfer (CO gains 0.06 $e$). Combining the Fig. 2(e) and (f), it is found that the slight charge transfer, which is the net result of the donation of CO lone$-$pair electrons to Pd$-$4$d$ orbital and back$-$donation of Pd$-$4$d$ electrons to the CO$-$2$\pi$$^{\rm \ast}$ states.[@abild2007co]
Interestingly, the adsorption strength of O${\rm _2}$/CO well correlates with the electron transfer amount from the catalyst to the adsorbate when comparing the two systems \[see Fig. 2 and Fig. S3, corresponding to the Pd@CN and Pt@CN, respectively\]. The more electron transfer from Pt@CN to O${\rm _2}$/CO \[0.6/0.08 $e$\], the stronger adsorption strength of O${\rm _2}$/CO with Pt/CN substrate \[$E{\rm _a}$(O${\rm _2}$/CO) = $-$1.45/$-$2.27 eV \]. The concern turns to that the high adsorption strength of CO molecule on Pt/CN inevitably leads to CO poison. Yet, the O${\rm _2}$$-$riched working environment is able to induce the Pt atom to bind with O${\rm _2}$ preferentially.[@liu2014co] In addition, the previous works concluded that the SACs, in which the adsorption strength of CO over active sites were in the same level with that of Pt/CN \[$E{\rm_a}$(CO) = $-$2.39/$-$2.14 eV for Pt/MoS${\rm _2}$ [@du2015mos]and Cu${\rm _2}$/C${\rm _2}$N,[@li2018cu] respectively\], possessed ideal efficiency for CO oxidation. Herein, Pt/CN is still a promising candidate for SACs in CO oxidation reaction.
We now consider the adsorption of atomic O and single CO${\rm _2}$ molecule over both Pd@CN and Pt@CN systems. As presented in Fig. S4, the distances of metal$-$O bond are all less than 2.0 Å, indicate the stronger chemical interaction between Pd/Pt and O atom comparing with that of O${\rm _2}$, in line with the larger $E{\rm_a}$. The adsorption strength of CO${\rm _2}$ over Pd/Pt@CN substrate is still a key factor to determine whether these two systems could avoid CO${\rm _2}$ poison. Conventionally, the CO${\rm _2}$ adsorption energy should be less than 0.5 eV.[@Deng2013Single] It can be seen in the Fig. S4, the CO${\rm _2}$ adsorption energies on Pd/Pt@CN are $-$0.11 and $-$0.48 eV, respectively. Besides, the CO${\rm _2}$ molecule stays above the Pd/Pt@CN monolayer without any distortion, and the lengths of two C$-$O bonds are all equal to that of free CO${\rm _2}$ molecule. Therefore, we conclude that the molecule CO${\rm _2}$ can be released readily once formed.
Prior to study the reaction mechanism, it is necessary to check the capture performance (co$-$adsorption of the reactants) over the substrate. According to the Fig. S4, two reactants (O${\rm _2}$/CO and CO) are injected simultaneously into the Pd/Pt@CN, and form the co$-$adsorption configurations, either O${\rm _2}$+CO or CO+CO. The co$-$adsorption energies for these two configurations are larger than that of single O${\rm _2}$/CO molecule over both Pd@CN and Pt@CN. This indicates that the Pd/Pt@CN can host two reactants favorably. Herein, all above discussions demonstrate that Pd/Pt@CN can act as potent SACs for CO oxidation.
CO Oxidation on Pd/Pt@CN
------------------------
Two well$-$established reaction mechanisms for CO oxidation over SACs, namely, ER (Eley$-$Rideal) and LH (Langmuir$-$Hinshelwood), as well as new TER (termolecular Eley$-$Rideal) are taken into consideration.[@Bleakley1999A; @Deng2013Single; @Mao2014A]
{width="12cm"}
According to the adsorption energies discussed above, although the Pd/Pt@CN sheets show stronger interaction with CO compared to O${\rm _2}$, the catalytic center can be pre$-$occupied by O${\rm _2}$ under O${\rm _2}$ rich working environment. So we investigate the oxidation of the first CO by pre$-$adsorbed O${\rm _2}$ along the ER path. Taking Pd@CN as an example, the configuration of physisorbed CO above the pre$-$adsorbed O${\rm _2}$ over Pd@CN was selected as initial state (IS)\[see Fig. 3(a)\], and a chemisorbed atomic O and a physisorbed CO${\rm _2}$ molecule as final state (FS). After the optimization, the O$-$O bond length in the IS is elongated from 1.23 [Å]{} [free molecule]{} to 1.37 [Å]{}, and hence, the adsorbed O${\rm _2}$ is sufficiently activated, while the CO is located above substrate with a long distance of 2.92 [Å]{} between C and O from O${\rm _2}$ and the adsorption energy of CO is 0.09 eV. In the first half reaction, CO binds with pre$-$adsorbed O${\rm _2}$ to form a carbonate$-$like intermediate state (MS, CO${\rm _3}$) with a high reaction barrier of 0.98 eV by passing over the transition state 1 (TS1). At TS1, the new C$-$O (here, O atom comes from O${\rm _2}$) starts to build and the O$-$O bond length is further stretched to 1.60 Å. In the second half reaction, due to the breaking of the C$-$O and Pd$-$O bonds, passing over the transition state 2 (TS2) would experience a lager energy barrier of 2.92 eV, finally a CO${\rm _2}$ molecule forms, leaving an atomic O absorbed over Pd atom. Even though the small adsorption energy of CO${\rm _2}$ on atomic O adsorbed Pd/Pt@CN (0.10/0.42 eV) indicates that the formed CO${\rm _2}$ can be easily released at the room temperature, and the entire reaction process is exothermic. However, the ER mechanism for CO oxidation is prohibited because of the considerably large reaction barriers ($\sim$ 3 eV) for both Pd and Pt embedded CN monolayer.
We then examine the CO oxidation over Pd/Pt@CN through the LH mechanism by simultaneously adsorbing O${\rm _2}$ and CO. As mentioned before, the stronger ability for the co$-$adsorption of O${\rm _2}$ and CO demonstrates the feasibility of LH mechanism. Fig. 3(b) depicts the reaction path and energy profile for CO oxidation on Pd/Pt@CN. At the IS, both O${\rm _2}$ and CO molecules bind to the Pd/Pt atom with end$-$on configurations, where O${\rm _2}$ and CO are parallel and tilted to the CN sheet, respectively. As evidenced by the elongation of bond length in Fig. S4, reactants are slightly activated. And then one O atom (coming from O${\rm _2}$) without binding to Pd/Pt atom will approach to C atom of CO, forming a peroxo$-$type OOCO (MS) with the further enlargement of O$-$O bond (reaching up to 1.59 [Å]{} for Pd and Pt@CN). In this first half process, passing over the TS1 needs to climb the energy barrier of 0.34 eV and 0.15 eV for Pd and Pt@CN, respectively. And in the second half process, the reaction continuously proceeds from MS to TS through TS2 with an energy barrier of 0.87 (Pd@CN) and 0.68 eV (Pt@CN). To break the O$-$O and Pd/Pt$-$C bonds, the value of energy barrier is much higher than that of the previous half reaction. Finally, a CO${\rm _2}$ molecule is formed, leaving an atomic O at the Pd/Pt. It was reported that the reaction could process readily with the barrier of 0.8 eV or less.[@Deng2013Single] Therefore, CO can be oxidized through the LH mechanism on Pt@CN with a relatively small energy barrier of 0.68 eV and a fully exothermic reaction energy of 1.52 eV. The left O in O${\rm _2}$ will continue to react with the second incoming CO to form another CO${\rm _2}$. As Fig. 3(d) indicates, the configurations of CO staying above the atomic O on Pd/Pt@CN and CO${\rm _2}$ absorbed Pd/Pt@CN are set as IS and FS, respectively. For Pt@CN, the reaction barrier for the formation of second CO${\rm _2}$ is only 0.5 eV and the reaction releases a large heat of 3.32 eV. Therefore, the CO oxidation reaction over Pt@CN through LH mechanism is completely feasible.
A previous study has shown that two CO molecules can assist O${\rm _2}$ scission and promote CO oxidation.[@liu2013co] Moreover, the co$-$adsorption of two CO molecules over Pd/Pt@CN is stronger than that of single one. So the new TER route is worthy of consideration. The reaction route and relative energy are given in Fig. 3(c). Here, the CO oxidation reaction starts from the co$-$adsorption of two CO molecules over Pd/Pt (shown in Fig. S4). At the IS, the O${\rm _2}$ is physisorbed over the pre$-$adsorbed CO molecules with elongation of O$-$O bond to 1.26 A for Pd/Pt@CN. To process, O${\rm _2}$ is inserted into two CO molecules, leading to the formation of OCOOCO (MS) over Pd/Pt. The first half reaction would easily occur with a low energy barrier of 0.45 (Pd/CN) and 0.54 eV (Pt@CN). To pass over the TS2, the O$-$O bond length would be further extended until it breakes. After that, OCOOCO is decomposed into two free CO${\rm _2}$ molecules synchronously with a reaction barrier of 0.48 (Pd@CN) and 1.18 eV (Pt@CN). Overall, the rate$-$limiting reaction barrier for Pd@CN is only 0.48 eV, lower than those on noble metal catalysts (0.53$-$1.01 eV).[@gong2004systematic; @eichler2002co] Additionally, the averaged adsorption energy for one CO${\rm _2}$ molecule in FS is 0.19 eV, which provides the possibility of spontaneous desorption from the catalyst Pd@CN.
Conclusion
==========
In summary, by means of DFT calculations, we systemically investigate the potential of Pd and Pt anchored porous CN monolayer as SACs for CO oxidation. Both Pd and Pt possess appreciable binding energies to the CN monolayer along with large diffusion barriers, prohibiting the aggregation of adatoms, thereby enhancing the durability of catalysts. Besides, the MD simulation verifies that Pd/Pt@CN show highly thermodynamic stability at room temperature. Furthermore, the charge density difference plots and spin$-$polarized local density of state for O${\rm _2}$ and CO adsorption on Pd/Pt@CN demonstrate that the substrate can effectively capture and activate the gases via the electron “acceptance$-$donation” process. Finally, the detailed mechanistic pathways of CO oxidation over Pd/Pt@CN are comparatively investigated along ER, LH, and TER mechanisms. For Pd@CN, TER mechanism is more preferable, while Pt@CN is inclined to promote the CO oxidation through LH mechanism, and the rate$-$limiting energy barriers are less than those of noble metals catalysts. We predict that transition$-$metal atoms embedded in the porous CN monolayer is a promising platform to realize SACs for CO oxidation, which is worthy of further experimentation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research is supported by the Natural Science Foundation of China (grant No. 11574167), the New Century 151 Talents Project of Zhejiang Province and the KC Wong Magna Foundation in Ningbo University.
Conflict of interest {#conflict-of-interest .unnumbered}
====================
The authors declare they have no conflict of interest
References {#references .unnumbered}
==========
|
---
author:
- 'R. J. Harris and J. R. Allington-Smith Centre for Advanced Instrumentation, Phyiscs Department, Durham University, South Rd, Durham, DH1 3LE, UK;'
bibliography:
- 'ref.bib'
title: Modelling the application of integrated photonic spectrographs to astronomy
---
Introduction
============
Spectroscopy is the most useful technique in modern Astronomy, addressing problems ranging from fundamental cosmology to exoplanets. In its simplest form a dispersive spectrograph contains four components. A slit to isolate the area to be dispersed, a collimator, a grating or prism to disperse the light and a camera and detector to record the intensity at each wavelength. In order to retain throughput the slitwidth of a spectrograph is usually matched to the FWHM of the seeing. If the system is not diffraction limited then the collimated beam must increase in size in order to maintain the same resolution [@Lee2000]. This means that the disperser must also grow in size. This leads physical problems such as stresses and flexure in the materials, along with the difficulties inherent in building these large monolithic structures. In order to use the same design principles as existing instruments more exotic materials and construction processes are needed, which drives the costs of building the instruments up. This leads to the cost of instruments scaling with the square of the telescope aperture or more [@Bland-Hawthorn2006]. Many ways to break this relationship have been considering including image slicing [@Allington-Smith2009] and adopting technologies developed by the photonics industry [@Bland-Hawthorn2006]. The solution considered in this paper makes use of the Arrayed Waveguide Gratings (AWGs) as the dispersive element. These form part of the Integrated Photonic Spectrograph (IPS) concept [@Bland-Hawthorn2010]. The IPS takes light from an input fibre which is usually matched to the seeing limit (as with conventional fibre fed instruments) and so supports many modes. As the input to the AWG must be single moded the light has to be split into individual single moded fibres by a photonic lantern [@Noordegraaf2010]. These are then fed into the AWGs which disperse the light into individual spectra. [@Bland-Hawthorn2010]. While the principles have been demonstrated [@Cvetojevic2009; @Cvetojevic2012], so far these have only used single or few modes from a single input, so have suffered from loss of throughput and coverage of field.
In this paper we shall look at whether photonic spectroscopy can compete with conventional instruments in terms of size. We will construct a toy model of conventional instruments and photonic ones in order to evaluate the relative volumes produced by these instruments.
Instrument size scale relationships
===================================
It has been stated in Astrophotonics papers that using AWGs breaks the telescope diameter, spectral resolving power relation[@Bland-Hawthorn2010]. The relation,
$$R = \frac{m \rho \lambda W}{\chi D_{T}} = \frac{2 \tan \gamma D_{col}}{\chi D_{T}}
\label{eqn:res}$$
holds for a long slit spectrograph using a diffraction grating as the dispersive element. Here *R* is the resolution of the instrument, m is the diffraction order, $\lambda$ is the wavelength, *W* is the length of intersection between the grating and collimated beam, $\chi$ is the angular slitwidth, $D_{T}$ is the diameter of the telescope, $\gamma$ is the blaze angle and $D_{col}$ the diameter of the collimated beam. It shows that for a given resolution, angular slitwidth, and blaze angle, that as the diameter of telescope increases the diameter of the collimated beam must increase, leading to a bigger instrument. Unlike a conventional spectrograph the AWG must be diffraction limited ($\lambda = \chi D_{T}$) due to its single mode nature, so the spectral resolving power can be shown to be of the form
$$R = \frac{mN}{C}.
\label{eqn:awg_res}$$
where *N* is the number of waveguides, *m* is the diffraction order and *C* errors from manufacturing [@Lawrence2010a]. It must be noted that this is for a single AWG. When telescope is not at the diffraction limit multiple AWGs (and hence modes) would be needed to conserve throughput.
The number of spatial modes, in a stepped index fibre can be approximated as
$$M = \frac{V_{fibre}^{2}}{4} .$$
The associated V parameter being given by
$$V_{fibre} = \frac{ \pi s \Theta}{\lambda} .$$
Where *s* is the diameter of the fibre, and $\Theta$ the numerical aperture at which the fibre is operated [@cheo]. This must be less than the maximum numerical aperture of the fibre. This can be modified so that the variables so they are same as Eq. (\[eqn:res\]) by taking
$$\begin{aligned}
\Theta &\approx& \frac{1}{2 F_{T}} \\
s &=& \chi f_{T} = \chi F_{T} D_{T} \\\end{aligned}$$
with $F_{T}$ being the focal ratio, $f_{T}$ the focal length of the telescope and s the diameter of the fibre (equivalent to the slitwidth in Eq. (\[eqn:res\])), yielding
$$M = \left( \frac{\pi \chi D_{T}}{4 \lambda} \right)^{2}.
\label{eqn:modes}$$
Therefore it can be seen that for each sampling element the number of modes increases as the square of the telescope diameter in a similar way to the number of slices at the diffraction limit $(\chi D_{T} / 1.22 \lambda )^{2}$. This means unless we will need to utilise more AWGs in order to maintain throughput, therefore the size of the IPS will still scale with telescope diameter.
Modelling
=========
We model both the standard instrument and its AWG equivalent. For our standard instrument we use a double pass echelle model similar to that developed in Ref. . We choose a variable focal length for the camera (and hence telescope) to keep the model simple and compact. For our AWG model we choose a reflective design similar to those described in Refs. and . The major difference between the models is the way they are scaled. All other parameters are based upon existing instruments for both models. As such extra components such as the optics, detector and housing are accounted for in the echelle model simply by using scaling factors. The instrument itself will have been optimised for size and structural integrity so it will represent the optimal instrument design. By comparison there are no full scale photonic spectrographs currently in existence. The current generation are prototypes, only sampling a very small area of the sky and not making use of enough AWGs to allow high throughput. Because of this we use a bottoms up approach when scaling our AWG model. We use an existing AWG obtained from Gemfire Livingstone and use its known size to scale our model. This does however mean the AWG model does not include the extra equipment that would be required for a full instrument, such as detectors and the housing of the instrument. We also assume all the AWGs in the final device can be stacked on top of each other, which in practise may not be possible.
The echelle Model
-----------------
The echelle model is adapted from that shown in Ref. . Here we will be calculating the model using the initial input parameters of $\chi$, R, $\lambda_{c}$, $\Delta \lambda_{FSR}$, $N_{y}$, $\rho$, where $\Delta \lambda_{FSR}$ is the free spectral range (FSR). As stated in Ref. modelling a spectrograph for monochromatic light is simple, but for a large wavelength range we need to add in extra parameters accounting for dispersion. For our model here we use $\pm \Phi/2$ to represent the extra beamspread due to the wavelength dispersion and $\pm \Psi/2$ to represent the dispersion due to conservation of etendue. Our toy model is based on an echelle spectrograph with double pass construction (see Fig. \[fig:echelle\]). For an unsliced spectrograph its dimensions are estimated as
$$\begin{aligned}
L_{x} &=& \left( D_{col} + \Phi \left( a_{ech} + D_{col}\tan \gamma \right) + a_{ech} \right)b_{ech} \\
L_{y} &=& \left( D_{col} + \Psi \left( a_{ech} + D_{col}\tan \gamma \right) + a_{ech} \right)b_{ech} \\
L_{z} &=& \left( D_{col}F_{c} + D_{col}\tan \gamma + 2a_{ech} \right) b_{ech}. \end{aligned}$$
Where $D_{col}$ is the diameter of the collimated beam, $\gamma$ the blaze angle and $F_{c}$ the focal ratio of the collimator and camera. We include $a_{ech}$ and $b_{ech}$ as an oversizing parameter in order to allow the model to be fitted to existing instruments, with $a_{ech}$ as the additive term and $b_{ech}$ the multiplicative. We consider Littrow configuration at blaze, so we can re-arrange Eq. (\[eqn:res\]) to give the diameter of the collimated beam
$$D_{col} = \frac{R \chi D_{t}}{2 \tan \gamma}.$$
The blaze angle in Littrow configuration is $\sin \gamma = \frac{\rho \lambda_{c} m}{2}$, and the FSR of a system is $\Delta \lambda_{FSR} = \frac{\lambda_{c}}{m}$. Thus
$$T = \tan \gamma = \tan \left( \sin^{-1} \left( \frac{\rho \lambda_{c}^{2}}{2 \Delta \lambda_{FSR}} \right) \right).$$
To work out the spatial beamspread we use conservation of etendue, yielding $$\Psi = N_{y}\chi \frac{D_{T}}{D_{col}}.$$ We now work out $\Phi$, using the FSR and $\lambda_{c}$. In order to calculate the total angle occupied on the detector by the spectrum we use the grating equation $\sin(\theta_{i}) + \sin(\theta_{o}) = m \rho \lambda$. Setting $\theta_{i}$ = $\gamma$ gives
$$\Phi = \theta(\lambda_{max}) - \theta(\lambda_{min}).$$
Where $\lambda_{max}$ and $\lambda_{min}$ are the maximum and minimum wavelengths that the system operates at (= $\lambda_{c}$ $\pm$ $\Delta_{\lambda}$/2). In order to find the minimum focal length for our system we need to make sure we can resolve each point in our spectra. We know that $\chi F_{c} D_{T} = n_{0} d_{p}$, where $n_{0}$ is the oversampling of the detector (we use 2.5 allowing for a non-gaussian shape in the point spread function (PSF)) and $d_{p}$ is the size of each pixel (our default is 13.5$\mu$m). From this we can calculate the minimum focal ratio of the camera $F_{c}$.
With all this in place we can rewrite the equations for the scale lengths of the instrument as
$$\begin{aligned}
L_{x} \left(\chi \right) &=& \left( \frac{R \chi D_{T}}{2T} + \Phi \left(a_{ech} + \frac{R \chi D_{T}}{2} \right) + a_{ech} \right) b_{ech} \\
L_{y} \left(\chi \right) &=& \left( \frac{R \chi D_{T}}{2T} + \Psi \left(a_{ech} + \frac{R \chi D_{T}}{2} \right) + a_{ech} \right) b_{ech} \\
L_{z} \left(\chi \right) &=& \left( \frac{R \chi D_{T} }{2} \left( \frac{F_{c}}{T} + 1 \right) + 2a_{ech} \right) b_{ech} .\end{aligned}$$
Finally to correctly sample the output we require one sampling point in the spatial direction (as we have fixed our slit the the FWHM already) and $n_{0}$ samples in our spectral direction, yielding
$$N_{pixels} = n_{0} N_{y} N_{\lambda} = \frac{\Delta \lambda_{FSR} n_{0}}{\Delta \lambda} = \frac{\Delta \lambda_{FSR} R n_{0}}{\lambda_{min}}.$$
The AWG Model
-------------
AWGs are usually purpose built and their design is hugely varied, especially when it comes to arrangement allowing the path difference between waveguides. To keep the toy model simple we have chosen a reflective AWG with Rowlands Circle arrangement for the Free Propagation Region (FPR) [@Peralta2003; @DePeralta2004]. We have removed the bend at the reflective end of the waveguide array for simplicity (see Fig. \[awg\_model\]). Because of this it looks almost identical to a double pass echelle spectrograph model described above.
Using the definitions in Fig. \[awg\_model\] we arrive at the following equations for the size of the model AWG
$$\begin{aligned}
L_{x} &=& \left( max(D,E) + a_{awg} \right)b_{awg} \\
L_{y} &=& (c_{awg} + w)b_{awg} \\
L_{z} &= & \left( a_{awg} + A + (N_{wg}-1)\Delta L \right)b_{awg}. \end{aligned}$$
Here $N_{wg}$ is the number of waveguides, *A* is the *x* length of the FPR, $\Delta L$ the length difference between adjacent waveguides to achieve the required order for a given $\lambda$, *D* is the length containing the waveguides (analogous to the illuminated length of the grating in the echelle model), E the *x* length of the detecting surface, *w* is the waveguide diameter. The oversizing parameters are $a_{awg}$, $b_{awg}$ and $c_{awg}$.
First we calculate the appropriate dispersion order we use the re-arranged FSR equation for an AWG
$$m = \frac{\lambda_{min}}{\Delta \lambda_{FSR}}.\\
\label{eqn:fsr}$$
Here *m* is the diffraction order and $n_{c}$ is the refractive index of the waveguides. This can be combined with Eq. (\[eqn:awg\_res\]) and $D = N / \rho$ yielding
$$D = \frac{C R}{m \rho}.$$
We can make use of the equation for free spatial range in an arrayed waveguide grating ($X_{FSR} = \lambda_{min} L_{F} \rho/ n_{s} $), where $n_{s}$ is the refractive index of the FPR and $L_{f}$ the length of the free space propagation region. Combining with geometrical arguments gives
$$E = L_{f} \sin \left( \frac{\lambda_{min} \rho}{n_{s}} \right).$$
In order to calculate $\Delta L$ we make use of the equation for calculating the central wavelength of the AWG
$$\Delta L = \frac{\lambda_{c} m}{n_{c}}
\label{eqn:delta_l}$$
$\lambda_{c}$ the central operating wavelength. *A* can be calculated using geometry and is
$$A = L_{f} \cos( \theta) = L_{f} \cos \left( \frac{N}{2\rho L_{f}} \right).
\label{A_eqn}$$
Where *N* is calculated using Eq. (\[eqn:awg\_res\]). In order to calculate $L_{f}$ we make use of the fact the imaging requires the number of detector pixels to be able to adequately sample the PSF at the resolution required (equivalent to sampling of the echelle model in Ref. ). To do this we take the dispersion relation
$$\left( \frac{\delta \lambda}{\delta x} \right) = \frac{n_{s}}{L_{f}m \rho}.
\label{eqn:dl_dx}$$
This can be combined with Eq. (\[eqn:fsr\]) to remove *m*. We take the equation for resolution $\delta \lambda$ = $\lambda_{min} / R$ and $\delta_{x} = n_{0} d_{p}$. , We use $\lambda_{min}$ in order to get the maximum $L_{f}$ This allows Eq. (\[eqn:dl\_dx\]) to become
$$L_{f} \geq \frac{n_{s} n_{0} d_{p} R \Delta \lambda_{FSR} }{\rho \lambda_{min}^{2}}.
\label{eqn:Lf}$$
Finally we can calculate the number of pixels we need for the required resolution
$$N_{pixels} = \frac{X_{FSR}}{n_{min} d_{p}} = \frac{\lambda_{c} L_{f} \rho}{ n_{s} d_{p}}.$$
Results
=======
We set the input parameters in Table \[tab:instruments\] to represent GNIRS [@2006MNRAS.371..380A], using those listed in Ref. as a guide. We have selected GNIRS as it is representative of an existing facility-class instrument. We then vary resolution, field of view (FOV) of the instrument, telescope diameter and seeing.
The results are presented in graphical form in Figs. \[fig:resolution\] to \[fig:seeing\_pixels\]. The immediate conclusion is that in all cases the AWG instrument will be smaller. This is slightly misleading though as we have not included detector volumes or instrument housing, which would increase the scale lengths, leading to a much closer comparison. In all cases the number of detector pixels is much higher for the AWG model, though if the input is close to the diffraction limit this number decreases, making the devices more comparable.
Figure \[fig:resolution\] shows varying the resolving power will produce similar scale length differences between the AWG and echelle models. This means there is no significant advantage for using the AWG instrument for low or high resolution spectroscopy. In Fig. \[fig:res\_pixels\] we see that using instruments with higher resolving power also results in a similar difference between the AWG and echelle model, suggesting that resolution will not be a major factor in deciding the use of the instrument.
Figure \[fig:fov\] suggests the size of the AWG instrument will have the greatest advantage for smaller fields of view. The reason for this is that for small fields of view the angular dependence of the echelle makes the overall instrument larger than the AWG. When the slit length is increased the increase in the angle for the echelle becomes smaller than the extra size of the extra AWGs. This is illustrated in Fig. \[fig:awg\_ech\_comp\]. Figure \[fig:fov\_pixels\] shows once again the number of detector pixels on the AWG instrument becomes larger as the FOV increases, but this is at the same rate as the echelle instrument.
Figure \[fig:d\_tel\] suggests increasing the size of the telescope will not make any significant difference to the size difference of the AWG and echelle models. This means the same size issues will be present for AWG instruments as conventional ones for ELTs using the model setup. We also see from Fig. \[fig:d\_tel\_pixels\] that as we increase the telescope diameter the number of pixels required increases with the number of propagating modes(see Eq. (\[eqn:modes\])).
Figure \[fig:seeing\] suggest that the relative sizes of the instruments is not highly dependent on seeing. This should be qualified though. If the input is at the diffraction limit there will be no need for a photonic lantern, or potentially a smaller one, leading to a smaller and less complicated instrument, if the input is far from the diffraction limit it will require a large photonic lantern. Figure \[fig:seeing\_pixels\] shows the number of detector pixels is comparable for diffraction limited inputs, but increases hugely as the seeing becomes worse, due to the extra modes (see \[eqn:modes\]).
Conclusion
==========
We have used simplified models for both a conventional echelle model and an AWG model to simulate the size and number of detector pixels of GNIRS on Gemini. We have then varied resolution, telescope diameter, sample size and seeing. Our results are presented in terms of the scale length (cube root of the volume) of the instrument and the number of detector pixels the instrument will require.
We find that the AWG model will be slightly smaller in size for all cases becoming much smaller when the FOV of the instrument is small. This is slightly misleading though as we have not included other components in the model (such as detectors and housing). This means the actual instrument will most likely be the same size if not larger than a conventional non-photonic counterpart. We also find this problem will be compounded for larger telescopes unless the input is very close to the diffraction limit. We also find that larger a larger FOV makes the AWG instrument less competitive in terms of size, meaning that AWG instruments may be suited to instruments with smaller fields of view, such as amateur instruments or for high resolution spectroscopy. Our results indicate the number of detector pixels will be substantially more for photonic instruments unless the input is very close to the diffraction limit. This suggests that either near diffraction limited Adaptive Optics or a space based mission would be required for optimal performance from photonic spectrographs.
In conclusion, our findings suggest that photonic spectroscopy would be most useful on small instruments with diffraction limited seeing. We find that the AWG instrument will be on the same magnitude in terms of size as a conventional spectrograph, but will have more detector pixels. This does not provide a comprehensive model as we have only studied photonic spectrographs with a single input. Recent Astrophotonics papers [@Bland-Hawthorn2010; @Cvetojevic2012] suggest that multiple inputs could be used with varying configurations. This would potentially reduce the size of the instrument, but would not reduce the number of detector pixels required unless all the spectra could be combined onto a single detector pixel. We intend to investigate this further with similar models in the near future.
Acknowledgements
================
R. Harris gratefully acknowledges support from the Science and Technology Facilities council. J.R. Allington-Smith wishes to thank the European Union for funding for the OPTICON Research Infrastructure for Optical/IR astronomy under Framework Programme 7. We also thank Robert Thomson for many useful discussions into Photonic devices.
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$\chi$ Slit Telescope R $\lambda_{c}$ $\Delta_{FSR}$ Vol $F_{tel}$/$F_{cam}$ S a b
(“) &Length (”) Diameter (m) (nm) (nm) $(m^{3})$ (m) (mm)
[0.3]{} [99]{} [8]{} [5900]{} [1650]{} [400]{} [2.00]{} [16]{} 1 100 1.95
2 500 1.1
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: Table of input parameters for GNIRS calibration, taken from Ref. and calculated using the model. These will be used as the default parameters for the models.[]{data-label="tab:instruments"}
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abstract: 'The coding and noncoding length sequences constructed from a complete genome are characterised by multifractal analysis. The dimension spectrum $D_{q}$ and its derivative, the ’analogous’ specific heat $C_{q}$, are calculated for the coding and noncoding length sequences of bacteria, where $q$ is the moment order of the partition sum of the sequences. From the shape of the $D_{q}$ and $C_{q}$ curves, it is seen that there exists a clear difference between the coding/noncoding length sequences of all organisms considered and a completely random sequence. The complexity of noncoding length sequences is higher than that of coding length sequences for bacteria. Almost all $D_{q}$ curves for coding length sequences are flat, so their multifractality is small whereas almost all $D_{q}$ curves for noncoding length sequences are multifractal-like. It is seen that the ’analogous’ specific heats of noncoding length sequences of bacteria have a rich variety of behaviour which is much more complex than that of coding length sequences. We propose to characterise the bacteria according to the types of the $C_{q}$ curves of their noncoding length sequences. This new type of classification allows a better understanding of the relationship among bacteria at the global gene level instead of nucleotide sequence level.'
author:
- |
Zu-Guo Yu$^{1,2}$[^1], Vo Anh$^{1}$ and Ka-Sing Lau$^3$\
[$^1$Centre in Statistical Science and Industrial Mathematics, Queensland University]{}\
[of Technology, GPO Box 2434, Brisbane, Q 4001, Australia.]{}\
[$^2$Department of Mathematics, Xiangtan University, Hunan 411105, P. R. China.[^2]]{}\
[$^3$Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong]{}
date: 24 April 2001
title: '[**Multifractal characterisation of length sequences of coding and noncoding segments in a complete genome**]{}'
---
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[**PACS**]{} numbers: 87.10+e, 47.53+n
[**Key words**]{}: Coding/noncoding segments, length sequence, complete genome, multifractal analysis, ’analogous’ specific heat.
Introduction
============
The rapidly accumulating complete genome sequences of bacteria and archaea provide a new type of information resource for understanding gene functions and evolution$^{\cite{gelfand00}}$.
One can study the DNA sequences with details by considering the order of four kinds of nucleotides which DNA is assembled, namely adenine ($a$), cytosine ($c$), guanine ($g$), and thymine ($t$).
There has been considerable interest in the finding of long-range correlation (LRC) in DNA sequences at this level. Li [*et al*]{}$^{\cite{li}} $ found that the spectral density of a DNA sequence containing mostly introns shows $1/f^{\beta }$ behaviour, which indicates the presence of LRC. The correlation properties of coding and noncoding DNA sequences were also studied by Peng [*et al*]{}$^{\cite{peng}}$ in their fractal landscape or DNA walk model. The DNA walk defined in Ref.[@peng] is that the walker steps “up” if a pyrimidine ($c$ or $t$) occurs at position $i$ along the DNA chain, while the walker steps “down” if a purine ($a$ or $g$) occurs at position $i$. Peng [*et al*]{}$^{\cite{peng}}$ discovered that there exists LRC in noncoding DNA sequences while the coding sequences correspond to a regular random walk. By doing a more detailed analysis, Chatzidimitriou-Dreismann and Larhammar$^{\cite{CDL93}}$ concluded that both coding and noncoding sequences exhibit LRC. A subsequent work by Prabhu and Claverie$^{\cite{PC92}}$ also substantially corroborates these results. If one considers more details by distinguishing $c$ from $t$ in pyrimidine, and $a$ from $g$ in purine (such as two or three-dimensional DNA walk model$^{
\cite{luo}}$ and maps given in Ref.[@YC]), then the presence of base correlation has been found even in coding sequences. In view of the controversy about the presence of correlation in all DNA or only in noncoding DNA, Buldyrev [*et al*]{}$^{\cite{Bul95}}$ showed the LRC appears mainly in noncoding DNA using all the DNA sequences available. Alternatively, Voss$^{\cite{voss}}$, based on equal-symbol correlation, showed a power law behaviour for the sequences studied regardless of the percent of intron contents. Investigations based on different models seem to suggest different results, as they all look into only a certain aspect of the entire DNA sequence$^{\cite{MNR}}$.
The avoided and under-represented strings in some bacterial complete genomes have been discussed$^{\cite{yhxc99,hlz98,hxyc99}}$. A time series model of CDS in complete genome has been proposed$^{\cite{YW99}}$. Vieira$^{\cite
{Vie99}}$ performed a low-frequency analysis of the complete DNA of 13 microbial genomes and found that their fractal behaviour does not always prevail through the entire chain and their autocorrelation functions have a rich variety of behaviours including the presence of anti-persistence.
For the importance of the numbers, sizes and ordering of genes along the chromosome, one can refer to Part 5 of Lewin$^{\cite{Lew97}}$. Here one may ignore the composition of the four kinds of bases in coding and noncoding segments and only considers the rough structure of the complete genome or long DNA sequences. Provata and Almirantis $^{\cite{PY}}$ proposed a fractal Cantor pattern of DNA. They map coding segments to filled regions and noncoding segments to empty regions of random Cantor set and then calculate the fractal dimension of the random fractal set. They found that the coding/noncoding partition in DNA sequences of lower organisms is homogeneous-like, while in the higher eucariotes the partition is fractal. This result seems too rough to distinguish bacteria because the fractal dimensions of bacteria they gave out are all the same.
Viewing from the level of structure, the complete genome of an organism is made up of coding and noncoding segments. Here the length of a coding/noncoding segment means the number of its bases. Based on the lengths of coding/noncoding segments in the complete genome, one can get two kinds of integer sequences by the following ways:
i\) Order all lengths of coding segments according to the order of coding segments in the complete genome. This integer sequence is named [*coding length sequence*]{}.
ii\) Order all lengths of noncoding segments according to the order of noncoding segments in the complete genome. This integer sequence is named [*noncoding length sequence*]{}.
Yu and Anh$^{\cite{YA00}}$ proposed a time series model for the length sequences of DNA. After calculating the correlation dimensions and Hurst exponents, it was found that one can get more information from this model than that of fractal Cantor pattern$^{\cite{PY}}$. The quantification of these correlations could give an insight into the role of the ordering of genes on the chromosome. Through detrended fluctuation analysis (DFA)$^{\cite
{GPH}}$ and spectral analysis, the LRC was found in these length sequences$^{
\cite{YAW01}}$.
The correlation dimension and Hurst exponent are parameters of global analysis. Global calculations neglect the fact that length sequences from a complete genome are highly inhomogeneous. Thus multifractal analysis is a useful way to characterise the spatial inhomogeneity of both theoretical and experimental fractal patterns$^{\cite{GP83}}$. It was initially proposed to treat turbulence data. In recent years it has been applied successfully in many different fields including time series analysis$^{\cite{pas97}}$ and financial modelling$^{\cite{can00,ATT00}}$. For DNA sequences, application of the multifractal technique seems rare (we have found only Berthelsen [*et al.*]{}$^{\cite{BGR94}})$. Recently, Yu [*et al*]{}$^{\cite{YA00a}}$ considered the multifractal property of the measure representation of a complete genome. In this paper, we pay more attention to the multifractal characterisation of the coding and noncoding length sequences.
Some sets of physical interest have a nonanalytic dependence of dimension spectrum $D_{q}$ on the $q$-moments of the partition sum of the sequences. Moreover, multifractality has a direct analogy to the phenomenon of phase transition in condensed-matter physics$^{\cite{KP87}}$. The existence and type of phase transitions might turn out to be a worthwhile characterisation of universality classes for the structures$^{\cite{Boj87}}$. The concept of phase transition in multifractal spectra was introduced in the study of logistic maps, Julia sets and other simple systems. Evidence of phase transition was found in the multifractal spectrum of diffusion-limited aggregation$^{\cite{LeS88}}$. By following the thermodynamic formulation of multifractal measures, where $q$ represents an analogous temperature, Canessa$^{\cite{can00}}$ applied a standard expression for the ’analogous’ specific heat and showed that its form resembles a classical phase transition at a critical point for financial time series.
In this paper we calculate the ’analogous’ specific heat of coding and noncoding length sequences. Our motivation to apply Canessa’s framework to characterise stochastic sequences is to see whether there is a similar type of phase transition in the coding and noncoding length sequences as in other time series. We show that based on the shape of the $C_{q}$ curves and associated type of phase transitions, one can discuss the classification of bacteria. This new type of classification allows to better understand the relationship among bacteria at the global gene level instead of nucleotide sequence level.
Multifractal analysis
=====================
Let $T_{t},\ t=1,2,\cdots ,N,$ be the length sequence of coding or noncoding segments in the complete genome of an organism. First we define $$F_{t}=T_{t}/(\sum_{j=1}^{N}T_{j})$$ to be the frequency of $T_{t}$. It follows that $\sum_{t}F_{t}=1$. Now we can define a measure $\mu $ on $[0,1[$ by $d\mu (x)=Y(x)dx$, where $$Y(x)=N\times F_{t},\ \ \mbox{when}\ \ x\in \lbrack \frac{t-1}{N},\frac{t}{N}[.$$ It is easy to see that $\int_{0}^{1}d\mu (x)=1$ and $\mu
([(t-1)/N,t/N[)=F_{t}$.
The most common numerical implementations of multifractal analysis are the so-called [*fixed-size box-counting algorithms*]{} $^{\cite{hjkps}}$. In the one-dimensional case, for a given measure $\mu $ with support $E\subset {\bf R}$, we consider the [*partition sum*]{} $$Z_{\epsilon }(q)=\sum_{\mu (B)\neq 0}[\mu (B)]^{q},$$ $q\in {\bf R}$, where the sum runs over all different nonempty boxes $B$ of a given side $\epsilon $ in a grid covering of the support $E$, that is, $$B=[k\epsilon ,(k+1)\epsilon \lbrack .$$ The scaling exponent $\tau (q)$ is defined by $$\tau (q)=\lim_{\epsilon \rightarrow 0}\frac{\log Z_{\epsilon }(q)}{\log
\epsilon }$$ and the generalized fractal dimensions of the measure are defined as $$D_{q}=\tau (q)/(q-1),\ \ \mbox{for}\ q\neq 1,$$ and $$D_{q}=\lim_{\epsilon \rightarrow 0}\frac{Z_{1,\epsilon }}{\log \epsilon },\
\ \mbox{for}\ q=1,$$ where $Z_{1,\epsilon }=\sum_{\mu (B)\neq 0}\mu (B)\log \mu (B)$. The generalized fractal dimensions are numerically estimated through a linear regression of $$\frac{1}{q-1}\log Z_{\epsilon }(q)$$ against $\log \epsilon $ for $q\neq 1$, and similarly through a linear regression of $Z_{1,\epsilon }$ against $\log \epsilon $ for $q=1$. $D_{1}$ is called the [*information dimension*]{} and $D_{2}$ the [*correlation dimension*]{}. The $D_{q}$ of the positive values of $q$ give relevance to the regions where the measure is large, i.e., to the coding or noncoding segments which are relatively long. The $D_{q}$ of the negative values of $q$ deal with the structure and the properties of the most rarefied regions of the measure, i.e. to the segments which are relatively short.
By following the thermodynamic formulation of multifractal measures, Canessa$^{\cite{can00}}$ derived an expression for the ’analogous’ specific heat as $$C_{q}\equiv -\frac{\partial ^{2}\tau (q)}{\partial q^{2}}\approx 2\tau
(q)-\tau (q+1)-\tau (q-1).$$ He showed that the form of $C_{q}$ resembles a classical phase transition at a critical point for financial time series. In the following we calculate the ’analogous’ specific heat of coding and noncoding length sequences for the first time. The types of phase transitions are helpful to discuss the classification of bacteria.
Data and results
=================
More than 31 bacterial complete genomes are now available in public databases. There are five Archaebacteria: [*Archaeoglobus fulgidus*]{} (aful), [*Pyrococcus abyssi*]{} (pabyssi), [*Methanococcus jannaschii*]{} (mjan), [*Aeropyrum pernix*]{} (aero) and [*Methanobacterium thermoautotrophicum*]{} (mthe); five Gram-positive Eubacteria: [*Mycobacterium tuberculosis*]{} (mtub), [*Mycoplasma pneumoniae*]{} (mpneu), [*Mycoplasma genitalium*]{} (mgen), [*Ureaplasma urealyticum*]{} (uure), and [*Bacillus subtilis*]{} (bsub). The others are Gram-negative Eubacteria, which consist of two Hyperthermophilic bacteria: [*Aquifex aeolicus*]{} (aquae) and [*Thermotoga maritima*]{} (tmar); three Chlamydia: [*Chlamydia trachomatisserovar*]{} (ctra), [*Chlamydia muridarum*]{} (ctraM), and [*Chlamydia pneumoniae*]{} (cpneu); two Spirochaete: [*Borrelia burgdorferi*]{} (bbur) and [*Treponema pallidum*]{} (tpal); one Cyanobacterium: [*Synechocystis sp. PCC6803*]{} (synecho); and thirteen Proteobacteria. The thirteen Proteobacteria are divided into four subdivisions, which are alpha subdivision: [*Rhizobium sp. NGR234*]{} (pNGR234) and [*Rickettsia prowazekii*]{} (rpxx); gamma subdivision: [*Escherichia coli*]{} (ecoli), [*Haemophilus influenzae*]{} (hinf), [*Xylella fastidiosa*]{} (xfas), [*Vibrio cholerae*]{} (vcho1), [*Pseudomonas aeruginosa*]{} (paer) and [*Buchnera sp. APS*]{} (buch); beta subdivision: [*Neisseria meningitidis MC58*]{} (nmen) and [*Neisseria meningitidis Z2491*]{} (nmenA); epsilon subdivision: [*Helicobacter pylori J99*]{} (hpyl99), [*Helicobacter pylori 26695*]{} (hpyl) and [*Campylobacter jejuni*]{} (cjej).
First we counted out the length of coding and noncoding segments in the complete genomes of the above bacteria and obtained the coding and noncoding length sequences of these organisms. For example, we give the coding and noncoding length sequences of [*Pseudomonas aeruginosa*]{} (paer) in Figure \[paerlength\].
Then we calculated the dimension spectra $D_{q}$ and ’analogous’ specific heat $C_{q}$ of the coding and noncoding length sequences of all the above bacteria according to the methods given in section 2. In order to show the difference between coding and noncoding length sequences, we give the $C_{q}$ curves of length sequences of all the above bacteria as Figure \[Cqfigure1\] (for 19 bacteria) and Figure \[Cqfigure2\] (for another 12 bacteria).
The hill behaviour of the dimension spectrum $D_{q}$ for $q<0$ is a well known fact when using the box-counting method$^{\cite{pas97,can00}}$. In Figures \[Dqfigure1\] and \[Dqfigure2\], we present $D_{q}$ of the coding or noncoding length sequences of all bacteria selected within the range $q\geq 0$.
Discussion and conclusions
==========================
If a length sequence is completely random, then our measure definition yields a uniform measure ($D_{q}=1,\ C_{q}=0$).
From the curves of $D_{q}$ and $C_{q}$, it is seen that there exists a clear difference between the coding/noncoding length sequences of all organisms considered here and the completely random sequence. Hence we can conclude that complete genomes are not random sequences. But the $D_{q}$ values of coding length sequences are closer to 1 than that of noncoding length sequences. In other words, noncoding length sequences are further away from a complete random sequence than coding length sequences. The property of the length sequences is same as that of the DNA sequences$^{\cite{peng}}$.
We also found that for each bacterium selected, the $D_{q}$ values for $q>0$ of a noncoding length sequence are smaller than those of a coding length sequence, but for $q<0$, the situation is reversed. It is well known that the dimension is a measure for complexity. Here the complexity of noncoding length sequences is higher than that of coding length sequences for bacteria.
From Figures \[Dqfigure1\] and \[Dqfigure2\], almost all $D_{q}$ curves for coding length sequences are flat, so their multifractality is not pronounced. On the other hand, almost all $D_{q}$ curves for noncoding length sequences are multifractal-like.
In our previous paper$^{\cite{YA00a}}$, we counted out all substrings with fixed length appearing in the complete genome and gave a measure representation of the complete genome. We found that the shape of the $C_{q}$ curves of all bacteria we selected are single-peaked. Hence this type of phase transition of the measure representation is not useful for classification of bacteria. On the other hand, from Figures \[Cqfigure1\] and \[Cqfigure2\], one can see that the ’analogous’ specific heats of noncoding length sequences of bacteria have a rich variety of behaviours which is much more complex than that of coding length sequences. Some have only one main single peak. In this class, some $C_{q}$ curves display a shoulder to the right of the main peak, some display a shoulder to the left of the main peak, and some have no shoulder, which resembles a classical (first-order)phase transition at a critical point. In another class, the $C_{q}$ curves display a balance double-peak. So this provides a useful tool for classification of bacteria according to the types of ’analogous’ specific heats of the noncoding length sequences. The relevant finding here is that noncoding length sequences display higher $C_{q}$ peak heights and clear double peaked structures than coding length sequences. This reveals different types of long-range correlations between the two classes of sequences. This new type of classification allows a better understanding of the relationship among bacteria at the global gene level instead of nucleotide sequence level. It can be useful to distinguish between sequence curves as given in the example of Figure \[paerlength\].
To conclude, multifractal analysis provides a simple yet powerful method to amplify the difference between a DNA length sequence and a random sequence. In particular, the multifractal characterisation given by the ’analogous’ specific heat allows to distinguish DNA length sequences in more details.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
One of the authors, Zu-Guo Yu, would like to express his thanks to Prof. Bai-lin Hao of Institute of Theoretical Physics of Chinese Academy of Science for introducing him into this field and continuous encouragement. The authors also thank Dr. Enrique Canessa for many good suggestions and comments to improve this paper. The research was partially supported by QUT’s Postdoctoral Research Support Grant No. 9900658.
[10]{}
=-0.15mm
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[^1]: Corresponding author, corresponding address: School of Mathematical Science, Queensland University of Technology, Garden Point Campus, GPO Box 2434, Brisbane, Q 4001, Australia. Tel.: +61 7 38645194, Fax: +61 7 38642310, e-mail: [email protected] or [email protected]
[^2]: This is the permanent corresponding address of Zu-Guo Yu.
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author:
- 'Yoshio <span style="font-variant:small-caps;">Kamiya</span>$^{1}$, Misato <span style="font-variant:small-caps;">Tani</span>$^{1}$, Sachio <span style="font-variant:small-caps;">Komamiya</span>$^{1}$, Guinyun <span style="font-variant:small-caps;">Kim</span>$^{2}$, and Kyungsuk <span style="font-variant:small-caps;">Kim</span>$^{2}$'
title: 'A Search for non-Newtonian force in a precision measurement of the scattering of slow neutrons in Xenon gas'
---
Introduction
============
“Is there any new force in addition to the four known fundamental forces” “Is the Newtonian inverse square law of gravity valid even in the sub-mm range?” These are still open questions, and some theories, such as the Kaluza-Klein theory with extra dimensions[@ref.KK], predict the existence of a fifth force or show certain modifications to the Newtonian law of gravity. A simple Yukawa-type parametrization of an additional scattering potential due to new physics is given by $$V_{new}(r) = -\frac{1}{4\pi} g^2 Q_1 Q_2 \cdot \frac{e^{-\mu r}}{r} \hspace{1.5mm},$$ where $g^2$ is a coupling strength, $Q_i$ are coupling charges, and $\mu$ is the mass of a mediating boson for the new force. Much progress for the fifth force or the non-newtonian force search in this parametrization has been made over the last 10 years and this field is still very active, especially for gravity-like interactions with coupling charge of mass number or macroscopic mass. The current world limits in $g^2$-$\mu$ or $g^2$-$\lambdabar$ space are shown in figure \[fig.comb\](a)[@limitA; @limitB; @limitC; @limitD; @limitE; @limitF; @limitG; @limitH; @limitI], where $\lambdabar = 1/\mu$ is the interaction range.
This additional scattering potential slightly modifies the scattering angle distribution of the neutron-atom interaction. We propose to search for new forces by evaluating a deviation of the angular distribution from the known scattering process. The scheme has a sensitivity to new forces with around 1 nm interaction range, which corresponds to a mass of several hundred eV (see also Fig.\[fig.comb\](a)). The current world limit at 1 nm range is about $7\times10^{-16}$, which is obtained by reviewing available neutron scattering data[@limitB]. Statistical uncertainty limits the sensitivity.
In this article, we review the interaction between a slow neutron and Xenon with an additional cross section due to a Yukawa-type potential. After that, we show details of a preliminary experiment which was performed in January 2013 with small statistics of 25 hours irradiation time, and the achievable sensitivity expected for a planned high statistics run.
![(a) Current wold limits for non-Newtonian force in $g^2$-$\lambdabar$ space [@limitA; @limitB; @limitC; @limitD; @limitE; @limitF; @limitG; @limitH; @limitI]. The experimental scheme for the limits A and B are neutron scattering, and the limits C to I are obtained by a macroscopic scheme using a torsion balance or cantilever. Theoretical expectations from the large extra demotions[@ref.KK] and extra $U(1)$ gauge boson with supersymmetric extensions of the standard model[@ref.uboson] are shown as hatched areas. (b,c,d) Simulated scattering angle distribution of nuclear scattering, higher order EM scattering, and additional Yukawa-type scattering ($\lambdabar = 1$ nm), respectively. The thermal motion of Xenon gas is considered in the simulation. []{data-label="fig.comb"}](./setup.comb.v4.eps){width="130mm"}
Review of slow neutron/Xenon interactions
=========================================
A neutron scattering length $b({\boldsymbol{q}})$ for a diamagnetic atom such a Xenon is derived from [@sears] as $$b({\boldsymbol{q}}) = b_c(q) + \frac{1}{\sqrt{I(I+1)}} {\boldsymbol{\sigma}}\cdot{\boldsymbol{b_i}}({\boldsymbol{q}})\cdot{\boldsymbol{I}} + ib_s(q){\boldsymbol{\sigma}}\cdot{\boldsymbol{\hat{n}}} \hspace{1.5mm} ,$$ where ${\boldsymbol{q}}$ is the momentum transfer vector, ${\boldsymbol{I}}$ is the nucleus spin, ${\boldsymbol{\sigma}}/2$ is the neutron spin, and ${\boldsymbol{\hat{n}}}$ is a unit vector perpendicular to the plane of scattering. $b_c$ in the first term represents coherent scattering, the tensor ${\boldsymbol{b_i}}$ in the second term is for incoherent scattering, and the last term of $b_s$ represents so-called Schwinger scattering, whose scattering plane is perpendicular to the one of the other scattering processes. For an unpolarized neutron beam and Xenon atomic gas target, the second and the third terms vanish. The scales of each term, $b_c$, ${\boldsymbol{b_i}}$, and $b_s$ for the Xenon atom are $\sim 5$ fm, $\sim 10^{-3}$ fm, and $\sim 10^{-3}$ fm, respectively. Therefore, even when the beam is polarized with a certain fraction, the second and third terms can still be neglected in the following discussion for our intended sensitivity.
The coherent term is divided into two interactions, nuclear scattering and higher order electromagnetic scattering. The scattering length of the Yukawa-type potential is calculated by the Fourier transformation under the Born approximation and it is added to the known scattering lengths when a new force exists. The differential cross section with the Yukawa-type scattering is described by the square of the total scattering length, and is written as $$\begin{aligned}
\frac{d\sigma}{d\Omega} & \simeq & (b_{Nc} + b_{p})^2 \hspace{1mm} (1 + 2\chi[1-f(q)] + 2\chi_y[(\frac{q}{\mu})^2 + 1]^{-1} ) \\
f(q) & = & [1 + 3(\frac{q}{q_0})^{2}]^{-0.5} \hspace{5mm} {}_{(q_0 = 7 {\rm \AA}^{-1} \hspace{0.5mm} {\rm for \hspace{0.5mm} Xe})} \\
\chi & \equiv & \frac{b_F + b_I}{b_{Nc} + b_{p}}Z \hspace{5mm} \sim 10^{-2}\\
\chi_{y} & \equiv & \frac{m_n}{2\pi} g^2 Q_1 Q_2 \frac{1}{(b_{Nc} + b_{p})\mu^2}\end{aligned}$$ where $b_{Nc}$ is the coherent nuclear scattering length, $b_P$ is the polarization scattering length, $b_F$ is the Foldy scattering length, $b_I$ is the intrinsic n-e scattering length, and $m_n$ is the neutron mass. $\chi$ and $\chi_y$ are small enough to omit their second order terms. The first term from nuclear scattering has no $q$ dependence, on the other hand the second term due to higher order electromagnetic scattering has $q$ dependence which is empirically described by the atomic form factor $f(q)$. By integrating with azimuthal angle and by taking into account the thermal motion of Xenon gas, the scattering angle $\theta$ distribution of each term is simulated as shown in figure \[fig.comb\](b), \[fig.comb\](c), \[fig.comb\](d) in the case of $\lambdabar=1$ nm. These distributions are clearly distinguished each other, and, thanks to this property, an analytical method using the shapes of the distributions can be adopted to avoid uncertainty on measuring absolute values such as the total coherent cross section. In later discussion, functions $h_1(\theta)$, $h_2(\theta)$, $h_3(\theta;\lambdabar)$ are used to express these simulated angular distributions shown in the figure \[fig.comb\](b), \[fig.comb\](c), \[fig.comb\](d), respectively.
Preliminary Experiment
======================
A preliminary experiment was performed in Jan. 2013 with a 25 hour irradiation of a Xenon sample. We use the 40 m SANS(Small Angle Neutron Scattering) beam line[@han] at the HANARO research reactor located at the Korean Atomic Energy Research Institute. Figure \[fig.exp\](a) shows schematic drawing of the experiment. The wavelength of neutrons was selected to be 5 $\AA$ with a 12% FWHM spread. The beam was collimated to have a 1 mrad divergence and the beam size was 12 mm in diameter. The beam intensity was $1 \times 10^{5}$ neutrons/sec. A chamber of 250 mm length was filled with Xenon gas at 1 atm pressure. The detector used in this experiment was a $^{3}$He filled MWPC with 5 mm wire spacing along horizontal and vertical axes, whose detection efficiency is around 80% for neutrons with 5 $\AA$ wavelength. The size of sensitive area is 1 m $\times$ 1 m square and the distance from the Xe chamber to the detector was 2.5 m. The accuracy on measurements of the scattering angle is mainly limited by the finite chamber length, and is better than 5%. The other sources, due to the beam size and detector spatial resolution, give a negligible contribution. The vacuum is required to be less than 1 Pa, considering the influence of scattering from residual gas. A scattering image is shown in figure \[fig.exp\](b).
![(a) Schematic drawing of the preliminary experiment. The total length from the velocity selector to the end of the vacuum vessel is 40 m. The detector can be moved in the vessel to adjust the distance from the sample. (b) Measured scattering image on the detector. The sensitive area is $ 1 \times 1$ m$^2$.[]{data-label="fig.exp"}](./fig2.v1.eps){width="110mm"}
In figure \[fig.fit\](a), two sets of scattering data with and without(background) Xenon gas are shown. After subtracting the background distribution, the angular distribution of the Xenon sample signal $g(\theta)$ is evaluated by a least squares method with a linear combination of $h_1(\theta), h_2(\theta), h_3(\theta)$ written as $$g(\theta) = (1-\alpha-\beta)h_1(\theta) + \alpha h_2(\theta) + \beta h_3(\theta; \lambdabar) \hspace{1mm},$$ where $\alpha$ and $\beta$ are the parameters to be estimated, describing the fractions of each term. The fitting range is from 0.02 to 0.18 mrad, to avoid the effects from the direct beam and the edge of the active area. Figure \[fig.fit\](b) shows the analyzed deviations from the known scattering processes, when the interaction range of a new force $\lambdabar$ is 1 nm. The results is consistent with the expectation with no additional forces.
![(a) Scattering angle distribution for Xenon sample with background (solid line) and background only (dashed line). (b) Estimated deviation from the known scattering processes when the range of new force $\lambdabar = 1$ nm. Data from 0.02 mrad to 1.8 mrad are used for fitting. (c) 95% CL Limits obtained the preliminarily experiment (solid curve) and expected sensitivity for high statistic runs (dashed curve).[]{data-label="fig.fit"}](./fig4.v2.eps){width="115mm"}
The 95% CL limits for a Yukawa-type, non-Neutonian force obtained are $g^2 < 4.5\times10^{-15}$ for $\lambdabar = 0.3$ nm, $g^2 < 2\times10^{-15}$ for $\lambdabar = 1$ nm, and $g^2 < 2.7\times10^{-16}$ for $\lambdabar = 70$ nm. In figure \[fig.fit\](c), the limit is shown as a solid curve. Each contribution to the systematic uncertainty $\delta_{g^2}$ in terms of $g^2$ estimation are evaluated using Monte Carlo pseudo-experiments: for $\lambdabar = 1$ nm, $\delta_{g^2} \sim 5\times10^{-17}$ from a neutron flux uncertainty of 0.5%; $\delta_{g^2} \sim 2\times10^{-17}$ from detection non-uniformity of 1%(RMS); $\delta_{g^2} \sim 2\times10^{-17}$ from the neutron energy determination within 3%; $\delta_{g^2} < 10^{-17}$ from the atomic form factor model validity within 5%; $\delta_{g^2} < 10^{-17}$ from temperature uncertainty of $\pm$ 6 K; $\delta_{g^2} < 10^{-17}$ from the interaction position uncertainty of 25 cm due to the chamber size; $\delta_{g^2} < 10^{-17}$ from the gas contamination of $10^{-5}$. The statistical uncertainty still dominats the experimental sensitivity of this scheme. We are planning to perform high statistics runs with two order of magnitude larger statistics than this preliminary experiment. The expected limits for this next experiment is shown as a dashed curve in figure \[fig.fit\](c).
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors are grateful to Dr. Tae-Hwan Kim and Dr. Young-Soo Han for their specialist support in maintaining beam conditions, and Dr. Jong-Dae Jang and Dr. Eun-Hye Kim for supporting beam operation. This work is supported by JSPS KAKENHI Grant No. 90434323.
[9]{} N. Arkani-Hamed, Savas Dimopoulos, and Gia Dvali, Phys. Lett. B **429**, (1998) 263; J. M. Overduin and P. S. Wesson, Phys. Reports **283**, (1997) 303. Y. N. Pokotilovski, Phys. Atom. Nucl. **69**, (2006) 924. V. V. Nesvizhevsky, G. Pignol, and K. V. Protasov, Phys. Rev. D **77**, (2008) 034020. M. Bordag, U. Mohideen, and V. M. Mostepanenko, Phys. Reports **353**, (2001) 1. R. S. Decca [*et al.*]{}, PRL **94**, (2005) 240401. A. O. Sushkov [*et al.*]{}, PRL **107**, (2011) 171101. M. Masuda and M. Sasaki, PRL **102**, (2011) 171101. A. A. Geraci [*et al.*]{}, Phys. Rev. D **78**, (2008) 022002. D. J. Kapner [*et al.*]{}, PRL **98**, (2007) 021101. S. Q. Yang [*et al.*]{}, PRL **108**, (2012) 081101. P. Fayet, Phys. Rev. D **75**, (2007) 115017. V. F. Sears, Phys. Pepoorts **141**, (1986) 281. Y. S. Han [*et al.*]{}, The 11th Japan-Korea Meeting on Neutron Science, I01 (2011).
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abstract: 'As font is one of the core design concepts, automatic font identification and similar font suggestion from an image or photo has been on the wish list of many designers. We study the Visual Font Recognition (VFR) problem [@LFE], and advance the state-of-the-art remarkably by developing the *DeepFont* system. First of all, we build up the first available large-scale VFR dataset, named *AdobeVFR*, consisting of both labeled synthetic data and partially labeled real-world data. Next, to combat the domain mismatch between available training and testing data, we introduce a Convolutional Neural Network (CNN) decomposition approach, using a domain adaptation technique based on a Stacked Convolutional Auto-Encoder (SCAE) that exploits a large corpus of unlabeled real-world text images combined with synthetic data preprocessed in a specific way. Moreover, we study a novel learning-based model compression approach, in order to reduce the DeepFont model size without sacrificing its performance. The DeepFont system achieves an accuracy of higher than 80% (top-5) on our collected dataset, and also produces a good font similarity measure for font selection and suggestion. We also achieve around 6 times compression of the model without any visible loss of recognition accuracy.'
author:
- |
Zhangyang Wang$^1$ Jianchao Yang$^3$ Hailin Jin$^2$ Eli Shechtman$^2$ Aseem Agarwala $^4$ Jonathan Brandt $^2$ Thomas S. Huang $^1$\
\
\
\
bibliography:
- 'sigproc.bib'
title: 'DeepFont: Identify Your Font from An Image'
---
Introduction
============
Typography is fundamental to graphic design. Graphic designers have the desire to identify the fonts they encounter in daily life for later use. While they might take a photo of the text of a particularly interesting font and seek out an expert to identify the font, the manual identification process is extremely tedious and error-prone. Several websites allow users to search and recognize fonts by font similarity, including Identifont, MyFonts, WhatTheFont, and Fontspring. All of them rely on tedious humans interactions and high-quality manual pre-processing of images, and the accuracies are still unsatisfactory. On the other hand, the majority of font selection interfaces in existing softwares are simple linear lists, while exhaustively exploring the entire space of fonts using an alphabetical listing is unrealistic for most users.
Effective automatic font identification from an image or photo could greatly ease the above difficulties, and facilitate font organization and selection during the design process. Such a Visual Font Recognition (VFR) problem is inherently difficult, as pointed out in [@LFE], due to the huge space of possible fonts (online repositories provide hundreds of thousands), the dynamic and open-ended properties of font classes, and the very subtle and character-dependent difference among fonts (letter endings, weights, slopes, etc.). More importantly, while the popular machine learning techniques are data-driven, collecting real-world data for a large collection of font classes turns out to be extremely difficult. Most attainable real-world text images do not have font label information, while the error-prone font labeling task requires font expertise that is out of reach of most people. The few previous approaches [@3; @4; @5; @6; @7; @8] are mostly from the document analysis standpoint, which only focus on a small number of font classes, and are highly sensitive to noise, blur, perspective distortions, and complex backgrounds. In [@LFE] the authors proposed a large-scale, learning-based solution without dependence on character segmentation or OCR. The core algorithm is built on local feature embedding, local feature metric learning and max-margin template selection. However, their results suggest that the robustness to real-world variations is unsatisfactory, and a higher recognition accuracy is still demanded.
\
Inspired by the great success achieved by deep learning models [@imagenet] in many other computer vision tasks, we develop a VFR system for the Roman alphabets, based on the Convolutional neural networks (CNN), named *DeepFont*. Without any dependence on character segmentation or content text, the DeepFont system obtains an impressive performance on our collected large real-word dataset, covering an extensive variety of font categories. Our technical contributions are listed below:
- **AdobeVFR Dataset** A large set of *labeled real-world* images as well as a large corpus of unlabeled real-world data are collected for both training and testing, which is the first of its kind and is publicly released soon. We also leverage a large training corpus of labeled synthetic data augmented in a specific way.
- **Domain Adapted CNN** It is very easy to generate lots of rendered font examples but very hard to obtain labeled real-world images for supervised training. This real-to-synthetic domain gap caused poor generalization to new real data in previous VFR methods [@LFE]. We address this *domain mismatch* problem by leveraging synthetic data to obtain effective classification features, while introducing a domain adaptation technique based on Stacked Convolutional Auto-Encoder (SCAE) with the help of unlabeled real-world data.
- **Learning-based Model Compression** We introduce a novel learning-based approach to obtain a *losslessly compressible model*, for a high compression ratio without sacrificing its performance. An exact low-rank constraint is enforced on the targeted weight matrix.
Fig. \[fig:show\] shows successful VFR examples using DeepFont. In (a)(b), given the real-world query images, top-5 font recognition results are listed, within which the ground truth font classes are marked out[^1]. More real-world examples are displayed in (c). Although accompanied with high levels of background clutters, size and ratio variations, as well as perspective distortions, they are all correctly recognized by the DeepFont system.
Dataset name Source Label? Purpose Size Class
------------------- -------- -------- --------- ------------ --------
VFRWild325 [@LFE] Real Y Test 325 93
VFR\_real\_test Real Y Test 4, 384 617
VFR\_real\_u Real N Train 197, 396 /
VFR\_syn\_train Syn Y Train 2,383, 000 2, 383
VFR\_syn\_val Syn Y Test 238, 300 2, 383
: Comparison of All VFR Datasets[]{data-label="data"}
Dataset
=======
Domain Mismatch between Synthetic and Real-World Data
-----------------------------------------------------
To apply machine learning to VFR problem, we require realistic text images with ground truth font labels. However, such data is scarce and expensive to obtain. Moreover, the training data requirement is vast, since there are hundreds of thousands of fonts in use for Roman characters alone. One way to overcome the training data challenge is to synthesize the training set by rendering text fragments for all the necessary fonts. However, to attain effective recognition models with this strategy, we must face the domain mismatch between synthetic and real-world text images [@LFE]. For example, it is common for designers to edit the spacing, aspect ratio or alignment of text arbitrarily, to make the text fit other design components. The result is that characters in real-world images are spaced, stretched and distorted in numerous ways. For example, Fig. \[fig:per\] (a) and (b) depict typical examples of *character spacing* and *aspect ratio* differences between (standard rendered) synthetic and real-world images. Other perturbations, such as background clutter, perspective distortion, noise, and blur, are also ubiquitous.
The AdobeVFR Dataset
--------------------
Collecting and labeling real-world examples is notoriously hard and thus a labeled real-world dataset has been absent for long. A small dataset *VFRWild325* was collected in [@LFE], consisting of 325 real-world text images and 93 classes. However, the small size puts its effectiveness in jeopardy.
Chen et. al. in [@LFE] selected 2,420 font classes to work on. We remove some script classes, ending up with a total of 2,383 font classes. We collected 201,780 text images from various typography forums, where people post these images seeking help from experts to identify the fonts. Most of them come with hand-annotated font labels which may be inaccurate. Unfortunately, only a very small portion of them fall into our list of 2,383 fonts. All images are first converted into gray scale. Those images with our target class labels are then selected and inspected by independent experts if their labels are correct. Images with verified labels are then manually cropped with tight bounding boxes and normalized proportionally in size, to be with the identical height of 105 pixels. Finally, we obtain 4,384 real-world test images with reliable labels, covering 617 classes (out of 2,383). Compared to the synthetic data, these images typically have much larger appearance variations caused by scaling, background clutter, lighting, noise, perspective distortions, and compression artifacts. Removing the 4,384 labeled images from the full set, we are left with 197,396 unlabeled real-world images which we denote as VFR\_real\_u.
To create a sufficiently large set of synthetic training data, we follow the same way in [@LFE] to render long English words sampled from a large corpus, and generate tightly cropped, gray-scale, and size-normalized text images. For each class, we assign 1,000 images for training, and 100 for validation, which are denoted as VFR\_syn\_train and VFR\_syn\_val, respectively. The entire **AdobeVFR dataset**, consisting of VFR\_real \_test, VFR\_real\_u, VFR\_syn\_train and VFR\_syn\_val, are made publicly available[^2]. The AdobeVFR dataset is the first large-scale benchmark set consisting of both synthetic and real-world text images, for the task of font recognition. To our best knowledge, so far VFR\_real\_test is the largest available set of real-world text images with reliable font label information (12.5 times larger than VFRWild325). The AdobeVFR dataset is super fine-grain, with highly subtle categorical variations, leading itself to a new challenging dataset for object recognition. Moreover, the substantial mismatch between synthetic and real-world data makes the AdobeVFR dataset an ideal subject for general domain adaption and transfer learning research. It also promotes the new problem area of understanding design styles with deep learning.
Synthetic Data Augmentation: A First Step to Reduce the Mismatch
----------------------------------------------------------------
Before feeding synthetic data into model training, it is popular to artificially augment training data using label-preserving transformations to reduce overfitting. In [@imagenet], the authors applied image translations and horizontal reflections to the training images, as well as altering the intensities of their RGB channels. The authors in [@LFE] added moderate distortions and corruptions to the synthetic text images:
- **1. Noise:** a small Gaussian noise with zero mean and standard deviation 3 is added to input
- **2. Blur:** a random Gaussian blur with standard deviation from 2.5 to 3.5 is added to input
- **3. Perspective Rotation:** a randomly-parameterized affine transformation is added to input
- **4. Shading:** the input background is filled with a gradient in illumination.
The above augmentations cover standard perturbations for general images, and are adopted by us. However, as a very particular type of images, text images have various real-world appearances caused by specific handlings. Based on the observations in Fig. \[fig:per\] , we identify two additional font-specific augmentation steps to our training data:
- **5. Variable Character Spacing:** when rendering each synthetic image, we set the character spacing (by pixel) to be a Gaussian random variable of mean 10 and standard deviation 40, bounded by \[0, 50\].
- **6. Variable Aspect Ratio:** Before cropping each image into a input patch, the image, with heigh fixed, is squeezed in width by a random ratio, drawn from a uniform distribution between $\frac{5}{6}$ and $\frac{7}{6}$.
Note that these steps are not useful for the method in [@LFE] because it exploits very localized features. However, as we show in our experiments, these steps lead to significant performance improvements in our DeepFont system. Overall, our data augmentation includes steps 1-6.
To leave a visual impression, we take the real-world image Fig. \[fig:per\] (a), and synthesize a series of images in Fig. \[fig:pre\], all with the same text but with different data augmentation ways. Specially, (a) is synthesized with no data augmentation; (b) is (a) with standard augmentation 1-4 added; (c) is synthesized with spacing and aspect ratio customized to be identical to those of Fig. \[fig:per\] (a); (d) adds standard augmentation 1-4 to (c). We input images (a)-(d) through the trained DeepFont model. For each image, we compare its layer-wise activations with those of the real image Fig. \[fig:per\] (a) feeding through the same model, by calculating the normalized MSEs. Fig. \[fig:pre\] (e) shows that those augmentations, especially the spacing and aspect ratio changes, reduce the gap between the feature hierarchies of real-world and synthetic data to a large extent. A few synthetic patches after full data augmentation 1-6 are displayed in Fig. \[fig:patch\]. It is observable that they possess a much more visually similar appearance to real-world data.
Domain Adapted CNN
==================
Domain Adaptation by CNN Decomposition and SCAE
-----------------------------------------------
Despite that data augmentations are helpful to reduce the domain mismatch, enumerating all possible real-world degradations is impossible, and may further introduce degradation bias in training. In the section, we propose a learning framework to leverage both synthetic and real-world data, using multi-layer CNN decomposition and SCAE-based domain adaptation. Our approach extends the domain adaptation method in [@sentiment] to extract low-level features that represent both the synthetic and real-world data. We employs a Convolutional Neural Network (CNN) architecture, which is further decomposed into two sub-networks: a “shared” low-level sub-network which is learned from the composite set of synthetic and real-world data, and a high-level sub-network that learns a deep classifier from the low-level features.
The basic CNN architecture is similar to the popular ImageNet structure [@imagenet], as in Fig. \[fig:CNN\]. The numbers along with the network pipeline specify the dimensions of outputs of corresponding layers. The input is a $105 \times 105$ patch sampled from a “normalized” image. Since a square window may not capture sufficient discriminative local structures, and is unlikely to catch high-level combinational features when two or more graphemes or letters are joined as a single glyph (e.g., ligatures), we introduce a **squeezing** operation [^3], that scales the width of the height-normalized image to be of a constant ratio relative to the height (2.5 in all our experiments). Note that the squeezing operation is equivalent to producing “long” rectangular input patches.
When the CNN model is trained fully on a synthetic dataset, it witnesses a significant performance drop when testing on real-world data, compared to when applied to another synthetic validation set. This also happens with other models such as in [@LFE], which uses training and testing sets of similar properties to ours. It alludes to discrepancies between the distributions of synthetic and real-world examples. we propose to decompose the $N$ CNN layers into two sub-networks to be learned sequentially:
- **Unsupervised cross-domain sub-network $\mathbf{C_u}$**, which consists of the first $K$ layers of CNN. It accounts for extracting low-level visual features shared by both synthetic and real-world data domains. $\mathbf{C_u}$ will be trained in a unsupervised way, using unlabeled data from both domains. It constitutes the crucial step that further minimizes the low-level feature gap, beyond the previous data augmentation efforts.
- **Supervised domain-specific sub-network $\mathbf{C_s}$**, which consists of the remaining $N-K$ layers. It accounts for learning higher-level discriminative features for classification, based on the shared features from $\mathbf{C_u}$. $\mathbf{C_s}$ will be trained in a supervised way, using labeled data from the synthetic domain only.
We show an example of the proposed CNN decomposition in Fig. \[fig:CNN\]. The $\mathbf{C_u}$ and $\mathbf{C_s}$ parts are marked by red and green colors, respectively, with $N=8$ and $K=2$. Note that the low-level shared features are implied to be independent of class labels. Therefore in order to address the open-ended problem of font classes, one may keep re-using the $\mathbf{C_u}$ sub-network, and only re-train the $\mathbf{C_s}$ part.
**Learning $\mathbf{C_u}$ from SCAE** Representative unsupervised feature learning methods, such as the Auto-Encoder and the Denoising Auto-Encoder, perform a greedy layer-wise pre-training of weights using unlabeled data alone followed by supervised fine-tuning ([@semi3]). However, they rely mostly on fully-connected models and ignore the 2D image structure. In [@SCAE], a Convolutional Auto-Encoder (CAE) was proposed to learn non-trivial features using a hierarchical unsupervised feature extractor that scales well to high-dimensional inputs. The CAE architecture is intuitively similar to the the conventional auto-encoders in [@AE], except for that their weights are shared among all locations in the input, preserving spatial locality. CAEs can be stacked to form a deep hierarchy called the Stacked Convolutional Auto-Encoder (SCAE), where each layer receives its input from a latent representation of the layer below. Fig. \[fig:CAE\] plots the SCAE architecture for our $K$ = 2 case.
**Training Details** We first train the SCAE on both synthetic and real-world data in a unsupervised way, with a learning rate of 0.01 (we do not anneal it through training). Mean Squared Error (MSE) is used as the loss function. After SCAE is learned, its Conv. Layers 1 and 2 are imported to the CNN in Fig. \[fig:CNN\], as the $\mathbf{C_u}$ sub-network and fixed. The $\mathbf{C_s}$ sub-network, based on the output by $\mathbf{C_u}$, is then trained in a supervised manner. We start with the learning rate at 0.01, and follow a common heuristic to manually divide the learning rate by 10 when the validation error rate stops decreasing with the current rate. The “dropout” technique is applied to fc6 and fc7 layers during training. Both $\mathbf{C_u}$ and $\mathbf{C_s}$ are trained with a default batch size of 128, momentum of 0.9 and weight decay of 0.0005. The network training is implemented using the CUDA ConvNet package [@imagenet], and runs on a workstation with 12 Intel Xeon 2.67GHz CPUs and 1 GTX680 GPU. It takes around 1 day to complete the entire training pipeline.
**Testing Details** We adopt multi-scale multi-view testing to improve the result robustness. For each test image, it is first normalized to 105 pixels in height, but squeezed in width by three different random ratios, all drawn from a uniform distribution between 1.5 and 3.5, matching the effects of squeezing and variable aspect ratio operations during training. Under each squeezed scale, five $105 \times 105$ patches are sampled at different random locations. That constitutes in total fifteen test patches, each of which comes with different aspect ratios and views, from one test image. As every single patch could produce a softmax vector through the trained CNN, we average all fifteen softmax vectors to determine the final classification result of the test image.
Connections to Previous Work
----------------------------
We are not the first to look into an essentially “hierarchical” deep architecture for domain adaption. In [@Ng], the proposed transfer learning approach relies on the unsupervised learning of representations. Bengio et. al hypothesized in [@Bengio09] that more levels of representation can give rise to more abstract, more general features of the raw input, and that the lower layers of the predictor constitute a hierarchy of features that can be **shared** across variants of the input distribution. The authors in [@sentiment] used data from the union of all domains to learn their shared features, which is different from many previous domain adaptation methods that focus on learning features in a unsupervised way from the target domain only. However, their entire network hierarchy is learned in a unsupervised fashion, except for a simple linear classier trained on top of the network, i.e., $K=N-1$. In [@Ngtext], the CNN learned a set of filters from raw images as the first layer, and those low-level filters are fixed when training higher layers of the same CNN, i.e., $K=1$. In other words, they either adopt a simple feature extractor ($K=1$), or apply a shallow classifier ($K=N-1$). Our CNN decomposition is different from prior work in that:
- Our feature extractor $\mathbf{C_u}$ and classier $\mathbf{C_s}$ are both deep sub-networks with more than one layer (both $K$ and $N-K$ are larger than 1), which means that both are able to perform more sophisticated learning. More evaluations can be found in Section 5.2.
- We learn “shared-feature” convolutional filters rather than fully-connected networks such as in [@sentiment], the former of which is more suitable for visual feature extractions.
The domain mismatch between synthetic and real-world data on the lower-level statistics can occur in more scenarios, such as real-world face recognition from rendered images or sketches, recognizing characters in real scenes with synthetic training, human pose estimation with synthetic images generated from 3D human body models. We conjecture that our framework can be applicable to those scenarios as well, where labeled real-world data is scarce but synthetic data can be easily rendered.
Learning-based Model Compression
================================
The architecture in Fig. \[fig:CNN\] contains a huge number of parameters. It is widely known that the deep models are heavily over-parameterized [@denil2013predicting] and thus those parameters can be compressed to reduce storage by exploring their structure. For a typical CNN, about 90% of the storage is taken up by the dense connected layers, which shall be our focus for mode compression.
One way to shrink the number of parameters is using matrix factorization [@denton2014exploiting]. Given the parameter $W \in R^{m \times n}$, we factorize it using singular-value decomposition (SVD): $$\begin{array}{l}\label{SVD}
W=USV^T
\end{array}$$ where $U \in R^{m \times m}$ and $V \in R^{n \times n}$ are two dense orthogonal matrices and $S \in R^{m \times n}$ is a diagonal matrix. To restore an approximate $W$ , we can utilize $\widetilde{U}$, $\widetilde{V}$ and $\widetilde{S}$, which denote the submatrices corresponding to the top $k$ singular vectors in $U$ and $V$ along with the top $k$ eigenvalue in $S$: $$\begin{array}{l}\label{SVDapprox}
\widetilde{W}=\widetilde{U}\widetilde{S}\widetilde{V}^T
\end{array}$$ The compression ratio given $m$, $n$, and $k$ is $\frac{k(m + n + 1)}{mn}$, which is very promising when $m, n \gg k$. However, the approximation of SVD is controlled by the decay along the eigenvalues in $S$. Even it is verified in Fig. \[rank\] that eigenvalues of weight matrices usually decay fast (the 6-th largest eigenvalue is already less than 10% of the largest one in magnitude), the truncation inevitably leads to information loss, and potential performance degradations, compared to the uncompressed model.
Instead of first training a model then lossy-compressing its parameters, we propose to directly learn a **losslessly compressible** model (the term “lossless” is referred as there is no further loss after a model is trained). Assuming the parameter matrix $W$ of a certain network layer, our goal is to make sure that its rank is **exactly no more than a small constant $k$**. In terms of implementation, in each iteration, an extra hard thresholding operation [@ADM] is executed on $W$ after it is updated by a conventional back propagation step: $$\begin{array}{l}\label{hard}
W_k=U\mathcal{T}_k(S)V^T
\end{array}$$ where $\mathcal{T}_k$ will keep the largest $k$ eigenvalues in $S$ while setting others to zeros. $W_k$ is best rank-$k$ approximation of $W$, as similarly in (\[SVDapprox\]). However, different from (\[SVDapprox\]), the proposed method incorporates low-rank approximation into model training and jointly optimize them as a whole, guaranteeing a rank-$k$ weight matrix that is ready to be compressed losslessly by applying (\[SVD\]). Note there are other alternatives, such as vector quantization methods [@gong2014compressing], that have been applied to compressing deep models with appealing performances. We will investigate utilizing them together to further compress our model in the future.
Experiments
===========
Analysis of Domain Mismatch
---------------------------
We first analyze the domain mismatch between synthetic and real-world data, and examine how our synthetic data augmentation can help. First we define five dataset variations generated from VFR\_syn\_train and VFR\_real\_u. These are denoted by the letters N, S, F, R and FR and are explained in Table \[five\].
--------- -------------------------------------------------------------- ------- ------ -------
Methods Training Data Train
N R
SCAE N [**N**]{}: VFR\_syn\_train, no data augmentation 0.02 3.54 31.28
SCAE S [**S**]{}: VFR\_syn\_train, standard augmentation 1-4 0.21 2.24 19.34
SCAE F [**F**]{}: VFR\_syn\_train, full augmentation 1-6 1.20 1.67 15.26
SCAE R [**R**]{}:VFR\_real\_u, real unlabeled dataset 9.64 5.73 10.87
SCAE FR [**FR**]{}: Combination of data from [**F**]{} and [**R**]{} 6.52 2.02 14.01
--------- -------------------------------------------------------------- ------- ------ -------
We train five separate SCAEs, all of the same architecture as in Fig. \[fig:CAE\], using the above five training data variants. The training and testing errors are all measured by relative MSEs (normalized by the total energy) and compared in Table 1. The testing errors are evaluated on both the unaugmented synthetic dataset N and the real-world dataset R. Ideally, the better the SCAE captures the features from a domain, the smaller the reconstruction error will be on that domain.
As revealed by the training errors, real-world data contains rich visual variations and is more difficult to fit. The sharp performance drop from N to R of SCAE N indicates that the convolutional features for synthetic and real data are quite different. This gap is reduced in SCAE S, and further in SCAE F, which validates the effectiveness of adding font-specific data augmentation steps. SCAE R fits the real-world data best, at the expense of a larger error on N. SCAE FR achieves an overall best reconstruction performance of both synthetic and real-world images.
Fig. \[areal\] shows an example patch from a real-world font image of highly textured characters, and its reconstruction outputs from all five models. The gradual visual variations across the results confirm the existence of a mismatch between synthetic and real-world data, and verify the benefit of data augmentation as well as learning shared features.
\
Analysis of Network Structure
-----------------------------
**Fixing Network Depth $N$.** Given a fixed network complexity ($N$ layers), one may ask about how to best decompose the hierarchy to maximize the overall classification performance on real-world data. Intuitively, we should have sufficient layers of lower-level feature extractors as well as enough subsequent layers for good classification of labeled data. Thus, the depth $K$ of $C_u$ should neither be too small nor too large.
K 0 1 2 3 4 5
----------------- ------- ------- ------- ------- ------- -------
Train 8.46 9.88 11.23 12.54 15.21 17.88
VFR\_real\_test 20.72 20.31 18.21 18.96 22.52 25.97
: Top-5 Testing Errors (%) for Different CNN Decompositions (Varying $K$, $N$ = 8)[]{data-label="decom"}
Table \[decom\] shows that while the *classification* training error increases with $K$, the testing error does not vary monotonically. The best performance is obtained with $K$ = 2 (3 slightly worse), where smaller or larger values of $K$ give substantially worse performance. When $K$ = 5, all layers are learned using SCAE, leading to the worst results. Rather than learning all hidden layers by unsupervised training, as suggested in [@sentiment] and other DL-based transfer learning work, our CNN decomposition reaches its optimal performance when higher-layer convolutional filters are still trained by supervised data. A visual inspection of reconstruction results of a real-world example in Fig. \[fig:patch0\], using SCAE FR with different $K$ values, shows that a larger $K$ causes less information loss during feature extraction and leads to a better reconstruction. But in the meantime, the classification result may turn worse since noise and irrelevant high frequency details (e.g. textures) might hamper recognition performance. The optimal $K$ =2 corresponds to a proper “content-aware” smoothening, filtering out “noisy” details while keeping recognizable structural properties of the font style.
**Fixing $C_s$ or $C_u$ Depth.** We investigate the influences of $K$ (the depth of $C_u$) when the depth of $C_s$ (e.g. $N-K$) keeps fixed. Table \[decomK\] reveals that a deeper $C_u$ contributes little to the results. Similar trends are observed when we fix $K$ and adjust $N$ (and thus the depth of$C_s$). Therefore, we choose $N$= 8, $K$=2 to be the default setting.
\[decomK\]
K 1 2 3 4
----------------- ------- ------- ------- -------
Train 11.46 11.23 10.84 10.86
VFR\_real\_test 21.58 18.21 18.15 18.24
: Top-5 Testing Errors (%) for Different CNN Decompositions (Varying $K$, $N$ = $K$ + 6)
Recognition Performances on VFR Datasets
----------------------------------------
We implemented and evaluated the local feature embedding-based algorithm (LFE) in [@LFE] as a baseline, and include the four different DeepFont models as specified in Table \[full\]. The first two models are trained in a fully supervised manner on $\mathbf{F}$, without any decomposition applied. For each of the later two models, its corresponding SCAE (SCAE FR for DeepFont CAE\_FR, and SCAE R for DeepFont CAE\_R) is first trained and then exports the first two convolutional layers to $\mathbf{C_u}$. All trained models are evaluated in term of top-1 and top-5 classification errors, on the VFR\_syn\_val dataset for validation purpose. Benefiting from large learning capacity, it is clear that DeepFont models fit synthetic data significantly better than LFE. Notably, the top-5 errors of all DeepFont models (except for DeepFont CAE\_R) reach zero on the validation set, which is quite impressive for such a fine-grain classification task.
We then compare DeepFont models with LFE on the original VFRWild325 dataset in [@LFE]. As seen from Table \[full\], while DeepFont S fits synthetic training data best, its performance is the poorest on real-world data, showing a severe overfitting. With two font-specific data augmentations added in training, the DeepFont F model adapts better to real-world data, outperforming LFE by roughly 8% in top-5 error. An additional gain of 2% is obtained when unlabeled real-world data is utilized in DeepFont CAE\_FR. Next, the DeepFont models are evaluated on the new VFR\_real\_test dataset, which is more extensive in size and class coverage. A large margin of around 5% in top-1 error is gained by DeepFont CAE\_FR model over the second best (DeepFont F), with its top-5 error as low as 18.21%. We will use DeepFont CAE\_FR as the default DeepFont model.
Although SCAE R has the best reconstruction result on real-world data on which it is trained, it has large training and testing errors on synthetic data. Since our supervised training relies fully on synthetic data, an effective feature extraction for synthetic data is also indispensable. The error rates of DeepFont CAE\_R are also worse than those of DeepFont CAE\_FR and even DeepFont F on the real-world data, due to the large mismatch between the low-level and high-level layers in the CNN.
Another interesting observation is that all methods get similar top-5 errors on VFRWild325 and VFR\_real\_test, showing their statistical similarity. However, the top-1 errors of DeepFont models on VFRWild325 are significantly higher than those on VFR\_real\_test, with a difference of up to 10%. In contrast, the top-1 error of LFE rises more than 13% on VFR\_real\_test than on VFRWild325. For the small VFRWild325, the recognition result is easily affected by “bad" examples (e.g, low resolution or highly compressed images) and class bias (less than 4% of all classes are covered). On the other hand, the larger VFR\_real\_test dataset dilutes the possible effect of outliers, and examines a lot more classes.
------------------ ---------------- ---------------- ------- ------- ------- --------------- --------------- --------------- ---------------
Methods Training
$\mathbf{C_u}$ $\mathbf{C_s}$ Error Top-1 Top-5 Top-1 Top-5 Top-1 Top-5
LFE / / / 26.50 6.55 44.13 30.25 57.44 32.69
DeepFont S / F 0.84 1.03 0 64.60 57.23 57.51 50.76
DeepFont F / F 8.46 7.40 0 43.10 22.47 33.30 20.72
DeepFont CAE\_FR FR F 11.23 6.58 0 [**38.15**]{} [**20.62**]{} [**28.58**]{} [**18.21**]{}
DeepFont CAE\_R R F 13.67 8.21 1.26 44.62 29.23 39.46 27.33
------------------ ---------------- ---------------- ------- ------- ------- --------------- --------------- --------------- ---------------
Fig. \[fig:fail\] lists some failure cases of DeepFont. For example, the top left image contains extra “fluff” decorations along text boundaries, which is nonexistent in the original fonts, that makes the algorithm incorrectly map it to some “artistic” fonts. Others are affected by 3-D effects, strong obstacles in foreground, and in background. Being considerably difficult to be adapted, those examples fail mostly because there are neither specific augmentation steps handling their effects, nor enough examples in VFR\_real\_u to extract corresponding robust features.
Evaluating Font Similarity using DeepFont
-----------------------------------------
There are a variety of font selection tasks with different goals and requirements. One designer may wish to match a font to the style of a particular image. Another may wish to find a free font which looks similar to a commercial font such as Helvetica. A third may simply be exploring a large set of fonts such as Adobe TypeKit or Google Web Fonts. Exhaustively exploring the entire space of fonts using an alphabetical listing is unrealistic for most users. The authors in [@exploratory] proposed to select fonts based on online crowdsourced attributes, and explore *font similarity*, from which a user is enabled to explore other visually similar fonts given a specific font. The font similarity measure is very helpful for font selection, organization, browsing, and suggestion.
Based on our DeepFont system, we are able to build up measures of font similarity. We use the $4096 \times 1$ outputs of the fc7 layer as the high-level feature vectors describing font visual appearances. We then extract such features from all samples in VFR\_syn\_val Dataset, obtaining 100 feature vectors per class. Next for each class, the 100 feature vectors is averaged to a representative vector. Finally, we calculate the Euclidean distance between the representative vectors of two font classes as their similarity measure. Visualized examples are demonstrated in Fig. \[fig:graph\]. For each example, the top is the query image of a known font class; the most similar fonts obtained by the font similarity measures are sorted below. Note that although the result fonts can belong to different font families from the query, they share identifiable visual similarities by human perception.
Although not numerically verified as in [@exploratory], the DeepFont results are qualitatively better when we look at the top-10 most similar fonts for a wide range of query fonts. The authors of [@exploratory] agree per personal communication with us.
DeepFont Model Compression
--------------------------
Since the fc6 layer takes up 85% of the total model size, we first focus on its compression. We start from a well-trained DeepFont model (DeepFont CAE\_FR), and continue tuning it with the hard thresholding (\[hard\]) applied to the fc6 parameter matrix $W$ in each iteration, until the training/validation errors reach the plateau again.
Table \[compress\] compares the DeepFont models compressed using conventional matrix factorization (denoted as the “lossy” method), and the proposed learning based method (denoted as the “lossless” method), under different compression ratios (fc6 and total size counted by parameter numbers). The last column of Table \[compress\] lists the top-5 testing errors (%) on VFR\_real\_test. We observe a consistent margin of the “lossless” method over its “lossy” counterpart, which becomes more significant when the compression ratio goes low (more than 1% when $k$ = 5). Notably, when $k$ = 100, the proposed “lossless” compression suffers no visible performance loss, while still maintaining a good compression ratio of 5.79.
\[compress\]
fc6 size Total size Ratio **Method** Error
--------- ---------- ------------ ------- ------------ -------
default NA NA 18.21
Lossy 20.67
Lossless 19.23
Lossy 19.25
Lossless 18.87
Lossy 19.04
Lossless 18.67
Lossy 18.68
Lossless 18.21
: Performance Comparisons of Lossy and Lossless Compression Approaches
In practice, it takes around 700 megabytes to store all the parameters in our uncompressed DeepFont model, which is quite huge to be embedded or downloaded into most customer softwares. More aggressively, we reduce the output sizes of both fc6 and fc7 to 2048, and further apply the proposed compression method ($k$ = 10) to the fc6 parameter matrix. The obtained “mini” model, with only 9, 477, 066 parameters and a high compression ratio of 18.73, becomes less than 40 megabytes in storage. Being portable even on mobiles, It manages to keep a top-5 error rate around 22%.
Conclusion
==========
In the paper, we develop the DeepFont system to remarkably advance the state-of-the-art in the VFR task. A large set of labeled real-world data as well as a large corpus of unlabeled real-world images is collected for both training and testing, which is the first of its kind and will be made publicly available soon. While relying on the learning capacity of CNN, we need to combat the mismatch between available training and testing data. The introduction of SCAE-based domain adaption helps our trained model achieve a higher than 80% top-5 accuracy. A novel lossless model compression is further applied to promote the model storage efficiency. The DeepFont system not only is effective for font recognition, but can also produce a font similarity measure for font selection and suggestion.
[^1]: Note that the texts are input manually for rendering purposes only. The font recognition process does not need any content information.
[^2]: http://www.atlaswang.com/deepfont.html
[^3]: Note squeezing is independent from the variable aspect ratio operation introduced in Section 2.3, as they are for different purposes.
|
---
abstract: 'We consider two quasi-exactly solvable problems in one dimension for which the Schr$\ddot{o}$dinger equation can be converted to Heun’s equation. We show that in neither case the Bender-Dunne polynomials form an orthogonal set. Using the anti-isopectral transformation we also discover a new quasi-exactly solvable problem and show that even in this case the polynomials do not form an orthogonal set.'
address: |
Institute of Physics, Sachivalaya Marg,\
Bhubaneswar-751005, India,\
Email: khare, [email protected]
author:
- Avinash Khare and Bhabani Prasad Mandal
title: ' DO QUASI-EXACTLY SOLVABLE SYSTEMS ALWAYS CORRESPOND TO ORTHOGONAL POLYNOMIALS? '
---
Some time ago, in a remarkable paper Bender and Dunne [@bd] showed that the eigenfunctions of the Schr$\ddot{o}$dinger equation for a quasi-exactly solvable ( QES ) problem is the generating function for a set of orthogonal polynomials $\{P_n(E)\}$ in the energy variable $E$. It was further shown that these polynomials satisfy the three-term recursion relation $$P_n(E) = EP_{n-1}(E) +C_nP_{n-2}(E)
\label{r1}$$ where $C_n$ is $E$ independent quantity. Using the well known theorem [@the; @ext], “ the [*necessary and sufficient*]{} condition for a family of polynomials $\{P_n\}$ (with degree $P_n= n $) to form an orthogonal polynomial system is that $\{P_n\}$ satisfy a three-term recursion relation of the form $$P_n(E) = \left ( A_n E +B_n \right ) P_{n-1}(E) + C_n P_{n-2}(E) \ \ \ \ \ n\ge 1
\label{r2}$$ where the coefficients $A_n , B_n $ and $C_n$ are independent of $E$, $A_n \neq 0 , C_1 =0 , C_n \neq 0 $ for $n\ge 1$" , it then followed that $\{P_n(E)\}$ for this problem forms an orthogonal set of polynomials with respect to some weight function, $w(E)$. Recently several authors have studied the Bender-Dunne polynomials in detail [@ext; @two; @two1; @bd1]. In fact it has been claimed that the Bender-Dunne construction is quite universal and valid for any quasi-exactly solvable model in both one as well as multi-dimensions.
The purpose of this note is to critically examine this assertion. In particular, we discuss two QES systems, one of which is in one dimension while the other one is an effective one dimensional system. We show in both the cases that though these problems are QES problems, the corresponding three-term recursion relation is not of the type as given in Eq. (\[r2\]), and hence the corresponding polynomials do not form an orthogonal set. Using the anti-isospectral transformation $x\rightarrow ix$ introduced recently by Krajewska [*et al.*]{}[@two1], we discover a new QES problem and show that even in this case the polynomials do not form an orthogonal set.
Consider the potential, $$V(x) = \frac{ \mu^2 \left [ 8\sinh^4 \frac{ \mu x}{2}-4(\frac{ 5}
{\epsilon ^2}-1)\sinh^2 \frac{ \mu x}{2}+2 \left (\frac{ 1}{\epsilon ^4}
-\frac{ 1}{\epsilon ^2}-2 \right ) \right ]}{ 8 \left [1+\frac{ 1}{\epsilon ^2}+\sinh^2
\frac{ \mu x}{2} \right ]^2}
\label{p1}$$ which arises in the context of the stability analysis around the kink solution for $\phi^6 $-type field theory in $1+1$ dimensions [@lee]. It has already been shown that it is an example of QES system in which both the ground and the second excited states are exactly known in the case $\epsilon ^2 =\frac{ 1}{2}$. Further it has also been shown that contrary to the usual belief [@book], this QES problem can not be generated by the quadratic elements of an enveloping $Sl(2)$ algebra [@jat]. A related fact is that the Schr$\ddot{o}$dinger equation for this case, after some transformations, can be written in the form of Heun’s equation which has four regular singular points [@lee]. We shall now show that in this case one can obtain a three-term recursion relation for the polynomials $\{P_n(E)\}$ which is different from the one given in Eq. (\[r2\]) and hence these polynomials do [*not*]{} form an orthogonal set even though it is a QES system.
Consider the Schr$\ddot{o}$dinger equation ( $\hbar = 2m =1$) $$\left [ -\frac{ d^2}{dx^2} + V(x) \right ]\psi(x) = E\psi(x)
\label{scw}$$ On making use of the transformation $$y =\frac{ \sinh^2 \frac{ \mu x}{2}}{\left (1+\frac{ 1}{\epsilon ^2 } + \sinh^2 \frac{ \mu x}{2} \right
) },\ \ \ \psi = (1-y)^s f , \ \ \ s=\left (1-\frac{ E}{\mu^2} \right )^{\frac{ 1}{2}}$$ the Schr$\ddot{o}$dinger equation reduces to the Heun’s equation, $$f^{\prime\prime}(y)+ \left [ \frac{ 1}{2y} + \frac{ 1+2s}{y-1} + \frac{ 1}{
2(y+\epsilon ^2)} \right ]f^{\prime}(y) + \frac{ ( \alpha\beta y
-q)}{y(y-1)(y+\epsilon ^2)}f(y)=0
\label{heu}$$ Here $$\alpha = - \frac{ 5}{2}-s; \ \ \beta = \frac{ 3}{2} -s ; \ \ q= (1-s^2)(1+\epsilon ^2)-
\frac{ 1}{2}s \epsilon ^2- \frac{ 1}{4}(1-2 \epsilon ^2).$$
On further substituting $$t = \left (y+\frac{ 1}{2} \right )^{\frac{ 1}{2}}$$ it is easy to show that $f(t)$ satisfies $$\begin{aligned}
(t^2-\epsilon ^2)(t^2-1-\epsilon ^2)f^{\prime\prime}(t) &&+ \left [
(t^2-1-\epsilon ^2) + 2(1+2s)(t^2-\epsilon ^2)\right
]tf^\prime(t)\hspace{.3in} \nonumber \\
&&+\left [4\alpha \beta t^2 -4(\alpha \beta \epsilon ^2+q) \right ] f(t) =0
\label{heu1}\end{aligned}$$ From this equation one can easily obtain the three-term recursion relation satisfied by the polynomials associated with this QES system. In particular, on substituting $$f(t) = \sum_{n=0}^{\infty} \frac{ Q_n(s) t^n}{n!}
\label{13}$$ in Eq. (\[heu1\]) it is easily shown that $Q_n$’s satisfy the three-term recursion relation $$\begin{aligned}
\epsilon ^2(\epsilon ^2+1)Q_{n+2}(s) &&- \left [(2 \epsilon ^2+1
)n^2 +(5 \epsilon ^2+2+4s \epsilon ^2)n -4s^2 +2s \epsilon ^2+3 -9 \epsilon ^2\right ]Q_n(s)
\nonumber\\&&+
n(n-1) \left [ n^2+n(4s-2)
+4s^2-4s -15 \right ]Q_{n-2}(s) =0\end{aligned}$$
Notice that the even and the odd $Q_n$’s are unrelated and we can obtain two separate recursion relations for them. In particular it is easily shown that for even and odd cases, the recursion relations respectively are ($m\ge 1$) $$\begin{aligned}
\epsilon ^2(\epsilon ^2+1)P_m(s) -&&\left [(8 \epsilon ^2+4)m^2
+(8s \epsilon ^2-6 \epsilon ^2-4)m -4s^2 -6s \epsilon ^2 +3- 11\epsilon ^2 \right
]P_{m-1}(s) \nonumber \\+&&(m-1)(2m-3)
\left [8m^2+(16s-24)m+8s^2-24s-14 \right ]P_{m-2}(s) =0
\label{rc1}\end{aligned}$$ and $$\begin{aligned}
\epsilon ^2(\epsilon ^2+1)P_m(s)-\left [(8 \epsilon ^2+4)m^2 +(8s \epsilon ^2
+2 \epsilon ^2)m -4s^2 -2s \epsilon ^2+ 2-12 \epsilon ^2 \right ]P_{m-1}(s)
+\nonumber \\ (m-1)(2m-1)
\left [8m^2+(16s-16)m+8s^2-16s-24 \right ]P_{m-2}(s) =0
\label{rc2}\end{aligned}$$ subject to the initial condition, $P_0(s) =1$.
We observe that neither of these three-term recursion relations are of the form as given by Eq. (\[r2\]) and hence we conclude that even though it is a QES problem, the corresponding polynomials $P_m(s)$ are not orthogonal. Note that the QES solution corresponding to the ground state is obtained from Eqs. (\[13\]) and (\[rc2\]) and in this case $s=1
(i.e.\ E=0)$ and $m=2$( irrespective of the value of $\epsilon $), while the solution corresponding to the second excited state is obtained from Eqs. (\[13\]) and (\[rc1\]) when $s=\frac{ 1}{2}
(i.e.\ E=\frac{ 3}{4}\mu^2)$ and $m=3$ only if $\epsilon ^2 =\frac{ 1}{2}$. Thus only for $\epsilon ^2 = \frac{ 1}{2},$ this is a QES problem.
Why is that in this case the polynomials do not form an orthogonal set? While we do not have a definite answer, we suspect that only for those QES systems for which the symmetry group is $Sl(2) $, the corresponding polynomials form an orthonormal set. In this context recall that for the above problem, the Hamiltonian can not be written in terms of the quadratic generators of $Sl(2)$ [@jat]. As a support to our conjecture, we offer another QES problem where also the Schr$\ddot{o}$dinger equation can be reduced to Heun’s equation with four regular singular points and hence the corresponding symmetry group is not $Sl(2)$. In that case also we find the three-term recursion relation satisfied by the polynomials and show that it is not of the type as given by Eq. (\[r2\]) and hence these polynomials also do not form an orthogonal set.
Recently Bhaduri [*et al.*]{} [@bha] have considered a two body problem characterized by $$H= -\frac{ 1}{2}( \vec{\nabla}_1^2 +\vec{\nabla}_2^2 ) + \frac{ 1}{2}
\left [ r_1^2+r_2^2 \right ]+ \frac{ g_1}{2}\left ( \frac{
r_1^2+r_2^2}{X^2} \right )$$ where $X = x_1y_2-x_2y_1$. They were able to solve the problem in hyperspherical coordinates, $ (R, \theta,
\phi, \psi)$. In particular they showed that the angular equation is $$\left ( \Lambda^2 + \frac{ 4g_1}{\sin^2(2 \theta )
} \right ) \Phi = \beta(\beta+2)\Phi \ \ \ \beta\geq 1$$ while the radial equation is of the type of harmonic oscillator in 4-dimensions. $$\frac{ d^2F}{dR^2} +\frac{ 3}{R}\frac{ dF}{dR} + \left ( 2E-R^2-\frac{
\beta(\beta+2)}{R^2} \right ) F = 0$$ Here $\Lambda^2$ is the Laplacian on the sphere $S^3$. On making the ansatz, $$\Phi(\theta, \phi,\psi) = P(x) e^{iq\phi}\ e^{il\psi}$$ where $q $ and $l$ are integers and $$P(x)= |x|^a(1-x)^b(1+x)^c \Theta^{a,b,c}(x)$$ it was shown that $\Theta(x)$ satisfies Heun’s equation, $$\begin{aligned}
(1-x^2)\frac{ d^2\Theta}{dx^2}&&+ 2 \left [ \frac{ a}{x}-(b-c)-(a+b+c+1)x \right ]
\frac{ d\Theta}{dx}\nonumber \\&&+ \left [\frac{ (\beta+1)^2}{4}-
(a+b+c+\frac{ 1}{2})^2+ \frac{ 2a(c-b)}{x}\right ] \Theta(x) =0
\label{heu2}\end{aligned}$$ Here $b= \frac{ |l+q|}{4}, \ \ c= \frac{|l-q|}{4}, \ a(a-1)= g_1 $. In order to obtain the recursion relation satisfied by the polynomials associated with this QES problem we substitute $$\Theta(x) = \sum_{n=0}^\infty \frac{ P_n(\beta) x^n}{n!}$$ in Eq. (\[heu2\]). It is easy to show that $P_n(\beta) $ satisfies the three-term recursion relation $$\begin{aligned}
(n+2a-1) P_n -&&2(b-c)(n-1+a)P_{n-1}\nonumber \\ +&& (n-1) \left [
\frac{ (\beta+1)^2}{4}- \left (a+b+c-\frac{ 3}{2} \right )^2 \right ]P_{n-2}
=0\end{aligned}$$ when $n\geq 1$ and $P_0 =1$. We again observe that the recursion relation is not of the type given by Eq. (\[r2\]) and hence it follows that the polynomials $P_n(\beta)$ do not form an orthogonal set. The QES solutions in this case have already been discussed in Ref. [@bha].
Summarizing, we have seen that there are QES problems in one dimension and effective one dimension where the symmetry group is not $Sl(2)$, the Schr$\ddot{o}$dinger equation get converted to Heun’s equation, and the corresponding polynomials do not form an orthogonal set. It will be interesting to find the symmetry group in these cases.
Before finishing this note we would like to point out a nontrivial application of the anti-isospectral transformation induced recently by Krajewska [*et al.*]{}[@two1]. In particular, applying it we are able to discover a new QES problem in quantum mechanics. Consider the potential given in Eq. \[p1\] and apply the transformation $x\rightarrow i\theta
$. We then obtain a new QES problem which has not been discussed in the literature so far. In particular the ground and second excited state energies of the new (periodic) potential $$V(\theta ) = -\frac{ \mu^2 \left [ 8\sin^4 \frac{ \mu \theta }{2}+4(\frac{ 5}
{\epsilon ^2}-1)\sin^2 \frac{ \mu \theta }{2}+2 \left (\frac{ 1}{\epsilon ^4}
-\frac{ 1}{\epsilon ^2}-2 \right ) \right ]}{ 8 \left [1+\frac{ 1}{\epsilon ^2}-\sin^2
\frac{ \mu \theta }{2} \right ]^2}
\label{p2}$$ are $E_0= - \frac{ 3}{4}\mu^2, E_2 =0$ in case $ \epsilon ^2= \frac{
1}{2}$. The corresponding eigenfunctions can easily be written down. Needless to say in this case also one can obtain a three-term recursion relation and show that it is not of the type as given by Eq. (\[r2\]). Thus this provides a third QES problem where the Bender-Dunne polynomials do not form an orthogonal set. The details will be given elsewhere [@qes2] where we will also discuss other applications of the anti-isospectral transformation.
C. M. Bender and G. V. Dunne, J. Math. Phys. [**37**]{}(1996) 6.
T.S. Chihara, [*An Introduction to Orthogonal Polynomials* ]{} , Gordon and Breach, New York, (1975); A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, [*Higher Transcendental Functions*]{}, Vol.II, McGraw-Hill , New York, (1953). F. Finkel, A. Gonzalez-Lopez And M. A. Rodriguez, J. Math. Phys. [**37**]{}(1996) 3954.
A. Krajewska, A. Ushveridze and Z. Walczak, Mod. Phys. Lett. [**A 12**]{}(1997) 1131.
A. Krajewska, A. Ushveridze and Z. Walczak, Mod. Phys. Lett. [**A 12**]{}(1997) 1225.
C. M. Bender, G. V. Dunne and M. Moshe, Phys. Rev. [**A 55**]{}(1997) 2625.
N. H. Christ and T.D. Lee Phys. Rev. [**D 12**]{}(1975) 1606. A. Ushveridze, [*Quasi-Exactly Solvable Models in Quantum Mechanics*]{}, Inst. of Physics Publishing, Bristol, (1994). D. P. Jatkar, C. Nagaraja Kumar and A. Khare, Phys. Lett. [**A 142**]{}(1989) 200.
R. K. Bhaduri, A. Khare, J. Law, M. V. N. Murthy and D. Sen , J. Phys. A : Math. Gen. [**30**]{}(1997) 2557.
A. Khare and B. P. Mandal ( In preparation).
|
---
abstract: 'The main goal of this work is to present a new model able to deal with potentially misreported continuous time series. The proposed model is able to handle the autocorrelation structure in continuous time series data, which might be partially or totally underreported or overreported. Its performance is illustrated through a comprehensive simulation study considering several autocorrelation structures and two real data applications on human papillomavirus incidence in Girona (Catalunya, Spain) and COVID-19 incidence in the Chinese region of Heilongjiang.'
author:
- 'David Moriña, Amanda Fernández-Fontelo, Alejandra Cabaña, Pedro Puig'
bibliography:
- 'refs.bib'
title: New statistical model for misreported data with application to current public health challenges
---
Introduction {#intro}
============
There is a growing interest in the last years to deal with data that is only partially registered or underreported in the time series literature. This phenomenon is very common in many fields, and has been previously explored by different approaches in epidemiology, social and biomedical research among many other contexts[@Bernard2014; @Arendt2013; @Rosenman2006; @Alfonso2015; @Winkelmann1996]. The sources and underlying mechanisms that cause the underreporting might differ depending on the particular data. Some authors consider a situation where the registry is updated with time and therefore the underreporting issue is mitigated[@Hohle2014]. That leads to temporary underreporting while this work is focused on permanent underreporting, where the registered data are never updated in order to become more accurate. From the methodological point of view, several alternatives have been explored, from Markov chain Monte-Carlo based methods[@Winkelmann1996] to recent discrete time series approaches[@Fernandez-Fontelo2016; @FernandezFontelo2019] and several attempts to estimate the degree of underreporting in several contexts have been done[@Gibbons2014], although there is a lack of models incorporating continuous time series structures and handling underreporting.
One of the fields where the interest in addressing the underreporting issues is higher is the epidemiology of infectious diseases. In the last few years, many approaches to deal with underreported data have been suggested with a growing level of sophistication from the usage of multiplication factors[@Stocks2018] to several Markov-based models[@Azmon2014; @Magal2018] or even spatio-temporal modelling[@Stoner2019]. Even a new R[@RCoreTeam2019] package able to fitting endemic-epidemic models based on approximative maximum likelihood to underreported count data has been recently published[@JohannesBracher2019].
Human papillomavirus (HPV) is one of the most prevalent sexually transmitted infections. It is so common that nearly all sexually active people have it at some point in their lives, according to the information provided by the Centre of Control of Diseases (CDC)[@Dunne2014]. Generally, the infection disappears on its own without inducing any health problem, but in some cases it can produce an abnormal growth of cells on the surface of the cervix that could potentially lead to cervical cancer. HPV infection is also related to several cancers (cervical, vulva, vagina, penis, anus, $\dots$) and other diseases like genital warts (GW). The fact that most cases of HPV infection are asymptomatic causes that public health registries might be potentially underestimating its incidence. The underreporting phenomenon in HPV data from the discrete time series point of view has been recently studied[@Fernandez-Fontelo2016].
There is an enormous global concern around 2019-novel coronavirus (COVID-19) infection in the last few months, leading the World Health Organization (WHO) to declare public health emergency[@Sohrabi2020]. As the symptoms of this infection can be easily confused with those of similar diseases like Middle East Respiratory Syndrome Coronavirus (MERS-CoV) or Severe Acute Respiratory Syndrome Coronavirus (SARS-CoV), its incidence has been notably underreported, especially at the beginning of the outbreak in Wuhan (Hubei province, China) by December 2019.
The characteristics of the proposed model and the methodology to estimate the parameters are detailed in Section *Model definition*. The application of the model on the HPV incidence data in Girona (Catalunya, Spain) in the period 2010-2014 is illustrated in Section *Example: HPV infection incidence* and on the COVID-19 incidence in Heilongjiang (China) is shown in Section *Example: Coronavirus COVID-19 incidence in the region of Heilongjiang*. Finally the results of a thorough simulation study are summarised in Section *Simulation study*.
Model definition {#model}
================
Consider an unobservable process with an AutoRegressive Moving Average ($ARMA(p, r)$) structure defined by $$\begin{array}{ll}\label{eq:ARMA}
X_t = \alpha_1 X_{t-1} + \ldots + \alpha_p X_{t-p} + \theta_1 \epsilon_{t-1} + \ldots + \\
+ \theta_r \epsilon_{t-r} + \epsilon_t,
\end{array}$$ where $\epsilon_t$ is a Gaussian white noise process with $\epsilon_t \sim N(\mu_{\epsilon}, \sigma_{\epsilon}^2)$. In our setting, this process cannot be directly observed, and all we can see is a part of it, expressed as $$\label{eq:model}
Y_t=\left\{
\begin{array}{ll}
X_t \text{ with probability } 1-\omega \\
q \cdot X_t \text{ with probability } \omega
\end{array}
\right.$$ The interpretation of the parameters in Eq. (\[eq:model\]) is straightforward: $q$ is the overall intensity of misreporting (if $0 < q < 1$ the observed process $Y_t$ would be underreported while if $q > 1$ the observed process $Y_t$ would be overreported). The parameter $\omega$ can be interpreted as the overall frequency of misreporting (proportion of misreported observations).
Model properties
----------------
Consider that the unobserved process $X_t$ follows an $ARMA(p, r)$ model as defined in Eq. (\[eq:ARMA\]). As can be seen in *Appendix 1*, the observed process has mean $\mathbb{E}(Y_t) = \frac{\mu_{\epsilon}}{1-\alpha_1 - \ldots - \alpha_p} \cdot \left(1 - \omega + q \cdot \omega \right)$ and variance $\mathbb{V}(Y_t) = \left(\left(\frac{\sigma_{\epsilon}^2 \cdot (1+\theta_1^2+ \ldots + \theta_r^2)}{1-\alpha_1^2- \ldots - \alpha_p^2}\right)+\frac{\mu^2_{\epsilon}}{(1-\alpha_1 - \ldots - \alpha_p)^2}\right) \cdot (1+\omega \cdot (q^2-1))
-\frac{\mu^2_{\epsilon}}{(1-\alpha_1 - \ldots - \alpha_p)^2} \cdot (1-\omega+q \cdot \omega)^2$. The autocorrelation function of the observed process can be written in terms of the features of the hidden process $X_t$ as $$\begin{array}{ll}\label{corr_Y}
\rho_{Y}(k) = \\
=\frac{V(X_t) (1-\omega+q \omega)^2}{(V(X_t)+E(X_t)^2) (1+\omega (q^2-1))-E(X_t)^2 (1-\omega + q \cdot \omega)^2} \cdot \rho_{X}(k) = \\
= c(\alpha_1, \ldots, \alpha_p, \theta_1, \ldots, \theta_r, \mu_{\epsilon}, \sigma^2_{\epsilon}, \omega, q) \cdot \rho_X(k),
\end{array}$$ where $\rho_X$ is the autocorrelation function of the unobserved process $X_t$.
A situation of particular interest is the case $\omega = 1$, meaning that all the observations might be underreported and that a simpler model for $Y_t$ excluding the parameter $\omega$ might be suitable $$Y_t = q \cdot X_t.$$
In this case, however, the observed process $Y_t$ would be a non-identifiable $ARMA(p, r)$ as parameter $q$ could not be estimated on the basis of the methodology described in the following section.
Estimation
----------
The likelihood function of the observed process $Y_t$ is not easily computable but the parameters of the model can be estimated by means of an iterative algorithm based on its marginal distribution, using R packages *mixtools*[@Benaglia2009a] and *forecast*[@Hyndman2008; @Hyndman2018]. The main steps are described in detail below:
1. Following Eq. (\[eq:model\]), the observed process $Y_t$ can be written as $Y_t = (1-Z_t) \cdot X_t + q \cdot Z_t \cdot X_t$, where $Z_t$ is an indicator of the underreported observations, following a Bernoulli distribution with probability of success $\omega$ $(Z_t \sim Bern(\omega))$, its marginal distribution is a mixture of two normal random variables $N \left(\mu, \sigma^2 \right)$ and $N \left(q \cdot \mu, q^2 \cdot \sigma^2 \right)$ respectively, where $\mu = \frac{\mu_{\epsilon}}{1-\alpha_1-\ldots-\alpha_p}$ and $\sigma^2=\frac{\sigma_{\epsilon}^2 \cdot (1+\theta_1^2 + \ldots + \theta_r^2)}{1-\alpha_1^2- \ldots - \alpha_p^2}$. This fact can be used to obtain initial estimates for $q$ and $\omega$. Using the E-M algorithm (specifically on the E-step), the package *mixtools* calculates the posterior probabilities (conditional on the data and the obtained estimates) of each observation to come from one of these two normals.
2. Using the indicator $\hat{Z_t}$ obtained in the previous step, the series is divided in two: One including the underreported observations (treating the non-underreported values as missing data) and another with the non underreported observations (treating the underreported values as missing data). An $ARIMA$ model is fitted to each of these two series and a new $\hat{q}$ is obtained by dividing the fitted means.
3. A mixture of two normals is fitted to the observed series $Y_t$ with mean and standard deviation fixed to the corresponding values obtained from the previous step, and a new $\omega$ is estimated.
4. Steps (ii) and (iii) are repeated until the quadratic distance between two consecutive iterations $(\hat{q}_i-\hat{q}_{i-1})^2+(\hat{\omega}_i-\hat{\omega}_{i-1})^2+\sum_j (\hat{\alpha}_{j_{i}}- \hat{\alpha}_{j_{i-1}})^2 +\\ +\sum_k (\hat{\theta}_{k_{i}}- \hat{\theta}_{k_{i-1}})^2$ is below a fixed tolerance level.
5. Once the parameter estimates are stable according to the previous criterion, the underlying process $X_t$ is reconstructed as $\hat{X_t}=(1-\hat{Z_t}) \cdot Y_t + \frac{1}{\hat{q}} \cdot \hat{Z_t} \cdot Y_t$, and an $ARIMA$ model is fitted to the reconstructed process to obtain $\hat{\alpha}_j$, $j=1, \ldots, p$, $\hat{\theta}_k$, $k=1, \ldots, r$ and $\hat{\sigma_{\epsilon}}^2$.
To account for potential trends or seasonal behaviour, covariates can be included in the described estimation process. Additionally, a parametric bootstrap procedure with 1000 replicates is used to estimate standard errors and build confidence intervals based on the percentiles of the distribution of the estimates. All the R codes used to estimate the parameters and build the confidence intervals are available as supplementary material.
\[results\]Results
==================
The results of the proposed methodology over a comprehensive simulation study and an application on two real data sets are shown in this Section.
\[simulation\]Simulation study
------------------------------
A thorough simulation study has been conducted to ensure that the model behaves as expected in several situations, including $AR(1)$, $MA(1)$ and $ARMA(1, 1)$ structures for the hidden process $X_t$ with values for the parameters $\alpha$, $\theta$, $q$ and $\omega$ ranging from 0.1 to 0.9 for each parameter. For each autocorrelation structure and parameters combination, a random sample of size $n = 1000$ has been generated using the function *arima.sim* from R package *forecast*[@Hyndman2008; @Hyndman2018]. Different sample sizes ($n = 50, 100, 500$) have also been analysed to study the impact of sample size on accuracy and the results are reported in the Supplementary Material. The complete tables with the estimates corresponding to each combination of parameters can be found in Table S4 to S6 $n = 50$ and $AR(1)$, $MA(1)$ and $ARMA(1, 1)$ respectively, Tables S7 to S9 for $n = 100$ and Tables S10 to S12 for $n = 500$, and the performance is summarised in Tables S13 to S15 for $n = 50, 100$ and $500$ respectively. Absolute average bias is similar regardless of the sample size, while average interval lengths (AIL) are higher and interval coverages are poorer (around 75% for $n = 50$) for lower sample sizes as could be expected. Several bootstrap sizes ($b = 20, 50, 100$) were also considered and the difference between them were negligible, so only results corresponding to $b = 100$ bootstrap replicates are reported. The average absolute bias, interval coverage and 95% confidence interval length are reported in Table \[tab:estim\_sim\]. These values are averaged over all combinations of parameters. Additionally, standard $AR(1)$, $MA(1)$ and $ARMA(1, 1)$ models were fitted to the same simulated series without accounting for their underreporting structure.
The complete table with the estimates corresponding to each combination of parameters is available as Supplementary Material (Tables S1, S2 and S3 for $AR(1)$, $MA(1)$ and $ARMA(1, 1)$ structures respectively).
Structure Parameter Bias AIL Coverage (%)
------------------ ---------------- ------------ ------------ --------------
$\hat{\alpha}$ 0.003 0.099 94.92%
$\hat{q}$ $<10^{-3}$ $<10^{-3}$ 95.47%
$\hat{\omega}$ -0.001 0.052 92.46%
Standard $AR(1)$ $\hat{\alpha}$ 0.501 0.124 0.69%
$\hat{\theta}$ $<10^{-3}$ 0.117 94.38%
$\hat{q}$ $<10^{-3}$ $<10^{-3}$ 94.38%
$\hat{\omega}$ $<10^{-3}$ 0.052 93.28%
Standard $MA(1)$ $\hat{\theta}$ 0.502 0.124 1.10%
$\hat{\alpha}$ 0.002 0.165 96.02%
$\hat{\theta}$ 0.007 0.210 96.56%
$\hat{q}$ $<10^{-3}$ 0.002 94.56%
$\hat{\omega}$ $<10^{-3}$ 0.059 93.22%
$\hat{\alpha}$ 0.456 3.558 59.08%
$\hat{\theta}$ 0.579 3.496 56.00%
: Model performance measures (average absolute bias, average interval length and average coverage) summary based on a simulation study.\[tab:estim\_sim\]
It is clear from Table \[tab:estim\_sim\] that ignoring the underreported nature of data (labeled as *Standard* models in the table) leads to highly biased estimates with extremely low coverage rates, even with larger average interval lengths. This is especially relevant when the intensity or frequency of underreported observations is high.
\[example\]Example: HPV infection incidence
-------------------------------------------
The series of weekly cases of HPV infection in Girona in the period 2010-2014 was previously analysed as a discrete INAR(1) hidden Markov process[@Fernandez-Fontelo2016]. In a similar way, we aim to analyse the corresponding series of incidence, and an AR process of order 1 seems to be adequate (see Figure \[fig:log\_rho\_k\]). According to Eq. (\[corr\_Y\]), the autocorrelation function of the observed process $Y_t$ when the hidden process $X_t$ has an AR(1) structure takes the form $\rho_Y(k) = c \cdot \alpha^k$, where $c = c(\alpha, \mu_{\epsilon}, \sigma^2_{\epsilon}, \omega, q) = \frac{(1-\omega+q \cdot \omega)^2 \cdot \sigma^2_{\epsilon}}{(1-\alpha^2) \cdot \left(\left( \frac{\sigma^2_{\epsilon}}{1-\alpha^2}+\frac{\mu^2_{\epsilon}}{(1-\alpha^2)}\right) \cdot (1+\omega \cdot (q^2-1)) - (1-\omega+q \cdot \omega)^2 \cdot \frac{\mu_{\epsilon}^2}{(1-\alpha)^2} \right)}$. In particular, in this case we can write $\log(\rho_Y(k)) = \log(c)+k \cdot \log(\alpha)$, so a statistically significant intercept of this linear regression model (estimating the parameters by ordinary least squares method) could be interpreted as an evidence of underreporting, as in this case ($p-value = 0.0014$). It is clear from Figure \[fig:log\_rho\_k\] that the estimated regression line does not cross the origin, so the behavior of the observed process is consistent with an underlying underreported AR(1) process.
![\[fig:log\_rho\_k\]Sample autocorrelation coefficients and estimated regression line of $\log(\rho_Y(k)) = \log(c)+k \cdot \log(\alpha)$.](log_rho.eps){height="26.00000%"}
By means of the estimation method described in Section *Estimation*, it can be seen that the estimated model for the hidden process is $X_t = 0.104 \cdot X_{t-1} + \epsilon_t$, being the observed process $Y_t$,
$$Y_t = \begin{cases}
X_t & \text{with probability } 0.202 \\
0.236 \cdot X_t & \text{with probability } 0.798
\end{cases}$$
The estimated parameters are reported in Table \[tab:estimHPV\].
Parameter Bootstrap mean Bootstrap SE
---------------- ---------------- --------------
$\hat{\alpha}$ 0.104 0.049
$\hat{\omega}$ 0.798 0.205
$\hat{q}$ 0.236 0.044
: Bootstrap means standard errors of the proposed model.\[tab:estimHPV\]
These results are highly consistent with those previously reported in the literature for the number of HPV cases obtained through a discrete time series approach[@Fernandez-Fontelo2016] and can be interpreted in a very straightforward way. Moreover, this new methodology can be used to model the incidence of the disease instead of the number of cases, accounting for potential changes in the underlying population.
The estimated intensity of underreporting is $\hat{q} = 0.236$, with 95% confidence interval $(0.150, 0.322)$. The registered and estimated evolution of HPV incidence within the study period (2010-2014) can be seen in Figure \[fig:HPV\_series\].
![\[fig:HPV\_series\]Registered and estimated HPV incidence in Girona in the period 2010-2014.](serie_HPV.eps){height="30.00000%" width="50.00000%"}
\[example2\]Example: Coronavirus COVID-19 incidence in the region of Heilongjiang
---------------------------------------------------------------------------------
COVID-19 is a betacoronavirus that affects the lower respiratory tract and often manifests as pneumonia in humans which was identified as the causative agent of an unprecedented outbreak of pneumonia in Wuhan City, Hubei province in China starting in December 2019[@Sohrabi2020]. Considering that many cases run without developing symptoms beyond those of MERS-CoV, SARS-CoV or pneumonia due to other causes, it is reasonable to assume that the incidence of this disease has been underregistered, especially at the beginning of the outbreak[@Zhao2020].
Heilongjiang is a province in north-east China. Although in general the behavior of this kind of diseases is far from being stationary, this province is far enough from the focus in Hubei province (south central China) so the incidence is much lower and less explosive, and in the study period of time (2020/01/22-2020/02/26) it can be considered stationary, as can be seen in Figure \[fig:COVID\_series\], which shows registered and estimated evolution of COVID-19 incidence within the considered period of time.
A disease with a similar behavior as the MERS-CoV has been modeled as an $ARMA(3,1)$[@Alkhamis2019], so we checked this model and similar ones. Probably due to the shortness of the available data this autoregressive structure was not observed and in our case the best performing model was an $MA(1)$ (AIC of -151.31 against -148.82 for the $ARMA(3, 1)$), consistently with the residuals profile shown in Figure \[fig:COVID\_residuals\], obtained from fitting an $MA(1)$ model to the most likely process $X_t$ reconstructed following step (v) in Section *Estimation*.
![\[fig:COVID\_residuals\]Residuals analysis of the residuals from an $MA(1)$ model.](residuals_COVID.eps){height="30.00000%"}
By means of the estimation method described in Section *Estimation*, it can be seen that the estimated model for the hidden process is $X_t = 0.481 \cdot \epsilon_{t-1} + \epsilon_t$, being the observed process $Y_t$,
$$Y_t = \begin{cases}
X_t & \text{with probability } 0.507 \\
0.195 \cdot X_t & \text{with probability } 0.493
\end{cases}$$
The estimated parameters are reported in Table \[tab:estimCOVID\].
Parameter Bootstrap mean Bootstrap SE
---------------- ---------------- --------------
$\hat{\theta}$ 0.481 0.179
$\hat{\omega}$ 0.493 0.168
$\hat{q}$ 0.195 0.089
: Bootstrap means standard errors of the proposed model.\[tab:estimCOVID\]
![\[fig:COVID\_series\]Registered and estimated COVID-19 incidence in the region of Heilongjiang in the period 2020/01/22-2020/02/26.](serie_COVID.eps){height="30.00000%" width="50.00000%"}
\[discussion\]Discussion
========================
In biomedical and epidemiological research, the usage of disease registries in order to analyse the impact and incidence of health issues is very common. However, the accuracy and data quality of such registries is in many cases at least doubtful. This is the case, for instance, for rare diseases[@Kodra2017] or health issues that clear asymptomatically in most cases like HPV infection. In the case of HPV incidence in Girona in the period 2010-2014, the registered weekly average is 0.17 cases per 100,000 individuals, while the reconstructed process has a weekly average of 0.51 cases per 100,000 individuals, showing that only 33% of the real incidence is recorded by the public health system. This result is very consistent with that of[@Fernandez-Fontelo2016], claiming that only 38% of the HPV cases were registered in the same area and period of time.
The Heilongjiang region COVID-19 data reveal that in average about 66% of the actual incidence in the period 2020/01/22-2020/02/26 was reported. The unavailable data estimated by the proposed methodology are crucial to provide public health decision-makers with reliable information, which can also be used to improve the accuracy of dynamic models aimed to estimate the spread of the disease[@Zhao2020].
The concerns around accuracy of registered data have recently led to the publication of recommendations to improve data collection to ensure accuracy of registries (see for instance[@Kodra2018; @Harkener2019]). Nonetheless, these recommendations are very recent and may be difficult for the public health services of many countries to fully implement them, due to operational or structural issues.
The proposed methodology is able to deal with underreported (or overreported) data in a very natural and straightforward way, estimating its intensity and frequency on a continuous time series, and allowing to reconstruct the most likely unobserved process. It is also flexible enough to handle covariates straightforwardly, and therefore it is simple to introduce trends or seasonality if necessary, so it can be useful in many contexts, where these issues might arise.
The simulation study shows that the proposed methodology behaves as expected and that the parameters used in the simulations, under different autocorrelation structures, are properly recovered, regardless of the intensity and frequency of the underreporting issues.
David Moriña acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445) and Fundación Santander Universidades. This work was partially supported by grant RTI2018-096072-B-I00 from the Spanish Ministry of Science and Innovation.
\[appendix\]Appendix 1
======================
The observed process $Y$ can be expressed as $Y_t = (1-Z_t) \cdot X_t + Z_t \cdot q \cdot X_t$, where $Z_t \sim Bern(\omega)$, and therefore, taking into account that $\mathbb{E}(X_t) = \frac{\mu_{\epsilon}}{1-\alpha_1 - \ldots - \alpha_p}$ and $\mathbb{V}(X_t) = \frac{\sigma_{\epsilon}^2 \cdot (1+\theta_1^2 + \ldots + \theta_r^2)}{1-\alpha_1^2- \ldots - \alpha_p^2}$ and that $X_t$ and $Z_t$ are independent,
$$\begin{array}{ll}
\mathbb{E}(Y_t) = \mathbb{E}(X_t) \cdot \mathbb{E}(Z_t+q \cdot (1-Z_t)) = \frac{\mu_{\epsilon}}{1-\alpha_1 - \ldots - \alpha_p} \cdot \\
\left(1-\omega + q \cdot \omega \right).
\end{array}$$
To compute the variance of the observed process $Y_t$ it is important to observe that $\mathbb{E}(Y_t^2) = \mathbb{E}(X_t^2) \cdot \mathbb{E}((1-Z_t+qZ_t)^2)$. As $\mathbb{E}(X_t^2)=\left(\frac{\mu_{\epsilon}}{1-\alpha_1-\ldots - \alpha_p}\right)^2 + \frac{\sigma_{\epsilon}^2 \cdot (1+\theta_1^2 + \ldots + \theta_r^2)}{1-\alpha_1^2 - \ldots - \alpha_p^2}$, we only need to compute $\mathbb{E}((1-Z_t+qZ_t)^2)$:
$$\begin{array}{ll}
(1-Z_t+qZ_t)^2=1+Z_t^2-2Z_t+q^2Z_t^2=\\
=1+Z_t(q^2-1),
\end{array}$$
and so $\mathbb{E}((1-Z_t+qZ_t)^2)=1+\omega \cdot (q^2-1)$. From here, $\mathbb{E}(Y_t^2)=\left(\left(\frac{\mu_{\epsilon}}{1-\alpha_1 - \ldots - \alpha_p}\right)^2 + \frac{\sigma_{\epsilon}^2 \cdot (1+\theta_1^2 + \ldots + \theta_r^2)}{1-\alpha_1^2- \ldots - \alpha_p^2}\right) \cdot \left(1+\omega \cdot (q^2-1) \right)$ and $(\mathbb{E}(Y_t))^2=\left(\frac{\mu_{\epsilon}}{1-\alpha_1 - \ldots - \alpha_p}\right)^2 \cdot \left(1-\omega + q\omega \right)^2$. Therefore,
$$\begin{array}{ll}
\mathbb{V}(Y_t)=\mathbb{E}(Y_t^2)-(\mathbb{E}(Y_t))^2=\\
\left(\left(\frac{\sigma_{\epsilon}^2 \cdot (1+\theta_1^2+ \ldots + \theta_r^2)}{1-\alpha_1^2- \ldots - \alpha_p^2}\right)+\frac{\mu^2_{\epsilon}}{(1-\alpha_1 - \ldots - \alpha_p)^2}\right) \cdot \\
\cdot (1+\omega \cdot (q^2-1))
-\frac{\mu^2_{\epsilon}}{(1-\alpha_1 - \ldots - \alpha_p)^2} \cdot (1-\omega+q \cdot \omega)^2.
\end{array}$$
The covariance of the hidden process $X_t$ can be written as[@Brockwell1991] $$\gamma_X(k) = \sigma_{\epsilon}^2 \sum_{j=0}^{\infty} \psi_j \psi_{j + \mid k \mid},$$ where $\psi(z) = \sum_{j=0}^{\infty} \psi_j z^j = \frac{\theta(z)}{\alpha(z)}$ for $\mid z \mid \leq 1$, with $\theta(z) = 1+\theta_1 \cdot z+\ldots+\theta_r \cdot z^r$ and $\alpha(z) = 1+\alpha_1 \cdot z+\ldots+\alpha_p \cdot z^p$. Considering $Z_t \sim Bern(\omega)$ we can write $$\begin{array}{ll}
E(Y_t Y_{t+k}) = E(X_t \cdot (1-Z_t)X_{t+k} \cdot (1-Z_{t+k})) + \\
+ E(X_t \cdot (1-Z_t)qX_{t+k} \cdot Z_{t+k})+\\
+E(X_{t+k} \cdot (1-Z_{t+k})qX_t \cdot Z_t)+\\
+E(qX_tqX_{t+k} \cdot Z_t Z_{t+k})
\end{array}$$ As $Z_t$ and $Z_{t+k}$ are independent of $\{X_j\}$ and both have expectation $\omega$, $$\begin{array}{ll}
E(X_t \cdot (1-Z_t)X_{t+k} (1-Z_{t+k}))=(1-\omega)^2 E(X_t X_{t+k}) \\
E(X_t \cdot (1-Z_t)qX_{t+k} \cdot Z_{t+k})=(1-\omega)\omega q E(X_t X_{t+k}) \\
E(X_{t+k} \cdot (1-Z_{t+k})qX_t \cdot Z_t)=(1-\omega)q\omega E(X_t X_{t+k})\\
E(qX_tqX_{t+k} \cdot Z_t Z_{t+k})=\omega^2 q^2 E(X_t X_{t+k}).
\end{array}$$ Therefore, $E(Y_t Y_{t+k})=(1-\omega+q \cdot \omega)^2 E(X_t X_{t+k})$. From here, the covariance of the observed process $Y_t$ is $$\begin{array}{ll}
\gamma_Y(k) = Cov(Y_t, Y_{t+k}) = (1-\omega+q \cdot \omega)^2 \cdot \\
\cdot \left(\gamma_X(k) + \frac{\mu_{\epsilon}^2}{(1-\alpha_1-\ldots-\alpha_p)^2} \right) - \\
\left(\frac{\mu_{\epsilon}}{1-\alpha_1-\ldots-\alpha_p} \right)^2 \cdot (1-\omega+q \cdot \omega)^2 = \\
= (1- \omega + q \cdot \omega)^2 \cdot \gamma_X(k).
\end{array}$$ Finally, the autocorrelation of $Y_t$ can be expressed as $$\begin{array}{ll}
\rho_Y(k)=\frac{\gamma_Y(k)}{V(Y_t)}=\\
=\frac{\gamma_{X}(k) \cdot (1-\omega+q \cdot \omega)^2}{(V(X_t)+E(X_t)^2) \cdot (1+\omega \cdot (q^2-1))-E(X_t)^2 \cdot (1-\omega + q \cdot \omega)^2}=\\
=\frac{V(X_t) \cdot \rho_{X}(k) \cdot (1-\omega+q \cdot \omega)^2}{(V(X_t)+E(X_t)^2) \cdot (1+\omega \cdot (q^2-1))-E(X_t)^2 \cdot (1-\omega + q \cdot \omega)^2},
\end{array}$$ where $\rho_X(k)$ is the autocorrelation function of the hidden process $X_t$.
|
---
abstract: 'The effect of noise is studied in one-dimensional maps undergoing transcritical, tangent, and pitchfork bifurcations. The attractors of the noiseless map become metastable states in the presence of noise. In the weak-noise limit, a symplectic two-dimensional map is associated with the original one-dimensional map. The consequences of their noninvertibility on the phase-space structures are discussed. Heteroclinic orbits are identified which play a key role in the determination of the escape rates from the metastable states. Near bifurcations, the critical slowing down justifies the use of a continuous-time approximation replacing maps by flows, which allows the analytic calculation of the escape rates. This method provides the universal scaling behavior of the escape rates at the bifurcations.'
author:
- Jonathan Demaeyer
- Pierre Gaspard
title: |
Noise-induced escape from bifurcating attractors:\
Symplectic approach in the weak-noise limit
---
Introduction \[intro\]
======================
At the macroscale, deterministic dynamics often emerges out of the random motion of the many subunits interacting in a complex system. In between the micro- and macroscales, the time evolution may admit a mesoscopic description in terms of stochastic processes in which the deterministic dynamics is affected by noise. In such systems, the deterministic dynamics rules the mean fields or slow modes emerging out of the many rapidly fluctuating degrees of freedom. At the mesoscopic scale, these latter can still manifest themselves and perturb the deterministic time evolution of the slow modes by fast additive or multiplicative noises [@vK81; @G04]. Such stochastic processes have been considered for Brownian motion [@L08], mesoscopic electric circuits [@N28] and superconducting junctions [@DEMCC87], mesoscopic lasers and other optoelectronic devices [@G91], fluctuating hydrodynamics [@LL90; @OS06], nucleation processes [@TO98], reaction rate theory [@HTB90], nonequilibrium chemical kinetics [@NP77; @G02], as well as collective behaviors such as synchronization, flocking or swarming in biology [@K84; @PRK01; @GC04; @LGNS04; @CDFSTB01; @DAGPD86; @YEECBKMS09].
The presence of noise may modify qualitatively the time evolution of systems. In particular, noise can activate the crossing of otherwise impenetrable barriers. Therefore, states which are stable with respect to the deterministic dynamics may become metastable under the effect of noise, as it is the case in nucleation processes or chemical reactions [@TO98; @HTB90; @NP77]. Instead of remaining forever in a stable state, the trajectory thus escapes from the metastable state with a rate, which corresponds to the lifetime of the system in the metastable state. The phenomenon of noise-induced escape is known in many systems including chemical reactions [@HTB90], nucleation processes [@TO98], Josephson tunnel junctions [@DEMCC87], semiconductor lasers [@HZRD00], or systems with chaotic attractors or fractal basin boundaries [@B89; @KG04; @KG04bis; @ES08; @BMLSMcC05; @SBLMcC05].
Analytical methods have been developed to evaluate the lifetime of metastable states, especially, in the weak-noise limit. In this limit, the lifetime is often observed to grow exponentially with the inverse of the noise amplitude, a behavior which is reminiscent of Arrhenius’ law in chemical kinetics. In microscopic chemical reactions, the noise is of thermal origin and its amplitude is controlled by the temperature so that the rate of hopping over the energy barrier is proportional to a Boltzmann factor at the ambient temperature. In macroscopic complex systems, the fluctuations come from the individual motions of the multiple active entities interacting in the system so that the noise amplitude is related to the size of the population of entities and has no thermal origin. In such complex systems, the analogue of the activation energy is given by the action functional ruling the stochastic process in the weak-noise limit.
If the noise is weak enough, a variational principle can be used to deduce a Lagrangian or Hamiltonian dynamical system from the action functional, as shown by Onsager and Machlup [@OM53], Freidlin and Wentzell [@FW84], and others [@MS96; @D99; @CSPVD99; @PVV01; @G02JSP]. Remarkably, this dynamical system is deterministic and symplectic, i.e., area-preserving in problems with one degree of freedom. In this respect, the method based on this Hamiltonian dynamics is called the symplectic approach. We notice that this approach is similar to the semiclassical method of quantum mechanics in the limit where the mechanical action is much larger than Planck’s constant. As in quantum mechanics, the action functional plays a key role in the evaluation of various properties of interest [@P88; @G90; @R03].
In the present paper, our purpose is to use the symplectic approach to calculate analytically the rate of noise-induced escape from attractors undergoing transcritical or pitchfork bifurcations [@N95]. Such bifurcations occur in dissipative deterministic systems controlled by tuning parameters.
At a transcritical bifurcation, two steady state cross each other and exchange their stability. On both sides of the bifurcation, the unstable state may form the boundary with respect to a remote attractor. In this regard, this is a [*boundary bifurcation*]{}. In the presence of noise, this boundary can be crossed and trajectories may escape towards the remote attractor. Near the bifurcation, the proximity of the boundary to the stable state can strongly enhance the escape rate in a way to be determined.
At a pitchfork bifurcation, a stable state becomes unstable, leading to the emergence of a pair of stable states. Therefore, the deterministic system becomes bistable with two attractors beyond the bifurcation. Noise will induce random across the unstable state forming the boundary between both attractors. Here also, the escape rate from each one of both attractors is deeply modified close to the bifurcation.
As we show in the present paper, the symplectic approach allows us to evaluate the effects of these bifurcations on the lifetimes of the metastable states in the weak-noise limit. To investigate these effects, we consider one-dimensional maps with additive independent Gaussian random variables at each time step. In the weak-noise limit, we obtain a symplectic area-preserving map associated with the action functional of this Gaussian stochastic process. Near the bifurcation, we show that an appropriate rescaling of time transforms the two-dimensional symplectic map into a continuous-time Hamiltonian system suitable for analytical calculations. In this way, the noise-induced escape rate can be evaluated analytically.
The paper is organized as follow. In Section \[nmap\], we briefly present the escape problem for a general one-dimensional noisy map. In Section \[pint\], we detail the path integral formalism for such noisy systems. Thereafter, we perform semiclassical calculation using the steepest-descent method in order to obtain the general expression of the aforementioned symplectic map. Its phase portraits are analyzed for a specific example. With the help of this analysis, we introduce an analytical model in Section \[amod\] which allows us to calculate an approximate but analytic expression for the analogue of the activation energy in our context. In Section \[res\], we present the results given by this model for several noisy maps and compare them with the results of Monte-Carlo simulations. Finally, we give our conclusions in the last section.
Noisy maps \[nmap\]
===================
Consider a general one-dimensional map $$\label{eqmap}
x_{n+1} = f(x_n)$$ from the set $\mathbb{R}$ of real numbers onto itself. Such maps are models of the Poincaré recurrence in strongly dissipative systems. We notice that the dynamics is deterministic in the sense that each initial condition generates a unique trajectory: $x_n=f^n(x_0)$. Typically, the trajectories of such deterministic dynamical systems tend to an attractor or escape to infinity (see Fig. \[dtraj\]). The attractors of one-dimensional maps can be stationary, periodic, or chaotic. The basin of attraction $B(\Gamma) \subset \mathbb{R}$ of an attractor $\Gamma$ is the set of all the initial conditions of trajectories converging to $\Gamma$.
![\[dtraj\] Phase space $\mathbb{R}$ versus time with typical trajectories of the logistic map $x_{n+1} = \mu x_n (1 - x_n)$ for the parameter value $\mu=1.9$. The attracting fixed point $\Gamma$ of coordinate $x_\Gamma = 1-1/\mu$ and the repulsive fixed point $\beta$ ($x_\beta = 0$) are plotted as straight lines. $\beta$ is one of the two boundaries of the basin of attraction $B(\Gamma) = [0,1]$ of the attractor $\Gamma$. The trajectories outside of $[0,1]$ tend to $-\infty$. Initial conditions $x_0>1$ are sent to the region $]-\infty,0[$ at the first step. Thereafter, these trajectories stay there and tend to $-\infty$.](trajl.pdf){width="45.00000%"}
![\[ntraj\] Same plot as Fig. \[dtraj\] but for the noisy logistic map (\[eqstoch\]) with $\mu=1.9$. For the sake of clarity, each trajectory is plotted each thirty steps. Each trajectory starts on the attracting fixed point $\Gamma$ and fluctuates for a while before crossing the boundary $\beta$ of the basin of attraction $B(\Gamma) = [0,1]$. After this crossing, the trajectory is driven to $-\infty$.](ntrajl.pdf){width="45.00000%"}
In some circumstances, the deterministic dynamics is affected by additive noise and the quantity $x$ is ruled by the following noisy map $$\label{eqstoch}
x_{n+1} = f(x_n) + \xi_n$$ where $\{\xi_n\}_{n=-\infty}^{+\infty}$ is a sequence of independent Gaussian random variables $$\begin{aligned}
\langle \xi_n \rangle & = & 0 \label{aver_G}\\
\langle \xi_n \xi_m \rangle & = & \varepsilon \, \delta_{nm} \label{corr_G}\end{aligned}$$ $\varepsilon$ being the amplitude of the noise and $\delta_{nm}$ the Kronecker symbol. The trajectories of this system are no longer deterministic since the trajectories $\{x_i\}_{i=0}^n$ depends on the sequence of random variables $\{\xi_i\}_{i=0}^{n}$ besides the initial condition $x_0$.
One of the consequences is that the noise perturbs the stability of the attractors. The boundaries of the basin of attraction $B(\Gamma)$ can be crossed and the trajectory may escape from attractors. Therefore, the attractors $\Gamma$ of the deterministic dynamics typically become metastable in the presence of noise. Noise-induced escape is illustrated in Fig. \[ntraj\] for a noisy map with an otherwise stable steady state.
Perron-Frobenius equation and operator
--------------------------------------
Although the trajectories of noisy maps are random, a deterministic description is recovered for the probability density $\rho_n(x)$ that the system is found in the position $x$ at the $n^{\rm th}$ time step. Indeed, the probability density is ruled by the Perron-Frobenius equation [^1] given by $$\label{PFnoise}
\rho_{n+1} (y) = \int K(x,y) \, \rho_n (x) \, \mathrm{d}x$$ in terms of a stochastic kernel $K(x,y)=g[y-f(x)]$ where $g(\xi)$ is the probability density of the noise [@LM85]. For the Gaussian noise (\[aver\_G\])-(\[corr\_G\]) of noise amplitude $\varepsilon$, this probability density is given by $$\label{stokerwn}
g(\xi)=\frac{1}{\sqrt{2 \pi \varepsilon}} \, \exp\left(-\frac{\xi^2}{2 \varepsilon}\right)$$
In the zero-noise limit $\varepsilon\to 0$, the Gaussian probability density tends to the Dirac delta distribution and the Perron-Frobenius equation of the deterministic map is recovered [@LM85]: $$\rho_{n+1} (y) = \int \delta\left[y-f(x)\right] \rho_n (x) \, \mathrm{d}x$$
We notice that, with or without noise, the Perron-Frobenius equation is defined by a linear operator $\hat P$, which preserves the non-negativity of the probability density as well as its normalization to unity: $\int \rho_n(x) \, \mathrm{d}x=1$. In order to analyze the long-time behavior of the probability density, an eigenvalue problem can be posed for the Perron-Frobenius operator $$\hat P \Psi_{\alpha}(x) = \chi_{\alpha} \Psi_{\alpha}(x)$$ Since the probability density is not expected to grow without bound, the eigenvalues should belong to the unit disk: $\vert\chi_{\alpha}\vert \leq 1$.
The border of the unit disk contains the single eigenvalue $\chi_0=1$ if the probability density converges to a unique stationary solution of the Perron-Frobenius equation. In this case, all the other eigenvalues are strictly inside the unit disk, $\vert\chi_{\alpha}\vert < 1$ for $\alpha\neq 0$. As a consequence, the components of the probability density corresponding to these other eigenvalues are exponentially damped in the long-time limit, allowing the convergence of the probability density to the eigenfunction $\Psi_0$ associated with the leading eigenvalue $\chi_0=1$.
In contrast, the probability density is expected to vanish if trajectories escape to infinity as observed in Fig. \[ntraj\]. In this further case, the border of the unit disk does not contain any eigenvalue, otherwise the asymptotic probability density would not be vanishing. Instead, the long-time behavior is dominated by the eigenvalue which is the closest to the border of the unit disk: $\chi_0 <1$ such that $\vert\chi_{\alpha}\vert < \vert\chi_0\vert$ for $\alpha\neq 0$. After $n$ iterates of the Perron-Frobenius operator, the probability density is damped by a factor $\chi_0^n$ so that we may expect an exponential decay in this situation. The corresponding eigenfunction $\Psi_0(x)$ is non-negative according to the Perron-Frobenius theorem.
Escape rate and mean escape time {#subsec:esc_rate}
--------------------------------
Thanks to the probability density $\rho_n(x)$, we can write down the probability that the escape from the attractor $\Gamma$ occurs at a time greater than $n$ in the noisy system (\[eqstoch\]): $$\label{surv_proba}
P(\{t>n\}) \equiv G_n = \int_{B(\Gamma)} \rho_n (x) \, \mathrm{d}x$$ where $B(\Gamma)$ is the basin of attraction of $\Gamma$. This probability is also called the survival probability. On the other hand, the probability to escape at a time $n$ is given by $$\label{tescdist}
P(\{t=n\}) \equiv P_n = G_{n-1} - G_n$$ We notice that the normalization condition $\sum_{n=1}^{\infty} P_n =G_0=1$ is satisfied.
If trajectories escape to infinity, the escape rate $\gamma$ of the attractor of $\Gamma$ is defined as the long-time limit of the exponential decay rate of the survival probability: $$\label{esc_rate}
\gamma \equiv \lim_{n\to\infty} -\frac{1}{n}\ln G_n$$ In this case, the escape can alternatively be defined in terms of the leading eigenvalue $\chi_0$ of the Perron-Frobenius operator. Since the probability density is damped as $\chi_0^n$ for $n\to\infty$, so does the survival probability (\[surv\_proba\]) and we infer that $$\label{esc_rate_chi_0}
\gamma = - \ln \chi_0$$ The inverse of the escape rate is the lifetime of the metastable state corresponding to the eigenfunction $\Psi_0(x)$: $$\tau \equiv \frac{1}{\gamma}$$
On the other hand, the [*mean escape time*]{} can be defined as the statistical average of the random time the trajectory escapes from the basin $B(\Gamma)$: $$\label{esc_time}
\langle t \rangle \equiv \sum_{n=1}^{\infty} n \, P_n = \sum_{n=0}^{\infty} G_n$$ If the decay is exponential in the long-time limit, $G_n \simeq {\rm e}^{-\gamma n}$, the mean escape time is given by $\langle t\rangle \simeq (1-{\rm e}^{-\gamma})^{-1}$. In the weak-noise limit where the escape rate is expected to vanish as $$\label{teedep}
\gamma \sim \exp \left(-\frac{W_0}{\varepsilon}\right)$$ for $\varepsilon\to 0$, the mean escape time can be related to the escape rate itself by the simple relation $$\label{rate_time}
\lim_{\gamma\to 0} \gamma \, \langle t \rangle = 1$$ Otherwise, the mean escape time is not simply given as the inverse of the escape rate because, in general, the mean escape time depends on the transients before the long-time exponential decay at the escape rate.
Relaxation rate and mean first-exit time {#first-exit_time}
----------------------------------------
If trajectories do not escape to infinity but randomly jump between two attractors of the deterministic dynamics, the probability density is expected to converge towards a stationary density given by the eigenfunction $\Psi_0(x)$ of the Perron-Frobenius operator corresponding to the eigenvalue $\chi_0=1$. In this case, the escape rate (\[esc\_rate\]) vanishes and the quantity of interest is given by the next-to-leading eigenvalue $\chi_1$ such that $\vert\chi_{\alpha}\vert < \chi_1 < 1$ for $\alpha\neq 0,1$ if this latter is unique. The eigenvalue $\chi_1$ controls the convergence of the probability density towards the stationary state. This convergence is exponential, which defines the relaxation rate $$\label{relax_rate}
\tilde\gamma = - \ln \chi_1$$ This relaxation rate can be interpreted as the rate of jumps between the two attractors due to the noise. In some cases, this relaxation rate can be defined in the presence of escape if there is a clear separation of time scales between both.
A mean first-exit time can also be defined as the statistical average of the random time of first exit from a given basin of attraction $B(\Gamma)$. In this scheme, the trajectories starts with an initial condition inside the basin $B(\Gamma)$, evolves in time according to the noisy map (\[eqstoch\]), and stops as soon as the trajectory escapes from the basin $B(\Gamma)$. The Perron-Frobenius operator $\hat Q$ of this first-exit process is defined by restricting the domain of integration to the basin $B(\Gamma)$: $$\label{PFnoise_Q}
\rho_{n+1} (y) = \int_{B(\Gamma)} K(x,y) \, \rho_n (x) \, \mathrm{d}x$$ For a bounded basin $B(\Gamma)$, all the eigenvalues of this further operator are strictly inside the unit disk and an escape rate $\gamma$ can be defined by Eq. (\[esc\_rate\_chi\_0\]) in terms of the leading eigenvalue $\chi_0$, as in the previous Subsection \[subsec:esc\_rate\]. Furthermore, a mean escape time $\langle t\rangle$ can be defined for the present process by Eq. (\[esc\_time\]). If the attractor $\Gamma$ of the deterministic dynamics is strictly inside its basin of attraction $B(\Gamma)$, we expect that, in the weak-noise limit, the relaxation rate should behave as $\tilde\gamma \sim \exp(-W_0/\varepsilon)$, for $\varepsilon\to 0$, and be related to the mean escape time $\langle t \rangle$ and the corresponding escape rate $\gamma$ according to $$\lim_{\tilde\gamma\to 0} \tilde\gamma \, \langle t \rangle = \lim_{\gamma\to 0} \gamma \, \langle t \rangle = 1$$
For both escape and relaxation, the determination of the lifetime of the metastable state thus goes by the evaluation of the quantity $W_0$ which is the analogue of the activation energy in systems with thermal noise.
Path integrals and the symplectic approach \[pint\]
===================================================
The aim of the present section is to obtain the symplectic map ruling the process in the weak-noise limit.
Path integrals
--------------
Starting with an initial density $\rho_0$ and iterating Eq. (\[PFnoise\]), one can obtain the probability density at time $n$ in the form $$\label{nevol}
\rho_n (x_n) = \int \mathrm{d} x_0 \, K_n (x_0,x_n) \, \rho_0 (x_0)$$ where $K_n$ is an integral kernel resulting from $n$ iterations. This kernel is the propagator associated with the time evolution of the stochastic process. For the Gaussian noise (\[stokerwn\]), this kernel is given by the multiple or path integral $$\begin{aligned}
\label{kerbb}
K_n (x_0,x_n) & = & \frac{1}{(\sqrt{2 \pi \varepsilon})^n} \int \mathrm{d} x_{n-1} \ldots \int \mathrm{d} x_1 \\ \nonumber
& & \times \exp \left[ - \frac{1}{\varepsilon} \, W_n (x_0,x_1,\ldots,x_{n-1},x_n)\right]\end{aligned}$$ where $$\label{fact}
W_n (x_0,\ldots,x_n) = \frac{1}{2} \sum_{i=0}^{n-1} \left[x_{i+1} - f(x_i)\right]^2$$ which defines the action functional of the Gaussian stochastic process.
Symplectic map \[semiapp\]
--------------------------
In the limit of weak noise $\varepsilon \to 0$, the propagator (\[kerbb\]) can be integrated by the steepest-descent method, which will select the extrema of the action functional (\[fact\]). These extrema are obtained by applying Hamilton’s variational principle for the action function (\[fact\]). The selected paths are thus the trajectories of a symplectic deterministic system [@FJ05] . With this Hamiltonian formalism in mind, the functional (\[fact\]) can be written in terms of a generating function $F$ of the first kind [@G50] $$\label{Wpos}
W_n (x_0,\ldots,x_n) = \sum_{i=0}^{n-1} F(x_i, x_{i+1})$$ where $$\label{gfunc}
F(x_i,x_{i+1}) = \frac{1}{2} \, \left[x_{i+1} - f(x_i) \right]^2$$ Introducing the momentum $p_i$ canonically conjugated to the position $x_i$, the evolution over one time step is conceived as a canonical transformation corresponding to this generating function $$\label{can_transf}
\left\{ \begin{array}{lcl}
p_{i+1} & = & \frac{\partial F(x_i,x_{i+1})}{\partial x_{i+1}} \\ \\
p_{i} & = & - \frac{\partial F(x_i,x_{i+1})}{\partial x_i}
\end{array} \right.$$ With the generating function (\[gfunc\]), we obtain the two-dimensional map $$\label{mapsymp}
\phi \; \left\{ \begin{array}{lcl}
x_{i+1} & = & f (x_i) + \frac{p_i}{f'(x_i)} \\ \\
p_{i+1} & = & \frac{p_i}{f'(x_i)}
\end{array} \right.$$ This map is symplectic because it preserves the symplectic differential two-form $$\mathrm{d}x_{i+1} \wedge \mathrm{d}p_{i+1} = \mathrm{d}x_i \wedge \mathrm{d}p_i$$ where $\wedge$ denotes the exterior or wedge product [@M92]. Accordingly, the Jacobian of the map takes the unit value, $\det D\phi=1$, so that the map is area-preserving in the two-dimensional phase space $\mathcal{M}=\{(x,p): x\in\mathbb{R},p\in\mathbb{R}\}$.
Such area-preserving maps are expected to be typically chaotic as it is the case for the standard map [@M92]. The multitude of possible trajectories reflects the complexity of the landscape of the multidimensional functional (\[Wpos\]). This landscape unveils many local extrema among which the local minima should be selected in order to guarantee the local normalization of the probability distribution underlying the propagator (\[kerbb\]).
We emphasize that the symplectic map (\[mapsymp\]) has special properties which are specific to the problem at hand.
First, the map (\[mapsymp\]) leaves invariant the subspace $p=0$ of the phase space $\mathcal{M}$. In this subspace, the two-dimensional map (\[mapsymp\]) reduces to the noiseless one-dimensional dissipative map $x_{i+1} = f (x_i)$. In consequence, the attracting fixed points of the original map $f$ are hyperbolic fixed points of the symplectic map (\[mapsymp\]) that lie in the subspace $p=0$. Note that the repulsive fixed points of the map $f$ are also hyperbolic fixed points of (\[mapsymp\]) and also lie on the subspace $p=0$.
Secondly, the symplectic map (\[mapsymp\]) is not uniquely invertible and may admit several inverses. This is the consequence of the noninvertible character of the one-dimensional map $f$ itself, which often has extrema where its derivative $f'(x)$ vanishes. Indeed, the inverses of the two-dimensional map (\[mapsymp\]) are directly associated with the inverses $f_j^{-1}$ of the one-dimensional map $f$, such that $f_j^{-1}\left[f(x)\right]=x$ for $j=1,2,...,m$. The inverses of the two-dimensional map (\[mapsymp\]) can thus be written as $$\label{invmapsymp}
\phi_j^{-1} \; \left\{ \begin{array}{lcl}
x_i& = & f_j^{-1}(x_{i+1} - p_{i+1}) \\ \\
p_i & = & p_{i+1} \, f'\left[ f_j^{-1}(x_{i+1} - p_{i+1}) \right]
\end{array} \right.$$ with $j=1,2,...,m$. The inverses of one-dimensional map become singular at the positions such that $f'(x_k)=0$ with $k=1,2,...,\tilde m$. These loci correspond to the borders of the domains where the inverses (\[invmapsymp\]) are defined: $$\label{border}
x-p=f(x_k) \qquad \mbox{such that} \qquad f'(x_k)=0$$ with $k=1,2,...,\tilde m$. For noninvertible two-dimensional maps such as Eq. (\[mapsymp\]) with a vanishing denominator, the concepts of focal point and prefocal curves (or lines) have been introduced [@BGM99; @BGM03]. They will be defined here below in the discussion of the phase-space structures generated by these maps, which are different in this regard from those of the invertible area-preserving maps such as the standard map [@M92].
Phase space\[phasespaces\]
--------------------------
This section is devoted to the phase-space structures generated by two-dimensional symplectic maps (\[mapsymp\]) associated with one-dimensional maps $f$ undergoing bifurcations. The study of typical phase portraits of these symplectic maps is important because the contributions to the propagator (\[kerbb\]) are evaluated in the weak-noise limit in terms of the trajectories of these maps. Moreover, statistical averages are related to the propagator and, as we shall see, the study of phase space gives an insight on their properties.
### Invariant subspace $p=0$
As mentioned here above, the straight line $p=0$ is an invariant subspace of the phase space $\mathcal{M}=\{(x,p): x\in\mathbb{R},p\in\mathbb{R}\}$. This line is the $x$-axis where the dynamics is ruled by the one-dimensional map $x_{i+1}=f(x_i)$ of the noiseless macroscopic system. The trajectories outside the invariant subspace are specifically associated with noise. This invariant subspace divides the phase space in two half planes and, therefore, constitutes a major organizing geometric structure.
### Fixed points
The symplectic map (\[mapsymp\]) has two types of fixed points.
The fixed points $x=f(x)$ of the one-dimensional map, which are found in the invariant subspace $p=0$. For these fixed points, the linearized map has the eigenvalues $\Lambda_+=f'(x)$ and $\Lambda_-=f'(x)^{-1}$. Accordingly, these fixed points are hyperbolic (or possibly parabolic in the cases of marginal stability). If the point is attracting (resp. repelling) for the one-dimensional map, the unstable (resp. stable) direction is transverse to the invariant subspace $p=0$.
On the other hand, the symplectic map may admit genuine fixed points outside the invariant subspace $p=0$ for $$\label{fixed_pts}
\left\{ \begin{array}{l}
p = x-f(x) \\ \\
f'(x) = 1
\end{array} \right.$$ These fixed points have eigenvalues given by the roots of the characteristic equation: $$\Lambda^2 + \left[ p \, f"(x) - 2\right] \Lambda + 1 = 0$$ so that they can be either elliptic or hyperbolic (or also parabolic in the marginal cases).
### Singular lines, focal points, and prefocal lines
Because of its vanishing denominators, the two-dimensional map (\[mapsymp\]) is not defined on the [*singular lines*]{} $$\delta_{{\rm s},k} = \{ (x_k,p): f'(x_k)=0, p\in\mathbb{R}\}$$ with $k=1,2,...,\tilde m$. The implications of these singular lines on the phase-space structures generated by the map have been discovered by Mira and coworkers who introduced the concepts of focal points and prefocal lines [@BGM99; @BGM03].
As long as the numerator corresponding to the denominator is not vanishing, the points of the singular lines are sent to infinity by the map (\[mapsymp\]). However, this might not be the case if the numerator vanishes with the denominator at the so-called [*focal points*]{} [@BGM99; @BGM03] $$\boldsymbol{z}_k = (x_k,p=0)$$
Since the map (\[mapsymp\]) displays an undetermined ratio $0/0$ at each focal point, the image of the focal point is not necessarily sent at infinity and we may wonder what is the phase-space structure corresponding to this image. To answer this question, let us consider a line $\gamma$ going through the focal point and defined by the following parametric equations: $$\label{gamma_line}
\gamma \; \left\{ \begin{array}{lcl}
x & = & x_k + \xi \\ \\
p & = & \alpha \, \xi + O(\xi^2)
\end{array} \right.$$ where $\xi$ is the parameter and $\alpha$ the slope of the line at the focal point. The image of this line is given by $$\label{image_gamma_line}
\phi(\gamma) \; \left\{ \begin{array}{lcl}
x' & = & f(x_k + \xi) + \frac{\alpha \, \xi + O(\xi^2)}{f'(x_k + \xi)} = f(x_k) + \frac{\alpha}{f"(x_k)} + O(\xi) \\ \\
p' & = & \frac{\alpha \, \xi + O(\xi^2)}{f'(x_k + \xi)} = \frac{\alpha}{f"(x_k)} + O(\xi)
\end{array} \right.$$ where we have expanded in powers of the parameter $\xi$ around $\xi=0$, using $f'(x_k)=0$, and assuming that $f"(x_k)\neq 0$. The image $\phi(\gamma)$ is a line for varying values of the parameter $\xi$. Taking the limit $\xi=0$, we obtain the image of the focal point: $$\label{image_focal_point}
\phi(\boldsymbol{z}_k) \; \left\{ \begin{array}{lcl}
x' & = & f(x_k) + \frac{\alpha}{f"(x_k)} \\ \\
p' & = & \frac{\alpha}{f"(x_k)}
\end{array} \right.$$ We notice that the image depends on the slope $\alpha$ given to the line $\gamma$ in Eq. (\[gamma\_line\]), whereupon the images are multiple and form a line called the [*prefocal line*]{}: $$\label{prefocal}
\delta_{\boldsymbol{z}_k}: \qquad x' = p' + f(x_k)$$ which coincides with the border (\[border\]) of the domains of definition of the inverse two-dimensional maps (\[invmapsymp\]). Accordingly, the prefocal line is mapped onto the focal point under the inverse map (\[invmapsymp\]) having the prefocal line in its domain of definition: $$\label{preimage_prefocal}
\phi_j^{-1} \left( \delta_{\boldsymbol{z}_k}\right) = \boldsymbol{z}_k$$ In this sense, the prefocal line constitutes the image of the focal point, $\delta_{\boldsymbol{z}_k} = \phi(\boldsymbol{z}_k)$, showing that a point can be mapped onto a line in such two-dimensional map (\[mapsymp\]) with a vanishing denominator [@BGM99; @BGM03].
### Global stable set and unstable manifold
The structures generated by the map $\phi$ in its phase space $\mathcal{M}$ can be analyzed in terms of the stable and unstable sets associated with the saddle points.
The unstable set ${\cal W}^{\rm u}(\boldsymbol{z})$ of the hyperbolic fixed point $\boldsymbol{z}=(x,p)$ consists in the set of points that converge to $\boldsymbol{z}$ under backward iterations of $\phi$. This can be expressed in terms of the union of the successive images of the local unstable manifold ${\cal W}_\mathrm{loc}^{\rm u}(\boldsymbol{z})$, i. e., the unstable set in a neighborhood of $\boldsymbol{z}$: $${\cal W}^{\rm u}(\boldsymbol{z}) = \bigcup_{n=1}^\infty \phi^n [{\cal W}_\mathrm{loc}^{\rm u}(\boldsymbol{z})]$$ Even if $\phi$ is not invertible, the images of the local manifold ${\cal W}_\mathrm{loc}^{\rm u}(\boldsymbol{z})$ will be uniquely determined. Accordingly, the set ${\cal W}^{\rm u}(\boldsymbol{z})$ is a manifold in the phase space and is called the *unstable manifold* [@EKO04].
On the other hand, the stable set of the hyperbolic fixed point $\boldsymbol{z}$ is defined as the set of points that converge to $\boldsymbol{z}$ under forward iterations of $\phi$. It can be obtained as the union of successive pre-images given by the possibly multiple *inverse maps* $\phi^{-1}_j$ acting on the local stable manifold ${\cal W}_\mathrm{loc}^{\rm s}(\boldsymbol{z})$ : $${\cal W}^{\rm s}(\boldsymbol{z}) = \bigcup_{n=1}^\infty \bigcup_j \phi^{-n}_j [{\cal W}_\mathrm{loc}^{\rm s}(\boldsymbol{z})]$$ If $\phi$ is invertible, this set is a manifold and one speaks about *the stable manifold* of $\boldsymbol{z}$. However, if there are multiple inverse maps, the global stable set may consist of disjoint pieces and is not a manifold [@EKO04; @BCGM96]. Since we here consider noninvertible maps $f$, the symplectic map (\[mapsymp\]) is noninvertible so that we speak about the *global stable set* (instead of *stable manifold*).
![\[expps26\] The phase space $\mathcal{M}$ of the symplectic map (\[mapsymp\]) associated with the exponential map $x_{n+1}=\mu x_n\exp(-x_n)$ with $\mu=2.6$. Global unstable set (in red) and global stable manifold (in black) of respectively the fixed point $\Gamma$ and the boundary $\beta$. At the intersections of the two sets, the heteroclinic trajectory (black cross) between these two fixed points. In light blue, some bounded trajectories inside the region delimited by the two sets.](expps26l.pdf){width="50.00000%"}
At the intersection of the global stable set and the unstable manifold associated with one and the same hyperbolic fixed point $\boldsymbol{z}$, we find the so-called [*homoclinic orbit*]{} ${\cal W}^{\rm s}(\boldsymbol{z})\cap {\cal W}^{\rm u}(\boldsymbol{z})$.
If the global stable set and the unstable manifold are attached to different fixed points, we are in the presence of a so-called [*heteroclinic orbit*]{} ${\cal W}^{\rm s}(\boldsymbol{z})\cap {\cal W}^{\rm u}(\boldsymbol{z}')$, which plays an important role in the following.
As an example, a typical phase portrait of the symplectic map (\[mapsymp\]) is depicted in Fig. \[expps26\] for the exponential map $x_{n+1} = \mu x_n \exp(-x_n)$. For the noisy exponential map with the selected value of the parameter $\mu$, the escape scenario is the same as the one detailed in Fig. \[ntraj\] for the logistic map. Starting from the attractive fixed point $x_\Gamma=\ln \mu$, trajectories escape after crossing of the boundary $x_\beta = 0$. Once this boundary is crossed, the trajectory tends to $-\infty$. We notice that the fixed points of the one-dimensional map correspond to the fixed points $\Gamma=(\ln\mu,0)$ and $\beta=(0,0)$ of the two-dimensional map. Moreover, the stable set ${\cal W}^{\rm s}(\Gamma)$ and the unstable set ${\cal W}^{\rm u}(\beta)$ are contained in the invariant subspace $p=0$. The construction of the unstable manifold ${\cal W}^{\rm u}(\Gamma)$ and the global stable set ${\cal W}^{\rm s}(\beta)$ seen in Fig. \[expps26\] have first required the determination of the manifolds directly emanating from the hyperbolic fixed points. This has been performed with an algorithm specially devoted to noninvertible maps and called the search circle algorithm [@EKO04; @KO98]. Thereafter, we were able to compute significant parts of the global invariant sets using backward iterations with $\phi^{-1}_j$ for the global stable set ${\cal W}^{\rm s}(\beta)$ and forward iterations with $\phi$ for the unstable manifold ${\cal W}^{\rm u}(\Gamma)$. We have also computed with precision the heteroclinic orbit ${\cal W}^{\rm s}(\beta)\cap{\cal W}^{\rm u}(\Gamma)$ using an algorithm described in Ref. [@Y98]. An attentive examination of Fig. \[expps26\] shows that the global stable set ${\cal W}^{\rm s}(\beta)$ of the boundary fixed point $\beta$ forms closed loops around the focal point $\boldsymbol{z}=(1,0)$. This feature is inherent to the map (\[mapsymp\]) which is not defined on the singular lines where the derivative of the one-dimensional map $f$ vanishes. For the exponential map, there is such a singular line at $x=1$ and its intersection with the subspace $p=0$ where there is a $0/0$ limit gives the focal point $\boldsymbol{z}=(1,0)$ where $f'(1)=0$. The closed loops of the global stable set ${\cal W}^{\rm s}(\beta)$ are attached to this focal point. This feature is related to the noninvertibility of $f$ for which the focal points divide the configuration space $x$ into different pre-image regions. Since the nondefinition sets are lines that divide the phase space into distinct regions, the only way for a curve that is forward invariant to cross these lines is to pass through focal points, as here observed for the global stable set ${\cal W}^{\rm s}(\beta)$. We refer the reader to the papers [@BGM99; @BGM03; @EKO05] for more information about this issue.
### Dynamics close to the bifurcation
Close to the bifurcation, the dynamics undergoes a critical slowing down. As a consequence, the symplectic map (\[mapsymp\]) performs small steps at each iteration in the phase-space domain where the fixed points bifurcate. It turns out that the dynamics of the map is similar to a continuous-time flow. In order to illustrate this phenomenon, we continue to consider the exponential map $x_{n+1} = \mu x_n \exp(-x_n)$ and we depict in Fig. \[exp1pp\] its phase portrait close to the transcritical bifurcation it undergoes at the parameter value $\mu=1$. At this bifurcation, the two fixed points $x_{\beta}=0$ and $x_{\Gamma}=\ln\mu$ cross each other and exchange their stability. The unstable of these fixed points is the boundary of the basin of attraction of the other so that a boundary bifurcation happens at the transcritical bifurcation. Below and above the bifurcation, an elliptic island exists in the lower half plane $p<0$ around the extra fixed point approximately located at $x\simeq (\mu-1)/2$ and $p\simeq-(\mu-1)^2/4$. At the bifurcation, the three fixed points coalesce at the origin with the surrounding island.
{width="90.00000%"}
Figures \[exp1pp\](a) and (f) depicts the phase portraits for values of $\mu$ far from the bifurcation where we observe that the global stable set and the unstable manifolds are well developed. The trajectories in light blue are trapped and form an elliptic island composed of invariant circles and chaotic zones of moderate extension. At these values, the system (\[mapsymp\]) is clearly not close to integrability since the global stable set and the unstable manifold form typical heteroclinic structures.
For values of $\mu$ closer to the bifurcation, Figs. \[exp1pp\](b) and (c) show that most of the trajectories of the elliptic islands are pretty regular lines bounded by the global stable set and the unstable manifold nearly forming a separatrix. The chaotic zones have disappeared on tiny scales in phase space. Hence, we can conclude that, close to the bifurcation, the system (\[mapsymp\]) tends to complete integrability.
It is interesting to understand how the transition occurs between the two situations. In Fig. \[exp1pp\](d), the phase space is depicted for an intermediate value between these of Figs. \[exp1pp\](c) and (f). Although the phase portrait is similar to the one in Fig. \[exp1pp\](c), Fig. \[exp1pp\](e) shows a zoom of the global stable set and the unstable manifold around the origin revealing heteroclinic structures typical of non-integrable dynamics.
We conclude that, as the bifurcation is approached, the slowing down of the dynamics tends to transform the discrete-time map into a continuous-time flow with one degree of freedom which is thus integrable at the bifurcation. This idea will be elaborated upon in the next section.
Continuous-time limit at bifurcation \[amod\]
=============================================
Hamiltonian flow
----------------
In this section, we establish the correspondence between the symplectic map (\[mapsymp\]) and a continuous-time flow with one degree of freedom in the neighborhood of the bifurcation. This correspondence provides us with a method to compute analytically the properties of the noisy map close to the bifurcation and, in particular, the rate of noise-induced escape from the attractor.
With this aim, we consider a general map of the form: $$\label{fanaexp}
f(x) = \sum_{i=0}^\infty c_i \, x^i$$ where the coefficients $c_i(\mu)$ depend on the parameter $\mu$. For the transcritical and pitchfork bifurcations, we take $c_0 = 0$ which implies that $x=0$ remains a fixed point across the bifurcation. For a bifurcation to happen, we must moreover suppose that the map $f$ possesses at least another fixed point. This means that the series (\[fanaexp\]) contains nonlinear terms. Let us suppose that $c_k$ with $k>1$ is the first non-vanishing coefficient besides $c_1$.
A bifurcation occurs if the fixed point $x=0$ looses or gains stability. This is the case at the value of $\mu$ where the linear stability eigenvalue of the map $f$ at $x=0$ is equal to unity. Since the linear stability of $x=0$ is $f'(0) = c_1$, the bifurcation occurs at the critical parameter value $\mu=\mu_{\rm c}$ where $c_1(\mu_{\rm c})=1$. Near the bifurcation, the dynamics of the map is slowed down since its slope is close to unity, $c_1\simeq 1$, which limits the iterations to small steps. Consistently, the other fixed points of the one-dimensional map are approximately given by $x\simeq [(1-c_1)/c_k]^{1/k}$, which coalesce onto the origin $x=0$ at the bifurcation.
In this regard, we can suppose that trajectories $\{x_n\}_{n=-\infty}^{+\infty}$ of the map can be seen as trajectories $x(t)$ of a continuous-time stochastic dynamical system with the time $t=n$. Since the time step is unity, $\Delta t=1$, differences become derivatives: $$\begin{aligned}
x_{n+1}-x_n = & = & \left[{\rm e}^{\partial_t} x(t)-x(t)\right]_{t=n} \nonumber \\
& = & \dot x + \frac{1}{2} \, \ddot x + \, \cdots\end{aligned}$$ The critical slowing down near the bifurcation justifies the truncation of this expansion to the term with the first derivative with respect to the continuous time $t=n$, which amounts to assimilate the difference between two successive time steps with a time derivative.
To implement this idea, the position $x_n$ at the previous step is subtracted from the noisy map (\[eqstoch\]), which is rewritten as $$x_{n+1} - x_n = f(x_n) - x_n + \xi_n$$ Introducing the function $$\label{ftrans}
g(x) \equiv f(x) -x$$ the noisy map can be replaced by the Ito stochastic differential equation $$\label{eqstoch2}
\frac{\mathrm{d}x}{\mathrm{d}t} = g (x) + \eta(t)$$ with the Gaussian white noise $$\begin{aligned}
\langle \eta(t) \rangle & = & 0 \\
\langle \eta(t) \eta(t') \rangle & = & \varepsilon \, \delta(t-t')\end{aligned}$$ We notice that the time integral of this noise defines the Wiener process ${\rm W}(t)=\int_0^t \eta(t')\mathrm{d}t'$. We emphasize the replacement of the noisy map by the stochastic differential equation is justified by the critical slowing down of the dynamics close to the bifurcation.
{width="\textwidth"}
In the weak-noise limit, this stochastic process can be analyzed thanks to the theory by Onsager and Machlup [@OM53] or Freidlin and Wentzell [@FW84] in terms of a Hamiltonian flow with one degree of freedom. Its Hamiltonian function is given by $$\label{Hamiltonian}
H= \frac{1}{2} \, p^2 + g(x) \, p$$ This Hamiltonian function rules the Hamilton-Jacobi equation derived from the Fokker-Planck equation associated to the equation (\[eqstoch2\]). The trajectories are now solutions of Hamilton’s equations $$\label{flowsymp}
\left\{ \begin{array}{lcl}
\dot x & = & g (x) + p \\ \\
\dot p & = & -g' (x) \, p
\end{array} \right.$$ where $p$ is the momentum variable canonically conjugated to the position $x$. We notice that, as for the symplectic map (\[mapsymp\]), the fixed points of the flow $\dot x = g(x)$ are hyperbolic fixed points of Eqs. (\[flowsymp\]) that lie on the invariant subspace $p=0$ of the phase space. At least, one of the fixed point $x_{\Gamma}$ of the flow $\dot x = g(x)$ is an attractor. Its basin of attraction may be limited by another fixed point $x_{\beta}$ which is unstable. For the exponential map, the phase portraits of this Hamiltonian flow and the corresponding symplectic map are depicted in Fig. \[ppcdexp1\].
The heteroclinic orbit and its action
-------------------------------------
For the stochastic process (\[eqstoch2\]), the attractor $x_{\Gamma}$ is found in a well of the kinetic potential defined by [@GT84; @GT85] $$\begin{aligned}
\label{apot}
U(x) & = & -\int g(x) \, \mathrm{d}x \nonumber \\
&=& \frac{x^2}{2} - \int f(x) \, \mathrm{d}x\end{aligned}$$ On the other hand, the unstable fixed point $x_{\beta}$ is the top of a barrier for the noise-induced escape from the potential well. The escape over this barrier is an activated process with an escape rate given by Eq. (\[teedep\]) where the constant $W_0$ is the action of a special trajectory of the Hamiltonian flow (\[flowsymp\]). This trajectory is given by the separatrix connecting the hyperbolic fixed point $\Gamma$ corresponding to the attractor to the other hyperbolic fixed point $\beta$ at the top of the barrier [@GT84; @GT85; @K88]. See Fig. \[ppcdexp1\](a). The parametric equation of this separatrix is $$p(x)=-2 \, g (x)$$ and the action of this solution between the hyperbolic points $(x_{\Gamma},0)$ and $(x_{\beta},0)$ is given by $$\label{actcont}
W_{\rm h}^{\rm c} = \int_{x_{\Gamma}}^{x_{\beta}} p(x) \, \mathrm{d}x$$ where the superscript ‘c’ stands to recall that this is the action of the continuous-time Hamiltonian system. This action yield the activation barrier for the escape problem in the stochastic process (\[eqstoch2\]) from the attracting fixed point $x_\Gamma$ through the boundary $x_\beta$, as if the crossing by the point $x_\beta$ was the only possible channel of escape from $x_\Gamma$. The integration of the function $g$ gives the final result: $$\label{aacten}
W_{\rm h}^{\rm c} = 2 \left[U(x_{\beta})-U(x_{\Gamma})\right]$$ in terms of the kinetic potential (\[apot\]).
![\[trajhetex\] Evolution of the heteroclinic orbit of the symplectic map (\[mapsymp\]) associated with the exponential map $x_{n+1} = \mu x_n \exp(-x_n)$ as the parameter $\mu$ varies from $0.4$ to $2$. Around the boundary bifurcation, the heteroclinic orbit tends to a continuous curve which is the separatrix of the Hamiltonian flow (\[flowsymp\]).](trajhetexl.pdf){width="45.00000%"}
We notice that the separatrix of the Hamiltonian flow (\[flowsymp\]) is the limit of the heteroclinic orbit ${\rm h}={\cal W}^{\rm s}(\beta)\cap{\cal W}^{\rm u}(\Gamma)$ between the global stable set of the boundary point $\beta$ and the unstable manifold of the attractor $\Gamma$. Therefore, we expect that the action (\[actcont\]) or (\[aacten\]) of the separatrix is the limit of the action of the heteroclinic orbit of the symplectic map evaluated by the sum: $$\label{h_fact}
W_{\rm h}= \frac{1}{2} \sum_{n=-\infty}^{+\infty} \left[x_{n+1} - f(x_n)\right]^2\Big\vert_{\rm h}$$ The approximation of the map action (\[h\_fact\]) by the flow action (\[aacten\]) is justified as the bifurcation is approached as seen in Fig. \[trajhetex\] where we observe that the heteroclinic orbit tends to the separatrix of the Hamiltonian flow (\[flowsymp\]).
Results \[res\]
===============
To investigate the relation between the *activation barrier* $W_0$ and the actions (\[aacten\]) or (\[h\_fact\]) for different types of bifurcations, we have carried out a Monte-Carlo simulation generating $2 \times 10^5$ escape events starting from the attractor $\Gamma$. We have computed the distributions of escape times for different noise amplitudes $\varepsilon$. As these distributions slowly decrease exponentially in time in the limit $\varepsilon\to 0$, the best estimator of their average is simply the statistical average (\[esc\_time\]) according to Eq. (\[rate\_time\]). In this way, we have obtained the mean escape time (\[esc\_time\]) as a function of $\varepsilon$. For $\varepsilon \to 0$, this function shows exponential increase according to Eq. (\[teedep\]). By fitting, we have computed the *activation barrier* $W_0$. It is important to notice that Eq. (\[teedep\]) is valid under the condition $\varepsilon << W_0$ on the values of the noise amplitude we had to consider.
In the case the one-dimensional map has several attractors, we have to consider the first-exit process from one basin of attraction as described in Subsection \[first-exit\_time\].
Transcritical bifurcation
-------------------------
In this section, we study two simple maps which undergo a transcritical bifurcation.
### Logistic map
We first consider the logistic map $$\label{logmap}
x_{n+1} = \mu \, x_n (1 - x_n)$$ which is a well-known model in the fields of hydrodynamics [@K89; @P91], glass formation dynamics [@BR05], and population dynamics [@N95; @HS98]. This map possesses two fixed points : $$\begin{aligned}
x_1 & = & 0 \label{logistic_x1}\\
x_2 & = & 1-1/\mu \label{logistic_x2}\end{aligned}$$ The logistic map undergoes a transcritical bifurcation at $\mu=1$ where these two fixed points exchange their stability. For $\mu < 1$, $x_1$ is stable while $x_2$ is unstable and forms with its non-trivial pre-image $f^{-1}(x_2)$ the boundaries of the basin of attraction of $x_1$: $B(x_1)=]x_2,f^{-1}(x_2)[$. For $\mu > 1$, $x_2$ is stable while $x_1$ is unstable and forms with its non-trivial pre-image the boundaries the basin of attraction of $x_2$: $B(x_2) = ]0,1[$. In both cases, the point at infinity is a second possible attractor for initial conditions outside the described basin of attraction.
![\[situ19\] Phase portrait of the symplectic map (\[mapsymp\]) associated with the logistic map (\[logmap\]) for $\mu=1.9$. Under this condition, the attractor corresponds to the fixed point $\Gamma=(x_2,0)$ with Eq. (\[logistic\_x2\]). The fixed point of the escape barrier is $\beta=(0,0)$. Note that manifolds of the global stable set have to pass by the focal point $\boldsymbol{z}=(1/2,0)$ to cross the nondefinition set given by the singular line $\{(1/2,p): \, p \in \mathbb{R}\}$. The heteroclinic orbit ${\rm h}={\cal W}^{\rm s}(\beta)\cap{\cal W}^{\rm u}(\Gamma)$ is depicted by crosses.](situ19l.pdf){width="50.00000%"}
![\[acthetlog\] Action of the heteroclinic orbit $W_{\rm h}$ and activation barrier $W_0$ (with error bars) from the Monte-Carlo simulation versus the parameter value $\mu$ for the logistic map (\[logmap\]). The blue hatched curve is the analytical action $W_{\rm h}^{\rm c}$ calculated near the transcritical bifurcation and given by Eq. (\[aelogi\]).](acthetlogl.pdf){width="45.00000%"}
For the noisy logistic map, the escape to infinity may occur via the two boundaries of the basin of attraction $B(\Gamma)$. Both boundaries correspond to the same fixed point $x_{\beta}$ since one of the boundaries is the non-trivial pre-image $f^{-1}(x_{\beta})$ of the other. The phase portrait of the symplectic map (\[mapsymp\]) is depicted in Fig. \[situ19\] where we observe similar structures as in Fig. \[expps26\] for the exponential map. In particular, an elliptic island surrounds the non-trivial fixed point at $x=(\mu-1)/(2\mu)$ and $p=-(\mu-1)^2/(4\mu)$. The symplectic map (\[mapsymp\]) has two inverse maps $\phi_{\pm}^{-1}$, which are defined in the domain $p \geq x-(\mu/4)$ limited by the prefocal line (\[prefocal\]). The map (\[mapsymp\]) is not defined on the singular line $x=1/2$ where $f'(x)=0$ and a focal point exists at $\boldsymbol{z}=(1/2,0)$. This focal point is the passage of the global stable set ${\cal W}^{\rm s}(\beta)$, as seen in Fig. \[situ19\]. The heteroclinic orbit ${\rm h}={\cal W}^{\rm s}(\beta)\cap{\cal W}^{\rm u}(\Gamma)$ connecting the fixed points $\Gamma$ to $\beta$ was constructed with an algorithm described in Ref. [@Y98]. This heteroclinic orbit effectively controls the process of noise-induced escape to infinity of trajectories issued from the attractor. Monte-Carlo simulations have allowed us to compute the escape rate and the activation barrier $W_0$, verifying in Fig. \[acthetlog\] that it coincides with the action of the heteroclinic orbit. Near the transcritical bifurcation, the continuous-time approximation provides us with the following analytical expression for this action: $$\label{aelogi}
W_{\rm h}^{\rm c} = \left| \frac{(\mu-1)^3}{3\mu^2} \right|$$ obtained from Eq. (\[aacten\]) with the potential (\[apot\]). This formula applies to both sides of the transcritical bifurcation at $\mu=1$ because of the absolute value $\vert\cdot\vert$. Indeed, the fixed points (\[logistic\_x1\]) and (\[logistic\_x2\]) exchange their role of attractor and barrier at the bifurcation. Figure \[acthetlog\] shows the excellent agreement between the activation barrier $W_0$ and Eq. (\[aelogi\]) as the transcritical bifurcation is approached in the limit $\vert\mu\vert\to 1$. At the bifurcation, the activation barrier vanishes as the cube of the control parameter $\Delta\mu=\mu-1$.
### Exponential map
The exponential map $$\label{exmap}
x_{n+1} = \mu \, x_n \exp\left(-x_n\right) \qquad \mbox{with} \quad \mu >0$$ shares several features with the logistic map. It has two fixed points: $$\begin{aligned}
x_1 & = & 0 \label{exp_x1}\\
x_2 & = & \ln\mu \label{exp_x2}\end{aligned}$$ which cross each other in a transcritical bifurcation at $\mu=1$. Moreover, the point at $x=-\infty$ is also attracting. For $0<\mu < 1$, $x_1$ is stable, $x_2$ is unstable, and the basin of attraction of $x_1$ is $B(x_1) = ]x_2,+\infty[$. For $\mu > 1$, $x_2$ is stable, $x_1$ is unstable, and the basin of attraction of $x_2$ is $B(x_2) = ]x_1,+\infty[$. Contrary to the logistic map, these basins of attraction are not compact and extend up to $x=+\infty$ where the exponential map vanishes and sends the points toward the fixed point $x_1=0$. The other half of the real axis is the basin of attraction of $x=-\infty$.
Phase portraits of the symplectic map (\[mapsymp\]) are depicted in Figs. \[expps26\] and \[exp1pp\], and discussed in Subsection \[phasespaces\]. The singular line is located at $x=1$ and the focal point at $\boldsymbol{z}=(1,0)$. According to Eq. (\[prefocal\]), the prefocal line is $x=p+\mu/{\rm e}$. The inverse (\[invmapsymp\]) is not defined below this prefocal line for $p>x-\mu/{\rm e}$.
![\[acthetex\] Action of the heteroclinic orbit $W_{\rm h}$ and activation barrier $W_0$ (with error bars) from the Monte-Carlo simulation versus the parameter value $\mu$ for the exponential map (\[exmap\]). The blue hatched curve is the analytical action $W_{\rm h}^{\rm c}$ calculated near the transcritical bifurcation and given by Eq. (\[aeexp1\]).](acthetexl.pdf){width="45.00000%"}
An analytical calculation of the activation barrier can be carried out around the transcritical bifurcation at $\mu=1$ by using Eq. (\[aacten\]) with the potential (\[apot\]) yielding $$\label{aeexp1}
W_{\rm h}^{\rm c} = \left| 2 \mu - 2 - 2 \ln\mu - (\ln\mu)^2 \right|$$ which applies below and above the bifurcation. As for the logistic map, the activation barrier vanishes as the cube of the control parameter $\Delta\mu=\mu-1$. This result is verified in Fig. \[acthetex\] by comparison with the activation barrier computed by Monte-Carlo simulations. A very good agreement is also observed with the action (\[h\_fact\]) of the heteroclinic orbit ${\rm h}={\cal W}^{\rm s}(\beta)\cap{\cal W}^{\rm u}(\Gamma)$. It show evidence that the heteroclinic orbits play the key role in the determination of the activation barrier for the noise-induced escape from the attractor.
Tangent bifurcation
-------------------
At the tangent bifurcation, a pair of stable and unstable fixed points emerge in the system. A one-dimensional map illustrating this bifurcation is the quadratic map $$\label{quadmap}
x_{n+1} = \mu + x_n - x_n^2$$ which has the fixed points $$x_{1,2} = \pm \sqrt{\mu}$$ for $\mu \geq 0$. $x_1$ is stable while $x_2$ is unstable. For $\mu>0$, the phase portrait of the symplectic map (\[mapsymp\]) is similar to the one of the logistic map but the elliptic island no longer exists for $\mu <0$ where all the trajectories of the noiseless map [quadmap]{} escape to infinity. For $\mu>0$, the escape form the basin of attraction of $x_1=x_{\Gamma}=+\sqrt{\mu}$ should be activated by the noise. Close to the tangent bifurcation at $\mu=0$, the critical slowing down justifies the treatment with the continuous-time approximation. Using Eq. (\[aacten\]) with the potential (\[apot\]), the activation barrier is here given by $$\label{aequadmap}
W_{\rm h}^{\rm c} = \frac{4}{3} \, \mu^{3/2}$$ Here, we recover the result obtained by Beale that the exponent $3/2$ is universal for the activation barrier close to a tangent bifurcation [@B89].
We notice that the quadratic map (\[quadmap\]) is transformed into the logistic map (\[logmap\]) by the changes $x\to \mu x + (1-\mu)/2$ and $\mu\to(\mu-1)^2/4$. This transformation is folding the parameter space two to one, therefore, mapping the transcritical bifurcation onto the tangent bifurcation. In this way, the exponent $3$ of the transcritical bifurcation is consistent with the exponent $3/2$ of the tangent bifurcation [@B89].
Pitchfork bifurcation
---------------------
In the pitchfork bifurcation, a fixed point which pre-exists to the bifurcation destabilizes and generated two new stable fixed points. These two new attractors are separated by the now unstable fixed point, which is the boundary between the two basins of attraction. At the bifurcation, the two attractors meet with this boundary so that the pitchfork bifurcation is a boundary bifurcation in this regard.
We are concerned by the first-exit process induced by the noise from one basin of attraction to the other across their common boundary, which constitutes a activation barrier between two wells for the kinetic potential (\[apot\]). As explained in Subsection \[first-exit\_time\], this first-exit process is similar to an escape process in the sense that its rate is of Arrhenius type and also given by Eq. (\[teedep\]) in the weak-noise limit. At the pitchfork bifurcation, we expect that the activation barrier $W_0$ should vanish in a way characteristic of the bifurcation. Two maps are investigated to determine this barrier.
### Cubic map
The cubic map $$\label{cubmap}
x_{n+1} = \mu x_n (1 - x_n^2)$$ undergoes a pitchfork bifurcation at the critical parameter value $\mu=1$ and presents bistability above this bifurcation for $\mu>1$. For our purposes, we consider positive values of the parameter $\mu>0$. The cubic map has one fixed point below the bifurcation and three above: $$\begin{aligned}
x_1 & = & 0 \label{cubic_x1}\\
x_{2,3} & = & \pm\sqrt{1-1/\mu} \qquad\mbox{if} \quad \mu>1 \label{cubic_x23}\end{aligned}$$ Moreover, the cubic map has an unstable period-two orbit at $x_{4,5}=\pm\sqrt{1+1/\mu}$. For $\mu<1$, the fixed point $x_1=0$ is an attractor and its basin of attraction is bordered by the period-two orbit: $B(x_1)=]x_5,x_4[$. For $\mu>1$, the fixed point $x_1=0$ becomes unstable and the fixed points $x_2$ and $x_3$ are two attractors with their basin of attraction given by $B(x_2)=]x_1=0,x_4[$ and $B(x_3)=]x_5,x_1=0[$, respectively. Outside the interval delimited by the period-two orbit, trajectories can escape to infinity, which is a further attractor of the cubic map. The symplectic map (\[mapsymp\]) associated with the cubic map has three hyperbolic fixed points in the invariant subspace $p=0$ for $\mu>1$. Now, there exist two elliptic islands symmetrically located around the origin. The origin $\beta=(0,0)$ stands for the barrier while $\Gamma=(x_2,0)$ for instance. A heteroclinic orbit ${\rm h}={\cal W}^{\rm s}(\beta)\cap{\cal W}^{\rm u}(\Gamma)$ connects the hyperbolic fixed point $\Gamma$ to $\beta$. The associated symplectic map (\[mapsymp\]) has two focal points $\boldsymbol{z}_{\pm}=(\pm 1/\sqrt{3},0)$ and two corresponding prefocal lines (\[prefocal\]).
For $\mu>1$, noise induces random jumps between the two basins $B(x_2)$ and $B(x_3)$, as well as escape toward infinity. The barrier between the two basins $B(x_2)$ and $B(x_3)$ is located at the origin $x_1=0$ and vanishes at the pitchfork bifurcation while the barrier for escaping toward infinity is the period-two orbit, which does not vanish around $\mu=1$. Accordingly, the escape events remain rare and negligible with respect to the first-exit events from one basin to the other. The difference of time scale between first-exit and escape events allows us to investigate the former without perturbation by the latter in Monte-Carlo simulations in the parametric domain of interest. In the weak-noise limit, the first-exit process is controlled by the heteroclinic orbit ${\rm h}={\cal W}^{\rm s}(\beta)\cap{\cal W}^{\rm u}(\Gamma)$. Using the continuous-time approximation, we can calculate analytically the activation barrier near the pitchfork bifurcation with Eq. (\[aacten\]) for the cubic map: $$\label{actancub}
W_{\rm h}^{\rm c} = \frac{1}{2 \mu} (\mu-1)^2$$ which holds for $\mu>1$. The excellent agreement with the results of Monte-Carlo simulations is shown in Fig. \[acthetcub\] where the action $W_{\rm h}$ of the heteroclinic orbit is also plotted. Here, the activation barrier vanishes as the square of the control parameter $\Delta\mu=\mu-1$.
![\[acthetcub\] Action of the heteroclinic orbit $W_{\rm h}$ and activation barrier $W_0$ (with error bars) from the Monte-Carlo simulation versus the parameter value $\mu$ for the cubic map (\[cubmap\]). The blue hatched curve is the analytical action $W_{\rm h}^{\rm c}$ calculated near the pitchfork bifurcation and given by Eq. (\[actancub\]).](acthetcubl.pdf){width="45.00000%"}
### Gaussian map
Another example of bistable system is given by the Gaussian map: $$\label{exp2}
x_{n+1} = \mu \, x_n \exp(-x_n^2) \qquad \mbox{with} \quad \mu >0$$ Its fixed points are $$\begin{aligned}
x_1 & = & 0 \label{Gauss_x1}\\
x_{2,3} & = & \pm\sqrt{\ln\mu} \qquad\mbox{if} \quad \mu>1 \label{Gauss_x23}\end{aligned}$$ The pitchfork bifurcation happens at $\mu=1$ and bistability manifests itself for $\mu>1$. Contrary to the cubic map, escape to infinity is not possible for this map, even in the presence of noise. In this regard, this model is very convenient to investigate the effects of noise on bistability. For $0<\mu<1$, the whole real line is the basin of attraction of the origin $x_1=0$. For $\mu>1$, the basins of attractions of the two attractors emerging at the pitchfork bifurcation are respectively $B(x_2)=]-\infty,0[$ and $B(x_3)=]0,+\infty[$. Here also, the symplectic map (\[mapsymp\]) has two focal points $\boldsymbol{z}_{\pm}=(\pm 1/\sqrt{2},0)$ and two corresponding prefocal lines (\[prefocal\]).
Taking $x_\beta=0$ and $x_\Gamma=x_2$ for instance and calculating the action of the heteroclinic orbit ${\rm h}={\cal W}^{\rm s}(\beta)\cap{\cal W}^{\rm u}(\Gamma)$ in the continuous-time approximation, we find $$\label{actanGauss}
W_{\rm h}^{\rm c} = \mu -1 - \ln\mu$$ which holds for $\mu>1$. Here again, the comparison with the results of Monte-Carlo simulations and the action of the symplectic map itself are excellent as shown in Fig. \[acthetex2\]. As for the cubic map, the activation barrier vanishes as the square of the control parameter $\Delta\mu=\mu-1$, which characterizes the pitchfork bifurcation.
![\[acthetex2\] Action of the heteroclinic orbit $W_{\rm h}$ and activation barrier $W_0$ (with error bars) from the Monte-Carlo simulation versus the parameter value $\mu$ for the Gaussian map (\[exp2\]). The blue hatched curve is the analytical action $W_{\rm h}^{\rm c}$ calculated near the pitchfork bifurcation and given by Eq. (\[actanGauss\]).](acthetexp2l.pdf){width="45.00000%"}
Conclusions \[conc\]
====================
In this paper, we have studied noise-induced escape or exit from bifurcating fixed points of one-dimensional maps. Such fixed points correspond to periodic orbits of strongly dissipative dynamical systems. In this study, we have considered the transcritical, tangent, and pitchfork bifurcations [@N95]. We followed a symplectic approach which applies in the weak-noise limit. For noise generated by independent Gaussian random variables, the time evolution can be expressed in terms of path integrals defined by an action functional. In the weak-noise limit, path integrals are dominated by the contributions of trajectories which are the extremals of the action functional. For the present discrete-time systems, these trajectories are ruled by a symplectic map. This two-dimensional area-preserving map reduces to the noiseless one-dimensional map on its one-dimensional phase space which is left invariant.
The noninvertibility of the one-dimensional map and the associated symplectic map is at the origin of special phase-space structures. In particular, the two-dimensional map is not defined on [*singular lines*]{}, except at their intersections with the invariant subspace. These intersections are the so-called [*focal points*]{} [@BGM99; @BGM03]. The multiple inverses of the symplectic map are defined on domains bordered by so-called [*prefocal lines*]{} [@BGM99; @BGM03]. Strangely enough, the noninvertibility has for consequence that each focal point is the pre-image of a corresponding prefocal line. Under such circumstances, invariant curves such as the stable and unstable sets may form loops attached to the focal points and they may be composed of several disjoint pieces, as here observed for the global stable set. Otherwise, the symplectic map forms typical phase-space structures such as elliptic islands surrounded by chaotic zones and homoclinic or heteroclinic tangles. Heteroclinic orbits have been identified which connect the hyperbolic fixed points corresponding to the attractor and the top of the barrier separating the basins of attraction.
Noise induces the escape or exit from each basin of attraction, changing the attractors of the noiseless map into metastable states. The rates of escape or first exit have Arrhenius’ dependence on the noise amplitude. In the present non-thermal stochastic systems, the analogue of the activation energy is given by the action of the heteroclinic orbit connecting the attractor to the top of the barrier. We have developed an analytical method to calculate these actions close to the bifurcations. Thanks to the critical slowing down near the bifurcation, the noisy map behaves as a continuous-time stochastic system, whereupon the corresponding symplectic map can be approximated by a Hamiltonian flow with one degree of freedom [@OM53; @FW84]. For this flow, the action of the heteroclinic orbit can be calculated analytically.
bifurcation action exponent
-- --------------- -- ----------------------------- -- ---------- -- -- -- -- --
transcritical $\sim\vert\Delta\mu\vert^3$ $3$
tangent $\sim\Delta\mu^{3/2}$ $3/2$
pitchfork $\sim\Delta\mu^2$ $2$
: Scaling behavior of the action $W_0$ of the activation barrier versus the control parameter $\Delta\mu\equiv\mu-\mu_{\rm c}$ and universal exponent for noise-induced escape or first exit from the attractor undergoing different bifurcations. The corresponding rate is given by $\gamma,\tilde\gamma \sim \exp(-W_0/\varepsilon)$ in terms of the noise amplitude $\varepsilon$. \[tab.bif\]
Applying this method to several one-dimensional noisy maps, we have shown how the activation barrier for noise-induced escape vanishes with the control parameter $\Delta\mu\equiv\mu-\mu_{\rm c}$ near the transcritical, tangent, and pitchfork bifurcations. At these bifurcations, the activation barrier scales as $W_0\sim \vert\Delta\mu\vert^{\alpha}$ with a universal exponent $\alpha$ characteristic of the bifurcation, as summarized in Table \[tab.bif\]. The analytical results are in excellent agreement with Monte-Carlo simulations.
In conclusion, the symplectic approach is a powerful method to deal with stochastic dissipative systems in the weak-noise limit and to delineate the universality of their properties near bifurcations.
$\ast \, \ast \, \ast$
The authors thank P. de Buyl, D. Andrieux and T. Gilbert for useful discussions and support.
[**Acknowledgments.**]{} This research is financially supported by the Belgian Federal Government (IAP project “NOSY"), the “Communauté française de Belgique” (contract “Actions de Recherche Concertées” No. 04/09-312), and the F.R.S.-FNRS Belgium (contract F. R. F. C. No. 2.4577.04).
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[^1]: When none are specified, the domain of integration is the whole state space. For example, here it is $\mathbb{R}$.
|
---
author:
- |
**H. Eshraghi\
[[email protected]]{}\
\
[**R. Hafezi**]{}\
[[email protected]]{}\
\
[**Sh. Salarian**]{}\
[[email protected]]{}\
[*Department of Mathematics, University of Isfahan, P.O.Box: 81746-73441,*]{}\
[*Isfahan, Iran and* ]{}\
[*School of Mathematics, Institute for Research in Fundamental Sciences (IPM),* ]{}\
[*P.O.Box: 19395-5746, Tehran, Iran.*]{}\
\
**Z.W. Li\
[[email protected]]{}\
[*Department of Mathematics, Jiangsu Normal University, Xuzhou 221116,*]{}\
[*People’s Republic of China*]{}****
title: '**Gorenstein Projective Modules Over Triangular Matrix Rings[^1]**'
---
[**Abstract.**]{} Let $R$ and $S$ be Artin algebras and $\Gamma$ be their triangular matrix extension via a bimodule ${}_S M_R$. We study totally acyclic complexes of projective $\Gamma$-modules and obtain a complete description of Gorenstein projective $\Gamma$-modules. We then use this to construct some examples of Cohen-Macaulay finite and virtually Gorenstein triangular matrix algebras.
[**2000 Mathematics Subject Classification:**]{} 18G25, 16G10.
[**Keywords:**]{} Gorenstein projective, totally acyclic, Cohen-Macaulay finite.
[Introduction]{}
================
Let $A$ be a not necessarily commutative ring with nonzero unity, ${{\rm{Mod}}}A$ (resp. mod$A$) be the category of all (resp. finitely generated) left $A$-modules, and ${{\rm{Proj}}}(A)$ (resp. [proj]{}($A$)) be the full subcategory of all (resp. finitely generated) projective $A$-modules. Following [@EJ], an $A$-module $X$ is said to be Gorenstein projective if it is a syzygy of a totally acyclic complex of projective $A$-modules, i.e. an exact complex of projective $A$-modules which will be left exact under applying ${{\rm{Hom}}}_A(-,\,P)$ for every projective $A$-module $P$. Throughout the paper ${{\rm{GProj}}}(A)$ (resp. [Gproj]{}($A$)) denotes the category of all (resp. finitely generated) Gorenstein projective $A$-modules; Gorenstein injective $A$-modules are defined dually.
Regardless of their role as a foundation of Gorenstein homological algebra, Gorenstein projective modules usually receive particular importance from several points of view. They are used in the theory of singularities (see e.g. [@Y] for a standard text) and in Tate cohomology of algebras ([@AM], [@B]). Also Cohen-Macaulay finite algebras, having recently been of particular attention ([@Bb], [@C], [@LZb]), are defined in terms of Gorenstein projective modules; recall that an Artin algebra $\Lambda$ is said to be of finite Cohen-Macaulay type (C.M. finite for short) provided it has only finitely many, up to isomorphism, indecomposable finitely generated Gorenstein projective modules, that is to say, the subcategory ${\rm Gproj}(\Lambda)$ is of finite type.
This article aims at studying Gorenstein projective modules over triangular matrix rings. Such modules have been studied in some special cases, see for example [@LZa], [@XZ], and [@Z]. The approach we will take here is to study totally acyclic complexes of projective modules and regard Gorenstein projective modules as their syzygies. So throughout the rest of this paper, we fix a formal triangular matrix ring $$\Gamma = \left(
\begin{array}{ccc}
R & 0 \\
M & S
\end{array} \right)$$ where $R$ and $S$ are two associative not necessarily commutative rings with unity and $_{S}M_R$ is an $S-R$ bimodule. It is well-known that when $R$ and $S$ are $k$-Artin algebras, $k$ being a commutative artinian ring, and $M$ is finitely generated over $k$ which acts centrally on $M$, then $\Gamma$ is a $k$-Artin algebra. Therefore whenever $\Gamma$ is discussed from representation-theoretic aspects, we are assuming that it is an Artin algebra.
We then will use these information to construct some accessible examples of Cohen-Macaulay finite Artin algebras; namely, Cohen-Macaulay finite triangular matrix Artin algebras. In particular, we will deal with how $\Gamma$ inherits C.M. finiteness from $R$ and $S$ and vice versa. Finally, Gorenstein projective $\Gamma$-modules are used to study how virtually Gorensteinness may transfer from $R$ and $S$ to $\Gamma$; this is a concept which is closely related to the notion of Cohen-Macaulay finiteness.
[Gorenstein Projective Modules]{}
=================================
Keep the notations of the previous section. Recall (e.g. from [@ARS]) that the category ${{\rm{Mod}}}\Gamma$ may be identified with the category $\mathfrak{C}$ consisting of all triples $(X,\,Y)_\varphi$ where $X\in{{\rm{Mod}}}R$, $Y\in{{\rm{Mod}}}S$ and $\varphi:M\otimes_R X{\longrightarrow}Y$ is an $S$-linear map. Using this, we define two evaluation functors $e^1:{{\rm{Mod}}}\Gamma{\longrightarrow}{{\rm{Mod}}}R$ and $e^2:{{\rm{Mod}}}\Gamma{\longrightarrow}{{\rm{Mod}}}S$ in the following way: $e^1((X, Y)_\varphi)=X$ and $e^2((X, Y)_\varphi)=Y$, for every $\Gamma$-module $(X, Y)_\varphi$ and with the obvious rules over the morphisms. It is well-known that these functors both admit left and right adjoints. The subscriptions $\lambda$ and $\rho$ are reserved respectively to denote the left and right adjoint functors.
Before defining these adjoint functors we need to recall that any $\Gamma$-module could also be represented as $(X, Y)_\varphi$ where $X\in{{\rm{Mod}}}R$, $Y\in{{\rm{Mod}}}S$, and $\varphi:X{\longrightarrow}{{\rm{Hom}}}_S(M,\,Y)$ is a $R$-linear map. Hence for every $R$-module $X$, $e^1_\lambda(X)=(X,\,M\otimes_R X)_1$ and $e^1_\rho(X)=(X,\,0)_0$. Similarly, for any $S$-module $Y$, $e^2_\lambda(Y)=(0,\,Y)_0$ and $e^2_\rho(Y)=({{\rm{Hom}}}_S(M,\,Y),\, Y)_1$. More briefly, there are two adjoint pair of functors $(e^i,\,e^i_\rho)$ and $(e^i_\lambda,\,e^i)$, $i=1,2$.
We need to extend these pairs to the corresponding homotopy categories. So we briefly recall that ${\mathbb{K}(R)}$, the homotopy category of $R$, has all complexes of $R$-modules as objects and its morphisms are the homotopy equivalence classes of the chain maps. ${{\mathbb{K}({{{\rm{Proj}}}} \ R)}}$ is the homotopy category formed by all complexes of projective $R$-modules and ${{\mathbb{K}_{\rm{ac}}({{{\rm{Proj}}}} \ R)}}$ (resp. ${{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ R)}}$) is the full subcategory of ${{\mathbb{K}({{{\rm{Proj}}}} \ R)}}$ of all acyclic (resp. totally acyclic) complexes. Note that they are standard examples of triangulated categories [@N] and that extending the aforementioned adjoint pairs to the homotopy categories, a procedure that may be done in a natural way, leads in triangulated functors denoted by $k^1$, $k^2$ (the extended evaluation functors) and $k^1_\lambda$, $k^1_\rho$, $k^2_\lambda$, and $k^2_\rho$ (their corresponding adjoints).
The following result is well-known; see e.g. [@ET Proposition 2.1].
\[TPrj\] A $\Gamma$-module $(X, Y)_\varphi$ is projective if and only if the following statements are satisfied.
- $\varphi:M\otimes_R X{\longrightarrow}Y$ is a $S$-monomorphism.
- $X\in{{\rm{Proj}}}(R)$ and ${{\rm{Coker}}}(\varphi)\in{{\rm{Proj}}}(S)$.
It should be pointed out that an analogous description for injective $\Gamma$-modules, which will be used in Theorem \[TGIG\], has been included in [@ET].
Let ${\mathcal{E}}\in{{\mathbb{K}({{{\rm{Proj}}}} \ \Gamma)}}$. According to the definitions, we represent ${\mathcal{E}}$ as $${\mathcal{E}}:\quad\cdots{\longrightarrow}({\mathcal{E}}_1^a,\,{\mathcal{E}}_1^b)_{\varphi_1}{\longrightarrow}({\mathcal{E}}_0^a,\,{\mathcal{E}}_0^b)_{\varphi_0}{\longrightarrow}({\mathcal{E}}_{-1}^a,\,{\mathcal{E}}_{-1}^b)_{\varphi_{-1}}{\longrightarrow}\cdots$$ where, for any integer $i$, $({\mathcal{E}}_i^a,\,{\mathcal{E}}_i^b)_{\varphi_i}$ is a projective $\Gamma$-module. Note that, in view of the above lemma, there exists for any integer $i$ a split exact sequence $0{\longrightarrow}M\otimes_R{\mathcal{E}}_i^a{\longrightarrow}{\mathcal{E}}_i^b{\longrightarrow}{{\rm{Coker}}}\varphi_i{\longrightarrow}0$. This induces a complex $$\cdots{\longrightarrow}{{\rm{Coker}}}(\varphi_1)
{\longrightarrow}{{\rm{Coker}}}(\varphi_0){\longrightarrow}{{\rm{Coker}}}(\varphi_{-1}){\longrightarrow}\cdots$$ which we denote by ${{\rm{Coker}}}{\mathcal{E}}$.
Before stating the following lemma, we recall that the symbol ${}^\perp$ is reserved to denote the (left and right) orthogonal classes with respect to the functor ${{\rm{Ext}}}^1(-, -)$. Also, ${\rm Add}({}_S M)$ contains all $S$-modules which are direct summands of arbitrary direct sums of copies of $M$.
\[LTAPRSG\]
Let ${\mathcal{E}}\in{{\mathbb{K}({{{\rm{Proj}}}} \ R)}}$ and ${\mathcal{E}}'\in{{\mathbb{K}({{{\rm{Proj}}}} \ S)}}$.
- Assume that the functor $M\otimes_R-$ takes every acyclic complex of projective $R$-modules to an acyclic complex of $S$-modules. Then ${\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ R)}}$ if and only if $k^1_\lambda({\mathcal{E}})\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ \Gamma)}}$.
- Assume ${\rm Add}({}_S M)\subseteq {{\rm{GProj}}}(S)^\perp$. Then ${\mathcal{E}}'\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ S)}}$ if and only if $k^2_\lambda({\mathcal{E}}')\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ \Gamma)}}$.
Since the functors $e^1_\lambda$ and $e^2_\lambda$ preserve projective modules, the proof follows easily from Lemma \[TPrj\] and the adjointness properties. Note that the assumption on $M$ in the first statement yields that $k^1_\lambda({\mathcal{E}})\in{{\mathbb{K}_{\rm{ac}}({{{\rm{Proj}}}} \ \Gamma)}}$ whenever ${\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ R)}}$. Likewise, in the second one, the assumption gives that when ${\mathcal{E}}'\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ S)}}$,then for any projective $R$-module $P$, the complex ${{\rm{Hom}}}_S({\mathcal{E}}', M\otimes_R P)$ is acyclic from which the desired result might be deduced in conjunction with Lemma \[TPrj\].
As established in the above lemma, the following statements are crucial throughout the rest of this paper:
$(1)$ [*The functor $M\otimes_R-$ takes every acyclic complex of projective $R$-modules to an acyclic complex of $S$-modules*]{}.
$(2)$ [*${\rm Add}({}_S M)\subseteq {{\rm{GProj}}}(S)^\perp=\{Y\in{{\rm{Mod}}}S: {{\rm{Ext}}}^1_S(G, Y)=0, \forall G\in{{\rm{GProj}}}(S)$}*]{}.
We just refer to them by mentioning their assigned numbers.
The following proposition provides a complete description of totally acyclic complexes of projective $\Gamma$-modules.
\[TTAP\] Let the statements $(1)$ and $(2)$ hold and let ${\mathcal{E}}\in{{\mathbb{K}({{{\rm{Proj}}}} \ \Gamma)}}$. Then ${\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ \Gamma)}}$ if and only if $k^1({\mathcal{E}})\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ R)}}$ and ${{\rm{Coker}}}{\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ S)}}$.
Assume first that ${\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ \Gamma)}}$. Then the complexes $k^1({\mathcal{E}})$ and $k^2({\mathcal{E}})$ are acyclic. Since, by $(1)$, $M\otimes k^1({\mathcal{E}})$ is acyclic, the acyclicity of ${{\rm{Coker}}}{\mathcal{E}}$ follows. Then there exists a triangle $$\xymatrix{k_{\lambda}^1 k^1({\mathcal{E}}) \ar[r] & {\mathcal{E}}\ar[r] & k^2_\lambda({{\rm{Coker}}}{\mathcal{E}})
\ar@{~>}[r] & } \ \ \ (*)$$ in ${{\mathbb{K}({{{\rm{Proj}}}} \ \Gamma)}}$. For any $Q\in{{\mathbb{K}({{{\rm{Proj}}}} \ S)}}$, it is clear that $k^2_\lambda(Q)\in{{\mathbb{K}({{{\rm{Proj}}}} \ \Gamma)}}$. Apply the homological functor ${{\rm{Hom}}}_{{{\mathbb{K}({{{\rm{Proj}}}} \ \Gamma)}}}(-,\,k^2_\lambda(Q))$ on $(*)$ to obtain the obvious long exact sequence. Since for any $R$-module $U$ and $S$-module $V$, ${{\rm{Hom}}}_{{{\mathbb{K}({{{\rm{Proj}}}} \ \Gamma)}}}(k_{\lambda}^1 k^1(U),\,k^2_\lambda(V))=0$, the assumption implies that ${{\rm{Coker}}}{\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ S)}}$ so, by Lemma \[LTAPRSG\], $k^2_\lambda({{\rm{Coker}}}{\mathcal{E}})\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ S)}}$. The rest of the claim follows from the above triangle and another application of Lemma \[LTAPRSG\]; the converse may be proved using a similar argument.
\[LELGP\]
- If the statement $(1)$ holds, then for any Gorenstein projective $R$-module $U$, $e^1_\lambda(U)$ is a Gorenstein projective $\Gamma$-module.
- If the statement $(2)$ holds, then for any Gorenstein projective $S$-module $V$, $e^2_\lambda(V)$ is a Gorenstein projective $\Gamma$-module.
Is a direct consequence of Lemma \[LTAPRSG\].
We are now in the position to prove the main result, which is a complete description of Gorenstein projective $\Gamma$-modules.
\[TGPG\] Suppose that both of the statements $(1)$ and $(2)$ are satisfied. Then a $\Gamma$-module $(X,\,Y)_\varphi$ is Gorenstein projective if and only if
- $X$ and ${{\rm{Coker}}}\varphi$ are Gorenstein projective respectively as $R$ and $S$-modules.
- $\varphi$ is a monomorphism.
The sufficiency follows from the exact sequence $$0{\longrightarrow}(X,\,M\otimes_R X)_1{\longrightarrow}(X,\,Y)_\varphi{\longrightarrow}(0,\,{{\rm{Coker}}}\varphi)_0{\longrightarrow}0\quad (**)$$ in conjunction with Lemma \[LELGP\]. If, conversely, $(X,\,Y)_\varphi$ is a Gorenstein projective $\Gamma$-module, then it is, say, the 0-th syzygy of some ${\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ \Gamma)}}$. Note that by Proposition \[TTAP\], ${{\rm{Coker}}}{\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ S)}}$ and $k^1({\mathcal{E}})\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ R)}}$. Moreover, $X=\ker(k^1({\mathcal{E}})_0{\rightarrow}k^1({\mathcal{E}})_{-1})$ and ${{\rm{Coker}}}\varphi=\ker(({{\rm{Coker}}}{\mathcal{E}})_0{\rightarrow}({{\rm{Coker}}}{\mathcal{E}})_{-1})$. Hence the statement $(a)$ follows. That $\varphi$ is a monomorphism follows because $M\otimes k^1({\mathcal{E}})$ is an exact complex.
In the sequel, we will proceed in the reverse direction. Namely, we will examine how close we are to the statements $(1)$ and $(2)$ by knowing that the above description for all Gorenstein projective $\Gamma$-modules is valid.
\[RCGP\] Suppose that all Gorenstein projective $\Gamma$-modules can be classified as mentioned in the above theorem. We first show that $(2)$ holds. To this end, it is enough to prove that for any index set $I$, $\oplus_I M \in {\rm (GProj(S))^\perp}$ or, equivalently, any short exact sequence $$0 {\longrightarrow}\oplus_I M {\stackrel}{\alpha}{\longrightarrow}D {\stackrel}{\beta}{\longrightarrow}G {\longrightarrow}0$$ with $G$ Gorenstein projective $S$-module splits. From the hypothesis, it follows that $(\oplus_I R, D)_\alpha$ is a Gorenstein projective $\Gamma$-module. So it is the 0-th syzygy of some ${\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ \Gamma)}}$. Hence we get a short exact sequence $$0 {\longrightarrow}(\oplus_I R, D)_\alpha {\stackrel}{f}{\longrightarrow}{\mathcal{E}}_0 {\stackrel}{g}{\longrightarrow}(K_1,K_2)_\gamma {\longrightarrow}0$$ in which $(K_1, K_2)_\gamma$ is also a Gorenstein projective $\Gamma$-module. In view of the fact that $K_1$ should be Gorenstein projective as $R$-module, one infers that the sequence $0 {\longrightarrow}\oplus_I R{\stackrel}{f^a} {\longrightarrow}{\mathcal{E}}^a_0 {\stackrel}{g^a}{\longrightarrow}K_1 {\longrightarrow}0$ splits. Now apply Lemma \[TPrj\] and the commutative exact diagram
$$\xymatrix{
0 \ar[r] & \oplus_I M \ar[r]^{1_M\otimes f^a} \ar[d]^{\alpha} & M\otimes {\mathcal{E}}^a_0 \ar[r]^{1_M\otimes g^a} \ar[d]^{\varphi} &
M\otimes K_1 \ar[r] \ar[d]^{\gamma} & 0 \\
0\ar[r] & D\ar[r]^{f^b} & {\mathcal{E}}^b_0 \ar[r]^{g^b} & K_2 \ar[r] & 0 }$$
to show that the sequence at the beginning of the argument is split.
Next, we prove that ${\rm Tor}_1^R(M,\,P)=0$ for any Gorenstein projective $R$-module $P$. So let $P$ be such a module. Then, by the hypothesis, $(P,\,M\otimes_R P)_1$ is Gorenstein projective and so is the 0-th syzygy of some ${\mathcal{E}}\in{{\mathbb{K}_{\rm{tac}}({{{\rm{Proj}}}} \ \Gamma)}}$. Hence we get a short exact sequence $$0{\longrightarrow}(K_1,\, K_2)_\gamma{\longrightarrow}{\mathcal{E}}_{-1}{\longrightarrow}(P,\,M\otimes_R P)_1{\longrightarrow}0$$ with $(K_1,\,K_2)_\gamma$ Gorenstein projective. This gives a commutative exact diagram
$$\xymatrix{
& M\otimes_R K_1\ar[r]^{\omega}\ar[d]^\gamma & M\otimes_R {\mathcal{E}}_{-1}^a\ar[r]\ar[d] & M\otimes_R P\ar[r]\ar[d]^{=} & 0\\
0\ar[r] & K_2\ar[r] & {\mathcal{E}}_{-1}^b\ar[r] & M\otimes_R P\ar[r] & 0.
}$$ Since, by the assumption, $\psi$ is a monomorphism it follows that $\omega$ is a monomorphism, implying that ${\rm Tor}_1^R(M,\,P)=0$. It is straight forward to check that this is equivalent to saying that $M\otimes_R-$ preserves every totally acyclic complex of projective $R$-modules.
Recall that the ${\rm T}_2$-extension of a ring $R$ is given by $${\rm T}_2(R) = \left(
\begin{array}{ccc}
R & 0 \\
R & R
\end{array} \right)$$ and that every module over ${\rm T}_2(R)$ is a homomorphism $X{\stackrel}{\varphi}{\longrightarrow}Y$ of $R$-modules. We include the following corollary of Theorem \[TGPG\].
A ${\rm T}_2(R)$-module $X{\stackrel}{\varphi}{\longrightarrow}Y$ is Gorenstein projective if and only if $X$ and ${{\rm{Coker}}}\varphi$ are both Gorenstein projective $R$-modules and $\varphi$ is a monomorphism.
It is worth noting that the dual version of the above arguments may be applied to deduce a corresponding classification for Gorenstein injective modules over $\Gamma$. So we skip the proofs and record the dual version of Theorem \[TGPG\]. Just note that the following conditions which are, in some sense, dual to $(1)$ and $(2)$ should be fulfilled.
$(3)$ [*The functor ${{\rm{Hom}}}_S(M, -)$ takes every acyclic complex of injective $S$-modules to an acyclic complex of $R$-modules*]{}.
$(4)$ [*${{\rm{Hom}}}_S(M, I)\in{}^\perp{\rm GInj}(R)$, for all injective $S$-module $I$*]{}.
\[TGIG\] Suppose that the statements $(3)$ and $(4)$ are satisfied. Then a $\Gamma$-module $(X,\, Y)_\varphi$ is Gorenstein injective if and only if
- $Y$ and $\ker\varphi$ are Gorenstein injective respectively as $S$ and $R$-modules.
- $\varphi:X{\longrightarrow}{{\rm{Hom}}}_S(M,\,Y)$ is an epimorphism.
A ${\rm T}_2(R)$-module $X{\stackrel}{\varphi}{\longrightarrow}Y$ is Gorenstein injective if and only if $Y$ and ${{\rm{Ker}}}\varphi$ are both Gorenstein injective $R$-modules and $\varphi$ is an epimorphism.
[C.M. Finiteness And Virtually Gorensteinness]{}
================================================
[*3.1. C.M. finiteness*]{}. We will now deal with how C.M. finiteness may transfer from the algebras $R$ and $S$ to the triangular matrix algebra $\Gamma$. We start with the following easy observation. Assume that $(1)$ and $(2)$ are satisfied and $\Gamma$ is of finite Cohen-Macaulay type. For an indecomposable finitely generated Gorenstein projective $R$-module $X$, it is routine to verify that $e^1_\lambda(X)=(X,\,M\otimes_R X)_{1}$ is an indecomposable finitely generated $\Gamma$-module which is Gorenstein projective by Lemma \[LELGP\]. Since $\Gamma$ is C.M. finite one has only finitely many, up to isomorphism, choices for $X$, that is, $R$ should be of finite Cohen-Macaulay type. A similar argument applies to deduce that $S$ is also of finite Cohen-Macaulay type.
Therefore $\Gamma$ C.M. finite implies $R$ and $S$ C.M. finite while there are examples for which the converse fails. For, let $k$ be an algebraically closed field and consider the self injective $k$-algebra $\Lambda_t=k[x]/(x^t)$ with $t>5$. By [@Bb Example 4.17], the ${\rm T}_2$-extension of $\Lambda_t$ $${\rm T}_2(\Lambda_t)=\left(
\begin{array}{ccc}
\Lambda_t& 0 \\
\Lambda_t & \Lambda_t
\end{array} \right)$$ is of infinite Cohen-Macaulay type while $\Lambda_t$ itself is an algebra even of finite representation type.
However the material prepared in the previous section makes it possible to overcome this defect in some cases. Recall that an Artin algebra $\Lambda$ is said to be [*Cohen-Macaulay free*]{} if ${\rm Gproj}(\Lambda)={\rm proj}(\Lambda)$.
\[TMR1\] Assume $(1)$ and $(2)$ hold.
- Let $R$ be C.M. free. Then $\Gamma$ is of finite Cohen-Macaulay type if and only if so is $S$.
- Let $S$ be C.M. free. Then $\Gamma$ is of finite Cohen-Macaulay type if and only if so is $R$.
- $\Gamma$ is C.M. free if and only if so are $R$ and $S$.
$(i)$ Let $(X,\,Y)_\varphi$ be an indecomposable finitely generated Gorenstein projective $\Gamma$-module. Consider the short exact sequence $(**)$ (in the proof of Theorem \[TGPG\]) in ${{\rm{Mod}}}\Gamma$. One may apply Theorem \[TGPG\] to deduce that $X\in{\rm Gproj}(R)={\rm proj}(R)$, that is, $(X,\, M\otimes_R X)_1$ is a finitely generated projective $\Gamma$-module by \[TPrj\]. On the other hand, it follows from Theorem \[TGPG\] that $(0,\,{{\rm{Coker}}}\varphi)_0\in{\rm Gproj}(\Gamma)$, i.e., the sequence $(**)$ splits. Since $(X,\,Y)_\varphi$ is indecomposable, either $(X,\,Y)_\varphi=(0,\,{{\rm{Coker}}}\varphi)_0$ or $(X,\,Y)_\varphi=(X,\,M\otimes_R X)_1$. It follows that in the first case ${{\rm{Coker}}}\varphi$ is an indecomposable finitely generated Gorenstein projective $S$-module while in the second one $X$ is an indecomposable finitely generated projective $R$-module. This gives the sufficiency; the necessity follows from the above remarks and the proof of $(ii)$ is similar. Also $(iii)$ can be deduced directly from Lemma \[TPrj\] and Theorem \[TGPG\].
Note that important examples arise when $R$ or $S$ are algebras of finite global dimension since for such algebras projectives and Gorenstein projectives coincide.
The following example illustrates an application of Theorem \[TGPG\].
\[EALL\] Let $A_n$ be the quiver $v_1{\rightarrow}\cdots{\rightarrow}v_n$. Then $RA_n$, the path algebra of $A_n$ over the Artin algebra $R$, is the lower triangular $n\times n$ matrix algebra defined inductively as $$T_n(R)=\left(
\begin{array}{ccc}
R & 0 \\
N & {\rm T}_{n-1}(R)
\end{array} \right)$$ where $N=R\oplus R\cdots\oplus R$ ($n-1$ copies), as a right $R$-module, whose left $RA_{n-1}={\rm T}_{n-1}(R)$-structure corresponds to the representation $R{\rightarrow}\cdots{\rightarrow}R$ of $A_{n-1}$ and ${\rm T}_1(R)=R$. Since $N$ is projective both as $R$- and ${\rm T}_{n-1}(R)$-module, from Theorem \[TGPG\] we deduce that any Gorenstein projective ${\rm T}_n(R)$-module is of the form $(X,\,Y)_\varphi$ where $X\in{{\rm{GProj}}}(R)$, $\varphi:N\otimes_R X{\longrightarrow}Y$ is a monomorphism with ${{\rm{Coker}}}\varphi\in{{\rm{GProj}}}({\rm T}_{n-1}(R))$. Assume now that ${\rm T}_{n-1}(R)$ is of finite Cohen-Macaulay type for some $n\geq 1$ and pick a module $Z$ with ${\rm add }(Z)={\rm Gproj}({\rm T}_{n-1}(R))$. Set $\Lambda={\rm End}(Z)^{{{\rm{op}}}}$. Following [@LZa], we let $\mathcal{C}$ be the subcategory of the morphism category of ${\rm proj}(\Lambda)$ consisting of all monomorphisms $f:P_2{\longrightarrow}P_1$ with ${{\rm{Coker}}}f\in {\rm proj}^{\leq 1} \Omega(\Lambda)$ where the latter is the subcategory of all torsionless modules of projective dimension at most 1; see pages 1809-1811 of [@LZa] for the details. Moreover, put $\mathcal{C'}$ be the subcategory of $\mathcal{C}$ consisting of all $f:P_2{\longrightarrow}P_1$ satisfying the extra assumption $P_2={{\rm{Hom}}}_{T_{n-1}(R)}(Z,\, N\otimes_R X)$ where $X$ is a Gorenstein projective $R$-module; it is closed under direct summands and the functor ${{\rm{Hom}}}_{{\rm T}_{n-1}(R)}(Z,\,-)$ provides an embedding ${\rm Gproj}({\rm T}_{n}(R))\hookrightarrow \mathcal{C'}$. Therefore by the proof of [@LZa Theorem 1.2] if ${\rm proj}^{\leq 1} \Omega(\Lambda)$ is of finite type, the subcategory $\mathcal{C'}$, and so ${\rm Gproj}({\rm T}_n(R))$ is of finite type, that is, ${\rm T}_n(R)$ is of finite Cohen-Macaulay type. This provides an inductive procedure to relate C.M. finiteness of ${\rm T}_n(R)$ to that of ${\rm T}_m(R)$ for $m<n$.
[*3.2. Virtually Gorensteinness*]{}. Recall that the concept of virtually Gorenstein algebras was defined in [@BR] as a natural generalization of Gorenstein algebras. By definition, an Artin algebra $\Lambda$ is said to be virtually Gorenstein provided ${{\rm{GProj}}}(\Lambda)^\perp={}^\perp{{\rm{GInj}}}(\Lambda)$ where ${{\rm{GInj}}}(\Lambda)$ is the subcategory of all Gorenstein injective $\Lambda$-modules and the orthogonal classes are defined with respect to ${{\rm{Ext}}}^1$. Such algebras have been illuminated to be in close relationship with C.M. finite algebras; see [@Bb] for more information. Theorem \[TGPG\] then prepares a good tool to prove the following theorem.
\[TMR2\] Assume that the statements $(1)$, $(2)$, $(3)$, and $(4)$ are all true. Then $\Gamma$ is virtually Gorenstein if and only if $R$ and $S$ are.
For a $\Gamma$-module $(X,\,Y)_\varphi$, we claim that $(X,\,Y)_\varphi\in{{\rm{GProj}}}(\Gamma)^\perp$ if and only if $X\in{{\rm{GProj}}}(R)^\perp$ and $Y\in{{\rm{GProj}}}(S)^\perp$. To see this, assume first that $(X,\,Y)_\varphi\in{{\rm{GProj}}}(\Gamma)^\perp$ and $U$ is a Gorenstein projective $R$-module. By \[LELGP\], $e^1_\lambda(U)=(U,\,M\otimes_R U)_1\in{{\rm{GProj}}}(\Gamma)$. Therefore $0={{\rm{Ext}}}^1_\Gamma(e^1_\lambda(U),\,(X,\,Y)_\varphi)\simeq{{\rm{Ext}}}^1_R(U,\,X)$ implying that $X\in{{\rm{GProj}}}(R)^\perp$. (Note that the latter isomorphism follows by considering a projective resolution of $U$ and applying the functor $e^1_\lambda$, which is exact in view of statement ($1$), to obtain a projective resolution of $e^1_\lambda(U)$.) The other statement follows similarly. The converse is an immediate consequence of the adjoint isomorphisms and the short exact sequence $(**)$ (in the proof of Theorem \[TGPG\]) which is valid for every Gorenstein projective $\Gamma$-module. One may apply a dual argument and Theorem \[TGIG\] to elicit a similar statement concerning ${}^\perp{{\rm{GInj}}}(\Gamma)$. These, in particular, yield that $\Gamma$ is virtually Gorenstein if and only if $R$ and $S$ are.
Let $R$ be an algebra. Clearly the ${\rm T}_2$-extension of $R$, ${\rm T}_2(R)$, satisfies the hypothesis of the above proposition. Hence, ${\rm T}_2(R)$ is virtually Gorenstein if and only if so is $R$. In particular, ${\rm T}_2(R)$ is virtually Gorenstein if $R$ is of finite representation type.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the referee for her/his useful comments. Sh. Salarian thanks the Center of Excellence for Mathematics (University of Isfahan). H. Eshraghi and R. Hafezi thank the Institute for Research in Fundamental Sciences (IPM).
[15]{}
\(3) (2002) 393-440. in: [Cambridge Studies in Advanced Mathematics,]{} vol. 36, Cambridge University Press, Cambridge, 1995. , 1987. (1) (2005) 137-211. (2) (2011) 1973-2019. (2007). (6) (2008) 2043-2050. vol. 30, Walter de Gruyter Co., Berlin, 2000. (2011) no. 3 611-624. (2010) 1802-1812. (2) (2010) 728-734. , Annals of Mathematics Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001. DOI: 10.1142/S0219498812500661. London math. Soc. Lecture Note Ser., vol 146, Cambridge University Press, Cambridge, 1990. (2011) 180-202.
[^1]: This research is supported in part by a grant from IPM (No.91130218)
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abstract: 'For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in $S^3$, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links. Precisely, if a link has homological width greater than 2 then the two-fold branched cover must have infinite fundamental group. As an illustration, we recover the fact that the figure eight knot does not admit finite fillings, and prove that a family of pretzel knots, including the $(-2,5,5)$-pretzel, do not admit finite fillings. Further examples are also explored in order to compare the obstructions given here with other surgery obstructions, in particular those arising from Heegaard-Floer homology. We also illustrate these obstructions on knots in manifolds other than $S^3$ by example, studying surgery on a knot in the Poincaré homology sphere. Finally, a characterization of the trivial knot – among strongly invertible knots – from Khovanov homology is given.'
address: 'Département de Mathématiques, Université du Québec à Montréal, Montréal Canada.'
author:
- Liam Watson
bibliography:
- '/Users/liam/Documents/Mathematics/Bibliography/ABC.bib'
- '/Users/liam/Documents/Mathematics/Bibliography/DEF.bib'
- '/Users/liam/Documents/Mathematics/Bibliography/GHI.bib'
- '/Users/liam/Documents/Mathematics/Bibliography/JKL.bib'
- '/Users/liam/Documents/Mathematics/Bibliography/MNO.bib'
- '/Users/liam/Documents/Mathematics/Bibliography/PQRS.bib'
- '/Users/liam/Documents/Mathematics/Bibliography/TUV.bib'
- '/Users/liam/Documents/Mathematics/Bibliography/WXYZ.bib'
date: 'First version: July 8, 2008. This version: May 15, 2009.'
title: Surgery obstructions from Khovanov homology
---
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